The geometry of arbitration

1 — the living law

μ = σ(α·ln r − ln β)

r = signal_phon ⁄ signal_ortho. μ is the weight given to the phon route. Move α (the sharpness of the decision) and β (the bias). The grey curve doesn't move: it's the (1,1) case, the law's self-dual point.

the ribbon

One surface, one edge

The ortho↔phon arbitration is a Möbius strip. Follow the two-tone edge: teal (ortho) passes through gold (phon) and returns onto itself — a single boundary, the two routes are inseparable. The purple line is the axis β=1; the two beads, the coupling by the involution r→1/r. Drag to rotate.

2 — what actually carries it

Two faces, each sovereign in its own domain

Win rate by route and regime — verified record 06/2026, 3 seeds. Ortho reigns over the known; phon lifts the out-of-lexicon by +14 to +18 points. The switch is real: no vassal face.

The measured surfaces

Real θ sweep (seed 12345, N=40). On the left the win rate, on the right the coherence 4·E[μ(1−μ)]. The outline marks (1,1); the purple border, the Möbius axis β=1.

The same surfaces, in relief

Win rate and coherence as terrain. The peak rises toward the phon corner; the coherence ridge runs along β=1 — the Möbius axis. The purple mast marks (1,1): on the ridge, never on the peak. Drag to rotate.

3 — the shape of the space

A hyperbolic geometry

The parameter space θ=(α, ln β) carries the Fisher metric g = E_r[μ(1−μ)·((ln r)², −ln r ; −ln r, 1)]. Each ellipse is the local metric; its hue follows g_ββ. The curvature is negative everywhere — along β=1 you slide freely (symmetry), stepping off it costs.

the same law, three engines

Arbitrating between two routes — everywhere

This law isn't specific to the Hangman. The same ortho ↔ phon / context arbitration governs three engines. It's the noisy-channel model (noisy channel, Kernighan–Church–Gale 1990): choose the word w = argmax P(w)·P(s|w)^λ — where λ sets the weight of one route against the other. λ is the α of the curve above; (1,1) is the neutral weight.

Honesty (bench 07/2026): on the corrector, the principled arbitration equals the hand-tuned thresholds — without beating them yet. The wall there is the spelling-channel model (frequency traps), not the weight. The form is right; the calibration remains open.

Where the geometry did break through a wall

On candidate choice, arbitration ties, without winning. But aimed at another corrector problem, the same law brought down a barrier reputed to be out of reach for a lightweight tool — recovering the subject of a mis-agreed verb when the word stays valid (« les enfants joue »). The literature makes it the central problem. No single subject cue clears it (the best tops out at 81 %), and adding them up throws it into disarray. What worked is exactly the geometry of this page : blending the subject routes weighted by their reliability — the peakedness of each distribution, the same μ = r/(1+r) as at the ribbon's edge — a small language model carrying the confidence gradient. On the residual the parser gives up on, the subject's number comes out at ~94–97 %. Shipped as vigilance (it suggests, doesn't rewrite — the zero false positives stay intact).

And part of the wall even falls affirmatively, by a principle simpler than arbitration : the subject isn't always read right before the verb. The sibling verb carries it (« mangent et dort »→dorment), the antecedent of a relative clause carries it (« que je vois joue »→jouent), a parenthetical merely interrupts it. Four screens, four rules anchored on the audible plural — zero false positives across 2,500 sentences and on devious traps. The frontier stays honest (the general « de N » complement remains in vigilance) — but the structurally recoverable part becomes a clean correction again. The geometry of arbitration didn't beat the candidate calibration ; it broke through a neighboring wall, reputed impassable, with the same idea : to cross is to weight by reliability, never to add up.