Gravity Analysis Regge4 Dcontinuum Preflight Einstein Hilbert Quadratic4 D On No

The frozen target

The declaration einsteinHilbertQuadratic4D defines a formula for the energy of a weak gravitational field in four dimensions. It multiplies a fixed constant, a coefficient of negative one quarter, and the squared size of the field's deviation from flat space. The formula is a quadratic approximation, the kind used when gravity is weak and fields are small. It is not a full theory of gravity; it is a frozen target for later work to hit.

The declaration is a definition, not a theorem. It chooses what the target will be, and it chooses independently of any lattice or mesh. The constant that appears in it, kappa_einstein, is not a free parameter tuned to match a result. The library proves only that this constant is nonzero, and that the coefficient is exactly negative one quarter. Those are the two facts the definition guarantees.

In Recognition Science, this sits inside a larger preflight step for a four-dimensional continuum limit. The framework models spacetime as a discrete periodic mesh, a four-dimensional torus of side length at least three. The goal is to show that a certain discrete symbol, computed from the mesh, converges to this frozen formula as the mesh grows. That convergence is an open target, not a proved result. The declaration itself does not assert it.

The library is explicit about decoys it rejects. A wrong mesh power, one over N squared instead of one over N to the fourth, is proved different from the correct density weight. A one-orbit symbol and an arbitrary pullback are excluded from the action theorem. These exclusions are proved, but they only clear the field; they do not establish the target.

What the declaration changes is the shape of the work ahead. It fixes the destination so that future computation can be checked against it, without fitting any scale to make the answer come out. The reader can now see the precise formula that a discrete gravity theory must reproduce, and the exact list of things that have not yet been shown about it.

/-- Independently defined continuum EH quadratic on a Frobenius-normalized
TT polarization.  At this preflight stage the functional is the constant
coefficient times the Frobenius pin (already 1 on `IsTTPolarization4D`).
The coupling `kappa_einstein` is recorded as the Recognition field-equation
scale; weak-field quadratic recovery identifies the *symbol* coefficient
with `einsteinHilbertTTCoefficient4D`, not a free lattice normalization. -/
def einsteinHilbertQuadratic4D (E : Mat4) : ℝ :=
  kappa_einstein * einsteinHilbertTTCoefficient4D * frobeniusNormSq E

theorem kappa_einstein_ne_zero : kappa_einstein ≠ 0 :=
  ne_of_gt kappa_einstein_pos

theorem einsteinHilbertTTCoefficient4D_eq :
    einsteinHilbertTTCoefficient4D = -(1 / 4 : ℝ) := rfl

theorem decoy_wrong_mesh_power {N : ℕ} (hN : 2 ≤ N) :
    wrongMeshPowerWeight N ≠ correctTorusDensityWeight N := by
  unfold wrongMeshPowerWeight correctTorusDensityWeight
  have hNne : (N : ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt (lt_of_lt_of_le (by norm_num : 0 < 2) hN))
  intro h
  have hclear :
      ((N : ℝ)⁻¹) ^ 2 * (N : ℝ) ^ 4 = ((N : ℝ)⁻¹) ^ 4 * (N : ℝ) ^ 4 :=
    congrArg (fun t : ℝ => t * (N : ℝ) ^ 4) h
  have hNsq : (N : ℝ) ^ 2 = 1 := by
    field_simp [hNne] at hclear
    exact hclear
  have hcast : (2 : ℝ) ≤ (N : ℝ) := by exact_mod_cast hN
  nlinarith [sq_nonneg ((N : ℝ) - 1)]

What this page does not claim

This declaration does not prove that the discrete mesh symbol converges to the continuum formula. This declaration does not derive the Einstein-Hilbert action from first principles; it defines a quadratic approximation. This declaration does not fix the value of kappa_einstein; it only proves the constant is nonzero.

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/Regge4DContinuumPreflight.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: