Gravity Continuum Manifold Emergence Weak Field Correction Bound

The weak-field bound

In general relativity, the presence of a gravitational field changes the distance between two nearby events. The simplest way to describe that change is the weak-field approximation, where the field is small and the metric is the flat Minkowski form plus a small correction. The Recognition Science framework, which starts from a discrete ledger of recognition events and derives continuum spacetime, proves a precise bound on how large that correction can be. The theorem weak_field_correction_bound states that for any value of the gravitational potential Φ and any coordinates (t, x, y, z), the absolute difference between the weak-field interval and the flat Minkowski interval is at most 2|Φ| times (t² + x² + y² + z²).

This is a formal statement, machine-checked in the framework's library of formal theorems. It is not a numerical approximation with a fitted parameter; it is an inequality that holds for all real inputs. The bound is linear in the potential Φ, which means that as the field grows, the possible deviation from flat spacetime grows in proportion. The theorem is part of a larger chain that derives the Lorentzian signature, the speed of light, and the Einstein field equations from the discrete ledger structure, but the bound itself is a standalone result about the weak-field interval.

What the theorem does not claim is that the weak-field interval is a good approximation to the full curved metric. It only bounds the difference between the weak-field interval and the flat Minkowski interval. It does not say that the weak-field approximation is valid for all potentials; the bound grows without limit as |Φ| grows, so the approximation is only useful when |Φ| is small. It also does not claim that the weak-field interval itself is the exact metric of any physical spacetime; it is a specific functional form, and the theorem only constrains its deviation from flatness.

/-- **THEOREM (Perturbation Bound)**: The metric correction is bounded by
    2|Φ| times the displacement norm-squared. -/
theorem weak_field_correction_bound (Φ t x y z : ℝ) :
    |weak_field_interval Φ t x y z - minkowski_form t x y z| ≤
    2 * |Φ| * (t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2) := by
  suffices h : |weak_field_interval Φ t x y z - minkowski_form t x y z| =
    2 * |Φ| * (t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2) from le_of_eq h
  have hdiff : weak_field_interval Φ t x y z - minkowski_form t x y z =
    -(2 * Φ * (t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2)) := by
    unfold weak_field_interval minkowski_form; ring
  rw [hdiff, abs_neg]
  have hS : (0 : ℝ) ≤ t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 := by positivity
  rw [show (2 : ℝ) * Φ * (t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2) =
    (2 * Φ) * (t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2) from by ring,
    abs_mul, abs_of_nonneg hS]
  congr 1
  rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)]

What this page does not claim

This does not claim that the weak-field interval is a valid approximation for large potentials. This does not claim that the weak-field interval is the exact metric of a physical spacetime. This does not claim that the bound itself is derived from the Einstein field equations.

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

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