Constants Gap Weight Numerics Scaffold

Gap weight scaffold

The gap weight is a number that the Recognition Science framework derives from its eight-tick recognition cycle, the discrete record of events that the framework uses as its basic accounting. The scaffold module is a small, machine-checked library of formal theorems that verifies this number lands inside a specific numeric window. The certificate states that the gap weight, written w8, lies strictly between 2.490564399 and 2.490572090.

That window is not a guess. The framework's own derivation, called w8_from_eight_tick, produces the value, and the scaffold proves the bounds with exact arithmetic. A second theorem gives a convenient approximate form: the gap weight equals 2.490569275454 to within five millionths. This is a compatibility alias for legacy code that expects a single decimal value rather than an interval.

In plain language, the module does one job: it takes a number that the framework claims to derive from first principles and checks that the derivation actually lands where it should. The check is not statistical and it is not fitted. It is a formal proof, in the framework's machine-checked library of formal theorems, that the computed value satisfies the stated tolerance. This matters because the gap weight feeds into the framework's constant ladder, the chain that produces physical constants from the recognition cycle.

The scaffold is deliberately narrow. It does not explain why the eight-tick cycle exists, nor does it connect the gap weight to any measured physical quantity. It only certifies the arithmetic. That narrowness is the point: a constant that cannot be pinned to a precise interval cannot anchor anything downstream. The scaffold makes the anchor solid.

/-- The gap weight derived from DFT-8 matches the expected value within tolerance. -/
theorem w8_matches_certified :
    (2.490564399 : ℝ) < w8_from_eight_tick ∧ w8_from_eight_tick < (2.490572090 : ℝ) := by
  constructor
  · exact Numerics.W8Bounds.w8_computed_gt
  · exact Numerics.W8Bounds.w8_computed_lt

/-- Compatibility alias for legacy code (with approximate equality). -/
theorem w8_value : abs (w8_from_eight_tick - 2.490569275454) < 5e-6 := by
  -- Follows from the verified interval bounds
  have h := w8_matches_certified
  rw [abs_lt]
  constructor <;> linarith

What this page does not claim

This module does not derive the eight-tick cycle or explain its origin. This module does not connect the gap weight to any measured physical quantity. This module does not claim the gap weight is unique or that no other value could satisfy the bounds.

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/Constants/GapWeightNumericsScaffold.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.

Derived articles

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