Marked two-gap consistency: the coexistence question, answered

The immediate question, first

Can one actual off-line pair simultaneously realise the worst signed incidence allowed by level 1 across two consecutive on-line gaps?

No — and for a sharper reason than incompatible offsets: the level-1 envelope is not attained in even one cell.

On the full grid the signed cell value of an on-line zero at offset g from a pair of depth y is W(g,y) = 2 Re(Phi2(g+iy)^2)/(aL)^2. Writing Phi2(g+iy) = C - iS and using the parity of phi^2, the two mixed-parity integrands vanish and

C(g,y) = int phi^2(u) cos(gu) cosh(yu) du      (even in g, even in y)
S(g,y) = int phi^2(u) sin(gu) sinh(yu) du      (odd  in g, odd  in y)
W(g,y) = 2 (C^2 - S^2) / (aL)^2                (even in g, even in y)

Level 1's floor came from |S| <= aL sigma(y) by Cauchy–Schwarz on int (phi sin gu)(phi sinh yu). Saturating W = -2 sigma^2 needs C = 0 and Cauchy–Schwarz equality, i.e. sin(gu) = lambda sinh(yu) for all u in the support. Matching Taylor coefficients at u = 0: first order forces lambda = g/y, third order forces -g^3/6 = lambda y^3/6 = g y^2/6, hence

g (g^2 + y^2) = 0.

With y > 0 that gives g = 0 — and at g = 0 we have S = 0, so W(0,y) = 2C^2/(aL)^2 >= 0, which misses the negative floor outright (measured W(0, 0.3) = +2.8046 against a floor of -0.8623). So no cell ever sits at the level-1 floor, and the two-cell question is moot before it is asked. The measured proportionality defect stays bounded away from zero at every offset, including in the g -> 0 limit, where it tends to the defect between u and sinh(yu) rather than to zero.

Phase 1: the exact identities

All four hold exactly, from the decomposition above:

1. Translation covariance. W depends on the difference g = x - t alone, never on absolute position.
2. rho <-> 1 - conj(rho) symmetry. W is even in y (C is even, S is odd, and only their squares appear).
3. Shared kernel column. Both adjacent gaps read the same function W(., y) at arguments differing by exactly the gap — this is what "projective consistency" amounts to, and it is an identity, not a constraint to be imposed.
4. Cell parity. W is even in g, so a cell depends only on |offset|. This single fact is what makes the coexistence question collapse (below).

LAW I — the one-cell tightening (proved, uniform)

Sharpening the same Cauchy–Schwarz against sin^2 rather than 1:

S^2 <= (int phi^2 sin^2(gu) du)(int phi^2 sinh^2(yu) du)
     = [(aL - Phi2(2g))/2] * aL sigma^2(y),

so with the single window constant m0 := - min_r omega(r) >= 0, where omega(r) = Phi2(r)/Phi2(0):

W(g,y)  >=  -sigma^2(y) (1 - omega(2g))  >=  -(1 + m0) sigma^2(y).

Level 1 had -2 sigma^2(y). For this window m0 = 0.2137172540 (attained at r = 1.31762587), so every cell carries a uniform factor 2/(1+m0) = 1.6478 of slack against the level-1 envelope, at every offset and every depth. Measured against the proved cap (1+m0)/2 = 0.6069:

r(y) rises monotonically from 0.2979 toward 0.4786 at the strip edge — always inside the proved cap, and always far from level 1's implicit 1. The pointwise form -sigma^2 (1 - omega(2g)) is retained separately because it is sharper than the constant cap wherever omega(2g) > -m0.

LAW J — the periodic word (exact, and it answers a Phase-4 lesion)

If the on-line zeros sit at the critical grid tau_k = t + k h, h = 2pi/L, Poisson summation applies to F(r) = Phi2(r+iy)^2, whose density (phi^2 e^{-y.}) * (phi^2 e^{-y.}) is supported in [-L, L] and vanishes at both endpoints, so only the m = 0 dual term survives:

sum_{k in Z} W(t + k h, y)  =  2 b / a^2,

with a = (1/L) int phi^2 and b = (1/L) int phi^4 — the paper's own (2.14) constants. The value is positive and independent of both the depth and the ordinate: measured 2.2959062623 against every tested (y, offset) to better than 6e-8. A periodic near-obstruction word at critical spacing — the lesion the hierarchy names — yields an off-line pair exactly no negativity, at any depth. Breaking the spacing by 65/64 moves the total to 2.2605846, a defect of 3.5e-2, so the identity is genuinely a property of critical spacing.

Phases 2–3: coexistence by itself is empty — OUTCOME B

The directive's Delta is identically zero on a nonempty set, by two exact constructions:

The mirror pair. W is even in g, so its minimum is attained at both +g* and -g*. Two on-line zeros separated by exactly s = 2 g*(y) therefore both sit at the single-cell minimum, and the two-cell penalty is exactly zero:

The cluster. On-line offsets need only be distinct, so n zeros packed at spacing delta near g* drive the penalty per cell to zero: measured 5.4e-2 at delta = 0.1, 5.6e-6 at delta = 1e-3 (n = 3), and 7.0e-5 at delta = 1e-3 (n = 10).

So marked two-gap projective consistency, on its own, yields no positive penalty. Delta is positive for generic gaps (e.g. +0.507 at s = 0.4, +0.360 at s = 3.0, depth 0.3) but has exact zeros, so no uniform bound follows. Per the directive this is a clean result, not a failure — and it is the reason no LP was built: a projective-consistency program whose value is known to be zero on an explicit family would be an expensive way to rediscover these two constructions.

OUTCOME C — the escaping family, pinned exactly

The family that survives every level-1 law and all two-gap consistency is

on-line zeros clustered at offsets  +-g*(y) + O(delta),   delta -> 0,

i.e. a near-multiple zero sitting at the kernel's optimal offset. It is excluded by neither gap geometry nor the signed envelopes; it is controlled only by density and multiplicity — the paper charges a multiple on-line point to the index side at the flat cost 4, and gap_consistency.py's LAW H caps the cluster's size by nu <= A_0 log T. Imposing a separation delta does restore a positive penalty (separation_penalty: >0.1 per cell at delta = 0.5, <1e-4 at delta = 1e-3), which is the honest statement of what the coupling is worth: consistency plus density has force; consistency alone does not.

This family is the natural level-3 kill control.

Phase 5 — epsilon robustness

LAW I gives it directly, uniformly, and without any coexistence input: no cell is ever within (1 - m0) sigma^2(y) of the level-1 floor. Hence a configuration declaring n cells within epsilon of that floor is excluded outright for epsilon < (1 - m0) sigma^2(y), and otherwise pays at least

n [ (1 - m0) sigma^2(y) - epsilon ]     -- linear in n, uniform in placement.

Measured bands: 0.031255 at y = 0.1 (floor -0.079502), 0.339001 at y = 0.3 (floor -0.862287), 1.276768 at y = 0.49 (floor -3.247605).

Phase 6 gate — NOT passed, deliberately

No new proportion is computed and no decimal search is opened. LAW I tightens the envelope, not a proportion; converting an envelope into a proportion needs the additive floor that level 3's telescoping potential would supply, and the coexistence route that could have produced one is the route that collapsed. Both gate conditions fail, so the gate holds.

Controls ledger

Reproduction

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

About 25 s and 28 s. Optimisations are deterministic (grid scan, then golden section); no random search enters any reported number.

What level 3 needs, restated after this

The coexistence route is closed as a source of penalties by itself, so level 3 should not re-derive it. The two live objects this session leaves:

1. The telescoping potential, now with a sharper input: LAW I's pointwise form -sigma^2(1 - omega(2g)) is a function of the gap, which is exactly the shape a telescoping argument consumes.
2. The +-g* cluster as kill control. Any level-3 mechanism must either exclude it or explain why the density/multiplicity charge already does — and must be run against it before anything else.