Add thermodynamic-foundations companion to the discontinuity analysis

[maresb]

A self-contained, first-principles derivation of the moist-parcel entropy that tcpyPI holds fixed on ascent (utilities.entropy_S), why it is conserved, and how the temperature inversion recovers T from it. Companion to discontinuity_analysis.tex; builds with latexmk -pdf in the same setup (now additionally needs the bbm LaTeX package, in TeX Live's fonts-extra collection).

Motivated by a series of questions probing the foundations of the parcel thermodynamics behind the pressure solver. Every formula is tied back to the specific function/constant it corresponds to in the source (constants.py, utilities.py, pi.py), and every figure is generated from those same formulas by scripts/foundations_figs.py — the inversion figure by running the solver's own transcribed algebra, cross-checked against the source docstring examples.

Contents

The statistical section runs in four sweeps: derivation → audit of the postulate → the canonical computation → the microcanonical mirror.

• Statistical origin — the operator picture (Z = Tr e^{-βĤ}, spectrum as the {E_i}, ensembles as Legendre transforms, the symmetric-function encoding of exchange statistics spelled out); the Boltzmann distribution derived from energy conservation with its four assumptions stated; the canonical partition function and indistinguishability; Sackur–Tetrode with every step, the fate of the two 5/2's, and the polyatomic generalisation to s = c_p ln T − R ln p.
• The reservoir from the inside — canonical typicality (Popescu–Short–Winter / Goldstein–Lebowitz–Tumulka–Zanghì): the partial trace defined via the product-operator rule, ρ_S = Tr_E |φ⟩⟨φ| exhibited explicitly for n oscillators (diagonal by the conservation law), the PSW bound in closed form (~e^{-nσ(μ)/2}, thermal to 28 digits by n = 100), the Planck limit recovering the reservoir section's Taylor truncation as an explicit O(1/n) step, the exceptional product-Fock states where typicality fails maximally, and why the integrable oscillator fails ergodicity/ETH — with an FPUT simulation showing the integrable freeze vs. mixing-plus-recurrence.
• The flat-prior audit — the equal-a-priori postulate demoted to a gauge choice: arbitrary weights on the shell leave all physics unchanged iff the bath-conditioned tilt obeys |d ln w̄/dε| ≪ β; a four-tier hierarchy (conserved-total tilt = exact gauge; generic randomness of any magnitude washed out by bath averaging, with the Haar weights of the typicality figure as the worked demonstration; smooth sub-β tilt = subleading; observable-conjugate tilt = a genuine temperature shift, the one dangerous prior).
• The S(E) ↔ Z(β) bridge (as capstone, after the canonical computation) — density of states vs. windowed count kept distinct; Z as Laplace transform / generating function ((1−x)^{−n} for the SHO); the thermodynamic limit as a Legendre transform; why Z is the elegant object (convolution → product dissolves the reservoir); non-concave S(E), Maxwell construction, and ensemble inequivalence; hard vs. soft constraints as Laplace duals guarded by the 1/√N and N_S/N_B separations of scale.
• Classical scaffolding — entropy as a state function via the 1/T integrating factor (Carathéodory).
• The parcel as a Dalton mixture — per-dry-air bookkeeping, partial pressures (ev/rv as mutual inverses), Gibbs additivity.
• Phase equilibrium — g_v = g_ℓ → Clausius–Clapeyron and the L_v/T vaporisation entropy; Kirchhoff's law → Lv; the CC integral → es_cc.
• Assembly — into Emanuel (1994) eq. 4.5.9 = entropy_S, term for term.
• Conservation — on the reversible adiabat (adiabatic ⇒ no flux, reversible ⇒ no production), with the Liouville / adiabatic-theorem microscopic reading, cross-linked to the typicality section as the two faces of the closed-system stance.
• Moist adiabat & inversion — ds=0 → the path; buoyancy and the CAPE integral in the exact form ∫ R_d ΔT_ρ d ln p the code accumulates; the well-posed inversion s(T,p)=s₀ ↦ T, whose monotonicity ∂s/∂T > 0 and guarded Newton step match solve_temperature_from_entropy — drawn with the code's own formulas.

Notes

• Base branch is discontinuity-analysis (where the writeup lives).
• Adds thermodynamic_foundations.tex and the compiled .pdf (28 pp.), eight figures under figures/fig_sf_* (pdf+png, matching how the discontinuity figures are tracked), and their generator scripts/foundations_figs.py. Build is clean: 0 overfull boxes, no undefined references.
• A "notational hazards" remark collects every cross-section symbol collision; the partial trace, trace norm, Haar measure, Lévy's lemma, and effective dimension are defined at first use.

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