KTheta
plain-language theorem explainer
KTheta sets the Recognition Science phase-failure floor at J(φ)/45, with φ the golden ratio self-similar fixed point. Phase-budget analyses in finite non-identity reciprocal ledgers cite this constant to bound aggregate costs of failed gates under the explicit floor hypothesis. The definition is a direct scalar division with no further computation.
Claim. Let $K_θ := J(φ)/45$, where $J$ is the J-cost function $J(x) = (x + x^{-1})/2 - 1$ and $φ$ is the self-similar fixed point satisfying the Recognition Composition Law.
background
The Uniform Failure Floor module introduces the phase-failure floor scale for the phase-budget engine. J-cost evaluates the function $J(x) = (x + x^{-1})/2 - 1$ (equivalently cosh(log x) - 1) at the golden ratio φ, the self-similar fixed point forced in the T5-T6 steps of the unified forcing chain. This scale enters the KThetaFailureFloorHypothesis structure, which assumes every failed gate in a ledger of size n satisfies cost at least KTheta.
proof idea
The definition is a direct one-line assignment of the J-cost evaluated at phi divided by the integer 45.
why it matters
This definition supplies the numerical scale for the KThetaFailureFloorHypothesis used in bounded_phase_visibility and failed_gate_count_bounded_at_KTheta. It anchors the uniform failure floor in the RS phase-budget engine, ensuring failed finite phase gates carry a minimum cost derived from the J-function at the self-similar point. The parent theorems apply this to prove bounded visibility for non-identity reciprocal ledgers and to bound total failure costs by the stable budget.
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