Noita partial-permutation overlap catalog

TL;DR. Chaining, restricted to substitutions within single repeated-plaintext alignments (where the GAK deck model guarantees one fixed partial permutation π = D₂∘D₁⁻¹ and no commutativity is assumed), is exactly consistent on the eyes: 47 regions, 616 entries, 0 conflicts (shuffled-null contradiction rate 15–25%; cycle checks 562/562). The four initial contradictions localize a plaintext edit at positions ~63–67 between E4 and E5; the stable offsets imply E4, W4, E5 carry the same text up to a few inserted characters. The 616 entries are key-independent constraints on any candidate (initial deck, per-letter shuffles, plaintext).

Context. Alphabet chaining on isomorph pairs established that the eyes are non-commutative; the working model is a GAK deck shuffle. Chaining on the eyes produced conflicts, but in S83 conflicts are ~100% expected from non-commutativity alone, so they discredited the chaining assumptions rather than the model. The open question was whether chaining admits a formulation that extracts valid constraints.

Reformulation. Under the deck model, one transformation survives non-commutativity: when two stretches encrypt the same plaintext, the two decks evolve in lockstep, so the substitution between the ciphertext stretches is a single fixed partial permutation π = D₂∘D₁⁻¹, valid across the whole stretch. Classic chaining failed because it composed transformations across different plaintext contexts, which is exactly where non-commutativity enters. The restriction is therefore: merge substitution pairs only when they share the same (message pair, position offset) alignment, i.e. plausibly one repeated phrase.

Procedure. Inputs: Toboter’s isomorph catalog and the known shared openings. Symbol→symbol pairs were collected per alignment and merged. Two consistency checks: (1) internal — a merged table is contradictory if one symbol requires two translations; (2) cycle — π(A→B) composed with π(B→C) compared against direct π(A→C) at aligned positions. Calibration: the identical procedure on per-message shuffled ciphertext (chance contradiction rate), and on synthetic deck-cipher messages with planted repeated phrases (pipeline reports zero conflicts when the model holds).

Results. 616 merged entries, 4 contradictions (0.8%) versus 15–25% on shuffled nulls; 562/562 cycle agreements. All four contradictions lie in one alignment (E4↔E5, offset +1) and resolve under a single positional split: the span is two same-text regions separated by a small plaintext edit at ~63–67. Final catalog: 47 regions, 616 entries, zero conflicts — the colored spans above. The stable offsets (+1, −1/−2, +3 over long spans) additionally imply E4, W4, E5 are one text up to a few inserted characters.