Gravity Analysis Regge Bloch Star Edge Origins4 D

Position-resolved star edges

In Recognition Science, gravity analysis works with a discretized four-dimensional spacetime built from hinges and edges. A star edge is a contribution associated with one edge of such a hinge, and its origin is the lattice position from which that contribution is drawn. The module ReggeBlochStarEdgeOrigins4D (the name is a file label, not a description of a physical theory) establishes, in machine-checked form, that these star-edge contributions can be tracked with their positions resolved, so that each contribution carries not only a class index and a weight but also a four-component origin vector.

The core objects are simple: a seed edge contribution is a triple of a class index (one of fifteen), a real weight, and a four-dimensional origin. For three of the six hinge orbit types, the module lists explicit seed contributions: 22 for orbit t12, 24 for t13, and 32 for t22. The other three orbit types are defined as identity transports of these, so t21 reuses t12 and t31 reuses t13, while t11 is empty. These lists are proved to have exactly those lengths by definition, a trivial but load-bearing fact: the tables land where the framework expects them.

The substantive content is a transport operation. A covering permutation moves a seed-frame origin to a transported slot, and the module defines how a contribution is evaluated at that transported slot with a phase factor. Two derived quantities follow: a position-resolved deficit dot product and a phase-squared sum for an m² truncation. The first is proved to be linear in the matrix argument, so scaling the matrix scales the deficit by the same factor. The second feeds into a slot coefficient, which is nonzero only when the orbit is active, and that coefficient sums over slots to an orbit moment.

In Recognition Science, the framework models this as a fold repair: the module fixes a prior gap where star edges carried lattice origins but the covering permutations did not transport both class index and origin into the deficit phase for non-t11 orbits. The status structure records three flags: tables landed, gap action recovery not flipped, and base0 half-repair forbidden. All three are proved by decision procedure. The plain-language consequence is that the four-dimensional gravity calculation now has a position-resolved accounting for star-edge contributions, with the transport and linearity properties checked by the machine-checked library of formal theorems.

The framework's library proves the linearity theorem and the status flags; it does not prove that this accounting matches any particular physical measurement. That comparison remains a target. The module is a structural step: it makes the origin tracking explicit and verifies the algebraic properties that the larger gravity analysis relies on.

/-- One seed-frame star edge contribution: class index, weight, origin. -/
structure SeedEdgeContrib where
  cls : Fin 15
  weight : ℝ
  origin : Wave4

theorem seedEdgeContribs_t12_length :
    seedEdgeContribs_t12.length = 22 := rfl

theorem seedEdgeContribs_t13_length :
    seedEdgeContribs_t13.length = 24 := rfl

theorem seedEdgeContribs_t22_length :
    seedEdgeContribs_t22.length = 32 := rfl

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)

theorem starEdgeOriginsStatus_flags :
    starEdgeOriginsStatus.tablesLanded = true ∧
      starEdgeOriginsStatus.gapActionRecovery = false ∧
        starEdgeOriginsStatus.base0Forbidden = true := by
  decide

What this page does not claim

This module does not prove that the position-resolved star-edge accounting matches any measured gravitational quantity. The name ReggeBlochStarEdgeOrigins4D is a file label, not a claim about Regge calculus or Bloch theory in conventional physics. The linearity theorem holds for the deficit dot product, not for the phase-squared m² truncation.

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)

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