RESULTS: theory worker, Weil positivity propagation across window size

1. Strongest candidate was C2 (Levy-Markov + pole); as a positivity/propagation mechanism it is now refuted (line 3), and no candidate survives. What stands is its structural part, Proposition M. Grade: ordinary argument, unreviewed (checked by nobody outside this session). With the pole term removed, the zeta Weil form on any window is exactly a jump Dirichlet form minus a constant C_ℓ. The jumps are the digamma Levy density ½K(|u|), K(x) = 2e^{−x/2}/(1 − e^{−2x}), plus jumps ±log n at rate Λ(n)/√n, killed outside the window; C_ℓ = log π − ψ(1/4) + Σ_{log n<ℓ} 2Λ(n)/√n up to terms that vanish as ℓ grows (exact form in §2.6). Λ ≥ 0 (Markov), together with the irreducibility hypothesis (K continuous and strictly positive on (0, ℓ]), makes the pole-free operator's ground state simple and a.e. positive on every window.
2. What it would give. Simplicity transfers to the even sector of the full form with the pole by rank-one interlacing. This needs neither condition (a) nor positivity. The bottom is min(r₁, μ_k) over pole-free even eigenvalues μ_k with an eigenvector orthogonal to the pole vector c = cosh((x−x₀)/2), where r₁ is the smallest secular root. Simplicity fails only on a tie r₁ = μ_k, or a degenerate such μ_k below r₁. Grade: derivation, not measured. Evenness (even bottom below odd bottom) is not obtained. Positivity reduces exactly to (a), at most one even Dirichlet eigenvalue below C_ℓ, plus (b), a scalar pole-capacity inequality (derivation). As a propagation lemma, (a)+(b) is a reformulation of RH (even sector), not progress in the mission's sense. The output is the unconditional Proposition M. Measured at (c, N) = (13, 24) and (31, 32): (b) is 1.1e6 and 1.6e6 times tighter than (a), and (a)'s own margin is 3.5e−37 and 6.5e−63. The Poincaré attempt on (a) (§6) is unresolved.
3. What refutes it: Epstein (1,1,6), a rival that satisfies C2's hypotheses, refutes C2 as a positivity/propagation mechanism (hardened, via numerics). On every window where it matters, Epstein has one pole, a positive Levy density (Γ(s) = sum of the a = 1/4 and 3/4 kernels), and Λ_Q(n) ≥ 0 for all n ≤ 47. That last fact is re-checked here in exact arithmetic (check J): for n ≤ 29 every nonzero Λ_Q(n) is enclosed strictly positive, and the first negative is n = 48. Its Weil form is nevertheless negative from c = 28 (odd sector) and c = 29.5 (even sector), at N = 64 and N = 128, on every half-integer window up to 48 (hardened ball LDLᵀ inertia; numerics commit 3a799d8, epstein_N64.json, epstein_N128.json, confirmed here). The Dedekind zeta of Q(√−23) is positive on all 93 windows with the same blocks (control). The argument principle finds an off-line zero at 0.953260 + 16.290216i (measured, dps 20), where the negative even ground state peaks (frequency 16.71). The odd sector goes negative first (28 < 29.5): concrete evidence that "even bottom below odd bottom" fails for a Markov + one-pole form. Proposition M (pole-free simplicity, and even-sector simplicity of the full form by interlacing) is a separate claim, not refuted by negativity, and stays graded as a derivation. Which condition breaks (numerics commit 3b76739, epstein_polefree.json, N = 64; inertias hardened, eigenvalues float-grade): even sector: the pole capacity (b) fails first, then (a). Corrected by numerics commit e404a81 (epstein_mu2_cross.json): at N = 128 the full even form turns negative in (29.3037, 29.3042], while the pole-free even form gains its second negative eigenvalue (μ₂ < 0) only in (29.3174, 29.3179]; at N = 64 the pair is 29.3413 and 29.3555. The half-unit grid I first read had both in one step, and my earlier "(a) fails" reading is withdrawn. Odd sector: the odd pole capacity. The pole-free odd form stays positive through c = 30 (μ₁ ≈ +0.065) while the full odd form is negative from (27.7412, 27.7417] at N = 128. So the first failure, the odd one, is the negative odd pole term, not the Markov part. DH and W_a remain outside C2's hypotheses (signed jumps and no pole; two poles). Separately: C1 (propagation through entering atoms) is refuted at every band N = 64 … 256. DH's sign flips at hardened brackets c*(N) ∈ [30.617, 30.818], inside the step (29, 31), where no atom with nonzero weight enters (numerics crossing.json; Λ_f(30) = 0 here). For the continuum form this holds provided the continuum crossing lies above c = 29: the N-ladder indicates about 30.61 (measured) but does not establish it. C3 (norm-level ground-state transport) is refuted by DH (measured).
4. New or known: the Markov reading of the full Weil form, primes included, and the prime-entry kink formula are original to this session. They were not in the 10 sources read, and one web search found nothing. Suzuki (arXiv:2606.09096 s5) uses a Dirichlet form for the archimedean part only, and only for small windows. The fixed-window dilation formula is Suzuki's s4.2. Novelty is not established: the search was shallow.
5. Next step. No propagation mechanism survives in the searched set. Refuted: C1, C3, and C2 as a mechanism. Obstructed: comparison-type Poincaré bounds (measured, §6). Named (exact, §7.1): the step that Epstein (1,1,6) and W_a fail and Dedekind ζ_{Q(√−23)} passes is the semilocal unitary decomposition (U-S). Its two parts are atoms only at prime powers (Epstein fails at n = 6) and unitary local roots, |s_k(p)| ≤ degree (Epstein fails at n = 8, W_a at n = 2). The candidate that uses it, C4 (§7), is the Connes-Consani semilocal Sonin program restricted to windows. It is not original, and the bounded attempt is unresolved and paused. No rival satisfying its hypotheses exists, so it can only be tested by building its first instance, S = {∞, 2} on windows c ∈ [2, 3). There the conclusion is already known (Zhu), and what must be built is the semilocal trace remainder. A side result: local-factor flattening in the collar is vacuous (ordinary argument). From numerics (commits 3b76739, e404a81): Epstein breaks the even pole capacity (b) first (29.3042) and (a) later (29.3179), and the odd pole capacity in the odd sector; ζ holds (a) on all 234 cells with μ₂/λ₂ = 0.639 … 0.643 (hardened inertias). The Epstein/Dedekind pair differs in the prime block, and check K shows that means support, the local power sums and unitarity together, not support alone. Bounded desk attempt (§8): no mechanism from (U-S) pins μ₂/λ₂ or bounds the boundary trace by the margin; attempt unresolved. What it found (derivation, measured to six digits): the Weil ground state is u/‖u‖ with u = (A°)^{−1}c, the floor is −Φ/(2‖u‖²), and μ₂/λ₂ = 1 − ⟨φ₁,e₂⟩². The even edge amplitude is fixed by the odd floor through an exact commutator identity (CCM/CvS), which is arithmetic-free and weakens as ℓ grows. It also corrected a precision artifact in §6.

Study of 2026-09-23, branch teal-sea/weil-propagation-theory. Nothing here is a claim about RH. Grades follow the AGENTS.md ladder: measured, hardened, or an ordinary mathematical argument with its review status stated. No kernel-checked statement is made. Checks: checks.py, raw numbers in checks.json (total runtime about 2 minutes).

0. Conventions

A window is an interval I of length ℓ in the additive variable x = log u. Then c = e^ℓ (the CCM cutoff), λ = √c (CCM), a = ℓ/2 (Suzuki), and Zhu's L equals a. Test functions f ∈ L²(I) have autocorrelation g(x) = ∫ f(y) f(y − x) dy, supported in [−ℓ, ℓ]. The Weil form, in the normalization of Zhu arXiv:2608.24827 eq. (2)-(3) and Lemma 2.5, is

Q(f) = P(f) + A(f) − Σ_{log n < ℓ} 2Λ(n) n^{-1/2} g(log n), P(f) = 2 ∫ g(x) cosh(x/2) dx (pole), A(f) = −[γ + log π + log(1 − e^{−2ℓ})] g(0)

• ∫_0^ℓ 2[e^{−2x} g(0) − e^{−x/2} g(x)] / (1 − e^{−2x}) dx.

Check E confirms the constants against zeta/weil.py, whose convention the lab calibrated independently. λ*(ℓ) is the infimum of Q(f)/‖f‖². λ_N(c) is the lowest even-sector eigenvalue of the CCM Galerkin matrix at band N (hunts/rogue_frontier/weil_trunc/galerkin.py).

Conversions for the landmarks: Yoshida's range is ℓ = log 2 (c = 2). Zhu's support-1.6 theorem is ℓ = 1.6 (c ≈ 4.95). The DH lattice crossing is c = 31 (ℓ = 3.434, Zhu L = 1.717).

1. Primary sources: what each proves about window dependence

Read in full or in the cited sections, 2026-09-23. Local PDFs were kept in the session scratchpad and are not committed.

Summary of the literature on the question asked. No source gives a mechanism by which positivity on one window forces positivity on a larger one. The window dependence that is established:

• monotone decrease of the infimum (nested domains);
• continuity (Suzuki);
• positivity on small windows (Yoshida, Bombieri, Connes-Consani, Suzuki);
• a finite reduction per window (Yoshida, Zhu);
• measured and conditional decay laws (Zhu);
• a detection threshold for off-line zeros (Bombieri §13, numerics).

No source assigns a positive role to an entering prime. Every program reports being stuck at the same place, the arithmetic content beyond a fixed window: the global trace formula, the prime places, the simplicity and proximity steps, and comb cancellation.

2. Derivations: how the form depends on the window

2.1 The form does not move; the domain grows

Q(f) = ⟨D, f ⋆ f̃⟩ for one fixed even distribution D on ℝ: the pole, the archimedean principal value, and atoms Λ(n)n^{−1/2} at ±log n. In §0, log(1 − e^{−2ℓ}) is the exact integral of the archimedean kernel beyond ℓ, where g = 0, so the ℓ in the formula is bookkeeping, not dependence. The window only restricts supp f. Hence H_ℓ ⊂ H_ℓ' gives λ*(ℓ') ≤ λ*(ℓ) (as in CCM (3.27)), and the question is purely whether a positive form extends from a subspace to a larger one.

2.2 Fixed-window form (the "archimedean change")

For f ∈ L²[−1, 1], ‖f‖ = 1, set f_a(x) = a^{−1/2} f(x/a) ∈ L²[−a, a] (unitary) and g = f ⋆ f̃ on [−2, 2]. Substituting x = ay:

Q(f_a) = 2a P₊(a) P₋(a) − γ − log π − log(1 − e^{−4a})

• ∫0^2 2a [e^{−2ay} − e^{−ay/2} g(y)] / (1 − e^{−2ay}) dy − Σ{log n < 2a} 2Λ(n) n^{−1/2} g(log n / a), P_±(a) = ∫ f(u) e^{±au/2} du.

This is the Weil-form version of Suzuki's (4.5). Check E: for the box f = 1_{[−1,1]}/√2 it equals 2a·W(h) of zeta.weil.fejer_pair(a), with |dev| = 5.4e−16, 8.6e−19, 2.0e−21 at a = 0.3, 0.8, 1.2 (dps 30; the three cells include 0, 3 and 5 prime powers).

In these coordinates, growing the window leaves the function space alone and changes the form smoothly:

• the pole weight changes;
• the archimedean kernel is rescaled (its leading effect is Suzuki's −log a shift);
• every existing atom drifts inward to log n/a;
• new atoms enter at the edge y = 2.

If the ground state φ_a is simple (so the envelope theorem applies), dλ*/da = ∂_a Q(φ_a) evaluated with the formula above. Grade: derivation. The differentiability assumptions are stated here and not proved.

2.3 New prime-power atoms: zero weight, then a slope kink

An atom at n enters at ℓ = log n, where g(ℓ) = 0: it enters with zero weight, so λ* stays continuous. For ℓ slightly above log n, g(log n) ≈ (ℓ − log n) f(0⁺) f(ℓ⁻). The slope therefore jumps:

Δ (dλ*/dℓ) at ℓ = log n = −2 Λ(n) n^{−1/2} φ(0) φ(ℓ) (continuum; φ the ground state on [0, ℓ]) Δ (dλ_N/dL) = −Λ(n) n^{−1/2} (2/L) (Σ_{|k|≤N} u_k)² (exact at finite N, CCM basis)

The finite-N line holds because the CCM entries of the atom, q_{jk}(log n) with L near log n, vanish at L = log n and have derivative exactly (2/L)·𝟙𝟙ᵀ (differentiate CCM (2.9)-(2.10) in L). Hellmann-Feynman then applies, assuming a simple eigenvalue.

Check C (ζ, N = 16, dps 45, one-sided second-order differences at h = 1e−6), measured jump against predicted jump:

For ζ, Λ ≥ 0 and the ground state is reflection-even (φ(0) = φ(ℓ)), so every entering prime power makes the decrease steeper; each entry adds 50-80% to the slope in these N = 16 cells. That magnitude is band-specific. The numerics worker finds the band-N edge amplitude μ₀ falling and κ rising with N, with κμ₀² stable. So the continuum edge value may vanish slowly, the finite-N formula (exact) is the operative one, and the continuum line is a limiting form whose edge factor must be read as the N-stable combination, not as a pointwise value. An entering prime never helps positivity at first order. For DH, atoms with Λ_f(n) < 0 do the opposite.

2.4 The admissible space: the collar criterion

Grow [0, ℓ] to [0, ℓ'] and split H_ℓ' = H_ℓ ⊕ C with collar C = L²(ℓ, ℓ']. If Q > 0 on H_ℓ, then

Q ≥ 0 on H_ℓ' ⟺ S := Q_CC − Q_CI Q_II^{−1} Q_IC ≥ 0 on C (exact) necessary: Q(φ_ℓ, h)² ≤ λ*(ℓ) · Q(h) for every h ∈ C (one-mode test)

since Q_II^{−1} ≥ φ_ℓφ_ℓᵀ/λ*(ℓ). The arithmetic sits in the blocks as follows:

• Q_CC involves shifts smaller than ℓ' − ℓ, so for steps shorter than log 2 it contains no atoms, only archimedean and pole terms.
• Q_CI carries the old atoms (shifts up to ℓ').
• Q_II^{−1} carries the interior.

The one-mode test needs every collar function's coupling to the ground state to be O(√λ*), and λ* ≈ exp(−2π² N/ln N) (Zhu). Grade: elementary linear algebra (ordinary argument).

2.5 The DH lattice step 30 → 31 adds no arithmetic

Check A/B (exact arithmetic, confirmed numerically):

• Λ_f(n) = 0 for every n divisible by 5. f is supported on the multiplicative semigroup {5 ∤ n}, and so is −f′/f. So Λ_f(30) = 0.
• Λ_f(31) = log 31 exactly (31 is prime and 31 ≡ 1 mod 5).
• At c = 31 the n = 31 atom sits at y = L, where every CCM entry vanishes (largest contribution 8.4e−50).

Between the windows log 30 and log 31, the set of atoms with nonzero weight is {n ≤ 29, 5 ∤ n} at both ends. The same holds for ζ (prime powers ≤ 29). The DH sign flip at (31, 60) happens with no new arithmetic entering. Only the window length changed (by 0.033), with the archimedean kernel on the new shell. (My milestone-1 note in PROGRESS.md said the entering atom was n = 31 with ζ's weight; that was incomplete. The effective entering set is empty.)

2.6 The Levy-Markov structure (the new structural fact)

Using the exact identity g(0) − g(x) = ½ ∫_ℝ (f(y) − f(y − x))² dy:

Q(f) = P(f) + 𝓔(f) − C_ℓ ‖f‖², 𝓔(f) = ½ ∫_0^ℓ K(x) ∫ℝ (f(y) − f(y−x))² dy dx + Σ{log n<ℓ} Λ(n) n^{−1/2} ∫_ℝ (f(y) − f(y − log n))² dy, K(x) = 2 e^{−x/2} / (1 − e^{−2x}) > 0, C_ℓ = γ + log π + log(1 − e^{−2ℓ}) + ∫_0^ℓ 2(e^{−x/2} − e^{−2x})/(1 − e^{−2x}) dx + A_ℓ, A_ℓ = Σ_{log n<ℓ} 2Λ(n) n^{−1/2}.

In Fourier form, 𝓔(f) = (1/2π) ∫ |F|² ψ_ℓ(t) dt with

ψ_ℓ(t) = ∫0^ℓ K(x)(1 − cos tx) dx + Σ{log n<ℓ} 2Λ(n) n^{−1/2} (1 − cos(t log n)).

That is a Levy-Khintchine exponent, and ψ_∞ minus its comb part is Re ψ(1/4 + it/2) − ψ(1/4), Gauss's representation as in Zhu (9). As ℓ → ∞, C_ℓ − A_ℓ → log π − ψ(1/4), which recovers Zhu's symbol Ψ_L = ψ_ℓ − C_ℓ.

Proposition M (ordinary argument, unreviewed; checked by nobody outside this session). Take ζ and any window I, and let Q° = Q − P (pole removed). The hypotheses used are:

• (H1) Λ(n) ≥ 0 for every n, so the jump measure (½K(|u|)du plus atoms Λ(n)n^{−1/2} at u = ±log n) is nonnegative;
• (H2) irreducibility: the digamma Levy density ½K(|u|) is continuous and strictly positive for 0 < |u| ≤ ℓ, so every pair of points of I is joined by a jump of positive intensity;
• (H3) the form domain is that of the archimedean part. P, the atoms and the constant are bounded on L²(I), and Q has discrete spectrum (CCM Thm 3.6; Suzuki s4.1 for the compact embedding).

Under (H1)-(H3), 𝓔 is an irreducible Dirichlet form on L²(I): a pure jump form, with killing from jumps that leave I. Q° = 𝓔 − C_ℓ. Consequences:

• Its semigroup is positivity improving (the Beurling-Deny criterion, then irreducibility, via the same Dirichlet-form route as Suzuki s5.2).
• Its lowest eigenvalue μ₁ is simple, and the eigenfunction is a.e. positive and reflection-even about the window centre.

In probability terms: the pole-free Weil operator on a window is the generator of the zeta jump process killed outside the window, shifted down by C_ℓ.

Corollary (sector reduction). The pole is P = 2⟨f, c⟩² on even f and −2⟨f, s⟩² on odd f, with c = cosh((x − x₀)/2), s = sinh((x − x₀)/2) about the centre x₀, and no cross term (Zhu Lemma 6.1). Rank-one interlacing then gives:

• μ₁(Q°) ≤ λ₁(Q_e) ≤ μ₂(Q°_e) ≤ λ₂(Q_e).
• Simplicity of the full form's even bottom (derivation, not measured). The eigenvalues of Q_e are the roots of the secular function f(λ) = 1 + 2 Σ_k ⟨c, e_k⟩² / (μ_k − λ) (one in each gap between consecutive μ_k with ⟨c, e_k⟩ ≠ 0), together with every μ_k whose eigenvector is orthogonal to c. μ₁ is simple and ⟨c, e₁⟩ > 0 (both e₁ and c are positive), so the smallest root lies strictly above μ₁ and is simple. Hence λ₁(Q_e) = min(r₁, μ_k : e_k ⊥ c), with r₁ the smallest root. Since μ₁ is simple with ⟨c, e₁⟩ > 0, every c-orthogonal μ_k has k ≥ 2. The nongeneric coincidence, exactly: either r₁ equals a c-orthogonal μ_k (a tie), or a degenerate c-orthogonal μ_k lies below r₁. If a simple c-orthogonal μ_k lies below r₁, the bottom is still simple, but its eigenvector e_k changes sign. Neither condition (a) nor positivity is needed. Measured: ⟨c, e₂⟩ is small but not nearly zero on its own scale. 2⟨c, e₂⟩² = 2.8e−74 at c = 13 (dps 120) and 9.0e−126 at c = 31 (dps 200), about 0.2 μ₂² (check L). An earlier dps-60/110 reading of 4.4e−63 / 4.2e−113 was a precision artifact. With λ₁(Q) = 3.1e−43 against μ₂ = 3.5e−37 at c = 13, the bottom is simple in both cells.
• Even-sector positivity: Q_e ≥ 0 ⟺ (a) μ₂(Q°_e) ≥ 0, and (b) μ₁ ≥ 0 or Φ_ℓ := 1 + 2⟨c, (A°_e)^{−1} c⟩ ≤ 0.
• Under RH: on every window the killed zeta process has at most one even Dirichlet eigenvalue below C_ℓ.
• Odd sector: μ₁(Q°_o) ≥ μ₂(Q°), because the Perron-Frobenius state is even, and Q_o ≥ 0 ⟺ Q°_o ≥ 0 and 1 − 2⟨s, (A°_o)^{−1} s⟩ ≥ 0.

Whether the even bottom lies below the odd one, the other half of CCM step 1, is not decided here for ζ. For a Markov + one-pole form it can fail: Epstein (1,1,6)'s odd sector goes negative at c = 28 while its even sector is still positive (it turns negative at c = 29.5), hardened at N = 64 and 128 (numerics commit 3a799d8). So on [28, 29.5) the global bottom is odd. Evenness for ζ therefore needs more than (H1)-(H3).

Check D (measured; CCM basis; ground-state profile on a 400-point grid):

In both cells the pole-free form has exactly one negative even eigenvalue, as Proposition M plus positivity require. μ₂(Q°) = 0.64·λ₂(Q), which is 1.1e6 and 1.6e6 times λ₁(Q). So the pole capacity (b) is the tight condition. But (a) is not easy either: its margin is 1e−37 absolute against an O(6) gap.

3. Numerical checks, summary

All in checks.py, raw values in checks.json. Grade: measured, float arithmetic at the stated dps. Sign statements for DH at c = 30, 31 are the lab's hardened ones (hunts/rogue_frontier/weil_trunc/RESULTS.md s8.1).

4. Candidate propagation lemmas

C1. Boundary-arithmetic propagation: refuted

Statement (as a class). If Q_D ≥ 0 on a window of length ℓ, and the atoms of D with log n ∈ [ℓ, ℓ') satisfy a local condition 𝒫 (for example: prime powers only, weight Λ(n)n^{−1/2} ≥ 0, a local Ramanujan bound as in docs/24), then Q_D ≥ 0 on the window of length ℓ'.

Would imply. RH by chaining from Zhu's ℓ = 1.6, provided ζ's entering atoms satisfy 𝒫.

Euler product enters only through 𝒫, on the new atoms.

DH test. Across c = 30 → 31 no atom gains weight (§2.5), so any 𝒫 holds vacuously. DH is positive at (30, N) for every N ≤ 128 (hardened) and negative at (31, 60) (hardened). Refuted at the lattice level. The numerics worker's bisection puts the flip strictly inside the step. Hardened brackets from their crossing.json (branch teal-sea/weil-propagation, commit 75865a7), positive at c_pos and negative at c_neg:

ζ is conclusively positive at every c_neg (ball LDLᵀ). By their nesting argument (Fact B, ordinary), DH's continuum form is negative for every c ≥ 30.617188.

For the continuum form, DH's floor is:

• continuous (Suzuki's Thm 1.3 argument uses only compact embedding and a finite prime sum, so it ports);
• positive on small windows (Bombieri Thm 12's argument ports, with the pole term simply absent);
• negative at log 31.

So a first crossing ℓ* ≤ log 30.617188 exists. DH has no atom with nonzero weight in (29, 31). If ℓ* > log 29, the step from ℓ* to log 30.617188 is atom-free, and C1 is refuted for the continuum form with no lattice caveat. The N-ladder of c*(N) extrapolates to about 30.61 (numerics, measured), which supports ℓ* > log 29 but does not establish it.

Kink formula (§2.3): for ζ, each entering atom only steepens the decrease. Scoped obstruction: the Euler product cannot drive propagation through the atoms that enter. It has to act through the interior block and the interior-collar coupling (§2.4).

Smallest check. Done: checks A and B here, and the numerics worker's crossing.json for the location.

Restatement of RH? Not applicable: refuted.

C2. Levy-Markov + pole capacity: refuted as a positivity mechanism by Epstein (1,1,6); Proposition M stands as a derivation

Statement. Proposition M and its Corollary (§2.6). As a propagation lemma: for all ℓ, (a) μ₂(Q°_{ℓ,e}) ≥ 0, which says the zeta jump process killed outside a window of length ℓ has at most one even Dirichlet eigenvalue below C_ℓ, and (b) Φ_ℓ ≤ 0.

Would imply. (a) and (b) for all ℓ ⟺ even-sector Weil positivity on all windows. Together with the odd-sector analogue, that is RH. The unconditional part (Proposition M) already gives a simple positive ground state of the pole-free operator on every window, and generic simplicity of the even Weil ground state on every window. That is half of CCM s8 step 1, which was previously known only for small windows (Suzuki Thm 1.4) and at support 1.6 (Zhu Thm 6.2).

Euler product enters as Λ(n) ≥ 0, the nonnegative Dirichlet coefficients of −ζ′/ζ, which come from the Euler product with all local parameters equal to 1. The same holds for Dedekind zeta functions, not for Dirichlet L-functions with complex characters, so Markov structure is not necessary for positivity. Then the pole pays for exactly one negative Markov direction (μ₁ = −5.85 at c = 13).

DH test. DH violates the hypothesis on every window with c > 3, since Λ_f(3) = −0.312: its jump measure is signed and there is no Markov structure. It also has no pole, so its analogue of (a)+(b) is μ₁ ≥ 0, which fails at c = 31 with exactly one negative eigenvalue. This means DH does not test C2 at all. It is not a pass. C2 has so far faced no rival that satisfies its hypotheses.

The lab's Euler-product rival does not test it either, and shows what is load-bearing. W_a(s) = ζ(s+a)ζ(s−a), a = 1/4 (zeta/epstein.py; it killed Euler-product positivity in hunts/epp_herglotz/RESULTS.md). Its zeros are ρ ± a, so its explicit formula is ζ's applied to g·2cosh(ax): D_{W_a}(x) = 2cosh(ax)·D_ζ(x) (derivation). Consequences:

• Its jump measure is K(|u|)cosh(au) plus atoms Λ(n)(n^a + n^{−a})n^{−1/2}, all nonnegative, so it is Markov: (H1) and (H2) hold.
• Its pole part is 2cosh((½+a)x) + 2cosh((½−a)x), i.e. two positive pole directions in the even sector.
• All its zeros are off its critical line. It is positive on small windows (Bombieri Thm 12's argument goes through, since the archimedean density only doubles near 0). By Weil's criterion applied to W_a (a standard argument, not checked for W_a here), it must turn negative on some window.

So Markov structure plus an Euler product plus a functional equation do not propagate positivity. C2 survives only through its one-pole hypothesis, which W_a violates.

The control that matches C2's hypotheses: Epstein Q = (1,1,6), discriminant −23. It has one pole, off-line zeros, a Γ(s) factor (positive Levy density), and Λ_Q(n) ≥ 0 for n ≤ 47 (first negative at n = 48). That is recorded earlier in hunts/epp_herglotz/RESULTS.md and re-measured here. Handed to the numerics worker by the supervisor, 2026-09-23. It is decisive only if its window form turns negative below c = 48. This tree does not locate its first off-line zero (hunts/gate5_p6_a/RESULTS.md), so the test may come back uninformative (crossing above 48).

Result (numerics commit 3a799d8; numbers confirmed in their JSONs, and Λ_Q ≥ 0 re-checked here exactly, check J). Epstein turns negative well below 48:

• odd sector from c = 28;
• even sector from c = 29.5;
• at N = 64 and 128, on every later window up to 48;
• hardened (conclusive ball LDLᵀ inertia).

On those windows only atoms n ≤ 29 are active, where Λ_Q ≥ 0 holds exactly. So Epstein satisfies (H1)-(H3) and has one pole, and its form still loses positivity. "Markov + one pole + Γ-class" does not propagate positivity: C2 as a mechanism is refuted. The Dedekind zeta of Q(√−23), built from the same blocks with an Euler product, stays positive on all 93 windows. That locates the missing ingredient in multiplicativity itself. The refutation touches neither Proposition M (a statement about simplicity, not sign) nor the reduction (a)+(b), which is an equivalence and so fails exactly when positivity does.

Which condition breaks (numerics commit 3b76739; their epstein_polefree.json at N = 64, ball LDLᵀ inertia conclusive at every cell, so hardened; the eigenvalues quoted are float-grade approximations; refined crossings from their epstein_crossing.json, hardened negative by two routes).

• Even sector: the pole capacity (b) fails first, then (a) (corrected; numerics commit e404a81, epstein_mu2_cross.json, bisection on ball LDLᵀ inertia to 2^-10). At N = 128 the full even form turns negative in (29.3037, 29.3042], but the killed Epstein jump process acquires its second even eigenvalue below C_ℓ (μ₂ < 0) only in (29.3174, 29.3179]. At N = 64 the pair is (29.3408, 29.3413] and (29.3550, 29.3555]. So on a window interval of width about 0.014 in c, (a) still holds and the single pole cannot pay for the one negative Markov direction; after that, Epstein also develops a second one. My first reading, "(a) fails", came from the half-unit grid, which puts both changes in the same step, and is withdrawn.
• Odd sector: the odd pole capacity fails, not Q°_o ≥ 0. The pole-free odd form is positive at every window through c = 30 and negative only by c = 34. The full odd form is negative from (27.7412, 27.7417] at N = 128. So Epstein's first failure, the odd one, is the negative odd pole term −2⟨f, s⟩² (the condition 1 − 2⟨s, (A°_o)^{−1} s⟩ ≥ 0); the Markov part is not what breaks first. The failure of "even bottom below odd bottom" recorded above is therefore caused by the pole.

The attribution to (b)-then-(a) and to the odd pole capacity uses the equivalences of §2.6 (derivation). The inertias are hardened. The numerics worker's reading (their RESULTS s9.3, commit e404a81) agrees.

ζ holds (a) with a fixed ratio (numerics e404a81, their s9.4, zeta_pole_N64.json and zeta_pole_N128.json, confirmed here). On 117 windows c ∈ [2, 60] at N = 64 and 128 (234 cells):

• the full even form is positive definite at every cell, with ball LDLᵀ inertia (N+1, 0);
• the pole-free even form has exactly one negative eigenvalue at every cell, inertia (N, 1). That is the shape Proposition M plus positivity require. Hardened.
• μ₂/λ₂ lies in [0.6394, 0.6430] for c ≥ 5 at both N (measured; 0.88 at c = 2).

§8 derives μ₂/λ₂ ≈ 1 − ⟨φ₁, e₂⟩² and checks it to 1e−6.

What the Epstein/Dedekind pair isolates. Numerics' reading (their s9.3) is that the two differ only in the prime block. Check K sharpens which features of the prime block differ, all exactly:

• Dedekind's atoms sit on prime powers; Epstein has composite atoms.
• The local power sums differ at the primes themselves (Λ_Q(2) = 0 against Λ_K(2) = 2 log 2).
• Epstein's towers violate the unitarity bound (s₃ = 6 at n = 8 and 27).

So the matched pair shows that the prime block decides the sign here. It does not isolate support alone from the other features of (U-S).

Any C2-based argument must therefore use more of the Euler product. Candidates: prime-power support, and the local-factor identity (Σ_k p^{−|k|/2} δ_{k log p}) ∗ Φ_p = (1 − 1/p) δ₀ with Φ_p = (1 + 1/p)δ₀ − p^{−1/2}(δ_{log p} + δ_{−log p}), which holds exactly for degree-1 local factors.

Smallest checks.

1. Done by numerics (commit 3a799d8): the Epstein (1,1,6) scan for c ≤ 48; see the result above.
2. Numerics (handed off): for ζ over c ∈ [2, 60] at converged N, μ₂(Q°_e), λ₁(Q_e), λ₂(Q_e) and Φ_ℓ. The question is whether μ₂(Q°) tracks λ₂(Q) at a fixed ratio (0.64 in both measured cells).
3. Sign-definiteness of the pole-free ground state at more cells.
4. Optional, cheap once (1) exists: W_a's window form, built from the same blocks (archimedean a' = 1/8 and 3/8, pole blocks for cosh((½ ± a)x), atoms Λ(n)(n^a + n^{−a})n^{−1/2}). It locates where two poles stop sufficing.

Restatement of RH? The propagation form (a)+(b) is an exact reformulation, with no new estimate. What is new is the structure: a Perron-Frobenius operator, and access to the spectral theory of killed Levy processes (ground-state transform, Poincaré inequalities). The one missing estimate is named: a spectral gap of the Doob-transformed killed process of at least C_ℓ − μ₁^{Dir}(ℓ). The measured margin of that inequality is doubly-exponentially small (3.5e−37 on a gap of 5.85 at c = 13). So any proof of it must be structural (an identity), not perturbative.

C3. Local ground-state transport: refuted at norm scale; restatement at eigenvalue scale

Statement (as a class). Positivity at ℓ plus a bound on local ground-state data at ℓ implies positivity at ℓ + δ. Examples of such data: ‖φ_{ℓ+δ} − T_δφ_ℓ‖ ≤ ε with T_δ the dilation; the boundary-mass ratio φ_ℓ(0)²/λ*(ℓ) ≤ R; CCM step 2's proximity ‖ξ_λ − k_λ‖ = O(λ^{−2}).

Would imply RH by chaining, if the data bound held uniformly.

Euler product enters nowhere explicitly. The CCM near-radical k_λ = E(h_λ) uses the Dirichlet series and Poisson summation, not the Euler product. The Mellin transform of E_a(h) is f(s)·Mh(s) for any Dirichlet series f, so a DH analogue exists (ordinary argument, sketched; the odd Hermite data for Γ((s+1)/2) are not worked out).

DH test (checks F, H).

• Across 30 → 31 (N = 60), DH's ground state moves 7.7e−3 in norm, against 4.4e−3 for ζ.
• Yet its λ flips sign.

The norm hypothesis holds with ε ≈ 1e−2, and the conclusion fails. Refuted at the lattice level for norm transport. For the boundary-mass version, see §5: at matched N = 128 it is a reformulation of the log-derivative of λ, not an independent datum. Quantitatively, the transported old ground state has Rayleigh quotient 5.2e−5 (DH) and 6.8e−8 (ζ) at the new window, 10²³ and 10⁶¹ times λ. A transport statement that carries positivity must be accurate to about √λ in norm (1e−14 for DH, 1e−34 for ζ), and at that accuracy it is a restatement of positivity.

The lab record agrees: at (31, 60) the sign is carried by a beam component of coefficient mass 7.8e−29 (weil_trunc RESULTS s8.3 item 3). So CCM's O(λ^{−2}) prolate proximity cannot fix a sign by proximity. It is aimed at zero convergence via Hurwitz, a different route.

By CvS applied to DH (no arithmetic needed), DH's simple-even ground states have transforms with only real zeros at every c. Since DH has off-line zeros, the DH analogue of CCM step 2 must fail as c grows (Hurwitz).

Smallest check. Count the real zeros of DH's ground-state transform F_v in the on-line gap (83.109, 87.647) around the off-line ordinate 85.699, at c = 29, 30, 31, 47 (band edge above 88 needs N ≥ 48 at c = 31). DH has no on-line zero there. A real zero of F_v inside the gap is a ghost forced by CvS reality, and its first appearance against c* tells whether the zero route "sees" the off-line zero before, at, or after the sign flip.

Restatement of RH? At norm scale: refuted. At eigenvalue scale: yes, a reformulation with no new estimate.

5. Does the Euler product bound the boundary trace? (the numerics worker's question)

The numerics worker measures an edge law dλ/dL = −κμ₀², with μ₀ = Σu_k the band-N edge amplitude (so φ_N(0)² = μ₀²/L) and κ ≈ 1.0 … 1.3. It holds through DH's crossing, with κμ₀² N-stable and κ alone not (numerics RESULTS s4 row 7, s5). They ask whether the Euler product, via the Markov structure of §2.6, bounds μ₀² ≲ λ/(κδ).

Matched-N data (their grid_*_N128.json on branch teal-sea/weil-propagation, commit 75865a7; λ₁ and μ₀ as stored; the ratio is my arithmetic on their numbers):

Readings:

1. With the edge law, μ₀²/λ = −(1/κ) d ln λ/dL. The ratio is the relative decay rate of λ. A bound μ₀² ≤ Rλ is the log-derivative bound that the numerics worker's candidate 6 already tested, and it is equivalent to a lower bound on λ. Reformulation, not an estimate (ordinary argument given the measured edge law).
2. DH's ratio sits below ζ's until c ≈ 30.4: 101 at c = 30.25 (their s3.3 table) against ζ's ≈ 190, and 226 at c = 30.5. It then diverges within the last ΔL ≈ 0.013 before the crossing. So it gives no usable early warning, and no threshold R separates ζ from DH ahead of time (measured). My N = 60 reading in check H ("no precursor at c ≤ 30") was correct but stopped short of c*(60).
3. Does Λ ≥ 0 bound it? No, not by itself. W_a has Λ ≥ 0, an Euler product and the functional equation, and it must cross somewhere (§4 C2). At a transversal crossing (μ₀ ≠ 0 where λ = 0, as the numerics worker observes for DH) the ratio diverges. So the Markov structure cannot bound μ₀²/λ unless the one-pole structure is used. Grade: ordinary argument, conditional on Weil's criterion for W_a and on transversality. DH lies outside the hypothesis (signed jumps). Epstein (1,1,6), which does satisfy Λ_Q ≥ 0 and has one pole on these windows, crosses zero at c = 28 (odd) and 29.5 (even) (hardened, numerics). At a transversal crossing μ₀²/λ diverges, so even Markov + one pole does not bound it (transversality not measured for Epstein).
4. What the Markov structure does control: the pole-free ground state φ°, which is positive and flat (min/max 0.71 … 0.80, max φ°² ≈ 1.2 … 1.4 × 1/ℓ). The Weil ground state is instead the resolvent (A° − λ)^{−1} c at a spectral parameter inside the doubly-exponentially thin gap (μ₁, μ₂). Its boundary trace relative to λ is set by the secular balance (b), not by Markovianity (derivation).

6. The Poincaré attempt for condition (a) (about one hour, supervisor-allocated)

Ground-state representation (exact, ordinary argument). Write 𝓔(f) = ½∬ j(y−z)(f(y)−f(z))² dy dz + ∫ κ f², and let φ° > 0 be the Perron-Frobenius ground state, 𝓔(φ°, ·) = m₁⟨φ°, ·⟩ with m₁ = μ₁ + C_ℓ. For f = φ°u the cross terms telescope:

𝓔(φ°u) − m₁‖φ°u‖² = ½ ∬ j(y−z) φ°(y) φ°(z) (u(y) − u(z))² dy dz.

Expand (φ°(y)u(y) − φ°(z)u(z))² = φ°(y)φ°(z)(u(y)−u(z))² + (φ°(y) − φ°(z))(φ°(y)u(y)² − φ°(z)u(z)²); the second group is 𝓔(φ°, φ°u²) = m₁‖φ°u‖². So μ₂ − μ₁ is the spectral gap of the Doob- transformed process (kernel j φ°φ°, reference measure π = φ°² dy), and

(a) ⟺ gap_e(ℓ) ≥ D_ℓ := −μ₁(Q°) (Poincaré inequality for the conditioned process).

The simplest comparison bound (computed, check I). Var_π(u) = ½∬φ°²(y)φ°²(z)(u(y)−u(z))² (π normalised). Discarding the atoms, which is legitimate since they are nonnegative jumps, gives

gap ≥ inf_{y,z} j(y−z)/(φ°(y)φ°(z)) ≥ (K(ℓ)/2) / max φ°².

The comparison bound is short by factors of 10 and 20. More to the point, the true gap exceeds the requirement by a relative 6e−38 and 7e−64. Scoped obstruction (measured, restricted class): any Poincaré or comparison inequality whose ratio to the true gap is bounded away from 1 by more than about 1e−37 (c = 13) cannot establish (a). Condition (a) is saturated to the eigenvalue scale, so it can only come from an identity that exhibits the saturating direction, not from an inequality with slack.

The saturating direction (measured, checks I and L; corrected). The Poincaré extremal is e₂, the second pole-free eigenvector. I first reported it orthogonal to the pole vector to 2⟨c, e₂⟩² = 4.4e−63 (c = 13) and 4.2e−113 (c = 31). Those values were precision artifacts (dps 60 and 110 cannot resolve them). At dps 120 and 200 the values are 2.8e−74 and 9.0e−126, about 0.2 μ₂², while ‖W e₂‖ is stable. So ⟨c, e₂⟩ scales like μ₂, which is what c = A°u with u = (A°)^{−1}c forces (⟨c, e₂⟩ = μ₂⟨u, e₂⟩). It is not evidence of a near-radical vector. The near-radical reading below is therefore withdrawn as evidence; it remains only a hypothesis. For the record, the hypothesis was: k_λ = E(h_λ) with ∫h = 0 (Riemann's vanishing-integral condition, CCM s7 Lemma 7.1) has transform nearly vanishing at ±i/2, and Q(k_λ) ≈ 0 (CCM s8 item 2; Connes 2602.04022 s6.4). The first pole-free eigenvector carries the pole almost entirely: 2⟨c, e₁⟩² = 5.883 against μ₁ = −5.850, so Q(e₁) = +0.033 (c = 13). The Weil ground state mixes the two (overlaps 0.52 with e₁, 0.60 with e₂).

Status (ALIGNMENT s5).

• Attempt unresolved: (a) was not established.
• Restricted class obstructed (measured, not proved): non-sharp comparison or Poincaré bounds.
• Paused by allocation after the allotted hour.
• Reduction obtained: (a) holds iff the near-null directions of Q° orthogonal to φ° have nonnegative energy. Whether e₂ is a CCM near-radical vector, which would make (a) CCM's missing step 2 at the eigenvalue scale, is an open hypothesis. The orthogonality first offered as support was a precision artifact (above).

7. C4: a candidate that uses multiplicativity itself (bounded attempt, about one hour)

Supervisor-allocated, 2026-09-23. The rule was: name the step that Epstein (1,1,6) and W_a fail and Dedekind ζ_{Q(√−23)} passes before developing anything, or stop.

7.1 The step, named first

(U-S) Semilocal unitary decomposition. On a window of length ℓ = log c:

1. the Weil distribution has atoms only at prime powers;
2. for each p, the tower Λ(p^k)/log p =: s_k(p) equals Σ_j α_{j,p}^k with every |α_{j,p}| = 1.

A necessary, exactly checkable consequence of (2) is |s_k(p)| ≤ d, the degree.

Grade: exact arithmetic, check K. Rational coefficients in the log-prime basis for Dedekind and Epstein, n ≤ 60. W_a is closed form from Λ_W(n) = Λ(n)(n^a + n^{−a}), zeta/epstein.py.

Independent check (numerics commit 8449c29, their RESULTS s10, us_check.py, us_check.json; confirmed here). A separate exact implementation (rational coefficients in the log-prime basis) over n ≤ 200:

• Dedekind Q(√−23): no composite atoms and no tower violation. Both parts of (U-S) pass.
• Epstein (1,1,6): 31 composite atoms, the first at n = 6. Its tower violations |s_k(p)| > 2 occur exactly at n = 8 (s₃(2) = 6) and n = 27 (s₃(3) = 6).
• It agrees with check K entry for entry on n ≤ 60.

Grade of the (U-S) coefficient statements: hardened for n ≤ 200 only (two independent exact implementations). Beyond n = 200 nothing is claimed, and the windows used here need only n < 48. This grade covers the coefficient check. It does not cover the C4 implication below, which stays at its own rung.

Where the step is used. Connes' semilocal trace formula (arXiv:math/9811068, s VII Thm 4, stated for any global field) writes Σ_{v∈S} W_v as the geometric side of the trace of the scaling action of the S-idele class group on L²(X_S), with S = {∞} ∪ {p ≤ c}. Part (1) of (U-S) is what lets the window's distribution be such a sum:

• Epstein fails here. Its composite atoms (6 = 2·3 appears as soon as c > 6) are not the local term of any place.

Part (2) is what makes the scaling action, twisted by the local characters, unitary. Only then is a Sonin-type term Tr(ϑ_S(g) Π ϑ_S(g)^) a nonnegative quantity for g ∗ g^:

• W_a fails here. Its local data are the non-unitary quasi-characters |·|_p^{±1/4}, and its form is 2B_ζ(e^{a·}g, e^{−a·}g) (§4 C2): a pairing of two different vectors, not a diagonal trace.
• Dedekind passes both. It is GL₁ over K with the trivial character, or over ℚ the unitary pair (1, χ_{−23}).

7.2 The candidate

C4 (semilocal Sonin positivity on windows). Let S ⊇ {∞} ∪ {p ≤ c} satisfy (U-S). Let g be supported in a window of length ℓ = log c, with ĝ vanishing at ±i/2 and 0. Then

W_S(g ∗ g^) ≥ Tr(ϑ_S(g) Π_S ϑ_S(g)^) − R_S(g),

where Π_S is a semilocal Sonin projection and R_S is a remainder of bounded rank.

• For S = {∞} (support in (1/2, 2)) this is Connes-Consani arXiv:2006.13771 Thm 6.11, with R = c|ĝ(0)|², 13 < c < 17. Its proof uses numerically identified prolate data (their s6).
• The step of §7.1 is used exactly once: the trace term is nonnegative because ϑ_S is unitary, and it equals the geometric side because the distribution is a sum over places.

Would give: positivity (with the vanishing conditions, which cost nothing for RH by their Appendix C) on every window covered by S. Adding places one at a time, this is their stated plan (Intro), which would give RH.

Not original. This is the Connes-Consani program restricted to windows. Added here: the rival accounting above, and §7.3.

Rival test: exhausted, not passed. DH, Epstein and W_a all fail (U-S) (Epstein's failure confirmed exactly to n ≤ 200 by numerics 8449c29; DH and W_a as in the §7.1 table), so none of them tests C4. C4 is not yet DH- or battery-tested. Any object that satisfies (U-S) together with a functional equation and one pole lies in the tempered, Selberg-class-like family, where RH is conjectured. So no rival that satisfies C4's hypotheses exists in this tree, and none is known. C4 can only be tested by building its first instance, not by a rival.

7.3 What the hour produced

1. Local-factor flattening in the collar is vacuous (ordinary argument, unreviewed; original). Grow [0, ℓ] to [0, ℓ + δ] and take δ ≤ log p ≤ ℓ. For a collar function h, put v = −p^{1/2} h(· + log p), which lies in H_ℓ because its support is inside (ℓ − log p, ℓ + δ − log p] ⊂ [0, ℓ]. Then m_p ∗ v = v + h, so m_p ∗ v ≡ h modulo H_ℓ. The Schur complement of §2.4 is intrinsic to the quotient H_{ℓ+δ}/H_ℓ, so S(m_p ∗ v) = S(h). The flattening identity (§4 C2), which collapses the p-tower, therefore adds no information about propagation. With C1, multiplicativity can act through neither the entering atoms nor the collar. It has to act through the spectral structure of the interior form, which is where C4 puts it.
2. At a finite place the Sonin structure factorizes, for product cutoffs (derivation, unreviewed). On ℚ_p with the self-dual measure, 1_{ℤ_p} is its own Fourier transform, so time and frequency limiting to ℤ_p are the same projection P_p. For P = P_∞ ⊗ P_T and P̂ = P̂_∞ ⊗ P_T (T = S ∖ {∞}, P_T = ⊗_{p∈T} P_p):

Π_S = S_∞ ⊗ P_T + 1 ⊗ (1 − P_T).

All the uncertainty (prolate) content is archimedean. The finite places act only through the quotient by Γ_S, the sum over S-units in the map E_S, which is where the Euler product enters. Caveat: Connes' semilocal formula (arXiv:2602.04022 (22)) cuts off by the global module |x|_S, not by product balls, and that does not factor. Which cutoff can carry a positivity mechanism is open.

1. First open instance located. S = {∞, 2}, windows c ∈ [2, 3). Weil positivity itself is already proved there (Zhu arXiv:2608.24827, support ≤ 1.6), so this instance tests the mechanism, not the conclusion. What is needed is the semilocal analogue of the Connes-Consani trace remainder δ(ρ) (their eqs. (8), (10)) and of its Toeplitz/prolate analysis (their s6). Not attempted within the bound.

Status (ALIGNMENT s5).

• Attempt unresolved; paused by allocation.
• The separating step is named and checked (exact).
• The candidate is an existing program with its first open instance located.
• No new estimate.
• Not a restatement of RH: C4 is a stronger structural inequality whose per-S instances are open. Only its union over all S implies RH.

8. Does (U-S) pin μ₂/λ₂ away from 0, or bound the boundary trace by the margin? (bounded desk attempt)

Supervisor-allocated, under 45 minutes, 2026-09-23. Even sector throughout. Notation: Q_e = A°_e + 2cc^T, pole-free eigenpairs (μ_k, e_k), u := (A°_e)^{−1} c, and Φ := 1 + 2⟨c, u⟩ (the pole capacity of (b)).

8.1 What the floor and μ₂/λ₂ are (derivation; measured to six digits)

Since c_k := ⟨c, e_k⟩ = μ_k⟨u, e_k⟩, the secular function of §2.6 is exactly

f(λ) = 1 + 2Σ_k c_k²/(μ_k − λ) = Φ + 2λ‖u‖² + 2λ² Σ_k ⟨u,e_k⟩²/(μ_k − λ).

In ζ's regime λ₁ ≪ μ₂ ≪ μ₃ (the first inequality measured, μ₂/λ₁ ≥ 3e5 for c ≥ 5; μ₃ ≫ μ₂ assumed, not measured), the leading orders give:

λ₁ ≈ −Φ / (2‖u‖²), φ₁ ≈ u/‖u‖, μ₂/λ₂ ≈ 1 − ⟨u,e₂⟩²/‖u‖² = 1 − ⟨φ₁, e₂⟩².

Check L (dps 120 and 200):

Readings:

• The Weil ground state is the pole-free resolvent of the pole vector, and the floor is the pole-capacity deficit over 2‖u‖². Condition (b) is the floor equation, which is why it is the tight condition for ζ.
• Numerics' constant μ₂/λ₂ = 0.639 … 0.643 on c ∈ [5, 60] (hardened inertias, measured ratio) says the ground state keeps about 36% of its squared mass on the second pole-free eigenvector at every window.
• Pinning μ₂/λ₂ ≥ κ > 0 is the same as ⟨φ₁, e₂⟩² ≤ 1 − κ.

8.2 The boundary trace: an exact identity, arithmetic-free

The full Weil matrix E in the basis {−N, …, N} has the CvS form q_ii = a_i, q_ij = (b_i − b_j)/(i − j) (CCM Lemma 5.1; CvS Prop 4.2). Hence DE − ED = |β⟩⟨η| − |η⟩⟨β|, with D = diag(n), η = (1, …, 1) and β = (b_n), b_n = n E_{0n} (CvS (5.3), CCM Lemma 5.2(ii)). For the even ground state (⟨β, φ₁⟩ = 0 by parity):

(E − λ₁) Dφ₁ = −μ₀ β, μ₀ = ⟨η, φ₁⟩ (the numerics worker's edge amplitude),

so

|μ₀| ≤ ‖Dφ₁‖ (λ₁^odd − λ₁) / |⟨f₁^odd, β⟩|.

The identity is CCM's Lemma 5.4(i) in shifted form and is not new. The bound is a one-line corollary (ordinary argument).

Check L:

• identity residual 2.3e−100 (c = 13) and 2.1e−167 (c = 31);
• the bound is saturated, bound/|μ₀| = 1.0 at both cells. So (E_odd − λ₁)^{−1}β is carried by the odd ground state, and Dφ₁ is proportional to f₁^odd to that accuracy. The derivative of the even ground state is the odd ground state (measured, not proved);
• values: μ₀ = 3.83e−21 and 5.05e−34; λ₁^odd/λ₁ = 1.8e3 and 2.1e3; ⟨f₁^odd, β⟩ = 1.9e−19 and 2.8e−32.

So the even boundary trace is fixed by the odd margin: μ₀ ≈ ‖Dφ₁‖(λ₁^odd − λ₁)/⟨f₁^odd, β⟩. With the edge law dλ₁/dL = −κμ₀², the relative decay rate of the even floor is κ‖Dφ₁‖²(λ₁^odd − λ₁)²/(λ₁⟨f₁^odd, β⟩²).

8.3 Answer: no mechanism found (attempt unresolved)

• Both targets are eigenvector-level functionals of the whole interior operator. They are the angle ⟨φ₁, e₂⟩ between the Weil ground state and the second pole-free eigenvector; the odd floor; and the overlap ⟨f₁^odd, β⟩. (U-S) constrains which atoms exist and the local structure of each prime tower. I found no argument that passes from the support of the jump measure to eigenvector angles or odd/even floor ratios at the eigenvalue scale.
• The identities of 8.1 and 8.2 are arithmetic-free. They hold for any real distribution of CvS type, DH included. So they cannot be the mechanism; they only relocate the question onto the quantities named.
• The odd-floor transfer points the wrong way. The odd/even floor ratio grows with the window (Zhu Table 1: about 10^{2.07L+1.32}; here 1.8e3 at c = 13 and 2.1e3 at c = 31). So bounding μ₀ by the odd margin gets weaker relative to λ₁ as ℓ grows, not stronger.
• Epstein, for contrast (numerics' epstein_polefree.json, float-grade): μ₂/λ₂ = 0.808, 0.758, 0.733 and 0.701 at c = 27, 28, 28.5 and 29, falling toward its (a)-failure at 29.318. There λ₁ is not ≪ μ₂, so 8.1's relation does not apply. No Dedekind series of μ₂/λ₂ exists, so whether (U-S) correlates with the ratio is untested.
• Smallest informative check (for numerics, if wanted): μ₂/λ₂ and λ₁^odd/λ₁ for Dedekind ζ_{Q(√−23)} and Epstein on c ∈ [10, 29] at matched N. If Dedekind tracks ζ's 0.64 while Epstein's ratio drifts well before its crossing, the ratio is a (U-S)-sensitive observable. That would still not be a mechanism.

Status (ALIGNMENT s5).

• Attempt unresolved. It is not obstructed: no theorem found rules out a (U-S) mechanism.
• Paused by allocation.
• Grades:
• 8.1: derivation to leading order (μ₃ ≫ μ₂ assumed), with measured agreement to six digits;
• 8.2: identity from CCM/CvS (not new) plus an ordinary-argument bound; the saturation is measured;
• 8.3's negative: a scoped statement about what was searched in the time allotted, not a proof.

A correction this round forced. §6's claim that e₂ is orthogonal to the pole to about 1e−63 was a precision artifact (§6, corrected). The true overlap scales like μ₂, as u = (A°)^{−1}c forces, so it offered no support for a near-radical reading.

9. Threads (observations, not pursued)

• Boundary mass. At fixed N the ratio φ_N(0)²/λ_N stays within 11-24 for ζ over λ from 1e−8 to 1e−60 (check G). §5 shows it is the log-rate of λ. A Hadamard-type domain derivative for the Weil operator, whose principal part is half the logarithmic Laplacian (Suzuki (4.6)), would explain why κμ₀² is N-stable while κ and μ₀ are not. The continuum trace should be a log-weighted boundary quantity. Not derived here, and no log-Laplacian source was read.
• Krein-Langer continuation. Suzuki identifies g_ζ as a screw function exactly under RH. Window positivity is then the Krein-Langer continuation problem from an interval, whose continuous Schur-parameter description is the natural home for any propagation law. Not worked out here.
• Bombieri's window-independent inertia (Thm 8) in finite zero models, and the fake-zero critical window (§13), are the closest existing results to "the ground state at ℓ vs ℓ + δ". The DH lattice curve first_neg_N(c) in weil_trunc s8.2 is a measured instance of §13's t_c.
• Yoshida 1992 unread. The primary text should be read before any claim that leans on the exact form of his result.

10. Grading summary

• Proposition M, its Corollary, and the collar criterion: ordinary arguments, unreviewed. They rely on standard theorems: Beurling-Deny, positivity-improving semigroups of irreducible Dirichlet forms, rank-one interlacing.
• Kink formula at finite N: ordinary argument plus a measured agreement to 1e−7 or better. The continuum version is a derivation under stated assumptions (simple ground state, continuity at the edges).
• DH arithmetic (Λ_f(5m) = 0, Λ_f(31) = log 31, zero edge weight): exact, numerically confirmed.
• All eigenvalue data in §2.6 and §3: measured at float grade, except the DH signs at c = 30, 31, which are hardened (lab record).
• C1 refuted at every N = 64 … 256 (hardened inputs, the numerics worker's brackets); continuum refutation conditional on ℓ* > log 29.
• C2 as a positivity/propagation mechanism: refuted by Epstein (1,1,6), a rival satisfying its hypotheses. Hardened negativity by numerics (3a799d8); Λ_Q ≥ 0 exact here (check J). Proposition M (simplicity) is a separate derivation, unreviewed, and is not affected. Which of (a)/(b) fails for Epstein: (a) in the even sector and the odd pole capacity in the odd sector (hardened inertia, numerics 3b76739; attribution by derivation).
• C3 refuted at norm scale (measured inputs). The boundary-mass variant is a reformulation (§5).
• C4 (§7): the separating step (U-S) is hardened for n ≤ 200 only: two independent exact implementations, check K here (n ≤ 60) and numerics' us_check.py (8449c29, n ≤ 200), agreeing on the overlap. The C4 implication is not raised by this. It remains an unresolved attempt, not DH- or battery-tested, because no available rival satisfies (U-S).
• C4, other parts: the separating step's role (§7.1) is ordinary argument. The collar-flattening negative and the finite-place factorization are ordinary arguments, unreviewed. The candidate is the Connes-Consani program, not original. Attempt unresolved, paused by allocation. No rival satisfying its hypotheses exists in the tree.
• §8: λ₁ = −Φ/(2‖u‖²), φ₁ = u/‖u‖ and μ₂/λ₂ = 1 − ⟨φ₁,e₂⟩² are derivations to leading order, measured to six digits at two cells. The commutator identity is CCM/CvS; the odd-margin bound on μ₀ is an ordinary argument, measured to be saturated. Answer to "does (U-S) pin μ₂/λ₂ or bound the boundary trace?": no mechanism found; attempt unresolved.
• §6: the ground-state representation is an ordinary argument. The comparison-bound shortfall is measured. The earlier orthogonality 2⟨c, e₂⟩² ≈ 1e−63 was a precision artifact, corrected by check L: ≈ 0.2 μ₂². The near-radical identification is an open hypothesis with no measured support.
• Originality: the Levy-Markov decomposition of the full Weil form and the kink formula were produced here. Novelty was searched only in the 10 sources above plus one web search, which found no match.

11. Reproduction

.venv/bin/python hunts/weil_propagation/theory/checks.py # all checks A-L, about 3 min, writes checks.json .venv/bin/python hunts/weil_propagation/theory/checks.py J K # only the named checks, merged into checks.json

• Check J: the Epstein (1,1,6) Λ_Q signs. check_J computes each Λ_Q(n) exactly as a rational combination of log p, from zeta.epstein.epstein_representation_count, and encloses the sign with mpmath.iv.
• Check K: the separating step (U-S) for Dedekind ζ_{Q(√−23)} and Epstein, in exact arithmetic.
• Runtime and dependencies: both run in about one second and use no Galerkin assembly.
• External inputs: the numerics worker's numbers quoted in §§3-5 and §7 are read from their branch teal-sea/weil-propagation (commits 75865a7 and 3a799d8), not recomputed here.