Gravity Cubic Regge Convergence Exponential Defeats Cubic

A growth comparison

In numerical analysis, when you approximate a continuous problem on a grid, the grid must resolve the features you care about. If a quantity can concentrate into a tiny region, the grid spacing must be fine enough to see it. The declaration exponential_defeats_cubic is a machine-checked theorem stating a precise version of that idea for a specific lattice: for any positive constant C, there exists a grid refinement level N such that C times N cubed is less than phi to the power N. Here phi is the golden ratio, approximately 1.618. The statement is purely about real numbers and natural numbers: exponential growth in N eventually overtakes any fixed polynomial growth in N, here the cubic N cubed. The proof is a direct appeal to a previously established growth bound in the framework's machine-checked library of formal theorems.

The inequality matters because the cubic lattice model of gravity under discussion has a resolution that grows like phi to the N, while the number of curvature concentration sites in three dimensions grows like N cubed. The theorem guarantees that, for a sufficiently fine grid, the lattice resolution outpaces the worst-case concentration of curvature. This is the load-bearing step in the argument that the model's convergence to a continuum theory is controlled. It is not a statement about physics directly; it is a statement about the growth rates of two sequences of numbers. The physical interpretation, that curvature cannot concentrate faster than the lattice can resolve, is a gloss on the mathematics, not a separate theorem.

The declaration does not claim that the lattice model of gravity converges to the Einstein-Hilbert action in all regimes. Convergence in the weak-field regime is established by a separate, unconditional error estimate, and convergence in the strong-field regime is conditional on a bounded-curvature assumption. The exponential growth theorem supplies one ingredient in the convergence certificate, not the whole certificate. It also does not claim that the golden ratio itself has any direct physical meaning in this context; it is simply the base of the exponential sequence that happens to arise from the framework's structure. The theorem is a fact about growth rates, and its role is to rule out a particular failure mode, not to prove that the model is correct.

/-- The growth hierarchy: exponential beats polynomial.
    For large enough N, φ^N > N³.
    This is proved in GrowthBounds; we record the consequence. -/
theorem exponential_defeats_cubic (C : ℝ) (_hC : 0 < C) :
    ∃ N : ℕ, C * (N : ℝ) ^ 3 < phi ^ N := by
  exact IndisputableMonolith.Foundation.GrowthBounds.phi_exp_defeats_cubic C _hC

What this page does not claim

The theorem does not establish convergence of the lattice model to the Einstein-Hilbert action in the strong-field regime. The theorem does not assign any direct physical meaning to the golden ratio in this context. The theorem does not prove that the lattice model is a correct description of gravity.

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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