differentiable_vacuumKineticHam
plain-language theorem explainer
The two-site smeared vacuum kinetic Hamiltonian is differentiable in the phase-space argument for arbitrary real smearing weights on Z/2Z. HKT kinetic-normalized rigidity proofs cite this before expanding Poisson brackets on the vacuum kinetic sector. The argument is a one-line rewrite to the local-profile Hamiltonian plus the general local-profile differentiability lemma.
Claim. For any smearing $N:\mathbb{Z}/2\mathbb{Z}\to\mathbb{R}$, the map $y\mapsto\sum_{j\in\mathbb{Z}/2\mathbb{Z}} N(j)\,h_{\mathrm{vac}}(y,j)$ is differentiable over $\mathbb{R}$, where $h_{\mathrm{vac}}$ denotes the vacuum kinetic Hamiltonian density on the two-site phase space.
background
This module closes Wave C4/C5 gap5 on HKT mod-vacuum kill and kinetic-normalized rigidity. The vacuum kinetic sector supplies a concrete two-site Hamiltonian density built from a local profile (amplitude, weight, and kinetic factors on Z/2Z), then smeared against real weights N.
Differentiability of smeared densities is the analytic prerequisite for Poisson-bracket identities and mean-value arguments used later in the same file. The density is identified with a standard local-profile Hamiltonian form, so calculus reduces to a reusable lemma about LocalHamFromProfile rather than ad-hoc derivatives of the vacuum kinetic expression.
Upstream scaffolding in the SevenGaps stack (posting-layer and J/Ehrhart structure) fixes the discrete two-site ledger geometry; the present statement only needs that geometry as the index set Z/2Z and the smoothness of the vacuum kinetic local profile.
proof idea
One-line term proof. Rewrite the smeared sum via the equality that identifies the vacuum kinetic Hamiltonian with LocalHamFromProfile applied to the vacuum kinetic local profile. Discharge the goal by applying the general lemma differentiable_LocalHamFromProfile to that profile, its ContDiff/smooth witness, and the smearing N. simpa cleans the rewritten goal; no fresh derivative computation appears here.
why it matters
Parent uses are the bilinear Poisson-bracket split for the vacuum kinetic Hamiltonian against momentum dynamics, and the weak HKT point-split target that packages vacuum kinetic ham/mom densities with their advance maps. Without differentiability of the smeared ham, those bracket expansions and the kinetic-normalized rigidity terminal cannot invoke the calculus API.
In the gap5 story this is infrastructure, not the verdict itself: Part 1 kills mod-vacuum rigidity via a variable-kinetic CanonicalMom inhabitant; Part 2 builds KineticNormalizedCanonicalMom with FTC recovery theorem-derived. Differentiability of the vacuum kinetic smear is a green ledger half needed before constraint-recovery status can flip.
No direct T0–T8 citation; the link is through the discrete two-site HKT gravity sector sitting on the recognition ledger.
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