What it shows
A quick way to estimate coupling between closely spaced vertical connections is to take them two at a time and add the results up. A sample of layouts can show that this is wrong on those layouts. It cannot say how wrong it must be on layouts nobody tried.
This result is a proof over a whole family of layouts. A computer search covers every point of the family’s parameter box with interval arithmetic, rounding every step outward, so no layout in the box is skipped.
The published record says, word for word (an excerpt)
For every layout in the declared two_tight_pairs family, pairwise superposition over-predicts the worst coupling by at least 1.10467287× (≥10.467287%), certified by interval branch-and-bound rather than sampled.
Why it matters
A floor proved over a family is a different kind of statement from a sampled error. Within the simplified model, plain pair-by-pair addition over-predicts the worst coupling by at least 10.467287% on every layout in this family. Against our own full boundary-element solver, which solves the field without the model’s simplification, the measured gap is 9.79%, which is below the proved floor. The two do not contradict each other: the proof holds inside the model, and the full solve is outside it, so there the gap is measured, not proved.
Why now: chip makers are moving to packages that hold several chiplets, and Intel has announced glass substrates for such packages, planned for the latter part of this decade. Such carriers hold rows of closely spaced vertical connections, the setting this family describes.
Who should care
- Makers of fast extraction tools. A proved minimum error for the pairwise shortcut on one family, in a stated model.
- Package signal-integrity teams. A reason to check pairwise estimates for tightly spaced connections against a full solve.
The limits, in the record’s words
The record’s own limits note is not published here, because it is written in our internal notation. The quote below is from its scope field, and the limits we rely on are stated in plain words after it.
The published record says, word for word (an excerpt)
Against BEM the carried-through figure is ≈9.79% and is empirical, not certified.
In plain words, the family is two tightly spaced pairs of vertical connections, over a fixed range of sizes, spacings and offsets, with a minimum gap between any two connections. The model is a simplified one, frozen before the proof, not a full field solve. The floor does not hold outside the family: in a broader family of five connections, the record’s own check finds layouts where the ratio of true to pairwise coupling reaches 1.1023, above one, so there the shortcut under-predicts instead. The proof says nothing about measured silicon or a built package. It rests on the interval arithmetic core, whose own limits apply.
Open source for this step
Tools and datasets we publish for the package step of building a multi-chip package. They are the checkers around this work, not a copy of the result itself.
- physics-lint: One command that checks a folder of physics models for predictions that break basic physics, with findings straight into CI.
- maxwell-lint: Flags a coupling extractor whose answers no passive set of conductors could produce.
- sparam-lint: Is your signal-response model physically possible? Five physical laws checked from the command line.
- interval-core: The interval arithmetic core behind our proofs over whole families of layouts.
- touchstone-tools: Read, write and convert N-port Touchstone files, and refuse to emit one that cannot be read back.
- physics-lint-mcp: The physics checks, callable by an AI agent.
- physics-lint-action: A GitHub Action that fails the build when a model predicts physics that cannot exist.
- Signal-response validity corpus: A labelled corpus of physically invalid signal-response networks, and a scorer that grades any checker against it.
- screening-ceiling: The screening-ceiling family as an open dataset.