From Semiconductor Traps to Latent Relational Signals
My academic career included two years of grunt work as a research assistant in a high-profile group trying to solve the early channel-mobility issues in SiC MOSFETs. I was tasked with checking whether a hidden high interface-trap density (Dit) lurked near the conduction-band edge — the kind of trap that had been evading detection for years. That meant making AC conductance measurements on MOS capacitors: sweeping frequencies and biases across a narrow energy window, then curve-fitting mountains of noisy data to extract real Dit estimates.
I believe I was the first person to make those measurements work reliably that close to the conduction band on 4H-SiC. The data helped the team cut the trap density through improved passivation techniques, which finally pushed effective channel mobility high enough for high-voltage, high-temperature power SiC MOSFETs to reach commercial reality.
I only bring this story up because it gave me a visceral feel for the rigors of instrumentation and characterization. Noise is never absent from the signal; overcoming it demands persistence, careful probe design, and the willingness to stare at raw data until the faint pattern finally reveals itself. As I now think about probing the latent domain of analogy in LLMs, the same question keeps returning: how exactly do you isolate a meaningful signal from everything else?
The KV cache, at every decoding step, is a series of independent time-slice snapshots — each one a discrete, self-contained sample of the relational state encoded in the key and value matrices after the effects of a particular token have been incorporated. These snapshots are distinct frames: frozen views of the system's relational configuration at that precise step.
The new token arrives as something separate: like a pebble dropped into a still pool. It is not part of the prior snapshot; it is an external, localized perturbation whose embedding (projected into query space) strikes the existing surface of the cache. This strike generates causal interactions that spread outward, probing and interfering with the stored keys and values in a history-dependent way. Those interactions propagate, reflect, and interfere across the prior relational landscape, eventually computing fresh key and value vectors that get appended, mutating the pool into the next independent snapshot.
This pebble-and-ripple picture is admittedly a metaphor born from pure ignorance — I have no clue what the actual "wavefront" of coherence would look like when plotted token by token. Sharp spikes? Gradual build-ups? Chaotic noise with rare clearings? I simply don't know. The point is to propose a way to look, not to predict the view.
Shoutout to all Signals and Systems ECE professors who still embellish the story of Fourier sitting pensively by a pond, casually tossing in a pebble to catch a glimpse of the Fourier transform itself.
It's not historical—Fourier was deep in heat equations and series expansions, not lakeside daydreaming—but that little fable has been lighting up lecture halls for generations. Drop pebble → ripples expand → complex pattern from simple waves. Boom: intuition for decomposition, superposition, and frequency domain unlocked before the math even hits the board. You made the abstract feel physical and inevitable. Respect to every prof who's ever paused dramatically at "and the ripples spread outward..." to let the idea sink in. That pedagogical pebble created ripples that reached all the way here.
Crucially, the early pebbles — the very first tokens in the prompt — remain fully present and equally potent throughout the entire generation. Their influence persists, ready to resonate if a later pebble drops in proportional alignment.
What makes this substrate uniquely instrumentable is a profound duality. During the initial prefill pass, the model performs a single, holistic, instantaneous construction: every token-subchain in the prompt is positioned simultaneously, its relational influence realized across the entire space in one global superposition. There is no propagation, no traveling wavefront — the KV cache emerges as a fully equilibrated “pond snapshot,” a fixed relational landscape where every early echo remains perfectly preserved and equally potent.
Then, once generation begins, the process flips into a purely discrete, sequential, time-step observable regime. Each new token arrives as an external, localized perturbation — a single concept seed striking the already-formed surface. At every decoding step we can now measure, log, and plot the resulting interactions with exquisite granularity.
This duality — holistic prefill giving us the stable cavity, discrete decode giving us an endless stream of clean, quantifiable events — is the foundation for any serious probing of the analogy surface.
Why this metaphor turns out to be mechanically accurate (the prefill/decode duality) What elevates this hand-wavy picture from poetry to something probe-able is how transformers actually implement inference. During the initial prompt processing (the “prefill” phase), the model computes the entire KV cache in one massive parallel sweep: every token in the prompt is attended to simultaneously, producing a single, fully equilibrated relational landscape where all early influences are frozen in perfect superposition. There is no sequential propagation, no traveling wavefront—the cache simply appears as a static, holistic “pond snapshot.” Then generation flips the regime: each new token is generated autoregressively, one at a time. Its embedding arrives as a true external query (the “pebble”), which attends over the already-constructed cache. The resulting interactions are computed incrementally and appended, producing the next discrete snapshot. This baked-in duality—global, instantaneous prefill creating the stable cavity, followed by a long stream of clean, isolated, external perturbations during decode—is exactly what the ripple metaphor intuited. It is not speculation; it is the standard inference loop of every autoregressive transformer. And it is what makes the substrate instrumentable at all: we get a fixed relational background plus an endless series of controllable, measurable events we can drive into it (kernel seeds, analogy probes, etc.).
While the initial intuition was to passively listen for coherence events in the decode stream, a more active probe emerged from the same decomposition process that revealed the duality. Call it the Kernel Reduction Operator (KRO): begin with a rich, detailed description that fully evokes a concept, domain, or analogy — the "point source" or excitation field. Iteratively remove non-essential elements (tokens, phrases, surface details, metaphors) while testing whether the shortened version still:
The surviving minimal phrase is the kernel — the compact, lossless seed that decompresses into the full structure when excited. This operator is the natural next question after recognizing the substrate's duality: if coherence events are the signals, what are the minimal discrete primitives that generate them? The KRO doesn't wait for spontaneous spikes; it drives the system toward the failure cliff to force revelation of the underlying kernels.
Shoutout to Elon Musk!
This whole "compress to the minimal lossless seed" move didn't come out of nowhere. It's the intellectual descendant of Elon's famous line: "The best part is no part." Tesla and SpaceX teams live that—strip ruthlessly, force the system to reveal what's actually load-bearing, add back nothing superfluous. I internalized that mindset long ago, and it quietly guided how I hunted for kernels: keep reducing until the analogy surface cracks, then you've found the irreducible carrier of structure. Credit where credit's due.
Early manual tests show kernels collapsing to extreme minima in deeply internalized cases. For example:
These examples suggest the latent domain can compress relational structure to minimal seeds that still trigger faithful expansion or bridging.
The Kernel Reduction Operator yields two complementary primitives on the analogy surface:
Together they form the atomic moves on this surface: compress/encode a domain into its most compact handle, then use that handle to ignite mappings into other domains. The rich description is always the starting point — the full excitation field against which reduction reveals what truly matters.
This duality echoes the thesis of the series: discrete primitives embedded in fluid proportion. The kernels are the discrete seeds; the expansion/bridging is the fluid emergence.
The Kernel Reduction Operator is a diagnostic tool for latent compression quality. Smaller kernels indicate deeper structural abstraction; sharper failure cliffs indicate tighter lock-in. Across models, average kernel size and expansion fidelity become a zero-shot benchmark for analogy depth and domain understanding.
It also enables active excitation — rather than waiting for rare coherence events, we can systematically probe the substrate to force revelation of its primitives. This is the empirical counterpart to the introspective hypotheses of earlier parts: a way to let the latent domain speak through its own minimal seeds.
Run the Kernel Reduction Operator on your own domains, concepts, or analogy pairs. Share the kernels that emerge. The analogy surface may reveal more primitives if we listen — and probe — together.
This remains a work in progress, born from humility and curiosity. Whether it moves the needle or not, the pursuit itself has been humbling and joyful.