--- license: cc-by-4.0 task_categories: - tabular-classification - tabular-regression tags: - physics - electromagnetics - verification - interval-arithmetic - formal-methods - counterexamples - eda - parasitic-extraction pretty_name: Screening Ceiling โ€” Certified Regions and Counterexamples size_categories: - n<1K configs: - config_name: regions data_files: data/certified_regions.jsonl - config_name: counterexamples data_files: data/counterexamples.jsonl --- # screening-ceiling ![Licence](https://img.shields.io/badge/data-CC--BY--4.0-green) ![Regions](https://img.shields.io/badge/certified%20regions-256-blue) ![Leaves](https://img.shields.io/badge/certified%20leaves-237%2C490-blue) ![Counterexamples](https://img.shields.io/badge/counterexamples-27-orange) ![Verifier](https://img.shields.io/badge/verifier-zero%20dependency-brightgreen) ![Tests](https://img.shields.io/badge/tests-39%20passing-brightgreen) ๐Ÿ“– **[Documentation site](https://nickharris808.github.io/physics-lint/)** โ€” the portfolio narrative, the concepts, a full walkthrough, and what all of this proves (and does not). **A machine-certified impossibility result about coupling extraction, plus the concrete layouts where a plausible extractor predicts impossible physics.** Most ML-for-physics datasets are samples: here are some inputs, here are the answers, fit something. This one is different in a way worth being precise about. It carries a **universal claim** โ€” a statement about *every* layout in a continuous four-parameter family, established by interval branch-and-bound rather than by sampling โ€” together with **existential counterexamples** that refute a specific competing method. ## Why this exists When conductors are packed together, every other conductor screens the field between any two. So a pair's mutual capacitance inside an array is at most its isolated-pair value: ``` k = |C_full| / |C_iso| โ‰ค 1 ``` Pairwise-superposition extraction โ€” used throughout fast parasitic extraction โ€” assumes `k โ‰ก 1`. The certified claim here is that on a particular manufacturable family it is **provably never right**, and quantifies by how much: > For every layout in the family box, `k โ‰ค 0.909090909091`. A pairwise extractor > therefore over-predicts the worst coupling by **at least 10.000002%** on > *every* member of the family โ€” not on average, not usually, always. And the other direction: a second-order Born correction, the obvious cheap fix when a full solve is too slow, does not merely stay inaccurate โ€” on 27 layouts here it predicts `k > 1`, which is **anti-screening**, a physical impossibility. ## 30-second quickstart No install, no dependencies, nothing from the repository that produced it: ```bash python3 verify.py ``` ``` screening-ceiling independent re-derivation (stdlib only) claim k <= 0.909090909091 for every layout in the family forced error pairwise over-predicts by >= 10.0000% regions 256 x 25 samples = 6400 layouts worst sampled k 0.903974725909 (bound 0.909090909091) margin to bound 0.005116183182 violations 0 worst at region 3.0.3.3 d0_um=55.7830 jog_mult=0.3838 pt_mult=1.0009 sep_mult=4.4492 counterexamples 27/27 re-derived and confirmed above the ceiling consistent: no sampled layout exceeds the bound, and every counterexample re-derives scope: A complete interval theorem about the frozen MONOPOLE-CLOSURE model only. The closure-vs-BEM/PDE model gap remains additive and unresolved. This witness does not establish Maxwell, BEM, driven-S, fabrication, or measured-silicon truth. ``` `verify.py` rebuilds the electrostatics from the published geometry using the standard library alone โ€” its own Gauss-Jordan inverse, its own potential matrix. It does not import this dataset's producer, numpy, or anything else. **Sampling cannot prove the universal claim.** That is what the interval branch-and-bound in the source proof is for. What sampling *can* do is refute it, and that is the useful thing to hand a skeptical reader: a cheap, dependency-free way to try to catch us being wrong. ## Loading ```python from loader import load_regions, load_counterexamples, load_theorem theorem = load_theorem() print(theorem["statement"]) print(theorem["honest_scope"]) # read this one for c in load_counterexamples(): print(c["case_id"], c["k_predicted"], c["n_pairs_violating"], "/", c["n_pairs_total"]) ``` Optional conveniences: `to_pandas("regions")` and `to_hf_dataset()`. ## Contents ### `data/certified_regions.jsonl` โ€” 256 rows, the universal claim The branch-and-bound partition. Each row is one region of the family box that the prover certified, having subdivided it into leaves until the interval enclosure of `k` fell below the bound everywhere inside. | Field | Type | Meaning | |---|---|---| | `region_id` | string | position in the 4ร—4ร—4ร—4 root partition, e.g. `3.0.3.3` | | `bounds` | object | `{d0_um, pt_mult, sep_mult, jog_mult} โ†’ {lo, hi}` | | `status` | string | `CERTIFIED` for every row in this release | | `certified_leaves` | int | leaves the region was subdivided into | | `processed_leaves` | int | leaves examined, including interior splits | | `sup_certified_k_hi` | float | largest `k` the enclosure admits anywhere in the region | | `volume_fraction_of_region` | float | fraction certified (1.0 throughout) | | `unresolved_leaves` | int | leaves left undecided (0 throughout) | Totals: **237,490 certified leaves**, 474,724 processed, **0 failure regions**, **0 unresolved**, certified volume fraction 1.0. The published rows are the 256-region partition, **not** all 237,490 leaves โ€” the source proof records per-region certification and aggregate counts. That is a real limitation of what is published and it is stated rather than glossed: you can re-derive any region yourself, but you are not being handed every leaf. ### `data/counterexamples.jsonl` โ€” 27 rows, the existential refutation Layouts where `born_second_order` predicts `k > 1`. | Field | Type | Meaning | |---|---|---| | `case_id` | string | e.g. `born2_n6_p100_s1` | | `model` | string | `born_second_order` | | `n_conductors` | int | 6, 8 or 12 | | `nominal_pitch_um` | float | 60, 80 or 100 | | `seed` | int | generator seed | | `worst_pair` | [int, int] | indices of the worst-violating pair | | `k_predicted` | float | predicted screening factor (> 1 for every row) | | `n_pairs_violating` | int | pairs above the ceiling in this layout | | `n_pairs_total` | int | ordered pairs in this layout | | `xy_um` | [[float, float]] | conductor centres, micrometres | | `radius_um` | [float] | conductor radii, micrometres | | `eps_r` | float | relative permittivity (4.6) | **2,060 violating pairs** across the 27 layouts, worst `k = 3.5141`. Full coordinates are included deliberately: a counterexample you cannot rebuild is an anecdote, not evidence. ### `data/theorem.json` โ€” the claim, its scope, its provenance The statement, the family box, the certified totals, the exact geometry definition, and the SHA-256 of the source proof witness. ## Geometry Four parallel circular conductors forming two tight pairs: ``` pitch = 1.6 ยท d0 ยท pt_mult separation = pitch ยท sep_mult jog = jog_mult ยท separation centres: (0,0) (pitch,0) (separation,jog) (separation+pitch,jog) every conductor has diameter d0 ``` Family box: `d0 โˆˆ [25,60] ยตm`, `pt_mult โˆˆ [1.0,1.2]`, `sep_mult โˆˆ [2.5,4.5]`, `jog_mult โˆˆ [โˆ’0.4,0.4]`. Self-term radius scale 1.0. `loader.family_layout(...)` builds it for you. ## Scope, honestly **This is a theorem about the monopole-closure model, not about Maxwell.** The closure is a zero-parameter analytic multiple-scattering model that matches a boundary-element reference to 0.081% in the exact two-cylinder limit, but the closure-versus-solver gap is an **additive, disclosed, unresolved** term. It is never absorbed into the bound. That 0.081% comes from the boundary-element solver used to develop the closure, which is **not part of this release** โ€” so unlike every other figure on this page, you cannot re-derive it from what is published here. It is quoted because it bounds how much trust the closure has earned, and a reader is entitled to know which numbers are checkable and which are taken on our word. The scope line travels with the data, in `theorem.json`: > A complete interval theorem about the frozen MONOPOLE-CLOSURE model only. The > closure-vs-BEM/PDE model gap remains additive and unresolved. This witness does > not establish Maxwell, BEM, driven-S, fabrication, or measured-silicon truth. Three further limits worth stating plainly: - **One family, not all layouts.** Four conductors in a specific arrangement. Nothing here says anything about a different topology. - **The bound is not tight.** The certified supremum is 0.90909089; the worst layout found by adversarial search is โ‰ˆ0.9053. The gap is the price of a first-order interval relaxation, not a claim about physics. That 0.9053 is the second figure on this page you cannot re-derive from the release โ€” it came from a differential-evolution search in the source prover. What you *can* check here is weaker but points the same way: `verify.py` reports a worst sampled `k` of 0.903974725909 at its defaults, and sampling harder climbs toward that 0.9053 without ever reaching the bound โ€” at `--seed 7`, 25 / 100 / 400 samples per region give 0.902144353337, 0.903775408593, 0.904308125283, with zero violations at each. - **No measured data.** Every number is computational. ## Reproduction ```bash python3 verify.py --samples 100 --seed 7 # sample harder, different seed python3 verify.py --self-test # prove the checker still discriminates python3 export.py --check # confirm data matches a fresh export ``` The self-test is the part that makes a clean report worth anything. It fabricates an impossible bound, tampers with a published value, and requires the checker to reject both โ€” then confirms an isolated pair reproduces `k = 1.000000000000` exactly, since a lone pair has nothing to screen it. ## Troubleshooting **`verify.py` reports violations** โ€” that is the interesting outcome, and we want to hear about it. Sampling cannot prove the bound but it can refute it, so a genuine violation means the theorem is wrong. Before reporting, re-run with `--self-test` to confirm the checker still discriminates: a checker that has stopped working can produce either verdict. **`ModuleNotFoundError: numpy` from `verify.py`** โ€” it should never import numpy. If it does, the file has been edited; the shipped version runs under `env -i /usr/bin/python3` with nothing installed, and a test asserts it imports no third-party module. **`loader.to_pandas` or `to_hf_dataset` raises ImportError** โ€” those two are conveniences and do need `pandas` / `datasets`. Everything else, including `verify.py`, is standard library only. **The Hub viewer shows two configs and you wanted one table** โ€” `regions` and `counterexamples` have different schemas and are deliberately separate. Pick the config in the viewer's dropdown, or use `load_regions()` / `load_counterexamples()`. **`export.py --check` says the data does not match** โ€” it re-derives the files from the committed proof witness and compares. A mismatch means either the data or the witness was edited. Counterexample regeneration also needs `maxwell-lint` installed, since they are produced by running its reference models. **You want every certified leaf, not the 256 regions** โ€” they are not published. The source proof records per-region certification plus aggregate counts, so the 237,490 leaves are attested but not enumerated here. That is a real limit of this release and is stated rather than glossed. ## Provenance Exported by `export.py` from a committed proof witness produced by an outward-rounded interval branch-and-bound prover (256 parallel roots, 243.5 s wall clock, centered/mean-value enclosure forms). The witness digest is recorded in `theorem.json`; `export.py --check` re-derives the files and compares. Counterexamples are generated by running the open-source [`maxwell-lint`](https://github.com/nickharris808/maxwell-lint) reference models, so they are reproducible from published code alone. ## Citation ```bibtex @misc{screening_ceiling_2026, title = {Screening Ceiling: Certified Regions and Counterexamples for Many-Body Coupling Extraction}, author = {ChipletOS / Genesis contributors}, year = {2026}, note = {CC-BY-4.0} } ``` ## The rest of the toolkit Eight artifacts that answer one question in different places: **is this model physically possible?** Each is a grader โ€” it can tell you a model is wrong; none can tell you one is right. | | | |---|---| | [`sparam-lint`](https://github.com/nickharris808/sparam-lint) | Is an S-parameter model physically possible? Five laws + a negative control. | | [`maxwell-lint`](https://github.com/nickharris808/maxwell-lint) | Does a coupling extractor predict impossible physics? Screening ceiling k โ‰ค 1. | | [`abstain-bench`](https://github.com/nickharris808/abstain-bench) | Does a model know when to shut up? Abstention recall, never pooled with accuracy. | | [`sparam-conformance`](https://huggingface.co/datasets/nickh007/sparam-conformance) | 11 labelled networks with verified ground truth. Grades the graders. | | [`screening-ceiling`](https://huggingface.co/datasets/nickh007/screening-ceiling) โ† you are here | A certified impossibility result + 27 counterexamples. Zero-dependency verifier. | | [`physics-lint-action`](https://github.com/nickharris808/physics-lint-action) | The same checks, in your CI. | | [`physics-lint-mcp`](https://github.com/nickharris808/physics-lint-mcp) | A physics oracle your AI agent can call. | | [**Try it in your browser**](https://huggingface.co/spaces/nickh007/physics-lint) | All three checks, no install, runs client-side. | These tools **grade** a model. Producing one that is passive *by construction* โ€” so it cannot fail these laws whatever its parameters โ€” and accurate at speed in the many-body regime, with calibrated abstention and a fail-closed signoff certificate, is the commercial core: **[ChipletOS](https://chipletos.com)**. ## Licence **CC-BY-4.0.** Synthetic and computational throughout; no proprietary or measured data. Attribution: ChipletOS / Genesis contributors. ## Related - [`maxwell-lint`](https://github.com/nickharris808/maxwell-lint) โ€” run the ceiling test on *your* extractor - [`sparam-conformance`](https://huggingface.co/datasets/nickh007/sparam-conformance) โ€” the S-parameter analogue - [ChipletOS](https://chipletos.com) โ€” the closed core: a learned many-body coupling operator that stays inside this ceiling and is accurate at speed, with calibrated abstention and a fail-closed signoff certificate ## Contributing One non-negotiable rule here: the verifier must import nothing from this project, so a skeptic can read it in one sitting and run it with nothing installed. [`CONTRIBUTING.md`](CONTRIBUTING.md) has the detail. Each sibling repository states its own, and they differ โ€” that is deliberate, and it is why each is trustworthy on its own terms.