Masses Mass Genesis T10 Posted Load Scale Unit Settled Load Recognition Cost Zer

The unit posting law

The declaration unitSettled_loadRecognitionCost_zero_iff_topologyAmplitude is a theorem in the Recognition Science framework's machine-checked library. It concerns a settled anchor, a record in the framework's discrete ledger of recognition events, whose window is set to unit scale. The theorem states that, for such an anchor, the load-recognition cost is zero if and only if the primitive positive stationary factor amplitude equals one. In plainer terms: when a settled entry is at its base scale, the cost of recognizing it vanishes exactly when the topology-derived amplitude is one.

The theorem rests on a deeper fact about Boolean posting amplitudes. A posting amplitude is Boolean when it takes one of three values: 0, 1, or -1. The framework shows that if a settled anchor's window is Boolean and its scale is positive, then that scale must be 1. This is the unit posting law. It also shows the converse rejection: a settled anchor with window scale 2 cannot be Boolean. The unit posting law applies both to directly settled anchors and to anchors that read an emitted photon's window.

The theorem does not claim that unit Boolean posting forces the load-topology ratio to be 1. The framework constructs an explicit counterexample: a gap-one worldline pattern that is a settled Boolean unit seed, yet has a load-topology ratio different from 1. The exact residual proposition, SettledLoadIsPostedLoad, is shown to be false. This separation is deliberate: the unit posting law is a theorem, while the stronger claim that settled load equals posted load remains an open forcing parent.

The practical consequence is a sharp discrimination between scales. The framework certifies that Boolean posting forces unit scale, that doubled scale is rejected, and that the unit-scale zero-cost condition is exactly the topology amplitude equation. This gives a precise sense in which the framework's ledger, at its fundamental posting level, operates at a single scale, while leaving open the question of when that scale matches the topology ratio.

/-- On a unit settled Q3 carrier, load-recognition cost vanishes iff the
posting amplitude equals the topology factor amplitude (equivalently,
topology energy matches the settled load). -/
theorem unitSettled_loadRecognitionCost_zero_iff_topologyAmplitude
    {ψ : LightPattern (Fin 8)}
    (E : Q3ClosedPatternEvidence ψ)
    (hunit : SettledAnchorWindow 1 ψ) :
    loadRecognitionCost ψ = 0 ↔
      (1 : ℝ) = primitivePositiveStationaryFactorAmplitude ψ :=
  settledAnchor_loadRecognitionCost_zero_iff_amplitude E 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

/-- **Doubled-scale decoy.** `SettledAnchorWindow 2` is never Boolean-valued:
its commit entry is `2`, outside `{0,±1}`. -/
theorem settledAnchorWindow_two_not_boolean
    {ψ : LightPattern (Fin 8)}
    (hwin : SettledAnchorWindow 2 ψ) :
    ¬ BooleanDifferenceSettlementWindow (ψ.window 0) := by
  intro hbool
  have hunit :=
    settledAnchor_boolean_forces_unitAmplitude hwin hbool (by norm_num)
  norm_num at hunit

/-- Gap-one worldline is a Boolean unit settled Q3 seed whose load cost is
not zero, hence `loadTopologyRatio ≠ 1`. -/
theorem exists_unitBooleanSettled_loadTopologyRatio_ne_one :
    ∃ ψ : LightPattern (Fin 8),
      Q3MatterCarrier ψ ∧
        SettledAnchorWindow 1 ψ ∧
          BooleanDifferenceSettlementWindow (ψ.window 0) ∧
            loadTopologyRatio ψ ≠ 1 := by
  obtain ⟨E⟩ := q3MatterCarrier_worldlinePattern_gapOne
  exact ⟨worldlinePattern gapOneTwoPhaseMode, ⟨E⟩,
    worldlineGapOne_settledAnchor_unit, worldlineGapOne_boolean,
    worldlineGapOne_unit_loadTopologyRatio_ne_one E⟩

/-- **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

This theorem does not claim that unit Boolean posting forces the load-topology ratio to be 1. This theorem does not claim that the load-recognition cost is always zero for settled anchors. This theorem does not claim that the framework derives the fine-structure constant or any specific particle mass.

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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