IndisputableMonolith.Materials.FatigueLifeFromPhiLadder
Module packaging fatigue-life certificates from the RS phi-ladder and J-cost. Materials theorists cite it to treat cyclic damage as discrete rung accumulation up to a fixed threshold. It defines a non-negative domain cost (vanishing at equilibrium), a positive canonical threshold, and an inhabited FatigueCert. Structure is definitional plus elementary positivity from Cost and Constants.
claimIntroduces a domain cost $C$ on material configurations with $C\ge 0$ and $C=0$ at equilibrium, a positive canonical threshold $\theta>0$, and a fatigue certificate asserting that cyclic loading crosses $\theta$ after a number of cycles fixed by the $\varphi$-ladder.
background
Recognition Science measures mismatch by the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$). The Cost import supplies this functional and its basic algebraic properties; Constants supplies the RS time quantum $\tau_0=1$ tick and the golden ratio $\varphi$ forced as the self-similar fixed point (T6).
In the materials reading, fatigue is accumulation of domain cost under cyclic loading until a canonical threshold is crossed. The phi-ladder discretizes admissible damage increments, so predicted life is a rung count rather than a continuum integral. The module lives in the Materials domain and imports only Mathlib, Constants, and Cost.
proof idea
Definition-and-certificate module, not a deep derivation. It defines domainCost, records non-negativity and vanishing at equilibrium, fixes a positive canonicalThreshold, then assembles the FatigueCert structure with an inhabited instance. Positivity and equilibrium facts are elementary consequences of Cost lemmas and arithmetic on $\varphi$; no forcing-chain or RCL argument is replayed here.
why it matters in Recognition Science
Gives the materials-side interface that turns phi-ladder discreteness into a usable fatigue-life certificate. The graph currently lists no downstream consumers, so the module is a leaf that closes the path from pure Cost theory into applied Materials predictions. It ties the T6 phi fixed point (and, indirectly, the eight-tick octave) to an engineering observable: cycle counts become integer combinations of rungs. Experimental S-N comparison and continuum Paris-law recovery remain outside the present declarations.
scope and limits
- Does not derive continuum Paris-law exponents from first principles.
- Does not claim quantitative agreement with measured S-N curves.
- Does not encode microstructure, grain size, or alloy chemistry.
- Does not replay the T0-T8 forcing chain beyond phi and J-cost.
- Does not fix units conversion from RS ticks to laboratory seconds.