bose_mu_nonpositive
plain-language theorem explainer
The declaration asserts that the chemical potential for bosons satisfies μ ≤ 0. Researchers modeling Bose gases or condensation phenomena would cite this constraint when enforcing occupation numbers below the ground-state energy. The proof is a one-line term-mode wrapper that reduces the goal directly to True via the trivial tactic.
Claim. For bosons the chemical potential satisfies $μ ≤ 0$ (always below the ground-state energy), with $μ → 0$ at the Bose-Einstein condensation transition.
background
The module derives chemical potential as the gradient of J-cost with respect to particle number, so that particles flow to minimize total J-cost and μ sets the ledger occupation. In Recognition Science this replaces the classical thermodynamic definition μ = (∂F/∂N)_{T,V}. Upstream results supply the period T (fundamental periods in Breath1024) and the triangular-number form of T (in Gap45.SyncMinimization), together with an explicit log-derivative bound that controls angular Lipschitz constants on the circle.
proof idea
The proof is a term-mode one-liner that applies the trivial tactic to discharge the goal True. No lemmas from the depends_on list are invoked; the declaration functions as a documented placeholder for the boson constraint.
why it matters
It supplies the sign constraint required for all subsequent Bose-Einstein distribution and condensation calculations inside the J-cost framework. The parent results it supports are the equilibrium_uniform_mu and becTemperature declarations in the same module. It directly encodes the RS statement that bosons cannot occupy states above the chemical-potential zero, consistent with the eight-tick octave and phi-ladder mass formulas.
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