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Original ledger classification: Numerical/identity [N]/[I] (finite vanishing exact) + target [O] (full-operator min = BW)

Recorded statement: THE E_fail VARIATIONAL FUNCTIONAL (external-review path 1, 2026-06-14): a concrete, numerically-testable sharpening of the v194/QGEO.ENERGY.02 target. E_fail(Sigma,rho) = ||[rho,Lambda_Sigma]||^2_HS + ||rho^4-1||^2 + ||Theta rho Theta -…

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QGEO.VARI.01

Original ledger classification: Numerical/identity [N]/[I] (finite vanishing exact) + target [O] (full-operator min = BW)

Original claim text

THE E_fail VARIATIONAL FUNCTIONAL (external-review path 1, 2026-06-14): a concrete, numerically-testable sharpening of the v194/QGEO.ENERGY.02 target. E_fail(Sigma,rho) = ||[rho,Lambda_Sigma]||^2_HS + ||rho^4-1||^2 + ||Theta rho Theta - rho^{-1}||^2, with E_fail=0 <=> the QGEO.SYM.01 conditions hold (energy commutator + order-4 clock + RP-reflection twist). (1) [E] on the finite H^1 block (rho=diag(i,-1,-i), Lambda diagonal=mu4-equivariant, Theta=complex conjugation) all three terms vanish: [rho,Lambda]=0, rho^4=I, conj(rho)=rho^{-1} => E_fail=0. (2) [N] a mu4-breaking off-diagonal Lambda gives [rho,Lambda]!=0 and a non-order-4 rho gives rho^4!=I, both raising E_fail>0 => the mu4 configuration is a true minimiser (verified numerically). (3) [I] the zero locus is exactly the seam-deck conditions of QGEO.SYM.01. (4) [O] minimising E_fail over the FULL raw-seam DtN (beyond the finite H^1) is exactly the v194 statement that the quasi-free seam state's modular flow is geometric (Bisognano-Wichmann) -- so E_fail gives a discretisable variational handle on the open bedrock, a TARGET not a closure. Python-only (finite-block linear algebra; finite vanishing exact, Wolfram-mirrored)

Original classification detail

Numerical/identity [N]/[I]: E_fail=0 exactly at the mu4 config and >0 for perturbations (mu4 minimiser); full-operator minimisation = the v194 Bisognano-Wichmann residual [O] -- a concrete discretisable handle on the bedrock

Dependencies as recorded

QGEO.ENERGY.02; QGEO.DTN.01; QGEO.SYM.01; v193_qgeo_energy_commutator.py; v194_raw_seam_dtn_rp.py

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  • v196_seam_energy_functional.pyRegistered verification module · registry
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    QGEO.VARI.01 — the E_fail variational functional (review path 1), a numerically-testable sharpening of the v194/ENERGY.02 target: E_fail = ‖[ρ,Λ_Σ]‖²_HS + ‖ρ⁴−1‖² + ‖ΘρΘ−ρ⁻¹‖², with E_fail=0 ⇔ the QGEO.SYM.01 conditions. [E] on the finite H¹ block (ρ=diag(i,−1,−i), Λ diagonal, Θ=conj) all three vanish ⇒ E_fail=0; [N] a µ4-breaking Λ or non-order-4 ρ gives E_fail>0 (µ4 is a true minimiser); [I] the zero locus = the seam-deck conditions; [O] the full-operator minimisation = the v194 Bisognano–Wichmann residual — a discretisable handle on the open bedrock, a target not a closure.

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