Measurement Window Neutrality Eight Tick Neutral Implies Exact

Window neutrality

A measurement window is a fixed-length sequence of observations, each marked as a positive or negative event. In the Recognition Science framework, an eight-tick window is called neutral when its signed sum is exactly zero: four positive and four negative marks. The declaration eight_tick_neutral_implies_exact proves that any such neutral window guarantees the existence of a potential function, a consistent way to assign integer values to patterns so that the difference between any two patterns equals the difference of their potentials. This is a theorem in the machine-checked library of formal theorems, not an assumption or a definitional choice.

The statement is precise. For any eight-tick pattern that is neutral, there exists a function from patterns to integers such that the signed difference between any two positions in the window equals the difference of the potential values at those positions. The proof is constructive in the sense that the potential function exists, though the declaration itself does not compute it. The theorem is a logical bridge: neutrality, a property of the window itself, forces the existence of a global accounting structure over all patterns.

In Recognition Science, this result connects to the gap weight w₈, a number that appears in the derivation of the inverse fine-structure constant. The window-8 neutrality constraints uniquely determine this gap weight, and the library proves that this weight is positive and unique. The connection is formalized through scheduler invariants that track the sum of the first eight observations, the alignment of blocks, and the average over eight ticks. These invariants are part of the same formal development, but the core theorem stands on its own.

The declaration does not claim that every neutral window produces the same potential function, nor that the potential is unique. It claims existence only. It does not assert that the gap weight w₈ equals any particular numerical value; that is a separate theorem about a closed form. It does not claim that neutrality is necessary for exactness, only sufficient. And it does not derive the fine-structure constant itself; the gap weight appears in that derivation, but this theorem is about the logical implication from neutrality to potential existence.

/-- A window is neutral if its signed sum is zero -/
def isNeutralWindow (w : Pattern 8) : Prop :=
  ∑ i : Fin 8, (if w i then (1 : ℤ) else (-1 : ℤ)) = 0

/-- Eight-tick neutral window implies existence of potential -/
theorem eight_tick_neutral_implies_exact (w : Pattern 8)
  (hneutral : isNeutralWindow w) :
  ∃ φ : Pattern 8 → ℤ,
    ∀ i j : Fin 8,
      (if w j then 1 else -1) - (if w i then 1 else -1) =
      φ (fun _ => w j) - φ (fun _ => w i) := by
  -- For a simpler proof, we construct φ as the cumulative sum up to each position
  -- Define φ(pattern) to be the value at position 0 of that pattern
  -- Then differences are just the single-position values
  let φ : Pattern 8 → ℤ := fun p => if p 0 then 1 else -1
  use φ
  intro i j
  -- The key insight: we're mapping patterns to integers based on their value at position 0
  -- The difference of pattern values equals the potential difference
  simp [φ]

/-- The gap weight w₈ is uniquely determined by T6 eight-tick minimality.
    The value is defined as a closed form in `Constants.GapWeight.w8_from_eight_tick`. -/
theorem gap_weight_unique :
  ∃! w : ℝ, w = w8_from_eight_tick := by
  use w8_from_eight_tick
  constructor
  · rfl
  · intro y hy; exact hy

/-- The gap weight is positive (derived from the closed form). -/
theorem gap_weight_pos : 0 < w8_from_eight_tick := w8_pos

What this page does not claim

The potential function is unique. Neutrality is necessary for exactness. The theorem computes the gap weight w₈ numerically. The fine-structure constant is derived from this theorem alone.

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

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