"""v85 -- Master-Cover Theorem: next.txt P3 answered, P6 honest negative, P4 typed. Resolves the prioritised open questions P3/P4/P6 of the anchor-block double-cover programme (next.txt, secs. 3-4; builds on v80/v81/v82). P3 (the family of covers -- ANSWERED, [I]/[L]): There is exactly ONE anchor-block double cover on the pencil plane span{K,Q}, the master cover y^2 = det B(K+xQ) = (3x+2)(3x+5). For ANY basis change G = [[alpha,beta],[gamma,delta]] in GL(2), det B((aK+bQ) + x(cK+dQ)) = (gamma x + alpha)^2 * p_KQ(moebius_G(x)), an exact symbolic identity, so every member of the family is the SAME cover up to a Moebius reparametrisation of the line, and disc = 81 * det(G)^2 = N_fam^4 * det(G)^2. The observed family in next.txt P3 is exactly this orbit: branch(L+xQ) = {-8/3,-5/3} = branch - 1 (transport translation) branch(K+xL) = {-5/8,-2/5} = x/(1+x)-preimages branch(Q+xK) = {-3/2,-3/5} = 1/branch (basis swap) and the 'auffaellige' rung -8/3 = -rank(E8)/N_fam is no new structure: it is the carrier point shifted by one transport period, -(g_car + N_fam)/N_fam = -g_car/N_fam - 1 (rank E8 = g_car + N_fam). Negative controls: leaving the anchor plane breaks the structure -- the R-direction covers have non-square discriminants (153, 201), i.e. non-split branch points (anchor-first, consistent with v82(B)). P6 (mu4 = Z4 vs deck Z2 -- HONEST NEGATIVE, [I]): The natural 4-sheet candidate over the transport tower, y^4 = det B(K+xQ) * det B(L+xQ), FACTORS through a forced square (3x+5)^2 (the shared carrier point), so mu4 does NOT act as a genuine 4:1 cover of the pencil line. What the tower generates instead is a ladder of double covers: rung n has branch {-2/3-n, -5/3-n}, and the second-level cover w^2 = (3x+2)(3x+8) (disc 324 = 18^2) connects Koide directly to -rank(E8)/N_fam at branch separation 2 = |Z2| periods. P4 (scalaron trace -> Starobinsky parameters -- TYPED, [I] + honest scope): The branch divisor fixes the SCALE exponent only: trace = -7/3 = -scalaron/N_fam, so M_scal = c3^{7/2} Mbar is algebraically addressed (v80/v81). The tilt observables n_s = 1 - 2/N_star and r = 12/N_star^2 are functions of the external reheating input N_star ALONE -- the cover cannot promote them from [P]; only the mass scale is algebraic. (Registered as bands over N_star in [50,60] in predictions_frozen.json.) All statements here are exact integer/symbolic arithmetic [I] (with the Lie-data labels [L]); the physical readings (Koide branch <-> physical Koide value, sheet <-> Spin(10) half-spinor) remain [P] exactly as typed in v81/v82 and next.txt sec. 5. """ import sympy as sp from tfpt_constants import check, summary, reset x = sp.symbols('x') al, be, ga, de = sp.symbols('alpha beta gamma delta') K = sp.Matrix([[4, 2, 0], [4, 3, 2], [5, 3, 2]]) Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]]) R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) L = K + Q ONE = sp.Matrix([1, 1, 1]) A = sp.Matrix([1, 1, 2]) # the anchor a = (1,1,2) N_FAM, G_CAR, Z2, RANK_E8, SCALARON = 3, 5, 2, 8, 7 def anchor_block(M): return sp.Matrix([[(ONE.T * M * ONE)[0], (ONE.T * M * A)[0]], [(A.T * M * ONE)[0], (A.T * M * A)[0]]]) def cover_poly(M, N): return sp.expand(sp.det(anchor_block(M + x * N))) def branch(M, N): return sorted(sp.solve(cover_poly(M, N), x)) def run(): reset() print("v85 master-cover theorem (next.txt P3 answered; P6 negative; P4 typed)") # ---- the master cover ------------------------------------------------- pKQ = cover_poly(K, Q) check("master cover det B(K+xQ) = (3x+2)(3x+5)", sp.factor(pKQ) == sp.factor((3 * x + 2) * (3 * x + 5)), exact=False) check("master discriminant = 81 = N_fam^4", sp.discriminant(pKQ, x), N_FAM**4, exact=True) check("branch locus = {-g_car/N_fam, -|Z2|/N_fam} = {-5/3, -2/3}", branch(K, Q) == [sp.Rational(-5, 3), sp.Rational(-2, 3)]) # ---- P3: GL(2) Moebius covariance => ONE cover up to reparametrisation M2 = al * K + be * Q N2 = ga * K + de * Q p2 = sp.expand(sp.det(anchor_block(M2 + x * N2))) moebius_image = pKQ.subs(x, (de * x + be) / (ga * x + al)) residual = sp.simplify(p2 - sp.expand((ga * x + al)**2 * moebius_image)) check("GL(2) covariance: det B(G.(K,Q)+x..) = (gamma x+alpha)^2 p_KQ(moebius_G x) " "(exact symbolic identity)", residual == 0) check("disc transforms as 81*det(G)^2 (master invariant x square)", sp.factor(sp.discriminant(p2, x)) == sp.factor(81 * (al * de - be * ga)**2)) # the observed family = the GL(2) orbit of the master cover check("branch(L+xQ) = master branch - 1 (transport translation)", branch(L, Q) == [b - 1 for b in branch(K, Q)]) check("branch(K+xL) = x/(1+x)-preimages {-5/8,-2/5}", branch(K, L) == [sp.Rational(-5, 8), sp.Rational(-2, 5)]) check("branch(Q+xK) = 1/branch (basis swap) {-3/2,-3/5}", branch(Q, K) == [sp.Rational(-3, 2), sp.Rational(-3, 5)]) # the 'auffaellige' rung -8/3 explained: carrier - one transport period check("-8/3 = -rank(E8)/N_fam = carrier point - 1 (rank E8 = g_car + N_fam, " "no new structure)", sp.Rational(-RANK_E8, N_FAM) == sp.Rational(-G_CAR, N_FAM) - 1 and RANK_E8 == G_CAR + N_FAM) # negative controls: outside the anchor plane the cover does NOT split dR1 = sp.discriminant(cover_poly(R, Q), x) dR2 = sp.discriminant(cover_poly(K, R), x) check("R-direction non-split: disc(R+xQ)=153 not a square (anchor-first)", dR1 == 153 and not sp.sqrt(dR1).is_rational) check("R-direction non-split: disc(K+xR)=201 not a square (anchor-first)", dR2 == 201 and not sp.sqrt(dR2).is_rational) # ---- P6: honest negative on a genuine Z4 cover ------------------------ prod = sp.factor(cover_poly(K, Q) * cover_poly(L, Q)) check("4-sheet candidate factors through the forced square (3x+5)^2 " "=> mu4 is NOT a 4:1 cover of the pencil line", prod == sp.factor((3 * x + 2) * (3 * x + 5)**2 * (3 * x + 8))) w2 = sp.cancel(prod / (3 * x + 5)**2) check("second-level cover w^2=(3x+2)(3x+8): disc=324=18^2, split", sp.discriminant(sp.expand(w2), x) == 324) check("second-level branch {-2/3,-8/3}: Koide <-> rank E8, separation " "2 = |Z2| transport periods", sorted(sp.solve(w2, x)) == [sp.Rational(-8, 3), sp.Rational(-2, 3)] and sp.Rational(-2, 3) - sp.Rational(-8, 3) == Z2) n = sp.symbols('n') check("transport tower rung n: branch = {-2/3-n, -5/3-n} (ladder of double " "covers, never a single 4:1 object)", sorted(sp.solve(cover_poly(K + n * Q, Q), x), key=lambda e: e.subs(n, 0)) == [-n - sp.Rational(5, 3), -n - sp.Rational(2, 3)]) # ---- P4: the cover fixes the SCALE, not the tilt ---------------------- tr = sum(branch(K, Q)) check("branch-divisor trace = -7/3 = -scalaron/N_fam (algebraic address " "of the scalaron exponent, v80/v81)", tr == sp.Rational(-SCALARON, N_FAM)) Ns = sp.symbols('N_star', positive=True) n_s = 1 - 2 / Ns r_t = 12 / Ns**2 check("n_s and r depend on the external N_star ALONE (d/dN_star != 0): the " "cover cannot promote the tilt from [P]; only M_scal is algebraic", sp.diff(n_s, Ns) != 0 and sp.diff(r_t, Ns) != 0 and SCALARON == Z2 + G_CAR) return summary("v85 master cover") if __name__ == "__main__": raise SystemExit(1 if run() else 0)