Arithmetic supersymmetry on X₀(143) × 143a1

Five interactive pictures of the note (29 Sep 2026). Every curve is drawn from the exact formulas in mtft (PV-01/PV-02); nothing is fitted. Class of each statement is marked.

1 · The five means are the BPS bound EXACT

A gauge block of bidegree (a, b) has two field strengths, x along X₀(143) and y along 143a1. At fixed volume the shape r = A_X/A_E slides (x, y) along the hyperbola xy = |ab| — its topological charge. Its Yang–Mills energy is x² + y², the circle through the point. The circle can only touch the hyperbola on the diagonal x = y: that tangency is rms ≥ gm, and for opposite-sign blocks it is the zero-slope, D-flat, supersymmetric point. Off the diagonal the Higgs mass² is the signed gap and (x − y)² is the D-term.

block shape r = A_X/A_E = 2.00

2 · Möbius is (−1)^F EXACT (Spector 1990)

Put a boson and a fermion on every prime. The graded count of states at level n is Σ_d|n μ(d): every excited level cancels exactly; only the vacuum n = 1 survives. Below: each level's divisors coloured by μ, then the Mertens function M(x) = Σ_n≤x μ(n) — the fermion-number imbalance — against ±√x. The prime number theorem is Σμ(n)/n = 0; RH is M(x) staying inside x^1/2+ε.

3 · Where the 12 lives, and what the twist does to it EXACT

11 and 13 mean to 12 because they are twin primes; what is unique to 143 among twin-prime levels is that this mean equals g − 1 = deg K^½, the degree of the spin structure. The spin twist (12, 0) makes every adjoint block vector-like; the supersymmetric twist O gives net chirality −24b = χ(X₀(143)) × (torus flux); the blow-up adds the exceptional flux e, net = e − 24b, and the lattice becomes odd (no spin structure at all).

4 · The blow-up escape: two ratios, four slope lines, one point EXACT (finite search)

On the product, every stack's slope-zero condition is a vertical line in the (r, ε/A_E) plane and one ratio can hit only one of them — the root of the trilemma. On the blow-up each condition is a line a + br + e t = 0. Left: a non-collinear D-flat Standard-Model solution from the search, four lines through one point inside the ample triangle. Right: the single-line-bundle family, D-flat along the whole line t = 1/3.

5 · The flux landscape: shape stabilised, volume not EXACT

E_split(r) = 4π²(19/r + 11r) for both S1 and S2; the HYM floor of each class is 4π²[Δ + A₁²/r + B₁²r]/6; the gap is the slope variance — the D-term energy. In the Einstein frame both classical terms fall with the volume, so nothing selects a cosmological constant (CC-34).