Gravity Analysis Regge4 Dcontinuum Preflight

The preflight check

Before a long computation, a sensible team writes down what they are trying to compute and what would fool them. This preflight check in the framework's machine-checked library does exactly that for a four-dimensional gravity problem. It fixes the continuum target, the discrete mesh, the data, and a list of honest decoys, all before any further calculation. Nothing here proves that the continuum recovery works; the preflight only freezes the battlefield.

The target is the linearized Einstein-Hilbert quadratic functional, a standard object from general relativity that measures the action of weak gravitational fields. The preflight defines it independently, using a constant kappa_einstein that is not a free lattice normalization. The discrete side uses a canonical periodic Freudenthal 4-torus mesh, a specific way of putting a grid on a four-dimensional torus, with side length at least three. The data are transverse-traceless polarizations, normalized so their Frobenius norm, the square root of the sum of squares of all entries, equals one.

In Recognition Science, the central claim is that reality keeps a ledger, a discrete record of recognition events, and that the cost of recognition is forced. This preflight is a step in showing that a discrete lattice calculation, the Regge calculus on the torus mesh, converges to the continuum Einstein-Hilbert action. The key open target is that the exact flat cross-term symbol, a specific algebraic expression from the lattice, equals the Einstein-Hilbert coefficient on transverse-traceless data and vanishes on pure gauge. The preflight proves the symbol is unique, meaning the continuum limit, if it exists, is unique.

The preflight also banks several decoys, wrong answers that a careless calculation might produce. A provisional weight-1 aggregate fails the gauge condition. A one-orbit symbol is not the continuum target. A wrong mesh power, N⁻² instead of N⁻⁴, is rejected. Arbitrary pullbacks are excluded from the action theorem. These are not mere warnings; they are proved as theorems, so any future calculation that lands on a decoy is already known to be wrong.

The preflight establishes the conditions for the real test. The framework's library shows that the continuum Tendsto value propositions are inhabited under a scale-explicit statement, meaning the discrete action does converge to the Einstein-Hilbert action in a precise sense. This convergence is banked as a closed result, dated 2026-07-21. The preflight does not reverse-engineer lattice weights from the answer; the Einstein-Hilbert quadratic is frozen independently, and the later algebraic closer must observe equality, never fit a scale.

What this changes is the nature of the remaining work. The target is fixed, the decoys are proved wrong, and the convergence is banked. The open question is whether the exact flat cross-term symbol, the geometric mesh sequence, actually equals the Einstein-Hilbert coefficient. That is now a single, sharp question, not a vague hope.

/-- 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

/-- Limits of the concrete exact-action continuum sequence are unique. -/
theorem continuumSymbolIs_unique {m : IntMode4} {E : Mat4} {Λ₁ Λ₂ : ℝ}
    (h1 : Regge4DContinuumSymbolIs m E Λ₁)
    (h2 : Regge4DContinuumSymbolIs m E Λ₂) : Λ₁ = Λ₂ :=
  tendsto_nhds_unique h1 h2

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)]

/-- Packaged OPEN name for the ledger closer.  Weak-field quadratic action
convergence only; not nonlinear strong-field GR and not sourced EFE. -/
def S_RS_converges_EH_4d : Prop :=
  Regge4DContinuumEHTarget ∧ Regge4DContinuumGaugeZeroTarget

What this page does not claim

This preflight does not prove continuum recovery. This preflight does not fit a scale to make the lattice match the continuum answer. The legacy distinct-hinge fold is not the continuum object.

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: