canonicalThreshold
plain-language theorem explainer
Defines the canonical astrophysical cost threshold as φ − 3/2 in RS-native units. Anyone working the white-dwarf radius match (φ^19 · 0.75 km ≈ 7000 km) or domain-cost comparisons in this module cites it. The body is a one-line real constant built from the forced golden ratio.
Claim. The canonical threshold is the real number $\varphi - 3/2$, where $\varphi$ is the golden ratio fixed by the Recognition self-similarity equation.
background
Recognition Science forces φ as the unique self-similar fixed point of the cost functional (forcing step T6). In RS-native units the same φ sets the mass ladder, the eight-tick octave, and several dimensionless thresholds.
This module treats white-dwarf radius as a structural prediction: φ^19 times a 0.75 km yardstick yields ≈ 7012 km, matching the observed ~7000 km scale. Domain costs (nonnegative real-valued penalties built from the J-cost) are compared against a fixed cutoff; the present definition supplies that cutoff.
The constant 3/2 is the ordinary rational offset that places the threshold just below φ − 1 ≈ 0.618, inside the open interval where small positive domain costs remain admissible.
proof idea
Pure definition: the real constant is introduced by the single equation canonicalThreshold := φ − 3/2. No lemmas or tactics are required; positivity is proved separately by the sibling lemma that φ > 3/2.
why it matters
Gives the numerical gate used by the white-dwarf structural certificate in RS Astrophysics Module 5. Downstream positivity and certificate inhabitants (canonicalThreshold_pos, RSAstro005Cert) rely on this exact value so that domain-cost comparisons stay inside the proved MATCH band φ^19 · 0.75 km ≈ 7000 km. It is a local calibration constant, not a new forcing step, but it anchors the module’s claim that the radius prediction is theorem-level rather than fitted.
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