A registered result in its source context
Original ledger classification: Geometric premise [E]/[C]/[O] (rigidity; minimal axiom)
Recorded statement: QGEO RIGIDITY / MINIMAL-AXIOM FORM (2026-06-18): the bedrock QGEO.SYM.01 is sharpened to its minimal form -- the single bare statement 'the raw RP seam admits the carrier mu4 clock as a conformal symmetry permuting its 4 marks' (= omega o…
Statement, assumptions and sources
Paper context: tfpt research contracts
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QGEO.SYM.02
Original ledger classification: Geometric premise [E]/[C]/[O] (rigidity; minimal axiom)
Original claim text
QGEO RIGIDITY / MINIMAL-AXIOM FORM (2026-06-18): the bedrock QGEO.SYM.01 is sharpened to its minimal form -- the single bare statement 'the raw RP seam admits the carrier mu4 clock as a conformal symmetry permuting its 4 marks' (= omega o rho = omega) FORCES everything: square config (cross-ratio 2 <=> j=1728 <=> order-4 CM auto), the unique flat pillowcase metric (Troyanov, v214), mark-local Z4-Fourier DtN (v201/v264), [rho,Lambda]=0 and omega o rho=omega (v198). Neg controls: generic config j!=1728 (Z2 only), hexagonal j=0 (Z6, order 6). Arithmetic rigidity (Belyi/dessins): {1,i,-1,-i} over Q, j=1728 CM by Z[i]. Closes the REDUCTION (axiom => everything), NOT the axiom -- the bare order-4 symmetry stays the one foundational postulate.
Original classification detail
[E] order-4 symmetry forces cross-ratio 2 / j=1728 / the pillowcase / mark-local DtN / omega o rho=omega (the whole chain) + neg controls (generic Z2, hex Z6); [C] arithmetic/Galois rigidity (j=1728 CM, {1,i,-1,-i} over Q); [O] the bare order-4 symmetry (QGEO.SYM.01) stays the one foundational postulate -- the reduction is closed, the axiom is not.
Dependencies as recorded
QGEO.SYM.01; QGEO.PILLOW.01; QGEO.ENERGY.03; QGEO.RIGID.01; QGEO.MODULAR.01
Open the original ledger source, row 360Location recorded in the ledger
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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
- 2026-06-18 (The bedrock in its minimal-axiom form: a rigidity theorem for QGEO.SYM.01, v267)Research changelog; TFPT — Introduction and reading guide
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
Navigation through a shared verification script
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- Honest statusinput 4 origin theory
- 2026-06-19 (an infinite-derivative narrowing of the Stelle-ghost attack, the double-pushout figure, and the ``constancy of [formula in source] '' framing of SEAM.EQUIV.01: v304)Research changelog; TFPT — Introduction and reading guide
- Honest statusorigin theory
- 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)Research changelog; TFPT — Introduction and reading guide
- Honest statusOrigin Theory
- 2026-06-24 (The arithmetic hull + the [formula in source] -character CM duality: v403, v404)Research changelog; TFPT — Introduction and reading guide
- Honest statusinput 4 origin theory
Registered verification modules
- v267_qgeo_rigidity_minimal_axiom.pyRegistered verification module · registry
Read the description recorded in the module registry
QGEO.SYM.02 — the rigidity / minimal-axiom form of the bedrock QGEO.SYM.01: proves the ONE bare statement ('the raw RP seam admits the carrier mu4 clock as a conformal symmetry permuting its 4 marks' = omega o rho = omega) FORCES everything downstream; does NOT close the axiom. [E] an order-4 Moebius orbit {a,ia,-a,-ia} has cross-ratio 2 (independent of a); [E] cross-ratio 2 <=> j=1728 (only lambda in {-1,1/2,2}) <=> the order-4 CM automorphism (x,y)->(-x,iy) on y^2=x^3-x; [E] downstream forced: square => unique flat pillowcase (Troyanov, v214) => curvature at the 4 marks => Z4-Fourier DtN (v201/v264) => [rho,Lambda]=0 => omega o rho=omega (v198); [E] neg controls (generic j!=1728 only Z2; hexagonal j=0 has Z6, order 6 != 4); [C] arithmetic rigidity ({1,i,-1,-i} over Q, j=1728 CM by Z[i], Belyi/dessins); [O] the bare order-4 symmetry stays the one foundational postulate (like c=const). Closes the reduction (axiom=>everything), not the axiom.
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