unity_unique_existent
Unity is the unique real number satisfying the existence predicate in the Recognition Science framework. Researchers working on the Law of Existence cite the result to close the uniqueness half of the cost-minimum characterization. The proof splits the biconditional via constructor and reduces each direction to the zero-defect lemma for positive arguments together with the explicit evaluation at unity.
claim$forall x in mathbb{R}, Exists(x) iff x = 1$, where $Exists(x)$ asserts $0 < x$ and $defect(x) = 0$.
background
The Law of Existence module supplies a literal formalization in which a ratio x exists precisely when its defect vanishes. The local Exists structure packages the two conditions positivity and zero defect. Upstream, the CostAxioms module gives the equivalent formulation Exists(x) := 0 < x wedge J(x) = 0 with J the cost function (x + x^{-1})/2 - 1. The sibling theorem defect_zero_iff_one states that for x > 0 one has defect(x) = 0 iff x = 1, while defect_at_one records the direct evaluation defect(1) = 0.
proof idea
The term proof introduces x and splits the biconditional with constructor. The forward direction extracts the defect_zero field of the Exists hypothesis and feeds it to the left-to-right direction of defect_zero_iff_one. The reverse direction substitutes x = 1 and assembles the Exists structure from one_pos together with defect_at_one.
why it matters in Recognition Science
The declaration completes the third key theorem listed in the module documentation, establishing that unity is the sole point satisfying the existence predicate. It thereby anchors the Law of Existence to the unique zero of the J-cost function. In the broader Recognition Science framework this confirms that the self-similar fixed point is realized only at unity.
scope and limits
- Does not extend the uniqueness claim to non-positive reals.
- Does not derive existence of any physical quantity.
- Does not connect the result to the eight-tick octave or spatial dimensions.
- Does not supply a numerical approximation or error bound.
formal statement (Lean)
89theorem unity_unique_existent : ∀ x : ℝ, Exists x ↔ x = 1 := by
proof body
Term-mode proof.
90 intro x
91 constructor
92 · intro ⟨hpos, hdef⟩; exact (defect_zero_iff_one hpos).mp hdef
93 · intro h; subst h; exact ⟨one_pos, defect_at_one⟩
94
95/-- Alias for `defect_at_one`. -/