Gravity Recognition Horizon3 From Jcost

The horizon as a cost level

In general relativity, an apparent horizon is the surface where light and matter can no longer move outward. Recognition Science offers a different picture. It models the horizon as a level surface of a cost function, the recognition cost, a number that measures how hard it is for reality to keep a discrete record of events at a given scale. The framework's central theorem forces this cost to take one specific form, J(x) = (x + 1/x)/2 - 1, and the horizon appears where that cost equals a fixed value.

The module in question, RecognitionHorizon3_FromJcost, contains a research note that gives the intended physical picture. It identifies the horizon with the surface where the cost function J reaches the value J(phi), with phi the golden ratio, about 1.618. The note ties this to a bounce rung, a proposed discrete step in the framework's account of a collapsing star. That part is a research note, not a proved result. The Lean code in the module proves something narrower and fully general: for any positive numbers m and e, the cost J(m/e) is never negative, and it is exactly zero when m equals e. It also proves that the threshold phi - 3/2, about 0.118, is positive.

What the module actually establishes, in plain language, is that the cost function has the two properties any sensible ledger needs. First, when the two quantities being compared are equal, the cost of recognizing them as the same is zero. Second, the cost of comparing any two positive quantities is never a negative number. These are the properties that let the framework treat the horizon as a threshold: a surface where the cost rises to a particular positive level, and beyond which it would have to grow without bound. The module certifies that such a threshold exists.

The gap between the research note and the theorem is the definition of m and e. The note wants m to be a mass and e an energy, but the Lean code defines the cost only as J(m/e) for arbitrary real numbers. Nothing in the module ties those numbers to gravity, to a star, or to a horizon. The proof that the cost has its two basic properties is complete. The proof that those properties describe an actual black hole horizon is not present in this file; it remains a target for future work.

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]

What this page does not claim

This module proves that a black hole horizon exists in the framework. The research note's identification of the horizon with the J(phi) level is a proved theorem. The module defines what m and e mean for a specific star or spacetime.

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/RecognitionHorizon3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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