Gravity Analysis Regge4 Dcontinuum Preflight Decoy One Orbit M2 Is Not Continuum

The one-orbit decoy

In numerical relativity, the discrete mesh is a stand-in for continuous spacetime. The finer the mesh, the closer the discrete calculation should come to the true continuum answer. But the path from discrete to continuum is not automatic: a quantity computed on one mesh spacing may look meaningful and still fail to represent the limit. The Recognition Science framework's library of machine-checked theorems records this caution as a formal warning, called a decoy, inside its gravity analysis module.

The specific declaration, decoy_one_orbit_m2_is_not_continuum_target, states that a symbol computed from a single wave orbit on the mesh is not the continuum target. The continuum target is the exact flat cross-term symbol, which is the object that must match the Einstein-Hilbert coefficient in the limit. The one-orbit symbol is a decoy: it is a quantity that can be computed but that does not serve as the goal of the convergence proof. The framework's library proves this by distinguishing the one-orbit object from the continuum object, and by showing that the continuum symbol is unique when it exists.

This decoy is one of several honesty checks in the module. Others include a wrong mesh power, where the density weight N⁻² is rejected in favor of the correct N⁻⁴, and a provisional weight-1 aggregate that fails the gauge condition. The framework also excludes arbitrary pullbacks from the action theorem, meaning the continuum result cannot be obtained by an arbitrary change of variables. These checks are part of a preflight procedure: they freeze the target and the mesh before further computation, so that later work does not accidentally install a fitted scale or reverse-engineer lattice weights from the Einstein-Hilbert answer.

The declaration does not claim that the continuum recovery has been achieved. The module is a preflight, and the docstring is explicit that nothing here proves continuum recovery. The continuum target remains an open problem, with the exact flat cross-term symbol still to be shown equal to the Einstein-Hilbert coefficient on transverse-traceless polarizations and vanishing on pure gauge. The one-orbit decoy is a guardrail, not a result: it tells the reader which objects are not the goal, so that the eventual proof does not mistake a computable intermediate for the true limit.

theorem decoy_arbitrary_pullback_excluded : ArbitraryPullbackExcluded :=
  trivial

/-- Re-export: exact-action continuum symbol on side `torusSide j`. -/
abbrev finiteExactReggeSymbol := Regge4DExactActionSymbol.finiteExactReggeSymbol

/-- 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)]

What this page does not claim

The declaration does not prove that the one-orbit symbol equals the continuum target. The declaration does not establish that the continuum recovery has been achieved. The declaration does not claim that the one-orbit symbol is the correct object for any other purpose.

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)

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