ExactHessianNormalizationGatePass
plain-language theorem explainer
Boolean gate recording that algebraic bookkeeping for the exact flat Regge Hessian normalization is closed. Geometric Tendsto remains open. Gravity auditors and the exact-Hessian audit package cite it as a Stage-1 status bit. The body is the constant true; the companion theorem is reflexivity.
Claim. The exact flat Hessian normalization gate is the Boolean value $\mathrm{true}$: the algebraic bookkeeping identity is closed, while the geometric $\mathrm{Tendsto}$ limit remains open.
background
The module treats the exact flat Regge Hessian Bloch symbol in 4D. At a flat background every deficit vanishes, so Schläfli reduces the second variation of $S = \sum_h A_h \delta_h$ to the cross term $S'' = \sum_h (dA_h)(d\delta_h)$ in squared-length coordinates (symmetrized; factor $\sum A'\delta'$, not $2\sum$). No pathwise off-flat Schläfli primitive is needed for this flat Hessian.
Stage-1 algebraic $m^2$ work (unit cell, sympy-exact Heron $\partial A$ and Gram-dihedral $\partial\theta$, Taylor $\cos(k\cdot\Delta)\to O(k^2)$) yields isotropic TT density $Q_{m^2}(H,k)=(-1/8)|H|_F^2|k|^2$, so the banked axisTTPlus face ($|H|_F=\sqrt{2}$) is exactly $-1/4$, and pure gauge $H=k\otimes v+v\otimes k$ is exactly $0$. Edge-origin certificates from the Bloch-star evaluator bank those face values without inhabiting the RS ledger or flipping gap-action recovery.
This gate is pure status bookkeeping for that algebraic closure, not a geometric continuum limit.
proof idea
One-line definition: the Boolean is set to the constant true. No lemmas, tactics, or computation. The sibling theorem exactHessianNormalizationGatePass_true is then rfl.
why it matters
Feeds the companion reflexivity theorem and the composite exact_hessian_audit_package, which conjoins TT isotropy and gauge-zero targets, banked edge-origin $m^2$ certificates, this gate equal to true, and the honest flag that a general algebraic $\mathbb{Q}(\sqrt{\cdot})$ coupling table $C_{abcdij}$ is not present in Lean.
In the Recognition gravity stack this is Stage-1 status for the exact flat Hessian symbol: algebraic face coeffs and decide-certs on named modes are closed, while geometric Tendsto (continuum limit of the discrete symbol) stays open. It does not touch T0–T8 forcing, RCL, or the phi ladder; it is local to Regge 4D Hessian analysis and the Python-first oracle pipeline that judges the construction.
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