2026-07-21 (XXI) ( v510 — the freedom attack on the v506/ v507 alignment bit promoted as SEAM.BIT.FREEDOM.01: the POSITION half of the bit is TOPOLOGY — the collar deck is a covering deck (the [formula in source] seam of the M\"obius double), and covering decks act FREELY; the DIHEDRAL THEOREM (on the seam circle the free involution is UNIQUE [formula in source] the antipode: [formula in source] involutions in [formula in source] , exactly [formula in source] free; [formula in source] complete census, brute force [formula in source] at [formula in source] ; quotient topology circle/antipode [formula in source] CIRCLE [formula in source] vs circle/reflection [formula in source] INTERVAL [formula in source] — the [formula in source] Gauss–Bonnet budget needs closed circles, so freeness IS the established P1 datum); NONSPLIT [formula in source] FREE over the complete [formula in source] -involution census in exact [formula in source] (free [formula in source] nonsplit with [formula in source] roots [formula in source] the [formula in source] tower; ALL [formula in source] reflections [formula in source] scalar [formula in source] , split, [formula in source] roots); the REAL-STRUCTURE classification (deck-commuting families [formula in source] exactly two; [formula in source] equatorial, deck FREE; silver [formula in source] real axis THROUGH the deck poles, NOT free — the edge class is the restriction of the BRANCHED pillowcase involution; hexagonal [formula in source] NO mark-fixing real structure, RP-incompatible); HARMONIC [formula in source] FREE [formula in source] exactly ( [formula in source] ); honest counterwitnesses (``nonsplit [formula in source] clock'' is FALSE: [formula in source] and [formula in source] with [formula in source] ); the alignment bit REDUCES from ``harmonic AND central'' to the square-modulus datum [formula in source] alone; Wolfram extension [formula in source] (verified engine run); NO marker moves — QGEO.REALIZE.01 stays / , AX.P2.01 stays an axiom, P2-R1 halved)
Research changelog · TFPT — Introduction and reading guide
v510_seam_bit_freedom.py ( SEAM.BIT.FREEDOM.01, 30 checks, [formula in source] s; sympy [formula in source] exact [formula in source] Fractions, no floats; deterministic). Part A (dihedral/fermionic): the involution census in [formula in source] on the refined [formula in source] -point seam circle ( [formula in source] : [formula in source] involutions, EXACTLY ONE free [formula in source] the antipodal shift by [fo
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documents/changelog/sections/005-source.tex · lines 186 to 242
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