Concentration inequalities for dependent sums and chaos

Use this skill to choose a concentration mechanism and to keep its hypotheses, norms, and losses aligned with the native problem. Its main purpose is to prevent a theorem for a stronger coefficient model or a lossy proxy from being reported as a theorem under pointwise stability.

Source of truth and synchronization

This route map is synchronized with dashboard v401, the P219 closure round, the sharp all-moment saturation round, and the separate small-failure stable-generalization campaign of the cleaned StabilityExploration campaign on 2026-07-23. Before doing project-specific work, read the current theorem-status ledger and the P219 closure packet. The complete reader-facing proofs are in main.tex and its listed input modules. Exact reusable counterexamples are in ROUTE_TOMBSTONES.md. Historical drafts and Git history are provenance only.

The universal linear rate is now proved. P219 closes P14 through the capacity-obstacle relocation, sparse obstacle support, and a support-localized (L^1)-(L^2) Hessian estimate. The route map below is retained as proof provenance and as a catalogue of reusable class theorems and method walls. Any phrases such as ``active target'' inside that pre-closure catalogue are historical unless a later post-closure project explicitly reopens them.

The full moment-order saturation is also closed. If (q=\min{p,n}), then the sharp worst-case envelope under the BKZ hypotheses is

||sum_i g_i||_p \asymp n beta q + M sqrt(n q).

For (p\ge n), the upper cap follows from (|g_i|\le M+(n-1)\beta), obtained by averaging the outside-coordinate telescoping bound at fixed (Z_i). The BKZ quadratic Rademacher example reaches (Mn+\beta n^2) on the all-positive atom of probability (2^{-n}), so the same family supplies the matching lower bound. This settles the (p\gg n) moment question. The distinct sharp small-failure-probability lower bound for actual uniformly stable algorithms is now also proved by the winner-margin construction described below.

The small-failure theorem is maintained separately in SMALL_DELTA_STABILITY_STATUS.md and small_delta_lower_bound.tex. Its exact native object is a bounded sample-dependent query

q_S : Z -> [0,L],   ||q_S-q_S'||_infinity <= gamma

under one sample replacement, with discrepancy P q_S-P_n q_S. The formerly unmatched parameter region was

L^2/(gamma^2 n) <= t <= L/gamma,   gamma >= L/n.

The first lower calibrator is the clipped signed-coordinate construction

G=(gamma/(2n)) sum_k |D_k| min(|D_k|,L/gamma),

which proves exactly why the elementary quadratic mechanism saturates at the sampling scale before entering that region. The decisive construction instead takes exponentially many independent Rademacher score coordinates and selects the unique empirical winner with amplitude equal to its truncated top-two margin. The map

W_r(x) = min(x_(1)-x_(2),r) e_w

at a unique winner, and zero at a tie, satisfies

||W_r(x)-W_r(y)||_1 <= 2 ||x-y||_infinity.

For a suitable number of scores, a winner above n/4 with margin 2r has probability at least 3^(-r)/20; the associated bounded stable query has gap at least gamma r/16. This proves the sharp lower scale

min(L, gamma t + L sqrt(t/n))

up to universal constants and confidence rescaling, and refutes a universal improvement to gamma + L sqrt(t/n).

On the class-theorem side, bounded range forces every symmetric additive query q_S=b+sum_i h(Z_i,.) to have actual stability at most L/n. Hence a sharp lower example must use nonlinear sample interactions. This theorem explains the structure of the winner-margin example; it does not extend to all bounded stable queries.

The final tree is deliberately compressed:

• one closed exact cube target: P14 set transport, proved by P219;

• two exact representations of that same target: P80 cut towers and the P83/P85 matched-section endpoint;

• one secondary explicit sufficient successor: P138 Euclidean row transport. If (t_i(f)=|\partial_i f-\mathbb E\partial_i f|_{\mathrm{KR}}), it asks for

  (sum_i t_i(1_A)^2)^(1/2) <= C sqrt(n) a log(e/a).
  

  This implies P14 by Cauchy--Schwarz and is equivalent to the anisotropic residual estimate

  ||sum_i Z_i q_i||_p
    <= C (p-1) sqrt(n) (sum_i Lip(q_i)^2)^(1/2).
  

  P89 implies P138, but twisted majority times a two-bit parity has bounded Euclidean row transport and P89 compatible cost of order sqrt(n). Therefore independent exact row transports cannot be glued back into one dimension-free compatible P89 flow; simultaneous compatibility is excess structure for the direct P138 route. No separation between P138 and P14 is known. P89 is now a stronger parked successor. Its fixed-orbit joint Calderón bilinear form is equivalent to P89, not a smaller active crux. P133 tangent recoupling is likewise equivalent to the centered two-scale endpoint and is parked. P134's bounded biased-sign preflight is also complete and parked: it proves the sharp (p=2) derivative estimate and the all-(p) common-phase class, but its scalar unsigned transfer loses (\sqrt n) exactly where cyclic phases cancel the true derivative, while complete quadratics saturate the proposed (p^2) scale. A purely skew five-row field with zero symmetric Jacobian nevertheless has increasing eighth moment under biased-sign interpolation, so symmetric-Jacobian domination is false; the full signed rectangle gate remains open. P135's fresh-generation audit adds three exact preflight walls: bounded-order labelled Fourier/dual-check data miss an (\Omega(n)) separation in exact P14 cost; the plain (L^1) norm of the exact scalar aggregate (\mathcal LP_{\ge2}f) loses a linear factor on antipodal and simplex codes, although the aggregate itself remains information-complete on the relevant degrees; and rectangle holonomy misses the additive transport baseline. A new direct state must see unbounded global section geometry and either preserve row-indexed phase or use a genuinely nonlocal, degree-sensitive functional of the scalar aggregate. P141 has now audited the three immediate P138 continuations. Exponentially sparse fixed-cardinality fibers satisfy P138, but positive graph assembly loses linearly on ordinary majority. Full random-deletion telescoping forces both negative savings with coefficient one and is then exactly the parent energy. Signed resolvent--carré repairs the P140 witness, but the exact cube chain rule uses edge-dependent divided differences and becomes P138 itself. Finally, thresholding long rows gives a valid diagnostic inequality, but separate entropy packing is false: an indexed-majority selector has arbitrarily many high rows and constant total entropy while its P138 ratio tends to zero. The shared latent majority transport is charged once per label. Do not continue with separate one-row entropy allocation, full-order scalar deletion, or an unweighted signed chain rule. P142 strengthens this wall. If

  e_i=D(nu||mu)-D(nu_-i||mu_-i),
  r_i=t_i/a,
  

  then the universal one-row theorem

  r_i^2 <= (n-1) L e_i / log 2
  

  holds, but an imbalanced equality of two majority signs has arbitrarily many high rows with sum e_i ~ sqrt(k) at fixed entropy while (K_n/a)^2 <= n/4. Thus even native deletion entropy cannot be summed rowwise. P143 has completed the bounded same-geometry preflight for the threshold-localized joint-rank estimate

  sum_(i in I_lambda) r_i^2
    <= C n L R_nu(I_lambda),
  R_nu(H)=D(nu||mu)-D(nu_(H^c)||mu_(H^c)).
  

  With h=|H|, d=n-h, R=R_nu(H), and S=log(1/a)-R, the proved conditional split is

  (sum_(i in H) r_i^2)^(1/2)
    <= C [sqrt(d R (1+S))
          + min(h sqrt(R), sqrt(h R)+h^(3/2) R)].
  

  Therefore joint rank closes when h^2 <= C nL or h^3 R <= C nL. Balanced monotone equality compositions obey the stronger bound

  sum_(i in I) r_i^2 <= (n/2) R_nu(I)
  

  for every I. The unrestricted internal field is nevertheless an arbitrary P138 residual on Q_h. Conditional descent gives

  sqrt(sum_(i in H) r_i^2)
    <= C_h sqrt(h L R)+C sqrt(d R (1+S)),
  

  whose cross term is not constant-preserving when h=n-1. Thus the universal joint-rank leaf is parked as circular.

  P144 completes the proposed structural walkback. Directed row-divergence Beckmann flow without reciprocity has the exact value

  2 K_n(f)
    = inf_U [sum_i (sum_(j!=i) ||U_ij||_1)^2]^(1/2).
  

  Its feasible set factors over rows, so this is P138 itself; reciprocity is P89, electrical selection loses the sparse scale, and common-base randomness glues every row optimum without contraction. Static global contact is also exact P138 duality:

  max_(mu(A)=a) K_H(1_A)/a
    = max_q AVaR_a(sum_(i in H) Z_i q_i).
  

  The KKT laws are only rowwise. For singleton or repetition-code contact,

  U=(S_H^2-h)/(2 sqrt(h)),
  M=S_H S_(H^c)/(2 sqrt(h))
  

  are reference-orthogonal but have aligned conditioned expectations. Therefore Hilbert, martingale, entropy-chain, and static-contact orthogonality do not produce a no-cross theorem. Universal no-cross is not a strict child because H=[n] is P138. The positive boundary is exact: every affine code satisfies

  sum_(i in H) r_i^2
    <= [h L R_nu(H)+d L^2]/[4 (log 2)^2].
  

  Park bare directed flow, static KKT/contact, and orthogonality-based no-cross. Re-enter only with genuinely new common-indicator integrability, second-order stability of global maximizers, or nonlinear entropy-weighted directed geometry. The surviving falsifier must have a dense, nonmonotone, non-affine, large high-row block.

  P145 completes that re-entry audit. For centered rows h_i=h_i(Z_-i), the laws partial_j h_i=partial_i h_j are equivalent to a unique common scalar potential Z_i h_i=D_i f_>=2; this exhausts linear common-origin information. Booleanity adds f^2=f and the pointwise same-orthant law D_i f=(2f-1)b_i/2, but parity refutes unsigned-curvature aggregation. At a global fixed-density maximizer, every normer lies in the hypersimplex normal cone and

  E[(1_A-1_B)F] >= ||v||_2^2/(2T).
  

  Opposite singleton maximizers share the same unique normers and have an exactly flat chord, so local second-order stability is parked. The exact lower-dimensional fiber ceiling is

  |I| <= C_h sqrt(h) (1+R).
  

  A constant-preserving induction would follow from the full gate

  2 I M + M^2 + sum_(j in Hc) r_j^2
    <= C_*^2 h (1+R)^2 - I^2 + C_0 d L^2,
  

  with C_0<=C_*^2, already for d=1. At best-constant calibration, repetition shows that O(dR) and O(dL) are insufficient and O(dL^2) has the required debt scale unless an independent constant gap supplies the slack; the removed-row energy cannot be omitted. Tilted thresholds have an exact positive conditional optimizer deficit away from repetition. The antipodal simplex-star family is a scalable, nonmonotone, non-affine all-high calibrator with every row

  r_i=((n-2)(n-3)+2)/(2n)
  

  and P138 ratio tending to 1/(2 log 2). Its exact dimension-six normers have nonlinear degrees 1,3,5 and all positive saturated cross contacts. Do not assume sparse contact, negative pairwise cancellation, or deficient Hessian interaction.

  P146 sharpens the codimension-one contract using the actual child optima. For sections A_r^+ and A_r^-, put

  E=K_n(1_A)^2,
  E_r^s=K_(n-1)(1_(A_r^s))^2 for s in {+,-},
  B_r=(sqrt(E_r^+)+sqrt(E_r^-))/(2a),
  Delta_r=E/a^2-B_r^2.
  

  Then the exact P139 signed-defect identity is

  Delta_r=r_r^2+D_r/a^2
    +(sqrt(E_r^+)-sqrt(E_r^-))^2/(4a^2).
  

  The sole active child is the adaptive theorem

  min_r Delta_r <= C_0 log(e/a)^2.
  

  It closes P138 by constant-preserving induction. Prescribed deletion is false: for balanced majority equality A={X S_h>=1}, deleting X has Delta_X asymptotic to a positive constant times h, whereas deleting any majority coordinate has Delta tending to 1/pi. Antipodal repetition forces C_0 at least 3/[4(log 2)^2] asymptotically. Generic averaging is circular because its first term is E/(na^2). Reopen only with a Boolean signed-defect coordinate-selection principle or a scalable counterexample for which min_r Delta_r/log(e/a)^2 diverges.

  P147 proves the first strict contraction inside this leaf. With

  c_ir=|fhat({i,r})|/a,
  c_r^2=sum_(i!=r)c_ir^2,
  

  the child-norm imbalance cancels the complete J-debt, and

  Delta_r <= r_r^2+c_r B_r+c_r^2/4,
  Delta_r <= r_r^2+sum_(i!=r)c_ir r_i,
  Delta_r <= r_r^2+c_r R,  R^2=sum_i r_i^2.
  

  Hypercontractivity gives sum_r c_r^2 <= 2 e^2 L^2. Under the lower-dimensional P138 theorem, this closes the induction whenever a fixed positive fraction of rows satisfy r_r <= lambda L. Therefore a remaining obstruction must be almost-all-high, level-two-pseudorandom, and simultaneously near equality in the exact binary KR chain rule. Ordinary spectral averaging of the weighted-star majorant leaves R^2/n+O(LR/sqrt(n)) and is circular. The next admissible moves are a rigidity/inverse theorem for simultaneous chain-rule near-equality or a high-degree counterexample to adaptive deletion.

  In particular, c_r=0 gives Delta_r <= r_r^2. Exact bent-quadratic, complete-quadratic, resilient-composition, cyclic-phase, and affine-code assays found no growing deletion debt. Treat these as counterexample diagnostics, not theorem evidence: a genuine level-two-free failure would need almost every row itself to be super-entropic.

  P148 audits equality rigidity. If U is the constant-vertical binary chain cost, t the exact parent row cost, and Omega=U^2-t^2, then every parent normer has horizontal child deficits satisfying

  delta_+ + delta_- <= 2 Omega/(U+t) <= Omega/t.
  

  Thus Omega=0 makes the normer hereditarily exact on both faces. Equality alone is nevertheless structure-blind: lifting an arbitrary Boolean g through disjoint parity blocks of size at least three gives c_r=S_r=0 for every physical coordinate, while every normalized row is at most 1/2. The high-row hypothesis is load-bearing. The active inverse problem is hereditary near-optimality plus row-normer oscillation much larger than L, not bare equality classification.

  P149 replaces raw oscillation by a gauge-invariant transport certificate. For one row/deletion pair, at least

  2t-q-2a ell-Omega/(epsilon t)
  

  units of child cost lie on pairs of length at least ell whose relative dual calibration is at least 1-epsilon. At ell=r_i/2, small q/t and Omega/t^2 force a positive fraction of the row cost onto Hamming intervals of dimension at least r_i/2 where the normer is nearly oriented distance. Small c_r and S_r make the number of exceptional high rows bounded independently of n. Raw oscillation is false as a rigidity state because dummy-coordinate affine gauges enlarge it arbitrarily with zero rerouting. The active lemma is simultaneous interval overlap or packing across the high rows of one common indicator.

  P150 gives an exact orientation export for any optimal row coupling:

  beta_i = t_i-(1/2)sum_(j!=i)|fhat({i,j})|
         = 2 sum_(j!=i) min(p_ij^+,p_ij^-).
  

  Thus beta_i is precisely bidirectional coordinate cancellation. It vanishes exactly when the signed affine level-two potential is an exact row normer. The zero-cancellation sector obeys P138 with constant e/sqrt(2), and generally

  K_n <= (e/sqrt(2))sqrt(n-1) a L + ||beta||_2.
  

  The target bound on ||beta||_2 is P138-equivalent, so this is structural information rather than a smaller universal endpoint. Combine it with P149: oppositely oriented exactly calibrated interval families cannot share an edge in that coordinate. The active target is simultaneous oriented packing across almost all high rows. The simplex-star has only beta_i/a=2/n, so the decomposition absorbs the nonlinear all-high calibrator. A balanced central-slice/remote-tail fiber construction has one row with beta_i/a growing as sqrt(log n), refuting every one-row beta_i <= C a L claim. Multirow common-indicator structure is essential.

  P151 gives the exact common-origin recursion

  beta_i(parent)
    = average_child beta_i + higher_order_exposure - sigma_i|r,
  

  and after a random deletion order

  beta_i = V_i-S_i.
  

  Here V_i is stopped absolute variation of conditional mixed derivatives and S_i is accumulated exact rerouting. The exact energy audit is

  sum_(i!=j) E M_ij^2 = sum_T |T| fhat(T)^2
                       = (1/4) sum_i Inf_i(f).
  

  Dropping S and bounding V positively loses sqrt(n) at balanced density. Plain Shapley variation, unsigned Hessian activity, and separate higher-order exposure are parked. Retain activity minus rerouting pathwise and jointly across rows.

  P152 subtracts the symmetric affine level-two potential from exact row normers. The residual potentials satisfy

  -1 <= epsilon_ij partial_j psi_i <= 0,
  epsilon_ij=epsilon_ji,
  

  and beta_i=E[h_i psi_i]. Primal-dually beta_i is minimum wrong-way transport relative to the symmetric orientation matrix. The associated gate is

  ||sum_i Z_i q_i||_p <= C p sqrt(n)
  

  for centered rows with -w_i <= epsilon_ij partial_j q_i <= 0 and sum w_i^2<=1. It implies P138. The p=2 constant sqrt(2) is proved, as are the full gate for at most p active rows, affine rows, the strip 2 <= p <= 2+c/log(n), and p>=n.

  P154 proves that this uniform sign-coherent gate is not strict. For arbitrary centered rows u_i of edge-Lipschitz scale w_i, set

  q_i = u_i - (w_i/2) sum_(j!=i) Z_j.
  

  Then -w_i <= partial_j q_i <= 0, while the two divergences differ by

  Q_w = (1/2) sum_(i<j) (w_i+w_j) Z_i Z_j,
  ||Q_w||_p <= p sqrt((n-1)/2).
  

  Conversely, sign-coherent rows have edge-Lipschitz scale at most 2w_i. Thus the sign gate is gauge-equivalent to the full arbitrary-row P138 theorem. Keep the exact common-indicator normal form, but do not attack the nonlinear intermediate-p strip as a contracted leaf. Return to P146 and the P149--P151 contact/activity-minus-rerouting structure.

  P153 closes an adversarial class. If A is invariant under translations by K and d=d_min(K^perp)>=3, exact row transport descends to the quotient by K_i^0. The quotient coordinate walk has conductance at least (d-1)/(n-1), so

  r_i <= min{ell_i/2, C (n-1)/(d-1) L},
  

  where ell_i is the least punctured weight of a kernel word through i.

  Therefore every arbitrary nonlinear union of K-cosets with d>=delta n satisfies P138. Random/high-dual-distance quotient-code counterexamples are closed, as are short-local-generator kernels. Only low-dual-distance, large-local-distance translation quotients and trivial-kernel transitive OA2 sets remain on that adversarial lane.

  P155 is the mandatory Hadamard calibrator. Puncturing the constant column of an order-N normalized Hadamard matrix gives an OA2 set with

  r_i = (N-2)/4, beta_i=t_i,
  

  in every row and P138 ratio tending 1/(4 log 2). Dense long bidirectional contact in every row is therefore compatible with P138, including exact trivial-kernel Paley instances. Do not charge raw interval count or total calibrated length.

  P156 is the two-point Hall-incidence wall. Two cyclic balanced OA2 sets in Q7 with trivial translation kernel have identical global and coordinate-oriented P-N, P-P, and N-N distance enumerators, but exact normalized rows 5/16 and 3/8. One- and two-point distance histograms cannot determine row transport. Retain higher-order Hall incidence or joint matching compatibility.

  P157 closes every one-pairing simplex phase. For an odd Boolean h on q=2^(m-1)-1 pair parities,

  A_(i,h) = {Z_i h((Z_v Z_(v+i))_pairs)=1}
  

  is balanced, simplex-invariant, and OA2, with exact energy

  R^2 = (1/4) W1(mu_h+,mu_h-)^2
        + (1/2) sum_p Inf_p(h)^2
      <= q(1+log 2)/2.
  

  Majority gives one row of order sqrt(n) but total energy asymptotic to n/(4 pi). Therefore one-row entropy control is false even in the sharp simplex quotient, while the whole one-pairing class satisfies P138. Attack only aggregate compatibility across incompatible projective pair decompositions.

  P158 is a proved identity/majorant and a refuted pair of gates. For the exact row-source laws and exact KR normers,

  H_i^2 = t_i^2/2
    + m_i^2 inf_exact(Var_(rho_i+) phi + Var_(rho_i-) phi),
  Delta_r <= 2 H_r^2/a^2 + sqrt(2) c_r H/a.
  

  Either entropy-scale control of the minimum majorant or H <= C sqrt(n) a L would imply P146, but both statements are false. On n=16r^2, the radial set of odd weights outside the central sqrt(n) band has constant density, every H_i^2 of order sqrt(n), and H^2 of order n^(3/2), while every true row KR cost is O(1) and P138 energy is O(n). A flux of order n^(-1/2) crossing a source-free plateau of length sqrt(n) forces an exact normer to accumulate a long slope; quadratic source variance charges its endpoint separation rather than the small transported flux. Retain the identity as a diagnostic, but do not reopen either P158 gate. A repair must be flux-weighted or truncated and must still retain cross-row common-indicator/Hall structure so that it does not collapse to P138 itself.

  P159 is the active dual/coarea preflight. Exact KR contact gives the nonnegative profile

  j_i(s)=m_i[rho_i+(phi_i>s)-rho_i-(phi_i>s)],
  integral j_i = t_i.
  

  If j_i^down is decreasing rearrangement, define

  F_n(A)=inf_exact integral_0^infinity
         (sum_i j_i^down(u)^2)^(1/2) du.
  

  Minkowski and a dyadic decomposition of integral extreme normers give

  K_n <= F_n <= C sqrt(log(en)) K_n.
  

  Native indicator profiles also satisfy the width envelope

  measure{s:j_i(s)>u} <= C sqrt(n log(C/u)).
  

  The open gate

  F_n(A) <= C sqrt(n) a L
  

  implies P146 with min_r Delta_r <= (C^2+sqrt(2)eC)L^2, hence P138. The P158 plateau contributes its true flux times length, not squared endpoint separation. Generic scale-separated signed rows separate F_n from K_n by an unbounded factor. More sharply, disjoint radial point-to-sphere blocks give the same sqrt(log n) loss for curl-free derivatives of one bounded edge-Lipschitz real scalar. Thus curl-freeness and bounded range do not suffice, but no indicator separation is known. Treat P159 as a sharply falsifiable Boolean preflight, not yet as a certified contraction. The missing Boolean lemma must remove the dyadic sqrt(log n) loss.

  P160 is the active primal/four-point sibling. For a coupling pi of the Jordan source laws, let q_ij=pi(X_j!=Y_j) and

  C_i=(m_i/a)^2 inf_pi sum_(j!=i) q_ij^2.
  

  Two independent pairs identify the unnormalized cost with expected reuse of discrepancy coordinates. Sion minimax gives

  sqrt(C_i)=(m_i/a) sup_(lambda>=0, ||lambda||_2<=1)
            W_(d_lambda)(rho_i+,rho_i-),
  

  while Cauchy gives r_i^2<=(n-1)C_i. Therefore the open gate

  sum_(i:r_i>L) C_i <= C L^2
  

  implies P138. This preserves the higher-order Hall incidence required by P156 and reduces the simplex remainder to collision packing across incompatible projective pairings.

  The Hilbert endpoint now proves the universal dense bound

  sum_i C_i <= (1-a)/(2a).
  

  Hence P138 is closed on every class whose density is bounded below. For H={i:r_i>L}, normalize maximizing anisotropic exact normers into the contact field G_H. Then E_A G_H=sqrt(sum_(i in H) C_i), and the sparse gate follows from the restricted moment estimate

  ||G_H||_p <= C p,  p=max(2,L).
  

  This is much narrower than the parked arbitrary-row P133 endpoint: only high exact contacts of one indicator are admitted, with coefficients fixed by their Hall costs.

  The sparse contact moment is proved under four structural conditions. If H is the high set and D_H has an arc j->i when psi_i depends on Z_j,

  ||G_H||_p <= C min(sqrt(p|H|), p sqrt(chi_ac(D_H))).
  

  For collision-minimizing incidences q_ij, the canonical KKT choice is

  lambda_ij=q_ij/||q_i||_2,
  w_ij=gamma_i lambda_ij=(m_i/a)q_ij/sqrt(S_H),
  sum_ij w_ij^2=1.
  

  If a feedback arc set R has weighted energy

  epsilon_R=sum_((j->i) in R) w_ij^2,
  

  then

  ||G_H||_p <= C[p+sqrt(p|H| epsilon_R)].
  

  Thus |H|=O(p), bounded acyclic partition number, or epsilon_R=O(p/|H|) closes P160. Conditional high-block degree at most one also closes by Khintchine plus Hanson--Wright, even with dense bidirected feedback.

  Fully nonlinear low-phase feedback also closes. If H is partitioned into s blocks, S_b is the sign sum on block b, and each row i in B_b depends on the high signs only through

  (S_1,...,S_b-Z_i,...,S_s),
  

  then conditioning on the Hamming-slice scores gives

  ||G_H||_p <= C[p+sqrt(p s)].
  

  Thus arbitrary row-dependent nonlinear functions of s=O(p) collective scores are good even with complete bidirected dependency and unbounded Walsh degree. Approximation error of weighted size epsilon adds only C sqrt(p|H| epsilon).

  A quotient-first common-channel theorem now closes overlapping phases as well. Remove the high-affine part and write eta_i for the nonlinear residual. If T is a randomized channel with averaged posterior chi-square information

  I_2=E_(W,T)||dP(Z_H|W,T)/dmu_H||_2^2 <= exp(kappa p),
  

  and Z_H' is an independent posterior replica, then

  ||G_H||_p <= C(1+exp(kappa/2))p
    + sqrt(|H|) [sum_i gamma_i^2
      ||eta_i(W,Z_H)-eta_i(W,Z_H')||_p^2]^(1/2).
  

  For deterministic T, I_2 is exactly the expected number of nonempty attained values; no minimum-fiber hypothesis is needed. Thus exact rank-O(p) affine syndromes close even when their parity checks overlap. The constructive crux is to find an exp[O(p)]-chi-square-information channel with weighted replica disagreement O(p/sqrt(|H|)).

  This certificate is not a consequence of rowwise contact. For the projective one-pairing normers, sqrt(8) psi_i are orthonormal and

  A_(p,T)^2 >= (1/4)[1-(I_2(T)-1)/|H|],
  || |H|^(-1/2) sum_i Z_i psi_i ||_2^2=3/8.
  

  Each row is an exact normer for a different majority indicator. Hence this is a proxy wall, not a P160 counterexample. Universality requires a new common-indicator inverse theorem forcing collision-weighted residual effective rank at most exp[O(p)], or a direct bypass of the agreement certificate.

  Coordinate transcripts do not provide that inverse theorem. If a depth-k adaptive transcript queries Q(T), its residual KKT edge energy is

  E_(p,T)^2 >= E_0-sum_(ell<=k) c_(ell)^2,
  c_j^2=sum_i gamma_i^2 lambda_ij^2.
  

  Uniform columns leave E_0(1-k/|H|), whereas closure needs O(p/|H|). Aggregate hashes are essential.

  Marginal collision is also insufficient for positive curvature coverage. In the same projective majority plan, each changed pair parity is lifted by flipping only one physical endpoint. Thus no same-plan discrepancy contains both endpoints of a pair, although those pairs are exactly the nonzero Hessian directions and both marginals are positive. A surviving curvature route must preserve signed cross-row phase. The natural within-one-plan quadratic state does not repair this: its transport cost is nonmetric, metric closure deletes off-diagonal terms, and its exact two-potential dual cannot assemble into E[h_i psi_i]=E[f Z_i psi_i]. Signed second moments are diagonal on the projective lift. Park second-moment joint-discrepancy states.

  P161 is now the primary strict child. On the fixed-density hypersimplex define

  c_i(f)=a^(-1) sup_(lambda,psi) E[h_i(f) psi],
  tau=L/sqrt(n-1),
  Q_tau(f)=||(c_i(f)-tau)_+||_2.
  

  This functional is convex, and

  sum_(r_i>L) C_i <= 2 Q_tau(f)^2+4L^2.
  

  A maximum may be chosen at an indicator A. With u_i=(c_i-tau)_+/Q_tau and exact potentials,

  G_tau=sum_i u_i Z_i psi_i,
  G_tau/a in partial Q_tau(A),
  E[(g-A)G_tau]<=0 for every equal-density g,
  E_A G_tau=Q_tau(A)+O(L),
  ||G_tau||_2^2<=1/2.
  

  More strongly, with I={i:c_i(A)>tau}, X_i=Z_i psi_i, and m_i(g)=a^(-1)E[gX_i], global extremality gives

  ||((m_i(g)-tau)_+)_(i in I)||_2 <= Q_tau(A)
  

  for every equal-density fractional g. If delta_i=m_i(g)-c_i(A), uniform convexity gives

  ||(delta_i)_+||_2^2
  <=(2 Q_tau(A)/a) E[(A-g)G_tau].
  

  Hence A is a top level set of its own contact field and simultaneously supports the full positive cone of its active contacts. The active target is Q_tau(A)<=C L only for these nonlinear fixed points. Point swaps lose the cardinality |A|, and top-level contact plus the L2 bound alone admits abstract spiky countermodels; use a macroscopic competitor. This is strictly stronger structural input than rowwise contact and excludes the projective proxy wall unless its rows can be realized by one such A.

  Two further common-indicator facts sharpen the surviving sector. If eta_i=P_(>=2)^H psi_i, xi_i=Z_i eta_i, sigma_i=||xi_i||_2, Y_i=xi_i/sigma_i, s_i=E_A xi_i/sigma_i, and R_ij=E[Y_iY_j], then

  s^T R^dagger s <= (1-a)/a,
  ||P_[0,lambda](R)s||_2^2 <= lambda(1-a)/a.
  

  This controls contact-visible spectral mass but retains the exact loss r_eff(R)<=|H|(1-a)/(a||s||_2^2). The cone certificate does not remove it: a common-atom character model saturates both inequalities while Q_tau>>L, although that abstract model is not an exact cube-KR row realization. Positive low-eigenvalue selection is therefore parked unless paired with new variance-sensitive concentration using row provenance.

  Also, for b_i=1_{f differs across its i-edge}, Booleanity gives

  E(D_iD_j f)^2=(1/4)E[b_i b_j].
  

  Exact potentials can be selected with dependency graph contained in the simultaneous-boundary graph. Bounded chromatic number therefore closes P160. A surviving obstruction must have high-chromatic boundary entanglement.

  The projective high-rank wall is now fully closed at the signed-cone level. For the projective one-pairing normers,

  ||sum_i u_i Z_i psi_i||_p <= C p       (||u||_2<=1).
  

  The proof uses heavy/light row splitting, exact order-three tensor partition norms, fixed-order decoupling, and Latala's Gaussian-chaos theorem. Therefore every indicator whose active exact contacts are these projective normers has Q_tau<=C L; uniform cubic thresholds and all weighted projective top-level analogues are benign. Do not treat high projective effective rank as a surviving P161 obstruction.

  There is a second strict closure localized to the actual nonlinear contact signal. Put eta_i=P_(>=2)^I psi_i and d_i=E_A[Z_i eta_i]. On every induced DAG V of the residual dependency graph and for every w supported on V,

  ||sum_(i in V) w_i Z_i eta_i||_p
  <= C[sqrt(p Var(sum_(i in V) w_i Z_i eta_i))+p||w||_2].
  

  Hence ||(d_i^+)_(i in V)||_2<=C L. The high-affine contact vector is also O(L) in ell2. If kappa_ac(d) is total positive nonlinear signal divided by its largest induced-DAG portion, then

  Q_tau(A) <= C(1+kappa_ac(d))L,
  kappa_ac(d) <= sqrt(chi_ac(D_eta)).
  

  This closes bounded signal-local acyclic ratio even when the unweighted graph has large coloring complexity. Random-order predictable projection yields the stronger degree-resolved curve

  bar d_i(t)=sum_(k>=2) t^k d_i^(k)/(k+1),
  ||bar d(t)||_2 <= C L.
  

  Hence bounded random-order recovery ratio closes P161, and Vandermonde inversion closes every fixed active-block residual degree without a graph hypothesis. A transverse anisotropic square-field theorem proves

  ||sum_i u_i Z_i psi_i||_p <= C_D p
  

  whenever the combined contact field has fixed aggregate Walsh degree D. Individual rows may have arbitrarily high degree and cancel in the aggregate. Thus a survivor must have unbounded active-block degree and unbounded aggregate degree, in addition to diffuse high-dichromatic positive signal. Literal complete bidirectedness is not necessary.

  Do not try to recover the endpoint signal from the scalar random-order curve alone. For D=4m+2, the positive Chebyshev profile F_D=(T_D+1)/2 has sup_[0,1] F_D<=1 but F_D'(1)+F_D(1)=D^2/2+1. This is an abstract degree-signal wall, not a cube-contact counterexample. Scalar Abel/Tauberian inversion must be replaced by an argument retaining common-indicator, Walsh-support, or exact Kantorovich provenance.

  Every cube translate of a P161 maximizer has nonpositive response in every fixed active contact coordinate. Hence so does every translation mixture, heat/noise smoothing, subgroup average, or radial interpolation; the uniform-convexity remainder is identically vacuous on those routes. P162 gives an exact threshold-band/lock analysis, but not an exhaustive dichotomy. For B_eta={|G_tau-theta|<=eta}, the localized rotation has an exact positive-orthant support formula and normal cost at most

  2 eta min(p_eta,q_eta).
  

  An affordable band with fixed-fraction rotation closes P161. If every active row is strictly locked, A={Z_i psi_i>s_i}, then

  ||(c_i-s_i)_i||_2^2 <= |I|/(|I|-1),
  

  and the remaining row is the weighted coordinate-code margin

  ||((s_i-tau)_+)_i||_2 <= C L.
  

  Balanced tie-free strict locks close. Repetition cylinders refute the implication from long exact KR intervals to positive rotation, while Gilbert--Varshamov codes refute replacing the row-dependent metrics by Hamming distance. More decisively, an abstract near-band spike model obeys the convex, top-level, cone-certificate, uniform-convexity, and L2 hypotheses while every individual support defect is one. Thus ``no macroscopic band rotation'' does not imply aggregate approximate locks from those abstract data: it loses the sharp factor sqrt(|I|). This is a proxy wall, not an exact cube/KR counterexample.

  The strict-lock remainder has an exact forbidden-difference Cayley representation, but its unrestricted independence-number/Delsarte gate is false on pinned faces. After separating the coordinates which vary on the code, put

  E_i={(x,y) in A^2:x_i!=y_i},
  rho_i^2=min_(p_i in Prob(E_i))
            sum_(j!=i) Pr_(p_i)(x_j!=y_j)^2.
  

  This is an exact fractional multicommodity code SOCP, with dual

  rho_i=max_(lambda_i>=0, ||lambda_i||_2<=1)
          min_((x,y) in E_i)
          sum_(j!=i) lambda_ij 1_(x_j!=y_j).
  

  Strict locks give 2(s_i-tau)_+<rho_i; pinned active coordinates cost O(L^2) by the face-codimension and subgaussian-tail bounds. The formerly proposed pure code gate

  sum_(i in V(A)) rho_i^2 <= C L^2
  

  would close every strict lock, but it is false. Repetition and parity calibrate benign sectors, and exhaustive tests through n=4 pass, but the outlier half-ball below gives the first failure in dimension five.

  Random priorities now give an exact nonlinear reduction. Let k=log_2(1/a) and let S(U) contain coordinates which are not the unique priority maximum of any incident difference support. Projection off S(U) is injective, so |S(U)|<=k. If q_i is the closest fractional link-incidence vector, ell_i=||q_i||_1, and p_i=Pr(i in S(U)), threshold integration gives

  rho_i^2 <= 2 p_i ell_i,
  sum_i rho_i^2 <= 2(n-1)k.
  

  For every binary linear code of parity-check rank r, circuit minimality makes every link size at most r and S(U) matroid- independent. Therefore

  sum_i rho_i^2 <= 2 r^2.
  

  The same holds for affine cosets and has the target quadratic scale. An arbitrary-hypergraph extension by independence number is false: all two-edges on m old vertices plus one redundant edge containing a new distinguished vertex have independence number two but distinguished row value rho_i^2=m. Matroid circuit descent is load-bearing.

  The proposed sharp nonlinear transfer

  rho_i^2 <= (k+1) p_i.
  

  would sum to k(k+1), but it is also false. For even m>=4, let

  A_m={(0,b):wt(b)<=m/2} union {(1,1^m)}.
  

  Its density tends to 1/4, so k<2, while the distinguished row has

  rho_i^2=m/4,
  p_i=m/[2(m+1)],
  ell_i=m/2.
  

  Already at m=4, rho_i^2>(k+1)p_i; asymptotically one row refutes both the pure code gate and the aggregate priority-length gate. The exact entropy--Shapley relaxation also fails: if d_i=E_U[1-H_2(X_i|X_{H_i(U)})], then p_i<=d_i<=1 and sum_i d_i=k, but rho_i^2 still grows like m.

  The missing datum is section mass. If the two section densities are alpha_0,alpha_1 and theta_i=min(alpha_0,alpha_1)/(alpha_0+alpha_1), applying bounded-difference concentration to the weighted distance from one section gives

  rho_i^2 <= log(1/(alpha_0 alpha_1))
            =2 log(1/a)+log(1/[4 theta_i(1-theta_i)]),
  theta_i rho_i^2 <= C L.
  

  Marginal entropy also pays all strict-lock thresholds with theta_i<=delta by O_delta(L^2). Thus only a quantitatively section-balanced code statement which retains the actual lock thresholds can be reopened. The explicit repaired candidate is

  sum_i theta_i rho_i^2 <= C k^2.
  

  It would close every strict lock after the near-pinned/balanced split. It holds with sharp constant one for affine codes, vanishes for coordinate-flipped down-sets, and is stable under Cartesian products of those classes. Exact tests through n=4 have sharp ratio one and the outlier wall has large slack, but the universal statement remains open.

  Do not try to prove it from the mass-weighted projection probability

  theta_i rho_i^2 <= C (k+1) p_i.
  

  This is false even for equal sections. Take antipodal Hamming balls of radii m/2-t in the two sections. Their separation is d=2t, and the distinguished row has

  theta_i=1/2,  rho_i^2=4t^2/m,  p_i=2t/(m+1).
  

  If t tends to infinity with t=o(sqrt(m)), then a->1/2, k->1, and the row ratio diverges like t/2, although the aggregate weighted value is only 2t^2/m->0. Exact support separation is too coarse.

  The sharper information-weighted assay

  theta_i rho_i^2 <= C k d_i,
  d_i=E_U[1-H_2(X_i|X_{H_i(U)})],
  sum_i d_i=k,
  

  has the exact erasure representation

  d_i=int_0^1 E[1-H_2(X_i|X_(R_t),R_t)] dt,
  J_i=int_0^1 E|E[(-1)^X_i|X_(R_t),R_t]|^2 dt,
  J_i/(2 log 2) <= d_i <= J_i,
  

  where R_t retains tail coordinates independently with probability t. For section projection counts N_b(S,z),

  E M_S^2
    =(1/|A|) sum_z (N_0(S,z)-N_1(S,z))^2
                      /(N_0(S,z)+N_1(S,z)).
  

  Thus the assay is exactly, up to constants, a beta-averaged triangular- discrimination comparison with the fractional cross-difference distance. It holds with constant one for affine codes and for every full Boolean graph {(f(y),y)}. If f has s relevant coordinates, its graph row has rho_i^2=1/s and d_i>=1/(s+1). The universal statement is still unproved. The count-level lift is exact: with

  I_S=(1/N) sum_z (N_0-N_1)^2/(N_0+N_1),
  Omega_S=(4/N) sum_z N_0 N_1/(N_0+N_1),
  

  one has I_S+Omega_S=1, J=sum_S w(S)I_S, and a harmonic common-fiber subprobability mixture of mass 1-J. But overlap counts do not choose economical geometry inside the fibers. On balanced parity, every proper projection has full overlap, while independent uniform fiber coupling gives each coordinate load tending to 1/4 and total square ~m/16; the optimum is 1/m. Pairing through one uniformly chosen unrevealed coordinate recovers that optimum.

  The surviving stronger assay is coherent beta-harmonic fiber selection: choose all common-fiber couplings jointly across scales so their final coordinate loads square-sum to O(k J_i/theta_i). This now has an exact convex dual. If alpha_(S,z) are the normalized harmonic common-fiber masses and

  d_(S,z)(lambda)
    =min_(x in B_z,y in C_z) sum_j lambda_j 1_(x_j != y_j),
  

  then

  sqrt(L_sel)
    =sup_(lambda>=0, ||lambda||_2<=1)
       sum_(S,z) alpha_(S,z) d_(S,z)(lambda).
  

  The same anisotropic direction lambda is load-bearing across every scale and fiber. Separate local optimization followed by square aggregation is not merely circular; it is false at target scale. On parity the average local squared norm is H_m/m, while k J_i/theta_i=2/(m+1), a logarithmic loss. The exact shared-direction quantity is 1/m, so parity remains benign. The stronger common-direction second-moment assay is

  sup_(lambda>=0, ||lambda||_2<=1)
    sum_(S,z) alpha_(S,z) d_(S,z)(lambda)^2
    <= C k J_i/theta_i.
  

  P171 proves this with constant one for every affine code, without a symmetry hypothesis. Run random-greedy on the parity-check matroid, use the final fundamental circuit, and swap the distinguished coordinate with a circuit element. If J is circuit nonavailability, the inclusion probabilities q_j satisfy sum q_j<=r and max q_j<=min(1,J/(1-J)).

  P172 closes a strictly larger class: every Boolean graph over every affine domain D=u+V, with arbitrary nonlinear labels. On a projection fiber use the quotient Fourier characters of V_S={v in V:v_S=0}. For each global character chi, form the virtual linear graph code

  L_chi={(ell_chi(v),v):v in V}.
  

  P171 bounds its available weighted circuit cost. Quotient Parseval then sums the characterwise charges to

  E[v_(S,z) d_(S,z)(lambda)^2]
    <= k min(J_i,1-J_i).
  

  After harmonic normalization this is at most k J_i/theta_i.

  Do not continue searching for asymmetric affine codes or nonlinear labels on affine supports; those classes are closed. For a native row, nonzero fractional cross-distance forces disjoint sections, hence a partial Boolean graph. The unresolved selected-lift profile is therefore a partial graph on a proper nonlinear domain. The missing mechanism is a replacement for common affine-coset Fourier decomposition and characterwise circuit descent. Do not equate maximal projected overlap, independent coupling, local-square Jensen aggregation, or independently optimized dual directions with a proof of the information-weighted gate. The unweighted code gate, its unweighted priority/entropy variants, mass-weighted support-priority repair, and rank-only repairs are tombstoned.

  Affine completion is now fully tombstoned. P173 refutes coefficient-one information/energy monotonicity on a three-point domain. More decisively, P174 takes

  D_m={0} union {x: |x|>=floor(m/4)}
  

  with the origin carrying the minority label. Its affine-hull density defect is exponentially small, but at the uniform direction every partial mixed fiber has distance r/sqrt(m), whereas every completed mixed fiber has distance 1/sqrt(m). The best signed completion defect is at least (r^2/m-1)m_D for every extension. Thus no universal constant repairs the transfer. The family is benign for the true target because theta_D=1/|D_m|; completion failed by discarding this section-mass normalization.

  Immediate or bounded-depth information swaps are also closed. On

  B={000,111}, C={001,010,101,110},
  

  both children of every coordinate retain the root 1:2 label ratio, so all one-step information increments vanish, while the root harmonic distance energy is 2/27. Orthogonal-array variants give the same wall at every fixed depth.

  A further global assay chooses all fiber pair laws jointly and defines coordinate loads q_j. The unweighted first proposal

  ||q||_1 <= C k
  

  is also refuted by the P174 radial family: every selected pair has at least r mismatches, so every selection has ||q||_1>=r while k->1. Even the weighted first bound is false: balanced antipodal balls have theta=1/2, k->1, and minimum cross-distance 2t->infinity, so every selection has ||q||_1>=2t. Separate universal load caps are closed.

  The final same-selection product preflight

  inf_sigma ||q(sigma)||_1 ||q(sigma)||_infinity
    <= C k J/theta.
  

  is also false. Add a tag bit to balanced antipodal balls. Every selected pair differs in the tag and at least 2t tails, so q_tag=1 and the product is at least 1+2t. An equivariant selection has q=(1,2t/n,...,2t/n) and quadratic load 1+4t^2/n -> 1, while the target right side stays above 2. Thus the actual l2 target has fixed slack. All l1/linfinity factorization routes are closed. The nonlinear selected lift has returned to its native quadratic/SOCP selection problem; any successor needs structure acting directly on that geometry.

  The raw positive-superlevel square self-bound

  sum_j (G_tau(z)-G_tau(z^(j)))_+^2
    <= C[(G_tau(z))_+ + L]
  

  is false from exact one-row provenance plus the fixed-contact certificates. On a projective one-pairing majority contact F=Z_i psi_i, the common indicator is {F>0}, the row is an exact KR normer, and the top-level cone and remainder hold. At the all-positive vertex, however,

  F=sqrt(h-1)/4,
  sum_j (F-F^(j))_+^2=(h-1)/4+1.
  

  The loss is sqrt(h). This is not a counterexample for global P161 maximizers; it proves that only a new maximizer-driven cross-row compensation theorem could rescue the square gate.

  The calibrated one-row replacement uses Phi(s)=s_+^2/(1+s_+):

  sum_j Phi(F-F^(j)) <= 2 F_+ + 1.
  

  The front flip has linear cost and the transverse flips use the native squared oscillation budget. The smaller live analytic question is a dimension-free assembly of this truncated carre, or an equivalent first-power nested-superlevel flux, for a full globally maximizing contact field.

  The next native moves are therefore a section-balanced, provenance-preserving lock theorem, the maximizer-only truncated-carre assembly, or a realizable unbounded-degree counterexample. Keep a P138-versus-P14 separation attack parallel because P161 is only a strict sufficient row.

  Distinct simplex-majority pairing phases have exact covariance asymptotic 8/(pi n) but Dirichlet overlap tending to 4/pi; the shared projective-line and full-parity modes must be quotiented first in a Dirichlet phase-rank diagnostic. For k=o(n) the phases still retain genuine L2 rank. The one-pairing distinguished collision is exactly q p_q^2/8 -> 1/(4pi). No mixed counterexample follows: stability of diffuse row couplings under simultaneous projective conditioning is the missing theorem.

  For the nonlinear remainder, define

  Theta_(p,H)^2=sum_i gamma_i^2 || |D_H^2 psi_i|_HS ||_p^2.
  

  Two cube Poincare inequalities give

  ||G_H||_p <= C[p+p sqrt(|H|) Theta_(p,H)].
  

  Exact normers are affine on every optimal transport interval, so this Hessian vanishes on contacted two-faces. The open crux is a collision-weighted contact-coverage theorem for the remaining nonlinear, many-row, irreducibly cyclic, high-phase-rank case. Do not replace it by an unweighted global Hessian bound without testing the P158 plateau.

  P136 sharpens the earlier admission test. Positive heat/Bessel smoothing of (\mathcal LP_{\ge2}f) followed by one scalar (L\log L) norm has the correct automatic indicator upper bound but loses (\sqrt n/\log n) against P14 on balanced majority. Independently optimizing Hall matchings at every distance threshold can be linearly smaller than row (W_1), while forcing one common matching reconstructs (W_1) exactly by layer cake. Ordinary coordinate compression is also nonmonotone already on four points in (Q_4). Reopen these directions only with signed/derivative-sensitive multiscale structure, an entropy-sized correction, or a genuinely simultaneous row repair. P137 audits exactly those immediate survivors. The exposed-face extension defect has a complementary-slack/Hall normal form, but local optimum values plus the defect reconstruct the original transport; splitting the P123 matched core into its Hall core and matching-incompatibility defect leaves two independent entropy bounds. The vertical Poisson square has the correct indicator (L\log L) budget, yet its dimension-free (L^2) normalization makes it smaller than P14 by (n^{1/4}) on majority. Therefore reject every dimension-free (L^2)-bounded scalar square as a direct P14 dominator. The half-derivative square has degree-k response sqrt(k)|F|/2, but no ambient normalization satisfies both contracts: full parity forces c_n=O(n^(-1/2)), while embedded two-bit parity forces c_n=Omega(1). Degree-adaptive squares, signed cross-scale pairings, and adaptive scale/test couplings remain open. The row-indexed survivor is precisely P138. The P103 transfer identity is proved, but P105 refutes its stronger scale-by-scale compatible variation bound. The SF-P89 predictable reveal-square estimate is a strict sufficient statement. P106 proves it for linear-hypergraph supports and for the combined P105 affine family. P107 proves weighted binary backward peeling and identifies the exact local condition as a Rudin-scale Lambda(p) estimate for every last-pivot residual spectrum. P108 proves that compatible potentials may have unbounded local Lambda(4) constants while their full reveal square is pointwise target-good. P109 gives the exact fresh-endpoint/inherited-coefficient reveal recursion and refutes the natural resolvent/P30 occupation lift at the fractional-cover scale; that lift can be large even when the scalar output vanishes. Thus local good-pivot deletion and pre-quotient occupation are both refuted. For a fixed/random order the invariant scalar bracket is the P82 chronological square object, while its Carleson dual is the desired norm restated. SF-P89 is parked unless a genuinely smaller quotient-sensitive adaptive state is found. P110 proves P89 for every binary quotient factor with at most O(p) singly occurring nonzero label classes, and gives the sharper all-moment square-root bound when every nonzero class repeats. It also refutes two proposed architectures: independently optimizing the P89 quotient on homogeneous levels can lose exponentially on a bent quadratic phase, and controlling the exact compatible matrix transform before taking its scalar trace costs a sharp extra sqrt(p). These are a class theorem and method walls, not a universal proof. P131 adds a rank--trace globalization wall: if the identity on the nonaffine scalar range is a finite signed mixture of binary-quotient conditional expectations, the total P110 component cost is at least n(1-(n+1)/2^n). Thus componentwise P110 plus triangle inequality cannot yield O(p) when p << n. This does not exclude adaptive, nonlinear, input-dependent, or martingale-difference quotient assemblies. P111 iterates repeated-label quotients as a reverse martingale: deterministic conditional subgaussian proxies square-sum with no collision-depth loss. P112 upgrades this to the conditional Bernstein class

  ||F_0||_p <= ||F_T||_p + C(sqrt(pV)+pL).
  

  The P111 rare-event family passes this sharper variance/peak gate. Nevertheless P113's depth-one repeated-core/leaf family has chord graph K_(N,2m), native budget n tau_2(c)^2=2mN, but deterministic conditional variance N^2. Thus worst-coset Bernstein transfer loses N/(2pm) even though the native P89 ratio is small. Keeping the bracket random either pays a generic maximal-increment loss or returns to SF-P89/P82; park the quotient-flag continuation. Independently, P112 proves every canonical Boolean corrected gradient P89-good, for all degrees, labels, and multiplicities. P114 proves the arbitrary-gauge half-space barrier and exact representative-blindness calibrations. P115 sharpens this to rho_n(F)>=S(S-1)/(2(n-1)sqrt(n)), adds an averaged-Hessian certificate, proves the exact symmetric real quotient (with a sharper Boolean consequence), and proves exact disjoint- output tensorization. Any Boolean pointwise survivor must have sublinear maximum sensitivity and |F(x)|/sqrt(E[k(k-1)]) tending to infinity along offending vertices. Full permutation symmetry and additive repetition of a fixed disjoint output cannot create the amplification. This bounded Boolean assay is parked; its fixed-output dual still has the full P89 ball. P116--P121 audit the universal fixed-orbit continuation. P116 defines an exact one-orbit envelope C<=O<=D and a correlated Gamma-moment multiplier hull. It contracts the false P103 norm by sqrt(M) on the full P105 affine-spread witness, but sigma=1 recovers P_(>=2) and hence P89 itself: this repairs a proxy without producing a smaller P89 crux. P117 proves that every unweighted separated L^r_mu(L^p) x L^(r')_mu(L^(p')) factorization loses an extra sqrt(n), uniformly for 1<=r<=infinity. P119 shows that positive adaptivity does not repair the wall: the optimal pointwise positive weight equals the absolute-product integral, which loses sqrt(n) on the simultaneous complete-quadratic/point-mass pair; positive stopped local Hölder sums have the same loss, while unrestricted signed stopping is tautological. P118 gives the exact signless-incidence factorization F=R^* I A, with squared support norm k/(2(k-1)) and transverse-gauge contraction at most 1/4; common-core, matching, quadratic, and degree-three tournament tests close pointwise and generic two-stage continuations. P120 strengthens P110's phase wall: aligned and quadratic-bent flat spectra have identical canonical unsigned per-support incidence data but exponentially separated native quotients and moments. Thus the active P89 proof must remain signed, quotient-aware, and genuinely bilinear before norming either heat-orbit leg. P116 and P118 are bounded parked assays, not new active leaves. A focused branch audit found no named post-P89 formulation with a proved strict contraction. One bounded assay remains under the existing P89 node: with the norming functional J_p F=sgn(F)|F|^(p-1)/||F||_p^(p-1), test

  V_p(F)=integral |E[(K_q F)(K_q J_p F)]| dmu
         <= C p sqrt(n) rho_n(F).
  

  Exact Calderón reproduction makes this sufficient for P89. It is a diagonal restriction of P116 and genuinely stronger than P89, but the norming-functional/extremizer reduction is generic until quotient KKT contacts produce a scale-variation contraction. It is exact at p=2, for homogeneous outputs, and for one-signed Fourier data at even p. Bounded bent/aligned, radial, design-like, random, and exact-LP tests found no counterexample. P121 completes the bounded analytic/KKT iteration. The norming map is surjective on dual classes modulo affine functions; global extremal coupling is a strict subset and yields singleton exposed faces plus exact sign-support, tight-cover, and proportional-degree contacts. Those contacts are complementary slackness for existing P89 duality and give no nonconstant-scale transition. For the normalized scalar contact polynomial P_F, shifted heat gives only |P_F(s)|<=1 and P_F(1)=1, while shifted Chebyshev polynomials satisfying those conditions have Gamma variation of order the square of their degree. This refutes the scalar heat-profile compression, not the full variation estimate, because the Chebyshev data are not shown realizable by cube extremizers. Treat P121 as PARKED: reopen only with a contact-set Gamma-multiplier contraction or deletion map, not with more static KKT or scalar heat arguments. P131 reopened exactly one bounded continuation under the same P89 node. At every genuine global extremizer, the tight chord-contact sets injectively sample the full nonaffine quotient-Hodge range through 2 D_i D_j L^(-1), forcing total normalized contact mass

  sum_{tight {i,j}} mu(E_ij) >= 4(1-(n+1)/2^n).
  

  The optimal dual flow defines a probability law Pi supported on signed tight contacts with first-endpoint marginal u_i^2. This is a strict extremizer restriction, not a moment estimate. A positive/Hadamard complete-quadratic pair has identical static Hilbert-endpoint contact data but high moments separated by sqrt(n). Therefore do not continue with unweighted contact counts, separate edgewise entropy, or p=2 flow magnitudes. P132 completes the permitted target-p preflight. Symmetrizing the optimal flow gives a contact-frame seminorm s_(p,barPi) with exact signed mean E_(barPi) T G=E[hG]/M_(p,n) and s(F)=1; an inverse bound

  ||G||_p <= C p sqrt(n) s_(p,barPi)(G)
  

  would imply P89. Existing KKT data do not supply it: normalized local perfect-matching primal/dual data satisfying the same contact identities support the flow only on the matching and annihilate the nonmatching quadratic Z_1 Z_3. They are not claimed globally extremal. At p=2, the complete quadratic has the exact frame weights (|S|-1)/(|S|(n-1)), so the full sqrt(n) scale is intrinsic. Ordinary Lewis weights, operator scaling, and Bourgain--Tzafriri selection see unsigned Gram geometry; isometric equal-leverage frames can still have zero signed first moment. For the actual analysis operator the missing signed angle is 1/M_(p,n), so the desired lower bound is P89 itself. Treat the contact branch as PARKED. Reopen only with a new quantitative consequence of global exposed-singleton extremality, not another KKT or frame reformulation. P133 is the post-contact branch audit. Do not promote the rowwise optimal-KR exponential-tail statement: after inactive rows are normalized, entropy duality and the residual moment theorem make it equivalent to P14. Do not continue the P90-coloring/conditional-Hanson--Wright proof template either. Coloring leaves at most one edge from each Walsh support and hence erases the compatible four-cycle identity; R=p=k disjoint rooted k-stars force a sqrt(k) loss in the reconstructed operator term. This is a method wall, not a counterexample to an uncolored compatible Hanson--Wright estimate.

  The separate sufficient program is the two-scale Bernoulli divergence

  ||sum_i Z_i u_i||_p
    <= C(sqrt(p) ||u||_(L^p(ell_2))
         + p ||D u||_(L^p(HS))).
  

  It is equivalent to the centered derivative-only form. Centered rows satisfy the exact range identity

  sum_i Z_i u_i
    = sum_(i!=j) Z_i Z_j (I+L_(-ij))^(-1) D_j u_i.
  

  Thus full parity refutes the unrestricted unsmoothed double divergence but not this compatible smoothed range. Direct tangent recoupling,

  ||sum_i Z_i u_i(Z)||_p
    <= ||sum_i eps_i u_i(Z)||_(L^p(Z,eps))
       + C p ||D u||_(L^p(HS)).
  

  is equivalent up to constants to the centered two-scale endpoint: conditional Khintchine and Hilbert-valued cube Poincare give one implication, while the endpoint gives recoupling because the tangent term is nonnegative. Do not promote it as a smaller crux.

  A column-coupled common-conductance primal is also parked. Its exact dual is a quotient of the pointwise column-ell2 norm by the annihilator of the componentwise edge-range and row-independence constraints; the raw max-column norm is only a lower dual subclass, not the polar. Balanced majority of disjoint pair-products separates this conductance functional from P138 by exactly sqrt(n), although it remains target-sharp. Do not promote the raw column estimate as an upper bound for the primal, and do not present the functional as a P138-level reformulation.

  P134 completed one local proof/falsifier preflight. With independent biased signs of mean t, put

  X_t = sum_i Y_i(t) Z_i u_i,
  M_p(t) = E |X_t|^p.
  

  Russo differentiation followed by pairing Z_i=+-1 gives an exact signed mixed-rectangle sum for M_p'(t). The required one-sided differential gate

  M_p'(t) <= C p^2 ||D u||_(L^p(HS)) M_p(t)^(1-1/p)
  

  would integrate to the centered endpoint. Absolute derivative control is false already for u_1=Z_2, u_2=-Z_1: then M_p(t)=2^(p-1)(1-t^2) and M_p'(t)=-2^p t. The negative derivative is harmless for the required upper bound. After preserving this sign, termwise vectorization of the cross-column half-slope still recreates the dimensionally false nonpredictable-column-divergence proxy.

  The completed audit proves

  M_2'(t) <= 2 ||D u||_(L^2(HS)) M_2(t)^(1/2)
  

  universally, with normalized constant (1/2) asymptotically sharp on full parity, and

  M_p'(t) <=
    2 p ||sum_i |u_i|||_p M_p(t)^(1-1/p)
  

  for every (p\ge2). The latter closes common-Walsh-phase rows but cannot be transferred to the Hilbert--Schmidt budget: on (u_i=Z_{i+1}) it loses (\sqrt n), whereas the exact derivative vanishes for every even (p<n). Complete quadratic rows give normalized positive derivative (||G^2-1||_p/(2p)) after (n\to\infty), tending to (1/e); hence the conjectured (p^2) scale is sharp. P134 is parked without a proof or counterexample to the universal one-sided gate. Reopen only with a new phase-sensitive cancellation principle for the total rectangle sum, not with unsigned majorization, vectorization, or another diagonal integration-by-parts pass. The neighboring Gaussian Meyer--Skorohod results are stored durably in literature/nualart2006malliavindivergence/ and literature/pisier1988meyer/; their derivation/rotation arguments are precedent, not cube transfer. Independently, interpolation closes P89 on 2 <= p <= 2+1/log(n), so a P89 falsifier must satisfy (p-2)log(n) -> infinity. P122 completes the bounded deletion/state audit of the parked P11/P82 invariant square. If (r) is last, its bracket obeys the exact recursion

  V_(pi'+r)(R) = V_(pi')(E_r R) + |partial_r R|^2,
  

  and order averaging gives the signed collision weight (|T\cap U|/|T\cup U|). Neither is a contraction: a three-coordinate two-root collision forces first-child row parameter (3\beta/2), Jensen controls the collision kernel in the wrong direction, and P108 gives

  Gamma(E_pi V_pi)/(beta^2 n^2 E_pi V_pi) ~ n/12
  

  at the all-positive vertex even though its true square is uniformly bounded. Exact even-moment diagrams also fail to tensorize their permutation charge: repeating one last-vertex constraint costs (1/|T|), not (|T|^{-m}). Kill noninflating scalar deletion, fixed-order or order-averaged carré self-bounds, collision-only upper states, and absolute diagrams. Park exact Bellman and signed-diagram forms as restatements. Reopen P11/P82 only with a signed scalar-output parent-to-child contraction; otherwise attack P14 directly. P123 gives the current direct-P14 normalization. For a set (B) of density (a), let (q_i) be the smaller front-bit mass and (\rho_i^\pm) the two conditional opposite-section laws. Then

  |T_n(1_B)/a - sum_i q_i W1(rho_i^+,rho_i^-)|
      <= sqrt(2n(n-1)) log(1/a).
  

  The error is proved at the target entropy scale, so the matched dependence endpoint is equivalent to P14, not a smaller leaf. Exact binary Wasserstein disintegration creates diagonal cross-corner transports under deletion. Dropping them fails on two-bit parity; retaining all corners reconstructs the original coupling LP; comparing corners separately to the ambient cube loses (\sqrt n) on majority. Scalar mutual information is metrically blind: parity and a majority graph can have identical scalar information but conditional (W_1) distances (1) and (\asymp\sqrt n). Two further exact adversaries are mandatory. For quadratic tails (A={(\sum_iZ_i)^2\ge t}),

  T_n(1_A) = L_2(1_A) = E[1_A((sum_i Z_i)^2-n)/2],
  

  while every coordinate section has the parent density; scalar deletion is therefore blind on the strongest known finite P14 calibrators. Conversely, a full-support binary code has

  T_n(1_C) = (mu(C)/2) sum_i(d_i-1),
  

  and simplex codes have (L_2=0) but ((T_n-L_2)/(n\mu(C))\to\infty). Kill any universal (T_n\le L_2+Cna) repair. A valid continuation needs a signed, cycle-coherent cross-corner state with a strict parent-to-child gain that retains both quadratic alignment and global section-support separation. A direct total-correlation assay has now been audited. If pi=product_i nu_i, oriented exact row KR potentials produce

  E_nu F_phi = M_n(B),
  E_pi F_phi = 0,
  D(nu||pi) = TC(nu).
  

  The linear claim M_n<=C n TC is false on a punctured subcube, where M_n has an extra factor m. The scale-compatible replacement

  M_n <= C n (sqrt(TC)+TC)
  

  would follow by entropy duality if one could select centered exact normers satisfying a product-law mgf estimate at the single scale

  lambda_D=(c/n) min(1,sqrt(TC)).
  

  Do not quantify over every exact normer. Without centering, additive constants make every dimension-only mgf envelope false. After centering, all-normer quantification is circular: on a coordinate halfcube every centered one-Lipschitz row function is an exact normer for a zero row, so the contact field contains arbitrary native residuals.

  The current canonical repair is strictly stronger than existential selection but avoids both walls. On the centered exposed KR face K_i^0, select the unique minimizer

  phi_i^o = argmin_(phi in K_i^0) E_(pi_-i) phi^2.
  

  Product-law centering removes additive gauge; if the row cost is zero, strict convexity selects phi_i^o=0, so the halfcube circularity disappears. For F^o=sum_i A_i phi_i^o, test

  log E_pi exp(lambda_D F^o) <= C lambda_D^2 n^2,
  lambda_D=(c/n) min(1,sqrt(TC)),  0<c<1.
  

  P177 now supplies proof-grade structure. For every centered one-Lipschitz selection, not only the canonical one,

  Var_pi(F) <= n(n-1)/2.
  

  Thus the target has the correct quadratic scale at the origin; the missing content begins in higher cumulants at distance 1/n. Canonical KKT gives an exact potential--flow identity. More sharply, every connected component of the saturated-edge graph of phi_i^o is centered separately. If all such component diameters are at most d, then

  log E_pi exp(lambda F^o) <= (d^2/2) lambda^2 n^2
  

  for every real lambda.

  The selector mgf is proved for every permutation-invariant set:

  log E_pi exp(lambda F^o) <= 256 lambda^2 n^2,
  |lambda| <= 1/(16n).
  

  This closes full parity, antipodal repetition, every radial shell union, and P158's radial plateau as selector classes. Repetition still shows that c=1 is a genuine radius-of-convergence wall. Certified exhaustive QP tests through dimension three find no violation, but are falsification evidence only.

  The coordinatewise-positive Green route is closed. It used

  G^o = L_pi^{-1} F^o,
  V_+^o = (1/2) sum_j E_j' [(Delta_j G^o)(Delta_j F^o)]_+.
  

  Both the tilted mean condition and P178's untilted exponential strengthening are valid sufficient statements, but their universal premises are false. On the canonical P158 plateau, with n=h^2=16r^2, the lower-tail gradients are

  Delta_k G^o = -2h-(-1)^k,
  Delta_k F^o = -2h-n(-1)^k.
  

  Thus one parity of edges has positive product of order h n, and on a fixed-probability binomial band

  V_+^o / n^2 >= h/8.
  

  Consequently E V_+^o/n^2 grows like sqrt(n) and every fixed exponential moment of V_+^o/n^2 diverges. The actual P177 mgf remains uniformly bounded on this family. Taking the positive part coordinate by coordinate has deleted essential alternating cancellation. Retire semigroup moment, self-bound, and KKT-to-codimension attacks on V_+.

  P179 is an exact signed-secant implication, but no longer a live universal premise. Set

  kappa(x)=(1-exp(-x))/x,
  W_t=(1/2) sum_j E_j'
      [(Delta_j G^o)(Delta_j F^o) kappa(t Delta_j F^o)],
  Q_t=exp(tF^o-m(t)) pi.
  

  Exchangeability gives the exact identity

  m'(t)=t E_(Q_t) W_t.
  

  Therefore the aggregate-positive estimate

  E_(Q_t) [W_t]_+ <= C_c n^2,   |t|<=c/n,
  

  implies the P177 one-scale mgf. P158 passes this condition for every fixed c<1, but universal validity is false even at t=0. For complement-symmetric Boolean shell bits with b_0=0,b_1=1, the exact prefix signs force every minimum-L^2 canonical KR slope to be

  phi_(r+1)-phi_r = 1-2 b_(r+1).
  

  Random central shell bits therefore create iid saturated slopes in the actual canonical selector. The exact radial Green expansion and Khintchine give

  E [W_0]_+ >= c n^(9/4).
  

  These sets are permutation invariant, so their true P177 mgf is uniformly bounded. P179 has lost essential cancellation only at the final positive truncation. Retire P179, its F-conditioned positive repair unless a separate asymptotic theorem is proved, and every claim that canonical KKT provenance alone forbids independent cross-scale radial slopes.

  P182 is the primary strict successor. For centered X, define

  b_+(X)=sup_(u>=0) E[(X-u)_+]/P(X>u),
  b_-(X)=b_+(-X),  b=max(b_+,b_-).
  

  Layer cake gives the exact recursion

  E|X|^k <= (k!/2) b^(k-2) Var(X),
  

  and hence

  log E exp(tX) <= Var(X)t^2/[2(1-b|t|)].
  

  Therefore b(F^o)<=C n, together with the P177 variance theorem, closes the one-scale mgf. Repetition forces C>=2. The statement is genuinely stronger than the mgf: a sufficiently rare scalar n^2 spike separates them.

  P182 is proved for every permutation-invariant canonical field:

  b_+(F^o), b_-(F^o) <= 10000 n.
  

  The proof is uniform in the common bias and in arbitrary radial one-Lipschitz slope patterns. It decomposes the field into a product term satisfying a quadratic binomial envelope plus an O(n) oscillatory perturbation. Every superlevel component is an interval; monotone adjacent binomial-mass ratios control its conditional overshoot. No monotonicity of the potential or its slopes is used.

  The nonradial interaction theorem is sharper. For

  D_i={j: Delta_j phi_i is not identically zero},
  

  hub conditioning gives an explicit certificate

  b_+(F), b_-(F) <= min_H Gamma(H) <= sum_i |D_i|.
  

  Thus O(n) total directed interaction incidence, bounded average row width, bounded interaction components, and fixed active-row width are closed, even without canonical selection. The fixed-k bound is k(n-1). A complementary weighted theorem uses

  ell_ij=sup |Delta_j phi_i|.
  

  Bounded two-sided row/column load gives P182. Exact dependency components of size at most sqrt(n) give P177, and so does any partition into sub-root blocks whose total weighted cross-load is O(n).

  Without any sparsity, a near-maximal residual creates a long weak-expansion ladder and a boundary band with large aggregate row amplitude. Product expansion lower-bounds the band mass; event-localized McDiarmid bounds its conditional row amplitude. The resulting bootstrap proves the strict universal estimate

  b_+(F), b_-(F) <= C n^(7/5)
  

  for every centered contact field. If b(F)=kappa n, some near-maximal band crosses Omega(kappa^2) rows and meets a canonical saturated component of diameter Omega(kappa^2). This is a genuine exponent and falsifier-profile contraction, not the desired linear closure.

  The current inverse-ladder package is idempotent at exponent 7/5. Optimizing the threshold approximation, using all ladder rungs, or feeding back the hub and long-component conclusions returns the same exponent. Its exact exponent formula is

  beta=(a+gamma+mu theta)/(2+nu theta),
  

  and a dense affine-row family makes the unsigned event-local amplitude estimate sharp at the critical mass scale. That family is not an arbitrary-row artifact: the antipodal repetition set has the unique centered exact canonical rows

  phi_i^o=(1/2) sum_(j!=i) Z_j,
  

  and saturates the same amplitude scale on an actual O(n)-width band of its contact field. Boolean common provenance, minimum-L2 selection, KKT contact, full saturation, and thin-band localization therefore do not improve the unsigned input. Repetition itself has linear mean residual life, so the missing estimate must also be aligned to a near-maximal residual band. Do not relaunch a scalar bootstrap without a genuinely new signed, residual-aligned estimate.

  Every centered row selection also has

  max_edge |Delta F| <= 3(n-1).
  

  Exhaustive canonical search through n=4 and a certified adversarial n=5 packet leave repetition extremal. This is evidence only. A failure must be robustly dense under the hub and weighted-partition certificates, and must sustain diffuse signed threshold contact across a long nested superlevel ladder. The exact next object is

  C_i(eps,u)
    =eps(1-p_i) E[phi_i
      (1_{Q_{i,s_i}}-1_{Q_{i,-s_i}})],
  Q={eps F>u},
  

  with target sum_i C_i <= (u+C n) P(Q). Taking absolute values returns to P14 and loses the threshold rebate. The contacts are not individually nonnegative: an exact canonical five-cube witness has a negative row at its unique maximizing prefix. What remains true is the transverse transfer

  sum_i (-C_i)_+
    <= E[1_Q sum_i 1_{X_i}|R_i|],
  

  so wrong-sign contact must be created by cross-row derivative interference on the same threshold boundary.

  Truncation exposes the exact signed first-power identity. For a_u=(eps F-u)_+,

  sum_i theta_i E[eps phi_i D_i a_u]
    = u E a_u + E a_u^2.
  

  Equivalently, this is the integral over levels of the signed P187 contacts. It retains precisely the threshold rebate lost by unsigned amplitude bounds. But do not take positive parts rowwise or fiberwise: canonical radial shell unions make that positive truncated flux at least order n^(9/4), while n E(F^o)_+=O(n^2) and the true P182 bound is linear. Cancellation must occur before positive localization or be charged by an equally signed transverse term.

  Raw matrix structure does not close that load. Every balanced Boolean graph has b_+,b_- <= n-1, even though explicit members have complete interaction and maximal signed-Jacobian stable rank. Parity makes the outward-drop and cross-interaction terms n^2 and n^2-n separately. Therefore stable rank, cut norm, pairwise phase, and separate operator control are parked. The primary continuation must retain the total threshold rebate and nonlinear common-source KR structure.

  Do not infer P182 from variance and edge oscillation alone. If H~Bin(n,1/2), a=ceil(sqrt(n log n)), and X=(H-n/2-a)_+ - E(H-n/2-a)_+, then Var(X)<=1, the edge oscillation is one, but b_+(X)>=c sqrt(n/log n). The surviving descent theorem must use exact common-indicator KR contact to exclude a scaled version of this moderate-deviation ramp. The active problem is the universal nonradial P182 bound under that provenance.

  Two parallel post-P179 attacks are admissible. First, seek an exponential n^2-scale bound for the predictable Doob bracket of F^o. Bennett's supermartingale and the universal canonical reveal bound |d_k|<=3(n-1) prove that this bracket estimate implies P177. It is proved for the entire permutation-invariant canonical class. Pointwise bracket control is false on repetition, whose maximum bracket is order n^3 but whose bracket has a uniform exponential moment at scale n^2. On the fair cube the bracket is exactly the old P82 chronological square, so canonical provenance must supply the genuinely new nonradial estimate. Direct local conditional-variance depletion is false already for an exact canonical three-coordinate set: every first reveal has zero bracket but unequal child variances. A time-reserved curvature hypothesis does imply the exponential bracket, but exact canonical radial shell unions violate that local hypothesis by an unbounded factor while their actual bracket remains good. The same wall survives every fixed correction by conditional row-square energy. Therefore only a rarity-weighted path argument or accumulated-bracket descent remains credible; worst-node variance depletion is closed.

  The exact accumulated object is now known. If

  Y_k=Var(F^o | F_k),
  Delta_k=Y_k^+-Y_k^-,
  e_k=Y_k-E[Y_k|F_(k-1)],
  

  then

  V=Y_0+sum_k e_k.
  

  With

  psi_p(x)=log((1-p)e^(-px)+p e^((1-p)x)),
  Psi_(rho,lambda)=sum_k psi_(p_k)(2 lambda Delta_k/n^2),
  

  one has the exact one-way reduction

  log E exp(lambda V/n^2)
    <= lambda Y_0/n^2 + (1/2) log E exp(Psi_(rho,lambda)).
  

  The square gate sum_k Delta_k^2/n^4 is a stronger fair-cube assay. Variance-monotone reveal trees satisfy the bracket gate; parity has zero cost and repetition satisfies the square gate. The universal Psi gate is valid but is not yet a strict contraction: no canonical bound or quantitative separation currently makes it easier than the bracket. Rarity of the single P185 node does not control the whole tilted path.

  An event-local entropy formulation does not shrink this debt. The bound

  E[Psi_(rho,lambda) | A]
    <= K + eta log(1/P(A)),   eta<1,
  

  for every terminal event implies log E exp(Psi)<=O(1). Conversely, a P192 exponential bound at parameter Lambda implies this condition at lambda=t Lambda with eta=t<1, because Psi_(rho,t Lambda)<=t Psi_(rho,Lambda). Thus path-space entropy is an exact rarity interpretation but is equivalent to P192 after parameter rescaling. Do not promote it without a strictly smaller canonical information functional.

  P190 and P192 admit one exact scalar profile synthesis. For

  K_X(u)=E(X-u)_+-(E X-u)_+,
  

  predictable reveal tents define a nonnegative random profile Lambda such that

  E Lambda(u)=K_X(u),
  2 integral Lambda(u) du = V
  

  pathwise. P190 is the upper-tail Hardy transform of E Lambda; P192 is the scalar-area projection of the associated profile martingale. This does not retain P190's row provenance and is not a contraction. The P185 three-point field has both child means zero, hence zero mean-splitting tent, but child variances 8/9 and 20/9, hence nonzero profile-shape area 2/3. Direct pointwise or area domination of fixed-mean shape exchange by mean splitting is closed. Only a global reach-weighted canonical bound on accumulated shape exchange remains conceivable in this language.

  The full-profile candidate makes that remaining direction precise. Put

  Q_k=2 integral |K_k^+-K_k^-|.
  

  Since |Delta_k|<=Q_k, an exponential bound for n^(-4) sum_k Q_k^2 implies the P192 square gate. An exact canonical Q_10 shell has zero tent and Delta_k=0 but Q_k=3/2, so local convex order, tent, and signed-area comparison are false even under exact provenance. A generic rare-trigger martingale with variance at most n^2 and reveal increments at most n makes the reach-weighted Q-gate diverge, so scalar profile/entropy/reach data cannot prove it. That martingale is not a canonical contact field; the full-profile gate is not refuted. Reopen it only with the original minimum-L2 rowwise KR/KKT currents retained inside the accumulated second-order transport estimate.

  A fixed-density native walkback also closes a common loop. At a global P14 maximizer every exact aggregate normer lies in the normal cone, so the set is a top-level set; maximizing its threshold plus overshoot is exactly the fixed-density P14/AVaR dual. The outer l1 norm is flat under tangent redistribution among positive row costs and has no Euclidean swap square. Static top-level contact is therefore an equivalent reformulation, not a contraction. Re-entry at P14 needs a dynamic Boolean exchange principle.

  The first three provenance-preserving continuations have now been audited. P197 expands the signed threshold flux into ordered row pairs:

  u E a_u + E a_u^2 = D_u - J_u.
  

  Antisymmetric curl cancels exactly and only symmetric row strain remains. Every negative front contact has a congestion-one local transfer, but canonical pairwise strain positivity and generic total positivity are false; the unrestricted all-cuts condition is the P182 second-residual target itself. P198 shows that parity and repetition have identical signed threshold interaction matrices but incompatible rebate accounting. Thus Hodge/cycle control and matrix-only signed expansion are closed. Re-enter the threshold-current route only with front anchors, threshold level, and original KKT source provenance retained simultaneously.

  P199 lifts the scalar threshold-profile descent to row-labelled currents c_(r,i) and areas d_(r,i), with

  Delta_r = sum_i d_(r,i).
  

  The lift resolves further into the original KKT source currents and excludes the generic rare-trigger architecture for n>=17. Its natural Hilbertian gate is nevertheless false on canonical repetition:

  E[n sum_k ||d_k||_2^2]
    = n^2(n-2)(n-1)(n+1)/3,
  

  one factor n too large. Continue only through signed cancellation, conditional low rank, or a smaller source-flow norm before squaring.

  P200 gives the first strict fixed-density walkback below the static AVaR equivalence. Put

  H_n(m)=(2^n/m) max_(|A|=m) T_n(1_A).
  

  If A is a maximizer and F_A is any exact aggregate normer, then H_n(m)=AVaR_(m/2^n)(F_A). The dyadic increment

  AVaR_a(F_A)-AVaR_(2a)(F_A) <= C n
  

  for a<=1/4 iterates directly to P14. An exterior edge injection A -> A^c proves this with C=1, because every admissible contact changes by at most 2(n-1) on a cube edge. Literal matching is a strong sufficient certificate, not the native target: Hall-deficient radial sets can already be exact local P14 maximizers. Bounded average Hamming length is also miscalibrated: deep exact quadratic tails force length sqrt(n/log n) while retaining the desired linear contact increment.

  P202 supplies the calibrated replacement. Give an oriented edge the cost

  c_F(u,v)=[F(u)-F(v)]_+
  

  and let D_F(A) be the minimum capped exterior path-transport cost. If at every sparse cardinality some P14 maximizer admits an exact normal field with D_F(A)<=C n, then P14 follows. This normer-descent exterior-flow condition allows long paths and charges only downward variation; the quadratic calibrator satisfies it with linear cost. Pair cost F(x)-F(y) without pathwise positive variation is exactly the dyadic AVaR gap and is circular. The active missing theorem must convert a weighted Hall/min-cost-flow obstruction into a global cardinality-preserving set exchange and control row-normer rotation or reoptimization. One-swap or another fixed-normer AVaR calculation is insufficient.

  P201 makes the rotation debt exact. For a same-cardinality exchange,

  T(B)-T(A)=E[(1_B-1_A)F_A]+sum_i R_i,
  

  where R_i>=0 is the slack of the new optimal row flow against the old exact potential. A Hall-rooted Q_6 swap has frozen gain -15/256, rotation reserve 35/512, and actual gain 5/512. Therefore frozen normal scores can miss a genuine improving exchange. A maximality-to-NDEF theorem must lower-bound the new-row flow slack from the deficient-flow certificate; formal first variation or normer uniqueness does not suffice.

  P206 gives the exact NDEF dual geometry. Dual potentials are graphwise isotone one-contractions of F; after scalar uncrossing their value is a weighted packing of nested F-oriented cuts with positive A-imbalance. Lexicographic minimization among maximizing set--normer pairs controls only the threshold plateau. Repetition has a unique top set and an optimal cut in the strict core. Pointwise weighted Hall at radius O(n) is false on deep quadratic tails even though their average NDEF cost is linear.

  P204 rewrites the P201 reserve as a row-residual min-cost flow with arc cost 1-(phi_i(u)-phi_i(v)). A singleton exact contact has raw dyadic gap zero and NDEF 2(n-1), while every singleton exchange has reserve exactly equal to frozen loss. Therefore a positive cut cannot by itself force improvement; a valid OCX/coupled-flow theorem must first pay an O(n) baseline and use only superlinear aggregate cut mass.

  P207 audits the first direct row-shadow shortcut. On a directed j-edge,

  F(z)-F(z^j)
    =2 z_j phi_j(z_-j)+(n-1)-sum_(i!=j) s_i(a_ij,b_ij).
  

  Aggregate descent therefore favors small sign-corrected forward residual slacks, and the direct term has no row-edge analogue. P208 proves that strict cheap-reachability closure turns every expensive reverse arc into a feasible residual cut certificate. Check the native normalization before using it: since

  h_i(B)=1/2 (g_i^B-E g_i^B),
  

  a point swap contributes 2^(-n), not 2^(-(n-1)), to one row cut. The corrected edge gain is

  T(B_e)-T(A) >= (|I_e|-c_e)/2^n-sum_i d_(i,e).
  

  It is vacuous on singleton/repetition first exits. P209 additionally refutes automatic one-signed-shadow extraction for one selected exact Q_4 pair/dual, without refuting full multi-shadow packing or existence-form OCX. Exact P205/P209 data remain finite evidence; ordinary concavity and log-concavity of the maximum envelope are refuted by a written Q_3 orbit calculation.

  P210 gives an independent direct walkback. For a maximizer A, a coordinate translate, collision I_j=A intersect tau_j A, and a fractional exterior relocation d,

  2 K_n(m)-K_n(2m)
    <= 4 n m/2^n + T_n(d-1_(I_j)).
  

  Hence T_n(d-1_(I_j))<=C n m/2^n for one admissible (j,d) at every sparse cardinality closes the P200 increment and P14. Low overlap merges with bounded Hamming assignment. P211 proves the genuinely large-overlap nearest-shell theorem for every sparse one-sided and symmetric two-sided threshold:

  T_n(d-1_(I_j)) <= 33 (n-1) mu(I_j).
  

  P213 gives the exact collision--exterior AVaR dual and the universal quadratic-overlap estimate

  inf_d T_n(d-1_(I_j)) <= (sqrt(3)/2) n sqrt(mu(I_j)).
  

  Thus overlap O(a^2) closes FCR, complementing the old O(a/n) regime. The hybrid near-maximal-influence condition

  min_j mu(A intersect tau_j A) <= C a (a+1/n)
  

  for one maximizer at every sparse cardinality would close P14, but remains open universally. P216 gives the exact erasure identity

  mu(A intersect tau_j A)/a = H(X_j | X_-j)/log(2)
  

  and the collision-sensitive row ceiling

  t_j(1_A) <= ((n-1)/2)(a-mu(A intersect tau_j A)).
  

  For a fixed-cardinality maximizer, comparison with the quadratic antipodal contact proves

  min_j mu(A intersect tau_j A) <= 4 a r_m/(n-1),
  

  where r_m is the least radius whose two antipodal Hamming balls contain m points. Hence NMI holds when r_m=O(1+na), including each fixed polynomial cardinality scale and densities bounded away from zero, with parameter-dependent constants. Growing-radius antipodal balls violate arbitrary-set NMI by ratio asymptotic to 2 r_m while having the quadratic field as an exact top-level normer. This threatens NMI, not FCR: P211 already proves FCR for those radial thresholds. One bad maximizing orbit would refute only an every-maximizer form; the existential P213 lemma fails only if every maximizer at a sequence of cardinalities is bad.

  P217 gives a canonical strict child of FCR. If u>=0 is the unique capacity-obstacle odometer for collision I and empty fibers E, then d=1_I-(I-P)u is admissible and

  C(I,E)
    <= T_s(d-1_I)
    <= (2/s) sum_(i!=j) E |partial_i partial_j u|.
  

  This reduces the CEAG supremum to the off-diagonal L^1 Hessian of one nonnegative obstacle solution. P218 proves the formerly open radial child. In cumulative radial flux variables,

  sum_(k=0)^(D-1) |D-2-2k| |Q_k| <= 2(D+1)a,
  

  and consequently the radial obstacle Hessian is at most 10Da. The proof uses contact-component envelopes and binomial hazard bounds.

  P219 then closes the nonradial problem by a different, shorter mechanism. Obstacle complementarity gives

  p=mu_s{u>0} <= 2a,
  ||L_s u||_2 <= (s/2) sqrt(2b),   b=mu_s(I)<=a.
  

  For every function supported on a set of mass p,

  sum_(r!=t) ||D_r D_t u||_1
    <= 2s sqrt(p) ||L_s u||_2.
  

  Indeed each mixed derivative is supported on four translates of supp(u), Cauchy--Schwarz localizes its L1 norm, and Walsh orthogonality gives

  sum_(r!=t) ||D_r D_t u||_2^2
    <= ||L_s u||_2^2.
  

  Therefore

  sum_(r!=t) E|partial_r partial_t u| <= 2s^2 a,
  T_s(d-1_I) <= 4sa.
  

  The lift to the ambient cube preserves T, so P210 gives

  H_n(m)-H_n(2m) <= 4n-2.
  

  P200 iteration proves P14, exact transport duality proves the Rademacher derivative-budget rate, and P88 gives the general-product BKZ rate.

  Keep the proof/wall distinction explicit. P219 does not bound total occupation and does not prove obstacle relocation is FCR-optimal. First-quartile thresholds still have E u/a of order s, and the exact radial Q_10 witness still makes heat balayage more than 2.28 times an explicit admissible competitor. Those facts refute the occupation and optimality proxies, not the sparse-support Hessian proof. Conversely, do not apply the P219 conclusion to an arbitrary Hessian field: the support bound and the source L2 bound come specifically from obstacle complementarity, capacity, and mass balance.

  P215 exhaustively certifies K_5(6)=1, one maximizing isometry orbit, and collision profile (4,4,0,0,0) for every maximizer. Thus the exact disjoint-coordinate evidence reaches size six in Q_5; sizes seven and eight are the next finite layers. P212/P214 refute both coordinate-compression monotonicity directions, even after minimizing the FCR defect over the coordinate. Do not use ordinary polarization to reduce FCR to radial sets.

  Second, test separate upper stop-loss domination of F^o and -F^o by a universal dilation of the repetition field (S_n^2-n)/2. Full convex order is false on the iid-shell witness, and ordinary coordinate compression is nonmonotone. Uniform Hanson--Wright for deterministic matrix families and fixed-degree higher-derivative inequalities do not transfer to the state-dependent, degree-n field; a viable theorem must retain pair-exclusion or equivalent canonical cancellation.

  The unaveraged double-KKT geometry is now closed. At a global mgf-minimizing selector the exact saddle is

  tau_tilde_i + b_i sigma_i + B q_i = 0.
  

  If H_i is a native optimal contact flow, then

  (b_i,q_i) -> (b_i-c,q_i+cH_i)
  

  is an exact gauge. Raw mass, overlap, and domination are therefore not invariant. Gauge fixing by the largest feasible multiplier yields a maximal upper-cut ratio gamma_i, and sum_i gamma_i W_i <= C t n^2 would imply the selected mgf. P158 refutes this proxy exponentially: its singleton tail cut has gamma_i(c/n) >= C_c exp(c sqrt(n)) while P179 stays bounded. Retire raw paired-flow, worst-cut, and codimension attacks. Only an averaged signed, mass-sensitive use of the exact saddle and common-density curl survives. P124 tests the exact signed binary-cycle continuation. Its common-order bicausal cost is a genuine restricted coupling class, dominates the P123 matched core, and passes parity, majority, twisted fibers, and quadratic tails. It still has total child coefficient one: opposite parity has zero cycle curvature, zero scalar entropy decrement, and two copies of the parent. A (1-1/m) deletion identity holds for the realized cost of one common optimal coupling, but it does not recurse on the original sections: conditioning on both exposed signs changes both tail marginals. On (P=\operatorname{Unif}{01,10}) and (Q=\operatorname{Unif}{00,01,10}), the weighted original-section cost is (1/3) while the deleted realized cost is (1/6). Park bicausal deletion. Reopen only with a signed state that controls these cross-conditioning marginal corrections at the target scale. P125 closes the natural order-flexible variants. Exogenous order mixtures minimize at one deterministic order; marginally faithful adaptive trees retain the coefficient-one Bellman recursion, and the same two-bit pair refutes their (1-1/m) continuation contraction. Endpoint-informed seeds recover every coupling. Plans bicausal in every order do contract under deletion, but opposite majority forces (K_{\rm all}\ge(m+1)(1+\lambda_m/m)/4) while (W_1=\lambda_m\le\sqrt m). Hence all-order faithfulness loses (\sqrt m). Do not reopen random/adaptive/all-order bicausal deletion without a new signed cross-row invariant. P126--P130 give the first all-order native affine/common-core contraction and its post-P130 branch audit. For an output F=chi_C H, projection onto the common-core character and permutation averaging give an exact quotient normal form: one scalar transfer row A, outside rows B_i, and precisely the core--core, core--outside, and outside--outside chord constraints. This is a genuine smaller state for the subclass, not a universal P89 reformulation. For the signed P105 affine packets, every four-block xor relation is paired; every genuine six-block relation is either a six-line affine-plane circuit or a level-two block with its five lines through a point. At length eight, a maximum-block cover proves that no distinct zero-xor support can meet a level m>=3. After repeated-pair reduction, the high levels therefore have exactly the joint eighth moments of independent Rademachers relative to the two low levels. Consequently arbitrary signed weights and any disjoint common core satisfy native P89 through p=8, uniformly in the number of levels. P129 then supplies two exact all-order reductions. Arbitrary coefficient phases are dominated with constant one in every even moment by the positive packet. At order s, every level above floor(log_2(s-1)) can be replaced exactly by independent block signs relative to the lower levels. Combining the remaining O(log s) levels by P106 and Cauchy--Schwarz proves

  ||sum_m t_m h_m||_s <= s sqrt(log_2(2s)) ||t||_2.
  

  This is substantive all-order subclass progress, not universal closure and not yet the linear rate. The exact residual is positive convolution flattening. Its correct combinatorial language is the weighted activity of inclusion-minimal binary incidence cycles (matroid circuits), not ordinary connected covers. P130 proves both directions: exponential primitive activity implies linear moments by the cosh--tanh expansion, while every primitive r-circuit contributes r! positive ordered terms, so linear moments imply exponential primitive activity. Thus AOE is an equivalent target up to constants, not a smaller crux.

  Do not project only to traces at one maximum block. With equal l2-normalized weights across the first m levels, the exact singleton-trace polynomial has root growth at least sqrt(m) after exterior parity is discarded. Also do not seek the one-level transition through ||Gamma||_q <= C(k+q) or through ||S_pi||_p <= C(1+sqrt(p/k)): a low-rank incidence-kernel atom on AG(2,4^m) refutes both by unbounded factors, for every fixed or predictable adaptive order, while remaining harmless for the terminal polynomial. These are proxy walls, not counterexamples to the affine moment estimate.

  The affine branch is parked after the bounded audit. Re-entry requires a strict l2-anchored exterior-parity estimate, for example

  sum_{C primitive, A(C)=A} prod_{B in C} u_B
  <= K^r u_A^2 (sum_B u_B^2)^((r-2)/2),
  

  proved through a full paired-incidence/nonbacktracking closure or an affine-code shortening recursion. Do not make either a live route before its bounded preflight proves a strict contraction. Do not substitute generic four-circuit freeness or a dissociated-set partition: APN Sidon caps have no nonpaired four-relations but exponentially large high moments, and a level-one affine packet requires order n dissociated classes. Also do not treat the subfields as a nested chain, condition on the full level-one algebra, reconstruct from one independent spread, use only whole-affine-block exchangeable flips, or close multiplication in the span of the original level sums. The gcd intersection law, deterministic level-one reconstruction, top-parity zero mode, and six/eight-point product orbits close those mechanisms exactly. The durable P124 literature packet confirms that no standard import fills this gap. Bicausal dynamic programming gives the fixed-order recursion without contraction; Marton/Dobrushin entropy contraction excludes hard parity constraints; trajectory-coupling theorems require a dependence matrix whose control is the missing geometry; and nonnegative weak-cost tensorization through the ambient reference repeats the majority-refuted triangle step. Do not open a generic Marton/Knothe/Bellman route without a new signed, order-flexible contraction. P111 also identifies the exact quotient-Hodge matrix H(F)=2(D_iD_jL^(-1)F)_(ij), but its natural bilinear estimate is P89 itself. Common-core fields refute local differential subordination, and Hutchinson, sparse-diagonal, and pointwise Schatten scalarizations fail at the target scale. These are strict class/structure gains and decisive branch walls, not universal closure. P87 mixed shifted Hessian is P89's stronger canonical-tensor child and also implies the stronger P22/P24 Green route, but P102 shows that the P100--P101 rooted phase of its fixed canonical representative is not invariant under the P89 zero-divergence quotient. Treat P87/P99 as stalled unless a new quotient-invariant gain appears. P131 also closes the most direct stochastic import: every exact linear full-space semigroup representation of the P87 pair trace has coefficient norm at least sqrt(n-1)/2; the natural shared-coefficient representation is sharp up to sqrt(2). A black-box full-space differential-subordination bound on one shared path therefore retains the forbidden sqrt(n) loss. This wall does not cover restricted gradient-range estimates, several coupled transforms, or nonlinear/input-dependent constructions. Inside P87, P97 gives the exact common-circle range and P98 proves its sharp half-derivative; P99 records a generic candidate random-insertion factorization but no structural contraction, so the coherent cycle--Riesz multiplier remains active; P78 is parked;

• one proved bridge: P88 dimension-neutral transfer from the Rademacher residual theorem to arbitrary product spaces;

• one parked native-compatible strengthening: P11/P82 random-order square function.

Do not create a new active branch merely by renaming P14, P22, or P11. A new branch must supply a strict gain, a counterexample, or a smaller native lemma with a decisive promote/kill test.

Native theorem and closure

The BKZ conjecture asks for

||sum_i g_i||_p <= C (p n beta + M sqrt(p n)),   p >= 2,

under

|E[g_i | Z_i]| <= M,
E[g_i | Z_{-i}] = 0,
changing Z_j, j != i, changes g_i by at most beta.

The independent one-coordinate main effects already give the M sqrt(p n) term. For Rademacher inputs, the residual problem is

R = sum_i Z_i q_i(Z_{-i}),
E q_i = 0,
||Delta_m q_i||_infinity <= beta/2  for m != i,

and the required estimate is

||R||_p <= C p n beta.

P219 proves this estimate through P14 and exact transport duality. P88 then transfers the Rademacher theorem to the general-product theorem, with only a universal change of constants. Thus the native BKZ linear rate is proved.

For the sharp all-moment formulation, put (q=\min{p,n}). The proved linear rate for (p\le n), the generic pointwise cap (|g_i|\le M+(n-1)\beta) for (p\ge n), and the saturated quadratic Rademacher example give

sup_(BKZ class) ||sum_i g_i||_p
  \asymp n beta q + M sqrt(n q).

Do not continue treating (p\gg n) as an open lower-bound regime. The separate small-(\delta) lower bound for learning-algorithm generalization remains open.

The proved P88 bridge is the dimension-neutral black-box implication

Rademacher derivative-budget rate
  => general-product residual rate.

Its proof uses one independent pair and one orientation sign per original coordinate. After centering the induced cube rows, the Rademacher theorem controls the centered part and bounded differences plus Khintchine control the empirical row means at cost (Cpn\beta). No atom count or encoding depth enters. Naive finite-atom or binary randomization remains invalid if it replaces the original coordinate count n by the number of auxiliary bits.

The bridge has been independently re-audited through v304. The exact orientation identity is obtained by swapping the two members of every pair according to the orientation sign; the product pair law is invariant under that swap. After averaging an outside orientation, replacing the whole outside pair changes a row mean by at most beta, not 2 beta. These are the two delicate constants to recheck in any future modification.

The two budgets must not be conflated

For

q_i = sum_A qhat_i(A) chi_A,

pointwise stability controls

||Delta_m q_i||_infinity
  = ||sum_(A contains m) qhat_i(A) chi_A||_infinity.

The coefficient-slice hypothesis instead controls

Lambda_(i,m) = sum_(A contains m) |qhat_i(A)|.

The coefficient condition implies the pointwise one, but the converse is false. Bent phases have bounded pointwise derivatives and exponentially large Fourier l1 slices. Therefore:

• coefficientwise martingale, cumulant, or decoupling estimates may prove a genuine stronger-model theorem;
• they do not prove the native BKZ rate without a new, separately audited bridge;
• taking Fourier absolute values before using pointwise cancellation is a known fatal move.

Exact native spine

P14: transport endpoint

For a cube function f, let

T_n(f) = sum_i ||partial_i f - E partial_i f||_(KR,n-1).

Exact duality and layer cake show, up to universal constants,

Rademacher derivative-budget rate
  <=> T_n(1_A) <= C n a log(e/a),

where a=min(mu(A),1-mu(A)). This set endpoint is the last exact native layer. It is proved by P219. P138 below remains an informative stronger strict sufficient formulation, but it is no longer load-bearing for the linear-rate theorem.

P80: exact cut-tower representation, no gain

Each row KR norm is exactly a maximum over an admissible nested cut tower, equivalently a finite maximum-weight-closure/min-cut problem. Summing rows gives a signed laminar-discrepancy formulation with the same endpoint constant as P14.

This preserves compatibility and is useful for exact falsification. It does not supply an entropy estimate or a smaller quantitative target. Treat P80 as a representation attached to P14, not as another open leaf.

P83/P85: matched sections, endpoint-equivalent

For B of density a<=1/2, let M(B) be the matched-section certificate defined in the paper. P83 gives the lower comparison and P85 proves

T_n(1_B)/a <= M(B)
  <= T_n(1_B)/a + 2 (n-1) log(1/a).

Hence the universal entropy-scale endpoint for M is equivalent to P14 up to constants. The coefficient of the unknown T_n/a on the right is one, so the comparison is not a contraction and cannot close the theorem by itself.

Two different losses must remain distinguished:

• transporting the two sections separately to the ambient cube loses Theta(sqrt(n)) on balanced majority;
• comparing P83 to exact P14 coordinate by coordinate loses Theta(sqrt(n)) in one row of the composed-majority example.

Neither example refutes the total P83 endpoint or P14. P85 works globally. The smaller Lipschitz constant of the cross-section Kantorovich potential does not produce a bias contraction: composed majority gives an unbounded violation if the marginal correction is deleted, and zero-barycenter quadratic thresholds make the proposed gain identically zero.

P86: nonlinear fixed-fiber closure

If a cube set has constant-cardinality fibers over a P14-good base, its P14 cost is bounded by the base matched certificate plus a target-scale fiber error. Fibers that are exponentially sparse in the added block preserve the P14 endpoint. In particular, graphs of arbitrary nonlinear vector-valued Boolean maps cannot amplify a P14 defect. This is a proved class closure, not a reduction of arbitrary sets to systematic form.

Do not extrapolate this theorem by graph-layer atomization or by summing transport between adjacent fibers. The first proxy assigns total exact graph cost k(k-1)/4 to the half-cube {x_1=1} x Q_m, although its native P14 cost is zero. The second loses Theta(sqrt(k)) on

B = {(x,y): H_k(x) P_m(y) = 1},

where H_k is odd-dimensional majority and P_m is base parity. Adjacent fibers are opposite majority halfspaces at Wasserstein distance E|S_k|, but the full cube routes every base row at cost one by flipping a different base coordinate. Exact block/entropy decompositions remain valid, but they retain lower-dimensional P14 with coefficient one plus a global interaction norm. Classify this as bookkeeping without descent. A viable fiber argument must preserve joint base-cube cycle coherence before taking KR norms or absolute values.

P74: exact chain rule with load-bearing child slack

The binary KR infimal-convolution identity yields

(n-1) T_n = n average(T_(n-1)) + L_2 - Sigma.

Keeping the child Bellman slack merely rewrites the parent target. Deleting it is false already for two-bit parity. Any inductive successor must add a genuinely stronger state; the exact recurrence alone is not descent.

One-way sufficient routes

P138: Euclidean row transport

For a cube function f, set

t_i(f) = ||partial_i f - E partial_i f||_(KR,n-1),
K_n(f) = (sum_i t_i(f)^2)^(1/2).

The set gate

K_n(1_A) <= C sqrt(n) a log(e/a)

is equivalent by layer cake and duality to

||sum_i Z_i q_i||_p
  <= C (p-1) sqrt(n) (sum_i Lip(q_i)^2)^(1/2).

Cauchy--Schwarz gives T_n(f)<=sqrt(n)K_n(f), while every compatible P89 flow gives K_n(f)<=C_n(f)/2. Hence

P87 => P89 => P138 => P14.

This is a strict functional contraction from P89. For odd m, put

F_m=Z_1 Z_2 sgn(sum_(r=3)^(m+2) Z_r).

Independent exact row flows give K_n(F_m)=O(1), whereas the rigid degree-three coefficients force C_n(F_m)>=c sqrt(m). Do not attempt to reconstruct a common P89 optimizer from the P138 row optimizers. The live mathematical target is still their anisotropic Euclidean moment gate, but the P141 audit leaves no immediate same-direction child. Mandatory tests are singleton, antipodal, full and fixed-degree parity, majority, quadratic tails, simplex/affine codes, this twisted-majority family, and the indexed-majority multiplexer below.

P139 narrows the adversary space. Every nonlinear systematic graph

A_F = {(F(y),y): y in Q_m}

of codimension k satisfies P138 uniformly, and every affine code has exact row energy

K_n(1_C) = (a/2) (sum_i (d_i-1)^2)^(1/2).

The diagnostic identity

R_138 = R_14 sqrt(n/N_eff),
N_eff = T_n^2/K_n^2,

shows that a remaining obstruction must make R_14 sqrt(n/N_eff) unbounded. Strong anisotropy may compensate even a vanishing P14 ratio, so do not add a near-extremality hypothesis. This is a class/adversary contraction, not a universal proof.

P141 adds one class theorem and four method walls. If every base fiber has fixed cardinality R and R <= 2^((1-delta)k), disjoint graph enumeration proves P138 with a constant depending only on delta. Positive graph assembly is not universal: every systematic graph contained in odd majority has more output than base coordinates, so graph bounds plus Minkowski lose Theta(n) although majority itself is P138-benign.

The exact thresholded diagnostic is

K_n(1_A)^2
  <= lambda^2 n a^2 L^2
     + a^2(n-1)/2 sum_(i:t_i>lambda a L) (ell-d_i),

where ell=log(1/a), L=1+ell, and d_i=D(nu_i||uniform). Do not conjecture a separate bound on the last sum. For k=2^q, a label V in Q_q, outputs U in Q_k, odd-majority input Y in Q_M, and

B = {U_(J(V)) = sgn(sum_j Y_j)},

all marginals are uniform and every output row has

t_(U_i)/a = E|S_M|/(2k).

Choosing M >> lambda^2 k^2 makes all k output rows high, so the separate entropy charge is k log 2, while the actual P138 ratio is O(k^(-1/2)). The proxy has charged one shared latent majority transport once per label. No fixed threshold or weak-type count repairs it. A joint entropy-rank quantity

R_nu(H)=D(nu||mu)-D(nu_(H^c)||mu_(H^c))

survives this witness but is only an unproved preflight direction, not a proved lemma.

P142 proves the exact individual deletion-entropy control

e_i = D(nu||mu)-D(nu_-i||mu_-i)
    = log 2-H_nu(Z_i|Z_-i),
(t_i/a)^2 <= (n-1) L e_i / log 2.

Do not sum these charges. For odd (k,M), the half-density set

B_(k,M)={Maj_k(U)=Maj_M(Y)}

has, for every (U_i),

t_(U_i)/a=p_(k-1) E|S_M|/2,
e_(U_i)=p_(k-1) log 2.

Taking (M/k) large makes all (k) rows high, but their deletion charges sum to order (\sqrt{k}), while the exact total energy satisfies ((K_n/a)^2\le n/4). This refutes deletion-entropy packing for every fixed threshold.

For (H\subseteq[n]), joint rank has the exact localized interpretation

R_nu(H)=D(nu||nu_(H^c) tensor mu_H),
(sum_(i in H) r_i^2)^(1/2)
  = sup |(E_nu-E_(nu_(H^c) tensor mu_H))
           sum_(i in H) Z_i q_i|.

The threshold-selected estimate

sum_(i in I_lambda) r_i^2
  <= C n L R_nu(I_lambda)

has completed its bounded P143 audit. If (h=|H|), (d=n-h), (R=R_\nu(H)), and (S=\log(1/a)-R), then

sqrt(sum_(i in H) r_i^2)
  <= C [sqrt(d R (1+S))
        + min(h sqrt(R),sqrt(h R)+h^(3/2)R)].

This proves the gate for (h^2\lesssim nL) or (h^3R\lesssim nL). Balanced monotone equality compositions satisfy

sum_(i in I) r_i^2 <= (n/2) R_nu(I)

for every subset (I). In general, however, the internally centered conditional field is exactly P138 on (Q_h). Importing lower-dimensional P138 produces a fiber term plus a latent term whose cross product is not constant-preserving. The full-block case is P138 itself, while parity refutes chain-increment allocation from submodularity. Park unrestricted joint rank; retain the proved regimes and the dense nonmonotone large-high-block falsifier profile.

P144 audits both proposed repairs. Bare directed Beckmann flow is exactly P138, while reciprocity is P89. Static global contact is the exact AVaR dual of P138 and has no cross-row KKT equation. Repetition-code contact makes the P143 internal and external fields orthogonal under the reference law but aligns their expectations under the conditioned law, so orthogonality does not remove the cross. Universal no-cross contains P138 at H=[n], although every affine code obeys

sum_(i in H) r_i^2
  <= [h L R_nu(H)+d L^2]/[4 (log 2)^2].

Park bare flow, static contact, and orthogonality-based no-cross as smaller routes. A future re-entry needs common-indicator integrability, second-order optimizer stability, or a nonlinear entropy-weighted directed constraint.

P145 completes the first two re-entry checks. Curl is the complete linear common-origin law; Booleanity adds nonlinear convolution and same-orthant gradient identities. Global maximizers satisfy the all-normers normal-cone and swap-square theorem, but opposite singleton maximizers have an exactly flat chord. Local curvature is therefore parked. The exact surviving induction contract is

|I| <= C_h sqrt(h)(1+R),

followed by the open full defect gate

2 I M + M^2 + sum_(j in Hc) r_j^2
  <= C_*^2 h(1+R)^2-I^2+C_0 dL^2.

Its d=1 case closes P138 with the same constant. Repetition forces the dL^2 order and the removed-row term. The antipodal simplex-star family forces dense, nonlinear, all-positive saturated contact while keeping the P138 ratio bounded. This full extension-defect inequality remains the active induction framework below P138; do not replace it by a weaker no-cross inequality which omits removed rows.

P146 isolates its sharpest normer-independent codimension-one child. With actual section energies E_r^+ and E_r^- and

B_r=(sqrt(E_r^+)+sqrt(E_r^-))/(2a),
Delta_r=E/a^2-B_r^2,

the active theorem is min_r Delta_r <= C log(e/a)^2. This closes P138 by induction. Balanced majority refutes choosing r in advance but has bounded debt on every majority coordinate. The averaged exact identity begins with E/(na^2), so scalar entropy or unsigned averaging is circular. The missing step is adaptive signed-defect coordinate selection.

The exact squared-row deletion law has a positive defect that exceeds every local one-step scalar entropy reserve by order n on antipodal pairs. Full random-restriction telescoping does not repair it: singleton and radius-one families force both negative savings with coefficient one, at which point the state is exactly K_n^2/n. The exact row-divergence quotient and the scalar integration-by-parts inequality remain valid, but the absolute resolvent--carre estimate is false: a complete quadratic plus scaled full parity loses sqrt(n) after the cross-degree product is put inside absolute values, while its actual P138 ratio is bounded. Retaining signs repairs that witness, but the cube chain rule then carries edge-dependent divided differences; the weighted estimate is P138 itself. The unweighted signed field can be negative and has no diffusion chain rule. Do not reopen absolute mixed-scale products, scalar full-order deletion, or separate high-row entropy packing.

P89: quotient before norm

For a cube function f, put

H_i(f) = D_i f - fhat({i}) Z_i.

Minimize over all symmetric compatible pair arrays

U_ij = U_ji,       D_i D_j U_ij = U_ij,
sum_(j != i) D_j U_ij = H_i(f),

and define

C_n(f) = inf_U [sum_i (sum_(j != i) ||U_ij||_1)^2]^(1/2).

The canonical choice U_ij=R_ij f is always admissible. P89 asks for

C_n(f) <= C sqrt(n)/(s-1) ||f||_s,                1<s<=2.

Its set form and strong-L^s form are equivalent to the exact dual

||sum_i psi_i||_p
  <= C (p-1) sqrt(n) tau_2(c),
c_ij = ||D_j psi_i + D_i psi_j||_infinity,

where D_i psi_i=psi_i, hat psi_i({i})=0, and

tau_2(c) = inf { ||u||_2 : u_i>=0, c_ij <= u_i+u_j }.

The exact Fourier Euler identity

sum_i psi_i
  = (L-I)^(-1)_(>=2) sum_(i<j)(D_j psi_i+D_i psi_j)

shows why the symmetrized derivative field, rather than the individual derivatives, is the native datum. Pair-array norm duality identifies its dual norm exactly with the Euclidean fractional vertex-cover norm tau_2.

The proved arrows are

P87 => P89 => P138 => P14.

The s=2 endpoint and dual range p>=n are proved. Singleton sets are asymptotically sharp, and on disjoint parity checks the optimized compatible flow gives C_n(1_A)=sqrt(n) a, avoiding the known divergent canonical Green/transport ratio. The live range is 2<p<n.

For a fixed cyclic order, an exact selector identity splits sum_i psi_i into an extreme-pair half and a co-interval half. Two nested predictable Rademacher inequalities prove the target bound for the extreme-pair half. The compatible co-interval estimate is quantitatively equivalent to P89. A parity selector reduces sufficiency to a matching field and its p=2 bound follows from averaging over every additive chord on each Walsh support. This is not yet a contraction: on every even Walsh level the matching selector equals the full P89 scalar field.

Never drop the common-potential form A_ij=D_j psi_i+D_i psi_j. A unit-edge arithmetic-progression array in the pair ranges violates the generic co-interval/matching estimate by sqrt(log n) in L2; the amplification is entirely in a forbidden curl component and disappears under the additive chord identities b_ij=a_i+a_j. Pointwise splitting according to a fractional vertex cover also destroys these identities. Every single-Walsh-support compatible field is target-good, so a real falsifier must mix supports and use cross-support L-infinity cancellation.

P90 now supplies large proved sectors. The P89 target holds when the output has binary basis degree at most two, partitions into O(p) binary-independent families, or has O(p^2) active characters. An optimized three-color restriction proves every homogeneous degree with universal constant 2e. Normalizing two color scales at their respective levels proves any two arbitrary degrees with constant 4e, independently of parity and separation; antipodal projection closes at most two even and two odd levels with constant 8e. Positive two-node quadrature closes every three-term arithmetic progression of output levels with universal constant 225. A matched-scale Neumann argument closes m 64-lacunary levels with constant 32 e m/15, uniformly in their locations. Bounded mixed degree and phase-aligned spectra of mean degree at least order n/p are also safe. P91 closes arbitrary-degree multi-core gradient--quadratic towers. P93 proves much more: for every fixed m, a confluent Muntz endpoint compactness theorem and finite-dimensional polar synthesis close every spectrum supported on any m arbitrary output levels, with a constant depending only on m. Colliding rates are handled by Hermite divided differences and separated log-scale clusters localize independently. A surviving obstruction must therefore have an unbounded number of interacting output levels. It must also remain nonlinear across every core/outside split and retain sign frustration, overlapping circuits, cross-edge L-infinity cancellation, and a heavy output.

The homogeneous coloring does not combine by scalar triangle estimates. Any signed mixture reproducing levels through degree D+2 has weighted variation at least 2 H_(D+1). Therefore seek cancellation across the color/scale parameter; do not report degreewise recombination as a dimension-free proof. On m 64-lacunary levels every exact scalar mixture has total variation greater than 1.14m, although their Calderon Gram matrix is almost diagonal. Scalar scale stopping is therefore provably linear in the number of separated levels. In particular the P93 constants produced by its scalar-synthesis proof cannot be uniform in m; positive lacunary coefficients force at least linear growth. This is a method wall, not a counterexample to P89.

The normalized color family has an exact Calderon isometry under dmu(q)=16 q log(1/q)/(1-q)^2 dq; on degree parameters r,s its Gram kernel is 4rs/(r+s)^2. This is the correct multilevel language, but do not stack a one-sided edge-square estimate with a generic reconstruction theorem. Geometrically separated independent degree levels force a sqrt(p) synthesis cost, while a single quadratic level forces the full p edge cost and has synthesis constant one. Their product p^(3/2) is artificial. The live P89 contract is the joint edge/test-function bilinear embedding under the Calderon reproducing formula, with moment budget allocated by scale.

Do not separate the dual test function through a pointwise Tent--Schatten functional and then integrate linearly in the color scale. At p=2, take disjoint Walsh supports with degree parameters r_m=4^(m-1). The input has L2 norm sqrt(m), whereas each Hessian block contributes a unit-mass scale bump to the operator-norm leg; the lacunary bumps occupy disjoint scale windows and accumulate as m. Thus the proposed test-only S2/S1 interpolation estimate is false. The compatible edge field and the test function must remain jointly coupled across scales.

Standard Clifford or Haagerup linearization does not create this coupling. For the Calderon embedding J, the Clifford field satisfies |gamma(Jf)|=|Jf|_(ell_2) I, so noncommutative Holder reduces exactly to the separated square-function product ||JF||_(L^p(ell_2)) ||Jh||_(L^(p')(ell_2)). The lacunary reconstruction witness then restores the same artificial p^(3/2) loss. Treat this as the existing separation wall in different notation, not as a new route.

Do not try to obtain that allocation from pointwise semigroup-variance BMO along a fixed radial noise filtration. For the compatible common-core field F=chi_C sum_(l in L) Z_l, with |C|=m and |L|=N, the chord graph is the unit K_(m,N) and n tau_2^2=mN, but at the all-one point its noise-semigroup variance is at least N^2/4. The BMO/native-scale ratio is therefore at least (1/2) sqrt(N/m). This kills pointwise radial/Carleson conditional-variance allocation only. An L^p square estimate, a spatial exposure of the core, or the joint test-function bilinear integral remains viable.

Optimizing a coordinate reveal order does not repair this endpoint. For the complete quadratic P89 output F=sum_(i<j) Z_i Z_j, every chord norm is one and sqrt(n) tau_2=n/2. For every order, reveal r=floor(2n/3) coordinates and condition on their all-positive atom. With m=n-r and U the unseen sign sum,

F-E_r F = r U + (U^2-m)/2,
E_r |F-E_r F|^2 = r^2 m + binom(m,2).

Thus the best-order coordinate-martingale BMO norm is order n^(3/2), an extra sqrt(n) over the native budget, even though the homogeneous quadratic class is P89-good. Treat every essential-supremum filtration BMO proposal as refuted unless it uses a genuinely different, moment-averaged state rather than merely choosing or randomizing the coordinate order.

Do not replace the compatible quotient by rowwise Hilbertization: the cube transportation space contains isometric l1 stars, which lose sqrt(n). Do not infer P89 from scalar L2/L-infinity interpolation: a compatible star forces the resulting n^(1-1/p) loss. These are mechanism failures, not counterexamples to P89.

Supportwise graph orthogonalization is also exhausted. Degree-regular selectors are exactly the P89 scalar in each realization; rooted stars lose sqrt(k) through degree fluctuation; and the Hamilton path is Q_pi=2F-W_pi, hence equivalent to the existing co-interval crux. A compatible common-core field shows that taking a local Korn trace after row summation loses sqrt(n), or sqrt(n/k) even relative to one derivative square tensor. A new route must use cancellation jointly across Walsh supports before scalar trace, or be genuinely non-graphical.

P110 sharpens both the quotient and matrix boundaries. For a binary quotient map Pi(x)=sum_i x_i gamma_i, let s be the number of nonzero label classes which occur exactly once. Quotient conditioning and within-class symmetrization prove

||F||_p <= C(1+K)(p-1) sqrt(n) tau_2(c),

whenever s<=Kp; if s=0, then ||F||_p<=sqrt(n) tau_2(c) for every p. A scalable quotient-code falsifier must therefore have s/p tending to infinity. Repeated-column amplification is not a survivor.

Do not turn this into a homogeneous atomization. For the even-dimensional bent quadratic phase g=(-1)^(sum_(i<j)x_i x_j) and F=L P_(>=2)g, the full quotient norm is at most sqrt(n)/2, while

[sum_(k>=2) rho_n(P_k F)^2]^(1/2) / rho_n(F)
  >= c 2^(n/2) n^(-1/4).

Thus even square-summing independently optimized homogeneous quotient norms destroys mixed-degree cancellation exponentially. Mixed-degree atoms remain logically possible.

The exact compatible matrix reconstruction

K(A)_(i,l) = sum_(j!=l) D_i D_j (L-I)^(-1)_(>=2) A_(j,l)

satisfies Tr K(A)=2F. On complete quadratic chaos,

||Tr K||_p >= c p n,
||K||_(L^p(S_2)) <= C n sqrt(p)

for a universal range p_0<=p<=cn. Therefore a separate trace after an universal separate trace comparison must lose at least sqrt(p). Appending it after an already order-p black-box matrix estimate cannot recover an order-p budget; its factor accounting has a p^(3/2)-scale lower requirement. A viable structure-sensitive matrix proof must couple transform and trace or improve the first-stage moment cost.

P111 gives the exact multistage quotient upgrade. At quotient stage t, write a_t for repeated-row cover mass, v_t for total singleton cover mass, b_t for the dimension of the repeated-label span, s_t for the number of singleton labels, and rho_t for the minimum cover mass of a repeated-label basis. The discarded pieces are reverse martingale differences and satisfy

||F_0||_p <= ||F_T||_p
  + C sqrt(p) [sum_t(a_t^2+b_t v_t^2+s_t^2 rho_t^2)]^(1/2).

Thus the terminal high-moment endpoint closes P89 whenever its active count is O(p) and the displayed square charge is O(p n tau_2(c)^2). The charges square-sum because the one-stage moment-generating bounds are conditional with deterministic proxies; never replace this by a triangle sum over flag depth.

This subgaussian gate is sufficient, not necessary. On the depth-one family with singleton labels g+a over H=F_2^b and doubled basis labels, the output is

F = 2^b 1_{0} - 1.

Its actual normalized P89 ratio is O(1/p), but every fractional cover gives a P111 bounded-difference charge whose normalized excess is at least sqrt(b/(16p)). The discarded increment lives on a rare point, which a worst-case derivative proxy does not see. P112 gives the correct general upgrade: if every reverse increment has conditional Bernstein parameters sigma_t^2 and L_t, then

||F_0||_p <= ||F_T||_p
  + C(sqrt(p sum_t sigma_t^2) + p max_t L_t).

The canonical choices are the essential supremum of the conditional variance and one third of the increment peak. They handle the preceding rare-event family, but they are not universally controlled by the original cover. For the P113 repeated-core/leaf potential,

n tau_2(c)^2 = 2mN,
E(Delta_0^2 | F_1) = (sum_(a=1)^N Z_a)^2,

so every deterministic variance parameter is at least N^2 and loses N/(2pm). The output itself satisfies the P89 target with room. Do not take conditional-variance suprema before the outer moment. Keeping the bracket in L^(p/2) is useful only if compatible structure also controls the maximal increment; generic Burkholder--Rosenthal fails here, and symmetric one-bit refinement returns to SF-P89/P82.

A separate P112 theorem closes every canonical Boolean corrected gradient: for f in {-1,1},

psi_i = D_i f - fhat({i}) chi_i

obeys

||sum_i psi_i||_p <= (2+sqrt(2)) sqrt(n) tau_2(c).

At each vertex the sensitive coordinates form a clique of unit chord constraints. This includes all Boolean quotient phases, degrees, label multiplicities, and the former quadratic-bent class. A Boolean-derived survivor must use a noncanonical compatible gauge with a substantially smaller chord cover.

P114 audits that caveat under the full zero-output gauge. If F=L P_(>=2) f, f is Boolean, and k_x is the local sensitivity, then every compatible representative satisfies

tau_2(c) >= (2 k_x - n)_+ / (2 sqrt(n)).

The proof uses the exact gauge normal form A_ij=2D_iD_j f+D_j r_i+D_i r_j, the pointwise identity sum_(i<j)(D_j r_i+D_i r_j)=0, and the Boolean face-sign trichotomy. Thus the desired pointwise scale holds whenever k_x>=3n/4. The remaining balanced Boolean--Korn region is strict but Boolean-only.

Do not infer that the canonical Boolean cover is the native quotient. Already at n=3 there is an exact Boolean output with tau_can=sqrt(2) and rho=2/sqrt(3). Its common-core extension has tau_can/rho=sqrt((m+1)/m), so this natural gap tends to one. The exact fixed-output dual keeps the full P89/P102 feasible ball and changes only the exposed Boolean objective. Re-dualizing it or maximizing tau_can/rho is not a new concentration route.

P115 completes the bounded follow-up. If S=max_x k_x, then every compatible representative satisfies

rho_n(L P_(>=2) f) >= S(S-1)/(2(n-1)sqrt(n)).

The proof pairs the pointwise Boolean two-face graph at a maximum-sensitivity vertex with P114's exact zero-gauge cancellation. A second, global gauge-annihilating pairing with C_ij=2D_iD_j f gives

rho_n(L P_(>=2) f) >= sqrt(E[k(k-1)]/(n-1)).

Full coordinate symmetrization forces the canonical allocation for every real permutation-invariant function (with a sharper Boolean consequence), and on disjoint scalar outputs the native quotient obeys the exact Euclidean tensor law

rho(F_I+F_J)^2 = rho(F_I)^2+rho(F_J)^2.

These are genuine quotient theorems, not canonical-cover estimates. They close all Boolean families with S>=delta n, all signed unions of Hamming layers, and additive repetition of a fixed disjoint output. They do not localize the offending value: a remaining counterexample sequence must have S/n tending to zero and |F(x)|/sqrt(E[k(k-1)]) tending to infinity. Park this Boolean-only branch unless a new localized gauge-annihilating certificate is supplied.

P116--P120 give the fixed-orbit follow-up. The orbit-absolute envelope

C_n(h) <= O_n(h) <= D_n(h)

has the exact representation

O_n(h) = sup_(||sigma||_infinity<=1) C_n(M_sigma h),

where the degree-k symbols are correlated Gamma moments and adjacent high-degree symbols differ by O(1/k). It contracts the full P105 affine-spread proxy by sqrt(M), but sigma=1 gives M_sigma=P_(>=2). Therefore it is not a smaller P89 obligation. More decisively, complete quadratic chaos and a normalized point mass prove that every unweighted separated conjugate mixed-norm factorization of the two fixed-orbit legs loses sqrt(n), for every time exponent. Any viable scale allocation must pair the two orbits before taking norms.

Positive joint reweighting is still too coarse. For the complete quadratic paired with a normalized point mass at p=n/4, the optimal positive density on the full (q,z) orbit space satisfies

inf_W ||W^(1/2)U||_2 ||W^(-1/2)V||_2 = integral |UV|

and the right side is at least c n^(5/2), versus native budget 2n^2. Positive stopped local Hölder sums inherit the same bound. Unrestricted signed partitions reduce to the one-cell exact pairing and hence merely restate P89. Therefore a viable stopping/corona state must be constrained, phase-sensitive, and cancellation-preserving. Modulus-only conclusions apply only on sign-closed input classes; do not overstate this wall against structure-sensitive Calderón-orbit forms.

The exact signless-incidence coordinate

F = R^* I A

has squared support norm k/(2(k-1)) and contracts the transverse gauge mode by at most 1/4. This is a useful Hilbert identity, not an all-moment gain. Common-core stars refute pointwise domination, matchings refute operator-norm-only input, and complete quadratics plus a degree-three tournament force both sqrt(p) moment stages. Keep the joint incidence estimate as a bounded assay only; generic two-stage Riesz estimates reopen a closed wall.

Unsigned supportwise spectral data are also insufficient. An aligned flat Walsh spectrum and a nondegenerate quadratic bent spectrum have identical canonical labelled support incidence, coefficient magnitudes, and radial Hodge data, but their exact native quotient and pointwise edge-Hilbert ratios are 2^(n/2-1). Retire only phase-blind/supportwise association-scheme, Grothendieck, and independent-randomization implementations. Some cross-support phase information is necessary, but the full signed Gram field is only one complete repair, not a uniquely forced one; quotient-aware and globally signed encodings remain open.

The same round sharpens the matrix audit. Right Helmholtz projection gives

H(F) = P_R K(A) = 2(D_iD_j L^(-1)F)_(ij),
Tr H(F)=2F,
||H(F)||_(L2(S2))=2||F||_2.

This removes all trace-null representative gauge before taking norms. Nevertheless the natural H(F)--H(h) pairing is exactly P89, while a common-core family makes both pointwise and coordinate carré-du-champ comparisons lose sqrt(N/k). Hutchinson scalarization reproduces the scalar G plus an off-diagonal chaos; unbiased sampling of n/p diagonals loses sqrt(p) on the complete quadratic; and disjoint Walsh characters refute the pointwise Schatten-to-scalar dual step. Use the quotient compression in future matrix work, but do not claim local differential subordination, standard scalarization, or a new contraction.

P87/P22/P24/P78: mixed and signed Green route

The canonical signed Green flow obeys

T_n(f) <= G_n(f).

Thus

G_n(1_A) <= C n a log(e/a)                         (P22)

would imply P14. P22 is equivalent up to universal constants to the global shifted-Hessian estimate

sum_(i != j) ||R_ij f||_1 <= C n/(s-1) ||f||_s,   1<s<=2,

and to its bounded two-index dual P24, with the shifted resolvent retained. These statements are stronger than P14, not equivalent to it.

The universal antipodal extremality statement P78 is open, but the implication

P78 => P22 <=> P24 => P14

is proved. P78 is proved for every affine set and every set of at most three vertices. A sequence with divergent normalized P22 ratio would have to be nonlinear, asymmetric, have density tending to zero, and have |A|=2^(n-o(n)). One-level atom pairing is sqrt(n) short in exactly that window, so a viable kernel proof needs multilevel cancellation stable under cluster merging.

P64 shows that G_n/T_n can diverge on disjoint parity checks while the P22 endpoint itself remains valid. This kills constant-factor Green/transport faithfulness, not P22. P89 repairs this particular gap by optimizing over compatible flows before taking row norms. The direct P78 extremality attack is operationally parked. P87 remains a valid stronger child of P89, but P102 shows that its canonical rooted phase is not quotient-invariant; treat this route as stalled unless a new quotient-preserving gain is found.

P87 asks for the exterior mixed endpoint

E_n(f) := (sum_(i != j) ||R_ij f||_1^2)^(1/2)
          <= C/(s-1) ||f||_s,                    1<s<=2.

Its set form, this L^s -> ell_2(L^1) form, and the exact dual

||sum_(i != j) R_ij a_ij||_p
  <= C(p-1) (sum_(i != j) ||a_ij||_infinity^2)^(1/2)

are equivalent up to universal constants. Cauchy--Schwarz across the ordered pairs supplies the single ambient factor n, so

P87 => P89 => P138 => P14,
P87 => P22 <=> P24 => P14.

At s=2 the exact square sum is at most 2 ||f||_2^2. In the bounded-field dual, the current concrete crux is a resolvent-compensated trace from a fully decoupled marked field to the coupled two-front field. It is bounded in L2 and L-infinity; ordinary interpolation on the marked subspace is not dimension-free, as a complemented first-chaos retract proves. The witness is mapped isometrically, so this kills only interpolation.

A random-linear-order wrap identity now proves the all-degree estimate one number-operator inverse below P87:

||L^(-1) S_b||_p <= C p beta_2(b),
beta_2(b) = (sum_(i != j) ||b_ij||_infinity^2)^(1/2).

It follows that outside degree at most D costs C p (D+2)^2 beta_2, while one homogeneous outside level of degree d costs C p (d+2) beta_2. Any counterexample therefore needs unbounded outside-degree complexity. The exact remaining operation is recovery of one number-operator derivative using cancellation among the complete pair field. Do not estimate the exact cyclic selector one order at a time: an aligned full-parity witness loses sqrt(n) in L2, even though its order average has constant scale. Scalar coloring mixtures also cannot close the gap; their total variation grows at least quadratically in the degree and no finite-total-variation selector reproduces all degrees.

The cyclic order average has a stronger square gate

H_b = (E_sigma |T_b^sigma|^2)^(1/2),
||H_b||_p <= C p beta_2(b).

This gate is sufficient for P87 and is proved when the output supports partition into at most p binary-independent families, for affine quadratic support clouds, at every fixed outside degree, and at arbitrary degree whenever one coordinate order places every dependency of b_ij before both endpoints. Two nested predictable Rademacher martingales prove the full cyclic-square estimate on this forward-dependency class. Pointwise order variance is not controlled: a one-core tournament loses order n, although its averaged square gate is target-good. Levelwise homogeneous aggregation is also false: odd majority on one edge has actual H_b<=1 but a degree-one row projection of L-infinity size Theta(sqrt(m)).

Generate the order by iid circle angles. Conditional on two endpoints, the mean arc selector on row b_ij is exactly T_q b_ij, with q the complementary arc length. Pairwise independence of distinct circle differences proves the pointwise identity

|S_b|^2 <= E_angle |M_b|^2 <= |S_b|^2+2 beta_2(b)^2.

Thus the positive-noise square is equivalent to P87, not a separate Carleson debt. The sole new cyclic-square obligation is the conditionally centered shared-angle fluctuation X_b in L^p(Z;L^2(angle)), only for 2<p<n. Its rows are not mutually independent, and pointwise Efron--Stein is unavailable.

Each fluctuation row nevertheless has an exact Bernoulli-restriction martingale telescope with total conditional energy at most ||b_ij||_infinity^2. P97 strictly sharpens this averaged, ordered fact. Along the symmetric radial path from the noise point (q,...,q) to the realized restriction, it gives

X_ij = sum_k (eta_ij,k-q_ij) Gamma_ij,k,
q_ij(1-q_ij) sum_k |Gamma_ij,k|^2
  <= 2 ||b_ij||_infinity^2

pointwise, with Gamma_ij,k independent of its current marked angle. The formula retains every Hoeffding order and uses no revealing order. Its innovation is the common-circle coboundary eta_ij,k-q_ij=rho_k(j)-rho_k(i). Fourier expansion in one marked angle has weights 1/(4 pi^2 m^2) of total mass 1/12 and turns the field into an endpoint graph divergence. The precise open operation is assembly of the jointly radial-path-generated coefficients over the shared marked angles with the Z moment outside the angle square. Generic centered-row/predictable assembly is false by sqrt(n), and pointwise incidence assembly fails on the tournament; neither wall refutes the generated common-circle field, for which common cycle weights telescope and the tournament has the stronger C sqrt(p) beta_2 bound. Even the exact normalized circle innovations plus pointwise row ell_2 energy are insufficient for arbitrary coefficients: nested intervals sharing one mark have positive covariance and aligned Walsh fronts of size N against budget sqrt(N). The radial-path range relation across all marked directions of each Walsh monomial is therefore load-bearing; never replace P97 by a generic row-energy lemma.

P98 proves that this generated range has a strict quantitative gain. On each full Walsh support S, the genuine angle operator satisfies

||A_S c||_(L2(angle)) <= 3 / sqrt(|S|-1) ||c||_2,

and Walsh orthogonality yields the sharp-scale global endpoint

||L_Z^(1/2) X_b||_(L2(Z,angle))
  <= (1+3/sqrt(2)) beta_2(b).

The |S|^(-1/2) scale is already sharp on one edge. In one row, the exact cross-support Gram law is

<Phi_(ij,A), Phi_(ij,B)>
  = |A intersect B|
    / [ (|A union B|+1) (|A|+|B|+1) ],

so disjoint outside supports are orthogonal. The tower also obeys the deletion identity

E_(Theta_k) Phi_(ij,A)
  = q_ij Phi_(ij,A minus {k}).

The same-row normalized tower factors exactly as a double-random-restriction square, equivalently

2 int_(0<v<u<1) T_u [
  sum_k |D_k (I+L)^(1/2) T_v g|^2
] dv du.

This is useful only before row aggregation. The normalized common-core star g_(0,l)=N^(-1/2) Z_1 has global angle square (N+1)/6 at the all-one cube point, while the sum of its individual-row angle squares is 1/3. Its actual mixed Lp norm is at most sqrt((p-1)/3). Thus pointwise aggregation of the exact rowwise conical squares is another lossy proxy; the global proof must retain cross-edge angle covariance coupled to the front signs.

Normalize with

g_ij = (I+L_(-ij))^(-1/2) b_ij.

Use the shifted resolvent, not the unshifted half inverse: it is the positive probability mixture

pi^(-1/2) int_0^infinity t^(-1/2) e^(-t)
  e^(-t L_(-ij)) dt,

so |g_ij|<=||b_ij||infinity pointwise. Exactly X_b=N g, where the coherent cycle--Riesz multiplier N multiplies outside degree d by sqrt(d+1) Phi(ij,A). Thus the current sufficient contract is

||N g||_(L^p_Z(L2_angle))
  <= C p ||(sum_(i!=j) |g_ij|^2)^(1/2)||_p.

This is one joint multiplier inequality, not two iterated first-order Riesz estimates. P98 supplies its dimension-free L2 support symbols, but generic block-multiplier promotion is false: artificial sign symbols on a bent phase have exponentially growing L^p norm. Standard coordinate-martingale and noise-semigroup BMO endpoints are also false by sqrt(n) on the target-good one-core tournament. These failures are lossy-proxy walls, not counterexamples to N. A proof must keep the actual radial deletion/insertion coherence and average rare configurations rather than take an essential supremum over stopping states.

P99 records one symmetric sufficient insertion architecture. Write each multiplier term as

a_alpha chi_(S_alpha) psi_alpha,   psi_alpha in L2(angle).

For a reveal set R and a fresh coordinate k, let V_(R,k) sum precisely the terms with k in S_alpha and S_alpha minus {k} subset R, after removing the final sign Z_k. For a uniform permutation pi,

V_pi^2 = sum_m ||V_(R_(m-1),pi_m)||_(L2(angle))^2

is the pathwise square of a conditionally symmetric Hilbert-valued martingale. The sharp martingale square theorem spends only sqrt(p), so the averaged candidate contract

(E_pi E_Z (V_pi^2)^(p/2))^(1/p)
  <= C sqrt(p) ||(sum_(i!=j) |g_ij|^2)^(1/2)||_p.

Do not replace this by the quadratic order average. Pointwise in Z, that average has the exact identity

E_pi V_pi^2
  = 2 int_0^infinity e^(-tL)
      [sum_k ||D_k e^(-tL) N g||_(L2(angle))^2] dt,

but Jensen runs in the wrong direction for the required p/2 moment. Pointwise reveal-state contraction is also false on the common-mark star. The factorization itself is generic for every Hilbert-valued Walsh polynomial, and the martingale argument needs only an infimum over reveal orders whereas P99 averages them all. Thus this is a candidate proof gate, not a strict reduction. A successful use would have to estimate the rare coherent fronts in mixed L^(p/2) using the signed cross-edge circle Gram matrix. P98 already closes p=2; the live range is 2<p<n.

P100 sharpens the endpoint-conditioning state below P99. For one fixed root and orientation, define

B_(j,A)^(r)
  = integral_0^1 t^r [1_(M_(j,A)<t)-t^|A|] dt.

Then outside-angle deletion is exactly

E_(Theta_y) B_(j,A)^(r) = B_(j,A minus {y})^(r+1),

and the same-root Gram law is

<B_(j,A)^(r),B_(j,C)^(s)>
  = |A intersect C|
    / [ (|A|+r+1)(|C|+s+1)
        (|A union C|+r+s+2) ].

The normalized reverse-martingale increment energies telescope without a harmonic loss. Endpoint averaging maps the generalized arc-power state q_e^r(I_(e,A)-q_e^|A|) into this rooted tower and gains the expected half derivative. On one full support, the two incident orientations have an exact Dirichlet-gap projection. In one row, the first boundary is also an exact shifted inverse-gradient conical square. Direct Minkowski and Riesz give a coarse O(p) bound; P101 below gives the sharp O(sqrt(p)) bound.

This does not close P99. The desired insertion-energy subcontract is O(sqrt(p)), but rowwise assembly is the already-refuted common-core-star proxy. The entire fresh-angle residual can be written as one signed interval covariance form

Delta_x = integral h_I(Theta_x) d mu_x(I),
E_(Theta_x)|Delta_x|^2
  = double_integral (|I intersect J|-|I||J|)
      d mu_x(I) conjugate(d mu_x(J)),

which retains endpoint/outside cross terms. A Bellman state containing only the original radial atom and the first centered maximum is exactly nonclosed: at outside degree two its first descendant leaves residual energy 1/720, one sixteenth of its norm. Deleting a root creates the new gap potential s^2-s+1/6. Thus the live state must retain the full power tower, signed interval covariance, and root-collision descendants. No L^(p/2)_Z inequality for that state is known.

P101 sharpens the quantitative boundary conclusion. If B_a denotes any family of rooted boundary operators and g_a their row inputs, then

|| (sum_a ||B_a g_a||_(L2(angle))^2)^(1/2) ||_p
  <= C sqrt(p) (sum_a ||g_a||_infinity^2)^(1/2).

After h_a=(I+L_a)^(-1/2)g_a, the sum of the exact P100 conical squares is pointwise dominated by the average coordinate-Doob square of the bounded Hilbert-valued terminal vector h=(h_a). Conditional future square energy is at most its terminal bound squared, so a stopping-time argument gives an exponential tail and the sharp sqrt(p) moment. Thus the earlier O(p) one-row estimate is lossy; same-root and external-direct-sum moment growth are closed at the scale P99 needs.

Do not infer the actual incident-boundary estimate from this direct sum. For a fresh endpoint x, the true field is

sum_(j!=x) Z_j (B_j^+ g_(xj) + B_j^- g_(jx)).

Each coefficient is independent of its own front sign but can depend on all the others, so this is a nonpredictable divergence. A second Khintchine or martingale estimate spends another sqrt(p). The lost phase is already full size on two coordinates:

B_(j,{k})^(0),+ = -B_(k,{j})^(0),+,
||B_(j,{k})^(0),+||_2^2=1/12.

The signed interval covariance has the exact phase-sensitive form

double_integral kappa(I,J) dmu(I) conjugate(dmu(J))
  = ||rho_mu||_(Hdot^(-1)(circle))^2,

where rho_mu is the distributional derivative of integral h_I dmu(I). Outside intervals give atomic graph-divergence charges, an endpoint residual gives delta_M-d u^(d-1)du, and the first root-collision descendant s^2-s+1/6 is twice the circle Green kernel.

P99 itself holds for every fixed reveal order on the generated one-core tournament. Its insertion square is a positive-semidefinite degree-two Rademacher form; the PSD fourth-moment identity and Bonami give

||V_pi||_p <= sqrt(p) B

uniformly in the order and dimension. Together with the exact common-mark star formula, this clears the two strongest rare-front/BMO adversaries but does not control general interacting root-collision states.

The all-degree forward theorem is now extended by the dependency-digraph audit. If deleting f coordinates makes the graph acyclic, conditioning on them gives

||T_b||_(L^p_Z(L^2_Theta)) <= C (p+f) beta_2.

If every strongly connected component has size at most s, block martingale concentration gives

||T_b||_(L^p_Z(L^2_Theta))
  <= C (p sqrt(s) + s sqrt(p)) beta_2.

Thus every fixed feedback number and every fixed SCC size is closed at arbitrary outside degree. P94 improves this structural boundary. If the dependency digraph partitions into r induced acyclic subdigraphs, two nested predictable Rademacher estimates on each color pair give

||T_b||_(L^p_Z(L^2_Theta))
  <= K_H^2 p sqrt(r(r+1)) beta_2.

The centered field costs at most twice this. This bounded-dichromatic closure applies even when SCC size and feedback number both diverge, as in the complete bidirected bipartite example. Consequently only arrays of unbounded dichromatic number remain in the universal cyclic subroute. Endpoint signs do not remove this loss. On the native array b_ij=chi_([r] minus {i,j}) for i<j, the dependency graph is complete bidirected and de-randomizing independent endpoint signs loses exactly sqrt(binomial(r,2)), even though the uncolored field is target-good. Vector endpoint colors have the same square-gain/reconstruction tradeoff. Any improvement must preserve the common-circle coboundary before color recombination. P95 closes the native P87 field on a genuine complete-bidirected outside-majority family with every odd output level active. The rows are the pair Hessian of the radial potential H(Sigma)=(Sigma |Sigma|-sgn(Sigma))/2, and the shifted-Hessian trace is exactly (1/2) L P_{>=2}H; hence ||S_b||_p <= 4p beta_2. This proves that neither unbounded dichromatic number nor unbounded level count alone is an obstruction, but it does not prove the centered common-circle gate. The latter remains supported only by finite angle diagnostics on this family.

Do not extend P95 by first taking the trace-preserving pair-Hessian projection. P96 identifies the unique supportwise orthogonal projection, but on a legal degree-four core/leaf array both its Hessian piece and its trace-null complement enlarge the native row-L-infinity square budget by order n. The input dependency graph is acyclic and its shifted trace is already target-good. The exact Hodge identity is useful structure, but the projection is a closed universal mechanism.

Do not linearize the common-circle covariance to its first Hodge layer. A degree-d monomial row on an arc of length q has variance q^d(1-q^d), whereas the singleton-mark layer has energy d q^(2d-1)(1-q). Their ratio diverges as q tends to zero. Thus all shared-mark Hoeffding orders must remain visible; this wall does not refute the generated telescope, integration before domination, or joint bilinear cancellation. Averaging forward projections over random orders has exact outside-degree multiplier 2/((d+1)(d+2)), so scalar inversion merely recreates a two-derivative loss.

There is an exact two-half-resolvent factorization into derivative-range first-order divergences. Applying a first-order Riesz theorem twice spends two first-order constants. Deleting the derivative-range condition to avoid one stage is invalid: Ivanisvili--Volberg's duality argument reduces that generic divergence estimate to Lamberton's false below-two vector-Riesz bound. Therefore attack the compensated two-front composition directly; do not cite a black-box first-order iteration as P87.

P11/P82: invariant random-order square function

For a uniform reveal order pi, the aggregate residual has exact Doob coefficient field A^pi+B^pi. The leading field is already controlled:

||A^pi||_(L^p(l2)) <= C n beta sqrt(p).

The correction gate

||B^pi||_(L^p(l2)) <= C n beta sqrt(p)

is equivalent, modulo that proved term, to the invariant square-function bound

||S_pi(R)||_p <= C n beta sqrt(p).

Either bound would imply the Rademacher rate, but the native scalar moment bound is not known to imply this sharper square-function estimate. The route is therefore sufficient and parked, not an exact alternative to P14.

The exact coherent/transverse formula is only an L2 identity. Separately projecting the two sectors need not preserve the native L-infinity derivative budget. Sparse source support, winner occupancy, dependency coloring, fractional conflict, and affine-control theorems are subclass closures beneath P11, not independent universal routes.

P81 proves that universal target-scale fractional conflict is false: its complexity is Theta(n^(3/2)) on a symmetric quadratic family whose actual correction remains at the target scale. A separate four-dimensional example shows that random stopping-line conditional expectation is not an L^p(l2) contraction. Neither example refutes P11.

P122 adds the exact last-deletion and signed collision laws, an exact (3/2) first-child obstruction, and the signed linear-extension diagram expansion. P108 refutes pointwise and natural averaged carré compression, including after averaging the reveal order. These are method walls, not a P11 counterexample: finite and scalable searches found no growing normalized P11 ratio, and the complete quadratic remains the strongest recorded calibration with exact (Q_2\to1/2). Do not promote this numerical evidence to a theorem.

Established theorem-level outputs

The current paper proves the following reusable results.

• P1--P2: for every centered Walsh polynomial,

  ||R||_p <= C (sigma sqrt(p) + p iota),
  

  with no degree dependence. This is a coefficient-budget theorem.

• P3: every native Rademacher residual has variance at most n(n-1) beta^2/2.

• P4: the stronger coefficientwise slice hypothesis gives the native target rate.

• P33: at p=3, the native BKZ logarithm improves to [log(en)(1+log log(en))]^(1/3) in the stated regime.

• P34: if every summand is additionally bounded by L, exchangeable-pair concentration gives

  ||sum_i g_i||_p <= C sqrt(p)(n sqrt(beta L)+L sqrt(n)).
  

• P35: the native Rademacher rate holds at every fixed aggregate Walsh degree, with a degree-dependent constant.

• P70: sparse Cartesian products of uniformly P14-good factors remain P14-good; products do not amplify a hidden P14 defect.

• P16/P29: permutation-invariant functions satisfy P14, and their signed Green flow is KR-optimal.

• P26: asymmetric product-Riesz families satisfy P24.

• P28: pairwise-independent random set membership satisfies the Green endpoint in expectation only.

• P77: every binary affine subspace or coset satisfies P22 and sharp P78.

• P84: exact two-atom Green cancellation, P78 for every three-point set, and the divergent-P22-ratio window.

• P217: the capacity-obstacle odometer is a canonical admissible collision relocation and satisfies the exact transfer

  T_s(d-1_I)
    <= (2/s) sum_(r!=t) E|partial_r partial_t u|.
  

  Total occupation is not a valid replacement, and obstacle relocation need not be FCR-optimal.

• P218: every sparse radial shell profile satisfies the weighted cumulative flux bound sum |D-2-2k||Q_k|<=2(D+1)a and hence the radial obstacle Hessian bound 10Da. This is a sharper one-dimensional structural theorem, not the universal proof mechanism.

• P219: complementarity gives mu{u>0}<=2a; support-localized Cauchy--Schwarz and the exact Walsh Hessian square sum give sum_(r!=t) E|partial_r partial_t u|<=2s^2a. Thus T_s(d-1_I)<=4sa, the ambient lift closes P210, and density doubling proves P14 and the Rademacher/general-product linear rate.

• Sharp all-moment saturation: with q=min(p,n), every BKZ family obeys ||sum_i g_i||_p<=C[n beta q+M sqrt(nq)], and the single quadratic Rademacher family matches this scale for every p>=2. For p>=n, use the all-positive atom for the lower bound and the implied generic pointwise cap for the upper bound.

• P86: sparse fixed-cardinality fibers over a P14-good base remain P14-good; arbitrary nonlinear graph extensions do not amplify a defect.

• P87: the mixed shifted-Hessian set, strong-L^s, and bounded-field dual formulations are equivalent; its exact L2 square sum is at most 2 ||f||_2^2; the degree-free once-smoothed trace is proved, and its cyclic square gate has binary-matroid, affine-quadratic, fixed-degree, and arbitrary-degree forward, bounded-feedback, and bounded-SCC dependency closures plus the P94 bounded-dichromatic closure and an exact random-arc/noise decomposition.

• P94: the P87 cyclic square gate holds when the coordinate-dependency digraph has dichromatic number r, with bound K_H^2 p sqrt(r(r+1)) beta_2; bounded dichromatic number strictly crosses the earlier bounded-feedback and bounded-SCC boundary.

• P95: the native P87 field holds with constant 4 on the complete odd-majority clique, whose dependency digraph is complete bidirected and whose output contains every odd degree from 3 through n; this is a radial-Hessian class theorem, not the universal centered square gate.

• P96: the unique trace-preserving pair-Hessian projection is exact in L2(ell_2) but unbounded in the native row-supremum budget; both projected pieces lose order n on a legal target-good forward array.

• P97: the centered common-circle row has an exact symmetric all-order radial-path representation with pointwise normalized coefficient energy at most 2 ||b_ij||_infinity^2; only assembly for the jointly generated coefficient range remains, and arbitrary coefficients with that row energy fail already in L2.

• P98: the genuine generated support operator has norm of order |S|^(-1/2), sharply; globally the half number-operator derivative is at most (1+3/sqrt(2)) beta_2. Its exact same-row Gram kernel and deletion hierarchy reduce the remaining centered gate to one normalized coherent cycle--Riesz all-moment multiplier. Generic bounded block symbols and standard BMO endpoints cannot promote this L2 result.

• P99: random coordinate insertion realizes that coherent multiplier as a conditionally symmetric Hilbert-valued martingale. The averaged insertion-square condition is sufficient and has an exact semigroup first-moment identity, but the construction is generic and stronger than the minimal infimum over orders. It is a candidate architecture, not a strict successor; Jensen and pointwise reveal-state bounds do not close the higher moment.

• P100: endpoint conditioning enters an exact rooted maximum-power tower, with same-root Gram/deletion identities, an orthogonal energy telescope, an exact incident-edge Dirichlet projection, and a one-row shifted conical-square factorization. This is proved state compression, not an all-moment estimate. The first-maximum-only state and scalar-energy Bellman functions are too small, while cross-root signed interval covariance remains open.

• P101: the P100 boundary has the sharp O(sqrt(p)) moment bound in one row and in arbitrary external Hilbert direct sums, by random-priority domination with a bounded-terminal Doob square. The covariance is exact circle Hdot^(-1) Green energy, and P99 holds for every order on the one-core tournament. The actual incident sum remains a nonpredictable cross-root divergence and is not controlled by the direct-sum theorem.

• P102: every compatible flow is the canonical shifted-Hessian flow plus a symmetric pair-range zero-divergence gauge, and compatible chord fields annihilate that gauge. Equal disjoint parity blocks make the canonical/optimal congestion ratio grow like the number of blocks. Thus canonical rooted phase is not a native P89 obstruction.

• P103/P105: delaying compatible-flow minimization until after Calderón scale gives the exact pairing

  |E Fh| <= tau_2(c) D_n(h),
  D_n(h) = int C_n((L-I)K_q h) dnu(q).
  

  Here K_q annihilates degrees at most one and the sufficient gate must be stated for every ambient test h:

  D_n(h) <= C p sqrt(n) ||h||_(p').
  

  Restricting to high-degree tests with their inherited norm leaves a quotient-duality gap. P105 proves the displayed gate false for every fixed p. Affine-spread Steiner designs of degrees k_m=4^m on one common n-coordinate set give, for h=sum_m h_m,

  ||h||_2 = sqrt(M),
  C_n(h) <= sqrt(4M(n-1)/3),
  D_n(h) >= c_0 M sqrt(n-1).
  

  Each component is a sharp homogeneous P89 L2 configuration, and all components even have the same uniform optimal fractional cover. The loss comes from choosing a different dual extremizer on each disjoint scale window. This refutes independent L1-in-scale quotient minimization, not P89. A single compatible potential and its fixed heat orbit reproduce C_n(h) exactly; preserve that orbit in every future Calderón argument.

• P104: same-core root-collision replication obeys the target O(sqrt(p)) moment bound, and the collision operator has sharp norm at most 1/sqrt(60) when distinct supports meet in at most one coordinate. Pointwise covariance domination is false, while dense overlaps of size at least two and cross-pair phases remain open. Keep this assay stalled, not as an extra active leaf.

• P137: the optimizer-extension defect has an exact complementary-slack/Hall normal form. Coordinate atoms recover the dependence remainder and radius atoms recover W1-H; for equal uniform sections the defect is exactly the incompatibility of maximum matchings across nested radius graphs. This compresses several walls but does not contract P14. The subordinated Poisson square satisfies the correct indicator LlogL-to-L1 budget and has exact L2 norm ||F||_2/2; balanced majority separates it from P14 by n^(1/4). Hence every dimension-free L2-normalized scalar square is retired as a direct dominator.

• P138: the Euclidean norm of the exact row KR transports is equivalent to the anisotropic residual gate and satisfies P89 => P138 => P14. Twisted majority times a two-bit parity separates P89 from P138 by sqrt(n), refuting dimension-free gluing of independent row optimizers into a compatible symmetric flow. A purely skew biased-sign field refutes symmetric-Jacobian domination, and the half-derivative Poisson square admits no fixed ambient normalization that has both the indicator budget and P14 domination. None of these walls refutes P138.

• P145: curl-free reconstruction exhausts linear common-indicator integrability, while opposite singleton maximizers park local second-order stability. Lower-dimensional P138 gives the coefficient-one fiber ceiling (C_h\sqrt h(1+R)). The full extension-defect inequality, including removed-row energy and an (O(dL^2)) remainder, is the sole active induction gate. At sharp calibration repetition forces this debt scale absent an independent constant gap; tilted thresholds and the antipodal simplex-star family are its mandatory positive calibrators.

• P146: actual-fiber deletion gives an exact signed coordinate defect. Prescribed deletion is false on balanced majority, but the adaptive gate min_r Delta_r <= C log(e/a)^2 survives repetition, parity, majority, and simplex-star and would close P138 by induction. Generic averaging retains the full normalized parent energy and is circular; seek signed Boolean coordinate selection or a growing best-coordinate counterexample.

• P147: child-norm imbalance cancels the full J-debt. The exact column and weighted-star majorants, together with the global level-two budget, close every fixed-positive-low-row regime. The residual profile is almost-all-high, level-two-pseudorandom, and chain-rule-near-extremal. Generic spectral averaging is circular; attack simultaneous near-equality rigidity or construct a genuine high-degree counterexample.

• P148: zero rerouting makes parent row normers hereditarily exact, but arbitrary parity-block lifts show that equality alone has no structural classification. These lifts have all rows at most 1/2. Equality-only classification is parked; use P149's source-weighted successor.

• P149: small level-two contact and rerouting force source-weighted calibrated long Hamming intervals for every high row outside a bounded exceptional set. Raw normer oscillation is gauge-dependent. Attack simultaneous interval overlap/packing, not generic concentration.

• P150: optimal row transport splits exactly into its affine level-two part and bidirectional orientation cancellation. The zero-cancellation sector is P138-good. The full cancellation norm is equivalent to P138; use oriented edge-disjointness together with P149 rather than renaming it.

• P151: bidirectional cancellation is exactly stopped absolute-Hessian variation minus accumulated rerouting. Its unsigned energy loses sqrt(n). Attack the signed activity--rerouting pairing only.

• P152: the signed residual lies in a symmetric sign-coherent derivative class and is exact wrong-way transport. Retain this common-indicator geometry, but P154 parks its unrestricted divergence gate.

• P153: nonlinear unions of cosets satisfy r_i <= min{ell_i/2, C(n-1)L/(d-1)}. Do not reopen high-dual-distance or short-local-generator quotient codes; test low-dual-distance, large-local-distance or trivial-kernel OA2 geometry.

• P154: the P152 uniform sign gate is affine-gauge equivalent to the full arbitrary-row P138 theorem. Do not reopen fixed-sign concentration without exact shared-indicator optimizer contact.

• P155: normalized Hadamard arrays have r_i=(N-2)/4 and bounded P138 ratio despite long bidirectional contact in every row. They are mandatory benign calibrators for overlap/packing arguments.

• P156: exact Q7 OA2 sets with identical two-point distance data have different row W1. Park association-scheme or histogram-only proxies; preserve higher-order Hall incidence.

• P157: every odd one-pairing simplex phase satisfies P138 by an exact transport-plus-influence formula. Its majority member refutes rowwise entropy control. The residual simplex theorem is aggregate across incompatible projective pairings.

• P158: exact-source quadratic spread gives a valid deletion majorant, but both proposed entropy-scale controls are refuted by the constant-density radial odd-tail family. It has H_i^2 much larger than one in every row while actual row transport stays bounded. Use it as a mandatory benign calibrator and do not reopen quadratic exact-source variance.

• P159: decreasing rearrangements of exact nonnegative contact-flux profiles define an L1(ell2) functional between K_n and sqrt(log(en)) K_n. Its target bound closes P146 and native profiles obey a subgaussian width envelope. The dyadic loss is sharp for one bounded curl-free real scalar, but indicator separation from P138 is unknown. Treat it as a Boolean preflight and attack the idempotent dyadic-scale upgrade.

• P160: high-row weak Hall collision has an exact anisotropic transport dual and implies P138. Its cost is four-point generator reuse, not a two-point distance histogram. The theorem sum C_i<=(1-a)/(2a) closes fixed positive density. Small/acyclic high blocks, low-energy feedback deletion, conditional high-block affinity, nonlinear O(p)-block score phases, and common posterior channels with chi-square information exp[O(p)] and weighted replica disagreement O(p/sqrt(|H|)) are also closed. The last theorem includes exact rank-O(p) overlapping affine syndromes. Rowwise projective normers refute deriving the channel from separate exact contacts. Bounded-chromatic simultaneous-boundary graphs are closed. Common-indicator spectral signal obeys s^T R^dagger s<=(1-a)/a but retains the |H| effective-rank loss. The full projective contact cone nevertheless has linear moments and closes. Variance-sensitive AVaR also closes bounded signal-local acyclic ratio. Random-order projection closes bounded recovery ratio and every fixed active-block degree; anisotropic square-field concentration independently closes fixed aggregate contact-field degree. The survivor is diffuse, high-dichromatic, and unbounded-degree in both senses.

• P161: the soft-threshold collision norm is convex and recovers the P160 high-row sum up to O(L^2). At a maximizing indicator, its normalized contact field is a subgradient, the indicator is a top level set of that same field, its conditional contact mean is Q_tau+O(L), and its L2 norm is at most 1/sqrt(2). The fixed active potentials also obey the exact cone-wide bound ||((a^(-1)E[gZ_i psi_i]-tau)_+)||_2<=Q_tau(A) for every equal-density fractional g, with a uniform-convexity remainder. Point swaps lose |A|; translation mixtures and heat smoothing have nonpositive response coordinatewise; cone plus spectral signal alone has an abstract common-atom wall. Projective contacts, bounded signal-local acyclic or random-order recovery ratio, bounded active-block degree, and bounded aggregate degree are proved benign. The remaining exact-contact row is diffuse and unbounded-degree. Do not return to arbitrary row fields, marginal curvature, quadratic joint-discrepancy states, or scalar random-order Tauberian recovery.

• P162: threshold-band redistribution has an exact support formula and normal-cost bound; a macroscopic rotation in an affordable band closes P161. Strict individual locks reduce, up to a universal additive term, to a weighted coordinate-code margin, and balanced tie-free locks close. These alternatives are not proved exhaustive. An abstract near-band spike model refutes deriving aggregate approximate locks from the convex/top-level data alone by a sharp sqrt(|I|) factor. Repetition refutes long-interval rotation and GV codes refute Hamming-only margins. The unrestricted Cayley/Delsarte independence gate is also false because pinned faces retain density 1/2 regardless of a charged coordinate. Random priorities prove the dimension-dependent 2(n-1)k bound and the target 2r^2 bound for codimension-r affine codes. But the outlier half-ball refutes both the universal fractional code-SOCP gate and rho_i^2<=(k+1)p_i at constant density; the SOCP forgot section mass. Near-pinned strict-lock thresholds are paid by marginal entropy. The repaired candidate sum_i theta_i rho_i^2<=Ck^2 would close the section-balanced remainder, but remains unproved; it is sharp for affine codes, zero for coordinate-flipped down-sets, and product-stable. Even after multiplying by theta_i, the rowwise priority shortcut is false on balanced antipodal Hamming balls. The information-weighted theta_i rho_i^2<=C k d_i is the smaller surviving assay; its exact harmonic overlap lift still needs coherent geometric selection inside the common projection fibers. The raw positive-superlevel square self-bound is false by sqrt(h) on an exact projective one-row KR/top-set contact satisfying the fixed-contact cone and remainder. It remains open only whether global P161 maximality forces cross-row compensation. The calibrated replacement is a truncated-carre or first-power nested-superlevel assembly.

• P163: for normalized exact row costs r_i=t_i/a, the separation ratio is exactly R_138/R_14=RMS(r)/mean(r)=sqrt(1+CV(r)^2). Transitive families have equality and uniformly macroscopic coordinate orbits give only a bounded separation. If R_14<=C, R_138=rho tends to infinity, and H={i:r_i>sqrt(rho)L}, then

  rho sqrt(nL) <= C |H| <= C n/sqrt(rho),
  R_nu(H) >= c n rho^2 L^2/|H|^2,
  rho^2 L/C <= max_i r_i <= sqrt(nL),
  rho^4 L <= C^2 n.
  

  Thus a separator needs a growing but vanishing, joint-entropy-rich mesoscopic spike block; neither finitely many spikes nor a macroscopic symmetric block can work. Products of P138-good factors remain P138-good with a factor-count-free constant, and first-order-resilient product profiles concatenate exactly. Sparse fixed-cardinality grafts, including every deterministic systematic graft, preserve P138 over a good core. A candidate must instead use variable/dense fibers or overlapping nonproduct constraints, with a multi-pairing construction in the mesoscopic window as the clean falsifier assay.

  The first dense-fiber class is now exact. For a surjective binary generator G and A={x:Gx in B_r}, deleting column g_i gives

  r_i=(1/2) W1_(d_i)(Unif(B_r),Unif(B_r+g_i))
  

  whenever the remaining columns span the quotient; d_i is their Cayley word metric. The nonsurjective formula uses the centered difference of the two section indicators on the leave-one-out span. When d_i is Hamming, the transport reduces exactly to two radial/binomial orbit coordinates.

  For G=[I_k|M], codensity ell=o(k), and m approximately equal high generator rows of size q, P14 and the mesoscopic entropy ceiling force

  q <= C k ell/m,
  q <= C sqrt(m ell).
  

  The only critical balance is

  m ~ k^(2/3) ell^(1/3),
  q ~ k^(1/3) ell^(2/3),
  R_138 ~ (k/ell)^(1/6).
  

  A separator must have full-span mixing of the ball but strong failure of mixing after almost every one-column deletion. Random dense blocks tend to mix both; structured blocks tend to have duplicates, short circuits, products, or insufficient joint entropy. Bounded exact tests through quotient dimension eight are benign, but this is not closure.

  This remains a falsifier-profile contraction, not a proof of P138. The scalable nonlinear antipodal three-pair family has unbounded Walsh degree but both normalized ratios tend to 1/(2 log 2), so it is benign.

• P106: if distinct active Walsh supports meet in at most one coordinate, then every fixed reveal order satisfies

  ||S_pi(F)||_p <= sqrt(p-1) ||F||_2.
  

  The P89 L2 endpoint converts this to the native SF-P89 scale. Levelwise assembly also proves the SF-P89 target on the full combined P105 affine design family, whose exact chord supremum charges the number of levels. This is stronger square-function structure on proved classes, not a universal last-pivot theorem.

• P107: write A_V(x)={S subset V:x in S} during backward coordinate deletion. If every selected pivot satisfies either

  (sum_(S in A_V(x)) |r_S|)^2 <= p sum_(S in A_V(x)) |r_S|^2
  

  or its residual incidence vectors 1_(S\{x}) are independent over F_2, then the reverse reveal order obeys

  ||S_pi(F)||_p <= sqrt(p) ||F||_2
                <= sqrt(2p(n-1)) tau_2(c).
  

  More generally it is enough that each residual frequency family have Lambda(p) constant O(sqrt(p)); the coefficient-specific version is enough for the given potential. A failed safe-deletion process leaves a hereditary local Lambda(p)-bad core. At constant one every vertex in that core has effective coefficient multiplicity >p and a residual binary circuit. This is a necessary falsifier profile, not a counterexample and not a derivation from the compatible chord budget.

• P108: the single-exceptional-shell compatible potential

  F_n = 2^(1-n) sum_(|S|>=2) chi_S
  

  has, at every first backward pivot,

  Q_x = 2^(1-n) (prod_(j!=x)(1+Z_j)-1),
  (||Q_x||_4/||Q_x||_2)^4
    = 2^(n-1)-2+1/(2^(n-1)-1).
  

  Nevertheless every predictable tree obeys

  S_T(F_n) < 2/sqrt(3) < 2 sqrt(n) tau_2(c)
  

  pointwise for n>=4. This exactly refutes universal local Lambda(4) deletion, not SF-P89. Large pivot spikes are nested on rare all-positive prefixes; worst-local summation is therefore unavailable.

• P109: if A_ij=Z_i Z_j a_ij are compatible chords, the invariant reveal coefficient at state (R,x) is exactly the joint sum of

  sum_(j in R) Z_j P_R B_xj
  

  and

  sum_({i,j} subset R) Z_i Z_j partial_x P_(R union {x}) B_ij,
  

  where B_ij=(I+L_-ij)^(-1)a_ij. The second, inherited-coefficient channel is load-bearing. The natural P30 occupation lift formed from the chords before quotienting is false: on the unit full-parity star its normalized loss is at least

  n^(1/2-1/p)/(2 sqrt(p)),
  

  and a transverse allocation has scalar output zero but occupation norm gtrsim n^(1-1/p) with tau_2(c)=O(1). Never replace the invariant scalar bracket by this pre-quotient field. Calling the scalar bracket's exact L^(p/2) dual ``global packing'' supplies no contraction by itself.

• P110: quotient-measurable P89 outputs with at most Kp singly occurring nonzero label classes satisfy the target with constant C(1+K), and fully repeated label systems satisfy the sharper all-moment square-root bound. Independent homogeneous quotient assembly can exceed the full quotient exponentially on a bent phase. The exact matrix reconstruction has trace 2F, but separating trace from its S_2 norm costs sqrt(p) on complete quadratic chaos. The first statement is a class theorem; the latter two are architecture walls.

• P111: successive repeated-label quotient increments form a reverse martingale and obey a depth-free conditional subgaussian estimate with square charge sum_t(a_t^2+b_t v_t^2+s_t^2 rho_t^2). The corresponding gate closes a strict zero-cover class with order-n initial singleton labels. Right-Hodge compression removes the matrix representative gauge exactly, but the resulting tensor is not locally subordinate to the chord field.

• P112: conditional Bernstein composition gives ||F_0||_p<=||F_T||_p+C(sqrt(pV)+pL) and handles P111's rare-event calibration. Every canonical Boolean corrected gradient is P89-good, uniformly over degree, quotient labels, and multiplicities. The exact joint-Hodge audit closes Hutchinson, sparse-diagonal, and pointwise Schatten scalarizations.

• P113: the depth-one repeated-core/leaf family has chord graph K_(N,2m) and native variance budget 2mN, but conditional-variance essential supremum N^2. This refutes deterministic Bernstein transfer from the original cover. Moment-averaged generic repair either loses through its maximal increment or returns to SF-P89/P82.

• P114--P115: arbitrary gauges for Boolean-generated outputs obey both the local half-space barrier and the maximum-sensitivity bound rho_n(F)>=S(S-1)/(2(n-1)sqrt(n)); the canonical Boolean Hessian also yields the gauge-annihilating average certificate rho_n(F)>=sqrt(E[k(k-1)]/(n-1)). Permutation-invariant Boolean outputs have an exact canonical quotient, and disjoint outputs tensorize exactly. Canonical sensitive cliques remain non-native. The sublinear-sensitivity localization remainder is parked as a strict special case.

• P116: the orbit-absolute envelope satisfies C<=O<=D and equals sup_sigma C(M_sigma h) for correlated Gamma-moment symbols. It repairs P103 on the P105 witness by sqrt(M), but its constant symbol is P89 itself, so it is not a contracted crux.

• P117: all unweighted separated fixed-orbit conjugate mixed norms lose an extra sqrt(n) at p=n/4; do not separate the edge and test legs before scale allocation.

• P118: F=R^* I A is an exact signless support-incidence factorization with a strict Hilbert contraction away from degree two. Its joint Cp edge-Hilbert estimate is sufficient but stronger than P89 and unproved; pointwise, operator-norm-only, and generic two-stage versions are closed.

• P119: arbitrary positive pointwise weights and positive stopped local Hölder sums still lose sqrt(n); unrestricted signed stopping is exactly the original pairing. Preserve signed spatial cancellation through a genuinely smaller state.

• P120: canonical unsigned per-support incidence/Hodge summaries are exponentially phase-blind on aligned versus bent flat spectra. This is a strengthened P110 wall, not a ban on quotient-aware or globally signed methods.

• P88: the Rademacher residual theorem and the general-product BKZ theorem are equivalent up to universal constants by two-copy orientation.

• P89: the compatible-flow endpoint and Euclidean fractional-vertex-cover dual are equivalent; P87 => P89 => P138 => P14, with the s=2 and dual p>=n ranges, the target all-p extreme-pair selector, the matching p=2 chord estimate, and the sharp singleton/disjoint-parity calibrations proved.

• P90: P89 is proved for low binary support complexity, every homogeneous degree with constant 2e, any two arbitrary degrees with constant 4e, every three-term arithmetic progression with constant 225, m 64-lacunary levels with constant 32em/15, at most two even plus two odd levels, bounded mixed degree, and sufficiently high-mean-degree phase-aligned spectra; the color family has an exact Calderon isometry.

• P91: arbitrary-degree multi-core gradient--quadratic towers satisfy P89 uniformly.

• P92: uniformly P89-good potentials on disjoint coordinate blocks glue without a block-count loss. With

  B_a = sqrt(|V_a|-1) tau_a,
  

  independent subexponential Bernstein gives

  ||sum_a F_a||_p
    <= C [sqrt(p) (sum_a B_a^2)^(1/2) + p max_a B_a],
  

  while the block-diagonal cover splits exactly as tau_global^2=sum_a tau_a^2. This permits arbitrarily many irregular global degree levels, but independence of the coordinate blocks is load-bearing.

• P93: for each fixed m, every P89 output supported on any m arbitrary degree levels satisfies the target with a constant C_m, uniformly in dimension, maximal degree, level locations, and separation. The proof uses a fixed-order Muntz endpoint inequality, confluent Newton compactification, and at most m+1 signed color scales with the native cover norm. Lacunary witnesses force C_m to grow at least linearly for scalar synthesis.

Finite enumeration remains evidence only even when it supports one of these universal conjectures.

Tool selection and accuracy walls

Tool	Reliable use	Wall to record before applying it
Martingale square functions	Predictable decompositions and sqrt(p) upgrades when the square field is controlled in the same native model	A scalar moment bound does not imply the stronger square-function bound; random stopping-line conditioning is not an L^p(l2) contraction; fixed radial pointwise variance allocation is false at the P89 scale on common-core stars; even the best coordinate order has BMO asymp n^(3/2) on the target-good complete quadratic chaos against native scale asymp n. P106 proves the L^p reveal square on linear supports and the combined affine stress family. P107 extends this by weighted backward peeling and local Lambda(p) spectra, but P108 refutes local good-pivot deletion universally. P109 refutes pre-quotient occupation. P113 additionally refutes deterministic quotient-flag variance transfer; generic random-bracket repair loses through its maximal increment, while symmetric refinement returns to P82. Only a smaller quotient-sensitive adaptive state would be new.
Bounded differences / Efron--Stein / Bernstein	Variance, subgaussian or subexponential concentration of a single stable functional, and depth-free conditional-mgf composition	It does not control Fourier l1 slices. Keep variance and peak separate, but never infer deterministic worst-coset variance from an average chord budget: P113 loses N/(2pm) on a native-benign common-core family. Conditional variance alone also misses rare spikes.
Hypercontractivity	Fixed degree, low-level Fourier mass, radial/binomial reductions	Degree growth or absolute aggregation can reintroduce logarithmic or polynomial losses. P110 shows that even native quotient optimization on each homogeneous level followed by ell_2 assembly can lose exponentially; mixed-degree cancellation must be retained
Decoupling / random selectors	Conditional expectations that leave genuinely independent front signs, with the reconstruction multiplier and selector probability both audited	P93 makes the optimized P89 color selector uniform on every fixed-cardinality degree spectrum, but its constant necessarily grows with the number of levels; scalar mixtures across many levels lose logarithmically on intervals and linearly on lacunary sets. P105 additionally refutes taking an independent optimal compatible-flow selector at every Calderón scale, even under one common fractional cover. Preserve one fixed compatible heat orbit in the joint bilinear contract. Degree-free front-sign decoupling loses sqrt(n) on full parity, endpoint-color randomization has an exact order-r reconstruction loss on a native complete-bidirected P87 array, and P161 random-order resolvents attenuate degree k by 1/(k+1). Chebyshev profiles show that even a positive bounded scale curve can hide endpoint signal of order D^2; scalar Tauberian recovery is not a degree-free continuation.
Green/Riesz resolvents	Canonical compatible flow, exact shifted-Hessian identities, and a nonempty feasible point for P89	Keep the shifted resolvent for P87. For P89 first quotient the symmetrized derivative field over all compatible pair flows; ambient Hilbert--Schmidt replacements attack a different proxy. P110's exact matrix reconstruction has trace 2F, but any universal separate trace comparison must lose at least sqrt(p). After P111 right-Hodge compression, P112 still closes Hutchinson, sparse-diagonal, and pointwise Schatten scalarizations.
KR transport / cut towers	Exact native duality, finite counterexample search, compatibility-preserving representations, P138 Euclidean row transport, and P89 pair-array duality	A second exact representation is not a quantitative estimate. Rowwise scalar summation loses sqrt(n), while P138 shows that Euclidean aggregation can be sufficient without P89 compatibility. Twisted majority refutes dimension-free gluing of the independent row optimizers back into one P89 flow.
Sparse-support obstacle Hessian	For the capacity-obstacle odometer, first use complementarity to prove mu{u>0}<=2a, then localize each mixed derivative to four support translates, square-sum by Walsh orthogonality, and use L_su=(s/2)(1_I-d)	The support and source bounds are obstacle-specific. Do not replace them by total occupation, which loses a factor s, or by a claim that heat balayage minimizes FCR, which is false. Do not export the conclusion to arbitrary Hessian fields without separately proving both sparse support and the source L2 estimate.
Transportation--entropy	Controlling one law against the ambient product law with the correct dimension factor	Applied after lossy section splitting it cannot restore cancellation already discarded
Stein / exchangeable pairs	Bounded-summand theorems with an explicit conditional variance proxy	The BKZ hypotheses imply only the generic bound `
Cumulants / diagrams	Fixed orders or absolutely summable coefficient models	Absolute diagram assembly can lose cancellation and does not give all-p control from finitely many orders

Mandatory adversaries

Test the actual target and every proxy on the relevant structured families.

• Full parity: coherent high degree; checks the resolvent and defeats degree-free front-sign decoupling.
• Bent phase: bounded pointwise derivatives with exponentially large Fourier l1 mass. The symmetric quadratic bent phase is also the mandatory P110 test for homogeneous quotient atomization: its degreewise native square budget is exponentially larger than its full quotient norm.
• Singleton: sparse endpoint and exact n^2 2^{-n} scale.
• Balanced majority and middle slices: radial positive controls for P14/P22, but destructive for separate-section and local entropy allocations.
• Twisted majority times a two-bit parity: separates P138's independent Euclidean row transports from P89's simultaneous compatible flow by sqrt(n). It is the mandatory gluing and row-indexed-state adversary.
• Disjoint parity checks: unbounded canonical Green/transport ratio while P22 remains within its entropy budget; P89 must and does repair this family by an optimized compatible flow of exact scale sqrt(n) a.
• Shared-core parity sunflowers and all affine systems: proved P22/P78 classes, not live falsifiers.
• Asymmetric product-Riesz sets and sparse Cartesian products: intermediate densities and product-depth calibration.
• Quadratic-chaos upper tails: strongest current finite P14 ratios under the script normalization.
• For all-moment sharpness, use the BKZ quadratic family g_i=M Z_i+(beta/2)Z_i sum_(j!=i)Z_j. Its moderate moments calibrate p<=n, while the all-positive atom of mass 2^(-n) calibrates every p>=n. Any proposed improvement for p>>n must first confront the resulting Mn+beta n^2 saturation.
• Positive versus Walsh--Hadamard complete quadratics: both have unit chord contacts, the same optimal cover, and identical separate p=2 edge-flow magnitude data, but their normalized moments at p=n/4 differ by sqrt(n). This is the mandatory test for any contact count, contact entropy, or edgewise uncertainty summary. Passing it requires target-p, jointly signed, flow-weighted information.
• Composed majority: one matched-section row is sqrt(n) larger than the corresponding exact transport row; tests false coordinatewise faithfulness.
• Symmetric quadratic P81 family: fractional conflict is maximally lossy while the true random-order correction is target-benign.
• Four-dimensional stopping-line field: defeats the false L^p(l2)-contraction shortcut.
• Dyadic mixed-degree staircases and quadratic-plus-parity mixtures: detect premature absolute aggregation across scales or degrees.
• Affine-spread Steiner resolutions at degrees (4,16,\ldots,4^M) on one common finite-field coordinate set: every component sharply calibrates P89 in (L^2), while disjoint Calderón windows make independent scale-by-scale quotient selection lose (\sqrt M). This is the mandatory P103/P105 proxy test; a common fractional cover does not repair it.
• For affine-incidence moment routes, first reduce an ordered even tuple to its odd-multiplicity support. Apply the maximum-block intersection cover before enumerating circuits. At order eight this isolates levels one and two and makes all higher levels relatively independent exactly. At general order it only reduces to logarithmically many low levels; a crude triangle inequality there is not an all-moment proof. Distinguish primitive binary circuits from geometric connectedness, retain exterior parity after every root trace, and never record primitive AOE as a strict reduction because it is equivalent to the positive moment target. The full coordinate carré and sharpened reveal square are scalably false on affine planes. Always test generic Sidon/Rudin proposals on APN Sidon caps and dissociated-coloring proposals on the full level-one affine packet.
• For SF-P89, reuse the complete quadratic, common-core, disjoint/full parity, Fano/affine-plane, and combined affine-spread tests. Complete quadratic refutes BMO but is target-scale for the reveal square. A meaningful new stress family must combine dense support intersections with load-bearing cancellation in the chord L-infinity norm and must survive the P107 deletion screen. At even p=2q, compute local zero-sum 2q-tuple weights, equivalently the coefficient-specific local Lambda(p) quotient, at every induced core, but also compute the full bracket: P108 shows that even an unbounded local quotient can be harmless when its high values are nested on rare prefixes. Testing only overlap, cancellation, or local Lambda(p) growth is no longer discriminating. Also test the P109 full-parity unit star and its transverse zero-output allocation. They separate the invariant scalar Doob square from every pre-quotient resolvent occupation field by a polynomial factor.
• For binary quotient-code proposals, record the multiplicities of nonzero label classes. P110 closes every model with at most O(p) singleton types, so a meaningful survivor must have s/p unbounded. Repeated-column replication alone is no longer a discriminating stress test.
• For quotient flags, test both P111's depth-one rare event and P113's repeated common core with independent leaves. The first distinguishes a Bernstein variance/peak split from a worst-case subgaussian charge. The second distinguishes valid conditional-mgf composition from the false deterministic transfer of worst-coset variance from the original cover. Report the actual output ratio, the random bracket in L^(p/2), and the maximal increment separately. Do not use canonical Boolean phases as prospective falsifiers: P112 closes every canonical Boolean corrected gradient, including all degrees and label multiplicities. For a noncanonical Boolean proposal, first apply P114--P115. Families with maximum sensitivity S>=delta n are already safe for every gauge. Full permutation symmetry is closed exactly, and additive repetition of a fixed disjoint output cannot amplify the ratio. Only sublinear-S examples with |F(x)|/sqrt(E[k(k-1)]) large test the unresolved pointwise claim. Report the native ratio itself, not only tau_can/rho, because finite optimization shows that gauge separation and target ratio need not correlate.
• For matrix routes, first apply the exact P111 right-Hodge compression. Then test the common-core leaf family, which defeats pointwise, carré-du-champ, and coordinate-martingale subordination by sqrt(N/k). Complete quadratic and P105 affine blocks have no removable gauge and remain mandatory tests of trace and scale separation. Also test: Hutchinson scalarization for circular reproduction of the scalar input; sampling n/p diagonals on complete quadratic for a sqrt(p) loss; and pointwise Schatten dualization on many disjoint Walsh characters.

For every claimed failure, print or prove both

actual target / target budget

and

proxy / proxy budget.

Growth of only the second ratio kills the proxy, not the theorem.

Literature imports

• Domelevo--Ivanisvili--Petermichl--Volberg26: the new commutative two-point Bellman proof gives the known dimension-free first-order Riesz bound only for (p\ge2). Its edge symmetrization is a useful method precedent, but black-box iteration returns to the audited ambient-Riesz wall and does not supply the P14 endpoint or simultaneous row geometry.
• Borichev--Volberg24: finite-cotype vector-valued cube concentration requires one Banach-valued map with a controlled pointwise strong gradient. P14 is a sum of rowwise transport optima and presently has no such encoding; packaging every optimizer merely reconstructs the gluing problem. Do not import the theorem without an explicit native map and dimension-uniform target geometry.

Use the durable packets in literature/ before searching again.

• DPMS94 and OZ16: decoupling directions and their degree dependence.

• Adamczak--Latala12: coupled/decoupled comparison and exact mixed-norm moment formulas through order three for tetrahedral chaoses. The durable packet is useful for auditing fixed-order tensor partitions, but it does not remove the unbounded-degree P161 reconstruction wall.

• Cassese26: a dimension-free variational Korn proof whose compatibility-preserving quadratic-variation idea is relevant, but whose diagonal strain is circular for P89; the natural local off-diagonal trace is refuted by the common-core field.

• Spector--Spector20 and Van Schaftingen13: Euclidean BMO Korn and canceling- operator endpoint theorems. The former uses the full symmetric gradient with a dimension-dependent constant; the latter gives a symbol-level L1 Sobolev criterion. Neither controls P89's off-diagonal Boolean quotient or its cross-support L-infinity cancellation.

• Junge--Mei--Parcet14: crossed-product tangent modules keep row/column geometry together, but the recorded constants grow like p^(3/2) and the general module norm does not preserve the scalar chord L-infinity cancellation. Use it as a language/method audit, not an order-p P89 theorem.

• Ben-Efraim--Lust-Piquard08: cube/CAR Poincare theory and its native row--column decomposition are useful precedents for quotienting before an embedding. Their input is the full one-index gradient, not P89's off-diagonal compatible chord quotient, and the CAR/Khintchine passage pays the extra sqrt(p). Use the durable packet as an ordering warning, not a P89 theorem.

• Cassese26 minimal Korn: the rank-one criterion rules out a black-box Korn inequality which measures only arbitrary off-diagonal matrices, because the unmeasured diagonal contains rank-one matrices. P89 removes the corresponding singleton sector and imposes additive chord identities, so this is not a theorem counterexample. Any Korn import must use those two structures before applying the matrix theorem.

• OSSS05 and Schramm--Steif10: adaptive revealment theorems are diagnostic analogues, not SF-P89 inputs. A full coordinate tree has revealment one; OSSS's generic real-valued squared-distance form also pays the path length, and Schramm--Steif controls terminal L2 Fourier mass rather than the L^(p/2) moment of the realized bracket. Read the durable packets before invoking decision-tree literature.

• Sherstov--Storozhenko--Wu23: their adaptive coordinate-decision-tree theorems give strong levelwise Fourier l1 bounds. For a full depth-n reveal, however, they are policy-blind and do not control the realized bracket or the compatible-chord quotient. This is not an SF-P89 input.

• Lai15 and Muller97: sharp general-filtration Bellman/Carleson embedding and fixed Haar-rearrangement boundedness both assume or characterize a Carleson packing constant. Neither derives that constant from tau_2(c) or chooses a sign-adaptive policy. Feeding squared reveal increments into Lai requires the already-refuted conditional future-energy/BMO gate; Muller's theorem is fixed and linear. Generic ``global Carleson packing'' is therefore not a new proof contract.

• Pisier16, Malliavin--Malliavin, and Horn: on the binary group, a character set is subgaussian exactly when it is a finite union of binary-independent sets, and Horn's rank-density formula gives the minimum number of such classes. This yields the native class bound

  ||S_pi(F)||_p <= sqrt(q_pi(p-1)) ||F||_2,
  

  where q_pi is the maximum last-pivot binary-matroid covering number. Use this as P107's structured certificate. P108 shows that bounded q_pi, or even a locally bounded Lambda(4) constant, is not necessary for a target-good global square function.

• Caro--West--Yuster11, Kallenberg05, and Pozdnyakov--Steele13: equitable hyperedge orientations, predictable sampling, and scalar random- permutation maximal inequalities are exact adjacent results but do not solve SF-P89. Respectively, they use an edgewise rather than global order, preserve selected-sign laws rather than order-dependent brackets, and treat value-independent scalar L2 maxima rather than adaptive L^(p/2) bracket moments. Consult their durable packets before reopening these directions.

• Chatterjee07: the exact exchangeable-pair lemma used by P34.

• Ivanisvili--Volberg22, DIPV26, and Chen--Dai26: derivative-range, dimension-free Riesz/Bellman, and randomized-projection results, with the exact exponent, norm-order, two-index, and resolvent mismatches with P87/P24 recorded.

• Ivanisvili--Xie--Zhang26: Beckmann--Talagrand results and why their Euclidean lower/quotient formulations do not supply P14's entrywise transport upper bound.

• Hytönen--Naor13: Pisier inequalities and their mismatch with the native P11/P14/P22 contracts.

• Naor16: the sharp-scaling metric X_p theorem controls balanced coordinate restrictions of one scalar function by its ordinary derivative norms. It suggests quotient-filtration architecture but does not convert P89's symmetric chord quotient to those derivatives, and its analytic p-dependence is not the desired linear one.

• Eskenazis--Ivanisvili20: low-degree, tail-space, and narrow-spectrum polynomial inequalities rigorously support the bounded-degree and narrow-band P89 sectors. Their degree or bandwidth dependence prevents universal mixed-degree assembly.

• Cano-Marmol--Conde-Alonso--Parcet24: balanced Fourier truncations over discrete groups and noncommutative X_p inequalities preserve row/column geometry, but still begin with one scalar/operator input and its ordinary cocycle derivatives. Passing from P89 chords to those derivatives loses the quotient and the known extra moment factor.

• Banuelos--Kim25: sharp O(p) probabilistic discrete second-order Riesz transforms are bounded matrix multipliers acting on a controlled scalar martingale. In P89 the chord field is the source datum and the scalar potential is unknown; using it as the martingale input is circular. Import only the joint martingale-transform architecture unless an exact chord-to-predictable-integrand factorization is supplied.

• BBLM05, APS22, and Fitzsimmons--Pitman99: historical P30/P51 imports; their exact translations and the reasons those retired proxy routes did not close are already recorded. Do not reopen them without one new native transfer lemma.

For new literature work, store metadata, source or durable extraction, and key-results.md under literature/<key>/, then regenerate the literature index. Do not leave theorem extraction only in /tmp.

Workflow

1. State the native hypotheses and every stronger auxiliary assumption.
2. Name the exact target and classify the route as equivalent, sufficient, necessary, special-case, or heuristic.
3. Locate the last exact arrow. If the current object is only a reformulation, return to that arrow before adding another proxy.
4. Record the tool's dimension, degree, exponent, norm, and envelope losses.
5. Test the mandatory adversaries that match those losses.
6. Distinguish failure of the actual target from failure of the tool.
7. Promote only a complete proof, exact counterexample, exact reformulation, or sharper falsifier profile.
8. Synchronize the proof spine, LINEAR_RATE_STATUS.md, dashboard.json, dashboard.html, relevant scripts, and this skill when the proof contract changes. Bump the dashboard only for such a change, not for repeated numerics or cosmetic prose.

Output standard

Every concentration proof report must end with four explicit lines:

Native theorem status: proved / open / refuted
Additional hypotheses used: ...
Strongest proved bound: ...
Remaining loss or exact counterexample: ...

Never claim the native rate from lower bounds, finite numerics, a stronger coefficient hypothesis, a route-equivalent representation, or the failure of a particular upper-bound method.