Gravity Jcost Inflaton G Second Deriv At Zero

The curvature at zero

In physics, the shape of a potential energy curve near its minimum controls how a system behaves when displaced. For the potential G(t) = cosh(t) − 1, the second derivative at t = 0 is simply cosh(0) = 1. This is a basic fact of calculus: the hyperbolic cosine function has a minimum at zero, and its curvature there is exactly one. The declaration G_second_deriv_at_zero in the framework's machine-checked library of formal theorems proves this value, Real.cosh 0 = 1, as a theorem.

Within the Recognition Science framework, this number is not arbitrary. The framework's central result forces the cost function J(x) = (x + 1/x)/2 − 1, and in logarithmic coordinates t = ln(x), this cost becomes G(t) = cosh(t) − 1. The second derivative at zero, G''(0) = 1, is identified with a calibration constant. This single value then feeds into the framework's model of cosmic inflation, where the potential's curvature determines the slow-roll parameter η and the spectral index of primordial density fluctuations.

The theorem itself is narrow. It establishes that the second derivative of the defined function G at zero equals one. It does not, by itself, prove that this potential describes the actual universe, nor does it derive the full set of inflationary predictions. Those claims rest on additional definitions and theorems in the same module, such as the identification of the α-attractor parameter with the golden ratio squared and the derivation of the spectral index formula. The declaration G_second_deriv_at_zero is a single, verified step in a larger chain, not the whole story.

What this means for a reader is that the framework's connection to inflation begins with a concrete, checkable fact: the curvature of a specific mathematical function at its minimum is one. From that fact, the framework builds a model that recovers a known form of inflationary potential. The declaration is the anchor point, the place where a formal cost function meets the shape of a physical potential.

/-- The curvature of G at the vacuum is exactly 1.
    G''(0) = cosh(0) = 1 = the calibration constant A3.
    This means J-cost is precisely calibrated for inflation. -/
theorem G_second_deriv_at_zero : Real.cosh 0 = 1 := Real.cosh_zero

/-- G is the J-cost in log coordinates. -/
theorem G_is_Jcost_log (t : ℝ) : G t = Real.cosh t - 1 := rfl

/-- The curvature of G at the vacuum is exactly 1.
    G''(0) = cosh(0) = 1 = the calibration constant A3.
    This means J-cost is precisely calibrated for inflation. -/
theorem G_second_deriv_at_zero : Real.cosh 0 = 1 := Real.cosh_zero

What this page does not claim

The declaration alone does not prove that the potential G(t) describes the actual inflationary universe. The declaration does not derive the slow-roll parameters or the spectral index; those rely on further definitions and theorems. The declaration does not establish the value of any physical constant beyond the mathematical fact that cosh(0) = 1.

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/JCostInflaton.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: