A registered result in its source context
Original ledger classification: Lean formalisation [E] (no prime-2-only attractor: involution eigenvalues +-1, projector {0,1}, none in (0,1), while 2/3 is a contraction rate; axiom-pure) -- machine-proves the v409 corollary
Recorded statement: LEAN 4 FORMALISATION of the RES.COXETER.SYMMETRY.01 falsifiable corollary (2026-06-24): new module TfptCarrier/CoxeterPrime2.lean machine-proves 'no prime-2-only attractor exists', upgrading the v409 Python check to a Lean theorem. [E]…
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Paper context: tfpt research contracts
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FORM.COXETER.PRIME2.01
Original ledger classification: Lean formalisation [E] (no prime-2-only attractor: involution eigenvalues +-1, projector {0,1}, none in (0,1), while 2/3 is a contraction rate; axiom-pure) -- machine-proves the v409 corollary
Original claim text
LEAN 4 FORMALISATION of the RES.COXETER.SYMMETRY.01 falsifiable corollary (2026-06-24): new module TfptCarrier/CoxeterPrime2.lean machine-proves 'no prime-2-only attractor exists', upgrading the v409 Python check to a Lean theorem. [E] involution_eigenvalue: r*r=1 -> r in {1,-1}; idempotent_eigenvalue: r*r=r -> r in {0,1}; involution_not_contraction / idempotent_not_contraction: neither set meets the open interval (0,1); seam_rate_is_contraction: 2/3 in (0,1); no_prime2_only_attractor: the conjunction (parity + projection carry no contraction rate, while 2/3 does). [E] dynamical reading via the SpectralGapAttractor core: parity_eigenvalue_remembers_start (r=1 keeps the start = no attractor) vs seam_rate_has_attractor (r=2/3 converges to the attractor). Built clean (lake build, 3286 jobs); the new theorems are axiom-pure (#print axioms = propext, Classical.choice, Quot.sound only), and #check (AuditCheck) + signature-lock (AuditContract) pass. Strengthens v409/RES.COXETER.SYMMETRY.01 to the machine-proof tier.
Original classification detail
[E] the involution/projector eigenvalue lemmas + the no-contraction-rate corollary + the dynamical reading, machine-proved in TfptCarrier/CoxeterPrime2.lean (axiom-pure: only the three standard kernel axioms). Upgrades v409 to the Lean tier; no new number.
Dependencies as recorded
RES.COXETER.SYMMETRY.01; COXETER.ATOMS.01; v394; v409
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- -1.2cm TFPT — Research Contracts for the Remaining Interfaces:\\v_ geo [formula in source] G_ net [formula in source] F_ transfer [formula in source] (U_ wall) [formula in source] (G_ metric) [formula in source] [formula in source] / _1/ _2/ _3 [formula in source] next)tfpt research contracts
- Front matter / supportResearch contracts
- -1.2cm TFPT — Research Contracts for the Remaining Interfaces:\\v_ geo [formula in source] G_ net [formula in source] F_ transfer [formula in source] (U_ wall) [formula in source] (G_ metric) [formula in source] [formula in source] / _1/ _2/ _3 [formula in source] next)tfpt research contracts
- 2026-06-24 (Two new tracked targets + axion-branch consistency: v408, v409)Research changelog; TFPT — Introduction and reading guide
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- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- The unique attractor: a gapped transport selects ONE fixed pointorigin theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- 2026-06-25 (The seam [formula in source] is the carrier's Galois group: v419 [formula in source] figure)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- The whole integer skeleton from one pair [formula in source]Origin Theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Registered verification modules
- v409_coxeter_prime2_lemma.pyRegistered verification module · frontier
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RES.COXETER.SYMMETRY.01 closed as a STRUCTURAL LEMMA: prime-2 is NECESSARY but NOT autonomous -- no prime-2-only attractor exists (resolves the v394/COXETER.ATOMS.01 open question's falsifiable corollary). [E] SHEET INVOLUTION: sigma=diag(1,-1,-1), sigma^2=I, spectrum {+1,-1} (parity). [E] SHEET PROJECTOR P=(I+sigma)/2=diag(1,0,0), P^2=P, spectrum {0,1} (projection). [E] NO CONTRACTION RATE: none of {-1,0,1} lies in (0,1), whereas the genuine rates 2/3 and (1/3)^6 do -- prime-2 alone gives parity/projection, never a nonzero rate. [E] FACTOR NOT RATE: 2/3=|Z2|/N_fam needs prime-3, (phi+2)/4=(phi+2)/|mu4| with |mu4|=2^2 needs the golden phi (prime-5). [E] 30=g_car(2 N_fam), carrier static, 2*3 dynamic. [C] prime-2 is the universal dynamical parity factor, necessary in every sheet-paired transport but not an autonomous attractor; dynamics begins after coupling to family-3 or carrier-5 (mu4=2^2 the Galois bridge). No new number.
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