continuumDiracDensity
plain-language theorem explainer
Continuum Dirac structure density for hypersurface deformations: the product of the lapse Wronskian (N M' − M N'), the dynamic structure factor G = 1 + q², and the momentum flux D = p q'. Continuum-limit arguments in the gravity seven-gaps stack cite it as the integrand target of scaled lattice bracket sums. Pure pointwise definition assembling three continuum densities already fixed upstream.
Claim. For real functions $N,M,q,p$, the continuum Dirac density at $t$ is $(N(t)M'(t)-M(t)N'(t))\cdot(1+q(t)^2)\cdot(p(t)q'(t))$, i.e. the product of the lapse Wronskian, the dynamic structure profile $G=1+q^2$, and the continuum momentum flux $D=p\,q'$.
background
Module Wave C2 R4 packages the sampled-lapse Wronskian rate-$h$ residual with the R2 lattice RHS shape and the R3 dynamic structure profile. The honest scaled continuum object is $n\cdot\Sigma$ of the lattice summands, converging to $\int (N M'-M N')\cdot G\cdot(p,q')$ on the unit interval; bare unscaled sums vanish and an $n^2$ prefactor would diverge.
Upstream pieces are fixed separately. The continuum momentum flux is $D(t)=p(t),q'(t)$. The dynamic structure profile is the continuum shape of the concrete dynamic inverse metric: $G(x)=1+(q x)^2$. The discrete identity bracket_HamDyn_HamDyn supplies the lattice summand shape $W_k,G_k,(\pi_{k+1}\Delta q_k)$ at $n=2$, with Wronskian and flux each $O(1/n)$ for $C^1$ data.
This definition is the pointwise continuum integrand those three factors assemble. It is not itself a limit theorem; the rate-$h$ analysis and Riemann packaging live in sibling results.
proof idea
Definition only: pointwise product of three already-named continuum densities. At each $t$, form the classical Wronskian $N(t)M'(t)-M(t)N'(t)$, multiply by the dynamic structure profile $1+(q t)^2$, then by the continuum momentum flux $p(t),q'(t)$. No tactics, no lemmas applied beyond those three constituent defs.
why it matters
Supplies the integrand for the shape continuum limit: scaled freestanding dynamic-bracket shape sums tend to $\int$ of this density (true Riemann/rate-$h$ theorem about the sampled sum, not a binding of the $n=2$-only Hamiltonian bracket). The repaired ledger terminal then binds the general-$n$ lattice bracket under 1-periodicity to the same integral, via that shape theorem plus Step-3 binding.
Also feeds the explicit decoy that the frozen-$G\equiv 1$ continuum integrand differs from the dynamic $G=1+q^2$ integrand (value 2 vs 1 at $t=1$ for $N\equiv 1$, $M=q=\mathrm{id}$, $p\equiv 1$). Gap-5 residual DAG entries name both the shape residual and the ledger-terminal residual in terms of this density.
Local honesty: does not by itself flip gap-5 constraint recovery (needs R6 as well); the ledger name for the full Dirac algebra continuum limit was held free until the general-$n$ HamDynN binding and periodic wrap treatment landed.
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