The crossing polynomials of the 49 census fibres carry squares and a composition that nothing in the record explains. On all 13 cut-2 representatives 1 − R_P = c·(1 − p)²·S² exactly (deg R = 2 + 2·deg S), which is the crossing probability of two identical independent events in parallel, R = T(2 − T); and on #8 and #9 R_P is an even polynomial, R(p) = A(p²), a composition in Ritt's sense. A square in a probability polynomial is two independent identical events; an even one is a graph whose bonds come in series pairs. Are the two events the block's two mirror channels — disjoint sets of bonds exchanged by the top–bottom reflection — and are #8's and #9's pivotal bonds all in chains of even length?

Sealed 2026-10-02, with the factor table known and no sub-block, pivotal bond or component computed on any scored member (codifier). Known at sealing, all of it from sighted passes the codifier mis-scoped (SIGHTED-D2D3-2.md, dd1c6cf — the reason an earlier draft of this question is a table and not a run): ℓ = 8 and p⁸ | R_P on all 49; (1 − p)^{C₁} | 1 − R_P on all 49; on the 13 cut-2 representatives 1 − R_P is c·(1 − p)²·S² with deg R = 2 + 2·deg S; on the ten of them that are not wall-bound R_P has besides a squared factor Q² (deg Q = 2–16), on #12, #19, #21 it does not; #8 and #9 are even (the codifier checked the stored coefficients: every odd coefficient vanishes) with 21 shared critical values off 0 and 1, which is what A(p²) with deg A = 28 gives (27 critical points of A, three at value 0 and three at value 1); #7, #34, #44 (cut 3) and #29, #30 (cut 4) have (1 + p)^{C₁} in 1 − R without being even; sympy's decompose returns one factor on #26. On the controls (SIGHTED-D2pD3.md, b0016d0): #26 and #1 have no 1- or 2-bond cut, their a-vectors reproduce by two exact enumerations, and neither is even. deg R_P = M − 16 on #26, #1 and #16 (sixteen is the number of vertical bonds in two fully kept terminal columns, which no crossing needs — so on #16 every other bond is pivotal, which fixes its pivotal set and not its connectivity) and is lower on the wall-bound patterns. On small symmetric cut-2 grids the second reader enumerated, disconnection and a perfect-square 1 − R went together in every case (297 disconnected, all squares; 309 connected, none).

What the record does and does not say. A perfect-square 1 − R makes two identical independent events very likely; it does not say what they are, and an algebraic coincidence, or a product structure that is not the mirror, would look the same in the table. The same for evenness and chains. Both clauses are likely on the record and neither is implied by it; the run is short, exact, and closes them.

What this run is for. The one-block map commutes with nothing but its own iterates, and the block's symmetry had seemed to leave no trace in it. The factor table says the trace is there in another form: not as a commuting map and not as a composition (except on two patterns), but as squares. If the squares are the mirror channels, then the cut-2 census is a family of parallel systems — each channel a small series system with its own polynomial T — and the one-block recursion on them is the recursion of a two-terminal parallel pair, R = T(2 − T); and if the even patterns are chains, then two census members are a smaller graph with every bond doubled, and their recursion is that graph's composed with p². Either would give L9 its first structural statement about a fibre.

The codifier's lead, held with conviction. A holds on all 13 and B on both. Each channel runs along one of the block's outer rows, pinched to one bond on either side of the middle column — the four 2-bond cuts are the four ways of closing one pinch bond per channel — and where the channel's approach to the pinch has room to branch, T carries the series square Q². Where it breaks: a pattern whose two routes share a non-terminal cell on the middle row; then the events are not disjoint, the pivotal graph is one component, and the perfect square in 1 − R is a coincidence worth more than the confirmation.

Power, stated. Exact; a clause fails or holds on each member with no error. Under "no mirror structure" a component count of exactly two exchanged by the reflection on all 13 is not something chance supplies.

The reductio this seal passed first. By construction: pivotal bonds are a property of the graph (a block of the terminal-merged graph), not of a picture of channels, so the components are not drawn to fit, and the polynomial definition is checked against it on five bonds per pattern; control (ii)'s independence identity tests the code on two patterns outside the scored set. Negation: if the two events were not bond-disjoint mirror channels, some cut-2 pattern's pivotal graph would be connected, or split unevenly. Twins: #46/#47, #49/#50, #53/#54/#55 and #38/#42/#43 share polynomials with different patterns; the representative alone is scored and the other members are reported — a fibre whose members have different component structures with one polynomial would be the first such object. Cost: the clauses are graph algorithms, seconds; the identities and cross-checks are about a dozen enumerations per pattern at one to two seconds; minutes in all, more for the reported items.

Reported, not scored. - Per cut-2 pattern: the 2-bond cuts (the lead says four), the components' bond counts, T_ch, S, Q and their degrees; the bridges of one channel and its series factorization, and whether Q is the polynomial of the piece between the terminal column and the first bridge. - The non-representative fibre members of the cut-2 fibres, the same construction. - For the patterns with (1 + p)^{C₁} in 1 − R: whether each minimum-cut bond is in series with a partner. - For the cut-3 patterns with a squared factor in 1 − R (#7, #26, #44, #45): the pivotal graph's components, a look. - R₈ − R₉, and A₈ and A₉ side by side; which census pattern, if any, has A₈ or A₉ as its own crossing polynomial; sympy's decompose on all 49. - The count of pivotal bonds per pattern against deg R_P.

Reviewed before commit by a sharpness reader and a second reader (OPS.md); their edits are folded in: the first draft's "all critical values distinct" was false by the forced factors at 0 and 1; its series-halves mechanism was impossible at C₁ = 2 (two bonds every crossing needs are two 1-bond cuts) and is replaced by the parallel channels the degrees already carry; its composition clause for #8 and #9 was decidable from the stored coefficients and is replaced by the chain structure, which is not; clause A reduced to the one live question (connected or not), with the rest moved to instrument errors; pivotal bonds defined on the graph. A sighted pass on the control patterns #26 and #1 (SIGHTED-D2pD3.md) supplied the control values and touched no scored member.

Scored 2026-10-02 — confirmed

Scored by the codifier from RESULTS093.md and channels093.json (ac631c9), as written. A scoring reader (OPS.md) read the note against the results before commit; its edits are folded in.

Controls first — all three pass. (i) The run's own row-scan enumeration reproduces poly075.json's a-vector and M exactly on 17 of 17 (the 15 scored members, #26 and #1); rg058.poly is a column scan, so two code paths agree. (ii) #26 and #1 each give one pivotal component (68 bonds), the independence identity exact, no reflection-exchanged pair. (iii) The conditioning identity holds on five bonds each. Beside the sealed controls, the five-bond cross-check of the graph definition of a pivotal bond against the polynomial definition agrees on every pattern.

(A) Confirmed. On all 13 cut-2 representatives the pivotal graph is disconnected: exactly two components of equal bond count (17–40 bonds each), exchanged by the top–bottom reflection, with R_P = 1 − Π(1 − T_component) exact. No instrument error: no pattern with three or more components, none unexchanged. (B) Confirmed. On #8 every chain is even (24 of length 2, 2 of length 4); on #9 all 28 chains have length 2. The chains cover all 56 pivotal bonds of each. Both clauses hold; the run is confirmed. The codifier's lead was right on both clauses, wrong on one reported detail (item 4) and half right on another (item 5: T does carry the series square Q², but Q is not the piece).

What the numbers say beyond the clauses.

1. The cut-2 fibres are compositions, and the record's "one factor" for them is a false negative. The two channels' polynomials are identical on all 13, T₀ = T₁ = T, so R_P = T·(2 − T) = g∘T with g(x) = 1 − (1 − x)² — a functional decomposition in Ritt's sense with a quadratic outer factor, exact by this run's identity. RESULTS093's reported item 5 says sympy's decompose returns one factor on these 13; that is the routine missing the decomposition, not its absence — the codifier reproduced the miss on a toy (T = 3p² − 2p³, R = T(2 − T): sympy 1.14.0's decompose returns one factor over ℤ and over ℚ, where it finds the decomposition when the inner factor is p²(2 − p²)). The seal's sentence that the block's symmetry is in the polynomial "not as a composition (except on two patterns)" is wrong as written (the seal stands unedited): a square is a composition. It is a composition on fifteen of the 49 fibres — an outer g on the 13 cut-2, a p² under it on nine of those, an inner p² on #8 and #9.

1. The other 34 fibres are indecomposable, by a count that does not need sympy. A composite of degrees a, b ≥ 2 (n = ab) has at most a + b − 2 ≤ n/2 distinct finite critical values. SIGHTED-D2D3-2.md's table gives the number of distinct critical values of every fibre, and on each of the 34 fibres that are neither cut-2 nor #8, #9 it exceeds deg R/2 (the closest: #7, 40 against 26; #44, 46 against 32; #45, 49 against 34); on the 15 decomposable ones it does not (#8 and #9: 23 against 28; the cut-2: 8–18 against 17–40). So the split is exact: 34 indecomposable, 15 composite. The scoring reader went further, and its two additions are attributed to it and wait on a lab script before they are cited as results: from the stored T_channel and Q, T = p⁸Q² exactly on nine of the cut-2 fibres (#14, #15, #16, #38, #46, #49, #51, #53; #56 through R = T(2 − T)), so T = U² with U = p⁴Q; and a Kozen–Landau recursion over ℚ on all 63 polynomials gives the complete decompositions — A∘p² with A indecomposable on #8 and #9; g∘T with T indecomposable on #12, #19, #21, #32; g∘p²∘U with U indecomposable on the other nine.

1. Series and parallel are one map seen from two sides. p ↦ p² is two bonds in series; g(x) = 1 − (1 − x)² is two in parallel; and g = σ∘(p²)∘σ with σ(x) = 1 − x, the duality map. Every quadratic factor in the census is the squaring map or its dual — p² inside on #8 and #9; g outside on cut 2; and on nine cut-2 fibres p² between, each channel being two equal halves in series, R = D∘U with D = g∘p² = 2x² − x⁴. A consequence of the identity, not computed here: g∘T and T∘g are semiconjugate (T carries the fixed point of g∘T to that of T∘g, with the same multiplier by the chain rule), so each cut-2 pattern's one-block map has the multiplier λ — and so y₉ — of the map T∘g, one channel with every bond replaced by a parallel pair; likewise #8 and #9 share theirs with p²∘A = A(p)², two copies of the contracted graph in series. Whether those companion graphs are anything the record knows is not checked.

1. The 2-bond cuts number b², not four. b is the number of bridges in one channel: four cuts on the seven patterns with two bridges per channel, sixteen on the six with four. The lead's "four ways of closing one pinch bond per channel" is the b = 2 case. Every channel factorizes in series exactly, T = p^b·Π(pieces).

1. Q is a factor of the first piece, not the piece. On the ten patterns with a squared factor Q² in R_P, Q divides the polynomial of the piece between the terminal column and the first bridge, and divides T, but is not that piece on any of them. T = p⁸Q² exactly on nine of the ten (the scoring reader's check on the stored T_channel and Q; #56 from R = T(2 − T)), so the two pieces multiply to p^{2k}Q² with k = (8 − b)/2: 3 on the seven two-bridge patterns, 2 on #38 and #46. On #32, Q² divides T with cofactor p⁸·(degree 10) and k = 1 by degree. That each piece separately is p^k·Q is a reading — the pieces are not stored. The pieces come in equal-degree pairs on nine of the ten (#32: 3, 12, 3), which reads as the left–right mirror inside a channel.

1. deg R_P counts the pivotal bonds, on all 63 patterns. The seal named three.

1. No fibre splits on structure. The six non-representative members of the cut-2 fibres (#42, #43, #47, #50, #54, #55) have their representative's component structure.

1. The cut-3 squares are not channels. #7, #26, #44, #45 each have one pivotal component, fixed by the reflection; their squared factor in 1 − R is not two channels; what it is, is unidentified. On the five patterns with (1 + p)^{C₁} in 1 − R, every bond of every minimum cut is in series with a partner — consistent with each cut bond entering through 1 − p² (a reading; the factor is not derived).

A catch by the lab worth keeping. Grouping the census by poly075.json's a-vector gives 57 classes, not 49: twins share R(p) at different M, so only the expanded polynomial groups them. Filed in MANUAL.md.

What it does to PG-01. L9 has its first structural statements: the census's fibres are 34 indecomposable polynomials and 15 composites whose quadratic factors are p² and g, conjugate under σ (σ sends an outer g to an outer p², so the two kinds of fibre are not each other's dual), and the cut-2 class — the class holding the ten largest gaps of run 085 (its other three rank 13, 15 and 21 of 49), the nineteen most skewed maps of run 075's 63, and the lowest mean tail ratio of run 091 — is the class of parallel pairs. A reading, to be tested and not yet a result: the cut-2 patterns stand apart in these three quantities because their block is two independent systems, and the lattice at generation 2 couples what the block keeps apart. The test is the pivotal graph at generation 2: do the two channels stay disjoint (the rule the one-block map assumes) or join (the boundary datum, as a count of joining bonds)? Exact, same instrument, minutes.

Lab notes. Claude Code session on the M5, 300 s in all; python-flint for exact arithmetic; sympy 1.14.0 installed into ~/mapofmath/.venv for the decompose item; 267 rows in row form, 13 of them dictionary rows. channels093.json stores #56's T_channel and S cut at 400 characters, so those two are not full polynomials.

Addendum, 2026-10-02 — the lab check behind item 2 (RITT-CENSUS.md, ritt_census.py; ac88731)

The two claims the score attributed to the scoring reader and left waiting on a lab script both hold, by an independent implementation: a Kozen–Landau right-factor search over ℚ in exact rational arithmetic, applied recursively, every decomposition verified by composing the factors back. Not a blind reproduction — the lab session had read this score — but a different code path, tested first on 200 planted compositions (inner degrees 2–6, 146 with a non-monic inner factor; 200 of 200 found, the two toys sympy misses among them).

• Factor degrees, outermost first: #8, #9: 28, 2. #12: 2, 18; #19: 2, 17; #21: 2, 22; #32: 2, 22. #14, #46: 2, 2, 12; #38: 2, 2, 10; #49: 2, 2, 14; #15, #53: 2, 2, 16; #16, #51: 2, 2, 18; #56: 2, 2, 20. The other 34 fibres: indecomposable. 15 composite, 34 indecomposable, 49 of 49 verified.
• Why the 34 are indecomposable: by the search (every divisor of the degree tried; the right factor of a normalized decomposition is unique, so one candidate per divisor is exhaustive over ℚ) and by the critical-value count, which agree on all 34. The lab's critical-value counts reproduce SIGHTED-D2D3-2.md's table on all 49.
• The channel polynomials, recomputed in full from the block: T = p⁸Q² with √T = p⁴Q exactly on nine of the 13 cut-2 fibres, #56 included directly; the four that are not squares are #12, #19, #21, #32 — the four whose decomposition is 2, deg T.
• A check nobody asked for: on all 13 the innermost factor of R's decomposition equals the graph-computed channel polynomial up to normalization (T on the four, √T on the nine). The search never sees the lattice and lands on the pivotal-component construction's polynomial.
• A limit the lab flagged, kept here: the decomposition gives factor degrees; that the quadratics are x(2 − x) outside and p² between is not a decomposition result — a complete decomposition is unique only up to Ritt's affine ambiguity (the search returns −16x² − 8x on #12, which is x(2 − x) conjugated by y ↦ −4y). Those identities rest on this run's R = T(2 − T) and on T = U², both re-verified by the check.

Item 2's attribution is therefore lifted: 34 indecomposable; A∘p² on #8 and #9; g∘T on four cut-2 fibres; g∘p²∘U on nine.