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Original repository path: verification/v369_qgamb_redundancy.py
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000001 | """v369 -- QGAMB.REDUNDANCY.01 (Track 2): the AMBIENT REDUNDANCY statement -- the
000002 | holographic route to Gravity-complete that SIDESTEPS building the general Euclidean QG
000003 | measure (C7/QG.AMB.01).
000004 | 
000005 | Strategy (the strongest move per the next-plan v2 Track 2 / the list.txt "Paper B"):
000006 | instead of constructing the non-perturbative ambient measure (the conformal-factor swamp,
000007 | v332/v334), prove that the ambient sector carries NO physical degrees of freedom relative
000008 | to the holomorphic boundary algebra -- so QG.AMB.01 is a non-fundamental CERTIFICATION
000009 | object (like a coordinate choice in GR), not missing dynamics.
000010 | 
000011 | The statement (the "Ambient Redundancy Theorem"):
000012 | 
000013 |     O_phys^bulk / Diff  ~=  A_Sigma^{mu4}                        (boundary reconstructibility)
000014 |     two ambient measures with the same boundary state are physically equivalent.
000015 | 
000016 | This module does NOT construct the measure and does NOT close SEAM.EQUIV.01; it ASSEMBLES
000017 | the established [E] discriminators that make the redundancy statement well-posed, and types
000018 | the theorem honestly as [C] (conditional on SEAM.EQUIV.01 + the Bisognano-Wichmann
000019 | intrinsicality of the raw collar). Exactly the style of v335 (keystone unification) and
000020 | v365 (one-loop saddle): a synthesis that sharpens a gate, fabricating nothing.
000021 | 
000022 |   [E] 1. HOLOMORPHIC => TRIVIAL SUPERSELECTION (DHR = Vec).  the number of DHR
000023 |         superselection sectors (primaries) of a chiral net equals |det Cartan|; for E8 it
000024 |         is 1 (ONE primary = the vacuum) so DHR((E8)_1) = Vec (trivial braided category),
000025 |         against det Cartan(D8/SO16) = 4 (four primaries).  A holomorphic boundary net has
000026 |         NO nontrivial superselection charges => no hidden boundary sector a bulk could carry
000027 |         invisibly.  (Computed from the Cartan matrices; cf. v83/v237/v308.)
000028 |   [E] 2. NO TORUS GROUND-STATE DEGENERACY.  the topological ground-state degeneracy on T^2
000029 |         equals the number of anyons = |det K| = 1 for E8 => a UNIQUE ground state => no
000030 |         topologically protected bulk d.o.f. invisible to the boundary.  (Same det K=1
000031 |         discriminator, the genus-1 obstruction of v344, here in its redundancy reading.)
000032 |   [E] 3. FINITE-RATE BOUNDARY (PETZ) RECOVERY.  the seam transfer spectrum
000033 |         {1,(2/3)^6,(1/3)^6}: the deviation (trace-zero) subspace contracts at rate
000034 |         (2/3)^6 = 64/729 < 1 (v221), so bulk perturbations are recovered by a boundary
000035 |         (Petz) map at a finite, gapped rate => the reconstruction map Rec_Sigma is bounded.
000036 |   [E] 4. GAP-DECOUPLING MARGIN.  Delta_eff = 6 ln(3/2) - 31/(4 pi^2) ~ 1.648 > 0
000037 |         (v76/v337): the un-built ambient sector is energetically SEPARATED from the
000038 |         admissible (boundary-reconstructible) sector => ambient fluctuations cannot inject
000039 |         new low-energy physical d.o.f. into A_Sigma.
000040 |   [C] 5. RECONSTRUCTION MAP FROM MODULAR COVARIANCE.  Bisognano-Wichmann / Tomita-Takesaki
000041 |         gives the modular flow as the geometric boost (v258/v309/v323); with holomorphy
000042 |         this supplies the canonical reconstruction Rec_Sigma: (bulk algebra) -> A_Sigma^{mu4}
000043 |         (cited; conditional).
000044 |   [C] 6. THE REDUNDANCY STATEMENT.  under SEAM.EQUIV.01 (boundary = (E8)_1) + BW: every
000045 |         gauge-invariant bulk observable is boundary-reconstructible, and ambient measures
000046 |         sharing the boundary state are physically equivalent => QG.AMB.01 carries no extra
000047 |         physical d.o.f.  Reframes "build the ambient measure" to "the ambient measure is a
000048 |         non-fundamental certification object" (the holographic No-Ambient route).
000049 |   [O] 7. RESIDUAL.  the theorem CONSUMES exactly two premises -- (a) SEAM.EQUIV.01 (the
000050 |         boundary IS (E8)_1; the S3 lattice input, v366) and (b) the BW intrinsicality of the
000051 |         raw collar state (v329 NEG).  It does NOT construct QG.AMB.01; it provides the
000052 |         ALTERNATIVE (redundancy) route to Gravity-complete, conditional on those two.
000053 | 
000054 | NET: Gravity-complete = Boundary-complete + Ambient-redundancy.  This module supplies the
000055 | "+ Ambient-redundancy" half as a [C] synthesis of established [E] facts.  Python-only
000056 | (sympy exact arithmetic + integer/lattice discriminators already in the Wolfram path);
000057 | the theorem itself is [C]/[O] so, like v335/v365/v366, this module is Python-only."""
000058 | import sympy as sp
000059 | 
000060 | from tfpt_constants import check, summary, reset, g_car, N_fam
000061 | 
000062 | 
000063 | def _cartan_det(n, edges):
000064 |     """Determinant of the Cartan matrix (2I - adjacency) of a simply-laced Dynkin
000065 |     diagram on n nodes with the given undirected edge list (1-based node labels)."""
000066 |     A = sp.zeros(n, n)
000067 |     for i in range(n):
000068 |         A[i, i] = 2
000069 |     for a, b in edges:
000070 |         A[a - 1, b - 1] = -1
000071 |         A[b - 1, a - 1] = -1
000072 |     return int(A.det()), A
000073 | 
000074 | 
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")
000151 | 
000152 | 
000153 | if __name__ == "__main__":
000154 |     raise SystemExit(1 if run() else 0)