Foundation Recognition Science Logo5 Rslogo5 Cert

The certificate's scope

The Recognition Science framework builds physics from a single cost function, a rule that assigns a nonnegative price to any comparison between two quantities. The declaration RSLogo5Cert is a machine-checked certificate: a packaged record that three properties of that cost function hold. The first says the cost is zero when the two quantities are equal. The second says the cost is never negative when both quantities are positive. The third says a particular constant, the golden ratio minus 1.5, is greater than zero. Each of these is proved in the framework's machine-checked library of formal theorems.

The certificate is deliberately thin. Its cost function is defined as J(x) = (x + 1/x)/2 - 1, applied to the ratio of two real numbers m and e. The certificate does not say what m and e are. It does not say they are masses, energies, or anything else. The documentation states this plainly: the proofs are general facts about the cost function, and they become a theorem about a subject only when someone defines m and e in that subject's terms. The certificate is a template, not a discovery about any particular physical system.

What the certificate does establish is that the framework's central object is well-behaved at its most basic level. The cost vanishes at equality, which means recognizing something as itself costs nothing. The cost is nonnegative, which means recognition never pays you. The golden-ratio threshold is positive, which is a small fact about where that famous constant sits relative to 1.5. These are the floorboards of the framework: before any physics can be built, the cost function must be shown to behave this way, and the certificate records that proof.

In Recognition Science, this curve J(x) is described as the entire content of the framework: one curve, all of physics. The certificate supports the first half of that slogan, the uniqueness of the curve, but not the second half. The step from a well-behaved cost function to the masses of particles or the number of spatial dimensions is made elsewhere in the library, by theorems that build on the cost function's properties. RSLogo5Cert itself does not take that step. It is the foundation stone, not the building.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The certificate does not claim that m and e represent any particular physical quantities. The certificate does not claim that the cost function alone produces any particle masses or dimensions. The certificate does not claim that the golden ratio threshold has any consequence beyond being positive.

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/Foundation/RecognitionScience_Logo5.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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