A registered result in its source context

FTR.DISC7.NORM.01

Original ledger classification: Identity [E] (exact norm/integrality/translation/discriminant arithmetic + negative controls -- sympy exact, Wolfram-mirrored) + [E, audit] (split prime 2, Heegner class number) + [C]/[O] (the arithmetic F_transfer frame: common Moebius action of the four solvers = preregistered next test; GATE.FTRANSFER unchanged)

Original claim text

THE SHEET TRANSFER IS A NORM LINE OF DISCRIMINANT -7 (2026-07-23; 12 checks, <1 s, sympy exact): the branch-preserving F_transfer path J -> K -> C -> F = M(1, -2..1) (FTR.PATH.01/v224, v412) carries the determinant D(t) = det M(1,t) = 4t^2 + 14t + 14 = ((4t+7)^2 + 7)/4 = N_{Q(sqrt-7)/Q}(alpha_t) with alpha_t = (4t+7+sqrt(-7))/2. [E] INTEGRALITY: -7 = 1 mod 4 and 4t+7 odd => alpha_t lies in the maximal order Z[(1+sqrt(-7))/2] for every integer t. [E] TRANSLATION ORBIT: alpha_{t+1} = alpha_t + 2 = alpha_t + |Z2| -- the whole discrete transfer path is ONE arithmetic translation orbit with norms (2,4,14,32) = (det J, det K, det C, det F) = (|Z2|, |mu4|, dim G2, det F). [E] CONSTANT DISCRIMINANT: the characteristic polynomial x^2 - (4t+7)x + D(t) of alpha_t has discriminant EXACTLY -7 for every t; vertex t = -7/4 with D_min = 7/4 > 0 (the branch-preserving path has no real zero); curvature D'' = 8 = rank E8 and sqrt(-disc) = sqrt(7) -- the sheet coefficients (8,7) reappear as (curvature, fixed quadratic discriminant), the scalaron 7 as arithmetic discriminant [C readings]. [E, audit] -7 = 1 mod 8 => 2 SPLITS in Q(sqrt-7) (class number 1, Heegner field) -- the J endpoint norm 2 = |Z2| is realised by the split prime above 2. [E] NEGATIVE CONTROLS: the winding axis det M(s,0) = 6s+8 is LINEAR (no norm form) and the anchor block det B(1,t) = 3t^2+21t+28 has POSITIVE discriminant 105 (real roots) -- only the sheet determinant is a complex norm line. [C]/[O] THE ARITHMETIC F_TRANSFER FRAME (the new solution perspective): the discrete data of the transfer path live in ONE quadratic order of discriminant -7; the preregistered next test is whether the four external transfer solvers (Koide, eta_B, axion, m_p/m_e) act on alpha_t by a COMMON fractional-linear action ('one functor, four interfaces' as an algebraic contract), with the kill criterion 'four incompatible arithmetic actions'; GATE.FTRANSFER unchanged, the external solver stays external. Discovery provenance: center_quotient_coxeter_modular_disc7_probe.py (76/76) in experiments/tfpt-discovery/ (2026-07-23). NO marker moves. [2026-08-01 XXIV, cross-reference audit: the preregistered 'common fractional-linear action' test is EXECUTED on its discrete half -- v571: the four discrete kernels lie on the alpha_t translation orbit and the cross-ratio 4/3 is Moebius-forced (y_F = 53 fit-free); v575: the continuous export is uniflow-degenerate (closure onto the single v425 flow); v578: each external instance is projective IN ITS OWN CLOCK and the joint obstruction is exactly the clock anchor -- so the residual of this preregistration is the same external-anchor-jet export that CONTRACT.F.01 fences; no independent open remains here.] [2026-08-02 XLI: the preregistered PGL2-action test EXECUTED in its literal group form by v632 -- ONE common parabolic action on the discrete half (exact intertwining with the base translation; CR = 4/3; the F corner 53 Moebius-forced), uniflow degeneration on the native continuous half (g_Boltzmann = g_pole identically, g_QCD exactly conjugate to the base translation, relic trivial); the kill criterion 'four incompatible arithmetic actions' is NOT met; see FTR.PGL2.01.]

Original classification detail

[E] the sheet path J->K->C->F is one translation orbit alpha_{t+1} = alpha_t + 2 in Z[(1+sqrt(-7))/2] with norms (2,4,14,32) and constant discriminant -7; [C]/[O] common-solver Moebius action = preregistered next test; no marker moves

Dependencies as recorded

FTR.PATH.01; DIAMOND.SHEET.SOURCE.J.01 (v412); DIAMOND.SHEET.CALCULUS.01 (v413); v224