Constants Gap Weight Numerics Scaffold W8 Matches Certified

The certified gap weight

The gap weight, a number the Recognition Science framework derives from its eight-tick recognition cycle, measures a discrete step in the framework's cost structure. The declaration w8_matches_certified is a machine-checked theorem stating that this weight, written w8_from_eight_tick, lies strictly between 2.490564399 and 2.490572090. The proof rests on two verified interval bounds, one giving the lower limit and one the upper, so the result is not an approximation but a certified range.

A companion theorem, w8_value, states that the same weight differs from 2.490569275454 by less than five millionths. This second statement follows directly from the first by a short algebraic argument, so it inherits the same machine-checked status. The two theorems together give a working value for the gap weight while the certified bounds remain the authoritative statement.

In Recognition Science, the framework models physical constants as consequences of its forcing chain rather than as free parameters. The gap weight here is one such derived quantity, tied to the eight-tick cycle that the framework's proofs force from the cost function. The certified bounds are the framework's way of stating the weight's value with a precise, checkable envelope.

What the declaration does not claim is just as important. It does not assert that the gap weight equals any exact closed-form expression, nor does it assign the weight a role in any physical measurement. The theorem is purely numerical: it certifies an interval. Whether this number corresponds to an observable quantity, and how it connects to the framework's particle mass ladder, remains a separate question the declaration does not address.

/-- 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

/-- 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

What this page does not claim

The gap weight equals exactly 2.490569275454 or any other single closed-form number. The gap weight has been measured in any experiment or linked to a specific physical constant. The declaration assigns the gap weight any role beyond its numerical interval in the framework.

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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