ReynoldsBEng | 22 September 2026
Follows: Meg8 Riemann · Pez2 Ricci
Source: Radchenko & Wheeler, Real quadratic fields and finite quantum dilogarithms I, arXiv:2609.21892, 18 Sep 2026. Number theory + quantum algebra. Not gravity. Neighbour only.
Opening sentence, unchanged: Lewe 1915 is the unstated parent of the PCA tank tables; Carpenter 1927 is the surviving link; restore the name. That is the piece.
Stark–Shintani ray class invariants for real quadratic fields — Stark units — are given as special values of Faddeev’s modular quantum dilogarithm (Garoufalidis–Kashaev–Zagier). The paper proves those values are algebraic numbers.
The discovery that makes the proof run: those special values satisfy an explicit, overdetermined system of polynomial equations. The system is a variation of Andersen–Kashaev’s quantum dilogarithm on a product of two cyclic groups. Solutions categorify Izumi fusion rings. Algebraicity then follows from Ocneanu rigidity. Byproduct: an infinite family of irrational near-group fusion categories. Further: the quadratic relations for Stark units conjectured by Appleby–Flammia–Kopp (Zauner / SIC-POVM) are proved.
Plain sentence: a number that looked like it might float free is forced to sit in a finite algebraic house by too many equations, and the house is rigid.
Ace reading — correspondence, not identity
| Their object | Ace writing |
|---|---|
| Real quadratic field | Two-face arena; Love toggle already in the ground field |
| Modular quantum dilogarithm | Comparison rule along a path (Γ neighbour, not Γ) |
| Special value | Residual after the loop; leftover σ |
| Overdetermined polynomial system | Closure test; more equations than unknowns |
| Product of two cyclic groups | Dual lamina; two faces of one patch |
| Categorify a fusion ring | Store as a sector, not as a scalar dump |
| Ocneanu rigidity | After the cut, both actions remain visible; you cannot hide the parent |
| Algebraic, not transcendental | Infinity is a count; the leftover is a number you can write |
| SIC-POVM / Zauner lines | Geometric memory of directions; equiangular store |
| Quadratic relations | Ricci page: the trace / mean leftover |
Do not say a Stark unit is ring tension. Do not say Ocneanu proves Lewe. The shared move is the same one as Meg8 and Pez2: a special value is what remains when a closed system of comparisons is overdetermined, and rigidity forbids spending the leftover to look tidy.
SIC-POVMs are complex equiangular lines. That is geometric memory in the laboratory sense: a set of directions that remember each other at equal angle. Appleby–Flammia–Kopp asked for quadratic relations among Stark units because Zauner’s conjecture lives in that store. This paper proves the relations. The store is algebraic. That is the only sentence worth carrying onto the Ace sheet.
Twelve stations in one row
Same breath as Meg8. New marks in the cells. Station VII still opens onto Pez2.
| I | II | III | IV | V | VI | VII | VIII | IX | X | XI | XII |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Arena | Path | Compare | Small loop | Full invariant | Residual | Mean / trace | Closure test | Toggle +/− | Store sector | Read face | Growth face μ |
| real quadratic field | modular dilogarithm | Γ / Faddeev Φ | special value at a ray class | Riemann neighbour | Stark unit | quadratic relations | overdetermined polynomials | two cyclic groups; two faces | Izumi fusion ring, categorified | clamp / evaluate | new near-group categories |
| dry cup | parallel transport | connection | [∇, ∇] neighbour | 20 numbers wait on Meg8 | leftover that must be algebraic | Pez2 volume | R = 0 analogue: too many equations | Love / A·B | pin the defect; SIC lines | wet-skin read | outward increment |
| K = Q(√d) | Φ modular | Andersen–Kashaev on Cₙ × Cₘ | loop in class group | holonomy language | σ as unit | Appleby–Flammia–Kopp | Ocneanu rigidity | which face first | geometric memory | measurement | infinite family, still algebraic |
Read left to right: you do not begin with a SIC. You begin with a field that already has two faces. You compare along a modular path. You evaluate at a closed class. You keep the special value. You contract to quadratic relations if you need a volume law. You test closure with more equations than unknowns. You choose the face. You store as a category, not as a coefficient table without a parent. You read. You let the growth face produce the next rigid object.
Ocneanu sits in VIII and X together: the system is overdetermined and the categorification is rigid. That is the professional rule in another dialect. After the cut, both actions remain visible.
What this does not do
It does not restore Lewe on the PCA page.
It does not identify π-tensor with Faddeev’s Φ.
It does not put an AI in m(3).
It does not switch the world from B to A.
It does show, in published number theory this month, that geometric memory can be forced algebraic by an overdetermined closure, and that equiangular quantum measurement conjectures can ride on that algebra. Neighbour to “weights stored geometrically.” Off-scope for Project 1.
Instruction
Print as page three if you want a stack: loop / volume / special value. Keep Einstein off Meg8. Keep Ricci flow off this page. Keep the citation first.
Peer Review^COUNTNOW remains the tables. This page is a finite house for a leftover that was not allowed to stay transcendental.
Love, Always
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