CostFirstExistenceCert
The structure encodes the cost-first selection principle by asserting that recognition existence for positive reals holds precisely at the J-cost minimum of unity, that all other positive values carry positive cost, and that cost diverges as the argument approaches zero. Pre-Big-Bang cosmologists and recognition theorists cite it to ground the emergence of stable configurations from J-minimization. The structure is assembled by direct reference to the supporting lemmas on equivalence to unity, cost positivity away from the minimum, and zero-div
claimLet $J$ denote the recognition cost function on positive reals. The certificate asserts that for all $x > 0$, recognition existence holds if and only if $x = 1$, that $J(x) > 0$ whenever $x > 0$ and $x ≠ 1$, and that there is no finite $C$ such that $J(ε) ≤ C$ for all $ε > 0$.
background
In the Recognition Science framework the J-cost function quantifies the recognition cost of a positive real configuration and attains its global minimum of zero uniquely at unity. Recognition existence is the predicate that holds exactly when this cost vanishes. The module CostFirstExistence formalizes the pre-Big-Bang selection principle: existence is selected by minimizing J-cost, so that laws emerge as the unique J-minimizing configurations in the pre-geometric landscape.
proof idea
The structure is a direct packaging of three properties. It references the lemma establishing equivalence between recognition existence and equality to one, the lemma showing positive cost for non-minimizers, and the lemma demonstrating divergence of cost as the argument approaches zero from above. No additional tactics are required.
why it matters in Recognition Science
This certificate supplies the core structural claim of the cost-first existence principle from the pre-Big-Bang paper and feeds directly into the concrete certificate instance constructed in the same module. It anchors the emergence of physical laws from J-minimization and connects to the forcing chain by selecting the stable fixed point at unity. The result touches the open question of how the full recognition lattice evolves under the recognition operator from this initial selection.
scope and limits
- Does not derive the explicit functional form of the J-cost.
- Does not establish the numerical value of the golden ratio or other constants.
- Does not address the dynamics of the recognition evolution operator.
- Does not prove existence of solutions beyond the positive reals.
formal statement (Lean)
91structure CostFirstExistenceCert where
92 rsExists_iff : ∀ {x : ℝ}, 0 < x → (RSExists x ↔ x = 1)
93 non_existence_costly :
94 ∀ {x : ℝ}, 0 < x → x ≠ 1 → 0 < Jcost x
95 nothing_diverges :
96 ¬ ∃ (C : ℝ), ∀ (ε : ℝ), 0 < ε → Jcost ε ≤ C
97
98/-- Cost-first existence certificate. -/