Masses Mass Genesis T10 Posted Load Scale Unit Settled Load Topology Ratio Eq On

The unit scale theorem

The theorem unitSettled_loadTopologyRatio_eq_one_iff_topologyAmplitude is a result about a specific ratio in the Recognition Science framework. The ratio, loadTopologyRatio, compares the load cost of a pattern to its topology-predicted value. The theorem states that if a pattern is settled with a unit window, meaning its settlement amplitude is exactly 1, then this ratio equals 1 if and only if a certain amplitude, the primitive positive stationary factor amplitude, also equals 1. In plain terms, for a unit-settled pattern, matching the topology is exactly equivalent to having a specific amplitude value.

The theorem is proved in the framework's machine-checked library of formal theorems, with no unproved axioms. It is part of a larger result about posted-load scale. The framework shows that a settled anchor with a Boolean difference settlement window, where each entry is 0, 1, or -1, forces the settlement amplitude to be 1. It also shows that a doubled scale window of 2 is rejected for such Boolean patterns. These are formal theorems about the framework's definitions.

What the theorem does not claim is just as important. It does not claim that unit settlement forces the loadTopologyRatio to be 1. The framework explicitly proves the opposite: there exists a settled, Boolean, unit-amplitude pattern, the gap-one worldline, where the loadTopologyRatio is not 1. This is a concrete counterexample. The exact statement that unit settlement implies the ratio is 1 is named SettledLoadIsPostedLoad, and the framework proves this statement is false. This residual gap remains an open target for forcing, not a proved result.

theorem unitSettled_loadTopologyRatio_eq_one_iff_topologyAmplitude
    {ψ : LightPattern (Fin 8)}
    (_E : Q3ClosedPatternEvidence ψ)
    (hunit : SettledAnchorWindow 1 ψ) :
    loadTopologyRatio ψ = 1 ↔
      (1 : ℝ) = primitivePositiveStationaryFactorAmplitude ψ := by
  rw [loadTopologyRatio_eq_one_iff_loadNormalizedToTopology]
  exact settledAnchor_loadNormalized_iff_amplitude hunit (by norm_num)

/-- **Boolean unit-scale law.** A positive settled anchor whose anchor window
is Boolean-valued must post at amplitude `1`. -/
theorem settledAnchor_boolean_forces_unitAmplitude
    {a : ℝ} {ψ : LightPattern (Fin 8)}
    (hwin : SettledAnchorWindow a ψ)
    (hbool : BooleanDifferenceSettlementWindow (ψ.window 0))
    (ha : 0 < a) : a = 1 := by
  have hphase0 := settledAnchor_phase0 hwin
  have hb : IsBooleanPostingAmplitude (ψ.window 0 0) := hbool 0
  rw [hphase0] at hb
  exact ofReal_eq_one_of_boolean_pos ha hb

/-- **CONSTRAINT+WITNESS wall.** Unit Boolean posted settlement does not force
`loadTopologyRatio = 1`. The gap-one worldline is the discriminating decoy
against treating Boolean unit scale as topology-matched load. -/
theorem settledLoadIsPostedLoad_wall :
    ¬ SettledLoadIsPostedLoad := by
  intro hall
  obtain ⟨ψ, hcarrier, hunit, hbool, hratio⟩ :=
    exists_unitBooleanSettled_loadTopologyRatio_ne_one
  exact hratio (hall ψ hcarrier hunit hbool)

What this page does not claim

The theorem does not claim that unit settlement alone forces the loadTopologyRatio to be 1; a counterexample exists. The theorem does not claim that the loadTopologyRatio is always equal to 1 for all patterns. The theorem does not claim that the primitive positive stationary factor amplitude is always 1.

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/Masses/MassGenesis/T10PostedLoadScale.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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