Gravity Gravity

The formal core

Gravity is the attraction between masses, described classically by Newton's inverse-square law and by Einstein's general relativity as the curvature of spacetime. In Recognition Science, the phrase gravity gravity names a different object: the cost, a nonnegative number that measures how far a recognition event is from perfect agreement, of recognizing one mass in the presence of another. The framework's central theorem forces the cost function to have the unique form J(x) = (x + 1/x)/2 - 1, so the cost of recognizing mass m against mass e is written J(m/e).

The machine-checked library of formal theorems proves three general facts about this cost. First, when the two masses are equal, the cost is exactly zero: recognizing a mass against itself costs nothing. Second, for positive masses, the cost is never negative; it is zero only at equality. Third, the quantity phi - 3/2, where phi is the golden ratio, is positive. That last fact is a threshold: it says the golden ratio exceeds 1.5, a property used elsewhere in the framework's forcing chain.

These three theorems are true for any positive real numbers m and e. The framework does not define what m and e are in gravitational terms; it does not say that m is a star's mass or e is a planet's. The definition domainCost simply applies the universal cost function to the ratio m/e. The framework is a template, shared verbatim with 2383 sibling modules, each waiting for a subject-specific definition of its two quantities.

In Recognition Science, the framework models gravity as a recognition process: the cost of recognizing one mass against another is forced by the same five conditions that force the cost function itself. The framework's contribution is to instantiate that general cost for a ratio of masses and to certify the three basic properties. What would turn this into a theorem about actual gravity is a definition of m and e in gravitational terms, such as masses of neutron stars or the ratio of a star's mass to a planet's. That definition is not present in this framework.

The research note attached to the framework records a numerical observation: for neutron star tidal deformability, the framework's phi-power ladder gives phi^13 ≈ 521, which sits within the measured range of 500 to 1000. That note is not a theorem; it is a research note recording where the idea was meant to go. The formal content proves only the three general facts, and the framework's own status line says it proves nothing specific to gravity.

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

This framework does not prove any gravity-specific law. The tidal deformability note is a research note, not a theorem. The framework does not derive the value of Newton's constant from this framework.

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/Gravity.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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