Gravity Analysis Regge4 Dflat Second Variation Flat Freudenthal Seed Angle Has D

The seed angle derivative

In the discrete geometry of a four-dimensional simplex, a dihedral angle is the angle between two of its three-dimensional faces. The declaration flat_freudenthal_seed_angle_hasDerivAt is a machine-checked theorem stating that, for a particular starting configuration (the flat Freudenthal seed), each of the ten such angles varies smoothly as its defining edge lengths are changed. Specifically, the derivative of the angle with respect to a path in edge-length space exists and equals a precomputed quantity called the angle kernel. This is a local, analytic fact: it says the angle responds predictably to infinitesimal changes, not that any global property of the geometry holds.

This theorem is one piece of a larger program in Recognition Science to show that a discrete, combinatorial model of spacetime (Regge calculus) reproduces the continuum theory of general relativity. The framework's machine-checked library of formal theorems has already proved that, in three dimensions, the analogous second variation of the action matches the expected continuum result. In four dimensions, the flat-seed angle derivative is a necessary ingredient for the same kind of match, but it is far from sufficient.

What the declaration does not claim is the full elevation of the nonlinear discrete action to its continuum counterpart. The theorem proves the derivative exists for the flat seed, and a companion theorem proves a directional version of the same fact. But the general, off-flat version of the Freudenthal simplex's Schläfli identity, which would be needed to close the gap, is explicitly marked as absent in the framework's own status flags. The framework records that the pathwise Schläfli property is false, the elevation to the candidate is open, and the recovery of the Einstein-Hilbert action is not achieved.

In plain terms, this theorem is a well-defined, checked calculation about a single angle's derivative in a special case. It is a building block, not the building. The gap between this local fact and the global conclusion about gravity is precisely what the framework declares open. A reader should understand that the existence of this derivative is proved, but that the physical payoff, the discrete theory's convergence to Einstein gravity, is not.

theorem flat_freudenthal_seed_angle_hasDerivAt (k : Fin 10) :
    HasDerivAt (fun t : ℝ => seedDihedralAngle (coordPath k t))
      (angleKernel k) (seedFlatSqEdges k) :=
  hasDerivAt_seedDihedralAngle_coord k

theorem flat_freudenthal_seed_angle_hasDerivAt (k : Fin 10) :
    HasDerivAt (fun t : ℝ => seedDihedralAngle (coordPath k t))
      (angleKernel k) (seedFlatSqEdges k) :=
  hasDerivAt_seedDihedralAngle_coord k

/-- Gate A2-style flat directional kill, re-exported for elevation wiring. -/
theorem flat_freudenthal_directional_schlaefli (v : Fin 10 → ℝ) :
    (∑ h : Fin 10, hingeAreaFlat h * flatDirectionalAngleDeriv v h) = 0 :=
  freudenthal4SimplexFlatDirectionalSchlaefli v

/-- Honesty: full pathwise absent; elevation OPEN; gap stays false. -/
theorem schlafli_does_not_flip_gap :
    regge4DFlatSecondVariationStatus.freudenthal4PathwiseSchlaefliPresent =
        false ∧
      regge4DFlatSecondVariationStatus.schlafliElevationOpen = true ∧
        regge4DFlatSecondVariationStatus.gapActionRecovery = false :=
  ⟨rfl, rfl, rfl⟩

What this page does not claim

The full nonlinear action's second variation equals the candidate in four dimensions. The discrete theory converges to Einstein gravity in four dimensions. The theorem holds for any non-flat Freudenthal simplex.

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/Regge4DFlatSecondVariation.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: