"""v499 -- P2.TYPING.01: the P2 weight-typing postulate HARDENED -- the anchor weights a = (1,1,2) typed as the Birkhoff-Grothendieck splitting exponents of the DELIGNE CANONICAL EXTENSION of the flavor connection. Exact sympy/integer arithmetic, no floats. v491 (P2.PARTITION.01) reduced g_car = 5 to "3 positive integers summing to 4"; its check-9 residual was the TYPING: why integers, why positive, why sum 4? This module shows that, given already-established [E] anchors, all three become THEOREMS of standard algebraic geometry: THE CONDITIONAL THEOREM (premises fenced below): Let V be the 3-dim generation module, identified with H^1(P^1 \\ mu4; C) (rank = #marks - 1 = 3; #marks = 4 by Gauss-Bonnet, v216), carrying a flat, irreducible, unitary (U) connection whose local monodromy at every mark is the cusp class (char poly lam^3 - 1, v117). Then the Deligne canonical extension E (residue eigenvalues in [0,1)) satisfies (i) rank E = 3 [rank of V] (ii) deg E = -sum_marks tr Res = -4 [Deligne; trace 1 per mark] (iii) h^0(E) = 0 [Mehta-Seshadri stability] (iv) E = O(-a1)+O(-a2)+O(-a3), a_i in Z [Birkhoff-Grothendieck] hence a_i >= 1, sum a_i = 4, {a} = {1,1,2} UNIQUE (v491 lemma), g_car = e2 = 5. [E] 1-3. PREMISE MAP. H^1 basis omega_k = z^(k-1)dz/(z^4-1), k = 1,2,3 (residues zeta^k/4, regular at infinity, deck characters i^k; k = 4 = trivial character ABSENT, dlog pole at infinity -- rank = #marks - 1 = 3, v137); the SAME 3 as coherent cohomology of L = O(-D), D = mu4: h^1(L) = 3 = N_fam, h^0(L) = 0, H^0(Omega^1(log D)) = H^0(O(2)) = 3 = the v137 basis, h^0(O(4)) = 5 = g_car (v228 index gate); cusp class => canonical residue exponents {0, 1/3, 2/3} unique in [0,1) (distinct mod Z => Deligne extension exists, unique, no Jordan ambiguity; v117). [E] 4-6. deg E = -4 AS A THEOREM. mu4-average lemma sum_k U^k X U^-k = 4 diag(X) (v115); residue spectrum {0,1/3,2/3} has trace 1 per mark, 4 marks (v216) => deg E = -4; par-deg = 0 (the v39 (U) consistency); infinity APPARENT (monodromy product = 1 exactly, (M0 U)^4 = 1, witness exponents diag(2,1,1) integer). [E] 7-9. POSITIVITY AS A THEOREM. Irreducibility EXACT from v117 (|| = 24 = |S4|, sum |chi|^2 = 24; S4 standard x sign -- replaces v38-type sampling); with (U) unitarity => Mehta-Seshadri stability => every section-saturation violates strict stability (all 324 cases par-deg >= 0) => h^0(E) = 0; BG dictionary h^0 = 0 <=> a_i >= 1 (exhaustive window) -- positivity IS h^0 = 0. [E] 10-11. UNIQUENESS + SELECTOR. {a in Z^3 : a_i >= 1, sum = 4} = {(1,1,2)} unique, e2 = 5 = g_car, e3 = 2 = |Z2|, e2 - e3 = 3 = N_fam (v491); Schur-Horn permutohedron of 4*spec(A0) = (8/3, 4/3, 0) has integer points exactly {(2,2,0), (2,1,1)} (the tfpt_2 'Global computation' pair); h^0 = 0 kills {0,2,2} (O(0) summand => h^0 = 1) -- the paper's 'balanced' heuristic is now TYPED as stability/positivity, and the two routes converge on {1,1,2}. [E] 12-15. NEGATIVE CONTROLS. Drop positivity -> 4 solutions (31 with negatives); deg -3 -> e2 = 3, deg -5 -> not unique; n-mark scan: {1,..,1,2} unique for EVERY n >= 3 but e2 = (n-2)(n+1)/2 = 5 ONLY at n = 4 -- the VALUE, not the uniqueness, is the n = 4 content; MS-rational weights as exponents -> e2 = 16/3 resp. 44/9, non-integer -- integrality lives on the BUNDLE side. [E] 16-18. THE h^1 DISCREPANCY, REAL + RESOLVED. h^1(O(-2)+O(-1)^2) = 1 != 3 (chi = deg + rk = -1): the naive typing "generations = H^1 of the rank-3 E" is FALSE; forcing it demands deg -6 and loses uniqueness ({1,1,4},{1,2,3},{2,2,2}, e2 never 5). Correct TWO-OBJECT typing: generations = H^1(L) = C^3 for the LINE bundle L = O(-D) (= H^0(Omega^1(log D)), the v137 basis); E carries them as its FIBER (rank 3) with the anchor as splitting type; shared invariants rank E = 3 = h^1(L), deg E = -4 = deg L, h^0 = 0 both. Three DIFFERENT H^1's coexist: 3 (generations, L-side), 1 (coherent h^1 of E), 6 = |R+(A3)| (local-system deformations); h^1(E(-D)) = 13 = Delta_Q recorded as curiosity. [E] 19-22. THE BISWAS COMPUTATION, EXPLICIT (independent convergence witness). The weight denominator is 3 = N_fam, NOT 2 = |Z2|: the Biswas cover is the Z3 cusp cover y^3 = (z-1)(z-i)(z+1)^2(z+i)^2 (genus 2 by RH, unramified over infinity; an all-equal branch choice is impossible, 4c != 0 mod 3), NOT the pillowcase double w^2 = z^4-1 (genus 1, j = 1728 -- the deck side). p_* O_Y = O + O(-2)^2 (chi cross-check -1 = -1); the equivariant model W = (chi^0+chi^1+chi^2) x O_Y (regular rep at every mark, weights {0,1/3,2/3}) has parabolic partner with underlying bundle O + O(-2)^2: rank 3 OK, deg -4 OK, par-deg 0 OK -- integrality AND the sum rule COME OUT of the correspondence -- but splitting {0,2,2} with h^0 = 1: exactly the UNSTABLE Schur-Horn companion. (1,1,2) is the Biswas image of the STABLE object; stability (irreducibility, v117) is the selector. [C]/[N]/[O] 23. HONEST RESIDUALS (P2 stays an axiom; no marker moves): R1 the module identification "generation space = this H^1 with this connection" [C] (QGEO.SYM territory); R2 unitarity (U) of the flavor monodromy [C] (v33/v40 premise); R3 the identification of the ANALYTIC monodromy with the exact W(A3) representation [N] (v117 check 4, ODE matching 1e-7). NON-CIRCULAR: neither v23's splitting-type input nor v115's diag(A0) = a/4 input is used; the chain is rank (topology) -> deg (residue trace) -> integrality (BG) -> positivity (stability) -> partition (v491); A0* appears only as the explicit consistency witness. The weight postulate shrinks from "positive integers summing to 4 (why?)" to "Deligne + BG + stability, given the cohomological typing". DIRECTION-REVERSAL NOTE (inherited from v491): tfpt_constants computes N_fam FROM g_car; here the arrow runs marks -> bundle -> partition -> g_car -- an over-determination check, not a circle, never double-counted in witness tallies. Status: [E] Formal/Identity (AG arithmetic: Deligne degree, BG dictionary, stability skeleton, Schur-Horn lattice points, Biswas cover -- sympy exact) + [C] R1/R2 (module identification, unitarity) + [N] R3 (v117 check 4). Sharpens v491 (types its check-9 residual); does NOT retire axiom P2; AX.P2.01 stays declared, QGEO.SYM stays [O]. Repo anchors: v216 (marks = 2chi), v137 (H^1 basis), v228 (RR index gate), v115 (mu4-average, A0*), v117 (exact monodromy, S4), v39/v33 (pardeg 0), v491 (partition), tfpt_2 'Global computation'. Python (sympy); Wolfram-mirrored (tfpt_readouts_extension.wl).""" from itertools import combinations_with_replacement, product import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam Z = sp.Symbol('z') I = sp.I # exact objects from the repo (v115 / v117) A0STAR = sp.Matrix([[sp.Rational(1, 2), sp.sqrt(2) / 6, 0], [sp.sqrt(2) / 6, sp.Rational(1, 4), sp.sqrt(5) / 12], [0, sp.sqrt(5) / 12, sp.Rational(1, 4)]]) M0 = sp.Matrix([[0, -(1 + I) / 2, (1 - I) / 2], [-(1 + I) / 2, -I / 2, sp.Rational(-1, 2)], [(1 - I) / 2, sp.Rational(-1, 2), I / 2]]) U = sp.diag(1, I, -I) MARKS = [sp.Integer(1), I, sp.Integer(-1), -I] WEIGHTS = [sp.Integer(0), sp.Rational(1, 3), sp.Rational(2, 3)] # cusp / MS weights def _e2(ms): x = list(ms) return sum(x[i] * x[j] for i in range(len(x)) for j in range(i + 1, len(x))) def _h0_line(d): """h^0(P^1, O(d))""" return d + 1 if d >= 0 else 0 def _h1_line(d): """h^1(P^1, O(d)) = h^0(O(-d-2)) (Serre)""" return -d - 1 if d <= -2 else 0 def _enumerate_group(gens): """BFS closure of (exact 3x3 sympy matrices, Gaussian-rational entries).""" def freeze(m): return tuple(sp.expand(x) for x in m) elems = {freeze(sp.eye(3)): sp.eye(3)} frontier = [sp.eye(3)] while frontier: new = [] for e in frontier: for g in gens: x = sp.expand(g * e) kx = freeze(x) if kx not in elems: elems[kx] = x new.append(x) frontier = new return list(elems.values()) def run(): reset() print("v499 P2.TYPING.01: the anchor (1,1,2) typed as Deligne/Birkhoff-Grothendieck " "splitting data (hardening the v491 weight-typing residual)") # ================================================================ A. premise map # 1. the generation H^1: basis omega_k, mu4 characters, trivial character absent chars, res_ok, reg_inf = [], True, True for k in (1, 2, 3, 4): om = Z ** (k - 1) / (Z ** 4 - 1) # character under the deck z -> iz (1-form transforms with the extra i from dz) rot = sp.simplify((I * Z) ** (k - 1) / ((I * Z) ** 4 - 1) * I / om) chars.append(sp.simplify(rot - I ** k)) ressum = sum(sp.residue(om, Z, zeta) for zeta in MARKS) for zeta in MARKS: if sp.simplify(sp.residue(om, Z, zeta) - zeta ** k / 4) != 0: res_ok = False if k <= 3 and sp.simplify(ressum) != 0: reg_inf = False # residue theorem: regular at infinity if k == 4 and sp.simplify(ressum + (-1)) != 0: reg_inf = False # res_infty(omega_4) = -1 != 0: dlog pole check("GENERATION H^1 [E, v137]: omega_k = z^(k-1)dz/(z^4-1), k = 1,2,3 span " "H^1(P^1 minus mu4) = C^3 (residues zeta^k/4, regular at infinity, deck " "characters i^k); k = 4 carries the TRIVIAL character and has res_inf = -1 " "(dlog pole) -- trivial character ABSENT, rank = #marks - 1 = 3", all(c == 0 for c in chars) and res_ok and reg_inf) # 2. the same 3 as coherent cohomology of the degree -4 line bundle L = O(-D) dims = (_h0_line(-4), _h1_line(-4), _h0_line(2), _h0_line(-2 + 4), _h0_line(4)) check("LINE-BUNDLE CARRIER [E, v228-type RR]: L = O(-D), deg -4: h^0 = %d, " "h^1 = %d = N_fam (RR chi = deg+1 = -3; Serre h^1(O(-4)) = h^0(O(2)) = %d); " "Omega^1(log D) = O(-2+4) = O(2): h^0 = %d = 3 (the v137 basis IS this " "space); function side h^0(O(+4)) = %d = 5 = g_car (v228 index gate)" % dims, dims == (0, N_fam, 3, 3, g_car)) # 3. the cusp class fixes the canonical residue exponents {0,1/3,2/3} in [0,1) lam = sp.Symbol('lam') cusp_ok = (sp.simplify(M0.H * M0) == sp.eye(3) and sp.simplify(M0 ** 3) == sp.eye(3) and sp.expand(M0.charpoly(lam).as_expr() - (lam ** 3 - 1)) == 0) expo = [sp.exp(2 * sp.pi * I * a) for a in WEIGHTS] spec_match = (all(sp.simplify(x ** 3 - 1) == 0 for x in expo) and len({sp.nsimplify(sp.expand_complex(x)) for x in expo}) == 3) distinct = all(sp.simplify(WEIGHTS[i] - WEIGHTS[j]) not in (0,) and not (WEIGHTS[i] - WEIGHTS[j]).is_integer for i in range(3) for j in range(3) if i != j) check("CUSP CLASS => CANONICAL EXPONENTS [E, v117]: M0 unitary, M0^3 = 1, char " "poly lam^3 - 1; the unique residue eigenvalues in [0,1) with exp(2 pi i a) " "= spec M0 are {0, 1/3, 2/3} (distinct mod Z => the Deligne canonical " "extension exists and is unique, no Jordan ambiguity)", cusp_ok and spec_match and distinct) # ============================================== B. deg E = -4 is a THEOREM # 4. mu4-average lemma, generic X (v115 check 1, replicated) xsym = sp.Matrix(3, 3, lambda i, j: sp.Symbol('x%d%d' % (i, j), complex=True)) ssum = sp.zeros(3) for k in range(4): uk = U ** k ssum += uk * xsym * uk.inv() check("MU4-AVERAGE LEMMA [E, v115]: sum_k U^k X U^-k = 4 diag(X) for generic 3x3 " "X (U = diag(1,i,-i)) -- the four residues A_k = U^k A0 U^-k are one mu4 " "orbit and their sum is diagonal", sp.simplify(sp.expand(ssum - 4 * sp.diag(xsym[0, 0], xsym[1, 1], xsym[2, 2]))) == sp.zeros(3)) # 5. Deligne degree bookkeeping on the explicit witness A0* char_ok = sp.simplify(A0STAR.charpoly(lam).as_expr() - lam * (lam - sp.Rational(1, 3)) * (lam - sp.Rational(2, 3))) == 0 traces = [sp.simplify((U ** k * A0STAR * (U ** k).inv()).trace()) for k in range(4)] deg_E = -sum(traces) pardeg = deg_E + 4 * sum(WEIGHTS) check("deg E = -4 [THEOREM given the cusp class]: residue spec {0,1/3,2/3} has " "trace 1 per mark; 4 marks (v216 Gauss-Bonnet) => deg E = -sum_k tr(Res_k) " "= %s (Deligne: deg of the canonical extension = -sum of residue traces); " "par-deg = deg + 4*(0+1/3+2/3) = %s (the v39 'pardeg 0' (U) consistency); " "witness A0* has char poly lam(lam-1/3)(lam-2/3) exactly" % (deg_E, pardeg), char_ok and traces == [1, 1, 1, 1] and deg_E == -4 and pardeg == 0) # 6. flatness across infinity + the integer exponents there (consistency route) prod_flat = sp.eye(3) for k in range(4): uk = U ** k prod_flat = sp.expand(prod_flat * (uk * M0 * uk.inv())) sumA = sp.simplify(sum((U ** k * A0STAR * (U ** k).inv() for k in range(4)), sp.zeros(3))) off_zero = all(sp.simplify(sumA[i, j]) == 0 for i in range(3) for j in range(3) if i != j) eig_inf = sorted([sumA[i, i] for i in range(3)], reverse=True) check("REGULAR INFINITY [E, v117/v115]: monodromy at infinity = " "prod_k U^k M0 U^-k = 1 exactly and (M0 U)^4 = 1 -- infinity is an " "APPARENT singularity (integer exponents); on the witness: sum_k A_k = " "4 diag(A0*) = diag(2,1,1), eigenvalues %s in Z^3, sum 4 -- consistent " "with BG integrality but NOT used for the uniqueness below" % (eig_inf,), sp.simplify(prod_flat) == sp.eye(3) and sp.simplify((M0 * U) ** 4) == sp.eye(3) and off_zero and eig_inf == [2, 1, 1]) # ============================================== C. h^0(E) = 0 is a THEOREM # 7. irreducibility is EXACT: has order 24 and sum |chi|^2 = |G| elems = _enumerate_group([U, M0]) chi2 = sp.Integer(0) for x in elems: tr = sp.expand(x.trace()) chi2 += sp.expand(tr * sp.conjugate(tr)) check("IRREDUCIBILITY EXACT [E, v117]: || = %d = |W(A3)| = |S4| and " "sum_g |tr g|^2 = %s = |G| -- the 3-dim monodromy representation is " "IRREDUCIBLE (standard x sign of S4); with (U) unitarity this is the " "Mehta-Seshadri stability input (replaces v38-type sampling)" % (len(elems), sp.simplify(chi2)), len(elems) == 24 and sp.simplify(chi2 - 24) == 0) # 8. stability => h^0 = 0 (the inequality skeleton, exhaustively) # a nonzero section O -> E saturates to a line bundle with deg d >= 0; its # parabolic degree is d + (one induced weight per mark) >= 0 = par-slope(E); # STRICT stability demands par-deg < 0 -- contradiction in EVERY case: viol = [] for d in range(0, 4): for wsel in product(WEIGHTS, repeat=4): pdeg = d + sum(wsel) if pdeg < 0: # would satisfy stability -> no clash viol.append((d, wsel)) check("STABILITY => h^0(E) = 0 [THEOREM given irreducibility + (U)]: every " "candidate section-saturation (deg d >= 0, any induced weights from " "{0,1/3,2/3} at the 4 marks) has par-deg = d + sum w >= 0 = par-slope(E) " "-- checked all %d cases, 0 satisfy the strict-stability bound par-deg < 0 " "=> no global sections (the v137 'trivial character absent' correlate, " "now typed)" % (4 * 3 ** 4), viol == []) # 9. the BG dictionary: h^0 = 0 <=> all splitting exponents >= 1 dict_ok = True for a in product(range(-3, 7), repeat=3): h0 = sum(_h0_line(-ai) for ai in a) if (h0 == 0) != all(ai >= 1 for ai in a): dict_ok = False check("BG DICTIONARY [E]: for E = O(-a1)+O(-a2)+O(-a3), h^0(E) = " "sum max(0, 1 - a_i) = 0 <=> a_i >= 1 for all i (window a_i in [-3,6], " "exhaustive) -- POSITIVITY of the anchor IS h^0 = 0; integrality itself is " "Birkhoff-Grothendieck (every holomorphic bundle on P^1 splits with integer " "degrees), automatic once E is typed as a bundle", dict_ok) # ==================================== D. uniqueness + convergence + controls # 10. the partition lemma under the full typing sols = [m for m in combinations_with_replacement(range(1, 11), 3) if sum(m) == 4] check("UNIQUENESS [E, v491]: {a in Z^3 : a_i >= 1 (h^0 = 0), sum a_i = 4 " "(deg E = -4), 3 parts (rank 3)} = %s -- unique; e2 = %d = g_car, " "e3 = %d = |Z2|, e2 - e3 = %d = N_fam" % (sols, _e2(sols[0]), sols[0][0] * sols[0][1] * sols[0][2], _e2(sols[0]) - 2), sols == [(1, 1, 2)] and _e2(sols[0]) == g_car and _e2(sols[0]) - sols[0][0] * sols[0][1] * sols[0][2] == N_fam) # 11. Schur-Horn route converges (the tfpt_2 theorem, with 'balance' TYPED) bound = [sp.Rational(8, 3), sp.Rational(4, 1)] # partial sums of 4*(2/3,1/3) sh = [] for a in product(range(0, 3), repeat=3): # diag in [0, 2/3] => a_i in {0,1,2} s = sorted(a, reverse=True) if sum(s) == 4 and s[0] <= bound[0] and s[0] + s[1] <= bound[1]: if tuple(s) not in sh: sh.append(tuple(s)) survivors = [s for s in sh if all(x >= 1 for x in s)] check("SCHUR-HORN CONVERGENCE [E, tfpt_2 'Global computation']: integer points " "of the permutohedron of 4*spec(A0) = (8/3, 4/3, 0) with sum 4 = %s " "(exactly the paper's two options); h^0 = 0 kills {2,2,0} (contains O(0), " "h^0 = 1) => %s -- the paper's 'balanced' selection is now TYPED as " "positivity/stability, and the two routes converge on {1,1,2}" % (sorted(sh), survivors), sorted(sh) == [(2, 1, 1), (2, 2, 0)] and survivors == [(2, 1, 1)]) # 12. negative control: drop positivity nc0 = [m for m in combinations_with_replacement(range(0, 5), 3) if sum(m) == 4] ncz = [m for m in combinations_with_replacement(range(-5, 10), 3) if sum(m) == 4] e2s0 = sorted(set(_e2(m) for m in nc0)) e2sz = set(_e2(m) for m in ncz) check("NEGATIVE CONTROL / POSITIVITY [E, v491]: allowing a_i = 0 gives %d " "solutions, e2 ambiguous %s; allowing negatives (window [-5,9]) gives %d " "solutions, %d distinct e2 values -- h^0 = 0 is load-bearing" % (len(nc0), e2s0, len(ncz), len(e2sz)), len(nc0) == 4 and e2s0 == [0, 3, 4, 5] and len(ncz) == 31 and len(e2sz) == 29 and 5 in e2sz) # 13. negative control: drop the degree fixing nc3 = [m for m in combinations_with_replacement(range(1, 11), 3) if sum(m) == 3] nc5 = [m for m in combinations_with_replacement(range(1, 11), 3) if sum(m) == 5] check("NEGATIVE CONTROL / DEGREE [E, v491]: deg -3 -> unique but e2 = %d != 5; " "deg -5 -> %d solutions (not unique) -- deg E = -4 = -#marks is " "load-bearing" % (_e2(nc3[0]), len(nc5)), len(nc3) == 1 and _e2(nc3[0]) == 3 and len(nc5) == 2) # 14. negative control: the mark count (n marks => n-1 parts summing to n) marks_scan = {} for n in range(3, 9): s = [m for m in combinations_with_replacement(range(1, n + 1), n - 1) if sum(m) == n] marks_scan[n] = (len(s), _e2(s[0])) formula_ok = all(marks_scan[n][1] == (n - 2) * (n + 1) // 2 for n in marks_scan) check("NEGATIVE CONTROL / MARK COUNT [E]: n marks => n-1 positive parts summing " "to n; the partition {1,..,1,2} is unique for EVERY n >= 3 (uniqueness is " "generic!), but e2 = (n-2)(n+1)/2 = %s -- e2 = 5 = g_car picks out n = 4 " "alone: the VALUE, not the uniqueness, is the n = 4 content" % ({n: v[1] for n, v in marks_scan.items()}), all(v[0] == 1 for v in marks_scan.values()) and formula_ok and [n for n, v in marks_scan.items() if v[1] == 5] == [4]) # 15. negative control: MS-rational weights typed directly as exponents w1 = [sp.Rational(1, 3)] * 3 w2 = [sp.Rational(1, 6), sp.Rational(1, 3), sp.Rational(1, 2)] e2w = [_e2([4 * x for x in w]) for w in (w1, w2)] check("NEGATIVE CONTROL / INTEGRALITY [E, v491]: typing the RATIONAL MS weights " "directly as exponents (scaled by 4) gives e2 = %s and %s, both != 5 and " "not integers -- integrality lives on the BUNDLE side (BG splitting of " "the Deligne extension), not on the parabolic side" % (e2w[0], e2w[1]), e2w == [sp.Rational(16, 3), sp.Rational(44, 9)]) # ==================================== E. the h^1 discrepancy, resolved # 16. h^1 of the anchor bundle is 1, not 3; the carrier of the generations is L a_anchor = (1, 1, 2) h1E = sum(_h1_line(-ai) for ai in a_anchor) chiE = sum((-ai) + 1 for ai in a_anchor) # RR: chi(O(d)) = d + 1 check("THE h^1 DISCREPANCY [E, resolved]: h^1(O(-2)+O(-1)^2) = sum(a_i - 1) = " "%d != 3 and chi(E) = deg + rk = %d (RR) -- the naive typing 'generation " "space = H^1(E)' is FALSE. Correct two-object typing: generations = " "H^1(L) = C^3 for L = O(-D); E carries them as its FIBER (rank 3) with " "the anchor as splitting type; shared invariants rank E = 3 = h^1(L), " "deg E = -4 = deg L, h^0(E) = 0 = h^0(L)" % (h1E, chiE), h1E == 1 and chiE == -1 and 3 == _h1_line(-4) and -4 == sum(-ai for ai in a_anchor) and sum(_h0_line(-ai) for ai in a_anchor) == 0 == _h0_line(-4)) # 17. negative control: forcing 'generations = H^1(rank-3 E)' breaks everything forced = [m for m in combinations_with_replacement(range(1, 8), 3) if sum(x - 1 for x in m) == 3] check("NEGATIVE CONTROL / NAIVE TYPING [E]: demanding h^1(E') = 3 with " "h^0(E') = 0 for a rank-3 E' forces sum a_i = 6, i.e. deg = -6 != -4, " "and the splitting is NOT unique: %s with e2 = %s -- never 5. The " "two-object typing is load-bearing" % (forced, [_e2(m) for m in forced]), forced == [(1, 1, 4), (1, 2, 3), (2, 2, 2)] and [_e2(m) for m in forced] == [9, 11, 12] and all(sum(m) == 6 for m in forced)) # 18. observations (exact, but NOT load-bearing -- recorded as curiosities) h1_EtwD = sum(_h1_line(-ai - 4) for ai in a_anchor) # E(-D) = O(-6)+O(-5)^2 h1_locsys = -3 * (2 - 4) # -chi = -rk*chi_top check("OBSERVATION [E arithmetic, curiosity, not load-bearing]: h^1(E(-D)) = " "5+4+4 = %d = Delta_Q (the 13 of v115's (8,0,5)/144 lemma), and h^1 of " "the rank-3 flavor LOCAL SYSTEM on P^1 minus mu4 is -chi = 3*(4-2) = %d " "= |R+(A3)| -- three DIFFERENT H^1's coexist: 3 (generations, L-side), " "1 (coherent h^1 of E), 6 (local-system deformations); the discrepancy " "came from conflating them" % (h1_EtwD, h1_locsys), h1_EtwD == 13 and h1_locsys == 6) # ==================================== F. the Biswas correspondence, explicit # 19. the pillowcase double (deck side): genus 1, j = 1728 -- NOT the weight cover g2 = sp.Rational(2 * 0 - 2) * 2 + 4 * (2 - 1) # RH: 2g-2 = N(2g_X-2)+sum(e-1) genus_z2 = (g2 + 2) / 2 lam_cr = sp.Rational(2) # cross-ratio of mu4 (v214) j_inv = 256 * (lam_cr ** 2 - lam_cr + 1) ** 3 / (lam_cr ** 2 * (lam_cr - 1) ** 2) check("DECK COVER [E, v214-type]: w^2 = z^4 - 1 (Z2, branched at mu4, deg 4 " "even => unramified over infinity): Riemann-Hurwitz genus = %s = 1, " "cross-ratio 2 => j = %s = 1728 (the tau = i pillowcase double) -- " "denominator 2 = |Z2|: this is the DECK-side cover, not the Biswas cover " "for the weights (denominator 3 = N_fam)" % (genus_z2, j_inv), genus_z2 == 1 and j_inv == 1728) # 20. the Z3 cusp cover exists only with asymmetric branch data (1,1,2,2) all_equal_bad = all((4 * c) % 3 != 0 for c in (1, 2)) m_branch = (1, 1, 2, 2) branch_ok = sum(m_branch) % 3 == 0 degq = sum(m_branch) # q = (z-1)(z-i)(z+1)^2(z+i)^2 unram_inf = degq % 3 == 0 g3 = 3 * (2 * 0 - 2) + 4 * (3 - 1) # 4 totally ramified points genus_z3 = sp.Rational(g3 + 2, 2) check("BISWAS COVER [E, explicit]: a Z3 cover killing the weight denominator " "3 = N_fam cannot have equal local monodromies (4c mod 3 != 0 for " "c = 1,2); the choice m = (1,1,2,2) works (sum 6 = 0 mod 3, deg q = 6 => " "unramified over infinity); Riemann-Hurwitz genus = %s = 2 -- the cusp " "cover Y: y^3 = (z-1)(z-i)(z+1)^2(z+i)^2 is a genus-2 curve with Z3 " "action, NOT the pillowcase double" % genus_z3, all_equal_bad and branch_ok and unram_inf and genus_z3 == 2) # 21. eigensheaf degrees of p_* O_Y by explicit pole bookkeeping + chi cross-check deg_V = {} for j in (1, 2): # allowed pole of g at mark i: floor(j*m_i/3); at infinity ord(y^j) = -2j deg_V[j] = sum((j * mi) // 3 for mi in m_branch) - 2 * j chi_sum = (0 + 1) + (deg_V[1] + 1) + (deg_V[2] + 1) # chi(O(d)) = d+1 on P^1 chi_Y = 1 - 2 # 1 - g(Y) check("EIGENSHEAVES [E, explicit]: p_* O_Y = O + V_1 + V_2 with " "deg V_j = sum_i floor(j m_i / 3) - 2j = (%d, %d) => V_1 = V_2 = O(-2); " "cross-check chi(p_* O_Y) = 1 - 1 - 1 = %d = chi(O_Y) = 1 - g(Y) = %d " "(Riemann-Roch on both sides of the cover agrees)" % (deg_V[1], deg_V[2], chi_sum, chi_Y), deg_V == {1: -2, 2: -2} and chi_sum == chi_Y == -1) # 22. the Biswas partner of the decomposable model W = (chi^0+chi^1+chi^2) x O_Y wt_sets_ok = all( sorted(sp.Rational(j * mi, 3) % 1 for j in range(3)) == sorted(WEIGHTS) for mi in m_branch) E_dec = (0, 2, 2) # O + O(-2) + O(-2), exponents deg_dec = -sum(E_dec) h0_dec = sum(_h0_line(-ai) for ai in E_dec) pardeg_dec = deg_dec + 4 * sum(WEIGHTS) check("BISWAS IMAGE [E, the independent witness]: W = (chi^0+chi^1+chi^2) x O_Y " "has rank 3, deg 0, isotropy = REGULAR rep at every ramification point " "(fractional weights {j m_i/3} = {0,1/3,2/3} at every mark, both m_i = 1 " "and 2); its parabolic partner has underlying bundle O + O(-2)^2: rank 3 " "OK, deg %d = -4 OK, par-deg %d = 0 OK -- integrality AND the sum rule " "COME OUT of the correspondence; but the splitting is {0,2,2} with " "h^0 = %d != 0: the decomposable (trivial-character) model is exactly the " "unstable Schur-Horn companion -- (1,1,2) is the Biswas image of the " "STABLE object, and stability (irreducibility, v117) is the selector" % (deg_dec, pardeg_dec, h0_dec), wt_sets_ok and deg_dec == -4 and pardeg_dec == 0 and h0_dec == 1) # ==================================== G. honest verdict # 23. residual map (recorded; P2 stays an axiom, no marker moves) check("HONEST RESIDUALS [C/N recorded]: THE CONDITIONAL THEOREM (standard AG, " "given [E] anchors): 'Given 4 marks (v216), rank 3 (v137), cusp class " "(v117), irreducibility (v117) and (U): Deligne + BG + stability force " "E = O(-1)^2 + O(-2), so a = {1,1,2} and g_car = e2 = 5.' RESIDUALS: R1 " "module identification 'generation space = this H^1 with this connection' " "[C] (QGEO.SYM territory, stays [O]); R2 unitarity (U) [C] (v33/v40); R3 " "analytic-monodromy identification [N] (v117 check 4). NON-CIRCULAR: " "neither v23's splitting-type input nor v115's diag(A0) = a/4 input is " "used; the weight postulate shrinks from 'positive integers summing to 4 " "(why?)' to 'Deligne + BG + stability, given the cohomological typing'. " "AX.P2.01 stays a declared axiom; direction reversal (marks -> bundle -> " "g_car) is an over-determination check, never double-counted", True) return summary("v499 P2.TYPING.01: the v491 weight-typing residual hardened -- rank 3 " "(topology) + deg E = -4 (Deligne residue trace) + integrality (Birkhoff-" "Grothendieck) + positivity (Mehta-Seshadri stability, h^0 = 0) type the anchor " "as E = O(-1)^2 + O(-2), so {1,1,2} and g_car = 5 follow; h^1 discrepancy " "resolved by the two-object typing (L = O(-D) carries the generations); Biswas " "Z3-cover computed explicitly (independent convergence, stability the selector); " "P2 stays an axiom (R1/R2 [C], R3 [N])") if __name__ == "__main__": raise SystemExit(1 if run() else 0)