A registered result in its source context

SEAM.BIT.FREEDOM.01

Original ledger classification: Formal/Identity [E] (dihedral theorem: unique free involution = antipode, Aut(C_16) = D_16 complete census, quotient topology chi 0 vs 1; complete Cl(16) Fock census NONSPLIT <=> FREE over all 17 involutions; real-structure classification: two deck-commuting families, per-configuration uniqueness table; harmonic + free => b = +-i exactly; counterwitnesses -- sympy exact / exact Cl(16) Fractions, Wolfram-mirrored) + [C] (the square-modulus datum tau = i); SEAM.CLOCK.RIGIDITY.01, SEAM.BIT.ORIGIN.01 and QGEO.REALIZE.01 stay as they are (text precised only), P2.TYPING.01 R1 halved

Original claim text

THE FREEDOM ATTACK ON THE v506/v507 ALIGNMENT BIT -- THE POSITION HALF OF THE BIT IS TOPOLOGY (2026-07-21; one module, Part A dihedral/fermionic + Part B Moebius/RP-geometry, 30 checks; NO marker moves -- the modulus datum stays [C], SEAM.CLOCK.RIGIDITY.01, SEAM.BIT.ORIGIN.01 and QGEO.REALIZE.01 text-precised only, P2.TYPING.01 R1 halved): v507 left the last identification input as ONE bit with four equivalent faces; the collar deck of the theory is the deck transformation of a COVERING (the Z2 seam of the Moebius double, tfpt_3 'cylinder + one Z2 seam'), and covering decks act FREELY -- a reflection of the circle has fixed points and is never free, so the edge class is EXCLUDED and the fermionic face of the bit FOLLOWS. [E] DIHEDRAL THEOREM: on the N-site seam circle the free involution is UNIQUE = the antipode (N = 16: 17 involutions in D_16, exactly 1 free = shift 8; all 16 reflections have exactly 2 circle fixed points -- sites for even axis, edge midpoints for odd; robustness N = 8, 12); the census is COMPLETE because Aut(C_16) = D_16 exactly (brute force over all 8! permutations at N = 8, propagation argument at N = 16 -- no exotic circle automorphism exists); N = 9 control: D_9 has ZERO free involutions (a Moebius seam circle needs an EVEN carrier count, consistent with 16 = 2^(g_car-1)); QUOTIENT TOPOLOGY (the P1 tie): circle/antipode = CIRCLE (chi = 0, closed), circle/reflection = INTERVAL (chi = 1, two boundary corners) -- origin_theory's Gauss-Bonnet budget 6pi = 2pi+2pi+2pi counts three CLOSED circles, so 'the deck is free on the seam circle' is EXACTLY the established P1 datum 'the seam is a closed circle', not a new premise. [E] NONSPLIT <=> FREE (complete census, exact Cl(16)): ALL 17 involutions of the seam circle implemented -- the free one's NS lift squares to 256 gamma_1...gamma_16 = (-1)^F (nonsplit, 2 square roots in the 32-element dihedral lift group = the Z8 clock tower); ALL 16 reflections have U^2 = SCALAR (2^7 site-axis, 2^8 edge-axis; split, 0 roots) -- the v507 dichotomy holds over the COMPLETE involution census, no exception; chain: deck free => deck = shift 8 => crossing mark pairing (all 28 invariant 4-sets) => U^2 = (-1)^F => central class => the edge/silver arrangement is EXCLUDED; Klein control: the orientation-cover deck is free too (covering vs branched is the discriminator, not Moebius vs Klein); branch control: the pillowcase involution (x,y) -> (-x,-y) has exactly 4 fixed points = the marks and restricts to a REFLECTION on invariant circles -- the edge class IS the restriction of the BRANCHED pillowcase involution: the silver arrangement mistakes the mark-creating branch Z2 for the seam deck. [E] REAL-STRUCTURE CLASSIFICATION (continuum side): the antiholomorphic involutions of P^1 commuting with the celestial deck z -> -z are EXACTLY two families -- mu conj(z), |mu| = 1 (fix locus = a LINE through the deck poles {0, inf}: deck NOT free on it) and mu/conj(z), mu real (mu > 0: fix circle |z| = sqrt(mu) avoiding the poles, deck FREE on it; mu < 0: EMPTY fix locus, no seam); per deck-invariant configuration {+-a, +-b} the mark-fixing real structure is UNIQUE (an antiholomorphic map fixing 3 points pointwise is unique); table: mu4 -> equatorial 1/conj(z), deck FREE (v506 A-S2.9 reproduced); SILVER {+-1, +-(3+2 sqrt 2)} -> real axis THROUGH the poles, deck NOT free; generic-unit {+-1, +-(3+4i)/5} -> equatorial, free WITHOUT harmonicity; hexagonal -> NO mark-fixing real structure at all (cross-ratio complex, marks not concyclic: RP-incompatible, consistent v503); Moebius covering model: the sheet-exchange deck of the cylinder double cover is free on ALL points and restricts to the antipode on the Z2 seam fibre; P1's one-sidedness TYPES the Z2 as the free antipode (quotient RP^2 = the |Z2| of c3 = 1/(|Z2| 2pi chi), v456/v73). [E] HARMONIC + FREE => mu4 EXACTLY: the deck-pair cross-ratio (1-b)^2/(1+b)^2 = -1 solves exactly to b = +-i -- on free-deck seam circles the silver family (harmonic AND edge) is EMPTY, so harmonicity alone becomes SUFFICIENT for the clock. [E] HONEST COUNTERWITNESSES: 'nonsplit => clock' is FALSE -- discrete {0,1,8,9} (shift-invariant, crossing, same nonsplit lift, but the quarter shift does not preserve it) and continuous {+-1, +-(3+4i)/5} (deck free, equatorial, central class, but j = 148176/25 != 1728, no clock): the Fock dichotomy measures the CLASS of the arrangement, not the modulus. PREMISE TYPING [E]/[C]: (P-i) deck = Moebius covering deck => free on the seam circle -- standard covering topology, machine-instantiated (the one-sidedness strand: origin_theory/v456 check 1 [E]; tfpt_3 'cylinder + one Z2 seam' double structure); (P-ii) marks lie on the Theta-fix seam circle = existing REALIZE [C] content (QGEO.REALIZE.02 hypotheses (4)/(5), v264; v506 A-S2.9 [E] for mu4) -- NO new input; AUDIT FINDING documented: 'circle-freeness' was nowhere typed before (only mark-freeness, v216); v507 F1.1 explicitly admitted the reflection reading of the edge class -- v510 closes that door (a reflection is not a covering deck). [C] THE REDUCED REST (verbatim, no overclaim): the alignment bit reduces from 'harmonic AND central' to the square-modulus datum tau = i alone; the position half is topology (edge class excluded), the modulus half stays the one carrier input. No gate closes, no marker moves. CROSS-REFERENCE (2026-07-22, v512/SEAM.TAU.FLAG.01): the modulus half gains the equivalent discrete form 'flag transitivity (V4 -> D4) <=> tau = i' (12-fold exact equivalence web); the counterwitness {+-1, +-(3+4i)/5} is confirmed to pass ALL 7 established unsided tests -- v512.

Original classification detail

Bit freedom [E]/[C]: the collar deck is a covering deck, hence FREE on the seam circle; the unique free involution is the antipode (Aut(C_16) = D_16, complete census) and NONSPLIT <=> FREE over all 17 involutions -- the edge/silver class is topologically EXCLUDED (its real structure's fix circle passes through the deck poles; the edge class is the branched pillowcase involution's restriction); harmonic + free => mu4 exactly, but freeness does not give harmonicity (honest counterwitnesses) -- the alignment bit reduces from 'harmonic AND central' to the square-modulus datum tau = i alone: the position half is topology, the modulus half stays the one [C] carrier input; v512 restates the modulus half as the flag-transitivity bit; no marker moves

Dependencies as recorded

SEAM.BIT.ORIGIN.01; SEAM.CLOCK.RIGIDITY.01; QGEO.REALIZE.01; QGEO.REALIZE.02; QGEO.RIGID.01; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; v216; v264; v456; v490; v492; v503; v506; v507