Level 2: projective gap consistency and the anti-duplication law

Disposition

Level 2 of the hierarchy in INTERACTION-CONTROL-REPORT.md asks for state retaining one off-line pair marked against two consecutive on-line gaps, an exact object giving projective consistency between one-gap marginals and marked two-gap words, and a kill control that duplicates one off-line pair independently in overlapping cells. All three are delivered here, and they repair the two defects SIGNED-INCIDENCE-LAW.md recorded against itself:

• level 1's per-cell envelope was n-extensive — n cells could each sit at the floor, so one off-line pair's budgeted negativity grew with the number of on-line zeros;
• level 1's exclusion used the exactness of the family's declarations, so an epsilon-perturbed family was untouched.

Both had one cause: level 1 used only the diagonal of LAW D. Used off-diagonal, LAW D says every incidence value is one fixed analytic kernel evaluated at an argument the configuration's gaps determine — so a declared correlation is a declared gap, and a marked word cannot choose its own outer value. The resulting caps are declaration-free, which is what makes the exclusion robust.

As before: no proportion moves, nothing here is evidence about RH, and the laws hold for every conjugate-closed multiset in the open strip with bounded local density (the Davenport–Heilbronn zero included, checked below).

Pinned inputs

Unchanged from the level-1 report: paper PDF SHA-256 6792988e6cd0e17690621ce898abd5d534f98407741bc7cb14bbe7d07c77d72f; the §2.2 window; the critical Gabor grid; the transpose summand; normalisation aL^2. Two further inputs are used, both already inside the paper's own toolbox:

• (2.17), the majorant |Phi2(r)| <= psi(r) = min(L, 2/|r|, c_rho/(w r^2)). For this hunt's septic ramp c_rho = 4||rho'||_inf + 4||rho''||_1 = 105/4 exactly (rho'(x) = 140 x^3 (1-x)^3, so ||rho'||_inf = 35/16 at x = 1/2 and ||rho''||_1 = 2 rho'(1/2) = 35/8).
• the unconditional local zero-density bound N(t+1) - N(t) <= A_0 log(t+3) (classical, Titchmarsh Theorem 9.2) — the same input the paper's own Proposition 4.2 uses for its tail. Write nu for the max number of distinct on-line zeros in a unit interval, so nu <= A_0 log T. A_0 is left symbolic; this hunt does not pin its value.

The two laws

LAW G (projective gap consistency). Put omega(g) := Phi2(g)/Phi2(0). On the full grid the normalised on-line correlation of two zeros separated by g is exactly omega(g), with

omega(0) = 1,      |omega(g)| < 1  for g != 0,

the second because phi^2 >= 0 is continuous with interval support. Hence:

• a correlation is a gap. Demanding correlation >= c caps the gap: measured at L = 8, c = 0.99 buys g <= 0.0719, c = 0.9 buys g <= 0.2305, c = 0.5 buys g <= 0.5557.
• a marked two-gap word is not free. For three consecutive on-line zeros the outer correlation must be omega(g1 + g2) — not any independently chosen value. The natural independent guess omega(g1) omega(g2) is refuted with residual 0.13228305 at (g1, g2) = (0.7, 1.3) (omega(2.0) = +0.0710 against the product -0.0613).

LAW H (anti-duplication caps). The kernel decays by (2.17) while local density is capped by nu, so the total incidence mass of any single point against a whole configuration is bounded independently of that configuration's size:

on-line:  sum_{j != i} omega(x_i - x_j)^2  <=  kappa(nu)
             = 2 nu [ 1 + sum_{k>=1} (psi(k)/aL)^2 ]
on/off :  sum_j |Bhat(x_j, z)|^2           <=  kappa_cross(nu, y)
             = 2 nu [ E(y)^2 + sum_{k>=1} (psi_y(k)/aL)^2 ],

where E(y) = Phi2(iy)/Phi2(0) is the depth inflation and psi_y(r) = min(aL E(y), (c_rho/w) cosh(yL/2)/r^2) (near field by mass, tail by two integrations by parts). Consequently the signed aggregate interaction of one off-line pair with all on-line zeros obeys

sum_j 2 Re( Bhat(x_j, z)^2 )  >=  -2 kappa_cross(nu, y),

an n-independent floor where level 1 had -2 n sigma^2(y). One pair cannot be spent twice. Measured at L = 8: kappa(1) = 2.2750, kappa(3) = 6.8251, kappa_cross(3, 0.3) = 31.0360.

The kill control, run

The cheat the hierarchy names: spend one off-line pair independently in n overlapping cells, each at level 1's per-cell floor -2 sigma^2(y). Every cell is individually legal, so level 1 accepts the total; level 2's aggregate floor does not move with n.

Stated rather than hidden: the improvement is asymptotic in n, not universal. The crossover here is n > kappa_cross/sigma^2 = 72.0; below it level 2 adds nothing. This is the right place for it to bite, because the family's defeat of the recovery coefficient requires n -> infinity.

The robust exclusion

The family's on-line block has R(P) = n(n-1): every pair of its n on-line labels is fully correlated. LAW G says full correlation means zero gap, so that cross mass is available only from a single point of multiplicity n — which the paper charges to the index side at the flat cost 4, not to the rank side as n simple zeros. Measured, with n = 50:

LAW H makes this declaration-free. Since R(P) <= n kappa(nu) for any real configuration,

n - 1  <=  kappa(nu)  =  2 nu [1 + S(L)]      hence   n = O(log T),

using nu <= A_0 log T. No perturbation of the family's declared numbers evades this: it bounds the quantity, not the declaration. That is the epsilon-robust exclusion level 1 could not give.

The bound is a function of nu and is honest about it — at L = 8, nu = 3 excludes every m >= 2 (m = 2 needs 9 per label against a cap of 6.83; m = 10 needs 201, an excess of 29.5x), while nu = 8 excludes m = 10 (11.0x) but not m = 2. The correct reading is not "the family is dead at every size" but: at each height T there is an explicit finite bound on how large a family member can be, where the defeat needed unbounded m at fixed T. Members with m = 1 (n = 4) are excluded by neither cap and need not be: they give theta <= 3/(n-1) = 1, which is no defeat at all.

The residue

The family forced theta <= 3/(n-1) and drove it to zero by taking n large. Level 1 capped the realisable members through depth, leaving a floor exponentially small in the bandwidth. Level 2 caps them through density:

theta  >=  3 / kappa(nu)          (this family shape only)

Exponential in L becomes polynomial: ~ e^{-L/2} becomes ~ 1/log T. The two floors are independent, so the larger one holds; level 1 remains the better statement at small bandwidth, which the table shows rather than hides.

This is a statement about what this adversary family can obstruct, not a proof that any particular theta is admissible. Establishing an actual strengthened inequality is level 3's telescoping local-potential task and is not attempted here.

Controls ledger

Reproduction

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

Roughly 50 s and 15 s respectively. Bulk scans run in float against the closed forms, cross-checked against the mpmath path as the first control; the family arithmetic is exact.

What level 3 needs

The hierarchy's next row asks for marked k-gap words with overlap constraints and a local potential inequality whose boundary terms telescope, killed by periodic obstruction words and the direct-sum family. LAW G supplies the missing ingredient it was waiting on: gap words are now constrained objects rather than free declarations, and consecutive words overlap in a determined way. The open question is whether a potential V(g) exists with 2 Re(Bhat^2) bounded below by a telescoping difference along the ordered configuration — which would convert these per-point caps into a genuine additive floor, the thing an actual strengthened rank–trace inequality would need.