canonicalThreshold
plain-language theorem explainer
Canonical RS compression threshold is the real number φ − 3/2, identical to the J-cost evaluated at the golden ratio. Information theorists working in the Recognition framework cite it as the excess bits per symbol of an RS-optimal code. The declaration is a pure definitional abbreviation of that closed form.
Claim. The canonical compression threshold is the real number $\varphi - 3/2$, where $\varphi$ denotes the golden ratio (self-similar fixed point of the Recognition forcing chain).
background
The module treats data-compression rates derived from the Recognition J-cost. In RS units an optimal code for a source of entropy $H$ bits per symbol carries average length $H + J(\varphi)$ rather than the Shannon length $H$; the additive overhead is the J-cost at the golden ratio.
The cost functional is $J(x) = (x + x^{-1})/2 - 1$. Substituting the golden ratio $\varphi$ (which satisfies $\varphi^{-1} = \varphi - 1$) yields the elementary identity $J(\varphi) = \varphi - 3/2$. Numerically this is approximately $0.118$ bits per symbol, matching the module’s stated overhead at $H = 1$.
The surrounding file imports only the global Constants and Cost modules, so $\varphi$ and $J$ are the ambient RS primitives; no further analytic hypotheses are required for the definition itself.
proof idea
Pure definitional abbreviation: the right-hand side is the closed-form evaluation of $J(\varphi)$. No tactics, lemmas, or proof terms are involved.
why it matters
Supplies the concrete numeric overhead that the module’s compression certificate and positivity lemmas (canonicalThreshold_pos, domainCost_nonneg, DataCompr3DeepCert) quantify. In the broader framework it realises the T5/T6 landmark: once J-uniqueness and the forcing of $\varphi$ are in hand, the excess length of any RS-optimal code is fixed at $J(\varphi) = \varphi - 3/2$. The value anchors the claim that RS coding sits a fixed $0.118$ bits above the Shannon limit, independent of source alphabet size.
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