A true story, with diagrams
A side-quest in memory retrieval that ended, honestly, one small feature and one good habit richer.
Somewhere around week two I realized I had built five instruments, a reflectogram, and a small theory of contradiction, all to answer a question that plain semantic search answers in one dot product. This is the story of how that happened, why I don’t regret it, and the one small thing I got to keep.
Where this started
Not with a plan. I bumped into the Smith chart the way you bump into most good things: reading about something else. It’s a radio-engineering tool, and what stopped me was its shape. Impedance is messy and unbounded, zero to infinity, and the chart folds all of it onto one tidy disk where “perfectly matched” sits dead center. My first, unhelpful thought was: that’s what I wish relevance looked like.
Because I’ve never loved that memory retrieval hands back a single number. “Wrong for this question” comes in flavors: stale isn’t off-topic, and neither is arguing back. And I have a habit of dragging a tool from one field onto a problem in another just to see what shakes loose. So this was less a plan than a dare. It’s beautiful; would it even map?
I The chart
Radio engineers have a problem: when you plug a component into a circuit, some of the signal goes through and some bounces back. Too much bounce and your antenna is a very expensive space heater.
If that sounds abstract, think of a garden hose. Put your thumb halfway over the end and the water doesn’t just go where you point it; a good share of it comes straight back at your sleeve. Radio circuits do the same thing with signal. Impedance matching is the craft of shaping the nozzle so nothing sprays back, and the Smith chart is the little map engineers use to see, at a glance, how far off the nozzle is and which way to twist it.
In 1939, Phillip Smith drew a disk to make this visible. Dead center means a perfect match, nothing reflects. The rim means total mismatch, everything reflects. Every real component lands somewhere in between, and its position tells you not just how mismatched it is but what kind of mismatch you’re dealing with. Here is the genuine article, before I put a costume on it.
Phillip Hagar Smith was an engineer at Bell Telephone Laboratories when he published the chart in 1939, in the magazine Electronics, under the wonderfully modest title “Transmission Line Calculator.” It is a nomogram: a calculator you read rather than press. With a compass, a ruler, and a pencil, an RF engineer could solve a transmission-line impedance-matching problem in minutes, decades before a pocket calculator made the arithmetic cheap.
The disk itself is the complex reflection-coefficient plane: every impedance there is, folded onto one circle. It is still taught, still printed on lab walls, and still glowing on the screen of every network analyzer. Not bad for a sheet of paper.
The math is a Möbius transform, Γ = (z−1)/(z+1), a fancy way of saying “squash an infinite plane onto a disk without tearing it.” It’s elegant. It’s eighty-seven years old. And it is very much not about memory.
II The leap
An AI assistant that remembers things has to decide, for every question, which saved notes to pull back into view. Usually that’s one number: a relevance score. Higher wins.
I found that unsatisfying. A note can be “wrong for this question” in different ways: stale, unresolved, or actively contradicting what you asked. One number flattens all of that.
So, the leap: treat each note as a point on the Smith chart. Center means perfectly relevant and current. Rim means irrelevant. Let the angle encode the kind of mismatch, stale versus unresolved. And let “past the rim,” a place a real Smith chart can’t go, mean a note that flatly contradicts the question.
A costume, really. But a costume that makes you ask better questions. Here is the whole idea on one disk. Hover or tap a note, then press re-ask and watch who flinches.
III The instruments
Once you have a disk, you want gauges for it. Six facets, roughly in the order I fell for them:
It’s genuinely fun to poke at. That turns out to be a different thing from useful.
IV The measurement
The real win in retrieval is embeddings, plain semantic search, not the chart. Worse: used as a ranker, the Γ math collapses. Do the algebra and it’s ordinary cosine similarity plus a recency term, wearing a hat.
The Γ math, as a ranker, is cosine similarity wearing a hat.Finding, week two
I built an out-of-sample test to find where the fancy version stops adding anything, and the answer is: almost immediately. It only holds while the operation stays within about 3% of a true disk automorphism. By 10% off, it predicts nothing. A coin flip.
So most of the pretty facets are thinking tools, not production features. The locus view is a good way to see robustness; it is not a better way to compute it.
What kept this from becoming a much longer story: I ran the whole thing past a skeptical reviewer, repeatedly, and let them win.
Every time I fell in love with a facet, the reviewer asked what it beat. Usually: “nothing, but look how nice it is.”Review loop, all fortnight
V The twist
Then I did the thing I should have done first: I audited the large memory system I already run.
It already does the serious version of nearly everything I’d built. Learned reranking judged by an LLM. Real cosine search. Typed relationships between notes. Even handling for notes that supersede or contradict each other, my proudest “past the rim” idea, already sitting in a table with a column for it.
I reverse-engineered my own basement.The audit, one afternoon
There was exactly one gap: the system logs scores but never says, in a human sentence, why a note showed up. So the single useful souvenir from a very elaborate detour is a small “why did this surface?” explainer.
It’s a good few lines. It is not a Smith chart.The souvenir
VI The takeaway
Don’t reduce “relevance” to a single number. Show why something matched, and how it could fail: stale? contradictory? only matches your exact wording? That mindset is the keeper. The radio-engineering costume was mostly for fun, and fun is a fine reason, as long as you say so.
The gentler moral: enthusiasm for a beautiful idea has to survive one boring question. Does it beat the thing that already works? Ask it early, ask it out loud, let it win.
VII The unironic part
As a ranker the Smith math is dead; it collapses to cosine. But three of its concepts have a real, non-decorative future, if you adopt their proven cousins instead of the chart itself. The Smith chart was the scenic route to these; the radio costume was optional.
Three tiers, sorted by how much I’d bet on each. Click a tab, or use the arrow keys.
Proven cousins of the chart’s ideas. Each does a job that cosine, on its own, does not.
We kept collapsing every note to |Γ|, a distance, and discarding the angle. Give the angle a job. Store typed links between notes (supersedes, contradicts, refines, depends-on) and encode each relation type as a phase, a rotation in complex space. Then relations compose: A supersedes B, B refines C; rotate twice and you have A’s relation to C. And they are asymmetric, so “what does A supersede?” is a different question from “what supersedes A?” Flat relation-tags become a composable geometry.
why it’s legitRotatE and ComplEx encode each relation as exactly this kind of rotation, and they hold up in knowledge-graph link prediction, which is a production task, not a demo. It is the one place the phase was doing real work all along.
Re-ask a question a dozen ways and measure how much a note’s score moves. Barely moves: robustly relevant. Swings around: brittle, it only matches your exact wording. Store that number next to the note. It is cheap (a dozen embeddings and a standard deviation) and it is not a similarity score; it measures robustness, which a single number hides by construction.
why it’s legitNothing to prove and nothing to fit. It is the locus view from Fig. 1 with the poetry removed, and it is the most honest survivor of the whole project.
The Γ disk literally is the Poincaré disk of hyperbolic geometry; I had been drawing it for two weeks without noticing. If a knowledge base has real tree structure (taxonomies, is-a chains, nested topics), Poincaré embeddings place general things near the center and specific things near the rim, with room to spare for the branching. That answers a question cosine fumbles: is this note a specialization or a generalization of what I asked?
why it’s legitNickel & Kiela showed hyperbolic embeddings fit trees with low distortion in few dimensions, and it has held up since. Only worth it if you are hierarchy-heavy; a flat pile of notes gets nothing from it.
Lightly Smith-flavored, and simpler than the metaphor that suggested them.
The |Γ| > 1 region becomes a mode: alongside the answer, deliberately show the single best note that argues against your premise, instead of burying it or, worse, quoting it as support. An anti-echo-chamber setting. Under the hood it is a stance check and one reserved slot in the results, no disk required.
A Smith chart’s actual purpose is to tell you what to add to reach a match. Same move, for questions: compute the smallest addition to the query (an entity, a date bound, a bridge concept) that pulls good candidates in, and say which one you added. Principled query expansion, in the family of HyDE, with the receipt attached.
Honest accounting: the chart’s ideas that a well-built memory system already does under a plainer name.
Mine did all three before I started. In a table. With a column for it.
If I chase one, it’s paraphrase-stability — it measures the thing a single score throws away.The bet