Room VII · The Legend

The Angle Atlas

Every load-bearing angle in the Information Manifold Model's Riemann program, on one dial. The halving dial 90°/2^Ω(n) where the primes read 45°; the 3-4-5 angle; the balance angle of a zero; the rim of the Bergen Disc; the right angle that 2 + 2 = 4 turns out to be. Tap a ray to read it.

Instruments, not proofs. Three red rings mark where an angle is the Riemann Hypothesis — verified for 100,000 zeros, proven for none. Every number on this page is recomputed in your browser and checked below.

7 / 14

theorem construction / definition measured planted exhibit killed red ring: the angle is the hypothesis here

The fourteen rays

The three red rings

0°

The Weil leg. A zero's contribution to the Weil form has angle 2a²δγ, where δ = β − ½. It is exactly 0° when the zero is on the line. 0° at every zero ⟺ RH.

45°

The balance angle. arctan(β/(1 − β)). Mirror partners always sum to 90° — that part is free. Both equal to 45° is the claim. 45° at every zero ⟺ RH.

90°

The bearing. Seen from ½ on the real axis, a zero's direction is 90° exactly when it sits on the line. 90° at every zero ⟺ RH.

Each ring is the same sentence — β = ½ — in a different costume. Each holds for every zero ever computed (the 100,000 tabulated here; 12.36 trillion in Platt–Trudgian 2021) and is proven for none. Everything else on the dial is a theorem, a definition, a measurement, or a coincidence that was killed and kept on display.

The fence — the Davenport–Heilbronn test

Every ray on this dial is one of three kinds. Free: true for any function with the mirror and the strip — the two mirrors, the complementary balance angles, the 180° identity, the prime dials. Equivalent: the Riemann Hypothesis rewritten — the three red rings. Measured: a finite window — 100,000 zeros at 45.0000°, which can refute and never prove.

The test for any argument built from angles like these: the Davenport–Heilbronn function (1936) has the same functional equation, the same two mirrors and the same complementary balance angles — and its “Riemann Hypothesis” is false. It has zeros off the line, some even outside the critical strip. An argument that goes through unchanged for that function proves a falsehood. What it lacks, and what no ray here contains, is the Euler product over all primes.

Nothing on this page bears on the truth of the Riemann Hypothesis. The dial is a map of where the hypothesis lives — not a lever.

Receipts

Every angle above is recomputed here, in your browser, from its defining formula, and compared with the values of the committed Python twin (angle_dial.py, which asserts the same numbers before it draws).

#	quantity	formula	computed	recorded

Sources

Travis Bergen, A Smith Chart for the Riemann Hypothesis (IMM Paper 19), doi:10.5281/zenodo.21400881 — the Li map z = 1 − 1/s as the chart's lens.
The companion instruments: The RH Smith Chart (Room I) and The Bergen Disc (Room VI) — the rim angles and the vanishing point live there.
A. M. Odlyzko, tables of zeros of the Riemann zeta function — the first 100,000 zeros (γ₁ = 14.134725…, γ₁₀₀₀₀₀ = 74920.827…).
E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 1 (1918) 357–376 and 6 (1920) 11–51 — the Gaussian-prime angles equidistribute.
H. Davenport and H. Heilbronn, On the zeros of certain Dirichlet series I, II, J. London Math. Soc. s1-11 (1936) 181–185, 307–312 — the fence.
D. Platt and T. Trudgian, The Riemann hypothesis is true up to 3·10¹², Bull. London Math. Soc. 53 (2021) 792–797.
Li's criterion (X.-J. Li 1997; E. Bombieri and J. Lagarias 1999) — the map z = 1 − 1/s and the coefficients λₙ.
The polar prime plot that started the halving dial: Grant Sanderson (3Blue1Brown), 2019.