Gravity Recognition Geodesic3 From Jcost Recog Geodesic3 Deep Cert

A certified cost function

A cost function assigns a number to a comparison. In Recognition Science, the cost of recognizing one thing as another is forced by a proved theorem to equal J(x) = (x + 1/x)/2 - 1. The declaration RecogGeodesic3DeepCert packages a machine-checked proof that this particular cost function has three basic properties. First, when the two inputs are equal, the cost is exactly zero. Second, for any two positive inputs, the cost is never negative. Third, a specific threshold value, the golden ratio minus 1.5, is greater than zero.

These three facts are proved for any positive real numbers. The proof does not depend on what the numbers represent. The declaration defines a cost as J applied to the ratio of two numbers, m over e, but it never defines what m and e are. The docstring is explicit: the module proves nothing specific to gravity, geodesics, or any physical subject, because the inputs are not connected to a physical model. The three properties hold for any positive ratio, whether the inputs are masses, distances, or prices.

In Recognition Science, the framework models a discrete record of events, called a ledger, where recognition events carry a forced cost. The framework's library of formal theorems proves the cost function's uniqueness and its basic analytic properties. What this declaration adds is a compact certificate: a structure that bundles the three proved facts into a single object, with a proof that such a certificate exists. It is a small, clean building block, not a physical result.

The research note attached to the module sketches an intended application: minimizing the integral of this cost along a path, with the zero-cost points as ground states and the golden-ratio threshold as a first departure from straight-line motion. That is a plan, not a proof. The declaration itself contains no geodesic equation, no manifold, and no physical law. It establishes only the three analytic facts about J, universally quantified over positive inputs.

What a reader can rely on is narrow but solid. The certificate proves that the forced cost function vanishes at equality, stays nonnegative for positive inputs, and has a positive threshold. What it does not prove is any connection to gravity, recognition manifolds, or geodesics. That connection remains open, awaiting a definition of m and e in physical terms.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty RecogGeodesic3DeepCert := ⟨cert⟩

What this page does not claim

The declaration does not define m and e in physical terms. The declaration does not prove any geodesic equation or manifold structure. The declaration does not establish any connection to gravity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/RecognitionGeodesic3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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