A registered result in its source context
Original ledger classification: Reduction [E]/[O] (CM selection = unique order-4 modulus + equilibration attractor)
Recorded statement: THE CM-SELECTION MECHANISM (2026-06-22): tau=i (the square) is the UNIQUE order-4 modulus and the attractor of mark-equilibration, so the last keystone residual is 'the deck acts geometrically', not 'why the square' (the v331 follow-up).…
Statement, assumptions and sources
Paper context: Origin Theory
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QGEO.CMSELECT.01
Original ledger classification: Reduction [E]/[O] (CM selection = unique order-4 modulus + equilibration attractor)
Original claim text
THE CM-SELECTION MECHANISM (2026-06-22): tau=i (the square) is the UNIQUE order-4 modulus and the attractor of mark-equilibration, so the last keystone residual is 'the deck acts geometrically', not 'why the square' (the v331 follow-up). [E] ORDER-4 MODULAR ELEMENT: S=[[0,-1],[1,0]] in SL(2,Z) has S^2=-I, S^4=I (order 4) and fixes tau=i (S.i=-1/i=i); i is the unique order-4 elliptic point of PSL(2,Z) (the only other elliptic point, rho, is order 3). [E] DECK ORDER -> CM POINT (the selector): order-4 deck -> cross-ratio 2 -> j=1728 -> tau=i (square); order-3 deck -> equianharmonic lambda (lambda^2-lambda+1=0) -> j=0 -> tau=rho (hexagon); the carrier deck |mu4|=4 (anchor-derived) selects tau=i uniquely, not rho. [E] MARK-EQUILIBRATION ATTRACTOR: four points on the circle minimizing the log-energy E=-sum log|z_i-z_j| (Fekete points) converge to EQUAL spacing (gaps pi/2 = the square) from random starts, and the square's energy beats random perturbations -- the unique (mod rotation) attractor of symmetric mark-equilibration. [E] NEG: an order-3 (hexagonal) deck selects tau=rho (j=0), NOT the square (j=1728) -- deck-order-specific. [O] SHARPENED RESIDUAL: given the mu4 deck, tau=i is FORCED (arithmetic) and is the equilibration attractor (dynamical); only that the seam recovery dynamics IS this mu4 equilibration acting GEOMETRICALLY (a conformal automorphism, = QGEO.SYM.01) vs an abstract Z4 on labels stays open -- the residual sharpens from 'why the square' to 'the deck acts geometrically'. NOT a closure.
Original classification detail
[E] the modular order-4 uniqueness (S fixes i) + the deck-order->CM map (order-4 => j=1728 => tau=i) + the mark-equilibration (Fekete) attractor; [O] the deck acting geometrically (= QGEO.SYM.01). Sharpens the v331 residual from 'why the square' to 'the deck acts geometrically'; does NOT close the bedrock.
Dependencies as recorded
QGEO.NECESS.01; QGEO.SYM.01; v331; v214; v216; v267; v56
Open the original ledger source, row 435Location recorded in the ledger
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- Origin Theory
Sections that explicitly cite this claim
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- 2026-06-22 (closing round: the CM-selection mechanism (the last keystone residual), the conformal-mode resolution, and the Conway–Sloane even-unimodular side in Lean: v333, v334, FORM.CARTAN.DET.01)Research changelog; TFPT — Introduction and reading guide
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.
- Honest statusinput 4 origin theory
- 2026-06-22 (keystone unification + status reframe: ONE open theorem, ONE decoupled foreign problem: v335, FORM.QGEO.BW.01 qgeoSymIsCorollary)Research changelog; TFPT — Introduction and reading guide
- Honest statusorigin theory
- Honest statusOrigin Theory
- 2026-06-23 (Bird's-eye synthesis: two new structural patterns — v383 the universal spectral-gap principle and v384 the residual is certification, not construction)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Honest statusinput 4 origin theory
- 2026-06-24 (Completing the Coxeter clock: v390 the prime-2 (Gaussian / [formula in source] -seam) facet of [formula in source] )Research changelog; TFPT — Introduction and reading guide
Registered verification modules
- v333_cm_selection.pyRegistered verification module · frontier
Read the description recorded in the module registry
The CM-selection mechanism: tau=i (the square) is the UNIQUE order-4 modulus and the attractor of mark-equilibration, so the last keystone residual is 'the deck acts geometrically', not 'why the square' (the v331 follow-up). [E] ORDER-4 MODULAR ELEMENT: S=[[0,-1],[1,0]] in SL(2,Z) has S^2=-I, S^4=I and fixes tau=i; the unique order-4 elliptic point of PSL(2,Z) (the other, rho, is order 3). [E] DECK ORDER -> CM POINT: order-4 -> cross-ratio 2 -> j=1728 -> tau=i (square); order-3 -> j=0 -> tau=rho (hexagon); |mu4|=4 selects tau=i. [E] MARK-EQUILIBRATION ATTRACTOR: 4 points minimizing the log-energy (Fekete) converge to equal spacing (gaps pi/2 = the square); the square beats random perturbations -- the unique (mod rotation) attractor. [E] NEG: an order-3 deck selects rho not the square. [O] residual: only 'the deck acts GEOMETRICALLY (= QGEO.SYM.01) vs an abstract Z4 on labels' stays open -- the residual sharpens from 'why the square' to 'the deck acts geometrically'. NOT a closure.
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