Gravity Analysis Regge Ttderivative Gate Flat Angle Jacobian Eq Dihedral Closed

The flat angle Jacobian

In the Recognition Science framework, a discrete record of physical events, the geometry of a single tetrahedron is a test case for how gravity might emerge from a lattice of simple shapes. The declaration flatAngleJacobian_eq_dihedralClosedDerivSq states that two ways of computing the same quantity are identical. One way is a table called the flat angle Jacobian, which lists how each of the six dihedral angles of a tetrahedron changes when one of its six squared edge lengths is varied, evaluated at the special flat configuration. The other way is the closed-form derivative of the dihedral angle formula, written directly from the geometry. The theorem proves these are the same by definition, a kernel-level equality in the machine-checked library of formal theorems.

The practical consequence is that the framework's gravity program can use either expression interchangeably. When a later stage needs the derivative of a dihedral angle, it can pick the form that is easier to compute or to reason about, knowing the two agree exactly. This is not a numerical approximation or a heuristic match; it is an identity that holds for every edge and every angle in the flat tetrahedron. The proof is pure algebra and analysis over the standard logical axioms, with no extra assumptions and no unverified steps.

What the theorem does not claim is broader. It does not say that the flat angle Jacobian is the correct description of gravity, nor that the continuum limit of the lattice action exists or equals a known value. It does not assert that second derivatives of the action profile exist, which is a later stage of the program. It does not even claim that the flat configuration is physically preferred, only that at this one geometric point the two derivative formulas coincide. The theorem is a local, exact identity about a single tetrahedron, not a statement about the whole lattice or about physics beyond this geometric fact.

/-- The shared stencil Jacobian is exactly the squared-edge closed-form
dihedral derivative of the Schläfli module at the flat tetrahedron
(kernel `rfl`; this ties the Gate-0c stencil to the proved Schläfli
machinery with zero transcription). -/
theorem flatAngleJacobian_eq_dihedralClosedDerivSq (f k : Fin 6) :
    flatAngleJacobian f k =
      Geometry.SchlaefliTetrahedronProof.dihedralClosedDerivSq freudenthalTet f k := rfl

/-- The shared stencil Jacobian is exactly the squared-edge closed-form
dihedral derivative of the Schläfli module at the flat tetrahedron
(kernel `rfl`; this ties the Gate-0c stencil to the proved Schläfli
machinery with zero transcription). -/
theorem flatAngleJacobian_eq_dihedralClosedDerivSq (f k : Fin 6) :
    flatAngleJacobian f k =
      Geometry.SchlaefliTetrahedronProof.dihedralClosedDerivSq freudenthalTet f k := rfl

What this page does not claim

The theorem does not prove that the continuum limit of the lattice action exists or equals a known value. The theorem does not assert that second derivatives of the action profile exist. The theorem does not claim that the flat configuration is physically preferred over any other.

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/ReggeTTDerivativeGate.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: