Relaxation-Driven Cyclic Cosmology: A Closing Report

Abstract

Relaxation-Driven Cyclic Cosmology: A Closing Report What was established, what was ruled out, and what remains open. The concluding entry of this series. Why this document exists Research programmes usually end without record. Positive results are published; the paths that were tried and abandoned are not, and the next person to have the same idea repeats the same work. This report exists to avoid that outcome for one specific line of investigation. The work began from a simple hypothesis: that dark matter is the gravitational shadow of a sector conjugate to our own, sharing one spacetime but interacting only through gravity. That hypothesis was tested and does not hold. This document records why, in enough detail that the arguments can be checked and reused. It is neither a defence of the framework nor a retraction of it. Earlier releases in this series remain available and are not withdrawn. The negative results are the substantial part Six mechanisms were examined quantitatively and found to fail. Each is set out with the argument that decides it, because each argument applies well beyond the particular framework considered here. Anyone building a two-sector cosmology will meet the same obstructions. The dark-radiation constraint and the dark-matter abundance cannot both be satisfied. They depend on the same sectoral temperature ratio through its fourth and third powers respectively, since radiation energy density scales with the fourth power of temperature while baryon number tracks photon number density, which scales with the third. The two requirements pull in opposite directions, and the gap exceeds two orders of magnitude in either direction. The relational reading fails on logic alone. If each sector's dark matter is the other's ordinary matter, one observer measures a ratio and the conjugate observer necessarily measures its reciprocal. Both match observation only if that ratio equals one. It does not. This argument uses no numbers beyond the observed value and no assumptions about particle content. Mirror baryons would form disks, not halos. Comparing cooling time to free-fall time across halo masses places them firmly in the cooling regime throughout the galactic range, precisely where the evidence for extended pressure-supported halos is strongest. Being colder does not help: the temperature of gas in a halo is set by the shared gravitational potential, not by the sector's own radiation temperature, and infalling gas is shock-heated to the same value regardless of where it came from. Supplying the full dark-matter budget would make mirror baryons denser than ordinary ones, and cooling time scales inversely with density, so they would cool faster, not slower. Distributed production at astrophysical horizons is far too weak. The curvature at a black hole horizon falls forty or more orders of magnitude below the scale where gravitational particle production becomes efficient. Independently, dark matter must be fully present at recombination, whereas the first black holes form hundreds of millions of years later. Energy cannot flow between the sectors. Black holes contain a fraction of a percent of the baryon density, against the seventy percent that a transfer explanation would require. The baryon density inferred from primordial nucleosynthesis and from the microwave background agree to better than one percent, constraining leakage far below what is needed. And exact conjugation symmetry forbids a net flux outright: every transfer has a conjugate transfer in the opposite direction at the same rate. Dark energy does not follow from the relaxation sector. Three separate mechanisms were examined and each requires a new, specifically tuned input to reproduce the observed value. What was established Six structural results survived scrutiny and are documented with proofs in the companion paper Open-System Dynamics Between CPT-Conjugate Sectors. Among them: that the two-sector structure is forced rather than chosen, since the relevant symmetry is an involution and no higher-order analogue exists; that the form of the reduced dynamics is fixed by theorem rather than modelling choice; that the most natural coupling operator provably produces no relaxation of the quantity it is meant to relax; that relaxation necessarily proceeds at two distinct rates in a fixed ratio; and that the coherence order parameter is a derived function rather than an adjustable constant. These results concern open quantum systems in a cosmological setting and stand independently of whether the framework describes our universe. The balance on the bipartite structure A mechanism that does supply the dark-matter abundance was identified: gravitational production of a superheavy, collisionless particle at a cosmological bounce. It resolves both the abundance problem and the clustering problem that defeats mirror baryons. But it does not require a conjugate sector. It would operate identically in a single-sector cosmology. What the two-sector structure contributes is that conjugation symmetry enforces equal production in both sectors, removing an obstruction that arises only because there are two sectors in the first place. What remains is an account of the thermodynamic arrow of time: global entropy constant, each sector's entropy increasing along its own time coordinate, the arrow a consequence of projection rather than a low-entropy initial condition requiring separate explanation. That is a genuine contribution to a genuine problem. Against it stand one additional parameter and a constraint forcing the conjugate sector to be substantially colder than ours. Whether that trade is worthwhile depends on how much weight one places on the arrow of time. It does not depend on dark matter, which was the original motivation and is no longer served. A note on method Several results in earlier versions of this work were stated as derivations but turned out, on checking, to be known measured values with plausible-looking expressions written around them, where the model parameters entering those expressions were never assigned numerical values. The pattern is recognizable once looked for: a quantity is said to follow from a relation, the relation contains parameters, and those parameters appear nowhere in the document. Any claimed derivation should be reproducible by substituting stated numbers into stated equations. Where it is not, it is not a derivation. The corrective that worked was mechanical rather than conceptual: computing every asserted number independently, and withdrawing claims that failed rather than reformulating them. What still stands One prediction distinguishable from standard cosmology and testable with foreseeable instruments: a suppression of the growth rate of cosmic structure by about one and a half percent, in which the coefficient is derived and the parameter entering it is fixed by the measured spectral index. Surveys now underway are expected to reach the required precision. This report makes no claim on the outcome. Michael Lehmann 79730 Murg/Germany mi.lehmann@gmx.de