Follow the source behind the result.

The registered Python source is shown as text. Displaying it does not execute the code.

"""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)

89e640abf05b9c86212b5e30c896e7f6458e7ad43660a507f62068452fc6b195

The public registry includes 116 document views, 6,693 section records and 985 physical LaTeX sources. Active publication documents, research material and archives retain their recorded roles. This is source navigation, not a new semantic review or proof run.