The census read without a question — the one-block fixed point overshoots every lattice threshold by a fixed amount set by the ring share of the pivotal set; the exponents fan around y₉ and are still falling everywhere

Why this exists. Filed 2026-10-01, the night run 083 was scored. Every carpet run since 051 has been read for the question it was sealed with. This probe is the first reading of the whole table with no question attached: every per-pattern number the record holds, joined across runs 068, 071, 073, 075, 080 and 083, and screened against every other, with the 14 named carpets (whose numbers live in different files from the census's 35) kept back as a hold-out for anything the 35 suggested. The join took a hundred lines of ad-hoc code; that cost is why docs/DATABASE.md is filed beside this.

The table. 49 distinct crossing polynomials with a measured 1/ν (run 080's completed census). For each: C₁, N₁, M, the one-block fixed point p*_map, λ, y₉, the pivotal sum and its ring share (068), the map's g₁ by depth and g_∞ (075), and — for the 35 census-measured members — the lattice threshold p̄ and width w at L = 81, 729, 6561, the two local slopes, the extrapolated p_∞ (068, 071, 080). For the 14 named carpets the same lattice numbers were recomputed from the stored sample files (053, 055, 051, 063) and used only as the hold-out. Every correlation below is Spearman.

What stands

1. The one-block fixed point lies above the lattice threshold on every pattern, at every size — 49 of 49. The gap G = p*_map − p̄ is 0.01 to 0.10 and, on cut 3 and 4 and the unbound cut-2 patterns, does not decay: #16 reads 0.074, 0.073, 0.075 at 81, 729, 6561 and 0.075 extrapolated; the five wall-bound cut-2 patterns (p*_map ≈ 0.93) are the only ones whose gap moves with L, shrinking (0.082 → 0.045 on #38). G is the difference of two L-independent numbers: the fixed point of R_P(p) = p and the lattice's p_c. The sign is the classical bias of a cell-renormalization crossing rule (a block counted open only when crossed is blind to connectivity that runs between blocks), so the sign is borrowed; what is new is what sets the size. Across the 35: ring share +0.83, C₁ −0.79, y₉ −0.81. On the hold-out of 14, ring share +0.83 and C₁ −0.13 (twelve of the fourteen are cut 4); among those twelve, ring share +0.73. So within a cut class the gap is ordered by how much of the crossing's pivotal set sits on the frame. Read with run 083 — the routes the recursion cannot represent are the majority on every lattice and run along block edges — the reading is: the fixed point overshoots by the connectivity the frame carries between blocks, and the ring share measures it.

2. Within cut 4 the lattice exponents read as fanning out with L, carried by three patterns; y₉ is the pivot, not the limit. The spread of the local slope across the 16 cut-4 patterns with sizes is 0.041 over 81→729 and 0.057 over 729→6561. The residual 1/ν − y₉ tracks the direction each pattern is still moving: +0.82 with the drift s₃₄ − s₂₃, +0.85 with s₃₄ itself. Rising ones sit above y₉, falling ones below. Honest size: #60 (+0.043), #63, #10 rise beyond the slope's error; the rest of the fan is at the noise level. (Cut 2: spread flat at 0.034, mean falling 0.32 → 0.28.)

3. The census is still falling. The local slope decreases between 729 and 6561 on 30 of the 35, and on 13 of the 14 named carpets, X's step sitting at the noise (−0.002 by the width, −0.008 by run 073's estimator, +0.005 by the lab database's recomputation): 43 or 44 of 49. With item 1: the threshold has converged on every pattern and the exponent on at most 5 of 49. Every quantity scored on an endpoint 1/ν is scored on a number that reads high census-wide, by an amount that differs per pattern (G4 of the panel gate already says this per case; this is the census-wide count).

4. The ring share alone orders 1/ν at −0.925 over the 49 (y₉: +0.948; λ with its ring part removed: +0.940). Not a new object — it is a reading of y₉ — but a one-number one: the exponent is low when the crossing's bottleneck is the frame.

What died on the hold-out

• The class offset of run 071's frozen rule as a wall effect. Residual against the wall guard (1 − p̄)/w: +0.67 over 35, +0.81 within cut 4. Leave-one-out fits of the residual on 35: class alone rms 0.024, class + ln N₁ 0.023, ln guard 0.027, drift 0.026, p*_map 0.027. Class is the better single predictor; the guard and the drift are its proxies. Dead as a replacement; alive only as item 2's within-class reading.
• "The drift's sign is set by the wall guard." The five risers all have guard above 20 and most fallers sit below; on the hold-out X, X1, H and X2 have guards 22–25 and all fall. Dead.
• "A rung-k fixed point closes the gap in item 1." Said in the look-back, killed by run 084's own stage-1 numbers before it was sealed: the rung-3 lattices' thresholds are 0.822, 0.848, 0.857 on T and 0.795, 0.826 on X by level — rising through the one-block fixed point on T (0.832, passed from the second level) and above it at every level on X (0.740), moving away from the carpet (0.787, 0.722). A rung is a subgraph of the carpet (bonds thinned to k per shared edge), so its threshold is higher, not closer. The ladder is an instrument for the order and the turn (084's clauses), not for the threshold.
• "The gap is the coarse lag." ⟨Δ⟩ = ⟨t − t_coarse⟩ from run 083 is −0.77w on T and −0.84w on X at 729, −0.83w and −0.97w at 6561: the lag scales with the width and vanishes with L, while G is fixed. A correlation of the two across the census would be the C₁ confound renamed. Not sealed. Corrected 2026-10-02 by run 085: the scaling half stands (⟨Δ⟩ ∝ w, slope −0.93, vanishing with L); the confound half was wrong — at L = 729, G_med is ordered by |⟨Δ⟩| at +0.98 over the 49 and +0.93 within cut 4, G ≈ 2.4·|⟨Δ⟩|₇₂₉ through the origin. The gap is read at each size by that size's lag, with a size-dependent constant (about 4.6 at 6561 on T and X, with 27% scatter at 729). Item 1's ring-share reading (+0.88 on the same 49) now has a lattice rival.

• "The gap is a duality defect of the block." Suggested the same night (the square is the null case of item 1: its 2×2 cell renormalization gives p* = 1/2 exactly because the square is self-dual), killed by the reductio before it was sealed. For the block with its left and right terminals, planar duality is an identity, R_G(p) = 1 − R_{G*}(1 − p), so the dual pattern's fixed point is 1 − p*(G) exactly and no block-level defect exists; on the lattice, p_c(carpet) + p_c(dual carpet) = 1 likewise, so the dual family has gap −G identically. What survives is a statement, not a run: G is odd under duality, self-duality forces it to zero, and 49 of 49 positive says every census carpet sits on the same side — the one-block rule undercounts the primal lattice's connectivity and overcounts its dual's. The ring share reads how far.

The combination sweep

Added after run 085 was drafted, on the owner's question about its clause A: which combinations of the table's quantities are nearly constant across patterns? Coefficient of variation over the 35, every ratio and product of 15 quantities. The low ones are definitional (y₉/λ, 0.03; p̄·M, the mean count of open bonds at crossing, 0.07) or restate the bands (1/ν per unit C₁, 0.12). One weak non-trivial hit: ring share / w₇₂₉, CV 0.155 against 0.39 and 0.50 for the parts, mean 24.6 — the width at 729 is about a twenty-fifth of the ring share. Held loosely; both parts order with 1/ν, so a shared dependence can make a ratio look constant. The sweep itself, with a shuffle null, is a saved query in docs/DATABASE.md.

The reductio this probe passed first

By construction: item 1's sign is the known bias of the rule, stated as borrowed; its ordering by ring share is not forced by C₁ (hold-out: −0.13 against +0.83) and not by y₉ within class. Item 2's fan is a spread of sd-level numbers with three patterns carrying it. Negation: if the fixed point's overshoot were a block-level constant of the rule (the same for every pattern), G would be flat across the census; it ranges 0.01–0.10 and orders at +0.83 with one quantity. If the exponents converged to class values the within-class spread would shrink with L; it reads wider, not narrower, with three patterns carrying the difference. What would kill the readings: a reseed of the five risers failing to rise (item 2); a pattern pair with equal ring share and gaps 0.03 apart (item 1); the rung thresholds at k = 7 recovering the carpet's only if k = 7 is the whole edge, which is the definition, not a finding.

An inkling from the owner (2026-10-01), recorded unjudged

From P-045: a block is a piece of a larger ensemble, and when the ensemble is closed the connections cancel the way interior faces cancel in Stokes, Gauss and Helmholtz — what survives is the boundary. On item 1: the one-block fixed point's overshoot G would be the non-cancelling boundary term of a decomposition whose faces are the holes at every scale; it vanishes on the square, where the decomposition closes (self-duality), and the ring share measures the surviving boundary. Not a seal; P-024's chase paragraph carries it beside the codimension candidate.

The choice, read as combinatorics (2026-10-02, owner's question; computed, not an inkling)

The census is not a sample of patterns; it is every pattern the construction rule allows. The 7×7 interior falls into ten D4 orbits — the centre (size 1), three diagonal orbits and three axis orbits (size 4 each), and three generic orbits (size 8). Seventeen removed cells with D4 symmetry means 17 = 1 + 4a + 8b, so the centre is removed in every one of the 63, and (a, b) ∈ {(4, 0), (2, 1), (0, 2)}: C(6,4) + C(6,2)·C(3,1) + C(3,2) = 15 + 45 + 3 = 63. Each pattern is one choice of orbits; the census is the weight-17 slice of the Boolean lattice on nine orbits, and its neighbours are swaps of equal weight (one 4-orbit for another; one 8-orbit for two 4-orbits). The polynomial twins of run 086 are swap-neighbours whose swap the crossing polynomial cannot see (#23/#24 swaps the diagonal-3 orbit for the axis-3 orbit).

The exponent is not additive over the choice. A linear model on the nine orbit indicators predicts 1/ν no better than its spread (leave-one-out rms 0.108 against sd 0.106 over the 49), and predicts y₉ no better either (0.094 against 0.091); y₉ alone gives 0.026, y₉ with class 0.0235. So the map from choice to exponent is rugged on the swap graph — a min over passages, not a sum over cells — and the crossing polynomial is the object that linearizes it: choice → R(p) (combinatorial, non-additive) → 1/ν (near-linear). Per-orbit marginal means say which cells matter most (removing the generic orbit nearest the corner, (1,2), raises the mean 1/ν of the patterns that remove it to 0.50; removing the second diagonal cell lowers it to 0.41), but no sum of them is the exponent. The integer invariants (C₁, N₁, M, the 17) are the first read of the choice; the constant-check sweep found only definitional constants among them (p̄·M, the mean count of open bonds at crossing). Inkling, held loosely: the right combinatorial coordinate is the monotone structure — which passages a choice leaves open — and C₁ is its first Boolean invariant; whether a second exists that the polynomial does not already carry is the eigenbasis question in combinatorial dress.

Pascal and Bernoulli, exactly (2026-10-02, owner's question). The block's threshold in the random bond order has an exact distribution: P(K ≤ k) = a_k / C(M, k), the crossing k-subsets over all k-subsets — the crossing polynomial's coefficients over Pascal's triangle. Computed for all 63 from poly075's a_k: the block's threshold has mean 0.63–0.76, standard deviation 0.085–0.100, skewness −0.74 to −0.05 (left-skewed at generation 1, where every lattice size from 81 up is right-skewed: the skewness changes sign between generation 1 and 2), excess kurtosis −0.32 to +0.55. The null against which all of this is read is the uniform threshold, whose cumulants are the Bernoulli numbers, κ_n = B_n/n (κ₂ = 1/12, g₂ = −6/5): structure concentrates the threshold three times tighter than Bernoulli and flips its tails. The block's mean and width order 1/ν at −0.79 and −0.80 over the 49 (the threshold's job, as p* does); its skewness only at +0.34. Euler–Maclaurin is the third exact entry: the finite-M corrections between the discrete order statistics and their continuous limits carry B₂, B₄, … in powers of 1/M. Analogies beyond these three are not in the record.

What it feeds

• Run 085 (sealed the same night): the coarse-crossing variant across the census at L = 729, with the lag in width units and the tie share as the clauses, and G reported against every per-sample quantity so item 1's reading gets its first look at a mechanism.
• PG-01 gains a line, L8, the threshold gap, with this probe as its lead and run 085 as its first run.
• Run 087, the five risers reseeded, is the test of item 2 (clauses on #60 alone and on the other four's mean); run 086 measures inside four polynomial fibres, the direct test of "how many boundary bits".
• docs/DATABASE.md: the join that produced this table, as a schema, so the next unasked reading is a query.