TFPT scientific source excerpt Original repository path: verification/v369_qgamb_redundancy.py Original file SHA-256: 89e640abf05b9c86212b5e30c896e7f6458e7ad43660a507f62068452fc6b195 Inclusive original lines: 75-150 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000075 | def run(): 000076 | reset() 000077 | print("v369 QGAMB.REDUNDANCY.01: the ambient measure is boundary-redundant " 000078 | "(the holographic route to Gravity-complete)") 000079 | 000080 | # --- Cartan-matrix discriminators (Bourbaki simply-laced diagrams) --------- 000081 | # E8: chain 1-3-4-5-6-7-8 with leg node 2 attached to node 4 000082 | detE8, _ = _cartan_det(8, [(1, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (2, 4)]) 000083 | # D8 (SO(16)): chain 1-2-3-4-5-6 with the fork nodes 7,8 both on node 6 000084 | detD8, _ = _cartan_det(8, [(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (6, 8)]) 000085 | 000086 | # 1. holomorphic => trivial DHR (Vec): #primaries = |det Cartan| 000087 | check("HOLOMORPHIC => DHR = Vec [E]: #DHR primaries = |det Cartan| -- E8 gives %d (ONE " 000088 | "primary = vacuum, trivial braided category) vs SO(16)/D8 = %d (four primaries); " 000089 | "a holomorphic boundary net has NO nontrivial superselection charges => no hidden " 000090 | "boundary sector a bulk could carry invisibly" % (detE8, detD8), 000091 | detE8 == 1 and detD8 == 4) 000092 | 000093 | # 2. no torus ground-state degeneracy: GSD(T^2) = #anyons = |det K| = 1 for E8 000094 | GSD_T2 = abs(detE8) 000095 | check("NO TORUS GSD [E]: ground-state degeneracy on T^2 = #anyons = |det K| = %d for E8 " 000096 | "(vs %d for SO(16)) => a UNIQUE ground state => no topologically protected bulk " 000097 | "d.o.f. invisible to the boundary (the genus-1 obstruction of v344, redundancy " 000098 | "reading)" % (GSD_T2, abs(detD8)), 000099 | GSD_T2 == 1) 000100 | 000101 | # 3. finite-rate Petz recovery on the deviation subspace: rate (2/3)^6 = 64/729 < 1 000102 | recov = sp.Rational(2, 3) ** 6 000103 | check("FINITE PETZ RECOVERY [E]: the seam transfer deviation subspace contracts at rate " 000104 | "(2/3)^6 = %s ~ %.4f < 1 (v221) => bulk perturbations are boundary-recoverable by " 000105 | "a Petz map at a finite gapped rate => Rec_Sigma is bounded" 000106 | % (recov, float(recov)), 000107 | recov == sp.Rational(64, 729) and 0 < float(recov) < 1) 000108 | 000109 | # 4. gap-decoupling margin Delta_eff > 0 (the ambient sector is energetically separated) 000110 | Delta = 6 * sp.log(sp.Rational(3, 2)) 000111 | backreaction = 2 * 248 * (sp.Rational(1, 1) / (8 * sp.pi)) ** 2 # 2*dim(E8)*c3^2 = 31/(4 pi^2) 000112 | Delta_eff = Delta - backreaction 000113 | check("GAP-DECOUPLING MARGIN [E]: Delta_eff = 6 ln(3/2) - 2*dim(E8)*c3^2 = " 000114 | "6 ln(3/2) - 31/(4 pi^2) ~ %.4f > 0 (v76/v337) => the un-built ambient sector is " 000115 | "energetically separated from the admissible (boundary-reconstructible) sector => " 000116 | "ambient fluctuations inject no new low-energy physical d.o.f. into A_Sigma" 000117 | % float(Delta_eff), 000118 | sp.simplify(backreaction - sp.Rational(31, 4) / sp.pi**2) == 0 and float(Delta_eff) > 1.6) 000119 | 000120 | # 5. reconstruction map from modular (Bisognano-Wichmann) covariance (cited, conditional) 000121 | check("RECONSTRUCTION MAP [C]: Bisognano-Wichmann / Tomita-Takesaki gives the modular " 000122 | "flow as the geometric boost (v258/v309/v323); with holomorphy this supplies the " 000123 | "canonical reconstruction Rec_Sigma: (bulk algebra) -> A_Sigma^{mu4} (cited)", True) 000124 | 000125 | # 6. THE REDUNDANCY STATEMENT (conditional on SEAM.EQUIV.01 + BW): the assembly is valid 000126 | # iff all four [E] discriminators above hold (trivial DHR, no GSD, finite recovery, 000127 | # positive gap margin) -- then ambient measures with the same boundary state are 000128 | # physically equivalent. 000129 | redundancy_well_posed = (detE8 == 1) and (GSD_T2 == 1) and (0 < float(recov) < 1) \ 000130 | and (float(Delta_eff) > 0) 000131 | check("AMBIENT REDUNDANCY [C]: under SEAM.EQUIV.01 (boundary = (E8)_1) + BW, every " 000132 | "gauge-invariant bulk observable is boundary-reconstructible (O_phys^bulk/Diff ~= " 000133 | "A_Sigma^{mu4}) and ambient measures sharing the boundary state are physically " 000134 | "equivalent => QG.AMB.01 carries NO extra physical d.o.f. (a certification object, " 000135 | "not missing dynamics) -- well-posed since DHR=Vec & no GSD & finite recovery & " 000136 | "gap margin>0", redundancy_well_posed) 000137 | 000138 | # 7. residual: the two consumed premises (does NOT close QG.AMB.01) 000139 | check("RESIDUAL [O]: the theorem consumes exactly TWO premises -- (a) SEAM.EQUIV.01 " 000140 | "(boundary IS (E8)_1; the S3 lattice input, v366) and (b) the BW intrinsicality of " 000141 | "the raw collar (v329 NEG); it does NOT construct QG.AMB.01 but provides the " 000142 | "ALTERNATIVE (holographic redundancy) route to Gravity-complete, conditional on " 000143 | "those two", True) 000144 | 000145 | return summary("v369 QGAMB.REDUNDANCY.01: the ambient QG measure (C7/QG.AMB.01) is " 000146 | "boundary-REDUNDANT -- holomorphic (E8)_1 gives det K=1 => DHR=Vec (no " 000147 | "superselection) + no torus GSD, finite Petz recovery (2/3)^6, and the gap " 000148 | "margin Delta_eff~1.648>0 make O_phys^bulk/Diff ~= A_Sigma^{mu4} well-posed; " 000149 | "so QG.AMB.01 is a non-fundamental certification object [C], conditional on " 000150 | "SEAM.EQUIV.01 + BW intrinsicality [O]. The holographic route to Gravity-complete")