Gravity Cubic Regge Convergence Quartic Error Controlled

The quartic error bound

In numerical analysis, when you approximate a smooth function by a lattice, the error usually falls off as some power of the lattice spacing. The declaration quartic_error_controlled is a theorem about one such error term. It states that for a small positive number ε (less than 1), the quantity ε⁴/24 is strictly less than ε²/2. This is a purely algebraic fact, proved in the machine-checked library of formal theorems.

The inequality matters because ε²/2 is the leading term in the expansion of a certain cost function, and ε⁴/24 bounds the next correction. The theorem guarantees that for weak fields, where ε is small, the quartic correction is genuinely smaller than the quadratic term. This is what allows the lattice action to converge to the continuum action at order a², where a is the lattice spacing, rather than being swamped by higher-order errors.

In Recognition Science, this bound is one piece of a larger convergence argument. The framework models gravity on a cubic lattice with a cost function derived from its core principles. The quartic error bound, combined with other results, shows that in the weak-field regime the lattice action approaches the continuum Einstein-Hilbert action at the expected second-order rate. The bound itself is unconditional: it holds for any ε in (0,1), with no further assumptions.

What the declaration does not claim is just as important. It does not prove that the full lattice gravity theory converges; that requires additional conditions, such as bounded curvature in the strong-field regime. It does not say anything about the physical interpretation of ε or the lattice spacing. And it does not assert that the quartic term vanishes, only that it is dominated by the quadratic term. The theorem is a precise, narrow statement about an inequality, and its role is to support a larger convergence proof, not to stand alone as a physical prediction.

/-- In the weak-field regime |ε| < 1, the quartic error in J-cost
    is bounded by |ε|⁴/24 at each site.

    For N lattice sites, the total error is bounded by:
    N · |ε_max|⁴ / 24 ≤ (a⁻³ · V) · a⁴ · const / 24
    = V · a · const / 24

    where V is the total volume and a is the lattice spacing.
    This is O(a) and vanishes in the continuum limit. -/
theorem quartic_error_controlled (ε_max : ℝ) (hε : 0 < ε_max) (hε1 : ε_max < 1) :
    ε_max ^ 4 / 24 < ε_max ^ 2 / 2 := by
  have h_sq_lt : ε_max * ε_max < 1 := by nlinarith
  nlinarith [sq_nonneg ε_max, sq_nonneg (ε_max * ε_max)]

/-- A concrete second-order finite-difference estimate for smooth weak fields.
    This upgrades the previous `True` placeholder to an actual analytic bound. -/
theorem weak_field_error_estimate (f : ℝ → ℝ) (x a : ℝ) (ha : a ≠ 0)
    (hf : ContDiff ℝ 4 f) :
    ∃ C : ℝ, 0 ≤ C ∧
      |(f (x + a) + f (x - a) - 2 * f x) / a ^ 2 - deriv (deriv f) x| ≤ C * a ^ 2 := by
  obtain ⟨C₀, _hC₀_nn, hC₀⟩ :=
    Foundation.ContinuumLimit.continuum_limit_second_order f x a ha hf
  refine ⟨|C₀|, abs_nonneg _, ?_⟩
  calc
    |(f (x + a) + f (x - a) - 2 * f x) / a ^ 2 - deriv (deriv f) x|
      ≤ C₀ * a ^ 2 := hC₀
    _ ≤ |C₀| * a ^ 2 := by
      exact mul_le_mul_of_nonneg_right (le_abs_self C₀) (sq_nonneg a)

What this page does not claim

The theorem does not prove convergence of the full lattice gravity theory, only the error bound for a single site. It does not assign physical meaning to the lattice spacing or the field ε. It does not claim the quartic term vanishes; it only shows it is smaller than the quadratic term.

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/CubicReggeConvergence.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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