Follow a result back to its evidence.

SEAM.BIT.FREEDOM.01: the freedom attack on the v506/v507 alignment bit -- the POSITION half of the bit is topology (Part A dihedral/Fock + Part B Moebius/RP in one module; NO marker moves -- the modulus datum stays [C]). Exact sympy + exact Cl(16) Fractions, no floats (30 checks, <5 s). [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: Aut(C_16) = D_16 exactly (brute force over all 8! permutations at N = 8, propagation at N = 16 -- no exotic circle automorphism); N = 9 control: D_9 has ZERO free involutions (even carrier count needed, consistent with 16 = 2^(g_car-1)); QUOTIENT TOPOLOGY (the P1 tie): circle/antipode = CIRCLE (chi = 0, closed), circle/reflection = INTERVAL (chi = 1, 2 boundary corners) -- origin_theory's 6pi = 2pi+2pi+2pi Gauss-Bonnet budget counts three CLOSED circles, so 'the deck is free on the seam circle' IS 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 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) -- 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 branched pillowcase involution's restriction: 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; mu < 0: empty fix locus, no seam); per deck-invariant configuration the mark-fixing real structure 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; 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, harmonicity alone becomes SUFFICIENT. [E] HONEST COUNTERWITNESSES: 'nonsplit => clock' is FALSE -- discrete {0,1,8,9} (shift-invariant, crossing, same nonsplit lift, no clock) and continuous {+-1, +-(3+4i)/5} (free, central, j = 148176/25 != 1728, no clock): the Fock dichotomy measures the CLASS of the arrangement, not the modulus. PREMISE TYPING: (P-i) deck = Moebius covering deck => free on the seam circle [standard covering topology, machine-instantiated; the one-sidedness strand origin_theory/v456]; (P-ii) marks on the Theta-fix seam circle = existing REALIZE [C] content (QGEO.REALIZE.02 hypotheses, v264; v506 A-S2.9 [E] for mu4) -- NO new input; audit finding: circle-freeness was nowhere typed before, v507 F1.1 explicitly admitted the reflection reading -- this door is now closed (a reflection is not a covering deck). [C] THE REDUCED REST (verbatim): 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. Exact (sympy + Cl(16) Fractions).

Linked claims

SEAM.BIT.FREEDOM.01

Paper sections

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