PG-01 — The carpet's critical behaviour at reachable sizes is set by an exact object the one-block recursion truncates

Rewritten 2026-10-25 (records audit R28_reading/audits/2026-10-25-records.md, fix 6; second reader's corrections from R28_reading/distilled/PG-01-2026-10-25.check.md applied) for the four runs scored since the 2026-10-24 text, which absorbed through 127: 131, 132, 135 and 136, all confirmed. Earlier versions are in git; the record is §2. It cites run files; no saved query set carries it yet. The 36 runs under PG-01 (075–136): confirmed 12, falsified 9, unanticipated 10, withdrawn 5. No run under PG-01 is sealed and open.

Survived. - The crossing polynomial as the ranker. Over 49 lattice members y₉ orders 1/ν at Spearman +0.948, ahead of C₁ (+0.883) and the generator's λ₂ (+0.820) (108). - In cut 2, the orbit where the wall enters the second ring. Midline entry sits above corner and off-axis entry on all 19 cut-2 patterns; of the six new, the nearest pair, #49 0.2922 ± 0.0035 and #42 0.2592 ± 0.0036, is eight errors apart, and y₉ ties #42 and #43 (107). Unanticipated only because the sealed line, 0.290, sat too close to #49. - T's fall in skewness, −0.091 ± 0.022 from 729 to 6561 on eleven files (092). A reading puts T's extra lean at 729 in the first crossing: the second crossing leans less there by −0.103 ± 0.023 and the same at 6561 (−0.0135 ± 0.0261) (109). - The algebra: all 13 cut-2 representatives are two mirror channels in parallel; 34 fibres are indecomposable (093). - Pendant parts separate twin pairs: four one-cell parts lower the threshold by 0.00193 ± 0.00020 on core A and 0.00260 ± 0.00020 on G (098). - The block transfer operator T_op, six scored facts. - What it sees. A single-cell flip changes T_op exactly when the cell lies on a route between two boundary cells: 129 visible, 6 invisible, none misclassified on nine generators the criterion was not formed on, the classes about 13 orders of magnitude apart (122). - Which flips it confuses. Two toggles give the same T_op when the kept-cell graph's network-reliability cores match with every boundary cell fixed (prune dead ends, merge series and parallel links, replace two-ended pockets by single links). On five blocks no operator run had built (20, 35, 36, 39, 44), all 490 comparisons fell on the predicted side: the five mirror-equal cells, (2,3) on each block, and the five equal pairs at 5.6e−17 to 2.2e−16, everything else at 3.03e−3 or more (132). That is the sufficient direction; necessity is unrefuted on 480 comparisons. - Where it is one-way. The one-way pairs are a cut U′_out × complement, count |U′_out|(511 − |U′_out|), with U′_out the right-column outputs a flood fill from the left cannot reach. The form held on all 62 scored census generators, though the sealed U failed on ten (127, falsified). The corrected flood-fill rule then held on 16 of 16 fresh blocks, including all eight with a kept column-7 cell cut off from the left, where the old rule misses every one (F04: 45,540 observed against the old rule's 64,860) (131). Where Q fails detailed balance is read from geometry alone, before any operator is built. - Reversibility, settled on the census. On the nine blocks with symmetric support (g57 to g63; fresh F14, F15), the curvature K = max|κ| of ln(Q[S,S′]/Q[S′,S]) over the faces of a cycle basis is 0.7046 (g57 to g60), 0.6857 (g61 to g63), 2.7835 (F14) and 2.8019 (F15), each far above the 1e−6 line (135). So E1, detailed balance in any diagonal metric, is absent on all 63 census generators: by one-way pairs on 56 (127) and by curvature on 7 (135). Beside the clause: 31 faces are exactly flat on every census host, all mapping to tree edges under the mirror σ; F14 and F15 break that mirror by 7.5e−3 and 1.8e−2 against 1e−14 on the census. - Along a row. Chaining T_op across one row of nine blocks falls short of the exact 9 × 81 strip by g_bf = 0.0132 (H), 0.0212 (U), 0.0142 (carpet U′), 0.0218 (pattern T) and 0.0341 (X) of a strip width (136). Clusters that cross a seam and come back barely move the strip's threshold. Rank 1 is within 0.1 of a width on all five (largest gap −0.051, carpet U′), rank 4 is exact to 1e−4, and ranks 1 and 2 agree exactly: with run 135's premise that rank 1 carries no curvature, the strip's threshold is set by one mode.

Died. Since 2026-10-24: none of the four new runs killed a clause. One guess and one framing died beside them. - The pre-registered guess that the strip sits at least 2 widths above the lattice at L = 81 on H, U and carpet U′ (2026-10-24, 17:15; P = 0.75) missed on carpet U′: 2.779 (H), 2.264 (U), 1.172 (carpet U′), converted by the codifier from RESULTS051.md:36 (136, reported). - "Rank-k T composed on the 9 × 9 arrangement at generation 2", named in the last text as the next seal, is not a defined calculation. T_op has no top or bottom sides, and along one row a cluster can cross a seam and come back, reaching cells T_op marks dry (exact for heights 1 and 2, not from height 3); the exact four-sided objects are out of reach (§2, 2026-10-24, the design reader, R28_reading/design-e4/REPORT.md). Its feasible form was run 136.

Dead before and unchanged: the census curvature at 2187 (104, withdrawn before any draw; 103 withdrawn on its seeds); the wall count as what sets the lean or the gap (109); the Chladni pair (108, reported); boundary isotropy as a D4 property, forced by the transpose, not hidden (119; §2, 2026-10-23); the blind chat's U from the right-column holes (127); the turn as a route (083), X as one map iterated (089), the coarse lag as a constant (085), three contacts per side (084, 094), the one-terminal table as sealed (097), one exponent per pattern (S2, struck after 096), the fan (§2, 2026-10-03).

The one object. The block transfer operator T_op at p*: 512 × 512, left-column subsets to right-column subsets, exact on all 63 generators. It is the left–right half of §1's boundary state, and the one-block map is its rank-1 part. Unscored readings: the rank-2 residual is 0.004–0.007 and the rank-4 residual 4e−4 to 8e−4 (docs/design/2026-10-19-transfer-operator-svd.md §7); carpet U′ (#12) is exactly rank 4 and pattern T (#26) exactly rank 10 (RESULTS-transfer-operator.md §2). Scored: what T_op sees (122), which toggles it confuses (132), where it runs one way (127, 131), that it fails detailed balance (E1) on every census generator (127, 135), and that along one row it is close to the strip's boundary state with one mode carrying the threshold (136). Not known: whether T_op reproduces the lattice. The strip is exact but is not the lattice: at L = 81 it sits 1.2 to 2.8 lattice widths above run 051's values (136, reported). So "that number is small" has one scored reading for it, in strip form (136), and three scored objects against it on the lattice (084, 094, 097).

What the next seal is for, and why it is next. E4 proper: a sampled block-row operator at L = 81, set against T_op at the renormalised p (§2, 2026-10-24, the composition question; TODO, E4 proper, "fire it after 136 has scored"). A tool chat builds that operator first. It is next because it is the first test of the conjecture's own claim on the lattice rather than on one block or one strip, and 136 has removed the come-back paths along a row as the obvious reason it would fail. The order of limits matters: a k that passes at 81 can fail at 6561, because w shrinks while a fixed truncation error does not (§2, same entry). Also owed on the operator, unsealed: the E3 premise (D4 commutes with T_op on one census host, and T_op's spectrum splits by D4 character; themes/symmetry.md §3, 2026-10-24, 20:45), and E2, which no host we have can force (themes/symmetry.md §3, 2026-10-22). TODO puts the Q3 seal before E4 proper (TODO:658) and the H8 reseal after the D4/I2 premise chat (TODO:651).

What would end the programme. Unchanged: no finite rung reproducing the carpet's numbers at L ≤ 6561, or a quantity computed with every link open ordering the patterns as well as the polynomial. Run 108 tried one of the second kind and it fell short: λ₂ of the kept-cell graph at +0.820 against y₉'s +0.948. The nearest test of the first is E4 proper.

Vocabulary. (1) Candidates (in a sealed clause of a scored run and here). Block transfer operator T_op, one-way pair and visible / invisible flip are in the §1 block (added 2026-10-25; T_op entered the glossary on 2026-10-24). New since the last text, each in a sealed clause and added to the §1 block on 2026-10-25 (docs/PROGRAMMES.md, Rules: a term enters the block the day it enters a sealed clause): network-reliability core (132's clause writes "the core"; the reader's word, R28_reading/formation/tblind/); K, the curvature max|κ| (135); strip, back-flow gap g_bf and strip width w_s (136; the clause writes "the strip's width"). Used, not yet in a clause: U′_out, the set (127, 131; question and statistic); face, cycle basis, curvature κ (135; question and statistic); back-flow as a word (136's question; the clause quantity is g_bf); rank-k truncation (136; question and reported). The 2026-10-24 candidates are ruled under Entry and ceiling below: λ₂ (108), second crossing t₂ and the crossing gap (109) and wedge (132's clause) enter the block; wall entry orbit is retired (107's clauses name patterns and the line 0.290, not the orbit). (2) Unused or circled: rung k and the ladder (idle since 094); the laggard gap (reported in 097, never a clause); the coarse lag (a quantity since S4 died); riser and fan (dead with L10); "composed on the arrangement" (circled for a week, then shown undefined; the strip replaced it). (3) Axes (docs/rung-vocabulary-v5.md): operator and one-way pair — no axis carries reversibility; the glossary's mathematical vocabulary does. Visible flip and the reliability core — the rigidity axis (§8.2) has isospectrality, not an invariant blind to a change in its object; the forgetting hierarchy's arrow 1 is nearest. Curvature on faces — no axis carries a cocycle; Stokes is a §8.1 bridge slot and a double-evaluation licence (reusable terms), not an axis coordinate. Back-flow — not placed. Entry orbit — the place axis (§4). λ₂, second crossing — not placed. (4) Enforced (struck on the word the clause was written in): 119's "removes D4" read as "breaks isotropy"; 108's "connected since the ring is kept", false on 19 of 63; 104's census "2187"; 127's "U from the right-column holes alone". None in 131–136. (5) Symbols and collisions. The glossary has already ruled on T and U′ (GLOSSARY, Collisions open: "Write the qualified form where two meet"; "Write carpet U′ / U′_out"); this text follows it. The rest are to resolve before the next seal. - T is pattern #26 and the operator (written T_op; "pattern T" where both meet). - U′ is pattern #12 (written carpet U′) and the unreachable-output set (written U′_out; 127, 131, 135). The §1 block's "the set U′" (2026-10-25) yields to the glossary's U′_out. - U is pattern #11 and the rule's set. - K is the kept-cell mask in every operator statistic (K = generator(idx), flipped K′) and the curvature max|κ| (135). - k is rank k (136) and rung k of the ladder. - σ is the singular values (σ₁…σ₄, 119, 132) and the mirror (135's "tree edges under σ"; Pσ in 119's reading). - F is the fresh masks F01–F16 (131, 135) and F₈/F₆₄ (095); PG-03's F̂ (137) is a third. - w is the lattice width and w_s the strip width (136); g_bf sits beside g₁, g₂. - λ is the one-block multiplier and the Laplacian's λ₂ (written λ₂(Lap) in the glossary). - Q is T_op on non-empty subsets (127), Q = 2.47 (086), PN-007's joint crossing probability and F₈/F₆₄ (095). - D_A is 098's p_med(#22) − p_med(#24) (beside D_G) and PG-03's comb statistic D_A (130). - gap is 109's crossing gap (t₂ − t)/w, L8's threshold gap, 094's gap′ and the laggard gap. - core is the ring core (098; §1, 2026-10-03), the network-reliability core (132) and PG-03's map core [R²(c), R(c)] (GLOSSARY, Collisions open, core). - generator is the census generator (used throughout this text) and the Synthesist's unit (GLOSSARY, Collisions open, generator). - orbit is 107's D4 orbit and PG-03's periodic orbit (GLOSSARY, Collisions open, orbit / class). - "rung" is the dictionary's coordinate, the ladder's rung k and the symmetry ladder's rung (119). (6) Drift: boundary state, k contact bits per side (2026-10-01) to 512 left-column subsets (2026-10-20); "disconnected", no crossing possible (design §7, 2026-10-20), isolated interior cells (108), interior components (127), not the separator (2026-10-24); λ₂, "second-smallest" against "smallest nonzero" (108); "hidden symmetry" to "forced" (2026-10-23); "composition" from the block on its own 9 × 9 arrangement (2026-10-20) to one row of nine blocks (136); "wall" from the lattice boundary p → 1 (§1, 2026-09-28) to the wall of holes, the C₁ cut (107, 108, 109). (7) Prose only: riser, fan, dial, the turn, carrier; straw (109), Chladni pair (108), "sees" (122, 132; the clause quantity is Δ_c), "Stokes" (135's from: line; the clause is K). Ruled to prose only today (Entry and ceiling): hump (with the turn), recorder, wall entry orbit. (8) Open claims in names: wall entry orbit (107's lead: the cut-2 form of the boundary state, untested across cuts; retired as a term, kept as the lead's name); carrier; network-reliability core (the sufficient direction is scored, necessity is not). U′_out under the corrected rule is no longer open (131). (9) Metaphors: Chladni figures (108; tested, wrong as written); the weakest link of the strongest wall, the straw (CASE-31, 109; tested, not held); "sees" (122; kept as shorthand for Δ); curvature and Stokes (135; imports a gauge picture in which a loop quantity is read through the faces it bounds; tested once, as Kolmogorov's criterion, and held); back-flow (136; imports a current; it is a threshold gap, kept as a name for g_bf). (10) Quantifiers: weakened — every generator connected to 44 of 63 (108), every symmetry-breaking flip visible to 152 of 266 (119), the rule's count on all 62 to 52 of 62 (127). Strengthened — the orbit separation, 13 to 19 (107); E1 absent on 56 of 63 to 63 of 63 (135); the corrected U′_out rule from 63 of 63 formed on the data to 16 of 16 out of sample (131). Entry and ceiling. The block was written on 2026-10-04; the 2026-10-24 text was the first rewrite to carry this paragraph, and it said that at the next rewrite each block term enters or is retired. This is that rewrite, and every term the 2026-10-24 text left open is ruled here (docs/PROGRAMMES.md, Distilled, fifth paragraph: "a term in the block through two rewrites enters or is retired, with the reason; there is no third option"). A term enters only if it names a computation (a function, a column or a query) and stands in a sealed clause of a scored run (Rules, A programme owns its vocabulary; the fold-in's part (1)); a term that meets both may still be retired, with the reason. Otherwise it is retired: from the block as a term, its computation kept under its own name, or to the prose-only list, which the same rule allows ("a prose-only term may stand in a lead or in Distilled and not in a clause"). Entry into the glossary takes the entry criterion's third condition as well (a second programme with the same meaning, or a panel reaching the object blind); it was checked term by term against PG-02, PG-03, PG-04, SHARED.md and CASE-29, and only census meets it. - Entered already: one-block map and block transfer operator (GLOSSARY, Promoted 2026-10-24). - Enter (block terms since 2026-10-04): - block seam — computed as strip.seam_B (the bond matrix across a seam, 2026-10-24); in the clauses of 095, 097 and 136. Third condition not met: PG-02's seam is another thing. - polynomial fibre — census068.json's crossing-polynomial coefficients, grouped; in 086's clause ("inside a fibre"). Third condition not met: PG-02's fibre is a level-map fibre, and CASE-29 gave the panel the twins, so the panel did not reach them blind. - twin pair — two members of one polynomial fibre (086's pair list from census068.json); in 086's clauses. Third condition not met. - ring core / pendant part / island — twins_core.py analyse; in the clauses of 098 ("the core", "pendant parts"), 086 ("island pairs") and 131 ("island masks"). Written "ring core" from today, beside the network-reliability core. Third condition not met: PG-03's core is the map's core. - census — the 63 generators of census068.json (census068.py, gen.py), 49 of them with a lattice 1/ν; in the clauses of 086 and 087. Third condition met: PG-03 uses it in the same sense (§1, Where it claims nothing and run 087's entry; its Distilled's end condition). It is proposed for the glossary. - Enter (2026-10-24 candidates): - λ₂ of the kept-cell graph's Laplacian — spectrum108.py; in 108's clause. Third condition not established: PG-02's λ₂ is a Schreier graph's, in prose only. - second crossing t₂, crossing gap (t₂ − t)/w — perc_cluster.c --second, analyze109.py (gap109.json); in 109's clauses. - wedge — the 15 cells (i, j), 0 ≤ i ≤ j ≤ 4 (vis122.py, tblind132.py); in 132's clause ("visible wedge cells"). - Retire, the computation kept under its own name: - wall — the block's sense, the boundary p → 1 (C₁, N₁), stands only in clauses of withdrawn runs (082, 103, 104). The one scored clause with the word (109, "minimum walls") means the wall of holes, a second sense. C₁ and N₁ stay in the symbol table. - mirror channel / coarse channel — neither stands in a clause. 093's clauses write the pivotal graph's components and "two bond-disjoint channels" (channels093.py); "coarse" is only in the questions of 083 and 085. Δ = t − t_coarse stays. - ring share — the column pattern.ring_share, in 086's question only. - step map — in 089's question only, and 089's reading died (X as one map iterated). - estimator — in no sealed clause. The procedure is OPS.md's (estimators; owner of local slope and pair slope, GLOSSARY, Promoted 2026-10-24), and the word belongs to docs/PROGRAMMES.md's methods programme. - ladder / rung k — it stands in 084's clause, but it has been idle since 094 and it collides three ways ((5)); rung084.json and rung094.json stay. - coarse lag (died with S4, run 085); laggard gap (never in a clause). - Retire to prose only: - hump / the turn — one block entry, in no sealed clause (077's question only). It describes data (g₁ by size), and it collides with PG-03's hump map. The turn was already prose only. - recorder — the name of an instrument (perc_blocks.c), not an object term; it stands in 097's falsifier as the instrument's name. - wall entry orbit — 107's lead's name, in its question only. - riser, fan, dial, carrier — prose only since 2026-10-04, unchanged. - Wait (first rewrite in the block): one-way pair and visible / invisible flip (in the block from 2026-10-25, so their ceiling counts from there); network-reliability core, K, strip, g_bf and w_s (added to the block today).