"""v96 -- Branch-Kernel Selection (next.txt P1 unblocked): the cover's own rank-1 kernels at the branch points define the sector -> sheet map. [I]/[L]. The P1 blocker was honest: "no independently motivated definition of the sector -> sheet map exists" (the row-projection candidate was rejected as number-fishing). This script supplies the missing definition from an object the cover itself produces: at each branch point of the anchor-block double cover y^2 = det B(K+xQ) (v81/v85) the 2x2 anchor block B is RANK 1, so it has a canonical right kernel (generation space) and left kernel (sector space). Pairing the established sector rows (v12/v46/v53: rows of K+xQ = (up, down, lepton)) with these forced kernels is the most natural Z2-capable quantity available -- nothing is chosen. RESULTS (all exact): [I] 1. KERNELS ARE INTEGER VECTORS. B(x) = [[15x+25, 16x+29],[21x+35, 23x+41]], det B = (3x+2)(3x+5). Koide x=-2/3: right kernel w = 11*1 - 9a = (2,2,-7), left kernel v = 7*1 - 5a = (2,2,-3); carrier x=-5/3: right kernel w = 1 = (1,1,1) (the democratic vector itself), left kernel v = 8*1 - 7a = (1,1,-6). [L] 2. COLLAPSE LEMMA: at a rank-1 branch point, P(x0).w is orthogonal to span{1,a}, hence proportional to the anti-symmetric direction of the anchor's two equal-weight slots: (-1,1,0). Verified at both branch points, on BOTH sides (sector rows and generation columns): P(-2/3).w = (20/3, -20/3, 0), P(-5/3).w = (-2/3, 2/3, 0), v^T.P(x0) prop. (-1,1,0) at both points. => up = -down (the deck-odd pair), lepton pairing = 0 (leptons sit ON the ramification -- consistent with Koide being leptonic); generation side: gen1 = -gen2, gen3 = 0. [I] 3. SECTOR ZERO LADDER on the pencil (exact linear forms): pairings with the Koide kernel: (4(2x+3), 10x, 3x+2), pairings with the carrier kernel: (2(2x+3), 5x+9, 2(3x+5)). lepton zeros = {-2/3, -5/3} = exactly the two branch points (the lepton pairings ARE the two factors of det B); up zeros = {-3/2} doubly = -N_fam/|Z2| (the v85 GL(2) rung): the entire up-row of P(-3/2) IS the odd direction (-1/2)(1,-1,0); down zeros = {0 (= K itself), -9/5 = -N_fam^2/g_car}. [I] 4. ORIENTATION: with kernels normalised to positive democratic component, Koide orients (up,down) = (+,-) and carrier (-,+): the deck translation x -> x+1 flips the up/down orientation; magnitude ratio (20/3)/(2/3) = 10 = A_Lambda. Convention-free invariant: f_up * f_down < 0 at both branch points. [L] 5. ANCHOR-PLACEMENT CONTROL: replacing the anchor by (2,1,1) moves the collapse direction to (0,1,-1) -- the anti-symmetric pair of THAT placement's equal slots (mechanism follows the anchor, anchor-first); placement (1,2,1) has disc 40, non-split: no rational kernels at all (matches the v82 splitting trichotomy). HONEST TYPING: items 1-5 are exact arithmetic plus a small structure lemma. The IDENTIFICATION of the up<->down transposition with the v92 chirality swap (the two Lagrangian glues) is the remaining [P] step -- sharpened and tested in v97. Audit-level integer readings (w.1 = -3 = -N_fam, w.a = -10 = -A_Lambda, v.a = -2 = -|Z2|, carrier v.1 = -4 = -|mu4|) are recorded, not promoted. """ import sympy as sp from tfpt_constants import check, summary, reset, N_fam, g_car R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]]) SIG = sp.diag(1, -1, -1) K = R + Q * SIG ONE = sp.Matrix([1, 1, 1]) A = sp.Matrix([1, 1, 2]) X = sp.symbols('x') P = K + X * Q W = sp.Matrix.hstack(ONE, A) B = W.T * P * W def branch_kernels(x0): """Right/left kernels of the rank-1 anchor block, canonically normalised (integer, gcd 1, democratic component positive).""" Bx = B.subs(X, x0) ker = Bx.nullspace()[0] ker = ker / sp.gcd(tuple(ker)) if ker[0] < 0: ker = -ker lker = Bx.T.nullspace()[0] lker = lker / sp.gcd(tuple(lker)) if lker[0] < 0: lker = -lker return W * ker, W * lker # generation-space w, sector-space v def run(): reset() print("v96 branch-kernel selection (P1: the cover defines the sector map)") # 0. the anchor block and its cover (cross-check with v80/v81) check("B(x) = [[15x+25,16x+29],[21x+35,23x+41]], det B = (3x+2)(3x+5)", B == sp.Matrix([[15 * X + 25, 16 * X + 29], [21 * X + 35, 23 * X + 41]]) and sp.factor(B.det()) == (3 * X + 2) * (3 * X + 5)) # 1. integer kernels at the branch points kx, kv = sp.Rational(-2, 3), sp.Rational(-5, 3) wK, vK = branch_kernels(kx) wC, vC = branch_kernels(kv) check("Koide x=-2/3: rank B = 1, right kernel w = 11*1-9a = (2,2,-7), " "left kernel v = 7*1-5a = (2,2,-3)", B.subs(X, kx).rank() == 1 and wK == sp.Matrix([2, 2, -7]) == 11 * ONE - 9 * A and vK == sp.Matrix([2, 2, -3]) == 7 * ONE - 5 * A) check("carrier x=-5/3: rank B = 1, right kernel w = 1 = (1,1,1) (the " "democratic vector), left kernel v = 8*1-7a = (1,1,-6)", B.subs(X, kv).rank() == 1 and wC == ONE and vC == sp.Matrix([1, 1, -6]) == 8 * ONE - 7 * A) check("AUDIT (recorded, not promoted): w_K.1 = -3 = -N_fam, w_K.a = -10 " "= -A_Lambda; v_K.a = -2 = -|Z2|; v_C.1 = -4 = -|mu4|, v_C.a = -10", (wK.T * ONE)[0] == -3 and (wK.T * A)[0] == -10 and (vK.T * A)[0] == -2 and (vC.T * ONE)[0] == -4 and (vC.T * A)[0] == -10) # 2. collapse lemma at both branch points, both sides d = sp.Matrix([-1, 1, 0]) fK = P.subs(X, kx) * wK fC = P.subs(X, kv) * wC check("COLLAPSE LEMMA (sector side): P(-2/3).w = (20/3)(1,-1,0), " "P(-5/3).w = (2/3)(-1,1,0) -- up = -down, lepton = 0 at BOTH " "branch points (leptons sit on the ramification)", fK == sp.Rational(-20, 3) * d and fC == sp.Rational(2, 3) * d) gK = (vK.T * P.subs(X, kx)).T gC = (vC.T * P.subs(X, kv)).T check("collapse lemma (generation side): v^T.P prop. (-1,1,0) at both " "points (gen1 = -gen2, gen3 = 0)", gK == -1 * d * -1 and gC == 2 * d) check("convention-free invariant: f_up * f_down < 0 at both branch " "points (up and down always on opposite sheets)", fK[0] * fK[1] < 0 and fC[0] * fC[1] < 0) check("ORIENTATION: canonical kernels give Koide (up,down) = (+,-) and " "carrier (-,+): the deck translation flips the pair; magnitude " "ratio = 10 = A_Lambda", fK[0] > 0 > fK[1] and fC[0] < 0 < fC[1] and fK[0] / fC[1] == 10) # 3. sector zero ladder fx_K = P * sp.Matrix([2, 2, -7]) fx_C = P * ONE check("sector pairing forms: Koide kernel -> (4(2x+3), 10x, 3x+2); " "carrier kernel -> (2(2x+3), 5x+9, 2(3x+5))", [sp.expand(e) for e in fx_K] == [8 * X + 12, 10 * X, 3 * X + 2] and [sp.expand(e) for e in fx_C] == [4 * X + 6, 5 * X + 9, 6 * X + 10]) check("LEPTON pairings = the two det-B factors: zeros exactly at the " "branch points {-2/3, -5/3} (the lepton sector carries the " "ramification)", sp.solve(fx_K[2], X) == [sp.Rational(-2, 3)] and sp.solve(fx_C[2], X) == [sp.Rational(-5, 3)]) check("UP zeros: x = -3/2 = -N_fam/|Z2| against BOTH kernels (a v85 " "GL(2) rung); the whole up-row of P(-3/2) = (-1/2)(1,-1,0) IS the " "odd direction", sp.solve(fx_K[0], X) == [sp.Rational(-3, 2)] and sp.solve(fx_C[0], X) == [sp.Rational(-3, 2)] and P.subs(X, sp.Rational(-3, 2))[0, :] == sp.Rational(-1, 2) * sp.Matrix([[1, -1, 0]])) check("DOWN zeros: x = 0 (= the mass operator K itself) and x = -9/5 = " "-N_fam^2/g_car", sp.solve(fx_K[1], X) == [0] and sp.solve(fx_C[1], X) == [sp.Rational(-9, 5)] and sp.Rational(-9, 5) == -sp.Rational(N_fam**2, g_car)) # 4. anchor-placement controls A2 = sp.Matrix([2, 1, 1]) W2 = sp.Matrix.hstack(ONE, A2) B2 = W2.T * P * W2 roots2 = sp.solve(B2.det(), X) ok2 = True for x0 in roots2: Bx = B2.subs(X, x0) ker = Bx.nullspace()[0] w = W2 * ker img = P.subs(X, x0) * w # collapse must be the anti-symmetric direction of slots {2,3} ok2 = ok2 and sp.simplify(img[0]) == 0 and \ sp.simplify(img[1] + img[2]) == 0 check("CONTROL anchor (2,1,1): det B = (x+2)(9x+11) (split, disc = 49 = " "scalaron^2, v82) and the collapse moves to (0,1,-1) -- the " "mechanism follows the anchor placement", sp.factor(B2.det()) == (X + 2) * (9 * X + 11) and ok2) A3v = sp.Matrix([1, 2, 1]) W3 = sp.Matrix.hstack(ONE, A3v) B3 = W3.T * P * W3 check("CONTROL anchor (1,2,1): disc = 40 NON-square (v82 trichotomy): " "no rational branch kernels at all", sp.discriminant(B3.det(), X) == 40 and not sp.sqrt(40).is_rational) check("STATUS: the sector -> sheet map is now DEFINED by the cover's own " "branch kernels [I]/[L]; the identification with the v92 chirality " "swap is the remaining [P] step (tested in v97)", True) return summary("v96 branch-kernel selection") if __name__ == "__main__": raise SystemExit(1 if run() else 0)