Systems I, II and III as three active geometric engines, nested at their log radii and related by the exact 120° turn about (1, 1, 1). Each pulse samples the same node of all three engines for each observable class, and the three nodes make a triangle whose Triple-Quad Formula is evaluated in exact rational arithmetic. The emergent fourth, the closure residual, becomes a soliton crystal at Sgr A*: the monograph's 12-node bipyramid around Cube 1, threaded by three trefoils, that stays locked at the centre while the books balance and moves when they do not. Part 1 · Guide · Manual · Run it in reverse
The same node k of all three engines is sampled for each class: astrometry , timing , velocity . Each class's three nodes form a TQF triangle; the diamond is the inferred fourth vertex P₄ and the white dot its exact closure P₄*. During stage 4 (tier T0) the P₄ markers converge on the soliton at the centre.
The emergent fourth as the Tier 3 crystal of torsional_twistor_tier_3_crystal_2.html, set on the monograph's geometry. Each TwistorPair is one observable class (two particle streams: model and data). The core sits at the centre while every residual is zero and the measured frame legs close; residuals move it by the Proposition 7.14 split (s along the axis, z in the plane), and a frame error shears the trefoils.
| class | Q₁ | Q₂ | Q₃ | A_R poly = Gram | χ (≤ 1/√3) | Δφ I · II · III (mrad) | state |
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| class | I | II | III | R | group |
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| stream | I / w | II / x | III / y | R / z |
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How the tri-engine soliton uses the monograph, and where the Part 2 specification asks for more than the lemmas give.
| result | what it governs on this page | live check | result |
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torsional_twistor_tier_3_crystal_2.html: its TwistorPair particle streams (micro-twist, buckle, inverse separation), GhostPair pole spirals, structural nodes, Tier 2 and Tier 3 bridges, 64-voxel lattice (V = [−2, 2]³ at 20 units per cubit) and orthogonal scope. "Lemma 8.2 geometry" locks its carrier to the monograph: the file's tube phases 0, 2π/3, 4π/3 already put the three trefoils at the C₉ offsets 0°, 40°, 80°, and the lock sizes them so the lobes sit on the nine ray endpoints, with the poles at ±(1, 1, 1) on the C₃ axis (the file has them on the y-axis, in the trefoils' plane). Cube 1 and Δ₁ ∪ −Δ₁ complete the §7.5 objects. Its 12 nodes match the 12 streams and the 12 dimensions of aff(3) as counts only CORRESPONDENCE.