t0_analytic_refinement_holds
plain-language theorem explainer
After the closed-form reciprocal cost is available, the analytic T0 surface still holds: some consistent configuration has cost zero, every contradiction is costly (or already a logical contradiction), and every consistent cost is nonnegative. Anyone citing the post-T5 analytic-refinements bridge needs this packing. The proof is a three-field structure constructor wiring the LogicFromCost zero-cost and positive-cost theorems to defect nonnegativity.
Claim. Once the closed-form cost $J$ is available, the analytic T0 refinement holds: there exists a consistent configuration $c$ with $\mathrm{cost}(c)=0$; for every contradiction configuration $c$, either $\mathrm{cost}(c)>0$ or $c$ is a logical contradiction; and for every consistent configuration $c$, $\mathrm{cost}(c)\ge 0$.
background
The Unified Forcing Chain module shows T0–T8 as forced from the cost foundation (Recognition Composition Law, normalization $F(1)=0$, calibration $F''(1)=1$). T0 is the claim that classical logic emerges from cost minimization: consistency is cheap, contradiction is expensive.
The structure being inhabited is the analytic refinement of that T0 surface. It keeps the older LogicFromCost payload (consistent configs, contradiction configs, and their costs) without forcing the pre-analytic chain to depend on the closed-form reciprocal cost $J(x)=(x+x^{-1})/2-1$. Costs are measured by the defect $\mathrm{defect}(x)=J(x)$, which vanishes at ratio $1$ and is nonnegative for positive arguments.
Upstream, consistent_zero_cost_possible builds a True-at-ratio-1 witness with cost zero; contradiction_positive_cost shows complementary ratios cannot both stabilize at defect zero without collapsing into a logical contradiction; defect_nonneg is the elementary positivity identity $(x+x^{-1})/2-1=((x-1)^2/x)/2\ge 0$.
proof idea
Term-mode structure constructor with three field assignments and no tactics.
consistency_cheapis exactlyLogicFromCost.consistent_zero_cost_possible(exists a consistent config at cost 0, the True-at-ratio-1 witness).contradiction_expensiveis exactlyLogicFromCost.contradiction_positive_cost(every contradiction config has positive cost or is already a logical contradiction).logic_emergentisfun c => LawOfExistence.defect_nonneg c.ratio_pos, applying nonnegativity of defect on the positive ratio of any consistent config.
No new algebra is done here; the declaration only packages those three prior results into the analytic T0 refinement Prop.
why it matters
In the Complete Inevitability Chain, T0 is the step "Logic ← Cost minimization (consistency is cheap)". The main T0 theorem establishes that emergence without the closed-form $J$; this declaration records that the older analytic surface still holds once $J$ is available, so downstream analytic scaffolding need not re-prove the cost-logic facts.
It is consumed by t5_to_analytic_refinements_bridge_holds, whose doc-comment states that the analytic-refinement bridge follows from T5 plus the closed-form identities of the analytic $J$-scaffolding. After T5 forces uniqueness of $J$, that bridge can safely reuse this packed T0 refinement rather than reopening LogicFromCost.
Framework landmark: T0 in the T-1…T8 forcing chain, sitting under T5 J-uniqueness and the Recognition Composition Law. No open scaffold remains; claim status is fully proved.
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