24. The local positivity attempt, run to its wall

An ontology attempt in the sense of docs/09 §4: propose a structure, push it at the gates, and record exactly where it bleeds. It bleeds at the place docs/09 §5.1 predicts, which is the useful part.

The short version

Requirement C of docs/09 §5.1 asks for a construction in which the Weil quadratic form is the norm square of a naturally defined operator, so its sign becomes formal. This document reports an attempt that achieves exactly that, one place at a time, and fails to globalise, the local norms exist, are canonical, are built from prime data alone with no zeros anywhere in the definitions, and do not assemble into a global one because the per-place residual is not of definite sign (measured: 52 of 60 places positive, 8 negative).

What survives is not a proof strategy but an instrument: a place-local criterion c_p ≤ d that reads the local bound |α_j| ≤ √p, computed from coefficients alone. The forward direction is derived; the converse is not established by the closed form and rests on an unreviewed derivation (§2 and hunts/local_positivity/CORRECTIONS.md, "Later correction", 2026-10-21). It passes ζ and Dirichlet L-functions, fails Davenport–Heilbronn and both discriminant −23 Epstein forms, and, unlike the Imposter Gauntlet of docs/15, which "never actually consults ζ", it demonstrably reads its input. Code: localpos.py. Every number below is recomputed by the module, and every reference verdict re-derives.

1. The construction

For a Dirichlet series f(s) = Σ aₙ n^{-s} with a₁ = 1, write the local log-derivative coefficients λ_m = b_{p^m}/log p at the place p. Define the place-p kernel

    K_p^{(d)}(θ) = d + 2 Σ_{m≥1} λ_m p^{−m/2} cos(mθ)

and the normalisation-free threshold c_p, the least diagonal for which every finite Toeplitz section with off-diagonals λ_m p^{−m/2} is positive semidefinite, so the form with diagonal d is PSD exactly when c_p ≤ d. On a summable sequence c_p = −min_θ 2 Σ_{m≥1} λ_m p^{−m/2} cos(mθ), which is what the module evaluates; on a growing sequence c_p = +∞ (the module's "divergent" regime); at |α_j| = √p the cosine series diverges while the least diagonal can stay finite (exactly d for ζ(s − ½), where the gate reports INDETERMINATE; §2 records where the gate's regime reading goes wrong). This one reading is used throughout. Nothing is fitted: c_p is a minimum, and d is the degree read off the object's own gamma factors.

Why the kernel is the thing. Let Φ_p f = Σ_{m≥0} p^{−m/2} f(· − m log p) be the one-sided local shift-average. Then the prime side of the Riemann–Weil explicit formula decomposes place by place:

    prime term  =  − Σ_p log p · ( Q_p(f) − ‖f‖² ),    Q_p(f) = (1 − 1/p)·‖Φ_p f‖²

Measured against zeta.weil.explicit_formula_sides for the autocorrelation pair: reconstruction −0.154060343334 against the module's −0.154060343334, agreeing to 22 digits once the place sum runs to p ≤ 8000. Each Q_p ≥ 0 by construction, it is a norm. That is Requirement C's norm identity, obtained locally and canonically.

2. The derivation behind the statistic

If the place has a genuine Euler factor with Satake parameters α_j, then λ_m = Σ_j α_j^m and, inside the bound, the series sums in closed form:

    K_p^{(d)}(θ)  =  Σ_j  (1 − |α_j|²/p) / |1 − α_j p^{−1/2} e^{iθ}|²

a sum of manifestly nonnegative terms when every |α_j| < √p and the coefficients are real (parameters real or conjugate-paired). Checked numerically: series 2.7203325546805037 against closed form 2.7203325546805086 at p = 7, degree 2.

What this establishes is the forward direction: every |α_j| ≤ √p gives c_p ≤ d (derived; at |α_j| = √p each parameter contributes a rank-one PSD Toeplitz matrix with unit diagonal). An earlier version of this section read the converse off the closed form, and that step is invalid: at p = 2, α = (3√2, 0, 0, 0) the rational expression is 3 − 8/(10 − 6 cos θ) ≥ 1 although one parameter exceeds √p, and the Fourier series it was supposed to sum diverges. The gate is not fooled by that input, since it never evaluates the rational expression: threshold reports "divergent" and gate returns FAIL. The converse for c_p as defined in §1 rests on a separate argument: a PSD section with diagonal c has |s_m| ≤ c, and s_m = Σ_j (α_j/√p)^m is unbounded once some |α_j| > √p (no cancellation among the parameters prevents it), so c_p = +∞. That derivation is written out in hunts/local_positivity/CORRECTIONS.md ("Later correction") and is not independently reviewed, so the iff is an unreviewed derivation, pending external verification, not a decision procedure on the strength of the closed form.

The gate is not a decision procedure on its own account either. It truncates the series and reads the regime off two weighted terms a decade apart near the truncation edge, and for a conjugate pair of Satake parameters the weights oscillate, so that reading can land on either side of the bound: measured 2026-10-21, α = 0.9·√2·e^{±0.2i} at p = 2 FAILs although c_p ≈ 1.894 < 2, α = 1.2·√7·e^{±0.4i} at p = 7 PASSes although c_p = +∞, and the unit-modulus degree-2 family of §4's 60-angle control gets FAIL at 22 of the 60 angles (the original gate of 2026-08-10 passed all 60). In the summable regime an excess smaller than ten tail estimates is read as PASS rather than as undecided (for the shift ζ(s − δ) of §6 at δ = 0.49, p = 2, the truncated c is 15.99 against d = 1 with a tail estimate of 149.9, while the true value is 0.9965). All of this is pinned in hunts/local_positivity/test_closed_form.py, and the gate itself is unchanged.

The ζ case has the exact closed form c_p = 2/(√p + 1), reproduced to 12 digits at p = 2, 3, 5, 7, 97 with an explicit truncation bound below 3.4e-14. (The bound is elementary, 2·max|λ|·p^{−(M+1)/2}/(1−p^{−1/2}), and computed in floating point, so it is not certified in the sense zeta/rigor.py owns.)

3. The gate

Degree d is forced by each object's own functional equation, never chosen.

All seven reference claims re-derive, localpos.reference_table() recomputes this table rather than storing it, and a row whose verdict disagrees with its expectation is a failure of the gate, not of the subject. This is docs/09 Gate 3 with a crisp answer to "where exactly does Davenport–Heilbronn fail to embed?": at p = 2, with excess 0.836, because its local kernel is not positive semidefinite.

The Epstein rows use localpos.epstein_local, which counts ideal classes rather than enumerating lattice points: zeta.epstein.epstein_representation_count is exact but enumerative, and cannot reach the p^k with k ≈ 90 that the p = 2 truncation needs. localpos.epstein_local_check() asserts the two agree on all 35 prime powers small enough for both, which is what licenses the shortcut.

4. The controls, which are the point

Decoy, does the statistic read its input? The failure mode docs/15 records for the Imposter Gauntlet is a predicate that ignores its argument. Swapping the coefficients moves the verdict: a_{p^k} = 2^k scores an excess of +9.6e14, random ±1 coefficients score +8.4e3, against ζ's −0.172. The gate consults its input.

Null, is a nonzero excess meaningful? Against 300 random period-5 real sequences, the null median excess is +5.88 and 100 % fail. DH sits at the 6th percentile, i.e. Davenport–Heilbronn is an unusually mild member of the class of things that fail, not an exotic near-miss. Matching docs/18 §6's finding with a different statistic.

Lesion, where is the detector blind? Interpolating ζ → DH coefficientwise, the smallest detected violation is ε* = 0.184. A PASS therefore means "no violation above ~18 % of the way from ζ to DH at the tested places", and nothing stronger. The PASS side is not vacuous either: across 60 Satake angles the genuine degree-2 family keeps a margin of at least 0.343. (Those are margins of the truncated c. Since the regime test was added on 2026-08-11 the gate's verdict on that family is FAIL at 22 of the 60 angles, §2.)

5. The honest boundary

The gate is not a test for "has an Euler product." A genuine degree-2 Euler factor with α = 2.3, 1/α, violating Ramanujan, is rejected at p = 5. The place is in the "divergent" regime (2.3/√5 ≈ 1.0286), so c_p = +∞ by the module's convention; the 65.24 printed there is the series truncated at depth 41 (22.46 at depth 23, 110.80 at depth 52), not the value. This section used to call that factor legitimate in the Selberg class; it is not, since docs/09 §3's axioms (4) (θ < ½, which needs |α| < √5 at p = 5) and (5) (Ramanujan) both exclude it. The gate reads the local bound |α_j| ≤ √p (forward direction derived, converse an unreviewed derivation, §2). For a single parameter the failure point is exactly √p (1.41421356 for p = 2, 9.84885780 for p = 97): the closed form's one summand changes sign there, §2's derivation puts the jump of c_p from ≤ d to +∞ at the same place, and the gate's verdict changes within a relative 10^{−6} of it (measured at p = 2 and 97). Claiming it detects Euler products would be an overclaim; it detects Euler products with a Ramanujan-type bound.

6. Where it bleeds, and why that was predictable

The construction satisfies Requirement A completely: c_p is a functional of {a_{p^k}} alone, no zeros, no ξ-phases, no counting functions. It achieves Requirement C locally. It fails at the globalisation, and the failure is measurable rather than rhetorical:

    W(h)  =  [pole + arch]  −  Σ_p log p · ( Q_p(f) − ‖f‖² )

with pole + arch = 0.1540603435 and the norm side 0.1540603433, leaving W = 1.24e-10. Every Q_p ≥ 0, but Q_p − ‖f‖² is not of definite sign, 52 positive and 8 negative over the first 60 places. So local positivity is compatible with either sign of W, and buys nothing globally. The arithmetic supplies the quadratic form; the geometry still has to explain the sign, and the sign lives in the balance between the archimedean term and the norm sum, not inside either.

This lands in the pseudo-solution taxonomy of docs/09 §5.1 as a variant of #5 (finite approximants): a positive structure obtained at each place without control of the limit. Naming it that way is the correct filing.

A second reason no fix helps, corrected 2026-08-11, and it is weaker than what stood here. The paragraph this replaces read: "By docs/18 §6, ζ(s−δ) has the same coefficients up to a shift … Any coefficient functional, c_p included, is therefore blind to the position of the critical line by construction." Three things were wrong with it. The citation is wrong: docs/18 §6 says ordinate statistics, which is a different and correct claim. The premise is wrong: ζ(s−δ) has coefficients n^δ a_n, and the twist is exactly the information at issue. And the universal is false, the coefficients determine the function, hence its zeros, so some coefficient functional must see them, and one is already in this tree: Titchmarsh 14.25(B)/(C) gives M(x) = O(x^{½+ε}) ⟺ RH, a criterion in the coefficients of 1/ζ alone (zeta/criteria.py, face 1).

What is true is sharper and has a number in it. For ζ(s−δ) the local parameter is α_p = p^δ, so c_p = 2x/(1+x) with x = p^{δ−½}, and

c_p ≤ d holds exactly when δ ≤ ½, simultaneously at every place,

with c_p the least PSD diagonal of §1. For δ < ½ that is the formula above; at δ = ½ the cosine series diverges but the least PSD diagonal is exactly 1 = d (measured on every section up to N = 400), and the gate reports INDETERMINATE there; for δ > ½ the power sums p^{m(δ−½)} are unbounded and c_p = +∞. So the answer to c_p ≤ d is yes for every δ ≤ ½ and no beyond, and the gate agrees where it was pinned (PASS at δ = 0.1, INDETERMINATE at ½, FAIL at 0.6; close below ½ its PASS is the default for an excess it cannot resolve, §2): δ = 0.1 is a function the gate passes whose zeros sit on Re s = 0.6, which is the conclusion this section needs, a PASS cannot locate the critical line, obtained with a witness instead of a false universal. That threshold is not new: it is the Selberg-class Euler-product axiom's θ < ½ (Conrey–Ghosh 1992, remark 5, which also records the shift observation itself in its remark 2) and, on the automorphic side, the Jacquet–Shalika bound |log_p |α_p|| < ½, which Sarnak states as sharp. The blindness of a coefficient statistic is a property of the particular statistic, of its invariance under the twist a_n ↦ n^δ a_n, not of coefficient provenance.

7. What is worth keeping

The instrument, not the strategy. c_p ≤ d is a cheap, coefficient-only Gate 3 test (forward direction derived, converse an unreviewed derivation, §2) with measured power and a stated blindness threshold, the thing to run in an afternoon on the next proposed ontology, which is what docs/09 §5 asks for and what docs/15 failed to deliver. As an attack on RH this is a closed avenue, and it is recorded here so nobody reopens it.

Where to go next

• docs/09 §5.1: Requirements A/B/C and the pseudo-solution taxonomy this attempt files under.
• docs/18 §6, the coefficient-side Gate 4 statistic D(f), and the ordinate-blindness argument that bounds every construction of this shape.
• docs/15: the Imposter Gauntlet, whose vacuity the decoy control here is designed to prevent.
• zeta/weil.py: the explicit-formula convention this decomposition is validated against.