Gravity Cubic Regge Convergence Phi Exponential Growth

Growth bound

The golden ratio, usually written φ, is the number that solves the equation φ² = φ + 1, which gives φ = (1 + √5)/2 ≈ 1.618. It is famous for appearing in pentagons, in the Fibonacci sequence, and in art and architecture. The declaration phi_exponential_growth in the Recognition Science framework's machine-checked library of formal theorems records a simple but powerful property: φ is greater than 1. This means that the sequence φ^N, where N is a positive whole number, grows without bound, and it grows faster than any polynomial in N. In particular, for any constant C, no matter how large, there comes a point where φ^N exceeds C times N³.

Why does this matter for gravity? In numerical approaches to gravity, one often approximates continuous spacetime by a fine lattice or grid. A central question is whether the approximate calculation converges to the true answer as the grid spacing shrinks. A known difficulty is that curvature can concentrate in small regions, and the lattice must be fine enough to resolve such concentrations. In three dimensions, the number of lattice points grows like N³ as the grid is refined. The bound φ^N > C·N³ says that the resolving power of the lattice, measured by φ^N, grows faster than the number of points needed to represent a concentration. Curvature cannot outrun the grid.

In Recognition Science, this bound is one of three structural advantages of a cubic lattice used for a discrete version of gravity. The other two are that all cubes are identical, so the shape quality is perfect, and that an eight-tick periodicity provides a natural cutoff for high-frequency modes. The growth bound is the piece that guarantees the lattice resolution outpaces any polynomial growth of curvature concentration. The framework's library proves the inequality directly from the definition of φ, and it uses it in a certificate that combines weak-field convergence, which holds unconditionally for smooth fields, with strong-field convergence, which holds when a curvature bound is assumed.

The declaration does not, by itself, prove that lattice gravity converges. It is one ingredient in a larger convergence argument. It does not specify how large N must be for the inequality to hold; it only says that such an N exists. It does not apply to grids with a different refinement scheme where resolution does not grow exponentially. And it makes no statement about the physical value of the lattice spacing or about whether the continuum limit is unique. The declaration is a number-theoretic fact about φ, with a consequence for resolution, not a complete theory of quantum gravity.

What the bound changes is the shape of a proof. Without it, a convergence argument for lattice gravity would need to control curvature concentrations by hand, case by case. With it, one has a uniform guarantee: no matter how sharply curvature piles up, a sufficiently refined lattice can always see it. That is the practical payoff of a simple inequality about a famous number.

/-- φ > 1, so φ^N → ∞ as N → ∞. -/
theorem phi_exponential_growth : 1 < phi := one_lt_phi

/-- 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

/-- The RS cubic lattice has unit shape quality. -/
theorem rs_cubic_shape_quality : cubic_shape_bound = 1 := rfl

What this page does not claim

The declaration does not prove that lattice gravity converges. The declaration does not give a concrete value of N for which the inequality holds. The declaration does not apply to lattices whose resolution does not grow exponentially.

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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