IndisputableMonolith.Materials.MagnetizationSaturationFromJCost
Materials module that ties magnetic domain energetics to the RS J-cost and packages a saturation certificate. Domain wall or misalignment cost is written as a nonnegative J-type functional of a local ratio, with a positive canonical threshold at which magnetization is declared saturated. Physicists working on RS condensed-matter predictions would cite the certificate and the nonnegativity lemmas. The file is mostly definitions plus short algebraic facts imported from Cost and Constants.
claimDefine a domain cost $C$ built from the RS cost $J$, prove $C\ge 0$ and evaluate it on the equality locus, fix a positive canonical threshold $\theta_*>0$, and package a saturation-magnetization certificate asserting that magnetization saturates once the domain cost crosses $\theta_*$.
background
Recognition Science measures mismatch by the unique symmetric cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law and the T5 uniqueness step. In materials, local spin or domain misalignment is treated as a positive real ratio; the module lifts that ratio into a domain cost built from $J$.
The module sits in the Materials domain and imports only Constants (RS-native units, including the tick $\tau_0$) and Cost (the $J$ infrastructure). Sibling objects name the pieces: domainCost and its evaluation/nonnegativity lemmas, a positive canonicalThreshold, and a SatMagnetizCert inhabited by a concrete certificate. No continuum micromagnetics or exchange stiffness is assumed; the cost is purely the RS scalar $J$ on a dimensionless ratio.
proof idea
Definition-heavy module with short supporting lemmas rather than a long derivation. Domain cost is defined by composing the imported $J$-cost with a domain ratio; nonnegativity and the on-equality evaluation are one-line consequences of the corresponding $J$ facts from Cost. The canonical threshold is a positive constant (positivity proved directly). The saturation certificate is a structure packing those ingredients; inhabitance is by exhibiting the canonical data. No deep tactic proof is required beyond algebra and the Cost API.
why it matters in Recognition Science
Gives the Materials layer a J-native notion of magnetic saturation instead of a phenomenological $M_s$. Downstream work that needs a certified saturation scale (domain energetics, hysteresis bounds, or RS-native material constants) can import the certificate rather than re-deriving nonnegativity of domain cost. The construction is consistent with the global forcing chain: cost uniqueness (T5) and the same $J$ used for mass and coupling ladders. Used-by edges are empty in the current graph, so this file is a leaf provider for later condensed-matter theorems rather than a step inside the T0–T8 chain itself.
scope and limits
- Does not derive exchange stiffness, anisotropy constants, or Landau–Lifshitz dynamics.
- Does not prove experimental $M_s$ values; only a J-cost certificate structure.
- Does not treat temperature, critical exponents, or renormalization-group flow.
- Does not claim uniqueness of the canonical threshold beyond the module’s definition.
- Does not connect to electromagnetic units beyond imported RS Constants.