RESULTS: replication + rigorization of the CvS/CCM truncated Weil form

Study of 2026-08-17, hunts/rogue_frontier/weil_trunc/. Sources, exact construction and claim inventory: SOURCE.md. Everything below is graded measured or hardened per the repository ladder; nothing here bears on RH (docs/08 discipline), and nothing in hunts/ is a result until it survives a battery it did not write. Raw numbers live in the JSON files beside this document; reproduction commands are at the end.

0. Source verification (the survey vs the record)

Both arXiv IDs the survey cited exist and are current work; the survey's attributions were scrambled (details in SOURCE.md s1). Correct picture: arXiv:2605.20224 and arXiv:2607.02828 are both by Akiva Groskin (May and July 2026, both revised 2026-08-14); the construction they implement and extend is Connes-van Suijlekom (arXiv:2511.23257, Prop 4.1) and Connes-Consani-Moscovici (arXiv:2511.22755); the open convergence question is Connes arXiv:2602.04022 s6 (Feb 2026). The headline numbers the survey quoted (2e-55 at c=13, 1.5e-168 at c=67, both at N=100) are real and sit in 2605.20224 Tables 3/20, not in 2607.02828. Survey verdict: not a hallucination, but sloppy provenance; every load-bearing fact checked out against the primary record.

1. Replication (independent implementation, no code consulted)

galerkin.py implements the truncation from the papers' definitions alone, via the CCM closed-form assembly Q = W02 - WR - Wp on the Fourier basis of L^2([0, L]), L = log c, with the archimedean entries re-derived from CCM (3.15) as digamma/trigamma closed forms plus geometric series in e^{-2L} (SOURCE.md s2). Validation gates, all passing (run_replication.py, replication.json):

Gate H matters most: a uniform error eps in the DH diagonal constant would shift every DH eigenvalue by exactly eps (Weyl), and the DH floors reported in s4 are ~1e-10; the gate validates that constant to ~1e-30, twenty orders below the smallest floor, via an identity whose integrand decays like r^-4 (so the window integral genuinely converges).

Headline cells (run_headline.py, headline.json), ours cutoff-free (T = infinity) vs the published T = 800 values:

Both lambda_min values land ABOVE the published finite-T numbers, which is not a discrepancy but the direction G2's tail-order theorem (Thm 3.2: lambda_j strictly increasing in T) requires; the gamma_1 errors land inside the published cluster for both cutoffs. Replication verdict: full. No defect was located in either posting's numbers at the cells we can reach; the only source blemishes found are two display-level constant slips in CCM eq. (4.4)/(4.14) that cancel in their own Prop 4.3 table (SOURCE.md s2), caught because we re-derived the entries from (3.15) instead of copying the table.

2. Enclosure-checked smallest eigenvalues (the rigorization)

enclosures.py mirrors the assembly in Arb ball arithmetic (python-flint, the same backend zeta.rigor standardizes on): rigorous special-function balls, geometric-series tails added as explicit ball radii, then three independent rigorous statements per cell: ball LDL^T inertia at shift 0, a [LDL-shift, Rayleigh] bracket, and python-flint acb_mat.eig (Rump) enclosures. All three agreed at every cell (enclosures.json).

Zeta, even sector (odd sector inertia also computed, all positive):

(20 zeta cells total over c in {6,9,13,19,29} x N in {4,8,16,32}; every cell conclusive, every eigenvalue of both parity sectors strictly positive, ball radii 1e-91 .. 1e-147.) The signs and magnitudes of lambda_min on this grid are therefore hardened: enclosure-carrying, two independent rigorous routes agreeing, on top of exact cited entry formulas. The G2 flagship-style statement "n_minus = 0 for the cutoff-free matrix" is reproduced here at every grid cell by the same LDL^T-in-balls method that paper reports at (100, 200).

3. The measured convergence law (grid.json, rates.json)

Grid: c in {6,7,9,11,13,17,19,23,29,31,37}, N in {4,8,16,24,32}, plus an N-ladder at c=13 up to N=64, cutoff-free entries, float eigensolver at cell-adaptive dps with a dps-130 mpmath.zetazero oracle.

N-law at fixed c (the Galerkin regime). At c = 13 the local log-log exponent p(N) = d log10 err_gamma1 / d log10 N is

N: 8 12 16 20 24 32 40 48 64 p: -40.7 -42.8 -46.3 -51.0 -48.2 -50.0 -45.7 -36.8 -18.4

i.e. err ~ N^p with p stabilizing near -49 +/- 3 on 16 <= N <= 40, then saturation toward the c = 13 floor (err 4.4e-55, lambda_min 6.3e-59 at N = 64, approaching our N = 100 values 2.6e-55 / 3.7e-59). Falsifiable statement: at c = 13 with cutoff-free entries, err_gamma1(N) on 16 <= N <= 40 is a power law with exponent -49 +/- 3; the same measurement at c = 19 or c = 29 (not yet saturated at N = 32) should show the same shape with a deeper floor. This complements G1: their s(c) exponents were measured near saturation at N in {40..100} with finite-T entries and their proposed Paley-Wiener mechanism for them is withdrawn in their v4; our window is the pre-saturation regime, where the exponent is far larger than their near-saturation values.

c-law. On the N = 32 row (which mixes N-limited and saturated cells), all five two-parameter smooth models we tried (power law in c, linear c, c/log c, sqrt(c) log c, log^2 c) fail with rms 3.3-5.8 log10 units, and a held-out fit misses the c = 37 cell by 7-13 orders. This echoes G1 s5.3, where all eight pre-registered smooth models fail at N = 100 over c in [13, 67]. The convergence law in c resists cheap parametrization in both datasets; that agreement is itself a replication datum.

The error/eigenvalue ratio err_gamma1 / lambda_min at N = 32 grows smoothly 3.4e3 -> 5.5e4 over c = 6 -> 37 (G1 report 7.0e3 -> 1.8e4 over c = 13 -> 67 at N = 100); same direction, same order.

4. The DH control (battery discipline; dh_control.json)

The construction ports to Davenport-Heilbronn with three structural changes, each dictated by the explicit formula (SOURCE.md s4): coefficient measure Lambda_f(n) on ALL n >= 2 (no Euler product; computed by the log-derivative recursion with kappa in closed form, validated against the lab's derived kappa to 1e-41), no pole block (f entire), and archimedean kernel a = 3/4 with conductor constant log(pi/5) (same shift Groskin's own chi_3 port uses, corroborating the port design). So the truncation does NOT structurally reject an RH-violating input; it runs happily on one.

Measured outcome, DH grid c in {6,9,13,19,29,37,47}, N in {4,8,16,32}:

1. Every DH cell is positive too , and at N = 32 this is enclosure-checked: inertia (33, 0) conclusive at every c, e.g. lambda_min(DH, 13, 32) = 1.0205115394980095e-10 +/- 8.9e-90. A finite window on Weil positivity does not see DH's RH-violation anywhere on this grid.
2. The DH ground state approximates DH's on-line zeros the way zeta's approximates gamma_k: at (29, 32), err(gamma_1^DH) = 7.3e-26; at (47, 32), 1.7e-36 (oracle: zeta.epstein.Z_dh bisection at dps 60, first ordinate 5.09415984457109492569879551708). The qualitative signature "truncated Weil form is positive and its ground state locates the zeros" is therefore NOT a zeta-specific behavior at these windows. By the battery rule (docs/09 gate #3), that qualitative signature alone distinguishes nothing.
3. What does differ is quantitative and large: where the N-saturated floor is reachable, DH's floor is dozens of orders of magnitude higher than zeta's at the same c:

(DH floors saturate by N = 16..32: the N = 16 -> 32 change is < 15%. All DH floor values are enclosure-checked at N = 32.) At c >= 19 the DH minimum has not yet saturated by N = 32 (4.9e-17 at 19, 7.5e-38 at 47, still falling), so only upper bounds are claimed there.

1. Attribution caveat, stated plainly: DH differs from zeta in three ways at once (RH-violation, no Euler product, no pole), so this experiment does NOT identify which property the floor gap tracks. It is the detector-strength confound of docs/22 / issue #21 in a new instrument. G1's chi_3 port (GRH-true, Euler product, no pole, odd character like DH) reaches 28-digit convergence, which weakly points away from "no pole" as the driver, but chi_3 vs DH still differs in two properties at once.
2. Thread raised, not pursued (mission scope): since Weil positivity is false for DH, the cutoff-free DH band minimum must go negative for some finite c (the band-limited family is asymptotically dense in the admissible class); our data say that c > 47 at N <= 32. Where the first negative appears, and whether the floor's decay accelerates as the band reaches DH's first off-line ordinate (t ~ 85.7), is a sharp, well-posed computational question this battery run surfaces. It belongs in an issue, not in this hunt.

5. Grading summary

• lambda_min signs and magnitudes on both grids (s2, s4 item 1/3): hardened (enclosure-carrying, independent rigorous routes agreeing).
• Replication matches (s1), rate measurements (s3), DH comparisons (s4): measured (float arithmetic under precision policies, high-precision oracles, printed-value cross-checks).
• Nothing here is kernel-checked; nothing here uses the reserved word, which belongs to zeta/rigor.py.

6. Caveats and incidents (recorded, not buried)

• Two oracle-precision defects were caught during the runs by the "identical floor across unrelated cells = artifact" reflex: (i) gamma errors floored at 7.7e-31 because the repo's dps-30 zero cache was used as oracle; (ii) after switching to a dps-130 oracle, errors floored at 9.71e-48 because the reference strings were parsed at the ambient dps of the first grid cell (47). Both fixed (run_grid.py parses inside mp.workdps(135)); the incident is why every floor in this document was re-derived after the fix and why the DH floors were additionally checked for c-dependence (a uniform-shift artifact cannot produce c-dependent floors).
• The published lambda_min values are finite-T (T = 800); ours are cutoff-free. They are different numbers by design and are compared only directionally (ours must sit above, and do).
• Zero-location errors use float eigenvectors (mpmath eigsy); only the eigenvalue statements carry enclosures. Extraction noise is bounded well below every reported error by the cell dps policy.
• The DH floor interpretation carries the attribution confound of s4 item 4, and DH floor values at c >= 19 are upper bounds only.
• The c = 13 N-ladder exponent p ~ -49 is a cutoff-free, pre-saturation measurement; it is not comparable 1:1 with G1's near-saturation finite-T s(c) values, and neither settles a mechanism (theirs is explicitly withdrawn in their v4).

7. Reproduction

From the repo root, in order (total ~25 min on this machine):

.venv/bin/python hunts/rogue_frontier/weil_trunc/run_replication.py .venv/bin/python hunts/rogue_frontier/weil_trunc/run_grid.py .venv/bin/python hunts/rogue_frontier/weil_trunc/run_enclosures.py .venv/bin/python hunts/rogue_frontier/weil_trunc/run_dh.py .venv/bin/python hunts/rogue_frontier/weil_trunc/run_headline.py .venv/bin/python hunts/rogue_frontier/weil_trunc/fit_rates.py

Requires: the repo venv (mpmath, numpy, python-flint), network access NOT required, writes only inside this directory. gammas_dps130.json is regenerated by twelve mpmath.zetazero calls at dps 130 if deleted.

8. Where DH's Weil-positivity failure first becomes visible (s4 item 5, pursued)

Follow-up study of 2026-08-17, run as its own tasked session; drivers dhneg_scan.py / dhneg_confirm.py / dhneg_localize.py, every data point in dhneg_scan.json, session log in dhneg_log.md. Question (raised in s4 item 5 and left there as a thread): Weil positivity is FALSE for DH, so the cutoff-free truncated form must eventually acquire a negative eigenvalue; where is the first (c, N)? The off-line pair is at rho = 0.80851718245663738555 + 85.69934848537759217193 i (zeta.epstein, pinned), delta = beta - 1/2 = 0.30851718...

8.1 The answer

The first negative cell on the integer lattice is (c*, N*) = (31, 60), even sector, and the sign statements below are hardened (enclosure- carrying, three independent rigorous routes, plus the independent-code mpmath scout):

The zeta control is the discrimination statement: the SAME truncation, at the SAME (c, N), is positive for zeta by a hundred orders of magnitude and negative for the structure-matched RH-violating rival. This is the first quantitative positivity-failure height for the 2026 truncated-form programme (CvS Prop 4.1 / CCM (3.10) / G1, G2) on an input that violates the RH analogue: the finite window stops being blind to DH's off-line zeros at c = 31, N = 60.

8.2 The transition curve (one factorization per c gives the whole N-ladder)

Method dividend that made the sweep cheap: the (N+1)-band even matrix is the leading principal submatrix of the Nmax-band one, and LDL^T pivots give the inertia of every leading principal submatrix at once (number of negative eigenvalues at band N = negative pivots among the first N+1). One conclusive ball factorization at Nmax = 128 per integer c in 6..60, prec 600, ~1 s each (dhneg_scan.json "ladders"):

• c <= 30: no negative eigenvalue at any N <= 128, either sector; c = 29 and c = 30 probed to N = 256, still conclusively positive. The band edge 2 pi N / L at those probes is ~470 >> 85.7, so this is NOT a reach limit: below c* the bandwidth L = log c is what is insufficient.
• c = 31: first negativity, N = 60 (band edge 109.8; odd sector still positive at N = 192; n_neg stays 1 through N = 192).
• c = 32..60: first_neg_N in 48..54, and the band edge at the crossing is 83.6 +- 2.5, i.e. the crossing happens as soon as the band reaches the off-line ordinate 85.699 (plus beam-shaping margin; at c = 31 the margin is largest, N* = 60 vs reach N = 47, where the analytic-continuation amplification e^{delta L} = c^0.3085 ~ 2.9 is weakest).
• c >= 44: a second negative eigenvalue appears by N = 128, at band edge ~114-138. The on-line cache has a second spacing gap 112.38..116.72, an argument-principle box-vs-line count on 112 < t < 117 gives box 4 vs line 2 (one more off-line pair), and a findroot polish inside that window lands on rho_2 = 0.6508300806 + 114.1633427308 i with |f(rho_2)| ~ 4e-42 (measured; delta_2 = 0.1508, weaker amplification, which is consistent with its higher c threshold).

Approach to negativity at c = 31 (even lam_min, enclosure mids; full trajectories for c in {13, 19, 29, 31, 32, 33, 37, 41, 43, 45, 47, 53} are in dhneg_scan.json "trajectories"):

N: 40 48 56 59 60 64 96 128 192 lam: +1.7e-29 +8.1e-30 +2.5e-30 +8.4e-31 -1.9e-31 -3.8e-30 -6.6e-30 -7.5e-30 -7.8e-30

Saturated depth vs c (N = 128 slice): +9.5e-29 (c=30), -7.5e-30 (31), -8.4e-28 (32), -2.0e-23 (33), -2.7e-7 (37), -3.4e-2 (41), -0.175 (43), -0.288 (45), -0.333 (47), -0.663 (53). The depth climbs ~28 orders of magnitude over 31 <= c <= 41: the threshold is sharp in c, and for c <= 30 the still-positive floors (e.g. 6.6e-28 at c = 29, N = 256) are falling only slowly in N.

8.3 Localization: the negativity is the off-line pair speaking

Three measured lines of evidence (float route, dps 40-60):

1. Deep cell (47, 64), lam_min = -0.3163: the eigenvector is a beam at the off-line ordinate. Peak mode k = 52 vs k_off = gamma_off L / (2 pi) = 52.5; 95.6% of the coefficient mass within +-6 of 85.699; |F_v| maximal at z = 84.5, inside the on-line gap 83.109..87.647 that contains the off-line ordinate. (That 4.5-wide gap, where DH's two would-be line zeros went off the line, is what a bandwidth-3.85 beam can thread; the mean on-line spacing there is 1.49.) The zeta control at this deep cell is also positive and conclusive: even inertia (65, 0) at prec 4200, lam_min(zeta, 47, 64) = +6.6006e-119, against DH's -0.3163 at the same cell.
2. Dictionary decomposition at the marginal cell (31, 60) (ported G2 Thm 2.5 zero-side sum; on-line ordinates from the lab cache, the 24 largest terms re-bisected with zeta.epstein.Z_dh at dps 50):

lam = -1.8739e-31 off-line quadruple 4 Re g_v(gamma_off - i delta) = -6.7350e-29 on-line partial sum (T <= 120, 64 terms, all >= 0) = +5.9537e-29 lam - quadruple = +6.7162e-29 (POSITIVE) tail model T > 120 (mean density) = +3.0e-30 second-pair quadruple = +7.6e-32 unexplained bookkeeping = ~4.7e-30 (7% of |quad|)

The quadruple term is 359x the eigenvalue and is the only negative entry in the decomposition: removing it flips the form value positive with a margin factor of ~14 over the bookkeeping slop. The largest single on-line term sits at gamma = 89.44, the nearest zero beyond the gap edge, exactly where beam leakage must land. The same decomposition at the deep cell (47, 64), dps 50: lam = -0.31630285, quadruple = -0.83065725 (again the only negative entry, 2.6x lam), on-line sum = +0.49306094, lam - quadruple = +0.51435440, bookkeeping closes to 2.2% (tail model 0.0098, residual 0.0213). Largest on-line term at gamma = 87.647, the other gap edge.

1. The marginal-cell shape: at (31, 60) the eigenvector is bulk low-frequency (quasi-ground-state) with a whisper of the 85.7 beam (coefficient mass 7.8e-29 near the target vs |lam| = 1.9e-31), which is why the dictionary, not the coefficient profile, carries the identification there; the second negative direction at (47, 128), itself a fresh marginal crossing at -4.4e-33, shows the same pattern aimed at the second gap.

8.4 Grading and caveats

• Hardened: every sign statement in 8.1; the transition curve first_neg_N(c) for c in 6..60 (each cell a conclusive-pivot ball factorization); the deep probes at c = 29, 30, 31; the trajectory values (eig enclosure mids with recorded radii, 1e-160s and below).
• Measured: the localization profiles, the dictionary decomposition (float cache ordinates for the 40 unrefined on-line zeros, a mean-density tail model, and the ported-dictionary assumption itself, corroborated here by the decomposition closing to 7%), rho_2, and the mechanism reading "the negativity is the first off-line pair". The composite claim takes this measured grade; the negativity itself does not depend on it.
• "First" is a lattice claim: first on integer c with N <= 128 (N <= 256 at c = 29, 30; N <= 192 at c = 31). The N -> infinity limit at fixed c <= 30 (the bandwidth-L Weil form) is not settled by these probes: lam_min was still slowly decreasing at the ceilings. What is settled is that the detector's first firing on the searched lattice is (31, 60), and that it fires there for the RH-violating input only.
• s4's grid statement "positive everywhere at N <= 32, c <= 47" stands; those windows never reached the off-line ordinate (band edge <= 52.2 < 85.7). The s4 item 4 attribution confound (DH differs from zeta in three properties at once) does not touch the negativity itself, which is a property of the form, but it also does not arise for the mechanism claim here: the dictionary term that flips the sign is computed at the off-line zero's own coordinates.

8.5 Reproduction

From the repo root (total ~45 min, writes only inside this directory):

.venv/bin/python hunts/rogue_frontier/weil_trunc/dhneg_scan.py # c-sweep 6..60, Nmax 128 .venv/bin/python hunts/rogue_frontier/weil_trunc/dhneg_confirm.py # deep probes, scouts, enclosure package, zeta control .venv/bin/python hunts/rogue_frontier/weil_trunc/dhneg_localize.py # localization + dictionary attribution