Level 7: the v-cell joint cap, the ladder correction, and the first reading

Disposition (three results, stated in the order they matter)

1. The joint cap exists and closes the entire swept density axis. The named theorem object of level 6b — an on-line-configuration-free upper bound on the joint profit against a pair cluster — is delivered twice over in v_certificate.py: the v-cell route (the spectral laws below) and the direct route, the level-4 chain counting dual on the damage field summed over pair centres with the positive part taken of the JOINT field. The final cap is the min of the two. At the level-6b pinch (nu_p = 1.5, y = 0.49, 15 pairs) the direct cap is 14.78 against a budget of 41.11: margin +26.33 (v-route: +3.75; greedy adversary: 5.00). Every lattice density nu_p in [0.75, 3], every mixed-depth battery shape, and every short-span cluster tested now closes — there is no uncovered configuration in the sweep. The verdict quality is measured (double-precision grids with one-sided structure); the arb pass is the named hardening.

1. The level-6a full-ladder claim is corrected, not extended. The full-ladder hardened penta scan (full_ladder_scan.py) — the compute level 6a deferred — FAILS between the probe depths: the measured optimal-split eta has a shallow spike (eta(0.02) = 0.4066 against a plateau of ~0.30) that the seven-point probe grid could not see, and honest cell widths dilate the caps. theta_full = 0.02 via the per-depth linear assembly is therefore withdrawn as a full-strip statement; what survives is the probe-grid statement plus the joint accounting, which does not use eta at all. First failed cell and the full scan record are below.

1. The reconnect's first reading is 0.6725009045 — a candidate, not a claim. With the retention theta_full = 0.02 fed through the one-sided ordered-gap floor (c_u = 5.02e-6 at nu_on = 0.6725007037), the candidate arithmetic gives +2.01e-7 over the pinned Theorem D constant. Every named unproven step is listed in reconnect.py; per the operating rules, no proportion is claimed to have moved.

The two exact laws behind the certificate

LAW N (windowed spectral floor). For n on-line points in an interval of length S and |v| <= pi/S:

|F_on(v)| >= n cos(vS/2),

because each phasor's argument about the window midpoint is within vS/2 of zero. Integrated against K over the floor region this forces on-line spectral mass kappa_00 n^2, so the internal mass obeys R >= kappa_00 n^2 - n: density is self-defeating pointwise in v — the fourth appearance of the self-defeat pattern (depth at level 3, on-line packing at level 4, pair stacking at level 6b, and now spectral concentration). Machine check: worst margin +1.07e-4 over 200 random windowed configurations (it is a proved inequality; the check is a typo control).

The safe cell. For on-line and pair ordinates in a common span-S window and |v| <= pi/(2S), arg(F_on conj(F_p)) lies in [-vS, vS], so the cross integrand is pointwise >= 0: the band where a dense pair cluster concentrates its spectral mass cannot be attacked. This is level 6b's chi collapse (0.15 -> 0.017) as a pointwise mechanism.

LAW M under the taper. int W_I dg = 2 * 4 pi b/(a^2 L) at every depth and every taper (only v = 0 survives the g-integral): the mean of the low-pass damage field is not attackable, only its band-limited ripple is. Measured: 3.6064 for v1 = 0.4 and 0.8 alike (2 x LAW M = 3.6064).

The cap, term by term

On-line zeros are split by distance from the pair window (near / mid / far; R >= R_near + R_mid + R_far drops only nonnegative terms), and the near cross by a smooth taper m_I + m_II = 1 (m_I = 1 on [0, v1/2], 0 beyond v1):

J_cap = DP_I zone I: chain DP on the low-pass field, charge rho (1-theta) K_delta

• M_II / ((1-rho)(1-theta)) zone II: per-cell Cauchy-Schwarz, telescoped to one pair-mass number
• (1-rho)(1-theta)/(4 kappa_00) the LAW N leak cap
• DP_mid + err_far measured mid field; psi_S tail

minimised over the (G_near, v1, rho) ladder — every choice one-sided, so the min is. At the pinch (v1 = 1.4, rho = 0.02, S = 13.33):

dp_I 0.58 + M_II 25.62 + leak 7.27 + dp_mid 3.22 + far 0.68 = 37.36 budget = slack 511.26 + T -470.16 = 41.11 margin +3.75

(the v-route's pinch verdict; the direct route reaches 14.78 there, and the final cap is the min of the two)

The pinch's spectral anatomy (from fourier_bridge.pinch_spectrum): the pair deficit lives at |v| in [3, 7] (peak -176 per bin), the mean at v ~ 0 (+59), and the certificate's accounting matches: the dense cluster's attackable mass M_II is 25.8 of a total pair mass of ~101 — the other ~75 hides in the coherent band where the safe cell and LAW M defend it.

Controls

The naive pointwise form is dead, and that record is kept

The directive's literal object — a fixed budget d(v) with |F_on + F_p|^2 - theta |F_on|^2 >= -d(v) pointwise for every admissible configuration — is defeated by the sub-critical on-line lattice (spacing tau < 2 pi / L = h, the critical spacing of LAW J): its nonzero spectral spikes leave the kernel band entirely, so on essentially all of (0, L] the diagonal-subtracted integrand sits near -(1-theta) N while any admissible budget is configuration-independent. The escaping v-band is the whole band minus the v ~ 0 spike, and the deficit grows linearly in N (pointwise_form_obstruction; the aliasing family of level 1, again). The delivered certificate survives the same family through LAW N: the concentration that empties the mid-band fills the floor region, and the n - kappa_00 n^2 credit caps the leak at (1-theta)/(4 kappa_00). The lesson is structural: the v-cell budget must be an integral-per-cell object with a spectral floor, not a pointwise envelope.

The sweep (joint verdicts, on-line-configuration-free)

Budget = sum slack + T_signed (exact); cap = min(direct, v-route); margins absolute. Every configuration closes, all via the direct route.

The v-route's own margins (closing nu_p >= 1.45, open below) and the separable margins (closing nu_p <= 1.05) are retained in the audit log as the two independent cross-checks; the direct route dominates both everywhere tested.

The arb pass (hardened_direct.py)

Gate-1 discipline applied to the direct dual at the binding configurations. The interval trap is recorded in the module docstring: a naive interval-argument Phi2 evaluation amplifies input radii by ~1.8e5 (the ramp^2 integer coefficients cancel analytically; interval radii add), so the hardened field uses the mean-value form — tight point balls for W and W' (Phi2' via index-shifted moment enclosures, radius ~2e-16, checked against finite differences) plus the global curvature blanket |W''| <= 2 L^2 E(y)^2, whose r^2 remainder is ~1e-3 per pair per 0.005-cell. Charges, DP inflation (1 + 1e-9), psi_S tail, slack and T_signed are all directed as at gate 1 (budget downward, cap upward).

The hardened caps are tighter than the float ones (fine 0.005 vs 0.0125: the resolution gain outweighs the directed inflation) — the opposite of the level-5 resolution scare, and a sign the float grid was the loose part, not the mathematics. The rest of the sweep remains float-measured; hardening it end to end is mechanical repetition of this pass.

Reading: the v-route closes nu_p >= 1.45, the separable accounting closes nu_p <= 1.05, and the band between them — the seam that was, for one session, a genuine two-sided hole — is closed by the direct route with wide margins (y = 0.49):

Short spans close too: span 6 at nu = 1.5 (+17.76 direct vs -1.09 v-route), span 4 at nu = 1.3 (+10.38). Sparse lattices close directly as well (nu = 0.75, y = 0.49: +63.68), so the direct route alone covers the axis; the v-route and the separable caps stand as independent cross-checks.

Why the direct route was almost missed, and what the seam taught. The direct object — "the level-4 machinery with the damage field summed over pair centres", named verbatim at level 6b — is tight only if the positive part is taken of the JOINT field: clipping per pair discards the coincident-pair shielding (a deep pair's own +2E(y)^2 hump protecting its neighbour's negative band) and inflates the field by an order of magnitude (at nu = 1.3, g = 4.65: joint field 0, per-pair-clipped field

4). That near-miss is the separable accounting's ghost, and it is now

a permanent control. The seam diagnosis (the v-route's Cauchy-Schwarz granting anti-alignment against the band-edge spike shoulder in v in [5, 7]) stands as the record of why the v-route is loose there — the mutual exclusion that protects the lattice is g-local shielding, visible to the direct field and invisible to band-wise moduli.

The extended greedy (sites to +-8 beyond the window, up to 100 zeros, min spacing 1/16) extracts 7.23 at nu = 1.3 against the direct cap 18.18 and budget 35.11: adversary, cap and budget are now in the right order with honest daylight at every tested density, and the control is a permanent test.

Instrument v2. After the first audit, the mid-zone cell margins were made per-cell (the blanket global-slope margin over ~500 cells was most of dp_mid at seam-scale budgets: 6.5 -> 3.2 at the pinch), and the direct route was added. The sweep table above and the audit log reflect the v2 instrument.

The full-ladder scan record (level 6a corrected)

220 depth cells (ratio 1.05 shallow, 1.012 from y = 0.09), hardened EnclosedPenta caps at theta = 0.02, eta measured-optimal on a ten-point grid, bracketed one-sided per cell. Verdict: OPEN — 45 cells fail, in two bands:

shallow y in [0.0095, 0.0501] margins -0.017 .. -0.056 (worst -0.0557 at [0.0293, 0.0307]; eta 0.37-0.41 vs 0.30) deep y in [0.4193, 0.4724] margins -0.002 .. -0.012 (cap/slack 0.699-0.709 + eta 0.303 at honest 1.2% cells)

Every other cell passes, with margins +0.02 (mid-deep) to +0.36 (y < 0.01, where eta = 0). The level-6a statement "theta_full = 0.02, hardened at all seven probe depths" was resolution-fragile in both bands: the probe grid could not see the shallow eta spike, and the deep probes (1%-wide cells, margin 9e-4 of slack) do not survive honest cell widths. theta_full via the per-depth linear assembly is accordingly reduced to the passing bands; the shallow failing band is covered at the joint level (shallow sandwich rows above), and the deep failing band coincides with the seam depths, where coverage is the certificate's (nu >= 1.5, span >= 10) plus the separable regime (nu <= 1.0) — the same named hole.

The eta spike mechanism: a shallow pair (y ~ 0.02) riding a deep lattice takes its split share of strong T-negativity against a slack of only 8 sigma^2 + 8 sigma^4 ~ 0.012; the optimal reweighting cannot push the ratio below ~0.41 because the deep partners' own budgets are nearly saturated. The joint accounting has no such bookkeeping: the shallow layer's spectral mass is coherent (cosh(0.02 v) ~ 1) and hides at v ~ 0. Joint verdict on the failing shape (shallow sandwich, y = 0.02 + 0.49): margin +169.85 at nu_p = 1.5 and +28.61 at nu_p = 1.0 (both close; at nu_p = 1.0 the separable accounting also holds it at +1.67 per slack). The eta band is a bookkeeping failure of the linear per-depth assembly, not a failure of theta_full's joint content on the tested shapes.

The reconnect: the first reading of the decimal

The 2026 clean kill established that the withdrawn 0.672529 chain needed exactly one missing input: an unconditional control on the hyperbolic off-line blocks. theta_full is that input, within its labels. The candidate arithmetic (reconnect.py, all floor minima one-sided, ladder converged at n = 600001 = 2400001):

c_u (one-sided, nu_on = 0.6725007037) = 0.00000502 candidate = 0.6725007037 + 2 * 0.02 * 0.00000502 = 0.6725009045 first reading: +2.01e-7 against the pinned constant

Controls: the hardened floor at CG's inputs is 0.00011287 <= CG's printed 0.00012636 (one-sided as it must be); the lambda_2 -> 2 lambda_1 lesion kills the floor exactly.

Named unproven steps, in severity order (the reading is a candidate until all close):

1. The transplant lemma: the paper's constant is linear in the retained on-line cross mass with coefficient 1, including the normalization match between the grid kernel omega^2 and the MT kernel g. This is the withdrawn chain's plumbing, audited at every gate except the one theta_full now supplies — but it has never been proved, only not refuted.
2. theta_full's own labels: the v-certificate is quadrature-measured (arb pass named); pair-side placement freeness rests on the swept families plus the stacking floor; the sparse regime rests on the level-6a caps whose full-ladder statement is the corrected one above.
3. Taper/truncation of the upstream count restated one-sided at the composed value.

Reproduction

.venv/bin/python hunts/frontier_math/v_certificate.py      # ~30 min
.venv/bin/python hunts/frontier_math/full_ladder_scan.py   # ~45 min
.venv/bin/python hunts/frontier_math/reconnect.py          # ~2 min
.venv/bin/python -m pytest -q -o addopts='' \
    hunts/frontier_math/test_v_certificate.py              # ~4 min