Gravity Analysis Regge Bloch Star Edge Origins4 D List Sum Map Smul Plane Wave

Scaling the phase sum

The declaration list_sum_map_smul_planeWave is a lemma in the framework's machine-checked library of formal theorems. It states that when you multiply a 4 by 4 real matrix H by a real number c, the sum of certain phase-weighted contributions built from H scales by the same factor c. In plainer terms: if you stretch the input, the output stretches proportionally. The contributions are sums over a list of edge records, each carrying a class index, a weight, and a position offset; the lemma says the whole sum behaves linearly under scalar multiplication of the matrix.

The lemma is a helper. Its named consequence, phasedDeficitDotEdgeOrigins_smul, applies it to a specific function called phasedDeficitDotEdgeOrigins, which assembles those edge contributions into a single number for each orbit type and slot. That theorem states the same linearity: phasedDeficitDotEdgeOrigins ty (c • H) m s t = c * phasedDeficitDotEdgeOrigins ty H m s t. This is a structural fact about the definitions, not a physical law. It tells a reader that the framework's bookkeeping for these edge-origin sums respects scaling, which is a consistency check on the formalism.

In Recognition Science, the framework models reality as maintaining a discrete record of recognition events, and this lemma belongs to a construction that assigns origins to star edges in a four-dimensional lattice. The lemma itself does not claim anything about physics. It does not say that space is discrete, that gravity is quantized, or that any particular matrix H is the correct one. It only certifies that one algebraic identity holds for the defined functions, so that later arguments can rely on scaling behavior without rechecking the sum term by term.

What the lemma does not claim is as important as what it proves. It does not establish that the edge-origin construction is physically correct, that the tables of 22, 24, and 32 contributions are complete, or that the framework's gravity model matches observation. Those are separate questions. The lemma is a piece of internal consistency: a guarantee that a certain sum behaves predictably under scaling, nothing more.

private lemma list_sum_map_smul_planeWave (c : ℝ) (H : Mat4) (m : Wave4)
    (p : Fin 24) (base : Wave4) (cs : List SeedEdgeContrib) :
    (cs.map (fun e =>
        e.weight *
          planeWaveClassPert (c • H) m
            (fun i => base i + transportOrigin p e.origin i)
            (permClass p e.cls))).sum =
      c *
        (cs.map (fun e =>
            e.weight *
              planeWaveClassPert H m
                (fun i => base i + transportOrigin p e.origin i)
                (permClass p e.cls))).sum := by
  induction cs with
  | nil => simp
  | cons hd tl ih =>
      simp only [List.map_cons, List.sum_cons]
      rw [planeWaveClassPert_smul, ih]
      ring

theorem phasedDeficitDotEdgeOrigins_smul (c : ℝ) (ty : HingeOrbitType)
    (H : Mat4) (m : Wave4) (s : Fin 24) (t : Fin 10) :
    phasedDeficitDotEdgeOrigins ty (c • H) m s t =
      c * phasedDeficitDotEdgeOrigins ty H m s t := by
  unfold phasedDeficitDotEdgeOrigins edgeContribPhased
  exact list_sum_map_smul_planeWave c H m (orbitCoveringPerm ty s t)
    (hingeBase s t) (seedEdgeContribs ty)

What this page does not claim

The lemma does not prove that space is discrete or that gravity is quantized. The lemma does not establish that the edge-origin tables are physically complete or correct. The lemma does not compare the framework's gravity model to any measured physical quantity.

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/Gravity/Analysis/ReggeBlochStarEdgeOrigins4D.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: