canonicalThreshold
plain-language theorem explainer
Defines the canonical real threshold as φ − 3/2, a fixed positive cutoff built from the golden ratio. Path-integral and domain-cost arguments in the RS v3 module cite it when separating stationary J=0 paths from suppressed off-path contributions. The body is a one-line arithmetic definition in terms of the global constant φ.
Claim. The canonical threshold is the real number $\varphi - 3/2$, where $\varphi$ is the golden-ratio fixed point of the Recognition self-similarity relation.
background
The module develops an RS reading of the path integral $Z = \int D[\phi],\exp(iS[\phi]/\hbar)$. On paths that sit at the J-cost minimum ($J=0$), the phase factor is stationary and $\exp(iS)=1$; off those paths the amplitude is damped by a factor $\exp(-J(\phi^k))<1$. Recognition therefore selects the $J=0$ sector.
The constant $\varphi$ is the unique self-similar fixed point forced by the T6 step of the unified forcing chain (the positive solution of $x=1+1/x$). The present definition subtracts the rational offset $3/2$ from $\varphi$, yielding a small positive real used as a numerical gate in the surrounding domain-cost and certification lemmas.
proof idea
Pure definition: the real constant is introduced by the arithmetic expression $\varphi - 3/2$. No proof obligations; positivity and downstream comparisons are handled by sibling lemmas such as the positivity certificate for this threshold.
why it matters
Supplies the explicit numerical cutoff that the RS path-integral v3 development uses when comparing domain cost against a fixed positive bar. In the broader framework it sits downstream of T6 (φ forced) and of the J-cost calculus (T5 uniqueness of $J(x)=(x+x^{-1})/2-1$). The module status is structural (zero sorry, zero axiom): this constant is part of that closed skeleton rather than an open hypothesis. It does not itself encode the Berry threshold $\varphi^{-1}$ or the eight-tick octave; those enter only through neighboring constructions.
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