nothing_cannot_exist
plain-language theorem explainer
The theorem shows that for any real bound C the defect functional exceeds C for all sufficiently small positive arguments. Researchers deriving existence from cost minimization in Recognition Science cite it to make precise the claim that the empty configuration carries infinite cost. The proof constructs an explicit epsilon equal to 1 over twice the absolute value of C plus two and then chains reciprocal inequalities with the definition of defect to reach the strict lower bound via linarith.
Claim. For any real number $C$ there exists $ε > 0$ such that for all $x > 0$ with $x < ε$ one has $C < defect(x)$, where $defect(x) := J(x)$ and $J(x) = (x + x^{-1})/2 - 1$.
background
The Law of Existence module formalizes the equivalence that a positive real exists precisely when its defect vanishes. The defect functional is introduced as defect(x) := J(x) for real x, with J the cost function that vanishes only at unity. This theorem appears explicitly among the module's key results together with defect_zero_iff_one and unity_unique_existent.
proof idea
The proof is a tactic-mode construction. It introduces ε := 1/(2*(|C|+2)), proves positivity, then shows that x < ε forces the reciprocal to exceed 2*(|C|+2). Substituting the definition of defect and applying the inequalities (x + x^{-1})/2 ≥ x^{-1}/2 together with |C| ≥ C yields the desired strict comparison by linarith.
why it matters
The result supplies the 'nothing costs infinity' clause inside the inevitability equivalence summary and the economic_inevitability theorem; it is also referenced directly in concrete_inevitability and in the logic-from-cost derivations. It thereby closes the cost-based half of the meta-principle that nothing cannot recognize itself, complementing the uniqueness of the phi fixed point.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.
papers checked against this theorem (showing 3 of 3)
-
Reward extrapolation in distillation lets students surpass teachers
"performing reward correction by choosing the reference model as the teacher's base model before RL yields a more accurate reward signal"
-
Unique reciprocal cost on ratios forces balanced discrete ledgers
"This cost exhibits reciprocity J(x)=J(x⁻¹), vanishes only at x=1, and diverges at boundary regimes x→0⁺ and x→∞"
-
Self-distillation turns feedback into denser RL signals
"SDPO treats the current model conditioned on feedback as a self-teacher and distills its feedback-informed next-token predictions back into the policy... leverages the model’s ability to retrospectively identify its own mistakes in-context."