Reviewed September 21, 2026 against the complete v1.37 manuscript and shifted_floor.py. The user has expressly authorized lambda=8 for this diagnostic. This review does not establish a cofinal bound or close G2.
The shifted complete-Schur construction is valid. A floor of -0.001 hardly changes the old scalar remote-cutoff cost. A fixed moderate floor of -8, however, makes the complete test inexpensive at the requested windows. This choice is justified by v1.36: a single finite uniform constant would suffice in a cofinal theorem. The numerical value need not be tiny.
The adopted construction holds delta=8, N=256, J=4096, r=16, tau=1/10, and Z=0 fixed across both parities and all three windows. Window-dependent coefficients, prime norm bounds, scalar tail weights, moments, and verification matrices must still be recomputed. This is a simpler use of an existing proof architecture, not a new independent arithmetic mechanism.
All six 160-bit complete runs saved PASS. The completed 256-bit replay in replay_b256.json reconstructs the same saved exact dyadic congruence and trial vectors and passes every complete proof gate. No fresh floating congruence proposal is generated by the replay.
Write W=[[F,B*],[B,T]] in an orthonormal head/tail split. For delta>=0,
F_delta=F+delta I, B_delta=B, T_delta=T+delta I.
For any finite trial solve Z,
K_delta=K_Z+delta(I+Z*Z),
R_delta=R-delta Z.
If T>=diag(g_n) and g_(N+1)+delta>0, then
T_delta^(-1)<=diag((g_n+delta)^(-1)).
The complete finite certificate is
L_delta=K_delta
-sum_(N<n<=J) R_delta,n* R_delta,n/(g_n+delta)
-U_J/(g_(J+1)+delta) >=0.
It proves W>=-delta I on the full closed form domain. Rows beyond the finite support of Z do not change under the shift, and neither does their moment majorant U_J. Shifting K while forgetting the finite residual change would be wrong for nonzero Z.
For the adopted Z=0 construction, K_delta=F+delta I and R_delta=B exactly. The implemented unchanged residual is therefore correct. The remote-moment theorem uses M=N because G is the orthonormal head embedding; its proof only needs M<J. The old presentation’s convenient intermediate strict inequality N<M is not essential when the solve is zero.
With c=0 the fresh scalar weights are
g_n=log(n/L)-E_L(2 pi n/L)-kappa_(lambda,N).
Their shifted positivity gate forces
N+1 > L exp(M_phi,lambda-delta).
More generally, for the older W-cD comparison, the exponent is (M_phi,lambda-delta)/(1-c). A fixed shift does not remove the asymptotic exponential-in-lambda cost of this scalar comparison. It multiplies its leading scale by exp(-delta/(1-c)).
Independent 192-bit Arb evaluation used the exact rational active-set decisions in prime_bound, c=0, both parity corrections, and the strict signs of the chosen integer and its predecessor. It gave the following first positive remote cutoffs:
| lambda | delta=0.001 even | delta=0.001 odd | delta=8 even | delta=8 odd |
|---|---|---|---|---|
| 5 | 1501 | 7217 | 1 | 4 |
| 6 | 7481 | 35985 | 3 | 14 |
| 8 | 124317 | 598024 | 43 | 204 |
The N=1 entry is the boundary of the formula’s admitted range N>=1. Every other listed predecessor had a strictly negative enclosure. These are tail gates only, not complete lower-floor certificates. In particular, delta=0.001 would leave the lambda=8 cost essentially unchanged from the old 124442/598623 scale.
The adopted fixed N=256 exceeds every delta=8 gate. This is the concrete computational benefit. It does not make N=256 a valid unbounded-window rule: the same formula eventually fails for every fixed N and delta.
The following parts of shifted_floor.py agree with the complete moment theorem:
As an independent indexing check, I expanded exact test coefficients through the full signed Fourier coordinates using the saved lambda=8 coefficient balls. Both parities agreed with actual_block at rows 0,1,2,5,6,17,4096 where applicable, and every moment through order 3 agreed with the directly summed full-coordinate moments, including the even zero column. These finite checks support the audit but do not replace the remote-moment proof.
All inspected coefficient caches record K=64 and the expected lambda, J, and precision. Since the cache filename does not include K, provenance checking remains necessary when reusing caches. The current files meet that condition. The fixed protocol also satisfies 1<=N<J, r>=1, tau>0, and positive shifted first-tail weight.
The floating inverse-Cholesky factor is only a proposal. Converting its IEEE values to exact Arb dyadics, checking triangularity and nonzero diagonal, and verifying all outward Gershgorin rows of C L_delta C* proves positive definiteness of the underlying exact lower matrix. An interval matrix need not itself be exactly symmetric entry-by-entry; it encloses the exact Hermitian expression, and the row absolute-value bound remains valid.
The displayed whitening row margins near one must not be presented as invariant certificate margins or as evidence that headroom is stable across windows. The congruence was constructed from L_delta itself, so values near one are expected by design.
The parent’s proposed separate generalized-margin lower bound is valid. If exact saved C satisfies
C L_delta C* >= m I,
C K_delta C* <= M I,
with certified m>0 and M>0, then
L_delta >= (m/M) K_delta,
1-lambda_max(U_delta,K_delta) >= m/M.
Here U_delta=K_delta-L_delta is the complete inverse-correction upper enclosure. Use a certified lower value for m and upper value for M. The resulting ratio is a lower bound on the coordinate-invariant margin; its tightness depends on the chosen bounding method. It is not the exact generalized eigenvalue.
A complete certified floor ell_lambda gives
ell_lambda <= lambda_min(W_lambda).
An exact finite-support vector v gives a complete Rayleigh upper bound
lambda_min(W_lambda) <= q_lambda[v]/||v||².
No omitted row enters that vector’s quadratic pairing, so an interval upper bound on this quotient is valid for the full operator. Combine the two parity lower bounds using their minimum; combine available Rayleigh upper bounds using their minimum.
The correct interpretation is:
The improvement in practical cost comes from using a fixed non-small additive floor permitted by v1.36. It does not repair the old lambda=5 inequality W>=10^(-8)D on Q_16, which remains disproved. That positive weighted metric and W>=-8I are different statements.
The frozen head/remote sizes, moment order, Young parameter and zero solve are unchanged across the three windows. The finite congruence witnesses naturally differ because the actual operators differ; generating and verifying each such witness is part of the fixed procedure, not a new analytic tuning rule.
No genuinely new signed arithmetic mechanism was found in this review. The existing complete Schur/moment argument becomes affordable for these finite floors. Its remaining asymptotic limitation is explicit: with delta=8 fixed, the old scalar remote comparison still requires exponentially growing cutoffs. A successful finite list supplies neither a uniform error estimate nor an unbounded-window certificate.
The final new_section.tex and research_report.md agree with the proved scope and the saved replay. In particular:
6094205ed4cd1feb634989147131ecb6a0ef5e154ac843a57d09cbb76b0b9f71.An independent complete witness comparison verified that all six archived C matrices and all six trial vectors equal the initial mantissa/exponent dyadics entry by entry; every stored integer mantissa has at most 53 bits and reconstructs its exact IEEE value without rounding. The archive SHA-256 also matches the completed replay record.
The downward support extension is valid. For any 1<lambda<=8, a smooth test supported in (-log(lambda),log(lambda)) is also an admissible test at lambda=8. Its ordinary norm and its global Weil form are unchanged by zero extension. The established compact smooth form core then extends the lower bound to the complete smaller-window form domain. The result is therefore W_lambda>=-8I for all 1<lambda<=8, not merely the three sample values. The small trial upper brackets are claimed only at the three computed windows, correctly.
The shifted-cutoff corollary is also correctly scoped. For the particular scalar comparison, bounded delta leaves the asymptotic lower cost liminf log((N+1)/L)/lambda>=1. A polynomial cutoff would require delta>=M_phi,lambda-O(log(lambda)). This is a restriction on that comparison’s positive-tail prerequisite; it is not a restriction on all possible signed metrics or on the true operator spectrum.
No blocking discrepancy was found in the final section or report. Their practical improvement and its limitation are both accurately stated: the same moderate floor is affordable over this bounded range, while a cofinal constant and the remaining signed arithmetic estimate are unproved.
No G2 gap was closed.