The signed on/off incidence law: level 1 of the hierarchy, delivered

Disposition

INTERACTION-CONTROL-REPORT.md closed with: "No decimal search should run before level 1 below has an unconditional zeta constraint that excludes the obstruction family." This report delivers that constraint. Three signed incidence laws — a diagonal law, a subgrid envelope, and a depth floor with a strip cap — hold unconditionally for the actual zeros of zeta read through the pinned paper's own window/grid interface, and every member of the exact obstruction family (both the scalar family and its moment-matched dilution) violates at least one of them at every bandwidth and every placement, with exact integer margins where the violation is sharpest.

Nothing here is evidence about RH, and no proportion moves. The laws hold for any conjugate-closed multiset in the open strip — the Davenport–Heilbronn off-line zero satisfies them too, as a structural lemma must (a lemma failing on the rival would be a bug, and passing distinguishes nothing). What changes is the adversary's freedom: the family that forced every universal recovery coefficient to zero is not realisable as incidence data of anything living in the strip.

Pinned inputs

• Paper PDF SHA-256: 6792988e6cd0e17690621ce898abd5d534f98407741bc7cb14bbe7d07c77d72f (re-downloaded this session; hash matches the one pinned by the two earlier audits).
• Local state at the start of this session: c35dc04.
• The paper's interface, as used here: window phi of its §2.2 (flat top, C³ ramps, supp phi = [-L/2, L/2], 1 <= w <= L/8), critical Gabor grid tau_k at spacing h = 2pi/L, evaluation vectors u_rho = (phihat(gamma_rho - tau_k))_k, summands m_rho u u^T (transpose, not conjugate transpose — the corrected reading that CLEAN-KILL-REPORT.md pinned), normalisation aL^2 (its (4.4)).
• Family under exclusion: interaction_obstruction.ScalarFamily(m) (one off-line pair with evaluation scalars im, -im against n = 2m^2 + 2 mutually-unit on-line labels) and MomentMatchedFamily(m, k, ·) (same bad block plus trace-repairing positive pairs and background dilution).

The three laws

Write Phi2 for the Fourier transform of phi^2, aL = Phi2(0), and for a point z = t - iy of the open strip (depth y = beta - 1/2, |y| < 1/2 unconditionally) let Bhat(z, z') = (aL^2)^{-1} sum_k phihat(z-tau_k) phihat(z'-tau_k) denote normalised bilinear incidence.

LAW D (diagonal law). For all complex z, z', the full-grid kernel is alias-free:

sum_{k in Z} phihat(z - tau_k) phihat(z' - tau_k) = L * Phi2(z - z').

The paper states its Lemma 2.2 for real arguments; the proof — Poisson summation plus the support observation that the only surviving dual frequency is 0, because H = phi_z * phi_{z'} is supported in [-L, L] and vanishes at the endpoints — never uses reality, and the paper's own Appendix B records numerical checks of the identity "at both real and complex arguments". The consequence it never draws: taking z' = z,

sum_{k in Z} phihat(z - tau_k)^2 = a L^2     for EVERY z in C.

The bilinear self-incidence of every zero equals +m_rho per unit multiplicity in the paper's normalisation — at every depth, at every position. (The quantity that explodes like X^{|2beta-1|} for deep pairs — the paper's Remark 5.10 — is the Hermitian mass sum |phihat|^2 = L*Phi2(2iy); the bilinear square is rigid.) A pair {rho, 1-conj(rho)} therefore contributes bilinear trace exactly +2 m_rho, like two on-line zeros. The paper's §6 says explicitly that "nothing is assumed about the traces of the individual pair blocks"; LAW D is that unretained datum, pinned.

LAW E (signed subgrid envelope). The imaginary mass is again alias-free: Im phihat(t - iy - r) as a function of real r is band-limited to [-L/2, L/2] with density phi(u) sinh(yu), so

sum_{k in Z} (Im phihat(t - iy - tau_k))^2
    = L * int phi^2(u) sinh^2(yu) du  =:  a L^2 sigma^2(y),

independent of t. Since every omitted term of LAW D's real part is (Re phihat)^2 >= 0, for EVERY subgrid S (any truncation, any placement) and every real ordinate x:

Re sum_{k in S} phihat(z - tau_k)^2   in  [-aL^2 sigma^2(y), aL^2 (1 + sigma^2(y))]
|Im sum_{k in S} phihat(x - tau_k) phihat(z - tau_k)|  <=  aL^2 sigma(y)

(the second by Cauchy–Schwarz against the two alias-free masses). Hence every normalised on/off cross cell obeys the rational-boundable envelope

2 Re( Bhat(x, z)^2 )  >=  -2 sigma^2(y),

and every pair's truncated bilinear trace is >= -2 m_rho sigma^2(y). This is the "rational lower envelope for 2 Re(B(x,z)^2) on every cell" that the hierarchy's level 1 required, and it is signed in the load-bearing direction: sigma(0) = 0, so negative incidence mass requires depth. On-line data cannot manufacture any of it, at any truncation.

LAW F (depth floor and strip cap). By convexity of sinh through the origin, sigma^2(y) <= sinh^2(yL/2). All zeros of zeta satisfy 0 < beta < 1 unconditionally (classical, de la Vallée Poussin), so y < 1/2 and

sigma(y) < sinh(L/4)      for every zero at bandwidth L.

An off-line pair whose cross column reaches |Im Bhat| = m needs sigma(y) >= m by LAW E, hence depth y >= y_min(m) := inf{y : sigma^2(y) >= m^2} — and is impossible at ANY placement once m >= sigma(1/2^-).

What is from the paper and what is new

From the paper: the window family, the critical grid, the transpose summand, the normalisation, Lemma 2.2 for real arguments, the parenthetical that pair traces are unretained, and the numerical observation (its Appendix B) that Lemma 2.2 holds at complex arguments. New here: the complex statement used structurally; the bilinear/Hermitian split (rigid diagonal vs X^{|2beta-1|} mass); the alias-free imaginary-mass identity; the subgrid two-sided envelope; the depth floor and cap; and the assembly into an exclusion of the pinned obstruction family. All of it is elementary — Poisson summation and support arithmetic — which is the point: it was available to the paper's interface all along and simply not retained.

The exclusion, wall by wall

Every member of the family dies on the first wall it hits; the table lists them in the order they bite. Margins are exact integers or measured values from incidence_law.py.

The moment-matched dilution reuses the identical bad block (MomentMatchedFamily(m, k, ·).bad == ScalarFamily(m), asserted in code), so W1–W4 transfer unchanged; the dilution repaired the aggregate trace identity, and none of these walls is aggregate.

Quantitative residue worth recording: the family's defeat of the recovery coefficient was theta <= 3/(2m^2+1) for every abstract m, driving theta to zero. With LAW E/F, members realisable at bandwidth L have m < sigma(1/2^-), so the defeat available from this family is capped at

theta  >=  3 / (2 sigma^2(1/2^-) + 1)

— measured 0.6729 at L = 8, 2.68e-2 at L = 16, 7.55e-4 at L = 24. Exponentially small in L, but nonzero, explicit, and now the adversary pays depth for every unit of negativity, which is the currency the level-2 mass accounting can charge (below).

Mass constraints, recorded and hedged

The hierarchy's level 1 also asks for "an unconditional mass constraint" on the depth cells. Two classical ones are recorded here for the level-2 LP; they are literature-pinned statements, hedged, not instrumented in this session, and none of the exclusion above uses them.

• Littlewood's lemma plus the unconditional second moment of zeta(1/2+it) gives sum_{T < gamma <= 2T} m_rho (beta - 1/2)^+ << T log log T; in kernel-depth units b = yL at bandwidth L = lambda l this reads sum m_rho b_rho <= (lambda/2)(log log T + O(1)) N(T, 2T) — the average depth per zero is O(log log T) kernel units.
• Zero-density (Ingham's N(sigma, T) << T^{3(1-sigma)/(2-sigma)} log^5 T suffices; sharper is known): cells at fixed depth y >= delta carry O(T^{1-c(delta)}) pairs — power-small mass exactly where LAW E permits the most negativity.

Together with LAW E these say: negative incidence mass requires depth, depth is scarce in aggregate, and fixed depth is power-scarce. That is the shape the level-2 "marked two-gap words with projective consistency" analysis needs.

What is not achieved, stated so it cannot be misread

• No proportion moves. The pinned 0.6725007... is untouched. The laws add retained state to the interface; converting that state into a positive theta for an actual strengthened rank–trace lemma is the open level-2+ task, and theta_recovery_floor decays with L, so nothing here even hints at a constant.
• The exclusion is of the exact pinned family. Epsilon-perturbed families (on-line incidences 1 - delta, pair diagonal -2m^2 + delta) are constrained by W1/W4 but not killed by W2/W3's exactness; a perturbation-robust exclusion is precisely the projective-consistency work of level 2, and pretending otherwise would repeat the mistake that CLEAN-KILL-REPORT.md exists to prevent. (Delivered 2026-08-11: LEVEL2-GAP-CONSISTENCY.md supplies it by bounding the quantity rather than the declaration — R(P) <= n kappa(nu) for any real configuration — and also removes the n-extensivity of the per-cell floor below.)
• The laws do not distinguish zeta from anything. They hold for every conjugate-closed multiset in the open strip (measured on the Davenport–Heilbronn off-line zero, defect 2.7e-14). Gate #3 does not apply because no RH-explaining structure is claimed — but it was checked anyway, in the direction it must hold.

Controls ledger

Reproduction

From the repository root:

.venv/bin/python hunts/frontier_math/incidence_law.py
.venv/bin/python -m pytest -q -o addopts='' \
    hunts/frontier_math/test_incidence_law.py

The first command prints the full audit (~100 s); the second runs the controls (~60 s). The family side of every exclusion check is integer or rational arithmetic; the kernel side is closed-form (the ramp is polynomial, so phihat and Phi2 are finite combinations of sin/cos values) evaluated in mpmath at stated precision with the ladders above.