Gravity Analysis Regge Exact Flat Hessian Symbol4 D Exact Hessian M2 Div Identit

The division identity

The declaration exactHessian_m2_div_identity is a theorem about arithmetic with real numbers. It states that for any real numbers a, r, and s, with s not equal to zero, the expression (a*s + r) / s equals a + r / s. In plain words, it is the standard rule for splitting a fraction whose numerator contains a common factor in one term: the part a*s / s simplifies to a, leaving the remainder r / s. The proof is a direct application of algebraic simplification, and the condition s ≠ 0 is what makes the division valid.

This identity is a ledger, a discrete record of events or in this case a formal mathematical statement, of a very specific and limited kind. It does not, by itself, say anything about the physics of gravity, the Regge calculus, or the behavior of a Hessian matrix. It is a piece of the formal infrastructure that the framework's machine-checked library of formal theorems uses to manipulate algebraic expressions. The theorem's entire content is the algebraic equivalence; it is a tool, not a result about the physical world.

What the surrounding module does is use this identity as a step within a larger, still incomplete, investigation. The module defines a model for a flat-space gravitational action, the exact flat Hessian symbol, a specific quadratic form on plane-wave edge strains. It records measured certificates from a Python oracle for certain modes, and it proves exact values for a few chosen faces: the coefficient for a unit Frobenius norm on transverse-traceless modes is exactly -1/8, and the gauge mode coefficient is exactly 0. These are discrete, bookkeeping facts, not a general continuum result.

The identity itself, however, is not the star. It is a lemma that helps rewrite expressions. The module is explicit about what remains open: there is no general algebraic table of coefficients, the ledger statement for convergence to the Einstein-Hilbert action is not inhabited, and the limit as the lattice spacing goes to zero is not proved. The theorem exactHessian_m2_div_identity establishes only the algebraic rule, and nothing more.

/-- Structural division identity used by the cosine two-jet Tendsto route:
`(a * s + r) / s = a + r / s` when `s ≠ 0`. -/
theorem exactHessian_m2_div_identity
    (a r s : ℝ) (hs : s ≠ 0) :
    (a * s + r) / s = a + r / s := by
  field_simp [hs]

/-- Structural division identity used by the cosine two-jet Tendsto route:
`(a * s + r) / s = a + r / s` when `s ≠ 0`. -/
theorem exactHessian_m2_div_identity
    (a r s : ℝ) (hs : s ≠ 0) :
    (a * s + r) / s = a + r / s := by
  field_simp [hs]

/-- Structural division identity used by the cosine two-jet Tendsto route:
`(a * s + r) / s = a + r / s` when `s ≠ 0`. -/
theorem exactHessian_m2_div_identity
    (a r s : ℝ) (hs : s ≠ 0) :
    (a * s + r) / s = a + r / s := by
  field_simp [hs]

What this page does not claim

This theorem does not prove any property of the Hessian matrix or the Regge action. This theorem does not establish the continuum limit or any convergence result. This theorem does not provide a general algebraic table of coefficients for all modes.

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/ReggeExactFlatHessianSymbol4D.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.

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