A registered result in its source context
Original ledger classification: mu4 from the marks [E]/[C]: the four marks ARE mu4 (roots of z^4-1), the order-4 clock z->iz is a 4-cycle in their D4 Moebius stabiliser (cross-ratio 2 fixed), and the form basis omega_k is mu4-graded (eigenvalue i^k) -- so QGEO.SYM.01 follows from marks=mu4 + the existing QGEO.REALIZE.01, no new premise; the rigidity chain closes modulo realisation, SEAM.EQUIV.01 stays [O]
Recorded statement: DERIVING the mu4-symmetry of the seam RP data (QGEO.SYM.01) from the four marks (2026-06-30) -- the LAST structural premise of the rigidity chain folded into the existing realisation premise. After v446 the rigidity residual was 'the seam…
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Paper context: Research contracts
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SEAM.RIGIDITY.MU4FROMMARKS.01
Original ledger classification: mu4 from the marks [E]/[C]: the four marks ARE mu4 (roots of z^4-1), the order-4 clock z->iz is a 4-cycle in their D4 Moebius stabiliser (cross-ratio 2 fixed), and the form basis omega_k is mu4-graded (eigenvalue i^k) -- so QGEO.SYM.01 follows from marks=mu4 + the existing QGEO.REALIZE.01, no new premise; the rigidity chain closes modulo realisation, SEAM.EQUIV.01 stays [O]
Original claim text
DERIVING the mu4-symmetry of the seam RP data (QGEO.SYM.01) from the four marks (2026-06-30) -- the LAST structural premise of the rigidity chain folded into the existing realisation premise. After v446 the rigidity residual was 'the seam RP data is mu4-symmetric' (QGEO.SYM.01); this module shows it is NOT an independent premise. [E] THE MARKS ARE mu4: the four Gauss-Bonnet marks are exactly {1,i,-1,-i} = roots of z^4-1 (v168/v216/v323). [E] THE CLOCK PERMUTES THE MARKS: z->iz is a 4-cycle of order 4 (1->i->-1->-i->1), fixing the configuration setwise (the order-4 clock of v445). [E] MOEBIUS STABILISER: z->iz is Moebius and preserves the marks' cross-ratio lambda=2 (in the D4 stabiliser, v168) -- the symmetry is geometric, not imposed. [E] THE FORM BASIS IS mu4-GRADED: omega_k=z^{k-1}dz/(z^4-1) -> i^k omega_k under z->iz (k=1,2,3 -> {i,-1,-i}, the three nontrivial mu4 characters; weights (1,2,3)=A3 exponents=Spec(Q_+)) -- the natural basis is mu4-graded, so any geometric datum is mu4-covariant. [C]/[O] SYNTHESIS: QGEO.SYM.01 ('the RP data is mu4-symmetric') is a CONSEQUENCE of marks=mu4 (this) + QGEO.REALIZE.01 (seam = the marked geometry, the EXISTING premise), NO new premise; the rigidity chain (this -> v446 -> v445) closes modulo QGEO.REALIZE.01, SEAM.EQUIV.01 stays [O].
Original classification detail
Identity
Dependencies as recorded
SEAM.EQUIV.01; SEAM.RIGIDITY.CLOCKINV.01; SEAM.RIGIDITY.FORCING.01; QGEO.SYM.01; QGEO.REALIZE.01; v446; v445; v168; v216; v323
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- Research contracts
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- 2026-06-30 ( v449– v453 + Lean + Wolfram — the edge CFT named end-to-end: uniform-in- [formula in source] convergence, three independent reads of [formula in source] , the Ising operator tower, the [formula in source] modular data, and [formula in source] -symmetry derived from the four marks)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
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- Research Contract 1 — [formula in source]tfpt research contracts
- OutcomeRed-team audit
- Front matter / supportResearch contracts
- Research Contract 1 — [formula in source]tfpt research contracts
Registered verification modules
- v453_seam_mu4_from_marks.pyRegistered verification module · frontier
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SEAM.RIGIDITY.MU4FROMMARKS.01: deriving QGEO.SYM.01 (mu4-symmetry of the seam RP data) from the four marks -- the last structural premise of the rigidity chain folded into the existing realisation premise. [E] THE MARKS ARE mu4: the four Gauss-Bonnet marks are exactly {1,i,-1,-i}=roots of z^4-1 (v168/v216/v323). [E] THE CLOCK PERMUTES THE MARKS: z->iz is a 4-cycle of order 4 (fixes the configuration setwise). [E] MOEBIUS STABILISER: z->iz preserves the cross-ratio 2 (in the D4 stabiliser, v168). [E] THE FORM BASIS IS mu4-GRADED: omega_k=z^{k-1}dz/(z^4-1)->i^k omega_k (k=1,2,3->{i,-1,-i}, weights (1,2,3)=A3 exponents). [C]/[O] QGEO.SYM.01 is a CONSEQUENCE of marks=mu4 + QGEO.REALIZE.01 (the EXISTING premise), NO new premise; the rigidity chain (this->v446->v445) closes modulo realisation, SEAM.EQUIV.01 stays [O]. Wolfram-mirrored.
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