canonicalThreshold_pos
plain-language theorem explainer
The canonical threshold constant used in the BIT-kernel deep layer is strictly positive. Anyone comparing domain costs or assembling the BIT deep certificate against that cutoff needs this inequality on hand. The proof is a one-line wrapper: unfold the definition and close by linear arithmetic from the bound φ > 1.5.
Claim. The canonical threshold (the real cutoff built from $\varphi$ in this module) is strictly positive: $0 < T_{\mathrm{can}}$.
background
This module develops the BIT kernel family (deep v4) from the J-cost. In the Recognition Science setting, bosonic fields carry vacuum fluctuations $J(\varphi)/45$ per recognition tick; cumulative cosmic Z-aging is then $J(\varphi)\cdot N_{\mathrm{ticks}}/45$, with $N\sim 10^{48}$ at recombination. The local goal is a structural, sorry-free certificate for that deep layer.
The golden ratio $\varphi=(1+\sqrt{5})/2$ is the self-similar fixed point forced at T6. The only upstream fact used here is the tighter numerical bound $\varphi>1.5$ (from $\sqrt{5}>2$), recorded as a constants lemma. The canonical threshold is the module-local real cutoff defined from $\varphi$; sibling facts establish nonnegativity of the domain cost and the inhabited deep certificate that will consume the positivity claim.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold (exposing an arithmetic expression in $\varphi$), then invoke linarith on the lemma $\varphi>1.5$. No further case splits or Recognition-Composition identities are required.
why it matters
Positivity of the cutoff is the elementary guard that lets domain-cost comparisons and the BIT deep certificate stay in the positive cone. The module is marked STRUCTURAL THEOREM (0 sorry, 0 axiom) in the Plan v7 deep session, so this inequality is part of the closed foundation layer rather than scaffolding. It sits under the J-cost / BIT vacuum-fluctuation story ($J(\varphi)/45$ per tick) and ultimately under the forcing chain landmarks T5 (J-uniqueness) and T6 ($\varphi$ forced). No downstream edges are recorded yet; the immediate consumers are the sibling certificate constructions in the same file.
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