clockRateBundle_of_rindler
plain-language theorem explainer
Every near-horizon Rindler normal form with surface gravity κ packages into the typed B3 clock-rate bundle: Euclidean angle advances at constant rate κ. Horizon holography and local recognition-cut arguments cite it to attach rate-only typing without importing period closure. The proof is a structure constructor that reuses the Rindler hypothesis and applies the Euclidean-angle derivative identity.
Claim. For every real $\kappa$, if the near-horizon geometry admits Rindler normal form with surface gravity $\kappa>0$, then the clock-rate bundle holds: the continued Euclidean angle $\theta=\kappa\tau_E$ satisfies $\frac{d\theta}{d\tau_E}=\kappa$ for all Euclidean time $\tau_E$.
background
This module isolates what panel B3 is allowed to deliver: near-horizon Rindler geometry makes the continued Euclidean angle advance at rate $\kappa$ per unit Euclidean time. It deliberately does not assert $2\pi$ period closure; that is B2 output via deficit-free period and turn-ratio unity. The legacy horizon-rate socket $\kappa=1/R$ for the Clausius bridge is a separate Schwarzschild normalization and must not be conflated with this rate bundle.
Near-horizon Rindler form is a model hypothesis: a Killing horizon with $\kappa>0$ admits adapted coordinates $(\rho,\tau)$ near the bifurcation surface with local metric coefficient $\kappa$ on the static Killing sector. Only existence of the rate parameter is recorded; no thermality, KMS, or entropy-area law is imported. The Euclidean angle after continuation is $\theta=\kappa\tau_E$. The clock-rate bundle is the typed package of that Rindler form together with the universal derivative identity $\partial_{\tau_E}\theta=\kappa$.
proof idea
Term-mode structure construction. The Rindler field of the bundle is the given near-horizon hypothesis. The rate field is filled by the already-proved derivative identity for the Euclidean angle at positive $\kappa$: for every Euclidean time $\tau_E$, the derivative of $\theta(\tau_E)=\kappa\tau_E$ equals $\kappa$. No further case analysis or period arithmetic appears.
why it matters
This is the forced-conditional bridge from raw Rindler form to the typed B3 rate bundle used throughout the holography stack. Downstream, every local recognition horizon context obtains its clock-rate bundle by applying this theorem to its Rindler field, so local cut arguments inherit rate-only typing without smuggling B2 period closure. In the Recognition framework it keeps LEG-B honest: B3 supplies $d\theta/d\tau_E=\kappa$ only; period and turn-ratio unity remain B2. That separation prevents conflating the rate bundle with the legacy Schwarzschild $\kappa=1/R$ socket used on the Clausius bridge.
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