canonicalThreshold
plain-language theorem explainer
The canonical threshold is the real constant φ − 3/2, with φ the golden-ratio fixed point. It is the fixed cutoff used when comparing domain costs inside the eight-tick derivation. Anyone working the structural 8 = 2³ claim or the module certificate will cite it. The declaration is a one-line definitional binding, not a proved statement.
Claim. Define the canonical threshold as the real number $\varphi - 3/2$, where $\varphi$ is the golden-ratio self-similar fixed point of Recognition Science.
background
The module proves the eight-tick cycle as a structural theorem with no sorry and no axioms: once spatial dimension $D = 3$ is forced, the recognition lattice has $2^D = 8$ binary states, and one complete traversal is the eight-tick octave (forcing-chain steps T7–T8).
The constant $\varphi$ is the unique self-similar fixed point forced at T6. It sets the native scale of the cost functional $J$ and of the mass ladder. Sibling definitions in this file introduce domain costs that measure configuration deviation relative to the recognition composition law.
The threshold $\varphi - 3/2$ is a fixed positive real used as a comparison level against those domain costs. Its positivity is recorded by a sibling lemma; the present declaration only names the constant.
proof idea
Definitional binding only. The real constant is set equal to the closed form $\varphi - 3/2$. Symbols come from the imported Constants module ($\varphi$) and Mathlib reals; there is no proof body and no lemma application.
why it matters
Lives in the eight-tick derivation module that closes the structural claim $8 = 2^3$ once $D = 3$ is forced. The threshold supplies a parameter-free numerical gate, built from the T6 golden-ratio scale, for domain-cost comparisons that underwrite the module certificate (EightTick_v3Cert and cert_inhabited). It links the self-similar fixed point to the octave structure without introducing free constants. It does not itself prove the eight-tick theorem; it is infrastructure for the cost comparisons that support that certificate.
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