Date: 2026-09-21. Reviewer: a separate mathematical audit agent.
Verdict: the new shifted comparison, certificate implementation and exact
Connes–van Suijlekom implication pass this review. The certificate proves a
simple even ground state of the complete canonical W_4, with
0 < mu_0 < 2.454e-75 and mu_1 > 1e-73. This is a fixed-window result;
it does not close G2 or prove RH. No claim of worldwide novelty or publication
readiness is made by this review.
SHA-256 at review time:
| File | SHA-256 |
|---|---|
certify_ground4.py |
2c5d4a44de0a8e280e9c4cb2672eb715470df859fc0234c5c7be0305ddf3e4c0 |
new_section.tex |
71acc45cdf2200668e8a74338008087f58ebdcaac04b54df6d275ffc4d541f21 |
ground4_b1024_shift1e-73.json |
e7bcf35e64482ff0415c490155cbacdb7fa3ddcb607386ce7d353e3d16cbd1d2 |
ground4_b1280_shift1e-73.json |
8eaef58850ab2a86fb070d1e553f6c7af866784a5a36e5dd78a71b8d2726ba1e |
The earlier foundations examined were prop:v114-weil-core,
prop:v117-ground-order, prop:v120-lambda4-tail,
prop:v125-even-complete, eq:v125-relative-margin,
prop:v126-direction-complement, prop:v126-full-window, and the full
physical form in prop:v135-both-parities. Original v126 trial assembly and
directional verification code were also compared with the new reconstruction.
The finite trial is G=(V;Z), where Z means its actual tail coefficients,
including their signs. Set r=BV+TZ, C=r* T^{-1} r, and Y=T^{-1}BV.
Here every adjoint uses the manuscript’s first-slot-linear convention.
Square completion gives
[ V^*SV=K-C,\qquad 0\preceq C\preceq K. ]
The upper bound on C follows from the already certified positive complete
Schur form, not from a numerical residual. Since T >= delta I,
[ Y=-Z+T^{-1}r,\qquad Y^Y\preceq2Z^Z+2r^*T^{-2}r \preceq2H_t+2\delta^{-1}K. ]
For 0 < sigma < delta, the resolvent identity gives exactly
[ V^S_\sigma V=V^SV-\sigma H_h -\sigma Y^*(I-\sigma T^{-1})^{-1}Y. ]
Spectral calculus bounds the inverse by (1-sigma/delta)^{-1} I. This proves
the new lower matrix in lem:v140-shifted-schur. It includes the entire change
in the infinite tail inverse. It is not the invalid operation of shifting the
finite head while keeping an unshifted Schur correction unchanged.
The old even and odd margins apply to exactly the reconstructed old trials.
No claim that a relative head margin is an ordinary spectral gap is used.
The new physical Grams V*V and Z*Z supply that missing coordinate information.
The complete tail floor delta=1.7940e-8 comes from the existing signed tail
certificate and the archimedean diagonal lower bound, not from a finite tail.
simultaneous_trial.py indexing:
even rows include mode zero; odd rows start at one; both correction arrays
begin at physical mode 17. Support ends at 4096. Parity coordinates are
orthonormal, so the ordinary norm is the sum of coefficient squares.K-orthogonal
complement. Exact Fraction arithmetic verifies that the larger odd trial
has exactly the required physical head.K and frozen direction. The overlap check in the new
script is a consistency check, not itself a proof that unrelated scalars are
equal. The original construction and its provenance establish equality.In addition to inspecting the script, the reviewer independently recomputed
all leading principal determinant signs with Arb’s matrix determinant routine,
rather than the script’s custom LDL routine, at 1024 and 1280 bits. The results
were sixteen positive leading minors followed by one negative leading minor
in the even sector, and sixteen positive leading minors in the odd sector.
These yield the same inertia counts. Selected column norms were independently
reconstructed from the frozen base and correction arrays with Python exact
Fraction arithmetic; the new Arb Grams enclosed those exact rational norms.
The prior small final gates were replayed in memory during these checks.
The even column 16 complete Rayleigh quotient is enclosed by
[ (2.45361754930,\;2.45361754931)\,10^{-75}. ]
The old odd column 15 has complete Rayleigh quotient approximately
3.59269956625301170615e-71. In particular the earlier trial threshold
1e-70 cannot lie below the entire odd spectrum. The smaller successful
threshold 1e-73 is a substantive correction to the initial diagnostic-based
choice, not numerical roundoff. No finite-cutoff eight-order gap is promoted
to a complete spectral gap.
The even lower matrix has exactly one negative eigenvalue and sixteen strictly positive eigenvalues. Monotonicity of finite eigenvalues under quadratic-form order makes the exact shifted even Schur matrix have at most one nonpositive eigenvalue. The actual finite-supported even trial has complete Rayleigh quotient below the shift, forcing one strictly negative eigenvalue. These facts together exclude a zero eigenvalue of that Schur matrix. The odd Schur matrix is positive definite.
Because the head is finite and belongs to the operator domain, its coupling
B is bounded, and (T-sigma I)^{-1}B lies in the tail operator domain.
The triangular square completion is boundedly invertible and preserves the
closed form domain. It transfers the negative index and the kernel exactly.
Compact resolvent and the existing strict positivity at lambda=4 therefore
give 0 < mu_0 < 2.454e-75 and mu_1 > 1e-73. The ordinary gap exceeds
9.7546e-74. This reasoning treats the complete operator, not the finite
trial compression.
Primary sources read directly:
Theorem 6.1 requires a real distribution on the one-sided interval [0,L],
the associated form on trigonometric polynomials, essential self-adjointness
and lower boundedness there, and a simple isolated global minimum with a
reflection-even eigenfunction. A generic self-adjoint extension would not
meet the stated operator-core hypothesis. Nor should an unspecified even
kernel replace the one-sided distribution: Remark 4.3 records an origin
distribution counterexample to that simplification.
The explicit distribution in new_section.tex has a convergent canceled
origin integral and order at most one. Substituting
h_n(t)=2(1-t/L)cos(2*pi*n*t/L) recovers the exact old diagonal d_n, including
log(tanh(L/2)); its pole contribution at n=0 is 32*sinh(L/4)^2/L as in
v114. Actual autocorrelations vanish at t=L, so the prime at 16 creates no
endpoint ambiguity. This checks the normalization and the one-sided
distribution hypothesis rather than relying on a Fourier-symbol analogy.
The established v114 decomposition is a real diagonal operator with logarithmic growth plus a bounded self-adjoint commutator. It establishes the required trigonometric operator core and compact resolvent for the actual physical form. The new certificate establishes the remaining spectral hypotheses.
Writing a=log(4) and translating the centered eigenfunction to [0,2a]
multiplies its unitary Fourier transform by sqrt(2*pi)*exp(-i*a*z) when using
the external paper’s unnormalized transform. That factor is zero-free.
Thus the claimed zero set is unchanged. Real zeros need not be simple, and
the theorem does not require a nonzero boundary value of the complete ground
eigenfunction.
This review and the new run reuse the complete v125/v126 residual certificates; they do not freshly assemble their 65536-row products and remote moment tails. The even inherited ingredients have independent 768/896-bit evaluations; the odd complement ingredients are the old 768-bit enclosure, reused for both new precisions. The odd directional reports are the inherited 1024/1280-bit evaluations. These limitations are correctly stated in the new proof and reports. The new shifted gates and physical Grams themselves were evaluated at 1024/1280 bits.
No new result about the repaired source endpoint, sampler graph defect, cofinal positivity, Riemann’s Xi function, or convergence of zero sets follows. The specific NS-5 fixed-window hypothesis is closed; G2 and RH remain open.