RESULTS: ground-state transport across window size (numerics worker)

Worker: numerics, branch teal-sea/weil-propagation, 2026-09-23. Scope and kill-control: ../MISSION.md, brief: BRIEF.md. Nothing here is a claim about RH. Every number quoted is in a JSON beside this file; summary.json collects the quoted ones (summarize.py).

0. Setup and the two exact facts everything leans on

Assembly: imported unchanged from hunts/rogue_frontier/weil_trunc (enclosures.BallTruncation, Arb; galerkin.Truncation, mpmath). Even sector throughout (the sector where DH goes negative). Window parameter c is an exact dyadic rational, so fractional windows mean the same thing in both assemblies (checked, s1). Ground pair: inverse iteration on the midpoint matrix, then every number about the resulting exact vector (Rayleigh quotient, residual, overlaps, energies) in balls. Helpers: wp_common.py.

Fact A (window-free form). The truncated form at window c on L²[0, L], L = log c, is the restriction of one window-independent quadratic form, Q(f) = W(f ⋆ f̃), where W is the explicit-formula functional (pole, archimedean, and −Σ Λ(n) n^{-1/2}(g(log n) + g(−log n))). The truncation only reflects that g = f ⋆ f̃ is supported in [−L, L], so only n ≤ c contribute. Q is translation invariant. This is how the construction is defined (weil_trunc SOURCE.md s2; DH: THEOREM_FEASIBILITY.md OBL-3).

Fact B (nesting). Extension by zero embeds L²[0, L] in L²[0, L′] isometrically and Q-preservingly for L < L′, and the band-N space V_N(c) sits inside L²[0, L]. Hence, with λ_∞(c) the infimum of Q/‖·‖² over the whole window:

λ(c, N) ≥ λ_∞(c) ≥ λ_∞(c′) for all c′ ≥ c and all N, λ(c, N) is nonincreasing in N at fixed c.

Two lines, ordinary argument, not externally reviewed. Consequence used below: a hardened negative cell (c, N) proves λ_∞(c′) < 0 for every c′ ≥ c. It also fixes the direction of the problem: positivity is inherited downward in window size for free, and a propagation mechanism has to go against that.

The two transports measured (both isometries of L²):

• Zero extension (ZE), centred: f unchanged, placed in the larger window. Q-preserving exactly (Fact A). Its entire transform F_v on the real line is literally unchanged (up to the L^{-1/2} normalisation), so ZE is the "common entire-function space" of the brief. In band-N coordinates the transported vector is the projection w = Gᵀv onto V_N(c′), G the closed-form centred Gram matrix (gram_zero_ext).
• Dilation (DIL): the same coefficient vector read in the c′ basis, f ↦ √(L/L′) f(·L/L′). This is the Galerkin-natural map and the one in which "archimedean change / new prime-power terms / rescaling" is a meaningful split. It rescales F_v(z) → F_v(zL′/L) up to normalisation, moving the real zeros of F_v off the ordinates they were tuned to.

1. Task 1: reproduction (repro.py, repro.json)

Hardened: the recorded signs and magnitudes, and that fractional c is handled identically by both assemblies. New cell on the way: zeta λ_min(31, 128) = 2.8557147127e-141 (Temple bracket at LDL precision 1600 bits; at 900 bits the ball LDL stalls at pivot 67, recorded as the reason ldl_prec exists).

2. Task 2: transport on a fine c-grid (transport.py, grid_*.json)

Grid c ∈ [29, 32] step 1/16 (49 windows), N ∈ {64, 128}, zeta and DH; pairs (c, c + h) for h ∈ {1/16, 1/4, 1}. Every DH grid λ carries a Temple bracket whose LDL step also proves it is the minimum (harden.py), zeta and DH alike: all 196 cells conclusive.

Beyond c = 31.5 the DH N = 128 form runs steeply into c = 32: λ falls from −2.7e-29 at 31.9375 to −8.4480e-28 at 32, continuously (λ(32 − 2^-20) = −8.4436e-28), with μ₀ growing to −3.4e-12, and there 1 − overlap reaches 2.3e-3. The table stops at 31.5 so that this does not dominate the ranges.

Readings (measured):

• ZE keeps the tuning, DIL destroys it. The dilated ground state's energy overshoots λ by 0.05·ΔL² (DH) or 2e-4·ΔL² (zeta): for h = 1/16 that is 2e-7 against λ ~ 1e-28 (DH) and 1e-9 against 1e-138 (zeta), i.e. 21 and 129 orders too coarse. The overshoot scales as ΔL²: C agrees to within 5% across h = 1/16, 1/4 and 1. The exact sandwich R_c′(v′) − R_c(v′) ≤ λ(c′) − λ(c) ≤ R_c′(v) − R_c(v) held at all 504 pairs (no rigorous violation), but both brackets are ~10^20 times |Δλ| wide or more, so the dilation picture cannot see the change it brackets.
• The ground state is continuous under ZE, for zeta and for DH through its crossing. DH N=128 across c* ≈ 30.65: 30.5 → 30.75 has 1 − overlap = 4.2e-9, and 30 → 31 has 9.0e-8. The negative-energy state at c > c* is the same bulk state continued, not a new state crossing in (λ₂ ≥ 1.1e-22 on c ≤ 31.5, gap ratio ≥ 1.3e7).
• At finite N the ZE projection is not exact, and for zeta the error swamps λ. For DH the projection's energy differs from λ(c) by up to 7e-29, the same size as the changes being measured; for zeta by up to 10^10 times λ (N = 128) and 10^13 times λ (N = 64). Checked rather than assumed (factA_check.py): projecting v(30, 64) into window 31 at N′ = 64 .. 1024, the deviation falls as −13.9%, −21.2%, −9.8%, −9.2%, −8.2%, −6.5%, −5.1% of λ(30) and equals the missing norm 1 − ‖Gᵀv‖² times a coefficient 7.5 .. 8.4 (N′ ≥ 256) that grows like log N′ (the archimedean weight at the band edge does the same). Two more DH cases give the same coefficient (7.6 to 7.7 at N′ = 256, 384). So the deviation is the truncated tail of a function with jumps, going to 0 as the missing norm does (Parseval), not a window-dependent term in the assembly: Fact A survives the basis-size check (measured). ZE energies are still not used below as numbers at finite N, only overlaps.

3. Task 3: what drives the change (decomposition)

3.1 First order: Hellmann-Feynman pieces in the dilation picture

dλ/dL = vᵀ(∂E/∂L)v at fixed coefficients, split into pole / arch / prime, by central differences with ε = 2^-64 (DH, 600 bits) or 2^-250 (zeta, 1600 bits), at every non-integer grid window (hf blocks).

Across the grid the pieces vary by less than 3% and cancel to 25 to 30 digits (DH) and 98 to 140 digits (zeta). No single term drives the ground-state energy: each piece is O(0.1) per unit L and the drive is a residual 26 to 136 orders smaller. The piece signs differ between zeta (primes push down with arch, the pole pushes up) and DH (primes push up against arch), but DH has no pole, so this is the three-way confound of weil_trunc s4 item 4 and says nothing about the Euler product.

3.2 New prime-power terms

• At first order they contribute exactly nothing at non-integer c (no index enters the stencil), and a new index n enters continuously (q_nm(L) = 0), so no jump in λ can come from one.
• Finite steps that cross an integer: on the new ground state (the lower bracket, which is the physically relevant one) the new-term energy is, for DH, between 1e-31 and 2e-25 in size and of either sign (zeta: 9e-103 to 4e-91, all negative, against |Δλ| ≤ 6e-101 at N = 64), e.g. DH 30.5 → 31.5 (Λ_f(31) enters): +4.3e-30 at N=128 and −2.1e-29 at N=64, against Δλ = −1.8e-29 and −2.5e-29. It is not systematically in the direction of the change.
• DH turns negative between integers. crossing.py, bisection over dyadic c ∈ [30, 31] to 2^-10, both ends hardened:

By Fact B the N = 256 row makes λ_∞,DH(c) < 0 for every c ≥ 30.617188, and windows in [30, 31) use exactly the coefficient set n ≤ 30. Hardened (negativity) plus the nesting argument. The positive side of each row is a finite-N statement only; λ(30, N) > 0 up to N = 256 in the record, and the c*(N) sequence extrapolates (c* ≈ c∞ + a N^{-p} through the bracket midpoints at N = 64, 128, 256: p = 2.50, c∞ = 30.610; the same fit predicts 30.686 and 30.624 at N = 96 and 192 against the measured 30.696 and 30.629) to c*(∞) ≈ 30.61 (measured). So "positivity can only be lost when a new coefficient enters" is false for DH at each N ≤ 256 tested (hardened at each N). For the continuum form only the upper side, c*∞ ≤ 30.617188, is established (hardened cell plus Fact B); the lower side, c*∞ > 30, is measured (finite-N positivity does not bound λ_∞ from below, Fact B points the other way), so the continuum statement is refuted only at the measured grade.

3.3 Across DH's crossing: what changes and what does not

DH, N = 128 (grid_dh_N128.json):

(Q₁ = 4 Re g_v(γ₁ − iδ₁) at the first off-line pair, S = Σ 2g_v(γ) over the 64 cached on-line ordinates below 120, rest = λ − Q₁ − Q₂ − S, i.e. on-line zeros above 120 plus farther off-line pairs. Float ordinates: measured.)

• Arithmetic side: nothing happens at c*. The pieces of 3.1 are smooth, dλ/dL is smooth and strictly negative through λ = 0 (a transversal crossing), the edge ratio is flat, and the ground state continues (s2).
• Zero side: the sign change is the balance of Q₁ against the on-line sum. Q₁ is not monotone in c (it passes through 0 near c ≈ 29.81, where |Q₁|/S ≈ 0.002, and was −1.1e-27 at c = 29.06 with λ still positive), and the crossing is where |Q₁| overtakes S + rest. That is information about where the zeros are, i.e. the RH-analogue itself, not a transport estimate.

4. Task 4: candidate relations, tested on DH

Candidate 7 is the closest thing to a transport relation the data show, so it deserves its reading. λ decreases at a rate set by the squared value of the ground state at the window's edge. DH's crossing is transversal because μ₀ ≠ 0 there (μ₀ ≈ 3.5e-14 at c*, so μ₀² ≈ 1.3e-27 while λ = 0). For positivity to propagate, the edge amplitude must go to zero at least as fast as the margin: μ₀² ≲ λ/(κδ) on each step. Stated without an estimate that is the same as positivity (λ reaches 0 only if μ₀²/λ blows up), so per the mission's kill conditions it is a reformulation, not progress. What it contributes is the target: an Euler-product bound on the boundary trace of the window ground state is what a propagation proof would need, and DH shows it cannot come from anything DH shares.

5. Caveats and incidents

• The shift-0 trap. Inverse iteration at shift 0 returns the eigenvalue of smallest magnitude. Just above c = 32 at N = 128 the eigenvalues +7.3e-28 and −2.06e-25 coexist, and the solver would have reported the first as λ_min. Caught while probing the steep drop into c = 32 (λ is continuous there: λ(32 − 2^-20) = −8.4436e-28, λ(32) = −8.4480e-28). Every grid cell was then re-checked by a Temple bracket whose LDL step proves the reported value is the minimum (harden.py); none of the 98 DH cells had been affected.
• κ is a basis-dependent split. At c = 13.5 (DH λ saturated at ≈ 2.1e-11), κ rises 1.04, 1.10, 1.28, 1.44, 1.70, 1.88 for N = 32 .. 256 while μ₀ falls from 3.19e-6 to 2.02e-6 and the product κμ₀² converges (1.06e-11, 8.5e-12, 8.0e-12, 7.8e-12, 7.7e-12, 7.7e-12). So the band-N edge amplitude is a proxy for a continuum boundary quantity that tends to zero slowly with N; the invariant is dλ/dL, and "κ ≈ 1.2" fails the basis-size check as a constant. Zeta on the same ladders: κ = 1.70, 1.52, 1.56, 1.46, 1.50 (c = 13.5, N = 64 .. 256) and 1.19, 1.20, 1.51, 1.45, 1.42 (c = 20.5), i.e. flatter than DH once saturated; at c = 30.5 zeta only saturates near N = 256 (λ = 2.6e-69, 5.0e-103, 1.7e-125, 8.0e-140, 3.1e-152, 6.6e-153 for N = 32 .. 256) and κ settles from scatter (3.55, 0.58, 2.34) to 1.15, 1.27, 1.31. The fine-grid zeta scatter (κ from 0.03 to 5.6 at N = 128) is the unsaturated regime.
• Zeta is far from N-saturation at c ≈ 30. λ(31, 60) = 4.8e-100, λ(31, 128) = 2.9e-141. Zeta numbers at fixed N describe the band-N form, whose c-dependence mixes window growth (down) with the band edge 2πN/L moving down (up): that is why zeta's λ(c, 64) is non-monotone in c and zeta's κ scatters. Comparisons with DH are at matched N.
• Zero-side split is float grade: 64 on-line ordinates from data/dh_zeros_online_T120.json (float64), ρ₁ to 50 digits, ρ₂ to 10; completeness of the list is not checked here (THEOREM_FEASIBILITY OBL-4/5). The rest column is not signed.
• Launch incident: the first launch of the transport runs used timeout, which macOS does not ship; nothing ran (RUNS.md).
• The HF stencil truncation O(ε²) is not inside the balls; precision_check.py moves ε by 2^-24 and doubles precision (s6).

6. Hardening coverage, precision response, basis-size response

• Hardening: all 196 grid cells (DH and zeta, N = 64 and 128) carry a conclusive Temple bracket whose ball LDLᵀ step also proves the reported eigenvalue is the minimum (hardened blocks; meta.hardened_all true in all four grids). Lower and upper ends agree to all 20 printed digits everywhere (residuals 1e-57 to 1e-181). The crossing table (s3.2) and its zeta controls are hardened as stated there.
• Precision response (precision_check.json; doubling the working precision, and moving the HF stencil ε by 2^-24, the one error not carried in a ball): largest relative change of λ, μ₀ and dλ/dL at DH (30.5, 128), DH (31.25, 128), zeta (30.5, 64), zeta (30.5, 128) is 8e-91 (precision) and 7e-13 (stencil; DH (31.25, 128), where dλ/dL is itself small, 5e-31). None of the quantities moves: they are not precision artifacts.
• Basis-size response: λ(c, N) ladders (edge_*.json, the record's ladders), c*(N) (s3.2), Fact A (s2) and κ (s5). The quantities used as findings (signs, c*(N) brackets, dλ/dL, overlaps) respond to N the way a converging quantity does; κ as a constant and the finite-N ZE energy do not, and are reported as basis-dependent.

7. The doors

Not a ceiling hunt, but the objects frozen here are the ones a follow-up would unfreeze:

1. Binding at c*: nothing on the arithmetic side binds; the sign is set by Q₁ against the on-line sum (zero side).
2. Frozen choices: the periodic Fourier basis on the window (its edge amplitude μ₀ is the only proxy for the boundary trace, and it is basis-dependent, s5); the even sector (DH's crossing is even; the odd sector stays positive at 31 in the record); the transport (ZE vs DIL, s2); dyadic grids at 1/16 and bisection to 2^-10.
3. Information class: every quantity measured reads only the form restricted to one window at a time. The edge-trace door needs a basis that resolves the boundary behaviour of the continuum ground state (for example functions vanishing at the edge to a chosen order, or a basis adapted to the log-symbol archimedean part), which is new information beyond the band-N family.

Proposed job (not run; too large for the laptop by the brief's limits). A boundary-trace study: for c on [29, 31] step 1/32 and DH and zeta, compute the window ground state in a basis of edge-vanishing functions (e.g. sin(πk y/L) products) at N up to 1024 with 3000 to 6000 bits for zeta, and measure whether the continuum boundary behaviour f_L(y) ~ A(L)·φ(y) near y = 0 has an amplitude A(L) with dλ/dL = −K·A(L)² for a basis-independent K. Estimate from the unit costs here (N = 1024 assembly 3 s at 400 bits, eigen-solve and LDL growing as N³): ~30 s per DH cell and ~10 min per zeta cell at 6000 bits, so ~130 cells ≈ 12 to 20 runner-hours: a GitHub Actions matrix of 20 jobs, one c-slice each, checkpointing per cell. Only worth funding if theory first says what A(L) should be compared against.

8. Reproduction

From the worktree root (all runs single-process, under 10 minutes each, estimates and actuals in RUNS.md):

P="$PWD/.venv/bin/python"; cd hunts/weil_propagation/numerics $P repro.py # task 1 for a in "dh 64" "dh 128" "zeta 64" "zeta 128"; do $P transport.py $a; done for a in "dh 64" "dh 128" "zeta 64" "zeta 128"; do $P harden.py $a; done $P crossing.py 64 96 128 192 256 $P edge.py dh; $P edge.py zeta $P precision_check.py $P factA_check.py $P summarize.py # summary.json

9. Theory handoff: C2 against Epstein (1,1,6), and zeta's pole-free spectrum

Handoff from the theory worker (branch teal-sea/weil-propagation-theory, commit 7d9636b, candidate C2 "Levy-Markov + pole"): C2 is outside DH's reach (DH's jump measure is signed at n = 3 and DH has no pole), so the live control is ζ_Q for Q = x² + xy + 6y² (discriminant −23, class number 3), which has a pole, off-line zeros, a Γ(s) factor with positive Lévy density, and Λ_Q(n) ≥ 0 for every n ≤ 47. Task A: its window floor on c ∈ [2, 48]. Task B: zeta's pole-free μ₂ against the full form's λ₁, λ₂ on c ∈ [2, 60]. Conventions as in the theory worker's s0; everything below is in the CCM basis of s0 here.

9.1 The Epstein assembly, and why it can be trusted

No new closed forms. The lab's completion is (√23/2π)^s Γ(s) ζ_Q(s) (zeta.epstein.epstein_completed, validated there by the class-group identity), which equals 23^{s/2} Γ_R(s) Γ_R(s+1) ζ_Q(s)/2. By the duplication formula its archimedean density is log 23 + [Re ψ(1/4 + ir/2) − log π] + [Re ψ(3/4 + ir/2) − log π], so the matrix is the weil_trunc zeta block (a = 1/4, pole block included; the completion has simple poles at 0 and 1) plus the DH archimedean block (a = 3/4) plus log(23/5)·I plus the prime block of Λ_Q (wp_common.epstein_matrices). Checks:

The control is the load-bearing one: an error in the shared archimedean, pole or conductor blocks would show in a GRH function with an Euler product, and it does not. What differs between the two runs is the prime block alone.

9.2 Task A result: the Epstein form goes negative well below c = 48

Grid c ∈ [2, 48] step 1/2, N = 64 and 128, both sectors (epstein_scan.py, epstein_N64.json, epstein_N128.json): ball LDLᵀ inertia conclusive at all 372 (cell, sector) pairs and a conclusive Temple bracket at every one. At both N the odd sector is negative at every grid window from c = 28 on and the even sector from c = 29.5 on, with exactly one negative eigenvalue per sector and no return to positivity on the grid. Bisection to 2^-10 with every step a conclusive ball LDLᵀ sign, and the negative end also carried by a ball Rayleigh upper bound < 0 (epstein_crossing.py):

By Fact B the N = 128 odd row makes the continuum Epstein form negative on every window c ≥ 27.741699, and every such window below 48 has Λ_Q(n) ≥ 0 for all n ≤ c. Depth grows to λ_min = −0.898 (even) and −1.065 (odd) at c = 48, N = 128. Grade: hardened (two rigorous routes per sign) plus the nesting argument.

Where the negative states point. The even negative ground states have their coefficient mass peaked at frequency 16.7 to 18.5 (c ≥ 29.5), and the odd ones at 15.3 to 16.6 once deep (c ≥ 34); the mode spacing there is 2π/L ≈ 1.6 to 1.9. At the marginal odd windows (c = 28 to 30) the state is the bulk low-frequency one (peak 1.9), as for DH at (31, 60). Independent check with code that shares nothing with the Galerkin route (epstein_offline.py: the lattice-sum completion and the lab's argument-principle counter): exactly one zero of the completed ζ_Q in [0.51, 1.3] × [14, 20], polished by findroot to

ρ₁ = 0.953260474794661 + 16.2902157203904 i, |Λ_Q(ρ₁)| = 1.4e-32 (dps 20),

an off-line zero at δ = β − 1/2 = 0.453, exactly where the beams point. Grade: the zero count is measured (float argument principle, integrality guard 1e-6); the root is measured.

Two more boxes, same route: no zeros in [0.51, 1.3] × [0.5, 7] and none in [0.51, 1.3] × [7, 14] (159 s and 273 s). So ρ₁ is the only zero of the completed ζ_Q with 0.51 ≤ σ ≤ 1.3 and 0.5 ≤ t ≤ 20: the lowest off-line zero in that strip, in the role DH's first pair at 85.7 played for DH (measured; zeros with 1/2 < σ < 0.51 or σ > 1.3 are outside what these boxes check).

9.3 Which part of C2 breaks (epstein_polefree.py, N = 64)

The theory worker's reduction: Q_e ≥ 0 ⟺ (a) μ₂(Q°_e) ≥ 0 and (b) the even pole capacity; Q_o ≥ 0 ⟺ Q°_o ≥ 0 and the odd pole capacity (the odd pole term is negative). Ball LDLᵀ inertia of the pole-free and full Epstein matrices, conclusive at every cell:

• Even sector: the pole capacity (b) fails first, then (a). The full even form turns negative at c = 29.3042 (N = 128; 29.3413 at N = 64), while the killed Epstein jump process acquires its second even eigenvalue below C_ℓ (μ₂ < 0) only at c = 29.3179 (N = 128; 29.3555 at N = 64), bisected on the pole-free inertia to 2^-10 (epstein_mu2_cross.py, epstein_mu2_cross.json). So on (29.304, 29.318] condition (a) still holds and the even pole capacity is what breaks; from 29.318 on, (a) fails too. Interlacing μ₁ ≤ λ₁ ≤ μ₂ holds at every cell. Markov structure plus one pole stops neither.
• Odd sector: the odd pole capacity fails first. The pole-free odd form stays positive until c ≈ 34 while the full odd form is negative from 27.74, so the odd crossing is the negative odd pole term; C2's Markov structure is not what is tested there.

Verdict on C2 (per the supervisor's criterion): refuted as a propagation mechanism. Every hypothesis it uses holds for ζ_Q on windows c < 48 (nonnegative jump measure, Γ-class Lévy density, one pole) and the form goes negative at c = 27.74 (odd) and 29.30 (even). Proposition M's unconditional structural consequences are not touched by this; only the use of "Markov + one pole" to carry positivity forward is. Any C2-based argument must use more of the Euler product than Λ ≥ 0 on the window.

What the Epstein/Dedekind pair isolates. The two runs share the completion, the pole, the Γ(s) Lévy density and Λ ≥ 0 on every window below 48; they differ only in the prime block. Dedekind's atoms sit on prime powers alone (Λ_K(p^k) = log p (1 + χ₋₂₃(p)^k)), as an Euler product forces. Epstein's vanish at 2, 3, 5, 7, 11 but sit on composites and on higher powers of those primes (Λ_Q(4) = 1.386, Λ_Q(6) = 3.584, Λ_Q(8) = 4.159, Λ_Q(9) = 2.197, Λ_Q(12) = 4.970). One form is positive on every window, the other negative from 27.74. That is the prime-power-support condition the theory worker named as the next thing C2 would need (its s4, C2 "DH test"), now measured as the difference between a positive and a negative case. It is one matched pair, so it shows the condition matters here; it does not show that it suffices.

9.4 Task B: zeta's pole-free μ₂ against λ₁, λ₂ (zeta_pole.py)

Even sector, c ∈ [2, 60] step 1/2 (117 windows), N = 64 and 128 (zeta_pole_N64.json, zeta_pole_N128.json). E is the full matrix, E° = E − W02 the pole-free one from the same assembly; λ₁, λ₂ of E and μ₁, μ₂ of E° as ball Rayleigh quotients of inverse-iteration vectors (μ₁ shifted at the lowest approximate eigenvalue).

• μ₂ tracks λ₂ at a nearly fixed ratio. μ₂/λ₂ = 0.6394 to 0.6430 on every window with c ≥ 5 at both N, drifting up slowly with c; N = 64 and N = 128 agree to 0.001 even where λ₂ itself differs by 64 orders of magnitude between the two bases (c = 60). So the ratio is basis-stable while the eigenvalues are nowhere near N-converged for c ≳ 15 (measured). It confirms the theory worker's 0.64 at c = 13 and 31 and extends it to [5, 60]. Below c = 5 the ratio is larger (0.88 at c = 2).
• Interlacing μ₁ ≤ λ₁ ≤ μ₂ ≤ λ₂ holds at all 234 (c, N) cells (midpoint comparison), and μ₂/λ₁ ranges from 4e2 (c = 2) to 2e8: condition (a)'s margin μ₂ sits 2.6 to 8.3 orders of magnitude above λ₁, so for zeta, as the theory worker measured at two cells, the even pole capacity (b) is the tight condition and (a) holds with room, on this whole range.
• Hardened at every one of the 234 cells: the full even form is positive definite (ball LDLᵀ inertia (N+1, 0)), and the pole-free even form has exactly one negative eigenvalue (inertia (N, 1)), which is the shape Proposition M plus positivity require. 1600 bits sufficed for c ≲ 30 at N = 128; the other cells were redone at 3200 bits (zeta_pole_fixup.py), all conclusive.
• Set against 9.3: Epstein breaks the condition that zeta satisfies with room here. Its μ₂ crosses zero at c = 29.318 (N = 128) while zeta's μ₂ stays at 0.64 λ₂ > 0. Whatever keeps zeta's μ₂ positive is not "Λ ≥ 0 on the window plus one pole", since Epstein has both.

Grade for 9.4: measured (ball upper bounds for all four eigenvalues, inverse-iteration vectors with residuals recorded; interlacing on midpoints), except the inertia statements, which are hardened where the ball LDLᵀ is conclusive.

9.5 Observations raised, not pursued

• zeta/epstein.py (claim_euler_product_positivity docstring) says the principal Epstein form fails at n = 48 "(−19.88)". The lab's own log_derivative_coefficients gives Λ_Q(48) = −7.742; −19.879 is Λ_Q(144), the most negative value for n ≤ 200. The first-failure index 48 is right; the quoted value belongs to another n. Issue candidate (outside this hunt's write scope).
• zeta_pole_N64_firstrun_mu1defect.json is the first Task B run at N = 64, kept as the record of a defect: μ₁ came from inverse iteration at a fixed shift −20, which converges slowly (ratio ≈ 0.7) and at small c landed on the wrong eigenvalue (μ₁ > 0 at c = 2 while the ball LDLᵀ said exactly one negative eigenvalue). Caught by the interlacing check (7 cells failed). The rerun uses the lowest approximate eigenvalue as the shift; λ₁, λ₂, μ₂ and every inertia in the first run were unaffected.
• The Dedekind control's floor sits far above zeta's: at (c, N) = (46, 64) its even λ_min is 1.4e-7 against zeta's 1.4e-118 (same N). Dedekind has everything C2 asks for plus a genuine Euler product, so the depth of zeta's floor is not explained by C2's ingredients either. A plausible reading (not tested): L(s, χ₋₂₃)'s own pole-free component sets the Dedekind floor, as DH's did at 1e-10 near c = 13, which would say the pole does the heavy lifting in zeta's depth. A thread for theory.
• Where ζ_Q's first off-line zero sits relative to all on-line zeros, and whether the even and odd crossings are both carried by ρ₁ through the dictionary (as DH's was by its first pair), is the natural next check. It needs Epstein on-line zeros to ~T = 60 and the odd-sector dictionary.

Reproduction (from this directory, each under 10 minutes):

$P epstein_scan.py 64; $P epstein_scan.py 128; $P epstein_scan.py 64 dedekind $P epstein_crossing.py; $P epstein_polefree.py; $P epstein_mu2_cross.py $P epstein_offline.py; $P epstein_offline.py 0.51,1.3,7,14; $P epstein_offline.py 0.51,1.3,0.5,7 $P zeta_pole.py 64; $P zeta_pole.py 128; $P zeta_pole_fixup.py 128

10. Theory's C4 prerequisite (U-S), checked from coefficients

Theory RESULTS s7.1 (branch teal-sea/weil-propagation-theory, commit

1. names the step Dedekind ζ_{Q(√−23)} passes and Epstein (1,1,6)

fails: (1) atoms only at prime powers; (2) unitary local roots, checkable as |s_k(p)| ≤ d with s_k(p) = Λ(p^k)/log p and d = 2. My own implementation (us_check.py, us_check.json) uses exact arithmetic. Every Λ(n) is kept as a rational combination of log p, so "Λ(n) = 0" means all rational coefficients vanish. It covers n ≤ 200, with coefficients from zeta.epstein.epstein_representation_count.

Grade: exact arithmetic, an independent implementation of theory's check K; it agrees with their table entry for entry on the overlap (n ≤ 60). Both failures sit inside Epstein's negative windows (c ≥ 27.74 contains n = 6, 8 and 27), consistent with the reading in s9.3. As theory notes, this makes (U-S) the separating step for this pair. It does not test C4, because no available rival satisfies (U-S).