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Original ledger classification: Structural lemma [E]/[C] (prime-2 necessary, not autonomous: the sheet involution has spectrum {+-1} and projector {0,1}, none in (0,1), so no prime-2-only attractor; reuses established atoms, no new number) -- closes the v394 open-question corollary

Recorded statement: CLOSED AS A STRUCTURAL LEMMA (2026-06-24, v409): prime-2 (the seam sheet involution) is NECESSARY but NOT autonomous, so no prime-2-only attractor exists -- resolving the v394 open question's falsifiable corollary. The facets are prime-2…

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Paper context: Origin Theory

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RES.COXETER.SYMMETRY.01

Original ledger classification: Structural lemma [E]/[C] (prime-2 necessary, not autonomous: the sheet involution has spectrum {+-1} and projector {0,1}, none in (0,1), so no prime-2-only attractor; reuses established atoms, no new number) -- closes the v394 open-question corollary

Original claim text

CLOSED AS A STRUCTURAL LEMMA (2026-06-24, v409): prime-2 (the seam sheet involution) is NECESSARY but NOT autonomous, so no prime-2-only attractor exists -- resolving the v394 open question's falsifiable corollary. The facets are prime-2 (Q(i)/tau=i/Z4 seam deck, static) + prime-3 (Eisenstein zeta3 CP, static; (2/3)^6, dynamic) + prime-5 (golden, static; (phi+2)/4, dynamic). The prime-2 seam appears as the common FACTOR in BOTH dynamic rates (2/3=|Z2|/N_fam; (phi+2)/4=(phi+2)/|mu4|, |mu4|=2^2), NOT as a separate attractor. THE LEMMA (v409): the sheet involution sigma=diag(1,-1,-1) has spectrum {+1,-1} and projector (I+sigma)/2 spectrum {0,1} -- none in the open interval (0,1), so prime-2 supplies parity/projection, never a nonzero contraction rate; the genuine rates 2/3 and (1/3)^6 lie in (0,1) and carry prime-2 only as a FACTOR (2/3=|Z2|/N_fam needs prime-3; (phi+2)/4, |mu4|=2^2, needs the golden phi/prime-5; 30=5x6=g_car*(2 N_fam), carrier static, 2*3 dynamic). [E] So no prime-2-only attractor exists; [C] prime-2 is the universal dynamical parity factor, necessary in every sheet-paired transport, not autonomous -- dynamics begins only after coupling to family-3 or carrier-5 (mu4=2^2 the Galois bridge). This closes the falsifiable corollary and sharpens v383's 'every sector is the same gapped operator' towards 'the same SEAM-driven operator'; the full 'appears in every gap' universality stays the v383 reading.

Original classification detail

[E] the sheet involution/projector spectra carry no contraction rate in (0,1), so no prime-2-only attractor exists; [C] prime-2 is the universal dynamical parity factor (necessary, not autonomous), dynamics needs coupling to family-3 or carrier-5. Closes the v394 open question's falsifiable corollary; the full 'appears in every gap' universality stays the v383 reading. No new number.

Dependencies as recorded

COXETER.ATOMS.01; v314; v383; v390; v394

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Registered verification modules

  • v394_coxeter_atoms.pyRegistered verification module · frontier
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    COXETER.ATOMS.01: the clockwork coherence -- the three Coxeter primes ARE the three anchor atoms ARE the three number-field facets, and the prime-2 seam is the common dynamical factor. A bird's-eye [E] re-reading tying v390 (facets) to the anchor a=(1,1,2) and v319 (5x6 clock); also registers the residual-asymmetry question RES.COXETER.SYMMETRY.01 as a tracked [O]. [E] a=(1,1,2): e3=2=|Z2|, p0=3=N_fam, e2=5=g_car, product 30=h(E8). [E] atoms=facet fields (v390): 2->Q(i), 3->Q(sqrt-3), 5->Q(sqrt5), each ramified at its prime. [E] chi=729/665=3^6/(3^6-2^6), (2/3)^6=2^6/3^6, exponent 6=2 N_fam. [E] both dynamic rates carry prime-2: 2/3=|Z2|/N_fam, (phi+2)/4=(phi+2)/|mu4|, |mu4|=4=2^2. [E] 30=5x6=g_car*(2 N_fam). [C] the seam is the universal dynamic engine. [O] RES.COXETER.SYMMETRY.01 registered, not closed.

  • 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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