Gravity Seven Gaps Three Pent Interior Hinge Witness Link Edges Eq Pent Residues

The hinge link as residuals

In the Recognition Science framework's combinatorial model of spacetime, a ledger (a discrete record of events and their connections) is built from four-dimensional blocks called pents. The declaration linkEdges_eq_pent_residues states a precise identity: the edges in the link of a shared hinge triangle are exactly the residual pairs left over when that hinge is removed from each pent. Concretely, for three pents sharing the hinge {0,1,2}, the link edges are {3,4}, {4,5}, and {3,5}, which are precisely the pairs of vertices remaining after subtracting the hinge from each pent.

This identity is a theorem, verified by the machine-checked library of formal theorems. It is a purely combinatorial fact about finite sets, not a statement about physical lengths or angles. The theorem establishes that each pent contributes exactly one edge to the link, and together these three edges form a cycle: 3 connected to 4, 4 to 5, and 5 back to 3. The hinge is therefore an interior hinge, meaning the dihedral angles around it close up in a cycle, which is the combinatorial requirement for calling a deficit angle an interior curvature quantity.

The declaration does not claim anything about the metric or causal structure of the pents. It does not provide edge-length assignments or angle values for the three glued pents. That metric compatibility question remains open and is a separate lane of investigation. The theorem is a structural result about incidence, not about geometry in the sense of distances or angles.

/-- THEOREM (by `decide`): the link edges are exactly the residual pairs
`P \ hinge` of the pents. -/
theorem linkEdges_eq_pent_residues :
    linkEdges = threePentComplex.image (fun P => P \ hinge) := by decide

/-- **THEOREM (main witness, cycle case)**: the residual link-edge set of
the three-pent complex satisfies `IsCycleLink` — the hinge `{0,1,2}` is a
GENUINE INTERIOR hinge.  At the incidence level this is exactly what a
Regge deficit `2π − Σθ` at the hinge requires: the dihedral angles close
up in a cycle around the hinge. -/
theorem threePent_hinge_is_interior :
    IsCycleLink (threePentComplex.image (fun P => P \ hinge)) := by
  unfold IsCycleLink
  decide

What this page does not claim

The declaration does not provide edge-length assignments or angle values for the three pents. The declaration does not establish causal-structure consistency for the glued pents. The declaration does not claim the metric compatibility question is settled.

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/SevenGaps/ThreePentInteriorHingeWitness.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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