A registered result in its source context
Original ledger classification: Formal
Recorded statement: LEAN: THE CARRIER CARTAN DETERMINANTS AS REAL Matrix.det COMPUTATIONS (2026-06-22). Lean 4 (TfptCarrier/CartanDeterminants.lean) grounds the carrier piece of the holomorphy/anyon discriminator in ACTUAL determinants of the integer Cartan…
Statement, assumptions and sources
Paper context: Origin Theory
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FORM.CARTAN.DET.01
Original ledger classification: Formal
Original claim text
LEAN: THE CARRIER CARTAN DETERMINANTS AS REAL Matrix.det COMPUTATIONS (2026-06-22). Lean 4 (TfptCarrier/CartanDeterminants.lean) grounds the carrier piece of the holomorphy/anyon discriminator in ACTUAL determinants of the integer Cartan matrices, proved (kernel decide, no extra axiom) rather than asserted: |det Cartan(A3)|=4=|mu4| (3x3, via Matrix.det_fin_three) and |det Cartan(D5)|=4 (5x5, via decide at maxRecDepth 4000), giving the carrier |det Gram(D5(+)A3)|=4*4=16=dim S+ (the v281 anyon count 16 -> the tower 16->4->1). For the rank-8 case the 8x8 Leibniz determinant is not kernel-decidable (Mathlib's own Mathlib.Data.Matrix.Cartan leaves E8_det=1 as a proof_wanted, proving only the smaller G2/F4 dets by decide), so instead the UNIMODULARITY -- the holomorphy discriminator |det Cartan(E8)|=1 (E8 has ONE anyon/primary, vs D8's four) -- is proved PRINCIPALLY and kernel-only: cartanE8 * cartanE8inv = 1 by decide on the 8x8 PRODUCT (~512 mults, feasible) gives IsUnit(cartanE8.det) via det(E8)*det(E8inv)=det(1)=1 (cartanE8_det_isUnit). The sign det=+1 then follows from positive-definiteness (standard). Built end-to-end (lake build) on Lean v4.29.1 + Mathlib. The VALUE is grounding BOTH the carrier anyon count AND the rank-8 holomorphy discriminator (E8 unimodular = one anyon) in real determinants/matrix algebra, no native_decide.
Original classification detail
[E] Lean-built: the carrier Cartan determinants |det A3|=4, |det D5|=4 => carrier index 16 as real Matrix.det computations (kernel decide, no native_decide axiom). Grounds the v281/v308 carrier anyon count AND the rank-8 holomorphy discriminator (|det Cartan(E8)|=1 = one anyon, via the integer inverse, kernel decide) in real determinants/matrix algebra; Mathlib leaves E8_det=1 (the sign) as proof_wanted, which positive-definiteness supplies.
Dependencies as recorded
SEAM.EQUIV.CHAIN.01; QGEO.KEYSTONE.01; v281; v325
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- Honest statusinput 4 origin theory
- Honest statusorigin theory
- 2026-06-22 (frontier round: the OS step discharged at the many-body level, the ambient measure built on the gap-decoupled sector, and the carrier Cartan determinants grounded in Lean: v329, v330, FORM.CARTAN.DET.01)Research changelog; TFPT — Introduction and reading guide
- 2026-06-22 (endgame round: the necessity of [formula in source] reduced to the order-4 CM selection, the QG metric sector characterized as the conformal-mode problem, [formula in source] and the [formula in source] commutation grounded in Lean: v331, v332, FORM.MU4COMM.01)Research changelog; TFPT — Introduction and reading guide
- Honest statusOrigin Theory
- Honest statusinput 4 origin theory
- 2026-06-22 (next-steps round: the one open lemma literature-anchored, the Decoupling Theorem, the holomorphy discriminator hardened in Lean, and the [formula in source] budget: v336, v337, v338, FORM.CARTAN.DET.01)Research changelog; TFPT — Introduction and reading guide
- 2026-06-22 (closing round: the CM-selection mechanism (the last keystone residual), the conformal-mode resolution, and the Conway–Sloane even-unimodular side in Lean: v333, v334, FORM.CARTAN.DET.01)Research changelog; TFPT — Introduction and reading guide
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- 2026-06-22 (keystone unification + status reframe: ONE open theorem, ONE decoupled foreign problem: v335, FORM.QGEO.BW.01 qgeoSymIsCorollary)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- 2026-06-22 (next-steps round: the keystone collapsed to one shared premise (Lean [formula in source] one citable theorem), the [formula in source] solver suite, the cusp weight derived from a minimal rewrite, and the [formula in source] pressure point: FORM.QGEO.BW.01, v325, v326, v327, v328)Research changelog; TFPT — Introduction and reading guide
Registered verification modules
- v325_pillowcase_keystone.pyRegistered verification module · frontier
Read the description recorded in the module registry
The seam keystone as ONE citable theorem: the raw seam state is the flat tau=i pillowcase state, with EXACTLY ONE open lemma. IF (raw RP-collar seam state = flat tau=i pillowcase / rotation-invariant state) THEN 4 square marks + mu4 deck + omega o rho=omega + holomorphic c=8 => (E8)_1, with the carrier recovery gap as input. Pieces (all exact): n=2chi(S^2)=4 (v216); cross-ratio 2 => j=1728 (v214/v267); flat orbifold S^2(2,2,2,2), chi_orb=0 (Troyanov); DtN sub-principal at m=0 mod 4 (v201/v280); det Cartan(E8)=1 vs D8=4 (v83/v143/v308); gap 6 ln(3/2)>0 (v76/v302). The ONE open lemma -- the raw collar state IS the flat tau=i pillowcase / rotation-invariant state -- is SHARED by both bedrock IDs (QGEO.SYM.01, SEAM.EQUIV.01; v323), machine-pinned in Lean (FORM.SEAMEQUIV.01 + FORM.QGEO.BW.01). An assembly/reduction certificate (like v176/v261), not a closure.
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