"""v369 -- QGAMB.REDUNDANCY.01 (Track 2): the AMBIENT REDUNDANCY statement -- the holographic route to Gravity-complete that SIDESTEPS building the general Euclidean QG measure (C7/QG.AMB.01). Strategy (the strongest move per the next-plan v2 Track 2 / the list.txt "Paper B"): instead of constructing the non-perturbative ambient measure (the conformal-factor swamp, v332/v334), prove that the ambient sector carries NO physical degrees of freedom relative to the holomorphic boundary algebra -- so QG.AMB.01 is a non-fundamental CERTIFICATION object (like a coordinate choice in GR), not missing dynamics. The statement (the "Ambient Redundancy Theorem"): O_phys^bulk / Diff ~= A_Sigma^{mu4} (boundary reconstructibility) two ambient measures with the same boundary state are physically equivalent. This module does NOT construct the measure and does NOT close SEAM.EQUIV.01; it ASSEMBLES the established [E] discriminators that make the redundancy statement well-posed, and types the theorem honestly as [C] (conditional on SEAM.EQUIV.01 + the Bisognano-Wichmann intrinsicality of the raw collar). Exactly the style of v335 (keystone unification) and v365 (one-loop saddle): a synthesis that sharpens a gate, fabricating nothing. [E] 1. HOLOMORPHIC => TRIVIAL SUPERSELECTION (DHR = Vec). the number of DHR superselection sectors (primaries) of a chiral net equals |det Cartan|; for E8 it is 1 (ONE primary = the vacuum) so DHR((E8)_1) = Vec (trivial braided category), against det Cartan(D8/SO16) = 4 (four primaries). A holomorphic boundary net has NO nontrivial superselection charges => no hidden boundary sector a bulk could carry invisibly. (Computed from the Cartan matrices; cf. v83/v237/v308.) [E] 2. NO TORUS GROUND-STATE DEGENERACY. the topological ground-state degeneracy on T^2 equals the number of anyons = |det K| = 1 for E8 => a UNIQUE ground state => no topologically protected bulk d.o.f. invisible to the boundary. (Same det K=1 discriminator, the genus-1 obstruction of v344, here in its redundancy reading.) [E] 3. FINITE-RATE BOUNDARY (PETZ) RECOVERY. the seam transfer spectrum {1,(2/3)^6,(1/3)^6}: the deviation (trace-zero) subspace contracts at rate (2/3)^6 = 64/729 < 1 (v221), so bulk perturbations are recovered by a boundary (Petz) map at a finite, gapped rate => the reconstruction map Rec_Sigma is bounded. [E] 4. GAP-DECOUPLING MARGIN. Delta_eff = 6 ln(3/2) - 31/(4 pi^2) ~ 1.648 > 0 (v76/v337): the un-built ambient sector is energetically SEPARATED from the admissible (boundary-reconstructible) sector => ambient fluctuations cannot inject new low-energy physical d.o.f. into A_Sigma. [C] 5. RECONSTRUCTION MAP FROM MODULAR COVARIANCE. Bisognano-Wichmann / Tomita-Takesaki gives the modular flow as the geometric boost (v258/v309/v323); with holomorphy this supplies the canonical reconstruction Rec_Sigma: (bulk algebra) -> A_Sigma^{mu4} (cited; conditional). [C] 6. THE REDUNDANCY STATEMENT. under SEAM.EQUIV.01 (boundary = (E8)_1) + BW: every gauge-invariant bulk observable is boundary-reconstructible, and ambient measures sharing the boundary state are physically equivalent => QG.AMB.01 carries no extra physical d.o.f. Reframes "build the ambient measure" to "the ambient measure is a non-fundamental certification object" (the holographic No-Ambient route). [O] 7. RESIDUAL. the theorem CONSUMES exactly two premises -- (a) SEAM.EQUIV.01 (the boundary IS (E8)_1; the S3 lattice input, v366) and (b) the BW intrinsicality of the raw collar state (v329 NEG). It does NOT construct QG.AMB.01; it provides the ALTERNATIVE (redundancy) route to Gravity-complete, conditional on those two. NET: Gravity-complete = Boundary-complete + Ambient-redundancy. This module supplies the "+ Ambient-redundancy" half as a [C] synthesis of established [E] facts. Python-only (sympy exact arithmetic + integer/lattice discriminators already in the Wolfram path); the theorem itself is [C]/[O] so, like v335/v365/v366, this module is Python-only.""" import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam def _cartan_det(n, edges): """Determinant of the Cartan matrix (2I - adjacency) of a simply-laced Dynkin diagram on n nodes with the given undirected edge list (1-based node labels).""" A = sp.zeros(n, n) for i in range(n): A[i, i] = 2 for a, b in edges: A[a - 1, b - 1] = -1 A[b - 1, a - 1] = -1 return int(A.det()), A def run(): reset() print("v369 QGAMB.REDUNDANCY.01: the ambient measure is boundary-redundant " "(the holographic route to Gravity-complete)") # --- Cartan-matrix discriminators (Bourbaki simply-laced diagrams) --------- # E8: chain 1-3-4-5-6-7-8 with leg node 2 attached to node 4 detE8, _ = _cartan_det(8, [(1, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (2, 4)]) # D8 (SO(16)): chain 1-2-3-4-5-6 with the fork nodes 7,8 both on node 6 detD8, _ = _cartan_det(8, [(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (6, 8)]) # 1. holomorphic => trivial DHR (Vec): #primaries = |det Cartan| check("HOLOMORPHIC => DHR = Vec [E]: #DHR primaries = |det Cartan| -- E8 gives %d (ONE " "primary = vacuum, trivial braided category) vs SO(16)/D8 = %d (four primaries); " "a holomorphic boundary net has NO nontrivial superselection charges => no hidden " "boundary sector a bulk could carry invisibly" % (detE8, detD8), detE8 == 1 and detD8 == 4) # 2. no torus ground-state degeneracy: GSD(T^2) = #anyons = |det K| = 1 for E8 GSD_T2 = abs(detE8) check("NO TORUS GSD [E]: ground-state degeneracy on T^2 = #anyons = |det K| = %d for E8 " "(vs %d for SO(16)) => a UNIQUE ground state => no topologically protected bulk " "d.o.f. invisible to the boundary (the genus-1 obstruction of v344, redundancy " "reading)" % (GSD_T2, abs(detD8)), GSD_T2 == 1) # 3. finite-rate Petz recovery on the deviation subspace: rate (2/3)^6 = 64/729 < 1 recov = sp.Rational(2, 3) ** 6 check("FINITE PETZ RECOVERY [E]: the seam transfer deviation subspace contracts at rate " "(2/3)^6 = %s ~ %.4f < 1 (v221) => bulk perturbations are boundary-recoverable by " "a Petz map at a finite gapped rate => Rec_Sigma is bounded" % (recov, float(recov)), recov == sp.Rational(64, 729) and 0 < float(recov) < 1) # 4. gap-decoupling margin Delta_eff > 0 (the ambient sector is energetically separated) Delta = 6 * sp.log(sp.Rational(3, 2)) backreaction = 2 * 248 * (sp.Rational(1, 1) / (8 * sp.pi)) ** 2 # 2*dim(E8)*c3^2 = 31/(4 pi^2) Delta_eff = Delta - backreaction check("GAP-DECOUPLING MARGIN [E]: Delta_eff = 6 ln(3/2) - 2*dim(E8)*c3^2 = " "6 ln(3/2) - 31/(4 pi^2) ~ %.4f > 0 (v76/v337) => the un-built ambient sector is " "energetically separated from the admissible (boundary-reconstructible) sector => " "ambient fluctuations inject no new low-energy physical d.o.f. into A_Sigma" % float(Delta_eff), sp.simplify(backreaction - sp.Rational(31, 4) / sp.pi**2) == 0 and float(Delta_eff) > 1.6) # 5. reconstruction map from modular (Bisognano-Wichmann) covariance (cited, conditional) check("RECONSTRUCTION MAP [C]: Bisognano-Wichmann / Tomita-Takesaki gives the modular " "flow as the geometric boost (v258/v309/v323); with holomorphy this supplies the " "canonical reconstruction Rec_Sigma: (bulk algebra) -> A_Sigma^{mu4} (cited)", True) # 6. THE REDUNDANCY STATEMENT (conditional on SEAM.EQUIV.01 + BW): the assembly is valid # iff all four [E] discriminators above hold (trivial DHR, no GSD, finite recovery, # positive gap margin) -- then ambient measures with the same boundary state are # physically equivalent. redundancy_well_posed = (detE8 == 1) and (GSD_T2 == 1) and (0 < float(recov) < 1) \ and (float(Delta_eff) > 0) check("AMBIENT REDUNDANCY [C]: under SEAM.EQUIV.01 (boundary = (E8)_1) + BW, every " "gauge-invariant bulk observable is boundary-reconstructible (O_phys^bulk/Diff ~= " "A_Sigma^{mu4}) and ambient measures sharing the boundary state are physically " "equivalent => QG.AMB.01 carries NO extra physical d.o.f. (a certification object, " "not missing dynamics) -- well-posed since DHR=Vec & no GSD & finite recovery & " "gap margin>0", redundancy_well_posed) # 7. residual: the two consumed premises (does NOT close QG.AMB.01) check("RESIDUAL [O]: the theorem consumes exactly TWO premises -- (a) SEAM.EQUIV.01 " "(boundary IS (E8)_1; the S3 lattice input, v366) and (b) the BW intrinsicality of " "the raw collar (v329 NEG); it does NOT construct QG.AMB.01 but provides the " "ALTERNATIVE (holographic redundancy) route to Gravity-complete, conditional on " "those two", True) return summary("v369 QGAMB.REDUNDANCY.01: the ambient QG measure (C7/QG.AMB.01) is " "boundary-REDUNDANT -- holomorphic (E8)_1 gives det K=1 => DHR=Vec (no " "superselection) + no torus GSD, finite Petz recovery (2/3)^6, and the gap " "margin Delta_eff~1.648>0 make O_phys^bulk/Diff ~= A_Sigma^{mu4} well-posed; " "so QG.AMB.01 is a non-fundamental certification object [C], conditional on " "SEAM.EQUIV.01 + BW intrinsicality [O]. The holographic route to Gravity-complete") if __name__ == "__main__": raise SystemExit(1 if run() else 0)