A registered result in its source context
Original ledger classification: Identity/Lattice [E]/[L] (cross-ratio 2 => j=1728 => order-4 CM clock) + premise [O] (canonical constant-curvature metric)
Recorded statement: PILLOWCASE REDUCTION OF QGEO.SYM.01 (2026-06-15): the two separately-tracked open QGEO residuals -- QGEO.ISO.01 (v180; the carrier clock is an order-4 ISOMETRY of the seam metric) and QGEO.SUBPRIN.01 (v201; the DtN sub-principal symbol is…
Statement, assumptions and sources
Paper context: tfpt research contracts
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QGEO.PILLOW.01
Original ledger classification: Identity/Lattice [E]/[L] (cross-ratio 2 => j=1728 => order-4 CM clock) + premise [O] (canonical constant-curvature metric)
Original claim text
PILLOWCASE REDUCTION OF QGEO.SYM.01 (2026-06-15): the two separately-tracked open QGEO residuals -- QGEO.ISO.01 (v180; the carrier clock is an order-4 ISOMETRY of the seam metric) and QGEO.SUBPRIN.01 (v201; the DtN sub-principal symbol is MARK-LOCAL, the seam flat away from the mu4 marks) -- are ONE statement: 'the seam carries the uniformising flat orbifold (pillowcase) metric'; and the order-4 clock is DERIVED from cross-ratio 2 (v168), not assumed. (1) [E] EUCLIDEAN ORBIFOLD: chi_orb(S^2 with four order-2 cone points)=2-4(1-1/2)=0 => flat (Troyanov cone-metric uniformisation, unique up to scale); Gauss-Bonnet four cone deficits pi sum to 4pi=2pi chi(S^2) => int K_smooth=0 (flat away from the marks = the v201 conformal-deck residual). (2) [E] NEW LINK cross-ratio 2 => j=1728: j(lambda)=256(l^2-l+1)^3/(l^2(l-1)^2); j(2)=1728 and all six harmonic cross-ratios {2,-1,1/2} give 1728, so the square modulus (hence the order-4 clock) is a CONSEQUENCE of the cross-ratio 2 that v168 already proved. (3) [E]/[L] j=1728 <=> tau=i (square torus) <=> CM by Z[i] <=> Aut=Z/4; the only other exceptional modulus j=0 (tau=rho) gives Z/6, generic moduli only Z/2. (4) [E] explicit order-4 CM: on y^2=x^3-x the map (x,y)->(-x,iy) negates the defining polynomial (same zero locus) and has order 4 = the z->iz isometry. (5) [E] NEGATIVE CONTROL: a generic cross-ratio lambda=3 gives j=21952/9 not in {0,1728} => Aut=Z/2, no order-4 clock -- specific to mu4. (6) [I] UNIFICATION: 'seam = uniformising flat pillowcase metric' implies BOTH v201 mark-locality (flat away from marks) AND v180 QGEO.ISO.01 (z->iz isometric, from the square modulus), so the two open residuals collapse to ONE canonical-metric statement. (7) [O] does NOT close QGEO.SYM.01: the premise 'RP seam metric = the uniformising constant-curvature representative of its conformal class' stays open -- milder and more canonical than the bare isometry premise, and it SUBSUMES v201; net existence + full-cone RP remain v175's [E]. Wolfram-mirrored (chi_orb=0, cross-ratio 2 => j=1728, harmonic orbit, lemniscatic CM order 4, neg control); Klein-J modular values mpmath-numerical
Original classification detail
Identity/Lattice [E]/[L] + premise [O]: cross-ratio 2 (v168) => j=1728 => square modulus => order-4 CM clock (DERIVED, not assumed); unifies the two open QGEO residuals (v180 ISO + v201 SUBPRIN) into ONE canonical flat-pillowcase-metric statement; does NOT close QGEO.SYM.01 (the constant-curvature representative choice stays open, milder/more canonical than the bare isometry premise)
Dependencies as recorded
QGEO.ISO.01; QGEO.SUBPRIN.01; QGEO.RIGID.01; v168_qgeo_rigidity.py; v180_clock_is_mobius.py; v201_seam_subprincipal_marks.py
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- Research contracts
Sections that explicitly cite this claim
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- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Research Contract 1 — [formula in source]tfpt research contracts
- 2026-06-15 (QGEO: the pillowcase reduction — two residuals become one canonical metric)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportResearch contracts
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- 2026-07-20 (V) ( v492 [formula in source] the new research contract CELEST.SEAM.01 — the celestial-holographic route: the [formula in source] [formula in source] -glue is the flat [formula in source] monodromy of the equivariant celestial chiral algebra on the [formula in source] ALE space, with the exact correction clock [formula in source] deck (spin bridge [formula in source] , [formula in source] ); verdict B, no gate closes, no marker moves, SEAM.EQUIV.01 stays )Research changelog; TFPT — Introduction and reading guide
- 2026-06-15 (QGEO: the bedrock recast as a falsifiable kill-test)Research changelog; TFPT — Introduction and reading guide
- Honest statusinput 4 origin theory
- 2026-06-18 (the two proof routes decomposed: Route (i) and Route (ii) into lemma chains, v284/ v285)Research changelog; TFPT — Introduction and reading guide
- Research Contract — CELEST.SEAM.01 (the celestial and twistor continuum route)tfpt research contracts
- Honest statusorigin theory
- Research Contract — CELEST.SEAM.01 (the celestial and twistor continuum route)tfpt research contracts
- 2026-07-20 (VII) ( v493 — WP2 of CELEST.SEAM.01 executed: the clock-invariant deformation [formula in source] is a PURE SCALE with the [formula in source] pillowcase frozen, the [formula in source] clock IS the Picard–Lefschetz/Coxeter monodromy of the seam deformation, and the BHS deformed-algebra pattern transfers [formula in source] with no sector leak; verdict B, contract updated to WP2 done, no gate closes, no marker moves, SEAM.EQUIV.01 and the v216 residual stay )Research changelog; TFPT — Introduction and reading guide
All 21 linked sections
- 2026-06-21 (TOE-roadmap solver round: the SEAM.EQUIV.01 keystone certificate, the modular bedrock, the carrier [formula in source] SM closure, the gap-decoupled measure, and the sharpened hypergraph substrate: v308– v312)
- The celestial and twistor continuum route
- 2026-06-22 (endgame round: the necessity of [formula in source] reduced to the order-4 CM selection, the QG metric sector characterized as the conformal-mode problem, [formula in source] and the [formula in source] commutation grounded in Lean: v331, v332, FORM.MU4COMM.01)
- Honest status
- 2026-06-24 (The arithmetic hull + the [formula in source] -character CM duality: v403, v404)
- Front matter / support
- Research Contract — CELEST.SEAM.01 (the celestial and twistor continuum route)
- The celestial and twistor continuum route
- Honest status
- 2026-06-17 (The Modular Spectral Closure: Dirac as covariance induction, the modular cutoff, the Kummer/K3 unification, and the assembly certificate, v258– v261)
- The celestial and twistor continuum route
- 2026-07-21 (XVII) ( v507 — the tautology attack on the v506 alignment bit, executed and refuted, promoted as SEAM.BIT.ORIGIN.01: the ORIGIN THEOREM — marks [formula in source] fixed points of the hyperelliptic involution [formula in source] the [formula in source] half-periods [formula in source] of the seam double, and EVERY mark-free collar deck IS a half-period translation (in all three stabiliser types [formula in source] the free involutions are exactly the three [formula in source] : the CLASS of the deck is forced, its POSITION is not); the CORE SOLVE (symbolic [formula in source] ): a marks-preserving root of [formula in source] exists [formula in source] (one per [formula in source] ; intrinsic: the deck pairs separate harmonically), union [formula in source] the harmonic orbit [formula in source] ; the EQUIVALENCE TETRAD ``Uhr existiert [formula in source] ( [formula in source] AND deck [formula in source] CM-fixed half-period) [formula in source] (Stab [formula in source] AND deck central) [formula in source] deck pairs harmonic [formula in source] NS Fock lift NONSPLIT''; the FERMION DICHOTOMY (the new physical face): central [formula in source] nonsplit vs edge [formula in source] SPLIT ( [formula in source] vs [formula in source] roots — the bit IS a Fidkowski–Kitaev-type extension class, distinguished but not derived); the silver set [formula in source] the in-family misalignment witness; Wolfram extension [formula in source] (verified engine run); NO marker moves — the bit stays , SEAM.CLOCK.RIGIDITY.01/ QGEO.REALIZE.01/ P2.TYPING.01 text precised only)
- 2026-06-18 (The bedrock in its minimal-axiom form: a rigidity theorem for QGEO.SYM.01, v267)
Registered verification modules
- v214_seam_pillowcase.pyRegistered verification module · registry
Read the description recorded in the module registry
the PILLOWCASE reduction of QGEO.SYM.01 (QGEO.PILLOW.01): the two separately-tracked open QGEO residuals — v180/QGEO.ISO.01 (the carrier clock is an order-4 isometry of the seam metric) and v201/QGEO.SUBPRIN.01 (the DtN sub-principal symbol is mark-local, the seam flat away from the μ4 marks) — are ONE statement, 'the seam carries the uniformising flat orbifold (pillowcase) metric', and the order-4 clock is DERIVED from cross-ratio 2 (v168), not assumed. A reduction + unification, NOT a closure. [E] χ_orb(S²(2,2,2,2))=2−4(1−1/2)=0 ⟹ flat (Troyanov cone-metric uniformisation, unique up to scale; Gauss–Bonnet four cone deficits π sum to 4π=2π·χ(S²) ⟹ ∫K_smooth=0, flat away from the marks). [E] NEW LINK: cross-ratio(μ4)=2 (v168) ⟹ j(λ)=256(λ²−λ+1)³/(λ²(λ−1)²) with j(2)=1728 (all six harmonic cross-ratios {2,−1,1/2} give 1728) ⟹ the square modulus, hence the order-4 clock, is a consequence of cross-ratio 2. [E]/[L] j=1728 ⟺ τ=i (square torus) ⟺ CM by ℤ[i] ⟺ Aut=ℤ/4 (only other exceptional modulus j=0 gives ℤ/6; generic ℤ/2). [E] explicit CM (x,y)→(−x,iy) on y²=x³−x has order 4 = z→iz. [E] negative control λ=3 ⟹ j=21952/9 ∉ {0,1728} ⟹ Aut=ℤ/2, no order-4 clock. [I] UNIFICATION: 'seam = uniformising flat pillowcase metric' implies BOTH v201 mark-locality AND v180 QGEO.ISO.01. [O] does NOT close QGEO.SYM.01 — the canonical constant-curvature-metric premise stays open, milder/more canonical than the bare isometry premise; net existence + full-cone RP remain v175's [E].
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