IndisputableMonolith.Materials.Polymer_Entanglement2
Polymer entanglement is treated as a Recognition-cost threshold on domain configurations. The module defines a nonnegative domain cost, a positive canonical threshold, and an inhabited certificate packaging that comparison. Materials workers in the RS ladder cite the certificate when tying chain topology to J-cost. Content is definitions plus elementary positivity; no forcing-chain argument.
claimThe module introduces a domain cost $C$ on polymer configurations, a canonical threshold $\theta>0$, and a certificate type asserting the entanglement criterion $C\ge\theta$ (in Recognition-native units built from the $J$-cost).
background
Recognition Science measures mismatch with the $J$-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), imported here via the Cost layer. Constants supplies the RS time quantum $\tau_0=1$ tick, fixing the native clock against which material rates are later compared.
In the materials domain one assigns to a polymer configuration a scalar domain cost built from that $J$-cost (or a close variant). Entanglement is then the event that this cost clears a fixed positive threshold. The module names that cost, proves it is nonnegative, names the threshold, proves it is positive, and packages the comparison as a certificate type with an inhabited instance.
No continuum polymer-field theory is assumed; the objects are discrete RS-native quantities on the $\phi$-ladder scale.
proof idea
Definition-and-lemma module, not a deep proof development. Domain cost is introduced as a def; equality-at-a-point and nonnegativity are short lemmas. Canonical threshold is a positive constant def with a positivity lemma. The certificate is a structure (or Prop bundle) whose inhabitant is constructed directly from those facts. No induction, no analysis beyond basic real inequalities from Mathlib and Cost.
why it matters in Recognition Science
Places polymer entanglement inside the same cost language used for the forcing chain (T5 $J$-uniqueness onward) and for mass/rung formulae. Downstream materials claims can quote the inhabited certificate rather than re-deriving nonnegativity of the domain cost or positivity of the threshold. No parent theorems are wired yet in the graph (used_by is empty); the module is a leaf that future entanglement or rheology results are expected to import. It does not itself touch T6--T8, RCL identities, or the $\alpha$ band.
scope and limits
- Does not derive entanglement from the forcing chain T0--T8.
- Does not prove a continuum polymer or reptation theorem.
- Does not fix numerical values of elastic moduli or critical molecular weight.
- Does not connect domain cost to measured scattering or rheology data.
- Does not claim uniqueness of the chosen threshold among all possible cutoffs.