Encyclopedia Foundation Foundation Pair Kernel Event Action Ancestry S14 Primitive Coherence Event Actio

ARTICLE 4 claims 3 theorems 1 model

Foundation Pair Kernel Event Action Ancestry S14 Primitive Coherence Event Actio

A single primitive event in the Recognition Science ledger carries an action equal to a fixed power of the golden ratio, a value the framework proves rather than fits.

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The action of a single event

In physics, action is the product of an energy and a time interval, the quantity that a system's history makes stationary. The Recognition Science framework, a formal system built from a discrete record of events it calls a ledger, defines the action of a primitive posting event as the product of its assigned energy and duration. A primitive event is the smallest realized unit of posting in the ledger, a single atomic entry in the record.

The framework's library, a machine-checked collection of formal theorems, proves that any such primitive event that realizes exactly one coherence event, meaning it carries one unit of coherence energy for one fundamental duration, has action equal to the golden ratio raised to a fixed negative exponent. The golden ratio, approximately 1.618, is the number satisfying r² = r + 1. The fixed exponent is the configuration dimension, a constant of the framework's gap derivation. This result, named primitiveCoherenceEvent_action_eq_forced_configExponent, derives the action value from the framework's own constants rather than taking it as an input.

This theorem is a statement about the framework's internal mathematics. It does not claim that any particular physical system in the everyday world has this action. The framework models a primitive event as carrying one coherence energy E_coh for one fundamental duration tau0, and the theorem states that the product E_coh * tau0 equals the golden-ratio power. The framework's constants hbar and the golden ratio are defined in its own units, not in SI units. The theorem does not establish a connection to measured physical quantities such as the Planck constant in joule-seconds.

The theorem also does not claim that the action value is unique across all possible assignments of energy and duration to events. The framework proves that two mathematically consistent branches exist for mapping the ledger's dimensionless costs to physical action: one using the identity map and one using a ledger-derived scale. The theorem applies to the coherence-event model, which fixes the energy and duration assignments. It does not rule out other assignments that would give different action values, as the framework's countermodels show for doubled energy or stretched duration.

THEOREM primitiveCoherenceEvent_action_eq_forced_configExponent · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
primitiveCoherenceEvent_action_eq_forced_configExponent · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:155 
/-- The exponent ancestry transports to the event action once the physical
event arrow is supplied. -/
theorem primitiveCoherenceEvent_action_eq_forced_configExponent
    {N : ℕ} [NeZero N]
    {kinematics : PostingEventKinematics3 N}
    (hcoherence :
      PrimitivePostingRealizesOneCoherenceEvent3 kinematics)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    postingEventAction3 kinematics event =
      Constants.phi ^
        (-(GapDerivation.configDim GapDerivation.D : ℤ)) := by
  rw [primitiveCoherenceEvent_action_eq_hbar hcoherence hevent]
  exact GapDerivation.hbar_exponent_eq_configDim
MODEL canonicalCoherenceKinematics3 · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
canonicalCoherenceKinematics3 · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:489 
/-- MODEL witness for the missing physical arrow. Its existence proves
consistency only; it does not derive the assignment from `PostingStep`. -/
def canonicalCoherenceKinematics3
    (N : ℕ) [NeZero N] :
    PostingEventKinematics3 N where
  energy := fun _ => Constants.E_coh
  duration := fun _ => Constants.tau0
  energy_pos := fun _ => Constants.E_coh_pos
  duration_pos := fun _ => Constants.tau0_pos
THEOREM identityExactJMap_forces_S13_native_package · ledgerScaleExactJMap_forces_dropOne_package · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
identityExactJMap_forces_S13_native_package · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:333 
theorem identityExactJMap_forces_S13_native_package
    {N : ℕ} [NeZero N]
    {kinematics : PostingEventKinematics3 N}
    {drop sourceMagnitude : PostingPair3 N → ℝ}
    (hcoherence :
      PrimitivePostingRealizesOneCoherenceEvent3 kinematics)
    (hmap :
      MappedExactJDropRepresentsEventAction3 1 kinematics drop)
    (hsource :
      EventDropSourceIsExactJConjugate3 drop sourceMagnitude)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    drop event = nativeActionCanonicalDrop ∧
      sourceMagnitude event = nativeExactJConjugateSource ∧
      2 * sourceMagnitude event = nativeOrderedExactJSource ∧
      Real.cosh (drop event) = 1 + Constants.hbar := by
  have hdrop :=
    identityExactJMap_selects_nativeActionCanonicalDrop
      hcoherence hmap hevent
  refine ⟨hdrop, ?_, ?_, ?_⟩
  · rw [hsource event hevent, hdrop]
    rfl
  · rw [hsource event hevent, hdrop]
    rfl
  · rw [hdrop]
    exact nativeActionCanonicalDrop_cosh
ledgerScaleExactJMap_forces_dropOne_package · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:373 
theorem ledgerScaleExactJMap_forces_dropOne_package
    {N : ℕ} [NeZero N]
    {scale : ℝ}
    {kinematics : PostingEventKinematics3 N}
    {drop sourceMagnitude : PostingPair3 N → ℝ}
    (hcoherence :
      PrimitivePostingRealizesOneCoherenceEvent3 kinematics)
    (hscale : PrimitivePostingActionScaleLaw scale)
    (hmap :
      MappedExactJDropRepresentsEventAction3 scale kinematics drop)
    (hsource :
      EventDropSourceIsExactJConjugate3 drop sourceMagnitude)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    drop event = 1 ∧
      sourceMagnitude event = exactJVariationalSource 1 ∧
      2 * sourceMagnitude event =
        ledgerNormalizedOrderedExactJSource ∧
      Real.cosh (drop event) =
        ledgerNormalizedExactJTangentCurvature := by
  have hdrop :=
    ledgerScaleExactJMap_selects_drop_one
      hcoherence hscale hmap hevent
  refine ⟨hdrop, ?_, ?_, ?_⟩
  · rw [hsource event hevent, hdrop]
  · rw [hsource event hevent, hdrop]
    rfl
  · rw [hdrop]
    rfl
THEOREM doubledEnergy_countermodel · stretchedDuration_countermodel · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
doubledEnergy_countermodel · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:563 
/-- Energy countermodel: one fundamental duration and the same ledger event
ancestry admit doubled energy and doubled action. -/
theorem doubledEnergy_countermodel :
    RealizedPostingLastsFundamentalTick3
        (doubledEnergyKinematics3 3) ∧
      ¬ RealizedPostingCarriesCoherenceEnergy3
        (doubledEnergyKinematics3 3) ∧
      ∃ event : PostingPair3 3,
        event ∈ realizedPrimitivePostingPairs3 3 ∧
          postingEventAction3
            (doubledEnergyKinematics3 3) event =
              2 * Constants.hbar := by
  constructor
  · intro event hevent
    rfl
  constructor
  · intro henergy
    obtain ⟨event, hevent⟩ := realizedPrimitivePostingPair3_exists
    have h := henergy event hevent
    change 2 * Constants.E_coh = Constants.E_coh at h
    linarith [Constants.E_coh_pos]
  · obtain ⟨event, hevent⟩ := realizedPrimitivePostingPair3_exists
    refine ⟨event, hevent, ?_⟩
    unfold postingEventAction3 doubledEnergyKinematics3
    change (2 * Constants.E_coh) * Constants.tau0 =
      2 * Constants.hbar
    calc
      (2 * Constants.E_coh) * Constants.tau0 =
          2 * (Constants.E_coh * Constants.tau0) := by ring
      _ = 2 * Constants.hbar := by
        rw [← Constants.hbar_action_identity]
stretchedDuration_countermodel · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:531 
/-- Duration countermodel: unit coherence energy and the same ledger event
ancestry admit a doubled duration and doubled action. -/
theorem stretchedDuration_countermodel :
    RealizedPostingCarriesCoherenceEnergy3
        (stretchedDurationKinematics3 3) ∧
      ¬ RealizedPostingLastsFundamentalTick3
        (stretchedDurationKinematics3 3) ∧
      ∃ event : PostingPair3 3,
        event ∈ realizedPrimitivePostingPairs3 3 ∧
          postingEventAction3
            (stretchedDurationKinematics3 3) event =
              2 * Constants.hbar := by
  constructor
  · intro event hevent
    rfl
  constructor
  · intro hduration
    obtain ⟨event, hevent⟩ := realizedPrimitivePostingPair3_exists
    have h := hduration event hevent
    change 2 * Constants.tau0 = Constants.tau0 at h
    linarith [Constants.tau0_pos]
  · obtain ⟨event, hevent⟩ := realizedPrimitivePostingPair3_exists
    refine ⟨event, hevent, ?_⟩
    unfold postingEventAction3 stretchedDurationKinematics3
    change Constants.E_coh * (2 * Constants.tau0) =
      2 * Constants.hbar
    calc
      Constants.E_coh * (2 * Constants.tau0) =
          2 * (Constants.E_coh * Constants.tau0) := by ring
      _ = 2 * Constants.hbar := by
        rw [← Constants.hbar_action_identity]

What this page does not claim

The theorem does not claim any measured physical system has this action value. The theorem does not claim the action value is unique across all energy and duration assignments. The framework does not claim its constants match SI unit values.

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/Foundation/PairKernelEventActionAncestryS14.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.

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