Gravity Cubic Regge Proof Taylor Coefficients Positive

The Taylor coefficients

The hyperbolic cosine, cosh(x) = (e^x + e^-x)/2, has a power series expansion cosh(x) = 1 + x^2/2 + x^4/24 + x^6/720 + ... . The declaration taylor_coefficients_positive proves that every coefficient in this series, from the x^2 term onward, is a positive real number. More precisely, for any positive integer n, the coefficient of x^(2n) equals 1/(2n)! and is strictly greater than zero. This is a theorem in the framework's machine-checked library of formal theorems, meaning it has been verified by a computer proof checker from the library's axioms.

The statement itself is elementary: the factorial in the denominator is always a positive integer, so its reciprocal is positive. The proof in the library is a one-line application of the positivity tactic. What makes the fact worth stating is its role in a larger argument. The series for cosh appears in the framework's model of gravity on a cubic lattice, where the cost of a recognition event between neighboring lattice points is J(exp(ε)) = cosh(ε) − 1. The positivity of the Taylor coefficients guarantees that the series has the alternating-sign structure needed for the convergence estimates in that model, and it supports the bound |J_log(ε) − ε²/2| ≤ |ε|⁴/20 for small ε.

In Recognition Science, the framework models spacetime as a discrete lattice of recognition events, and the cost function J is forced by the framework's axioms to take the form J(x) = (x + 1/x)/2 − 1. On a cubic lattice, this cost becomes cosh(ε) − 1, and the Euler-Lagrange equation for the action built from this cost involves the hyperbolic sine. The linearization of that equation at zero uses sinh'(0) = cosh(0) = 1, which is another theorem in the same library. The positivity of the Taylor coefficients is a supporting lemma in the proof that the lattice action converges to the continuum action at order a², where a is the lattice spacing.

The declaration does not claim that the Taylor series converges for all real ε; that is a separate fact about cosh, also known. It does not claim that the positivity of the coefficients alone proves the convergence of the lattice model; that requires the full chain of theorems in the library, including the action bound and the continuum limit. It does not claim that the coefficients have any special arithmetic property beyond positivity, such as being prime or having a closed form in terms of π. The declaration is a narrow, verified lemma: the coefficients of the cosh series are positive, and that is all it establishes.

/-- All Taylor coefficients 1/(2n)! are positive — the expansion has
    no sign-changing terms. This ensures monotonic convergence. -/
theorem taylor_coefficients_positive (n : ℕ) (_hn : 1 ≤ n) :
    (0 : ℝ) < 1 / (Nat.factorial (2 * n) : ℝ) := by positivity

/-- All Taylor coefficients 1/(2n)! are positive — the expansion has
    no sign-changing terms. This ensures monotonic convergence. -/
theorem taylor_coefficients_positive (n : ℕ) (_hn : 1 ≤ n) :
    (0 : ℝ) < 1 / (Nat.factorial (2 * n) : ℝ) := by positivity

/-- The J-cost action on each bond approximates the quadratic action
    with error ≤ |ε|⁴/20. This is J_log_quadratic_approx. -/
theorem action_per_bond (ε : ℝ) (hε : |ε| < 1) :
    |J_log ε - ε ^ 2 / 2| ≤ |ε| ^ 4 / 20 :=
  J_log_quadratic_approx ε hε

What this page does not claim

The declaration does not prove the Taylor series converges for all real ε. The declaration does not by itself prove the convergence of the lattice model to continuum gravity. The declaration does not assert any arithmetic property of the coefficients beyond positivity.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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