The Saturation Theorem: Axiomatic Constraints on Quantum Gravity from Cosmological Relaxation

DRAFT version. Subject to revisions. We formulate four minimal axioms — finite geometric capacity (A), cooperative reinforcement with antisymmetric extension (B/B'), monotone cosmic control (C), and renormalization-group composition with boundary regularity (D) — that any quantum theory of gravity must satisfy if its macroscopic limit reproduces general relativity with a bounded cosmological relaxation mechanism. The corrected Saturation Theorem is two-step. Axioms A--D determine a one-dimensional Abel-coordinate / boundary-collar structure for cosmological saturation dynamics. The hyperbolic tangent profile follows only after adding the projective boundary-quotient normalization D', namely that the quotient R(x) = (1 + x)/(1 - x) is multiplicative under composition. With D', the canonical representative is S(g) = S_max tanh(alpha u(g)), up to monotone reparametrization. Without D', smooth boundary regularity alone allows non-projective Abel collars and does not uniquely select arctanh. This revision therefore replaces the older shift-regularity uniqueness claim with a more precise guardrail: smooth scale coherence gives the Abel/collar class; the projective quotient selects the tanh normal form. The counter-collar family phi_epsilon(x) = arctanh(x) + epsilon x is explicitly documented as the reason D' must be separated from Axiom D. We compare the corrected axiom set across six quantum-gravity programs: Loop Quantum Gravity, asymptotic safety, string/holographic approaches, causal dynamical triangulations, causal set theory, and noncommutative geometry. The result is a conditional no-go theorem for one-dimensional cosmological saturation channels rather than an unconditional UV-completion theorem. RELATIONS Related: Paper VI — QG-CRM: Ultraviolet Completion (10.5281/zenodo.19352448, DRAFT). The companion paper addresses the open question left by Paper V: which UV completion selects the coefficients and the projective boundary quotient? In the corrected v4.0 reading, Paper VI is not just a parameter-selection follow-up; it must also explain why the projective quotient D' is physically natural. CHANGELOG Version 4.0 changes (2026-05-17): Major mathematical guardrail revision. The previous description that A--D or bare smooth boundary regularity uniquely force arctanh/tanh has been corrected. The theorem now distinguishes A--D => Abel/boundary-collar structure from A--D + D' => projective tanh normal form. A new projective boundary-quotient axiom D' is stated, the phi_epsilon counter-collar is documented, and the discussion, limitations, predictions, and conclusion are recalibrated. A combined EN/DE PDF is added alongside the separate English and German PDFs. The German version was checked for real umlauts. Version 3.0 changes (2026-03-27): Added complete German translation (Paper5_DE.pdf). Version 2.0 changes (2026-03-26): Major revision after 7-step adversarial review chain. Added the earlier shift-regularity proof attempt, Axiom B' as an explicit sub-axiom, monotonicity corollary, boundary-regularity condition, QG verification table, CDT as sixth QG program, Swampland and EFT-of-DE connections, and verified references. The v4.0 revision supersedes the old shift-regularity uniqueness claim. FRAMEWORK CRM Papers I-IV: doi:10.5281/zenodo.18728935Code and paper sources: github.com/research-line/crm-cosmology/tree/main/papers/extensions