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IndisputableMonolith.Foundation.SimplicialLedger.InteriorFlat

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This module introduces FlatInteriorMetric to equip 3-simplices in the simplicial ledger with a choice of Euclidean or Minkowski signature together with a flatness property. Workers on discrete Regge calculus or the J-cost to continuum bridge would cite the definition to enforce local isometry to flat space. The module supplies only definitions and a property; no proofs are present.

claimA FlatInteriorMetric on a 3-simplex consists of a signature $s \in \{-1,+1\}$, a Cayley-Menger positivity witness on the edge lengths, and the Regge axiom asserting that the simplex interior is isometric to the standard flat simplex of signature $s$.

background

The module belongs to Foundation.SimplicialLedger, which represents the ledger as a simplicial 3-complex via a coordinate-free sheaf that unifies local and global J-cost variations. It imports Constants, where the fundamental RS time quantum satisfies $\tau_0 = 1$ tick, and ContinuumBridge, whose doc-comment states that the J-cost functional on the simplicial ledger is the Regge action (up to normalization by $\kappa = 8\phi^5$) and that J-cost stationarity yields the Regge equations.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

FlatInteriorMetric supplies the local flatness structure required for the simplicial ledger to interface with the continuum limit. It feeds the argument in ContinuumBridge that J-cost stationarity produces the Regge equations, thereby supporting the Recognition Science derivation of Einstein's field equations from the single functional equation via the eight-tick octave and D=3.

scope and limits

depends on (3)

Lean names referenced from this declaration's body.

declarations in this module (16)