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Original ledger classification: Numerical sign [N] (argmin = mu4 clock) + open full minimisation [O]

Recorded statement: THE DISCRETISED VARIATIONAL TEST (work package B, external-review, 2026-06-14): v196 showed E_fail=0 at the mu4 config on the fixed H^1 block; here the clock angle is a FREE parameter and E_fail is MINIMISED over it. (1) [N] taking…

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QGEO.VARI.02

Original ledger classification: Numerical sign [N] (argmin = mu4 clock) + open full minimisation [O]

Original claim text

THE DISCRETISED VARIATIONAL TEST (work package B, external-review, 2026-06-14): v196 showed E_fail=0 at the mu4 config on the fixed H^1 block; here the clock angle is a FREE parameter and E_fail is MINIMISED over it. (1) [N] taking rho(theta)=diag(e^{i m theta}) on the H^1 generators m=(1,2,3) and E_fail(theta)=||rho(theta)^4-1||^2 + faithfulness penalty (penalising coincident characters), a scan over theta in (0,pi) has its UNIQUE minimum at theta=pi/2 (E_fail=0): there rho^4=1 AND the three characters i,-1,-i are distinct = the (1,2,3)=A3-exponent grading. (2) [N] decoys lose: theta->0 is trivial (one character), theta->pi gives only two distinct characters (m=1,3 coincide at -1), generic pi/3 and 2pi/3 give E_fail~6 -- so argmin E_fail = the mu4 normal form, no other clock reproduces the faithful order-4 mu4 grading. (3) [E] consistent with v196/v199: theta=pi/2 is the clock z|->iz whose principal-symbol commutator vanishes (v198) and whose H^1 covariance is mu4-equivariant (v177); E_fail=0 there reproduces the v196 finite-block vanishing. (4) [O] this is a numerical sign on the finite H^1 grading, not the full operator-valued minimisation over the raw-seam DtN on the continuum (the v199 bounded-sub-principal residual); strengthens the case that the variational principle selects mu4 but does not close it. Python-only (numerical variational scan)

Original classification detail

Numerical [N]: argmin E_fail over the clock angle = theta=pi/2 (the mu4 clock z|->iz), decoys lose; a numerical sign that the variational principle selects mu4; full operator-valued minimisation stays the v199 residual [O]

Dependencies as recorded

QGEO.VARI.01; QGEO.STATE.01; v196_seam_energy_functional.py; v199_seam_state_invariance.py

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  • v200_seam_variational_scan.pyRegistered verification module · registry
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    QGEO.VARI.02 (work package B) — the discretised variational test: v196 had E_fail=0 at the μ4 config on the fixed block; here the clock angle is free and E_fail is minimised over it. [N] scanning E_fail(θ)=‖ρ(θ)⁴−1‖²+faithfulness over θ∈(0,π) gives the unique minimum at θ=π/2 (E_fail=0: order-4 AND three distinct characters i,−1,−i = the (1,2,3) grading); decoys (0, π, π/3, 2π/3) all lose. So argmin E_fail = the μ4 normal form. [E] consistent with v196/v199/v198. [O] a numerical sign on the finite H¹ grading, not the full operator-valued minimisation (the v199 residual).

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