Cosmology Vacuum Uniformity Passive Fraction Lt One

A fraction less than one

In the Recognition Science framework, the vacuum is not empty space but a record of events. The framework models the universe's ledger as a three-dimensional grid of voxels, each holding a cost value. A recognition event, a discrete act of accounting, carries a forced cost given by J(x) = (x + 1/x)/2 - 1. Some of these events are phase-locked, meaning they are committed entries whose maintenance cost is zero. The fraction of such locked modes is a number the framework derives from a combinatorial budget of the carrier grid.

The declaration passive_fraction_lt_one states that this fraction, called passiveFraction, is less than one. Formally, it proves that 11/16 < 1. This is a theorem in the machine-checked library of formal theorems, meaning the inequality is verified by a computer from the definitions alone, with no physical measurement involved. It is a small piece of arithmetic: the fraction 11/16 is simply less than 1. The theorem does not say what the fraction is, only that it is not the whole.

The companion theorem passive_fraction_pos proves that the fraction is greater than zero. Together, the two inequalities place the passive fraction strictly between 0 and 1. This matters because the framework defines the active fraction as 1 minus the passive fraction, and the sum of the two is proved to equal 1. A passive fraction of exactly 1 would leave no room for active modes, and a fraction of 0 would mean no locked entries at all. The inequalities guarantee that both kinds of modes exist in the model.

In Recognition Science, this fraction feeds into a larger claim about the vacuum. The framework proves that the vacuum energy density is spatially uniform, because the fraction is a property of the grid's symmetry, not of any particular location. The inequality itself, however, is only about the number 11/16. It does not by itself establish uniformity, nor does it connect the fraction to any measured value of the cosmological constant. Those are separate steps in the framework's argument, and they are not part of this declaration.

theorem passive_fraction_lt_one : passiveFraction < 1 := by
  unfold passiveFraction; norm_num

theorem passive_fraction_lt_one : passiveFraction < 1 := by
  unfold passiveFraction; norm_num

theorem passive_fraction_pos : passiveFraction > 0 := by
  unfold passiveFraction; norm_num

What this page does not claim

The inequality does not establish that the vacuum energy density is spatially uniform. The theorem does not connect the fraction 11/16 to any measured value of the cosmological constant. The declaration does not claim that the passive fraction is the only component of the vacuum energy.

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

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