Nothing Is Lost: The Conservation Axiom for a Cyclic Universe (Phase 13 of the Unified Loop Framework)

The Unified Loop Framework's first twelve phases reported sixteen quantitative results matching observed phenomena across atomic structure, stellar lifecycles, compact objects, and cosmology, with quantum-scale entanglement as the foundational primitive. This paper formalises the foundational axiom from which those phases derive: configurational information cannot be created and cannot be destroyed, only redistributed across cycles, sectors, and scales. From this single principle eleven theorems follow. The axiom supplies mechanisms for results mainstream physics treats as axiomatic: the factor 4 in Bekenstein-Hawking entropy, the Second Law as cumulative entanglement stretching, and the black-hole information paradox as configurational redistribution rather than loss, and it resolves the Tolman entropy problem of cyclic cosmology through void-and-Legacy shedding. It predicts a universal brake-loosening signature preceding topological reorganisation at every scale; reframes the dark sector on a single derived timeline, the dark-matter cycle count following from an entanglement-decoherence limit while dark energy is cumulative saved time that dominates only in the present late epoch; establishes a three-level nested Russian Dolls coupling of cosmic, galactic, and planetary scales; and identifies the mirror sector as the imagination-accessible partner of waking experience. The paper states falsifiable predictions testing the coupling and the brake-loosening signature, and reports all results at honest strength, distinguishing the few genuine derivations from broader consistency-level support, with the dark-sector and coupling claims stated as order-and-direction rather than precision fits. Five supporting computational files accompany this deposit and reproduce the quantitative results documented in §11: entropyₚroblemₜest. py (Tolman entropy fixed-point Monte Carlo), bekensteinₕawkingᵥ3ₚlanckᵢdentification. py (Bekenstein-Hawking factor 4 derivation), sigmabcrossₛcaleᵥ4ₙeffcoupling. py (cross-scale shape ratio validation), smbhᵣecoilframeworkₜestᵥ3. py (SMBH recoil bimodal distribution test), and test7Aconsistencyᵣeport. md (cross-phase consistency audit).