"""v533 -- The sheet-transfer determinant is a norm line in Q(sqrt(-7)): the branch-preserving F_transfer path J -> K -> C -> F (v224/v412) carries the quadratic form D(t) = det M(1,t) = 4t^2 + 14t + 14 = N(alpha_t) with alpha_t = (4t+7+sqrt(-7))/2 an ALGEBRAIC INTEGER and the path a pure TRANSLATION alpha_{t+1} = alpha_t + 2 of constant discriminant -7. [E] exact algebra; the arithmetic F_transfer reading is [C]/[O]. [E] 1. THE NORM IDENTITY. On the sheet axis M(1,t) = R + Q diag(1,t,t): D(t) = det M(1,t) = 4t^2 + 14t + 14 = ((4t+7)^2 + 7)/4 = N_{Q(sqrt-7)/Q}(alpha_t), alpha_t = (4t+7+sqrt(-7))/2. Since -7 = 1 mod 4 and 4t+7 is odd, alpha_t lies in the maximal order O_K = Z[(1+sqrt(-7))/2] for every integer t. [E] 2. THE PATH IS A TRANSLATION. alpha_{t+1} = alpha_t + 2 = alpha_t + |Z2|: the whole discrete transfer path J,K,C,F = M(1,-2..1) is one arithmetic translation orbit in O_K with norms N(alpha_t) = (2, 4, 14, 32) = (det J, det K, det C, det F) = (|Z2|, |mu4|, dim G2, det F) at t = -2,-1,0,1. [E] 3. CONSTANT DISCRIMINANT. The characteristic polynomial of alpha_t, x^2 - (4t+7) x + D(t), has discriminant EXACTLY -7 for every t; the parabola vertex sits at t = -7/4 with D_min = 7/4 > 0 -- the branch-preserving path has no real zero. Curvature D'' = 8 = rank E8; the two sheet coefficients (8, 7) reappear as (curvature, sqrt(-disc)) -- the scalaron 7 is the fixed quadratic discriminant of the transfer. [E] 4. ARITHMETIC ANCHORS (audit). -7 = 1 mod 8, so 2 SPLITS in Q(sqrt-7) -- the J endpoint norm 2 = |Z2| is realised by the split prime above 2 (possible only because h(Q(sqrt-7)) = 1, a Heegner field); det B(1,t) = 3t^2+21t+28 has POSITIVE discriminant 105 (real roots) and the winding axis det M(s,0) = 6s+8 is linear: ONLY the sheet determinant is a complex norm line. [C/O] 5. F_TRANSFER READING. v224 typed F_transfer as the K -> C -> F sheet geodesic [C]; this module adds the exact arithmetic frame: the discrete data of the path live in ONE quadratic order of discriminant -7. Whether the four external transfer solvers (Koide, eta_B, axion, m_p/m_e) act on alpha_t by a COMMON fractional-linear action -- the 'one functor, four interfaces' contract -- stays OPEN [O]; this is the preregistered next test, with the kill criterion 'four incompatible arithmetic actions'. HONEST TYPING. Identities 1-4 are exact [E]; the Lie/atom readings of the norms are the v224/v412 corner determinants (cross-cited); item 5 is typed [C]/[O] -- no continuous transfer is claimed, GATE.FTRANSFER stays where it is. Mirrored in wolfram/tfpt_readouts_extension.wl. """ import sympy as sp from tfpt_constants import check, summary, reset, rankE8 R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]]) ONE = sp.Matrix([1, 1, 1]) A = sp.Matrix([1, 1, 2]) Z2 = 2 MU4 = 4 DIM_G2 = 14 def run(): reset() print("v533 sheet transfer = disc -7 norm line (J->K->C->F translation)") t, x, s = sp.symbols('t x s') Mt = R + Q * sp.diag(1, t, t) D = sp.expand(Mt.det()) J, K, C, F = [R + Q * sp.diag(1, n, n) for n in (-2, -1, 0, 1)] # 1. norm identity check("D(t) = det M(1,t) = 4t^2+14t+14 (v224) = ((4t+7)^2+7)/4 EXACTLY", D == 4 * t**2 + 14 * t + 14 and sp.expand(D - ((4 * t + 7)**2 + 7) / 4) == 0) s7 = sp.sqrt(-7) alpha = (4 * t + 7 + s7) / 2 alpha_bar = (4 * t + 7 - s7) / 2 check("NORM IDENTITY [E]: N(alpha_t) = alpha_t alpha_t-bar = D(t) with " "alpha_t = (4t+7+sqrt(-7))/2", sp.expand(alpha * alpha_bar - D) == 0) check("INTEGRALITY [E]: -7 = 1 mod 4 and 4t+7 odd => alpha_t in " "O_K = Z[(1+sqrt(-7))/2] for every integer t", (-7) % 4 == 1 and all((4 * n + 7) % 2 == 1 for n in range(-3, 3))) # 2. translation orbit check("TRANSLATION [E]: alpha_{t+1} = alpha_t + 2 = alpha_t + |Z2| " "(the whole sheet path is ONE arithmetic translation orbit)", sp.expand(alpha.subs(t, t + 1) - alpha - Z2) == 0) norms = [D.subs(t, n) for n in (-2, -1, 0, 1)] check("NORMS [E]: N(alpha_t) at t = -2..1 = (2,4,14,32) = " "(det J, det K, det C, det F) = (|Z2|,|mu4|,dim G2,det F)", norms == [2, 4, 14, 32] and [J.det(), K.det(), C.det(), F.det()] == [2, 4, 14, 32] and norms[0] == Z2 and norms[1] == MU4 and norms[2] == DIM_G2) # 3. constant discriminant mp = x**2 - (4 * t + 7) * x + D check("CONSTANT DISCRIMINANT [E]: char poly x^2-(4t+7)x+D(t) has " "discriminant -7 for EVERY t", sp.expand(sp.discriminant(mp, x)) == -7) check("VERTEX [E]: t = -7/4, D_min = 7/4 > 0 -- the branch-preserving " "path has NO real zero", sp.solve(sp.diff(D, t), t) == [sp.Rational(-7, 4)] and D.subs(t, sp.Rational(-7, 4)) == sp.Rational(7, 4)) check("COEFFICIENT READINGS [C]: curvature D'' = 8 = rank E8; " "sqrt(-disc) = sqrt(7) -- the scalaron 7 as the fixed quadratic " "discriminant of the transfer", sp.diff(D, t, 2) == rankE8 and -(-7) == 7) # 4. arithmetic anchors + negative controls check("ARITHMETIC ANCHOR [E, audit]: -7 = 1 mod 8 => 2 splits in " "Q(sqrt-7); the J endpoint norm 2 = |Z2| is the split prime above " "2 (h(Q(sqrt-7)) = 1, Heegner)", (-7) % 8 == 1 and norms[0] == 2) detwind = sp.expand((R + Q * sp.diag(s, 0, 0)).det()) check("NEG CONTROL [E]: winding axis det M(s,0) = 6s+8 is LINEAR -- " "no norm form on the winding direction", detwind == 6 * s + 8) B = sp.Matrix([[(ONE.T * Mt * ONE)[0], (ONE.T * Mt * A)[0]], [(A.T * Mt * ONE)[0], (A.T * Mt * A)[0]]]) dB = sp.expand(B.det()) check("NEG CONTROL [E]: 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", dB == 3 * t**2 + 21 * t + 28 and sp.discriminant(dB, t) == 105 > 0) # 5. typed reading check("F_TRANSFER FRAME [C]/[O]: the discrete path data live in ONE " "quadratic order of discriminant -7; whether the four external " "solvers share a common fractional-linear action on alpha_t stays " "OPEN (preregistered next test; kill = four incompatible actions)", True) return summary("v533 disc -7 norm line (norms 2,4,14,32; " "alpha_{t+1} = alpha_t + 2)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)