canonicalThreshold
plain-language theorem explainer
Defines the canonical real threshold φ − 3/2 used in the RS gravity structural layer for gap-45 (minimum stable self-reference rung at D = 3). Gravity and ladder analyses cite it as the fixed comparison value against domain cost. The body is a one-line constant definition from the golden ratio.
Claim. The canonical threshold is the real number $\varphi - \tfrac{3}{2}$, where $\varphi$ is the golden-ratio fixed point of the Recognition self-similarity relation.
background
Recognition Science forces $\varphi$ as the unique self-similar scale factor (forcing chain T6) and $D = 3$ spatial dimensions (T8). In that setting the structural gap identity $D^2(D+2) = 9\cdot 5 = 45$ marks the minimum rung for stable self-reference.
This module packages that gap-45 layer as a structural theorem block (no sorry, no axioms). The cost side of the story comes from the imported Cost and Constants modules: domain cost is a nonnegative real functional on the recognition geometry, and thresholds are ordinary reals against which that cost is compared.
The constant $\varphi - 3/2$ is the fixed numerical cut used in those comparisons. Numerically $\varphi \approx 1.618$, so the threshold sits near $0.118$, well below the Berry creation scale $\varphi^{-1}$ and inside the small-cost regime of the $\varphi$-ladder.
proof idea
Pure definition: the real constant is introduced as $\varphi - 3/2$ with no proof obligations. Positivity and any comparison lemmas live in sibling declarations (for example the positivity certificate for this same constant).
why it matters
Gap-45 is the structural minimum rung for stable self-reference once $D = 3$ is forced. A single named threshold keeps every domain-cost inequality in this gravity module aligned to the same RS-native cut rather than ad hoc numerics.
Downstream certificates in the same file (the structural 004 certificate and its inhabited wrapper) are the natural consumers: they package nonnegativity of domain cost and positivity of this threshold into a reusable gravity lemma bundle. The definition therefore sits under the gravity domain of the Recognition framework, tying the T6/T8 landmarks to concrete cost comparisons without opening new axioms.
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