48. The Weil functional's degeneracy: three measurements and a constraint on proof strategy

Hunts #155, #156 and #157. The records are hunts/arch_cover/RESULTS.md, hunts/weil_gram/RESULTS.md and hunts/weil_split/RESULTS.md. This page is the front door. It quotes those records and does not recompute them. The pins that keep the records from drifting are hunts/arch_cover/test_arch_cover.py, hunts/weil_gram/test_weil_gram.py and hunts/weil_split/test_weil_split.py.

Nothing here is evidence for the Riemann Hypothesis. A finite computation is not a uniform statement about RH (docs/08). A hunt measures. It does not claim. This page does not promote a claim. Grade of the page: measured, the grade of the weakest step below. No human has reviewed the hunts or this page.

A prior-art scan, docs/reviews/2026-10-30-weil-degeneracy-prior-art.md, compared the landed statements with sources it opened. Its three verdicts are PARTIAL, PARTIAL and PARTIAL. This page follows that scan's closing list. It claims nothing as new to the literature. The scan did not cover Undermind, did not open Weil's 1952 paper or Yoshida's 1992 paper, and used Google Scholar only for the two result pages named in the memo.

1. The object

In this repository's convention (docs/10, implemented in zeta/weil.py),

    W(h) = pole + arch + prime,
    pole = h(i/2) + h(-i/2),
    arch = (1/2pi) int h(r) [Re psi(1/4 + ir/2) - log pi] dr,
    prime = -2 sum_n Lambda(n) n^{-1/2} g(log n),

and W(h) = sum_rho h(gamma_rho) for every admissible h. Conventions differ across the literature, so the module was calibrated by comparing the two sides (docs/10). This page does not re-derive that calibration.

For h = |f^|^2 the test function is non-negative on the real line. When the ordinates in the sum are real, the zero side is a sum of non-negative terms, and it does not cancel. The cancellation that produces a small W then sits on the arithmetic side: three terms, each typically of size O(1), adding to that non-negative number. Off the line an ordinate is not real, h is no longer a square there, and the zero side need not be non-negative. That is the criterion, not a fact the measurements assume away. The zero table used below places its zeros on the line only up to its height.

The criterion, in this convention, is that W(h) >= 0 for every admissible h of positive type if and only if RH (docs/07, docs/10). Bombieri's Clay notes, as read in the scan, state the same criterion as negativity of their right-hand side, for every autocorrelation whose test function has vanishing integrals against dx/x and against dx. The displays are not the same, and this page does not equate them. The admissible class differs across sources. docs/10 records that, and neither that document nor this one re-derives the minimal hypotheses.

The balance is the explicit formula: one sum, three kinds of term, required to come out non-negative on autocorrelations, with either sign allowed on each term. It is not a requirement that the archimedean term cover the prime side. Hunt #155 measured a comb whose prime term is +0.0174 (two teeth at log 2, sigma = 0.01): on that row the prime sum helps. Connes and Consani, arXiv:2106.01715, as read in the scan, record the opposite numerical fact from any such covering demand. Past log 2 the archimedean contribution alone "ceases to be positive", and "the positivity is restored by adding the contribution of the prime 2."

Grouping the prime side by place is already in the sources the scan opened. Connes and Consani write a local term

    W_p(F) = (log p) sum_{m=1}^infty p^{-m/2} (F(p^m) + F(p^{-m}))

and a quadratic form as pole terms minus an archimedean distribution minus sum_p W_p, labeled as Bombieri's form of the explicit formula. Bombieri's Clay notes say: "The terms in the right-hand side of the equation can be written as a sum of terms of local nature, associated to the absolute values of the underlying number field, or function field in the case of curves over a field of positive characteristic." That sentence supports a place split. It does not write the shift-norm cost in section 2.3. That cost is not presented here as standard notation.

2. Three measurements

2.1 Hunt #155 (arch_cover): how deep the three pieces cancel

Grade: measured. No proof, no review. Two families: a Gaussian h = e^{-a r^2}, and a comb. The printed balance table rounds. The digit-by-digit comparison with the closed form is in section 4(b) of the hunt record. An enclosed scan also ran a Fejer grid. Those rows are not in the printed table, so this page does not quote them.

Gaussian rows, quoted from section 3 of hunts/arch_cover/RESULTS.md:

On the full table, comb rows included, the zero side and the arithmetic side agree between 1e-22 and 1e-32. The prime term grows with a on the Gaussian (0.116 at 0.05, 0.603 at 0.2, 1.240 at 0.65) while the zero sum collapses onto the first zero. The hard regime is inside the family, not outside it.

The cancellation depth is

    depth = log10( max(|pole|, |arch|, |prime|) / |W| ).

On the Gaussian, once one zero dominates, the zero side is 2 e^{-a gamma_1^2} times a factor 1 + O(e^{-a(gamma_2^2 - gamma_1^2)}), and the largest piece is pole = 2 e^{a/4}. The hunt record therefore states

    depth = a (gamma_1^2 + 1/4) / ln 10 = 86.88 a,

with gamma_1^2 = 199.79. Measured against that form, to every digit the hunt's results.json and a direct sum over the zeros carry:

The section 3 table prints the same rows more coarsely (4.34, 17.38, 39.1) and does not include a = 0.3. The statistic carries no information beyond h(gamma_1) against the pole. It measures how far the test function sits from the first zero.

The form first written was depth = a gamma_1^2 / ln 10, which drops the pole's e^{a/4}. The hunt's kill condition, that the closed form fail to reproduce a measured row to the printed digits, fired against that form (17.38 measured against 17.35 at a = 0.2) and was not recorded until landing. The corrected form reproduces every row. That firing is a defect of the first write-up, recorded in section 5.

Normalized by ||f||^2 = (1/2 pi) int h = 1/(2 sqrt(pi a)), the ratio W / ||f||^2 along this family is about 4 sqrt(pi a) e^{-a gamma_1^2}, and it tends to 0. That ratio is what the hunt wrote. The scan did not find it outside this repository. This page does not cite it as a known formula.

The inequality this page relies on is the one the ratio tending to 0 gives: no inequality W(|f^|^2) >= c ||f||^2 with a fixed c > 0 holds over the admissible class. The hunt record reads the same family as the bottom of the form being 0, approached rather than attained away from the zeros. A proof must be an identity, or an inequality whose constant degenerates as the test function concentrates away from the zeros. Section 3 keeps this separate from the window theorems, and records what the scan did find.

The enclosed instrument has a floor near 33 digits, set by prime truncation, not by precision. At a = 0.65 the enclosure is [-7.12e-33, +6.36e-34], with sign 0. The binding term is prime_tail_bound = 7.76e-33 at the n_max = 2e6 cap. The value from the zeros there is 7.98e-57. A report that the enclosure was identical at 256 and 512 bits is unpinned: only the 256-bit run is committed (rows.jsonl).

2.2 Hunt #156 (weil_gram): the bottom of the form on Gaussian spans

Grade: measured. Eigenvalues are not enclosed in ball arithmetic.

lambda_min on a span is the minimum of W(|f^|^2) / ||f||^2 over that span, and W(|f^|^2) = sum_rho h(gamma_rho). It is not a squared distance to functions vanishing at the zeros. Reading the zero side as a sum of |F(gamma)|^2 presupposes that the zeros lie on the line, which the zero table guarantees only to its height.

On a purely broad Gaussian span, refined by adding sigma = 0.05, then 0.25, then 0.30, section 3 of hunts/weil_gram/RESULTS.md gives:

The arithmetic side and the zero side agree at every printed digit. Every element is individually healthy: W(h_jj) at sigma = 0.30 is 3.10e-8, and at sigma = 0.05 it is 2.75. The bottom still falls by 19 orders from k = 2 to k = 6. At k = 6 the bottom eigenvector is concentrated on sigma = 0.30 with a -0.075 admixture of sigma = 0.25. The hunt record describes the descent as geometric in the dimension, roughly a factor 1e-6 per added element at this end of the range. That geometric drop was not found by the scan outside this repository. It is not Zhu's window law: different quotient, different parameter (section 3).

Adding one concentrated element is a different drop, the single-element degeneracy of hunt #155 restated on the form. From +3.574e-7 on the broad span the bottom falls to +3.917e-25 once sigma = 0.45 is included, and the bottom eigenvector is the coordinate vector of that element.

The plus2 row, which adds sigma = 0.60, is not a measurement. The (0.60, 0.60) entry has prime_tail_bound 1.2e-33 against a value 1.16e-31, 1 percent truncation uncertainty, and the eigenvalue sits below that uncertainty. The two sides disagree because the eigenvalue is far below the entry uncertainty. What is established is only lambda_min(plus2) < 1e-30. The unresolved digits are not quoted here.

A file first written as a dps 90 run was computed at dps 50: run_base took its precision default from a module constant, and main did not pass the requested precision. It was rerun at dps 90 at landing (3 min 45 s). No eigenvalue changed at any printed digit. The rerun is a null test of the arithmetic side. Entry residuals sit at the zero table's 30-digit floor, about 1e-32, and the check cannot confirm entries below that floor.

Consequence, scoped as the hunt record scopes it after the landing correction. A uniform positive gap does not survive enlargement of these subspaces. An argument that needs lambda_min bounded below independently of the subspace cannot work on these spans: the gap is created by the choice of subspace and destroyed by adding dimensions. Nonnegativity on each subspace of an exhausting sequence does pass to the limit. That weaker route is untouched by the measurement.

2.3 Hunt #157 (weil_split): the prime side split by place, on rectangles

Grade: measured, with one exact identity: the place sum against the prime term, checked to the float-sum floor.

For f = b^{-1/2} 1_{[-b/2, b/2]}, so ||f||^2 = 1, the shift Phi_p f = sum_{m >= 0} p^{-m/2} f(. - m log p) has finitely many overlapping terms. The hunt writes

    ||Phi_p f||^2 = (1/b) sum_{m,n >= 0} p^{-(m+n)/2} (b - |m-n| log p)_+
    Q_p = (1 - 1/p) ||Phi_p f||^2
    cost_p = log p (Q_p - 1)

cost_p is an exact finite closed form. It carries log p, p^{-1/2} and b. It is not a rational number. It is not standard notation. It is a rewriting of the von Mangoldt sum for an autocorrelation, already checked in this repository: docs/24 keeps the general factor Q_p - ||f||^2, and hunts/weil_split specialises to ||f||^2 = 1, which is why the cost is log p (Q_p - 1) on this family. The scan did not find this rewriting outside the repository.

sum_p cost_p = -prime_term was checked to 1e-16, the float-sum floor, at every b tested. Only primes with log p < b contribute, that is p <= e^b. The identity holds place by place by construction: expanding the overlap sum gives cost_p = 2 log p sum_{d >= 1} p^{-d/2} g(d log p), the p-part of the prime term, so the check verifies the code, not mathematics. A referee recomputation (hunts/weil_split/REFEREE.md) agrees at 50 digits, and hunts/weil_split/test_weil_split.py pins it.

Quoted from section 2 of hunts/weil_split/RESULTS.md:

cover in that table is (pole + arch) / cost, computed as (W_from_zeros - prime_term) / sum_p cost_p, not from the archimedean quadrature. By construction

    cover - 1 = W_zeros / sum_p cost_p,

with W_zeros = sum h(gamma). On this family h >= 0, and the zeros in the table are on the line, so W_zeros >= 0 inside that table's range. The margin is W normalised by the prime cost, not an independent statement about the archimedean term. Its decay says W grows more slowly than the prime side. From b = 2 to b = 8 the margin falls from 3.04e-2 to 1.59e-4, though not step by step: on hunt #158's 0.25 grid it rises on 5 of 24 steps. The local exponents between rows run from 2.2 to 5.6, and the overall decay is about b^{-3.8}. The identity cover - 1 = W_zeros / sum_p cost_p is true by the definition of cover, so there is nothing for a literature search to find; the scan did not find the measured decay outside the repository.

The cost spreads. At b = 2 the top three places carry share 0.977 (the hunt record names p = 2, 3, 5 as carrying 98 percent). At b = 8 there are 429 places and the top three carry 0.126. On this family there is no finite set of important primes doing the work, and no bounded slack.

What does exist in the literature, and is not this margin, is the semilocal reduction of Connes, Consani and Moscovici, arXiv:2310.18423, as read in the scan: a property P(n) using only the Euler factors for primes smaller than n, such that P(n) for all n is equivalent to RH, because Weil positivity on test functions supported in [1/n, n] is that criterion. More places enter as the support widens, and the criterion is the whole sequence. That is not a decaying covering margin.

The archimedean quadrature on this family agrees with the zero side only to between 1e-3 and 8e-3: three digits. h decays like 1/r^2, which is what compact support of f costs. Anyone quoting the quadrature value of pole + arch is quoting three-digit arithmetic. The cover column is printed to nine digits because it uses the zero side and the exact prime side. The two differ by the quad-vs-zero column.

3. What a proof through this functional would have to be

The constraint, stated so it does not outrun the records:

There is no inequality W(|f^|^2) >= c ||f||^2 with a fixed c > 0 over the admissible class. On the Gaussian family the ratio tends to 0 (hunt #155). On growing Gaussian spans a uniform positive gap does not survive enlargement (hunt #156). A proof through this functional has to be an identity, or a decomposition whose constant degenerates with the test function.

The weaker route is untouched. Nonnegativity on each subspace of an exhausting sequence passes to the limit. That is the shape of the criterion. It does not need a gap.

This is not a statement the laboratory is presenting as its own. The scan's verdict on the question is PARTIAL.

Bombieri, "Remarks on Weil's quadratic functional in the theory of prime numbers, I", Atti Accad. Naz. Lincei, Rend. Lincei Mat. Appl. (9) 11 (2000), 183-233, as the scan records it: the quadratic functional is positive semidefinite if and only if RH, it attains its minimum on the unit sphere of L2 functions supported in a given interval, and it is positive definite for sufficiently small support. The small-support half is attributed there to Yoshida, "On Hermitian forms attached to zeta functions", Adv. Stud. Pure Math. 21 (1992), 281-325. Yoshida's paper was not opened. Zhu, arXiv:2608.24827, quotes the classical range as support in [-(log 2)/2, (log 2)/2], attributed to Yoshida and, by a different method, to Connes and Consani. Small-support positivity is cited here only as Bombieri and Zhu state it. Bombieri's attainment on each fixed window is not a uniform gap. Nonnegativity on every compact window is the criterion. It does not require a constant c > 0 independent of the window.

Connes and Consani, arXiv:2106.01715, as the scan records them: if RH holds, their window form is strictly positive, so the radical on each finite window is {0}, and yet one can build a test function on that window with the form much smaller than the square of the norm. They report a smallest positive eigenvalue 2.389 x 10^{-48} at lambda^2 = 11. That is a reported eigenvalue, not a theorem that the infimum over the whole admissible class is 0, and it is not a geometric drop per added Gaussian.

Zhu, arXiv:2608.24827, defines lambda^*(L) as the infimum of Q(f) / ||f||_2^2 for support in [-L, L]. Under RH, lambda^*(L) <= exp(-L e^L) for all large L, by an explicit construction. He writes that the computation turns "the folklore that positivity gets exponentially hard" into measured constants and a barrier. That folklore sentence is his. The conditional bound is a citation for no fixed c > 0 on the union of the windows, because every window sits inside the admissible class and the bound tends to 0. It is conditional on RH. If RH is false, Weil's criterion already says the form takes a negative value, so no positive c exists either. The scan did not find that dichotomy written as one lemma. Zhu's unconditional upper bounds at specific L, down to 3.2 x 10^{-283} at L = 2 in the abstract the scan read, show the window constant is tiny at those L. They are not themselves a proof that the infimum over all L is 0.

Keep the two directions separate, as hunt #156 section 4 does. Nonnegativity on each subspace of an exhausting sequence is the criterion and passes to the limit. A uniform positive gap does not survive enlargement: under RH that is Zhu's theorem for windows, and the scan did not find it as a theorem about Gaussian spans. The Gaussian ratio in section 2.1 is not cited here as a known formula.

4. Prior art, in the scan's own verdict words

The scan is docs/reviews/2026-10-30-weil-degeneracy-prior-art.md. Verdict words, quoted exactly:

1. PARTIAL. The split of the explicit formula into one local term per place is in the sources opened. The L2 shift-norm identity is not.
2. PARTIAL. The three-term balance is the explicit formula. The covering-margin identity was not found. The disowned framing, that the archimedean term must dominate the sum of local costs, is not what the opened sources say.
3. PARTIAL. A falling window constant, and the passage of nonnegativity rather than of a uniform constant, are in the sources opened. The Gaussian ratio, and a geometric drop per added Gaussian, were not found.

No verdict in that scan is KNOWN, and none is NOT FOUND. Where a statement above is the explicit formula, a place split, a window infimum, or the passage of nonnegativity to the limit, the citation is to the source the scan opened, not to these hunts. Where a statement is the shift-norm cost, the rectangle margin, the Gaussian ratio, or the geometric drop on Gaussian spans, the page says the scan did not find it outside this repository. That is a limit of the scan, not a claim that the statement is unknown.

The scan did not cover Undermind. The site answered, and the page was a login wall, so no query was run. It did not open Weil 1952 or Yoshida 1992. Google Scholar was used only for the two result pages named in the memo. Neither Google Scholar nor the arXiv and zbMATH queries recorded there is a complete literature search.

5. Scope and defects

The page takes the grade of its weakest step. The place-sum identity is a finite check at the float-sum floor. The strategic reading also uses eigenvalues that are not enclosed, and a cover column whose quadrature sibling has only three digits. Grade: measured. Nothing under the three hunt directories is a result, and this page does not make one of them.

Defects recorded on the landed hunt records, carried here rather than left in the directories:

• Eigenvalues in hunt #156 are computed in mpmath, not enclosed. The generalised eigenproblem is badly conditioned by construction: entries span 1e-31 to 1. Nothing in that hunt is enclosed.
• The plus2 row is unresolved. The only established bound is lambda_min(plus2) < 1e-30. The unresolved digits are not quoted on this page.
• The archimedean quadrature in hunt #157 agrees with the zero side only to between 1e-3 and 8e-3. Three digits. The printed cover column does not use that quadrature. Quoting the quadrature value of pole + arch is quoting three-digit arithmetic.
• The families are narrow. Hunt #155: finite families, finite grids, no statement uniform in f. Hunt #156: Gaussians centred at the origin. Shifts and non-Gaussian elements were not measured. Hunt #157: one rectangle family. The widest member still has h of slow decay.
• No human has reviewed any of the three hunts, or this page.
• The enclosed instrument in hunt #155 has a floor near 33 digits, set by the prime-tail cap n_max = 2e6 (prime_tail_bound = 7.76e-33 at a = 0.65), against a value from the zeros of 7.98e-57. Adding bits does not move that floor. The floor is a measured optimum at one cap, not a proved ceiling: candidate, the Gaussian family at a <= 0.65, information class the prime sum truncated at n_max, proof status none.
• A statement that the a = 0.65 enclosure was identical at 256 and 512 bits is unpinned. Only the 256-bit run is committed.
• The first results_dps90.json in hunt #156 was computed at dps 50. It was rerun at dps 90 at landing. No eigenvalue changed at any printed digit. The arithmetic side is more accurate than the zero-table check, which cannot confirm entries below about 1e-32.
• The depth law was first written as a gamma_1^2 / ln 10, dropping the pole's e^{a/4}. The hunt's kill condition fired against that form (17.38 measured against 17.35 at a = 0.2) and was not recorded until landing. The corrected form is the one in section 2.1.
• The archimedean term has no second backend in this tree. enclose_weil_functional reports that a two-backend cross-check is unavailable, because mpmath.iv has no enclosed quadrature. A quadrature error in the archimedean term would look like W moving. The hunt record calls this the largest unmitigated risk in every enclosed number it produced.
• Hunt #157's cover uses the zero side for W, so it inherits the zero table's 30-digit accuracy and the range gamma < 10^4. The zero tail bound is reported by explicit_formula_sides and is negligible for these b, but it is not zero.
• Q_p > ||f||^2 for every place measured in hunt #157, so every cost_p > 0 there. That is the opposite sign to reading Q_p as the norm the local-positivity gate was about. The gate in docs/24 applies to Toeplitz sections of local coefficients, not to ||Phi_p f||^2 of a fixed f. The two should not be conflated.
• lambda_min > 0 at every measured k is consistent with RH and with its negation on other spaces. It is not evidence about RH in either direction.