A registered result in its source context
Original ledger classification: Reduction [E]/[O] (necessity of H = the order-4 CM selection)
Recorded statement: THE NECESSITY OF H REDUCED (2026-06-22): rotation-invariance of the seam DtN is EQUIVALENT to the four marks sitting at the square (tau=i), so the one open lemma is exactly the order-4 CM selection (the necessity companion to v329's…
Statement, assumptions and sources
Paper context: Origin Theory
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QGEO.NECESS.01
Original ledger classification: Reduction [E]/[O] (necessity of H = the order-4 CM selection)
Original claim text
THE NECESSITY OF H REDUCED (2026-06-22): rotation-invariance of the seam DtN is EQUIVALENT to the four marks sitting at the square (tau=i), so the one open lemma is exactly the order-4 CM selection (the necessity companion to v329's sufficiency). [E] SQUARE => [rho,Lambda]=0: with marks at {0,pi/2,pi,3pi/2} the curvature is Z4-symmetric (Fourier support 0 mod 4), M_f couples only modes differing by 4Z, [rho,Lambda]=0 (the v201/v210 sufficiency, recomputed as a DtN |k|+M_f operator). [E] NECESSITY: moving a mark off the square breaks Z4 and gives [rho,Lambda]!=0 -- so rotation-invariance of the DtN is EQUIVALENT to the square configuration, not merely implied. [E] ORDER-4 CM: the cross-ratio of the square points (1,i,-1,-i) is 2 => j=1728, the lemniscatic CM modulus tau=i with an order-4 automorphism -- the UNIQUE 4-config carrying the mu4 deck (v214/v267); so H <=> square <=> order-4 CM <=> mu4. [E] NEG: a Z2 (period-2) mark set commutes with rho^2 but not rho. [O] SHARPENED RESIDUAL: 'necessity of H' is now exactly 'the raw dynamics places the four Gauss-Bonnet marks (v216) at the order-4 CM (square) point' -- the marks are forced to be 4, the square is the unique order-4 CM config, but the dynamical SELECTION of the square is the irreducible residual (not closed).
Original classification detail
[E] the DtN equivalence (rotation-invariance <=> square marks, both directions) + the order-4 CM identification (cross-ratio 2 => j=1728); [O] the dynamical selection of the square (the residual of H). Reduces the necessity direction to one named configuration; does NOT close the bedrock.
Dependencies as recorded
QGEO.OSGAP.01; QGEO.KEYSTONE.01; QGEO.SUBPRIN.01; v329; v201; v210; v214; v216; v267
Open the original ledger source, row 432Location recorded in the ledger
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- Origin Theory
Sections that explicitly cite this claim
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- 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
Navigation through a shared verification script
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- Honest statusinput 4 origin theory
- The unique attractor: a gapped transport selects ONE fixed pointorigin theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- Honest statusorigin theory
- Honest statusOrigin Theory
- 2026-06-22 (can the one open arrow be fully solved? the closure-mode classification: the flat [formula in source] spherical base locus, four modes, no free theorem – only rigidity or axiom [formula in source] : v347)Research changelog; TFPT — Introduction and reading guide
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
All 10 linked sections
Registered verification modules
- v331_necessity_of_H.pyRegistered verification module · frontier
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The NECESSITY of H reduced: rotation-invariance of the seam DtN is EQUIVALENT to the four marks at the square (tau=i), so the one open lemma is exactly the order-4 CM selection (the necessity companion to v329's sufficiency). [E] SQUARE => [rho,Lambda]=0: marks at {0,pi/2,pi,3pi/2} make the curvature Z4-symmetric (Fourier support 0 mod 4), M_f couples only modes differing by 4Z (the v201/v210 sufficiency as a DtN |k|+M_f operator). [E] NECESSITY: a mark off the square breaks Z4 and gives [rho,Lambda]!=0 -- equivalence, not just implication. [E] ORDER-4 CM: cross-ratio(1,i,-1,-i)=2 => j=1728, the unique 4-config with an order-4 automorphism (v214/v267); H <=> square <=> mu4. [E] NEG: a Z2 mark set commutes with rho^2 but not rho. [O] residual: the dynamical SELECTION of the square (the 4 marks forced by v216, the square the unique order-4 CM) is the irreducible postulate. Does NOT close the bedrock.
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