The Breath Cannot Close: Why a Bounded Dark-Energy Sinusoid Forecloses a Cyclic Cosmology

"The Breathing Universe" staked dark-energy density as a single sinusoid in e-folds and left matter and radiation exactly standard. That leaves its governing equation, H^2 = (8πG/3)ρ, diverging at both ends of the history it describes: in the deep past, and again after the forced turnaround at N_t ≈ 0.517, once contraction reasserts matter domination. This note set out to remove both by importing loop quantum cosmology's non-singular correction. The closure fails, and the reason it fails is the result. The obstruction is an exponent race, and it is structural rather than technical. In contraction everything scales as a power of the scale factor and the steepest term wins. Shear scales as a^-6. A fluid with equation of state w > 1 scales as a^-3(1+w) and therefore outruns shear — which is why viable cyclic models use ekpyrotic slow contraction, and why Ijjas and Steinhardt's scenario can report no big crunch and no mixmaster behaviour. The staked model has no such term and cannot acquire one: its dark energy is bounded by construction, matter is standard at a^-3, radiation standard at a^-4. The steepest term the stake owns is a^-4. Nothing in it outruns anisotropy. Three failures follow, in order. Shear overtakes matter at a ≈ 10^-3 to 10^-6, giving BKL chaos. Black hole horizons fill space at a ≈ 10^-11 — black hole mass is conserved, so the cosmic density of black-hole mass climbs as a^-3 while the density inside a horizon is fixed by mass alone — at ρ ≈ 10^6 kg/m^3, roughly a white dwarf envelope, with no quantum gravity involved. The bounce would require ρ_c at a ≈ 10^-41: sixty-nine e-folds and ninety orders of magnitude in density past where the homogeneous description already ended. Nothing rescues it, and the staked model's own numbers are why. Black holes cannot evaporate in the time available: the turnaround is 15 Gyr away and the crunch 44 Gyr away, against evaporation times of 10^70 years for a 10-solar-mass hole and 10^94 years for a 10^9-solar-mass hole. Structure already dominates the entropy budget by sixteen orders of magnitude today, before contraction begins. The close turnaround — the feature that makes the parent paper testable and near-term — is precisely what denies the closure any time to work, where long-cycle models have room. A sharper statement of the same finding: at a ≈ 10^-11 going backward the universe is a hot homogeneous plasma at T ≈ 23 MeV with no structure in it; at a ≈ 10^-11 going forward it is filled with black holes and has no homogeneous description. Same scale factor, same F(a), utterly different universe. The stake makes the geometry time-symmetric about the turnaround; it does not make the history time-symmetric, because entropy is not a function of the scale factor. This recovers the position of "Entropy on the Spiral" — the arrow is expenditure, not geometry. The result generalises past this model: no cosmology whose steepest contraction term is shallower than a^-6 can be made cyclic, which covers the bounded-oscillating-dark-energy family, including the sign-switching and graduated dark-energy programmes the parent paper names as its nearest cousins. Two independent results in the literature point the same way and are recorded — the exclusion of symmetric dark-matter/dark-energy bounces on horizon-crossing grounds, and the requirement that a matter bounce in LQC have ρ_c ~ 10^-9 ρ_Pl to match the observed scalar amplitude — but neither is needed, because the obstruction here arrives first. Nothing in this note reaches the low-entropy dawn of "A Point on the Helix of Time": the backward branch is homogeneous throughout, so the dawn stays adopted, not derived, exactly as that paper left it. Nothing here alters the kill criterion of "The Breathing Universe", which lives at z ≲ 4 and does not overlap either bounce. The parent Euler Universe cosmology remains falsified by four independent tests and is not revived. This is a negative result, and it is stated as one. The stake is a bet on bounded dark energy, and bounded is exactly what forbids the steep contraction a cyclic cosmology needs. The two are incompatible in principle, not in detail. Claude (Anthropic) was used as an analytical and writing tool and is not credited as an author.