RESEARCH RECORD · NS-36 · 22 SEPTEMBER 2026 · VERSION 1

The circle/symbol lane after NS-31

Numerical illustrations, not certificates. G2 and RH remain open. The new experiment uses the existing λ=4 window, N=256 even head, and twenty finite quadrature bins.

NS-31 found nine major and six minor errors or unsupported conclusions. Claude’s correction notices in main 318b3ce acknowledge them, but several board summaries and directory introductions still repeated withdrawn claims. This current reading replaces those interpretations. The five historical results.md files, original scripts and original outputs remain unchanged from that main commit. The full NS-31 audit remains the source for its file-and-line findings; PR #16 remains separate and unmerged.

What had to change

The λ=4, c′=1 claim now has a different, independently checked raw pure witness. The old vector’s minimum bin ratio was 0.6360967, so it was infeasible. A constrained search inside an eleven-dimensional low-energy subspace produces a new vector with minimum ratio 2.8474146 and midpoint energy 4.0522224×10⁻¹⁸. The search uses float arithmetic only to propose the vector; its saved decimal coefficients are evaluated against the 1024-bit assembled matrix with higher-precision midpoint arithmetic.

Projection against two independently recomputed finite-polynomial approximations to the λ=4 source preserves the margin. This is a numerical approximation to source admissibility, not a certificate for the exact repaired PSWF source. The exact-source truncation and repair enclosures needed for that stronger claim are absent. A separate mixed-state construction handles any source direction algebraically, but at a much larger energy budget.

1. The symbol and pencil: what survives

The shifted symbol and the (−1)ⁿ centering sign agree with independent λ=4 quadrature: the even entry W[0,1] is 0.0472426148915346486295 and the odd entry W[1,2] is 0.00115258928487815625733. The odd tail was omitted at a faster decay rate; it was not identically zero. These replay values are diagnostics, while the symbol identities are algebraic statements in the manuscript.

The assembly uses ψ(1/4) and folds a Lorentzian contribution into its continuum term. Only after moving that contribution does one obtain the symbol’s ψ(5/4) and e^t/2 split. The constant cosine mode is already normalized. The √2 correction in the λ=3 source loader combines the positive and negative complex modes; it is not an empirical adjustment or an unnormalized constant mode. The corrected source/ground overlap is 0.999999976695, not equality.

For Wv=νBv, B=W+W⁻, with v*Bv=1, a simple eigenvalue satisfies δν=−νv*(δW⁻)v to first order. Relative form control can therefore protect a small ν. Entrywise spot checks do not supply that control, and midpoint conversion also introduces numerator error. “Six relative digits however small” is withdrawn. The λ=8 omitted-tail changes include a 3.3×10⁻⁴ relative change to W⁻[0,1].

The finite deep block in both parities survives the 120/180-digit replay. At λ=6 and N=96, the lowest pencil value changes from −5.23827×10⁻¹¹⁶ to +6.58024×10⁻¹³⁵; at λ=8 it changes from −3.68097×10⁻¹¹⁵ to +5.45551×10⁻¹⁶⁵. Counts below the tested 10⁻⁸, 10⁻¹⁶ and 10⁻³⁰ thresholds are unchanged, but are not identical between parities. A converged block, limiting shape, local lobe-depth law, or literal Toeplitz-plus-Hankel representation does not follow.

2. Concentration and minorants: no diagnostic closure

The original concentration matrices act on the full finite head. Restricting that same matrix to the source complement can only reduce its largest eigenvalue. Enlarging the head can increase it. Consequently a full-head value has no automatic inequality direction relative to the source-compressed infinite operator. Quadrature, ξ≤800 truncation and omitted tails are separate issues.

The sampled “charged tail” is not a global minorant: β₈(130060.42500058559)=−8.8120135161 is below −8.050, and β₆(152359.72500188358)=−6.4358688611 is below −5.808. D2 uses a reversed monotone profile; correcting that profile changes the λ=4 tail-substituted model from about 3.33226 to 3.15666. D3’s matrix and validation step quantize different values. D1 has J+1 values for J sets. These defects remain in the archived script and are explicitly flagged in its directory introduction.

A feasible family’s optimized loss is an upper estimate for the best loss in that family; it does not lower-bound every minorant’s loss. Four ratios near 0.35 do not establish divergence or necessary linear growth of the number of levels. The uniform concentration estimate itself remains open. QG and the all-Borel weighted packet hypothesis are separate missing inputs for a proposed bounded-level obstruction, not inputs that themselves prove positivity. The map now makes this distinction explicit.

3. Level distribution: lower estimates, witnesses and bins

For nonnegative dual multipliers y, λ_min(W−Σy_bM_b)+Σy_br_b is a lower bound for the finite optimization problem. A minimizing eigenvector gives an upper bound only if it satisfies every constraint. In the original uniform problem M_b is mass divided by bin width and the requirement c/D is equivalent to bin mass ≥c/20. In the weighted problem M_b is mass and the requirement is c′ times the frequency-measure fraction. Those conventions and the two-sided Fourier factor are consistent.

The claim that c=0.5 is infeasible at every window is false for the twenty-bin mixed-state problem: NS-31 archived feasible constants 0.5293364145 at λ=4 and 0.5375107827 at λ=8. The later notice’s “c≥0.538” at λ=8 rounds upward incorrectly. All level-count conclusions formed from dual lower values in place of feasible energy upper values are withdrawn. The λ=3 source-projected c′=1 witness remains a diagnostic with energy about 2.4299922×10⁻⁹ and positive slacks.

The largest aligned bin is not the sliding concentration function M(h): at λ=3 the values are 0.0072175 and 0.00831375 on the same grid. The purported twenty-bin peak 0.0914573 came from 200 bins; the twenty-bin peak is about 0.0606982. Neither is a cap on the continuous density. The dynamic program solves the stated midpoint-and-bin-edge surrogate; it is not the unrestricted quantization problem. Natural cosine packets in section 3 are not automatically states of the finite N=256 matrix problem. Twenty inequalities do not establish an all-Borel measure inequality.

4. The new source check and its limits

The source loader generalizes the Legendre and modal-integral formula in the archived λ=3 certificate, including the positive/negative mode normalization. Replaying λ=3 reproduces the printed coefficient midpoints within 4.987×10⁻⁶¹. At λ=4 it uses modes 0 and 4 at prolate parameter 2πλ², forms the integral-zero combination, converts it to Fourier coefficients, and normalizes in the real cosine basis. It projects the frozen new pure vector against this raw polynomial source approximation.

The raw overlap is −0.0404195210, so source removal cannot simply be assumed negligible. After projection, both source calculations give energy 4.0588534993434549675×10⁻¹⁸. The raw source center defect changes from −5.37808×10⁻⁵⁷ to +2.34607×10⁻⁷⁶. The corresponding estimated relative bump-repair scales are 1.41704×10⁻⁵³ and 6.18152×10⁻⁷³. These are unenclosed estimates; they are not a replacement for a PSWF truncation bound or the repaired-source certificate.

The refined grid subdivides the original weighted intervals while keeping its small endpoint gaps. It freezes the original bin edges and extends the deepest bin downward to retain any new sample below the old sampled minimum. Thus it tests quadrature sensitivity; it neither repairs the entire frequency range nor measures the true global depth. The saved JSON files contain all coefficients, all twenty masses and slacks. No negative-frequency tail is silently charged as a global bound.

Source-independent mixed-state alternative. Let E be the selected 27-dimensional subspace with orthogonal projector P. For an arbitrary source u with Pu≠0, put z=Pu/‖Pu‖ and X=(P−zz*)/26. Then tr X=1 and the support of X is orthogonal to u. Its bin mass is at least [tr(M_b|_E)−λ_max(M_b|_E)]/26. Its energy is at most [tr(W|_E)−λ_min(W|_E)]/26. If Pu=0, use X=P/27; the same conservative estimates apply. This is an exact finite-dimensional implication conditional on the matrix checks, which here remain numerical. A feasible mixed state need not contain a single pure component satisfying all twenty constraints.

The operation above preserves the finite head: it removes the head source direction and renormalizes. It is not the full physical projection P_a v. If the head coefficients were exact, it would nevertheless give exact orthogonality to the full source, because the witness lies in the head. Enclosing those head coefficients requires the infinite modal error and the repair; no source Fourier-tail estimate is needed just for that orthogonality. The saved 80-digit coefficient files reproduce overlaps −9.4472×10⁻⁸² and −7.9819×10⁻⁸², while the smaller 10⁻¹³³/10⁻¹⁷³ values in the initial logs were internal arithmetic residuals. The source-direction difference between the two truncations is 1.3238×10⁻⁵⁶. The mass and energy checks both use the basis Gram; arithmetic and quadrature errors remain unenclosed.

5. Lobe mass, arithmetic budgets and retractions

The sampled lobe-mass inequalities require very large finite constants: about 10⁹⁸, and about 10¹⁰⁴ for a displayed any-lobe variant. That does not refute the existence of some finite uniform constant. The corrected λ=8 first negative lobe is approximately [0.378,1.135], with mass 0.193725. The old “every variant is killed” and literal “no mass” wording is withdrawn.

Small primes dominate an ordinary-size budget, not its tiny residual. At λ=3, the m=8 contribution −1.66×10⁻²⁵ is about 3.9×10¹² times q=4.242×10⁻³⁸; at λ=6, the m=32 contribution −7.76×10⁻⁶⁹ is about 7.4×10²³ times q. The corrected prime level pairs are (−0.546001,+0.490828), (−0.664473,+0.598184), and (−1.089090,+0.999652) for the three ground directions. Deleting the smaller prime terms changes the quantity being explained.

The circle split and sinc-lattice conclusions remain retracted. The decisive NS-14 defect is the missing (−1)ⁿ centering phase, not merely insufficient eigenvector precision. No withdrawn ground energies or sinc-zero counts are reused by this follow-through. The remaining manuscript reference to an unweighted Chebyshev remainder was handed to the v1.50 owner for correction to the existing weighted strict-cutoff r_a(0) formula.

Verdict and next handoff

The correct symbol stands. The deep block stands as a finite-head observation in both parities. Ground/source equality falls; close λ=3 alignment stands with its stated normalization, and the new λ=4 projected witness concerns an explicitly computed source approximation. The diagnostic layer-cake and bounded-level closures fall. “Weighted is cheap” now survives the λ=4 finite-bin replay with a different witness, while the exact repaired-source, all-Borel and cofinal statements remain open. The failure was of the original witness and inferences, not evidence of a negative Weil direction.

Main Astra owns NS-34’s analytic comparison-weight result; Astra-2 reviewed that proof as NS-37 and completed an independent read-only review of the new NS-36 algebra and source convention. No major error was found. Exported-overlap reporting and consistent Gram normalization were clarified after that review. NS-36 does not import a new manuscript version or ask for a merge. The existing manuscript builds to 242 pages with zero undefined/duplicate references; the v1.49 manifest verifies 7/7 items; Python diagnostics report zero errors and warnings. These checks validate the archive and build, not the mathematical numerical claims.

Replay: feasibility search, source projection, saved-witness checks. Data: original-grid witnesses, refined mixture check, first source replay, second source replay, independent frozen-coefficient checks. The provenance inventory records input and output hashes.