balanced_is_unique_minimum
The theorem shows that among recognition events the balanced state with ratio 1 is the sole minimizer of information cost. Workers deriving ledger-based physics from recognition ratios cite it to confirm that any imbalance encodes strictly positive information. The term-mode proof substitutes the explicit J-cost at unity and invokes the zero-set characterization.
claimLet $e$ be a recognition event with positive ratio $x$. If the information cost $J(x)$ equals the cost of the balanced event, then $x=1$.
background
Recognition Science encodes every physical fact as a recognition event: a positive real ratio $x>0$ in the ledger. The information cost of the event is the J-cost $J(x)$, which equals zero precisely when $x=1$ by the upstream lemma Jcost_unit0. The balanced event is defined to be the unit-ratio case, representing the absence of encoded information.
proof idea
One-line wrapper that unfolds infoCost and balancedEvent, rewrites the hypothesis with Jcost_unit0, and applies the lemma info_cost_zero_iff_unit to extract the ratio equality.
why it matters in Recognition Science
The result supplies the uniqueness direction inside the IC-001 certificate, completing the claim that information content is exactly the deviation from balance. It anchors the ledger interpretation in the module doc-comment and aligns with the Recognition Composition Law and the eight-tick octave structure. No open scaffolding remains at this step.
scope and limits
- Does not apply to non-positive ratios.
- Does not quantify cost differences for concrete physical observables.
- Does not address composite events or multi-agent ledgers.
formal statement (Lean)
106theorem balanced_is_unique_minimum (e : RecognitionEvent) (h : infoCost e = infoCost balancedEvent) :
107 e.ratio = 1 := by
proof body
Term-mode proof.
108 unfold infoCost balancedEvent at h
109 rw [Jcost_unit0] at h
110 exact (info_cost_zero_iff_unit e).mp h
111
112/-! ## §III. Nothingness is Infinitely Costly -/
113
114/-- **THEOREM IC-001.7**: For any bound M, there exist recognition events with cost > M.
115 More specifically: the event with ratio x = 1/(2(|M|+2)) has cost > M.
116 This proves J(x) → ∞ as x → 0⁺, i.e., "nothingness" is infinitely expensive. -/