IndisputableMonolith.Numerics.IntervalProofs
The Numerics.IntervalProofs module supplies six supporting lemmas on exponential monotonicity, power monotonicity, the quadratic equation for phi, and explicit numerical bounds on phi and log phi. Researchers verifying phi-ladder mass formulas or J-cost evaluations in Recognition Science cite these for machine-checked interval arithmetic. Proofs proceed via direct application of Mathlib intervalCases, linarith, and normNum tactics.
claimLemmas establishing that $e^x$ is strictly increasing on $mathbb{R}$, that $x mapsto x^r$ is strictly increasing for $r > 0$, that $phi$ satisfies $phi^2 = phi + 1$, and that $phi$ and $log phi$ lie in explicit closed intervals around their known values.
background
The module sits in the numerics domain and draws on Mathlib results for real analysis, exponentials, logarithms, square roots, and the golden-ratio definition. It supplies the concrete inequalities needed to close numerical steps in the forcing chain after T6 (phi as self-similar fixed point) and before mass-ladder evaluations. No new mathematical objects are defined; the lemmas simply make the imported analytic facts usable inside interval proofs.
proof idea
Each of the six lemmas is proved independently by a short tactic script that invokes intervalCases to split real intervals, followed by normNum or linarith to discharge the resulting arithmetic goals. No inductive arguments or external libraries beyond the listed Mathlib imports are required.
why it matters in Recognition Science
The lemmas feed numerical verification steps that appear in downstream phi-ladder and alpha-band calculations inside the UnifiedForcingChain. They close the gap between the abstract RCL identity and concrete bounds on constants such as $hbar = phi^{-5}$ and $G = phi^5 / pi$.
scope and limits
- Does not prove general properties of the exponential beyond strict increase.
- Does not supply bounds on any constant other than phi and log phi.
- Does not contain definitions or theorems about physical units.
- Does not interface with the J-cost or defectDist functions directly.
- Does not rely on any hypothesis_interface or scaffolding.