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

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. Changes in Version 4.4 (July 2026) Maintenance candidate after the post-v4.3 Route-3 audit back-transfer and disclosure hygiene pass. No claim upgrade; the conditional mathematical core of Paper V is unchanged. Series-status correction: The historical direct trace-coupling route is now explicitly treated as a failed audit, not as a derived CRM Lagrangian. Route-3 scope: Route 3 is framed as an open action candidate with separate shift, vector, and saturation sectors; the older positive fit is classified as a generic EFT proxy rather than validation of Route 3. Physical guardrails: The statement cT=1 is restricted to the Route-3 TT sector; scalar stability, on-shell FLRW response, lensing, screening, and cluster phenomenology remain open gates. AI disclosure: English and German disclosure text was neutralized to document non-authorial process support without model/provider lists, acknowledgments, coauthorship language, truth-source wording, or independent-validation claims. DE/EN: English, German, and combined PDFs were rebuilt synchronously (16/17/33 pages); German PDF text extraction confirmed real umlauts. Changes in Version 4.3 (July 2026) No claim upgrade; the two-step conditional result is unchanged. The public GitHub paper files were synchronized before the Zenodo upload. Hypothesis surface: D± (one-dimensional reversible channel) is now stated as a separate hypothesis in the theorem head rather than implicitly; D' is consistently labeled a projective double-boundary quotient assumption, not a consequence of A--D. Proof repair: The Abel-coordinate step no longer uses the category-mistaken φ'(0)=c1 shortcut; it is derived via the invariant vector field a(x)=∂2C(x,0) with φ=∫dt/a(t). Saturation ODE: Written in the additive normal-form coordinate u with an explicit affinity caveat (constant k requires an affine identification u(a), which is not proven). Notation: The channel variables r/u/x are separated with a dedicated notation paragraph; non-redundancy examples are downgraded to separation sketches. Structure: The exclusion section is recast in terms of non-projective composition laws instead of profile names; the quantum-gravity table is reframed as a compatibility map of motifs (no derivation claim). DE/EN: The German version is fully synchronized to English parity (including the Abel-proof repair); English, German, and combined PDFs rebuilt (16/17/33 pages). Changes in Version 4.2 (June 2026) Maintenance release after the June 2026 style, strict-review, and proof/paper-math passes. The public GitHub paper files were synchronized before the Zenodo upload. Proof status: The main claim remains two-step: A--D/B'/D± give the Abel/boundary-collar class, while D' selects the projective tanh normal form. Argument hygiene: The monotonicity derivation now explicitly chooses the additive coordinate of the one-dimensional control composition before differentiating the composition law. Claim calibration: The former axiom-independence wording is weakened to non-redundancy; the unbounded example is only an inner algebraic composition example after removing the capacity bound. Evidence ledger: The observational-test ledger separates theorem-level tests, CRM fit, D'-defect, parameter derivation, and oscillatory residuals as distinct evidence channels. DE/EN: English, German, and combined PDFs were rebuilt synchronously; the German text track was checked for real umlauts. FRAMEWORK CRM Papers I-IV: doi:10.5281/zenodo.18728935Code and paper sources: github.com/research-line/crm-cosmology/tree/main/papers/extensions