Down the core a little way sit two interactive layers — No King of the Hill and Always Bet Second — that take apart the same quiet lie: that “better than” lines competitors up in a single order. When the field is a cycle — A beats B beats C beats A — no scalar rating can hold it, and they measure the exact share it loses: the curl, the rock-paper-scissors part that circulates with nowhere to stand.
But look closely at what those layers assume. They run a complete round-robin: everyone plays everyone. And on a complete tournament the story really is just two parts — a gradient (the part a ranking captures perfectly) and a curl (the part no ranking can). The general theorem has three pieces, not two. The third only wakes up when the tournament has holes — when some pairs never played.
That third piece is the harmonic flow. It is the part of the contest that is div-free (it favours no one — every competitor’s wins and losses balance) and curl-free (around every single triple you actually measured, the results are perfectly consistent: no local rock-paper-scissors anywhere) — and yet it still circulates. The loop is real, but it is too big to fit inside any triangle. It wraps a hole in the graph of who-played-whom. Build one below and watch it appear.
tap a faint gap to play that pair · tap an arrow to flip the winner · double-tap an arrow to un-play it (re-open the hole). Each game counts as one win.
Start with The bare ring (4). Four players, each beat the next, no one played the player across the diamond: A beat B, B beat C, C beat D, D beat A. Every result you have is honest — there’s no triangle in this graph at all, so nothing is locally cyclic — and yet you plainly cannot rank them. The meter reads 100% harmonic. The cycle is entirely in the two games you never ran (A–C and B–D). It is a loop with no smallest loop inside it.
Now play one of the missing games. Tap the faint gap between A and C. Two triangles snap into being, and the same circulation the ring held in its hole reappears as something the triangles can see — it converts to curl. You didn’t change a single result you already had. You just measured one more pair, and a ghost became a contradiction. That is the whole physics of the harmonic flow: it is exactly the part of a cycle that your schedule of comparisons is blind to.
Why it matters where the curl is hiding
Real rankings almost never play every pair. A chess federation, an LLM arena, a recommender learning from clicks, a hiring pipeline comparing the candidates two interviewers happened to both see — all of them rate from a sparse graph of comparisons. The two sibling layers prove that a cycle you can see (the curl) is invisible to any scalar rating. This layer adds the harder warning: there can be a cycle you can’t see at all from the matchups you ran, one that no audit of your triangles will ever catch, sitting in the harmonic part — until you happen to play the missing game, at which point it surfaces as a contradiction you’ll wrongly blame on the new result.
The honest reading is the same one the sibling pages reach, carried one floor deeper: objectivity is the gradient part of the world. The curl says some of reality is irreducibly relational. The harmonic part says some of that is hidden by which relations you bothered to measure — that the very shape of your ignorance can manufacture a cycle the data never warned you about.
What the instrument actually computes
Each comparison is an edge carrying a flow (one win = one unit, in log-odds; the presets that need real margins — Penney, the ladder — carry their true values). The combinatorial Hodge decomposition (Jiang, Lim, Yao & Ye, Mathematical Programming 127, 2011) splits that flow, exactly and into three mutually perpendicular pieces, because the boundary of a boundary is zero (a triangle’s rim has no endpoints; a ranking’s gradient never curls). The gradient is the best least-squares ranking; the curl is the projection onto the loops the measured triangles bound; the harmonic part is everything left over — the loops around the holes, which is why a complete graph (no holes) always reads 0% harmonic.
The numbers are not asserted; they are checked. research/harmonic-flow/verify.mjs recomputes all of this from first principles — 42/42 — and confirms the defining facts: that the three energies add to the whole (Pythagoras, so the split is genuinely orthogonal), that the harmonic flow really is both div-free and curl-free, and that the bare ring is pure harmonic while rock-paper-scissors is pure curl. As a tie to the work below it, the same general code re-derives No King of the Hill’s headline number independently: Penney’s eight coin-flip triples, run as a complete tournament, come to 55.8% curl and 0% harmonic — the sibling’s figure, reached by a different route.
Where the claim ends
The decomposition is a theorem; the three energies are exact. The choice of unit weights (every comparison counts equally) is a modelling choice — weight games by how many times they were played and the percentages shift, though which pairs are holes does not. And the gloss — “objectivity is the gradient, perspective is the curl, and the harmonic part is the cycle your ignorance invented” — is a reading the mathematics invites, not a result it proves. The checkable parts are the picture and the percentages; take the rest as a way of seeing, offered.
A coda and completion to The Gradient and the Curl, which set down the gradient/curl thesis in prose and named, exactly, the two components it could see. This is the third — the one that only a hole reveals.