meshHingeKappa_source_dominated
plain-language theorem explainer
The constant unit hinge coupling κ ≡ 1 meets the source-dominated admissibility bound against the mesh geometric deficit: |κ(h)·δ(h)| ≤ 4·(π/2) at every real carrier point. Gravity analysts closing Wave B residual R2 cite this as the real-admissibility half of the hinge-kappa identification. The proof collapses via κ = 1 to the banked |δ| ≤ 2π estimate and matches the channel-scale product by ring arithmetic.
Claim. For every real carrier point $h$, $\lvert \kappa(h)\,\delta(h)\rvert \le N_{\mathrm{ch}}\, s$, where $\kappa\equiv 1$ is the unit hinge coupling, $\delta$ is the mesh geometric deficit (star deficit from R1), $N_{\mathrm{ch}}=4$ is the bridge channel count, and $s=\pi/2>0$ is the mesh scale.
background
Wave B residual R2 packages hinge-level constitutive data for the Recognition mesh in 4D Regge geometry. The carrier was already reshaped in R1 from an abstract hinge type to $\mathbb{R}$, with geometric deficit identified to the star deficit of the 4D hinge kernel. Flat angle sum is $2\pi$; the exact-$J$/true-Regge-Hessian mesh context is conjoined as in R1.
The coupling here is not a free field: it is the banked unit coupling $\kappa(h)=1$ from the concrete stationarity-bridge pattern (every configuration has kappa equal to one), definitionally free of $x$-ratio and log. Channels equal 4 (bridge channel count); mesh scale is $\pi/2>0$. Source-dominated admissibility is the real inequality $|\kappa,\delta|\le$ channels $\times$ mesh scale, the shape required by the deficit-source constitutive coupling interface.
Upstream, the absolute bound $|\delta(h)|\le 2\pi$ comes from $|\arcsin|\le\pi/2$ applied to the star-deficit construction. That estimate, together with positivity of channels and scale, is what makes the unit coupling admissible without further constitutive input.
proof idea
Introduce the carrier point $h$. Rewrite with the sibling fact that hinge kappa is identically one, then cancel the factor of one. Invoke the upstream absolute bound $|\mathrm{meshGeometricDeficit}, h|\le 2\pi$. Unfold channels and mesh scale to the concrete constants $4$ and $\pi/2$. Prove by ring that $4\cdot(\pi/2)=2\pi$, rewrite the deficit bound into that product form, and close by exact application of the rewritten inequality. No case splits and no analysis beyond the banked arcsin bound.
why it matters
Closes the real-admissibility clause of typed residual R2 (hinge kappa identified with source-dominated shape, no $x$-ratio). The sibling closure theorem assembles kappa, nontriviality, this bound, exact-$J$ Hessian equality, and flat angle sum $2\pi$ into the full residual inhabitant. Downstream, the dual-entry coupling definition reuses the same channels, kappa, deficit, and scale as the banked R1/R2 legs of a DeficitSourceConstitutiveCoupling on $\mathbb{R}$.
In the QG Wave B attack on the residual DAG, this is the theorem content behind the unit-coupling naming: not a free constitutive field, but a proved source-dominated inequality against star geometry. Continuum Einstein-scale join (Einstein kappa versus this hinge-local unit coupling) remains open. The result does not flip the gap-1 bridge-derived flag, does not yet inhabit the full signed constitutive structure (needs R3 source strength), and does not evade the no-bare-ledger-selector constraint on signed sources.
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