The Low-Entropy Dawn: Running a Breathing Universe Back to the Beginning

The third of the Euler Cosmos sequence, and a continuation of A Point on the Helix of Time. Where the first paper extracts the one prediction the data can refute and the second follows the breathing universe to its end, this paper runs the same time-symmetric cosmology back to the beginning. Its governing constraint is stated at the outset: the bounded dark-energy winding is dynamically silent at the origin, so the framework imports the early universe from standard cosmology and derives none of it. What it contributes is a map and a confrontation. It walks the inherited history — recombination, nucleosynthesis, the radiation era, and inflation — checking at each step that the Euler form is consistent with it; it shows that the thermodynamic arrow survives every geometric and kinematic reversal the trilogy can apply, and so is a low-entropy boundary condition rather than a feature of the expansion, the geometry, or the winding; and it finds that the classical (FRW) description reaches some ninety-one orders of magnitude deeper in density into the smooth beginning than into the clumped end of the same geometry — the arrow of time measured as the depth to which the equations survive. The low-entropy dawn and the past-incompleteness that forces a genuine beginning are read as one unreduced given, mirror to the second paper's forced, bounce-free end. The paper makes no new falsifiable prediction and says so plainly. Claude (Anthropic) was used as an analytical and writing tool and is not credited as an author.