Foundation Recognition Science Summary3

What the summary module proves

Recognition Science studies the idea that reality keeps a discrete record of events, and that the cost of adding a new event to that record is forced by simple consistency conditions. The module named foundation recognition science summary3 is a small, machine-checked certificate. It does not add new physics. It records three basic facts about the cost function that the framework places at its foundation, and it packs those facts into a single reusable object.

The cost function is defined as J(x) = (x + 1/x)/2 - 1. In the module, the cost of recognizing one quantity m against a reference e is written as J(m/e). The first theorem states that the cost is zero when m equals e: recognizing a thing against itself costs nothing. The second states that the cost is never negative for positive inputs: recognition never yields a negative debt. The third states that the quantity phi - 3/2 is positive, where phi is the golden ratio, roughly 1.618. That positive gap is called the canonical threshold in the module.

These three facts are assembled into a structure called RSSummary3Cert, and the module proves that this structure is inhabited, meaning the certificate can be constructed. The certificate exists so that other parts of the framework can rely on these three facts without re-proving them. The module itself is deliberately generic: it proves facts about the cost function, not about any specific physical system. A separate research note in the module says that turning this into a statement about a particular subject would require defining what m and e mean in that subject's terms.

The larger framework claims that the same cost function, forced by four axioms, leads to the golden ratio, an eight-tick recognition cycle, and three spatial dimensions. This summary module does not prove any of that. It is a foundation stone: it certifies the basic behavior of the cost function so that later modules can build on it. A reader who wants the full chain of derivations should look at the unified forcing chain, not at this certificate.

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 RSSummary3Cert := ⟨cert⟩

What this page does not claim

This module does not prove that the golden ratio, the eight-tick cycle, or three spatial dimensions are forced. This module does not derive any physical constant. This module does not define what m and e mean for any specific subject.

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/RecognitionScienceSummary3.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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