IndisputableMonolith.Materials.ShapeMemoryAlloyFromJCost
Models shape-memory alloy (SMA) switching as a threshold crossing of the Recognition Science J-cost on a two-domain configuration space. Materials theorists working in the RS stack would cite the certificate that the canonical threshold is positive and that a non-negative domain cost attains equality at the identity. The module is mostly definitional: it packages a cost, a threshold, and an inhabited certificate record.
claimOn a two-domain configuration, define a non-negative domain cost $C$ built from the RS $J$-cost $J(x)=(x+x^{-1})/2-1$, a positive canonical threshold $\theta_*>0$, and a certificate record asserting $C\ge 0$, $C(e)=0$ at the identity configuration, and that SMA switching is the event $C\ge\theta_*$.
background
Recognition Science derives dynamics from a unique cost $J$ fixed by the Recognition Composition Law and the forcing chain (T5: $J(x)=\cosh(\log x)-1$). In materials language, a shape-memory alloy has two metastable domains (austenite/martensite analogues) whose relative mismatch is scored by that same $J$.
This module sits in the Materials domain and imports only Constants (RS time quantum $\tau_0$) and Cost (the $J$ infrastructure). It introduces a domain cost on configurations, proves non-negativity and vanishing at the equal-domain point, and fixes a positive canonical threshold against which switching is declared.
The local objects are therefore: a domain cost functional, its value and non-negativity lemmas, a positive threshold, and an SMA certificate record bundling those facts.
proof idea
Definition-heavy module with short supporting lemmas rather than a deep proof tree. The domain cost is defined from $J$; equality at the identity and non-negativity are immediate from the corresponding properties of $J$. The canonical threshold is a positive constant (positivity is a one-line fact). The SMA certificate is a structure packing those lemmas; inhabitance is by direct construction of the record.
why it matters in Recognition Science
Gives the RS materials stack a first concrete alloy model: SMA hysteresis as $J$-cost threshold crossing, not as an ad-hoc Landau free energy. No downstream consumers are wired yet in the graph (used_by empty), so the module is a leaf that future microstructure or phase-boundary theorems can import. It ties materials phenomenology to the same T5 $J$-uniqueness used for particles and constants, keeping the cost functional universal across domains.
scope and limits
- Does not derive experimental SMA transition temperatures or hysteresis widths.
- Does not model multi-variant martensite, grain boundaries, or continuum elasticity.
- Does not prove uniqueness of the threshold beyond the chosen canonical positive constant.
- Does not connect to the eight-tick octave or $D=3$ forcing steps.
- Does not supply dynamics or time evolution, only a static cost-threshold certificate.