IndisputableMonolith.Foundation.MaxwellDemon2DeepFromJCost
Module packaging a Maxwell-demon obstruction derived from the RS J-cost: a nonnegative domain cost, a positive canonical threshold, and an inhabited certificate that the demon cannot beat that threshold. Physicists citing RS thermodynamics or Landauer-style bounds from the cost functional would use it. Structure is definitions plus elementary nonnegativity and positivity lemmas feeding a cert record.
claimFrom the RS cost $J$, the module defines a domain cost $C(\cdot)\ge 0$, a canonical threshold $\theta>0$, and an inhabited Maxwell-demon certificate asserting that any two-deep demon protocol is bounded below by $\theta$ in $J$-units (with $\tau_0=1$ tick).
background
Recognition Science measures mismatch by the unique cost $J$ fixed in the forcing chain (T5): $J(x)=(x+x^{-1})/2-1$. The Cost import supplies that functional; Constants fixes the RS time quantum $\tau_0=1$ tick, so thresholds are dimensionless in tick units.
This Foundation module sits at the interface between that cost and a Maxwell-demon thought experiment: a device that would sort or extract work by discriminating microstates without paying the corresponding recognition cost. The siblings introduce a domain-level cost (nonnegative, and agreeing with evaluation at equality cases), then a canonical positive threshold against which demon protocols are judged.
The certificate bundle MaxwellDemon2DeepCert packages those inequalities into a single inhabited record, so downstream thermodynamics arguments can assume the obstruction as a unit rather than re-proving nonnegativity of $J$-derived costs.
proof idea
Definition-heavy module, not a single deep theorem. It introduces domainCost and proves elementary facts (domainCost_at_eq, domainCost_nonneg) from the imported Cost layer. canonicalThreshold is defined and shown positive. Those pieces are assembled into the certificate structure MaxwellDemon2DeepCert, with cert and cert_inhabited witnessing that the record is realizable. No long tactic scripts: nonnegativity and positivity are short lemmas; the cert is a structure pack.
why it matters in Recognition Science
Gives Foundation a reusable Maxwell-demon obstruction tied directly to $J$, rather than to classical Shannon entropy alone. That matters for RS claims that thermodynamic irreversibility is forced by recognition cost (T5 J-uniqueness and the RCL), including Landauer-type bounds in RS-native units ($\hbar=\varphi^{-5}$, tick $\tau_0=1$).
used_by is empty in the graph snapshot, so this module is presently a leaf certificate: it closes a local demon argument and stands ready for thermodynamics or measurement modules to import the inhabited cert. It does not itself advance T6--T8 (phi, eight-tick, $D=3$), but supplies a cost-side lemma those geometric steps can cite when discussing work extraction or sorting over the eight-tick octave.
scope and limits
- Does not derive J-uniqueness; assumes Cost/Constants.
- Does not model a concrete physical demon apparatus or experimental protocol.
- Does not prove classical Landauer bounds in SI units; works in RS tick units.
- Does not feed T6--T8 (phi fixed point, eight-tick, D=3) directly.
- Does not claim a numerical alpha or mass-ladder consequence.