IndisputableMonolith.Materials.Domain_Wall_Width2
Formalizes domain-wall energetics for magnetic materials in RS-native units: a domain cost built from the J-cost, a positive canonical threshold, and a Néel–Bloch certificate for admissible wall profiles. Materials workers on the RS ladder cite the cost nonnegativity and certificate inhabitance facts. The file is definition-heavy with short positivity and equality lemmas over imported Constants and Cost.
claimDefines a domain cost $C$ on wall configurations, proves $C\ge 0$ and an evaluation identity, fixes a canonical threshold $\theta_*>0$, and packages a Néel–Bloch certificate asserting admissible domain walls meet the RS cost bound tied to $J$.
background
Recognition Science treats condensed-matter scales through the same J-cost that forces the forcing chain (T5: $J(x)=(x+x^{-1})/2-1$). The Cost import supplies that functional; Constants supplies the RS tick $\tau_0=1$ used to normalize lengths and times.
Magnetic domain walls (Bloch vs Néel) are modeled as interfaces whose excess energy is measured by a domain cost assembled from $J$. A canonical threshold marks the scale at which a wall is energetically preferred; the Néel–Bloch certificate is a Prop-level witness that a configuration stays inside the admissible band.
Sibling definitions in the module are exactly those objects: the cost, its pointwise evaluation and nonnegativity, the threshold and its positivity, and the certificate together with an inhabitance fact.
proof idea
Definition module with short supporting lemmas. domainCost is introduced from the imported J-cost; domainCost_at_eq is an evaluation identity; domainCost_nonneg and canonicalThreshold_pos are elementary nonnegativity/positivity arguments. NeelBlochCert and cert package the admissibility Prop; cert_inhabited supplies a witness. No deep tactic scripts: structure is def-plus-positivity over Cost and Constants.
why it matters in Recognition Science
Places magnetic domain-wall width inside the Materials domain of the RS monolith so later gap and rung arguments can quote a certified cost bound rather than an external continuum model. Feeds any downstream materials theorems that need a nonnegative interface cost or a Bloch/Néel admissibility witness (none are wired yet in the graph). Ties wall energetics to the same $J$ that appears in T5 and the Recognition Composition Law, keeping materials scales on the phi-ladder rather than fitted continuum parameters.
scope and limits
- Does not derive a numerical wall width in SI units.
- Does not prove uniqueness of Néel versus Bloch structure.
- Does not connect the certificate to experimental hysteresis data.
- Does not discharge continuum micromagnetic PDEs.
- Does not depend on the eight-tick or D=3 forcing steps directly.