Constants Gap Weight Formula Geometric Weight Pos

The positivity lemma

In the framework's machine-checked library of formal theorems, recognition (a discrete record of events and their costs) gives rise to an eight-tick cycle. The library studies a candidate weight for one gap in that cycle. The weight is built from two factors: an oscillation term, the square of a sine at a frequency tied to the tick index, and a decay term, the golden ratio raised to a negative power. The declaration geometricWeight_pos proves a simple fact about that formula: for any tick whose index is not zero, the weight is strictly greater than zero. It also proves the weight is never negative for any tick at all. That is the entire content of the lemma.

Why does positivity matter? The weight is meant to contribute to a larger sum, a candidate for the gap weight used in the framework's account of the fine-structure constant. If the weight could go negative, the sum could cancel itself unpredictably. The lemma rules out that failure mode for this particular formula. It is a small, local guarantee, not a claim about the physical world. It says nothing about whether the formula is correct, whether it matches measurement, or whether it is the weight the framework actually uses. The library is explicit on that last point: the formula is a scaffold, a candidate, and it is not currently proven to equal the certified weight used in the alpha pipeline.

The proof is short and mechanical. For the zero index, the weight is defined to be zero. For every other index, the sine squared term is non-negative and the golden-ratio decay term is positive because the golden ratio itself is positive. A product of a non-negative number and a positive number is positive. The lemma's proof in the library is exactly that argument, written in a form a machine can check. It is a theorem in the sense that it is fully verified, but its scope is narrow: it is a fact about a definition, not a discovery about nature.

What the lemma does not claim is as important as what it does. It does not claim the candidate weight is the true gap weight. It does not claim the candidate sum matches the measured fine-structure constant. It does not claim the golden ratio appears in nature because of this formula. All of those are separate questions, and the library marks them as open or as empirical checks. The lemma is a piece of bookkeeping: it keeps the candidate formula honest by ensuring its terms stay positive. That is all, and that is enough for the role it plays.

/-- geometricWeight is positive for neutral modes. -/
lemma geometricWeight_pos {k : Fin 8} (hk : k.val ≠ 0) : 0 < geometricWeight k := by
  unfold geometricWeight
  simp only [hk, ↓reduceIte]
  apply mul_pos
  · apply sq_pos_of_pos
    apply Real.sin_pos_of_pos_of_lt_pi
    · have hk_pos : 0 < k.val := Nat.pos_of_ne_zero hk
      positivity
    · have h1 : k.val ≤ 7 := Nat.lt_succ_iff.mp k.isLt
      have h2 : (k.val : ℝ) ≤ 7 := by exact Nat.cast_le.mpr h1
      calc (k.val : ℝ) * Real.pi / 8
          ≤ 7 * Real.pi / 8 := by nlinarith [Real.pi_pos]
        _ < Real.pi := by nlinarith [Real.pi_pos]
  · exact zpow_pos phi_pos _

/-- geometricWeight is non-negative. -/
lemma geometricWeight_nonneg (k : Fin 8) : 0 ≤ geometricWeight k := by
  unfold geometricWeight
  split_ifs with h
  · exact le_refl 0
  · apply mul_nonneg
    · exact sq_nonneg _
    · exact zpow_nonneg (le_of_lt phi_pos) _

/-- A DFT-based candidate weight (scaffold).

This is *not* currently proven to match the certified `Constants.w8_from_eight_tick`
used by the α pipeline (see `Constants/GapWeight.lean`). -/
noncomputable def w8_dft_candidate : ℝ :=
  Finset.sum (Finset.filter (· ≠ 0) Finset.univ) fun k =>
    phiDFTAmplitude k * geometricWeight k

What this page does not claim

The lemma does not claim the candidate weight is the true gap weight. It does not claim the candidate sum matches the measured fine-structure constant. It does not claim the golden ratio appears in nature because of this formula.

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/GapWeight/Formula.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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