costFirstExistenceCert
plain-language theorem explainer
The cost-first existence certificate packages the three structural properties of the J-cost function on positive reals. Researchers formalizing pre-Big-Bang selection principles would cite it to ground the definition of recognition existence. It is assembled as a structure instance that directly references the supporting theorems on equivalence to unity, positive cost away from unity, and unboundedness near zero.
Claim. The cost-first existence certificate is the structure asserting that for all $x > 0$, RSExists$(x)$ holds if and only if $x = 1$, that $Jcost(x) > 0$ whenever $x > 0$ and $x ≠ 1$, and that $Jcost$ is unbounded below on $(0, ∞)$.
background
In the Cost-First Existence module the recognition existence predicate RSExists holds precisely when the J-cost vanishes. The J-cost function Jcost measures the recognition cost of a positive real configuration $x$, with its unique minimum at $x = 1$. The upstream theorem rsExists_iff_one proves the equivalence RSExists $x$ ↔ $x = 1$ for $x > 0$. The theorem non_existence_has_positive_cost shows that any deviation from unity incurs positive cost, while divergence_at_zero_direction establishes that cost diverges as $x$ approaches zero.
proof idea
This is a one-line structure instance that supplies the three fields of CostFirstExistenceCert by referencing the theorems rsExists_iff_one, non_existence_has_positive_cost, and divergence_at_zero_direction.
why it matters
This declaration packages the foundational properties needed to ground the cost-first selection principle in the Recognition Science framework. It directly implements the claim that laws emerge from J-minimizing configurations, as described in the module documentation. It closes the structural definition of RSExists and supports the T0-T8 forcing chain by establishing the cost minimum at unity.
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