Spacetime Crystallisation: The Convergent Sequence as Relaxation Spectrum

The convergent denominators of the Collatz rotation angle α = log₂(3) − 1 form the sequence 1, 1, 2, 5, 12, 41, 53, 306, ... Each convergent is a resonance point where the cumulative rotation nearly closes, releasing lattice tension and creating a window in which standing wave structure can precipitate. We define lattice tension τ(q) as the distance of the cumulative rotation from the nearest integer. At convergent denominators, τ reaches local minima forming a damped envelope: 0.415, 0.170, 0.075, 0.020, 0.017, 0.003, ... approaching zero. Between convergents, tension rises as Fibonacci growth drives lattice expansion; at each convergent, tension drops as Collatz stabilises new structure. The matter phase gap (V = 12 to nucleation prime 41) spans 29 rotation steps — exactly Collatz(43), uniquely determined by the partnership prime (the only number below 100 with trajectory length 29). Within this gap, the electron precipitates at F = 20, splitting it into 8 = F₆ = χ³ and 21 = F₈ = R₃, where R₃ is simultaneously the eighth Fibonacci number and the third Collatz runway — the exact point where growth hands off to stability. Protons crystallise at convergent resonances (deep coupling, heavy); electrons crystallise between convergents (weak coupling, light). The number of formation zones (rotation steps where tension falls below a precipitation threshold) grows at each convergent as the expanding lattice provides more nucleation sites. Tension exhaustion (τ → 0) marks the asymptotic completion of the lattice. The convergent sequence is not a temporal timeline but a relaxation spectrum: the series of damped oscillations by which the lattice approaches equilibrium, with thermodynamic ordering (high-tension modes relax first). Results are classified as DERIVED (tension envelope, formation zones), OBSERVED (matter gap = Collatz(43), p < 0.002), SUGGESTIVE (other gaps, recapitulation), and INTERPRETIVE (crystallisation front, cosmological analogy). Fourth of four papers connecting the Collatz conjecture to icosahedral geometry. Version 1.1: Addresses all 10 adversarial review issues. Introduces lattice tension model (damped oscillator with growing formation zones). Restructured gaps table with honest uniqueness assessment. Removed incorrect quantitative mass comparison (18.2 vs 1836). Temporal language replaced with relaxation/thermodynamic framing. Four-tier result classification added. References updated with DOIs for Papers 88 (doi:10.5281/zenodo.18674522) and 89 (doi:10.5281/zenodo.18674329).