Volume IVa is the first of two parts of Volume IV of the programme Thermodynamics of Arithmetic Action. It collects TAA Lemmas 48-65; the companion Volume IVb contains Lemmas 66-85. The parts form one volume, split only because of the archive's 5 MB file limit. Volumes I-III (ai.viXra:2610.0045v2, 2610.0054, 2610.0052) recast the Riemann Hypothesis (RH) as the absence of negative-norm ("ghost") directions for the Weil form, measured that obstruction, and asked which structure could supply the positivity. Volume IVa tests three families of candidates. Polarizations as vacua and complex structures: Kramers-Kronig causality, the Betti-de Rham comparison, spinor vacua and Krein collisions, the Hardy-space vacuum and its ghost charge, and the complex structure Frobenius chooses for itself. Saddles and positivity ladders: saddles of |zeta| and xi', the Weil positivity ladder, the derivative tower of xi, and ghosts as dents in a flat cone metric. The prime side: class numbers, a visibility criterion for zeros, helicity and the Moebius function, local versus global positivity, and the field with one element. Certified computer-assisted results concern the Davenport-Heilbronn function (a zero-free box, and a lower bound 0.230607 for its de Bruijn-Newman-type constant) and truncated Weil forms (eigenvalue bounds; the Davenport-Heilbronn form is certified indefinite for A >= 3.43). A statement at the start lists exactly what is claimed across both parts, and where. Formalization and computations were AI-assisted, and this is disclosed. RH is NOT proved: no chapter supplies the missing positivity for zeta; most identify why a candidate cannot.