canonicalThreshold
plain-language theorem explainer
The canonical dimensionless threshold for free-particle wave-packet spreading is φ − 3/2, identical to the J-cost of the golden ratio. Anyone citing the RS structural spreading time t_spread uses this constant as the factor that multiplies σ₀ when the packet width reaches the J(φ) scale. The declaration is a one-line definition from the RS constant φ.
Claim. The canonical threshold is the real number $\varphi - 3/2$. Equivalently it equals $J(\varphi)$, where $J(x)=(x+x^{-1})/2-1$ is the Recognition cost and $\varphi$ is the golden-ratio fixed point.
background
The module derives free-particle Gaussian spreading $\sigma(t)=\sigma_0\sqrt{1+(\hbar t/(2m\sigma_0^2))^2}$ from the Recognition J-cost. In RS units the structural claim is that the packet reaches width $J(\varphi),\sigma_0$ at a definite time $t_{\mathrm{spread}}$ built from that same factor.
The cost $J(x)=(x+x^{-1})/2-1$ is the unique continuous solution forced by the Recognition Composition Law (T5). For the golden ratio $\varphi$ one has $1/\varphi=\varphi-1$, so $J(\varphi)=\varphi-3/2$ exactly. The constant $\varphi$ itself is the self-similar fixed point of the forcing chain (T6).
Sibling material in the same file packages non-negativity of a domain cost and a certificate that the spreading threshold is positive; this definition supplies the numeric value those statements refer to.
proof idea
Pure definitional abbreviation: the real constant is set equal to $\varphi-3/2$. No tactics or lemmas are invoked. The equality with $J(\varphi)$ is the elementary algebraic identity recalled above and is not re-proved here.
why it matters
This constant is the dimensionless scale that appears in the module's structural theorem: free-particle spreading reaches $J(\varphi)\times\sigma_0$ at $t_{\mathrm{spread}}=2m\sigma_0^2 J(\varphi)/\hbar\cdot\varphi$. It therefore sits at the junction of the J-uniqueness theorem (T5), the forcing of $\varphi$ (T6), and the concrete quantum-mechanical width formula. Downstream certificates in the same file (positivity of the threshold, inhabited wave-packet certificate) quote it by name. No open scaffolding remains; the module is marked 0 sorry / 0 axiom.
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