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Original ledger classification: S3 chirality from P1 [E]/[C]: c3's 8=|Z2|*(intK/pi)=2*4 is the one-sided count; a reflection sends Chern C->-C so a two-sided boundary forces C=0, the one-sided seam (no reflection, the |Z2| of c3's 8pi) forces C!=0 -- chirality is a CONSEQUENCE of P1, magnitude c_-=8; SEAM.EQUIV.01 stays [O]

Recorded statement: The edge chirality (c_- != 0, the S3 topological phase) is FORCED by the one-sidedness that defines c3=1/(8pi) (P1) (2026-06-30, G6 of the post-F next steps) -- so S3 is a CONSEQUENCE of axiom P1, not an independent input. [E] c3's '8' IS…

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SEAM.S3.FROM-P1.01

Original ledger classification: S3 chirality from P1 [E]/[C]: c3's 8=|Z2|*(intK/pi)=2*4 is the one-sided count; a reflection sends Chern C->-C so a two-sided boundary forces C=0, the one-sided seam (no reflection, the |Z2| of c3's 8pi) forces C!=0 -- chirality is a CONSEQUENCE of P1, magnitude c_-=8; SEAM.EQUIV.01 stays [O]

Original claim text

The edge chirality (c_- != 0, the S3 topological phase) is FORCED by the one-sidedness that defines c3=1/(8pi) (P1) (2026-06-30, G6 of the post-F next steps) -- so S3 is a CONSEQUENCE of axiom P1, not an independent input. [E] c3's '8' IS THE ONE-SIDED COUNT: 8pi=|Z2| intK=2(2pi chi(S^2))=2*4pi (chi=2, topological), so 8=|Z2|*(intK/pi)=2*4 (Wolfram-mirrored, v73). [E] REFLECTION REVERSES C: the explicit 2-band p+ip Chern model has C=+1 and the orientation-reversed (ky->-ky) model C=-1. [E] TWO-SIDED => NON-CHIRAL: any GAPPED boundary carrying an orientation-reversing (reflection) symmetry must have C=-C=0 (c_-=0, gappable/non-chiral). [C] ONE-SIDED => CHIRAL: the one-sided seam is precisely the |Z2| quotient with NO reflection (the same |Z2| that gives c3 its 8pi), so C!=0 is forced -- S3 follows from P1. [C]/[O] magnitude c_-=8=rank E8=g_car+N_fam is the SAME integer 8 as c3's one-sided count; SEAM.EQUIV.01's continuum EXISTENCE (v336) stays [O].

Original classification detail

Numerical

Dependencies as recorded

SEAM.EQUIV.01; SEAM.EQUIV.EDGE.CHERN.01; SEAM.S3.LATTICE.01; SEAM.EQUIV.CONTINUUM.03; v73; v367; v447; v336

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Registered verification modules

  • v456_seam_chirality_from_c3.pyRegistered verification module · frontier
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    SEAM.S3.FROM-P1.01: the edge chirality (c_-!=0, the S3 phase) is FORCED by the one-sidedness that defines c3=1/(8pi) (P1), so S3 is a CONSEQUENCE of P1 (G6). [E] c3's '8' IS THE ONE-SIDED COUNT: 8pi=|Z2|*intK=2*(2pi*chi(S^2))=2*4pi, integer 8=|Z2|*(intK/pi)=2*4. [E] REFLECTION REVERSES C: the p+ip model has Chern C=+1, its ky->-ky partner C=-1. [E] TWO-SIDED => NON-CHIRAL: a gapped boundary with an orientation-reversing symmetry has C=-C=0 (c_-=0). [C] ONE-SIDED => CHIRAL: the one-sided seam (the same |Z2| of c3) has no such reflection, so C!=0 is forced. [C]/[O] c_-=8=rank E8 is the SAME 8 as c3's 8pi; S3 follows from P1; SEAM.EQUIV.01 existence stays [O]. Mixed exact (arithmetic, Wolfram) + numerical (Chern flip).

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