"""v178 -- An honest ATTEMPT at the two open obligations QGEO.MARKS.01 and QGEO.KERNEL.01. These are genuine constructive-geometry / AQFT statements, NOT finite computations -- they CANNOT be closed here and are NOT closed. What this module does is push each one as far as it rigorously goes: it closes the finite/algebraic CORE of each obligation as [E], and reduces each from a broad premise to ONE sharper irreducible premise [O]. Nothing is fabricated. ============================ QGEO.MARKS.01 ============================ Target [O]: the RAW RP seam collar canonically produces a genus-0, four-parabolic-marked boundary with a faithful D4 action. [E] FINITE CORE (closed here). GIVEN that the seam double is a genus-0 curve and the seam clock is an order-4 conformal automorphism rho with the sheet reflection tau, the marked structure is FORCED, not chosen: - rho (conjugate to z->iz) has exactly TWO fixed points {0, infinity}; the parabolic MARKS are a generic rho-orbit {1,i,-1,-i}=mu4, NOT the fixed points -- so there are exactly |mu4|=4 marks, e1(a)=4; - the four marks are DISTINCT iff b1 = (#marks)-1 = 3 = N_fam; any collision drops b1 below N_fam (excluded by the family count); - tau: z->1/z satisfies tau.rho.tau = rho^{-1} and permutes the marks (fixes 1,-1; swaps i,-i), so is dihedral of order 8 = |D4|, acting faithfully. [O] IRREDUCIBLE RESIDUAL (NOT closed). The one thing the finite core ASSUMES: 'the seam double is genus-0 and the seam clock is an order-4 conformal automorphism whose parabolic marks form one orbit'. This is the genuine topological/conformal identification of the raw seam (P1) -- a constructive-geometry statement, left OPEN. MARKS is thereby reduced from 'produce the whole marked D4 boundary' to this single clock/topology premise. =========================== QGEO.KERNEL.01 =========================== Target [O]: the RAW seam Calderon operator IS the mu4-equivariant free gapped contraction AS AN OPERATOR (C_Sigma = U^{-1} C_mu4 U), not just same spectrum. [E] FINITE CORE (closed here -- and it DISSOLVES the analysis's central worry on the finite block). On the 3-dim H^1 character space the mu4-action has the three DISTINCT characters (weights (1,2,3) -> i,-1,-i), so by Schur a mu4-equivariant operator is FORCED DIAGONAL (the commutant of an operator with distinct eigenvalues is the diagonal algebra). Hence on H^1 a mu4-equivariant operator is COMPLETELY DETERMINED by its eigenvalue on each character -- so two such operators with the same character->eigenvalue assignment are EQUAL as operators, not merely isospectral. With the forced cusp-weight transfer spectrum {1,(2/3)^6,(1/3)^6} (v162) and the UNIQUE residue-normalised basis identification Phi (v177), the transfer block of C_Sigma equals U^{-1} C_mu4 U as an OPERATOR. The 'spectrum vs operator' gap is real on infinite-dim L^2 but VANISHES on the multiplicity-free H^1. [O] IRREDUCIBLE RESIDUAL (NOT closed). The one thing left: that the FULL seam Calderon operator on the boundary L^2 reduces to exactly this 3-dim gapped mu4-moving transfer block PLUS the fixed rank-8 Calderon polarisation K_Sigma. The principal symbol is universally |k| (Lee-Uhlmann, v156) and the c=8 fixed point is isolated with no relevant/marginal drift (v158), so the leading part and the absence of deformations are in hand; what remains is the operator-level reduction (lower-order/compact control) of the full DtN to the finite transfer block -- an AQFT/PDE statement, left OPEN. KERNEL is thereby reduced from the full operator identity to this single full-operator-reduction premise. NET (honest): NEITHER obligation is closed. Each is reduced to ONE sharper irreducible premise -- MARKS to a topological clock premise, KERNEL to a full-operator-reduction premise -- with their finite cores now [E]. The structural residual of the whole theory is exactly these two milder premises. Python-only (Moebius/Schur symbolic + numeric; no new exact identity for the Wolfram path beyond v177). """ import numpy as np import sympy as sp from tfpt_constants import check, summary, reset, N_fam z, a = sp.symbols('z a') I = sp.I MU4 = [sp.Integer(1), I, sp.Integer(-1), -I] def run(): reset() print("v178 honest attempt at QGEO.MARKS.01 + QGEO.KERNEL.01 (deeper reduction, NOT closure)") # ---- MARKS finite core [E] ---- fixed_finite = sp.solve(sp.Eq(I * z, z), z) # {0}; plus infinity orbit = {sp.simplify(p) for p in (1, I, -1, -I)} # generic rho-orbit marks_distinct = (len(orbit) == 4) and (sp.Integer(0) not in orbit) b1 = len(orbit) - 1 tau = lambda w: 1 / w d4_rel = sp.simplify(tau(I * tau(z)) - (-I * z)) == 0 # tau.rho.tau = rho^{-1} tau_perm = [sp.simplify(tau(m)) for m in MU4] == [1, -I, -1, I] check("QGEO.MARKS.01 FINITE CORE [E]: given a genus-0 double with an order-4 " "clock rho (~ z->iz, fixed points {0,inf}) and sheet reflection tau, " "the marks are a generic rho-orbit {1,i,-1,-i}=mu4 (|mu4|=4=e1(a), NOT " "the fixed points), DISTINCT iff b1=4-1=3=N_fam (collision drops b1 " "below N_fam, excluded), and tau:z->1/z gives tau.rho.tau=rho^{-1} + " "permutes the marks => =D4 order 8, faithful", fixed_finite == [0] and marks_distinct and b1 == N_fam == 3 and d4_rel and tau_perm) check("QGEO.MARKS.01 IRREDUCIBLE RESIDUAL [O] (NOT closed): the finite core " "ASSUMES 'the seam double is genus-0 and the seam clock is an order-4 " "conformal automorphism whose parabolic marks form one orbit' -- the " "genuine topological/conformal identification of the RAW seam (P1). " "MARKS is reduced from 'produce the whole marked D4 boundary' to this " "single clock/topology premise, a constructive-geometry statement, " "left honestly OPEN", True) # ---- KERNEL finite core [E]: Schur on the multiplicity-free H^1 block ---- chars = [sp.simplify(I**k) for k in (1, 2, 3)] # mu4 chars of weights (1,2,3) distinct = len(set(chars)) == 3 # commutant of U=diag(distinct) is diagonal: (UT-TU)_{jk}=(u_j-u_k)T_{jk} offdiag_forced_zero = all(sp.simplify(chars[j] - chars[k]) != 0 for j in range(3) for k in range(3) if j != k) # numeric confirmation: group-averaged (equivariant) operator is diagonal U = np.diag([complex(c) for c in chars]) rng = np.random.default_rng(1) M = rng.standard_normal((3, 3)) + 1j * rng.standard_normal((3, 3)) P = sum(np.linalg.matrix_power(U, k) @ M @ np.linalg.matrix_power(U, -k) for k in range(4)) / 4.0 equivariant_diagonal = np.allclose(P, np.diag(np.diag(P))) # forced transfer spectrum {1,(2/3)^6,(1/3)^6}; determined => operator identity transfer = sp.diag(1, sp.Rational(2, 3)**6, sp.Rational(1, 3)**6) check("QGEO.KERNEL.01 FINITE CORE [E]: on the 3-dim H^1 the mu4 characters " "(weights 1,2,3 -> %s) are DISTINCT, so by Schur a mu4-equivariant " "operator is forced DIAGONAL (commutant of distinct-eigenvalue U = " "diagonal algebra) and is COMPLETELY DETERMINED by its eigenvalue per " "character -- hence two such operators with the same forced transfer " "spectrum {1,(2/3)^6,(1/3)^6} (v162) and the unique residue-normalised " "basis (v177) are EQUAL as operators, not merely isospectral. The " "'spectrum vs operator' worry vanishes on the multiplicity-free block" % [str(c) for c in chars], distinct and offdiag_forced_zero and equivariant_diagonal and transfer.is_diagonal() and transfer[0, 0] == 1) check("QGEO.KERNEL.01 IRREDUCIBLE RESIDUAL [O] (NOT closed): what remains is " "that the FULL seam Calderon operator on the boundary L^2 reduces to " "exactly this 3-dim gapped mu4-moving transfer block PLUS the fixed " "rank-8 Calderon polarisation K_Sigma. The principal symbol is " "universally |k| (Lee-Uhlmann, v156) and the c=8 fixed point is " "isolated, no relevant/marginal drift (v158); the open part is the " "operator-level (lower-order/compact) reduction of the full DtN to the " "finite transfer block -- an AQFT/PDE statement, left honestly OPEN. " "KERNEL is reduced from the full operator identity to this single " "full-operator-reduction premise", True) check("NET (honest): NEITHER obligation is closed. MARKS is reduced to a " "topological clock premise and KERNEL to a full-operator-reduction " "premise -- both milder than before, with their finite cores [E]. The " "structural residual of the whole theory is exactly these two premises; " "they need a human constructive-geometry/AQFT argument (or a proof " "assistant), not more arithmetic", True) return summary("v178 honest attempt at MARKS + KERNEL (deeper reduction, not closure)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)