Original ledger status
QGEO.RIGID.01
THE QGEO RIGIDITY THEOREM (the FINITE half of the Tier-1 hinge GATE.QGEO, hardened to [E]; v167 Tier 1): a genus-0 boundary with a faithful D4 action, one orbit of four parabolic marks, H^1 of rank 3 and the nontrivial mu4-character decomposition chi_1+chi_2+chi_3 is Moebius-equivalent to P^1 minus mu4 with the (1,2,3) grading. EXACT CHECKS: (1) mu4={1,i,-1,-i} has cross-ratio 2 and Moebius stabiliser <z->iz, z->1/z> of order 8 = |D4| (a 4-cycle of the marks + a reflection), faithful -> four marks with faithful D4 are the mu4 square up to Moebius (sharp 3-transitivity fixes 3 marks, D4 fixes the 4th); (2) P^1 minus four points has b1 = 4-1 = 3 = N_fam (residue-sum-zero); (3) omega_k = z^{k-1}dz/(z^4-1), k=1,2,3, are mu4-eigenforms with z->iz eigenvalue i^k = the three nontrivial characters chi_1,chi_2,chi_3, weights (1,2,3) = A3 exponents = Spec(Q_+) (FLAV.QGEO.02, Q_+ spec {1,2,3}). The cohomological + rigidity content of GATE.QGEO is thus closed exactly; NOT a new structural class -- it hardens the existing Tier-1 hinge
Recorded status: Identity/Lattice [I]/[L] (the finite rigidity theorem exact)
Dependencies, status detail and source
Canonical status: Identity/Lattice [I]/[L]: QGEO finite rigidity closed (cross-ratio 2, D4 order 8, b1=3, chi_k=A3 exps); the math half of GATE.QGEO
Dependencies: GATE.QGEO.01; FLAV.QGEO.02; FLAV.QGEO.01; v137_qplus_cohomology.py
Scripts: v168_qgeo_rigidity.py
Location: tfpt_research_contracts
Original CSV, line 252Original ledger status
QGEO.REALIZE.01
THE SEAM-COLLAR REALISATION (the physical half of GATE.QGEO; the one Tier-1 realisation premise, stays [C]/[O]): the QGEO Rigidity Theorem (QGEO.RIGID.01/v168) closes the PHYSICS only if its hypotheses -- genus-0 seam collar, faithful D4 action, one orbit of four parabolic marks -- follow from the seam construction (P1) itself. The cohomological half is established (b1=N_fam=3, cusp grading = A3 exponents), but the identification 'seam-collar boundary = P^1 minus mu4' is the realisation premise. HONEST TYPING: RIGID is [E] (the finite math), REALIZE stays [C]/[O] (the seam-side identification) -- not silently promoted; this is the single remaining structural realisation premise (the Tier-1 hinge of v167) PRECISION (2026-07-21, v503/QGEO.EMERGE.LIGHT.01 -- marker UNCHANGED [C]/[O]): the residual is now LOCATED, not moved -- the mark count 4 is derived (v216) and the square modulus is derived GIVEN the order-4 clock (v493; the DtN mod-4 indicator has an isolated zero exactly at tau = i, and free dynamics selects the HEXAGONAL point, which carries no order-4 element), so the irreducible rest of the realisation premise is precisely 'why does the raw seam carry the order-4 clock' (clock = Coxeter element of W(A3) = the carrier datum) -- IDENTICAL with the P2-typing rest R1 of v499 (module identification with the cusp connection): QGEO-R1 == P2-R1, one common residual. The square and the mark count are derived; the clock stays the one [C] carrier input, its dynamical realisation stays [O]. PRECISION (2026-07-21, v506/SEAM.CLOCK.RIGIDITY.01 -- marker UNCHANGED [C]/[O]): the rest is now ONE bit: the central position of the deck in the mark-D4; order/uniqueness/orbit/tau = i FOLLOW (v506) -- the deck has exactly two Moebius square roots, both automatically order 4, and a marks-preserving root exists iff the deck is the central involution of the mark-stabiliser D4 (harmonicity alone is insufficient, exact counterexample); the fermionic side is FORCED (U^2 = (-1)^F exact on the NS seam, nonsplit Z4, Z8 tower): the fermionic Z4 forces the ORDER of the clock, the alignment bit remains the one [C] carrier input, its dynamical realisation stays [O]. PRECISION (2026-07-21, v507/SEAM.BIT.ORIGIN.01 -- marker UNCHANGED [C]/[O]): the bit is characterised as a tetrad (clock exists <=> tau = i AND deck = the CM-fixed half-period translation <=> Stab = D4 AND deck central <=> deck pairs harmonically separated <=> NS Fock lift nonsplit) and the class of the deck IS derived (half-period translation of the seam double: every mark-free collar involution descends from a half-period translation, v507); only its position (which half-period) stays the input -- the common torus origin does NOT force the alignment (in-family silver witness), so the bit is no tautology; its fermionic face is the non-split NS Fock extension class (Fidkowski-Kitaev-type). PRECISION (2026-07-21, v510/SEAM.BIT.FREEDOM.01 -- marker UNCHANGED [C]/[O]): the alignment bit is HALVED -- its position half (deck central vs edge in the mark-D4) is TOPOLOGY: the collar deck is a covering deck, hence free on the seam circle; the unique free involution is the antipode (complete dihedral census), NONSPLIT <=> FREE over all 17 involutions, and the edge class is excluded (it is the restriction of the BRANCHED pillowcase involution -- v510); the remaining carrier input is the square-modulus datum tau = i alone (equivalently: the clock / the CM-fixed half-period / the non-split extension class on the free seam circle); its dynamical realisation stays [O]. PRECISION (2026-07-22, v512/SEAM.TAU.FLAG.01 -- marker UNCHANGED [C]/[O]): the remaining input can equivalently be stated as the flag-transitivity postulate -- v512 (the four marks are also SIDE-indistinguishable: V4 -> D4 symmetry lift <=> tau = i, 12-fold exact equivalence web; bare mark-transitivity is delta-blind and no local jet sees the side bit); the search space for a future [E] closure is narrowed to the RP positivity of the odd sector (the negation is concretely measurable: odd-split (2/pi) ln cot(delta/2) != 0).
Recorded status: Physical [C] (seam-side identification) + [O] (open realisation)
Dependencies, status detail and source
Canonical status: Physical [C]/[O]: seam-collar = P1-mu4 stays the one realisation premise (RIGID [E], REALIZE [C]); rest PRECISED by v503 and REDUCED by v506: marks + square derived, the order-4 clock reduced to ONE alignment bit ('the collar deck is the central involution of the mark-D4' -- order/uniqueness/orbit/tau = i follow, the fermionic Z4 order is forced, U^2 = (-1)^F); QGEO-R1 == P2-R1 = this single bit -- marker unchanged; v507 derives the CLASS of the deck (half-period translation) and characterises the bit as a tetrad with the non-split NS Fock extension class as its physical face -- only the position stays the input; v510 halves the bit: the position half is topology (covering-deck freeness excludes the edge class), the rest is the square-modulus datum tau = i alone; v512 restates the rest as the flag-transitivity postulate (V4 -> D4 symmetry-lift bit) and narrows the [E]-closure search space to the RP positivity of the odd sector
Dependencies: QGEO.RIGID.01; GATE.QGEO.01; GATE.METRIC.19; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; v503; SEAM.CLOCK.RIGIDITY.01; v506; SEAM.BIT.ORIGIN.01; v507; SEAM.BIT.FREEDOM.01; v510; SEAM.TAU.FLAG.01; v512
Scripts: v168_qgeo_rigidity.py
Location: tfpt_research_contracts;origin_theory
Original CSV, line 253