The Poincaré Homology Sphere and the Fine-Structure Constant: What Follows from a Point Being Here, with the Status of Every Step

This text follows a single definition — a point is here: from itself, surrounded by others — along one thread, and marks at every step what carries the step: a theorem (cited, or a uniqueness by exclusion in which every alternative is named and shown to be a mark), a name (a derived thing given the word already defined elsewhere by the same phrase; the text counts three), or a reading (what the structure says from a position, marked so that it can be read against what is found). Behind every step stand one definition of existence — the heres, each from itself, and nothing else — and one principle, that nothing is without a ground, which every exclusion applies. The thread derives, by exclusion: the golden proportion and the finiteness of the whole; the two self-references, algebraic and transcendental; the circle as the only unmarked closing and the residue theorem as uniqueness; the sphere; S³ as the only sphere above one dimension that is a group; the six as the complexification sl(2,C) of the three directions, with the Killing form of signature (3,3); the four as one height and three runs, with (1,3) inherited and Lorentz from the order kept; the binary icosahedral group as the unique catalogue of relations between heres (no axis, no trace), so that the horizon at every point is the Poincaré sphere, whose fundamental group is the group of ways; the five voices with φ in the character table; the spectral gap λ₁ = 168. What each voice weighs is read from the finite cellular table of the ways: the Reidemeister torsions 1 ∓ 1/√5, 5/3, 3/2 exactly, with no spectrum; and the five spectral determinants are derived as class sums of five class values, each obtained by Hurwitz-zeta regularisation, in an alphabet of Clausen values at the angles of the ways and logarithms of 2, 3, 5 and φ, with no e. The winding of a here through its 360 distinctions at the golden step, less what its two ends already are to each other, closes in 360/φ² − 2/φ³ = 137.0356 turns; what one turn delivers to that round is named α, and the measured value stands 2.7 ppm above it, the position's remainder. From there: light as the phase of the voice of the directions with Bianchi and Gauss as its laws; the two hands and the side the folding took; the running of the reading while the structure stands, every coefficient named; the Pati–Salam family with its right side folded; three generations as the three heights a position reads as one; the ground tone as the unlabelled voice with its gap and ladder; the twenty seats and the theorems on them; Koide's relation as the cone — no privilege between line and circle — holding where a triplet is read alone, and hence for the neutrinos: normal ordering, m₁ ≈ 0.36 meV, Σm ≈ 59 meV, Majorana; the composition with π/10 as toll over budget, marked as reading; and the epoch from the topology, as a reading: a closed sphere whose read scale is the reach of its reading, while its radius stays the position's, hence a ∝ t — H₀t₀ = 1, q₀ = 0, the capacity 3H²/c² at every instant with Λ its dark share, fractions fixed — with the two Friedmann readings beside it and nature to decide. Four of the five axioms of quantum mechanics are theorems of the six, the fifth is the token; entanglement is sharing at no distance; there is no dynamics, because the eye is never in the picture. What can fall is listed with what carries it. An appendix, "The seams", lists section by section where the thread bends. Supplementary code reproduces every finite computation. This deposit supersedes the derivation status of the author's earlier deposits for the questions it treats, and says where.