Foundation Hierarchy Realization From Scale Realized Closed Scale Ratio Step

The ratio step

A geometric sequence is one where each term is a fixed multiple of the previous one: 1, 2, 4, 8, and so on. The ratio between consecutive terms is constant, here 2. The theorem realized_closed_scale_ratio_step states that if a system's observed values, recorded at successive ticks, follow such a geometric sequence, then the ratio of any two consecutive observed values is exactly that fixed ratio. This is a direct consequence of the sequence's definition, not an additional assumption.

In the Recognition Science framework, a ledger is a discrete record of events. The framework models a system's behavior as a sequence of observed values, one per tick. The theorem applies when this sequence is geometric, meaning each observed value is a constant multiple of the one before it. Under that condition, the ratio of any two consecutive observed values equals that constant multiple. This is the ratio step: it pins down the local scaling behavior of the system.

The theorem is proved in the framework's machine-checked library of formal theorems. It is a conditional result: it holds for any system that satisfies the premise of having a geometric sequence of observed values. The proof is short and relies on the definition of a geometric sequence and basic arithmetic. The theorem does not claim that any particular physical system actually exhibits such a sequence; that is a separate question about existence, which the framework leaves open.

The ratio step is a building block. From it, the framework derives two further properties for such a system: ratio self-similarity, meaning the ratio of consecutive observed values is the same at every step, and additive posting, meaning the observed value at the third tick is the sum of the values at the first two ticks. These follow once the geometric sequence is also closed under a composition operation, a condition that is part of the framework's model. The ratio step itself does not establish these; it only establishes the constant ratio.

What the theorem does not claim is equally important. It does not assert that any system in the real world realizes a geometric sequence. It does not claim that the ratio is the golden ratio or any specific number; that would require additional structure. It does not claim that the additive posting property holds without the closure condition. The theorem is a precise, conditional statement about what follows from a geometric sequence of observed values, nothing more.

/-- The realized orbit has constant adjacent ratio. -/
theorem realized_closed_scale_ratio_step
    (F : ClosedObservableFramework) (H : RealizedClosedScaleModel F) (k : ℕ) :
    F.r (F.T^[k + 1] H.baseState) / F.r (F.T^[k] H.baseState) = H.scales.ratio := by
  rw [H.realize (k + 1), H.realize k]
  have ha : H.amplitude ≠ 0 := ne_of_gt H.amplitude_pos
  calc
    H.amplitude * H.scales.scale (k + 1) / (H.amplitude * H.scales.scale k)
      = H.scales.scale (k + 1) / H.scales.scale k := by
          rw [mul_div_mul_left _ _ ha]
    _ = H.scales.ratio := scale_step_ratio H.scales k

/-- Therefore the realized orbit satisfies ratio self-similarity. -/
theorem ratio_self_similar_of_realized_closed_scale
    (F : ClosedObservableFramework) (H : RealizedClosedScaleModel F) :
    ∀ k,
      F.r (F.T^[k + 2] H.baseState) / F.r (F.T^[k + 1] H.baseState) =
        F.r (F.T^[k + 1] H.baseState) / F.r (F.T^[k] H.baseState) := by
  intro k
  rw [realized_closed_scale_ratio_step F H (k + 1), realized_closed_scale_ratio_step F H k]

/-- Closure of the earlier geometric scale sequence yields additive
posting on the realized orbit. -/
theorem additive_posting_of_realized_closed_scale
    (F : ClosedObservableFramework) (H : RealizedClosedScaleModel F) :
    F.r (F.T^[2] H.baseState) =
      F.r (F.T^[1] H.baseState) + F.r (F.T^[0] H.baseState) := by
  have hclosed : H.scales.scale 0 + H.scales.scale 1 = H.scales.scale 2 := by
    simpa [GeometricScaleSequence.isClosed, ledgerCompose] using H.scales_closed
  have hclosed' : H.scales.scale 2 = H.scales.scale 1 + H.scales.scale 0 := by
    linarith
  rw [H.realize 2, H.realize 1, H.realize 0]
  rw [hclosed']
  ring

What this page does not claim

The theorem does not claim that any physical system realizes a geometric sequence. The theorem does not claim the ratio is any specific number, such as the golden ratio. The theorem does not claim additive posting holds without the closure condition.

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

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