Exact Symbolic Certification Data for "A Fixed-Level Certification Dichotomy for an Infinite Family of Frustration-Free Spin Chains"

Reproduction package containing the exact symbolic pipeline, generated JSON block data, LDLTLDL^{\mathsf T}LDLT positivity certificates, Sturm verification data, independent checking scripts, and SHA-256 digests supporting the computer-assisted results of the accompanying manuscript. This repository contains the complete computational record supporting the computer-assisted parts of the paper "A fixed-level certification dichotomy for an infinite family of frustration-free spin chains" (G. Lecomte, 2026). All computations are carried out in exact rational arithmetic; no floating-point value enters any stated result. Floating point appears only in the independent ring diagonalizations, which serve as cross-checks. Contents. pipeline.py implements the six-stage symbolic pipeline described in Appendix C of the paper: (1) construction of the interaction h_T over Q from the orthonormal kernel family; (2) assembly of A_5(T) column by column from sparse exact actions of the three window operators; (3) generation of the parametric superpotential Y_T(delta) as an exact dictionary of entries in Q[delta]; (4) grouping of the |S|^5 words by run skeleton, with an explicit leakage test verifying that no matrix element of M = A_5 - delta h_1 + D(Y_T) joins two distinct skeletons; (5) exact block extraction in a letter-independent basis order, deduplicated by the decorated data of the partition lemma, with an empirical verification that equal decorated data always yields identical blocks; (6) certification of every distinct block. run_all.sh is a one-command reproduction script: it re-runs the pipeline, re-runs the independent verification scripts, and re-checks all SHA-256 digests. independent_checks/ contains three verification scripts written independently of the pipeline. verify_blocks.py rebuilds the model from scratch in SymPy and reproduces every block displayed in the paper: the one-wall rank-one factorization, both triples of the terminated-wall family, the six-dimensional two-wall block with its determinant and minors, the forbidden two-run diagonals for three decorated cases, the five-run scalar, the residual matrices R_3^(0) and R_4^(0) with the characteristic polynomial (1/2)(lambda-5)(2 lambda^3 - 13 lambda^2 + 20 lambda - 3), and a full skeleton-leakage scan on a four-letter graph. verify_aba.py checks the sigma = aba three-run residual. verify_rings.py numerically diagonalizes the periodic chains of the directed 2- and 3-cycles for m = 3 to 8 and confirms the closed-form gaps, the forbidden-subspace minimum, and that the kernel is spanned by the uniform words (for the 3-cycle the closed form is claimed in the paper for m >= 5 only; the m = 3, 4 rows are boundary data). output/blocks.json lists every distinct decorated block of the catalogue: exact entries in Q[delta] (degree at most 2, stored as fraction coefficient lists), dimension, decorated key, and the exact characteristic polynomial det(lambda I - B) over Q[delta]. output/certificates.json contains, for every block, the Sturm certification record for each coefficient of det(lambda I + B) on the interval [0, 1/48] (stripped delta power, exact root count, exact value at the endpoint), and exact rational LDL^T factorizations at delta = 0 and delta = 1/48, with zero pivots allowed only against zero columns, as required for semidefinite factorizations. output/occurrences.json maps each (catalogue graph, skeleton) pair to its block. output/summary.json records the global verdicts. SHA256SUMS lists the digests of all files; COMMIT.txt records the git commit hash of this tree. Scope. The certified catalogue is the internally generated catalogue of the paper: the two 2-letter graphs, all 7 non-empty forward-edge patterns on three ordered letters, and all 63 such patterns on four ordered letters, with in-degree flags computed inside each catalogue graph. The external in-degree decorations are not swept independently; this matches the scope stated in the paper. The structural constant min{1/48, 1/sqrt(12H)} of the paper is proved analytically and does not rely on this catalogue, which serves as an independent cross-check. Results. 72 graphs; 66,277 columns scanned; skeleton leakage: zero; partition-lemma check: passed on all occurrences; 5,622 distinct decorated blocks; maximal block dimension: six; every block certified positive semidefinite on [0, 1/48]. Outputs are byte-deterministic across runs. Wall-clock on a single x86-64 core: about 7 seconds for stages 1 to 5 and 24 seconds for stage 6. Requirements. Python 3, SymPy >= 1.12, NumPy >= 1.24. Reproduce with: pip install -r requirements.txt && ./run_all.sh. Expected final line: ALL CHECKS PASSED.