In 2020 I was staring at a hexagonal wire-mesh ceiling when an idea arrived with the force of certainty: a neural network built on hexagonal connectivity — six neighbours, knowledge spreading like waves — might be the right substrate for genuine machine intelligence. I carried that conviction for six years, through a computer science degree, a postgraduate certificate in AI, and a career building software. Last year I finally did the only honest thing you can do with a conviction like that.
I tried to kill it.
What follows is what more than two dozen controlled experiments — on a GTX 1070, free cloud GPUs, and three of IBM’s quantum computers — actually established. It is not the story I wanted to tell. It is better.
One rule governed everything: no result gets written down before its run completes. Early on I caught myself once writing numbers for an experiment that was still running — anticipated numbers, plausible numbers. When the run finished, reality was backwards from what I’d written. I corrected it, logged the incident, and treated it as the most important data point of the project: conviction will fabricate evidence if you let it. Everything below survived that rule.
The core belief — hexagonal connectivity helps a network learn — turned out to be cleanly, repeatedly false. Across every regime I could construct: backpropagation, frozen reservoirs, local Hebbian rules, local online prediction, grid-cell-style scaffolds, message-passing networks at matched parameters with degree-matched random controls — the honeycomb ties or loses to the square lattice. The reason is almost embarrassing in hindsight: a trainable network routes around its wiring. Performance tracked node degree, not geometry. The square kept winning.
I could have kept tuning until a hexagon won something. That’s not an experiment, that’s p-hacking. The null is the result.
One thing survived learning, because it isn’t wiring — it’s a constraint on the function class. The hexagon has 6-fold rotational symmetry, and weight-tying across that symmetry group (group-equivariant convolution, exact on the true hex lattice — equivariance error 0.000000) is something backpropagation cannot route around. At matched parameters the full 6-fold model beat 3-fold, 2-fold, and plain CNNs monotonically — but only when data was scarce and the symmetry genuinely present. On real bike-share demand grids it won the cold-start forecasting task in 10 out of 10 city-seasons. On a benchmark whose symmetry is only approximate, hard equivariance actively hurt. A real gift, with sharp edges: it pays if and only if the symmetry is there.
Fixed-physics propagation on the honeycomb genuinely is exceptional: classical waves stay ~28% more isotropic than on a square grid; quantum walks 46–76% more — the Dirac-cone physics that makes graphene famous. For a while this looked like the vindication. It isn’t, and seeing why took me the longest:
Simulating a lattice’s physics on a digital computer in order to beat digital deep learning is structurally impossible — the simulation costs more than the physics it simulates.
If the hexagon’s physics is the gift, the gift only exists where the physics is real.
So I went to real physics: IBM’s superconducting quantum processors, free tier, ten minutes of hardware time per month, simulator-gated before every submission. The honeycomb did not disappoint there.
None of this is new physics — it’s textbook theory, and groups with far more qubits have gone further. What it is, is measured, by one person, honestly, on real hardware and a nine-year-old GPU, with controls and error bars.
The arc had one more turn in it. This month I went to the one place where hexagons and learning genuinely share a workplace: decoding quantum error-correcting codes. A decoder is a learning system that reads a quantum machine’s error alarms and must name the fault, fast, forever — exactly the job neural networks are being recruited for across the field, and one where labelled training data from real devices is desperately scarce.
What I found, verified to the same standard as everything above: the square-lattice surface code — the industry default — provably cannot hand its lattice symmetry to a learner at circuit level. Across 999 candidate symmetry maps and 1,728 gate schedules (including schedules co-designed to restore the symmetry), not one survives compilation — a check on the mirror axis would need a gate order equal to its own reflection, and no order is. The honeycomb Floquet code — built on the same three-bonds-per-site property that makes the Kitaev model possible — keeps its full rotation symmetry exactly, under the most naive compilation, because weight-2 checks leave the schedule nothing to break. And the symmetry it keeps is strange: rotating the lattice 120° swaps the two stored qubits — solved and verified mechanism-by-mechanism from the code’s own error model, at two lattice sizes.
Build that exact symmetry into a neural decoder and something happens that I had never seen in six years of hexagon experiments. Two identical 63,000-parameter networks, same training data, same everything: the plain one never learns the honeycomb task at all — it sits at the guess rate through 64,000 training examples, and stays there even given 22× more parameters — while the symmetry-aware twin reaches 81%, and with 1,000 examples already beats the plain network given 256,000. Not a speed-up. A switch. The first time in this whole story that hexagonal structure beat the square at a learning task — and it happened precisely the way verdicts one to four said it would have to: not by wiring a network hexagonally, but through real structure — a code’s exact, physics-level symmetry — that gradient descent cannot route around and cannot invent from data.
You can play against the trained network in your
browser — it ships with this site — install the solver that found the symmetry
(pip install demsym), and a preprint with every logged number is in
preparation.
The conviction was wrong as stated and right in a way I couldn’t have articulated in 2020. Intelligence did not live in the geometry — every learning system I built was supremely indifferent to it. But the hexagon really is special: it’s why grid cells in your brain are hexagonal, why graphene has Dirac cones, why the only exactly-solvable topological spin liquid lives on a honeycomb — and why the one error-correcting code that hands its full symmetry to a neural decoder is a honeycomb too. Its specialness belongs to physics and topology, not to gradient descent. The early note I wrote — “perhaps it requires qubits” — turned out to be the most accurate sentence in six years of notes.
I think there’s a general lesson, and it’s not about hexagons. Most technical convictions are never given a fair trial — they’re either abandoned untested or defended unfalsifiably. The trial is the valuable part. I falsified the thing I most wanted to be true, found the narrow places it was true, and can now tell you exactly where the line is. The discipline that drew that line — controls, matched parameters, sim-gating, never writing a number before reading the log — is the same discipline I bring to everything I build.