21 September 2026. Reviewed against complete manuscript v1.14 and the candidate files operator_residual_derivation.md and sharp_arch_tail.md. No main-manuscript edits were made.
Both proposed arguments passed internal analytic review. The first closes the ordinary-norm source/Rayleigh/residual prerequisites for the actual repaired two-mode source and its unchanged literal Fourier projection. The second gives a positive entire omitted Fourier complement at lambda=3 already beyond index 256. Neither supplies a signed head/complement Schur bound, a ground-state overlap lower bound, an endpoint-normalized graph estimate, or G2.
The conclusion checked is
[ c_0\le|p_\lambda|,|P_{N_\lambda}p_\lambda|\le C, \qquad |\mathsf W_\lambda p_\lambda|+ |\mathsf W_\lambda P_{N_\lambda}p_\lambda| \le C\lambda^6e^{-c/3}, ]
where c=2 pi lambda^2, L=2 log lambda, and N_lambda=ceil(lambda^8(1+L)). Constants and the eventual threshold are asymptotic, not numerically certified. The finite compression inherits the estimate by orthogonal projection.
The decisive point is that the manuscript’s Sobolev estimate explicitly holds for every periodic H1 test with one constant independent of the test and lambda. Taking its dual supremum over unit vectors in E_M gives a factor O(1+M/L+sqrt(M/L)). Summing separate per-mode estimates would introduce an unnecessary sqrt(M) and would not yield the claimed exponent at the chosen split. The submitted proof uses the correct supremum.
The following details were checked:
The limiting source is exactly
[ h_\infty(u)=\frac{4\pi}{\sqrt3}u^2(2\pi u^2-3)e^{-\pi u^2}. ]
Its two moments vanish, its Fourier eigenvalue is +1, and its co-Poisson output is inversion even. On the window the summand error is at most C lambda^-1 u^-1/2. Squaring and integrating du/u gives O(lambda^-1); Gaussian tails and inversion complete the claimed global L2 convergence O(lambda^-1/2).
For e=(I-P_N)p, the accepted v1.14 coefficient tail and the actual logarithmic diagonal give
[ |\mathsf W_\lambda e| \le C|B_\lambda|\lambda\sqrt{L/N} [\lambda+1+\log(2+N/L)] \le C|B_\lambda|\lambda^{-2}. ]
The bounded commutator has norm O(lambda), by the already established Chebyshev prime-mass estimate. The coefficient and graph bounds concern the complete operator, including its distinct logarithmic diagonal.
| For either ordinary unit-normalized candidate, this establishes alpha=<Wu,u>->0 and | (W-alpha)u | ->0. It does not identify which eigenvalue is approximated. |
All constants in sharp_arch_tail.md were checked. In particular:
| The sine-series comparison first uses the decreasing majorant t/(a_k^2+t^2); the exponential correction is at most R_L/t. This gives | S_n-(pi/4)sgn n | <=C_L/ | t_n | . |
| Binet’s digamma formula after shifting z to z+1 has denominator imaginary part 5t/4. With integral u/(exp(2 pi u)-1) du=1/24, its remainder is at most 1/(15t). The reciprocal terms contribute 7/(2t^2). The trigamma and finite-L series yield exactly the remaining terms in E_L(t). The logarithmic diagonal constant is log( | n | /L), with no missing 2 pi. |
The scalar Arb evaluation was rerun at 256 bits using the existing runtime dependency. It passed all strict inequalities:
[ \Gamma_{3,256}^{\rm sharp}>0.0148, \quad \Gamma_{3,256}^{\rm even}>1.5867, \quad \Gamma_{3,512}^{\rm sharp}>0.7085. ]
The enclosed values begin 0.0148329203130568, 1.5867887543730771, and 0.7085077224770671, respectively. This rerun uses the same Arb implementation, not independent software verification. It certifies the scalar analytic constants; it is not a certification of the head matrix, its inverse action, or the signed Schur correction.
| Let u_lambda be the normalized actual source, phi_lambda a normalized lowest eigenvector, kappa_lambda= | <u_lambda,phi_lambda> | , and epsilon_lambda=C lambda^6 exp(-c/3). The spectral equation gives the exact inequality |
[ |\lambda_0(\lambda)|\,\kappa_\lambda \le|\mathsf W_\lambda u_\lambda| \le\epsilon_\lambda. ]
Thus an independent estimate kappa_lambda/epsilon_lambda->infinity would imply lambda_0(lambda)->0. Together with domain-inclusion monotonicity, this would force nonnegative lowest values on every fixed window. Conversely, if lambda_0(lambda)<=-delta for a fixed delta>0, then kappa_lambda<=epsilon_lambda/delta. A negative ground direction can therefore be exponentially close to orthogonal to the constructed source without contradicting the new residual theorem.
No such independent lower overlap bound is established here. The elementary model diag(-1,epsilon_lambda), with the candidate equal to the second coordinate vector, satisfies arbitrarily small ordinary residual and Rayleigh value while its lowest-state overlap is zero. It is a logical warning, not a counterexample to the arithmetic operator. The overlap/signed-Schur problem remains the sign problem; it is not a negligible normalization detail.
Dividing the accepted residual estimate by the physical endpoint instead gives an exponentially growing upper bound proportional to exp(2c/3) times a polynomial. No endpoint-normalized compatibility estimate follows.
A separate route used the physical PSWF operator H_lambda=H_0+lambda^-2 V, with H_0=-d^2+4 pi^2 u^2 and V=(u^2 d/du)’. Harmonic ladder algebra gives V=(a^4+(a*)^4-2N^2-2N-3)/4. Its first corrections are w_0=-sqrt(6)H_4/(32 pi) and w_4=sqrt(6)H_0/(32 pi)-sqrt(105)H_8/(16 pi). Finite Gaussian quasimodes would then provide exact radical comparison sources with absolutely summable exterior couplings.
That backup was not promoted to a theorem or completed: the submitted output-splitting proof is stronger, shorter, and already uses established manuscript inputs. No additional quasimode or spectral-gap claim is needed for the accepted result. The direct exterior-tail approach for the actual prolate source remains unsuitable for separate unsigned prime/pole estimates, whose naive norms diverge.
At lambda=3, use the now positive full complement beyond N=256 in the exact signed Schur matrix F-B* T^(-1) B, with certified inverse-action residuals and both parity sectors. Positivity of a smaller finite section or the source residual must not replace this correction. A successful fixed-window certificate would still need a separate argument for growing windows before contributing a G2 conclusion.