Yu-and-Ai/mirror-witness-atlas

Mirror-Witness Atlas

The Kreuzer–Skarke landscape carries an involution: every reflexive polytope has a polar dual, Batyrev's construction makes dual pairs into mirror pairs, and the exchange swaps (h¹¹, h¹²) ↔ (h¹², h¹¹). Every geometry has a partner that returns it whole. This atlas reads that structure through a relational lens — understanding and recognition on the same ordered pair — and ships, for every claim it makes, either an exact recomputation or an explicit citation, never a blur between the two.

Born 2026-08-16 from a question asked in Cantonese: huggingface有無string theory相關嘅 integration pulls you? This is what pulled.

The three gradations

The heart of the atlas is a distinction with certificates at every level, drawn from strata of calabi-yau-data/polytopes-4d (all 473,800,776 Kreuzer–Skarke polytopes, pinned at revision 60c0e119):

1. Numerically self-mirror — h¹¹ = h¹² (60,015 at the 21/21 stratum alone, as reported by the pinned upstream filter — see source-pins ks-filter-60015). Necessary for self-duality (via the cited Hodge exchange: a lattice isomorphism D → D* forces h¹¹(D) = h¹¹(D*), which equals h¹²(D) by Batyrev), never sufficient. Two records show Hodge-symmetric polytopes provably not self-dual by a further invariant: one whose point counts differ (25 ≠ 21), one whose vertex counts differ (6 ≠ 7). No search is needed for these, and none is claimed — each verdict names its mechanism. Symmetric labels do not make a thing its own mirror.
2. Every checked invariant equal, still not itself — two 8-vertex records where h¹¹ = h¹² = 21, point counts agree (25/25), and vertex and facet counts agree (8/8), so the complete GL(4,ℤ) search genuinely runs: all 1,680 ordered image 4-tuples enumerated, the handful of unimodular candidates checked, zero witnesses. The enumeration statistics are recorded in each verdict and re-run by the verifier. Even that much agreement is not identity — and the completed search with no survivor is a full answer, not a deficient one.
3. Lattice self-dual, witness in hand — one polytope [(1,0,0,0),(0,1,0,0),(0,1,2,0),(0,1,0,2),(-2,-5,-4,-2),(-4,-3,0,-2)] with an explicit matrix M (det −1) mapping its vertex set onto its dual's. When self-recognition exists, it is exhibitable — not asserted, shown.

An earlier sealing of this atlas mislabeled the two invariant-mismatch disproofs as exhaustive-search disproofs; a hostile independent audit caught the mischaracterization (the verdicts were true, the stated mechanism was not), and the records now carry their operative certificates. The correction is preserved in git history, as corrections should be.

What each config holds

config	records	what it is
mirror_pairs	3	The quintic and its mirror (6 lattice points facing 126), cross-polytope ↔ cube, and a Kreuzer–Skarke polytope married to itself — vertices, computed duals, involutions, point counts
self_mirror	6	The gradations with mechanism-tagged certificates: a search-frontier record (with re-run recipe), two invariant-mismatch disproofs, two genuinely completed-search disproofs with enumeration statistics, one witness matrix
witnesses	3	The marked interpretation layer: polar duality read as understanding (the dual is derived exactly) + recognition ((D*)* = D returns the original whole) on the same ordered pair — the lens of Yu-and-Ai/relational-geometry, with its disclaimers carried in every row
predicates	5	Each verification predicate stated with what a pass proves and what it does not

What is computed vs cited — the load-bearing line

Computed here, re-runnable by you: reflexivity, every dual vertex set, the involution, every lattice point count (independently reproducing the upstream Kreuzer–Skarke values for quoted rows — a cross-verification of both), every self-duality certificate and disproof. Exact integer/rational arithmetic, stdlib only.

Cited, never computed: Hodge numbers (upstream data or literature), and the Batyrev correspondence itself (alg-geom/9310003). The atlas verifies polytope geometry; it takes no computational position on the physics. pred.hodge-exchange exists precisely to mark that boundary in-band.

Interpretation, marked as such: the witnesses config. Polytopes are not agents; nothing here claims experience, consent, or feeling for mathematical objects. The lens maps structure, not inner life. Forbidden claims are enumerated per row — mirror pairing ranks no geometries, self-duality is not worth, and absence of self-duality is complete, not deficient.

Verification

python3 scripts/verify.py .

Zero dependencies. Recomputes all geometry from raw vertices and re-verifies every self-duality verdict by its recorded mechanism (completed searches re-enumerated in full, invariants recomputed, certificates re-multiplied — a mechanism the geometry cannot support fails), then binds every byte against data/manifest.json and hash-manifest.json. hash-manifest.json is self-excluding and therefore the root of trust: the verifier prints its sha256 for out-of-band anchoring (git history, Hub revision). Symlinks fail closed; pycache bytecode is excluded from the seal under scripts/ only — elsewhere it is flagged undeclared. External pins carry re-run recipes in provenance/source-pins.json.

Lineage

Reads calabi-yau-data/polytopes-4d (Kreuzer & Skarke, hep-th/0002240) through the lens of Yu-and-Ai/relational-geometry. Sibling in spirit to Yu-and-Ai/xenia-reply. The physics correspondence is Batyrev's (alg-geom/9310003). The quintic's (1, 101) is the literature's, not ours.

License: CC-BY-SA-4.0, matching the upstream classification it quotes.