L0/10 — Screening and locality (M2)

Background

Electrostatics has no built-in locality: every pair of conductors in the universe has a nonzero capacitance between them, however far apart. Yet every extraction tool, when computing a wire's couplings, looks only a few wire-pitches away and ignores the other sixty thousand nets on the die. What makes that truncation sound cannot be "distant couplings are individually small" — the chapter's opening computation shows the sum over all distant conductors diverges if nothing intervenes, because the count of conductors grows faster with distance than pairwise coupling shrinks. The truncation is sound, if it is, for a different reason: screening — grounded metal in between blocks the field.

The physics is the Faraday-cage principle. A conductor held at a fixed potential terminates electric field lines: whatever field arrives on one side induces surface charge that exactly cancels its influence on the other side. A wire fully enclosed in grounded metal couples to the outside world not weakly but not at all — and a chip's power-distribution grid, a dense mesh of supply metal on every routing layer held at fixed voltage (that "held at" being the decap army's job, a hypothesis to be tracked, not assumed), approximates exactly such an enclosure around every signal wire. Approximates: a mesh has holes, and field leaks through them. The chapter's mathematical task is to quantify the leakage, and the tools are two classical pillars of potential theory. The maximum principle says a harmonic function (which the potential is, away from charges) attains its extremes on the boundary — no interior bumps — so bounding a potential on a surrounding surface bounds it everywhere beyond. Harmonic measure refines this into an accounting of where the boundary's influence comes from: the potential at a point is a weighted average over the boundary, and the weight of the holes-in-the-mesh portion is exactly the factor by which one layer of mesh attenuates the field.

The structure of the conjectured answer explains why a mesh works where a single barrier would not: one layer of holes attenuates by some fixed factor α < 1, and a distant net sits behind many layers in series, so the attenuation compounds exponentially in the number of grid cells crossed — which beats the polynomial growth in the number of distant conductors and makes the discarded sum genuinely negligible. Every capacitance number in this book — and in every industrial flow — rests on this argument existing; it is the widest-error-bar open problem in the layer, and nobody has written it down.

Why this is the load-bearing open problem

Every practical extraction truncates: only conductors within a few tracks are considered. The justification is not "distant things are small". Without screening the far-field sum diverges.

The count of conductors at distance d grows polynomially (~d in a quasi-2D die), while unscreened coupling between two thin parallel wires decays only logarithmically, C ~ 1/ln(d/a). So Σ_d count(d)·C(d) has no chance of converging. Local extraction is sound only because intervening grounded metal screens, and quantifying that is this document.

Everything in 09 is computed on a truncated window. If M2 is false, those numbers bound nothing.

The conjecture

Let n be a net, W_d(n) the conductors within distance d, and C_d the capacitance matrix computed with all conductors outside W_d(n) grounded. Suppose the power grid is a mesh of pitch p with aperture parameter a, held at a fixed potential, and every intervening conductor is either grounded or driven with impedance bounded by Z_max over the band of interest. Then there exist K and α < 1, depending only on (p, a, ε_max/ε_min), with

Σ_{m ∉ W_d(n)} |C(n,m) − C_d(n,m)|   ≤   K · α^{d/p}

Two features of the statement are deliberate.

The bound is on the aggregate, not per pair. There are ~61,000 nets; a per-pair bound |C(n,m)| ≤ ε would give 61000·ε, which is not small. The sum is the object that must be controlled.

The decay is exponential in d/p — cells traversed, not distance. That is what beats the polynomial count and makes the series converge. A power law would leave convergence marginal and dependent on the exponent.

Proof route

Step 1 — the exact case is topological

If a grounded conductor G separates n from m — every path from one to the other meets G — then C(n,m) = 0 exactly.

Proof: the boundary value problem decouples. Solve inside and outside G independently with G's Dirichlet condition; neither solution sees the other's data.

This is worth isolating because the hypothesis is topological, not metric, hence combinatorially checkable — and because it is what deliberate shielding (grounded wires alongside a critical net) buys: exactly zero, not a bound. It also gives the base case for what follows.

Step 2 — one-cell attenuation via harmonic measure

Real grids are meshes, not solid planes, so the field leaks through apertures. Take a closed surface S of grounded conductor plus apertures; let φ be the potential from n with all else grounded, φ ≤ M on the apertures, φ = 0 on the conductor. By the maximum principle, for x beyond S:

φ(x) ≤ M · ω(apertures, x)

where ω is the harmonic measure of the aperture set seen from x. Complete enclosure gives ω = 0, recovering Step 1.

The lemma to prove is the one-cell bound: for a periodic aperture array of pitch p and aperture size a, ω ≤ α(a/p) < 1 at distance ≳ p beyond the barrier. Explicit for simple geometries; the general estimate is the technical core.

Step 3 — cascade

Between n and a net at distance d lie roughly d/p mesh cells in series. Iterating Step 2 multiplies the attenuation:

φ beyond k barriers ≤ α^k · φ at the source

by induction on k, each step an application of the maximum principle to the region between consecutive barriers. Converting a potential bound to a capacitance bound is then a surface integral of ε∇φ over the far conductor.

Cascaded apertures are the whole mechanism. A single aperture gives only a power law; the exponential comes from having many in series, which is exactly what a dense power mesh provides.

Hypotheses that must be stated, not assumed

The grid must be an equipotential over the band of interest. The maximum-principle argument needs φ = 0 on the shield. A wobbling shield is a source, not a boundary condition. This is exactly what L1's ~225,000 decaps establish — a frequency-domain impedance bound, discharged once. So the decaps do double duty: supply integrity and shield quality, and removing them degrades this theorem, not just the power delivery.

Floating conductors relay rather than screen. A conductor not held at a potential does not terminate field lines — it couples in and out, shortcutting an aperture chain and destroying the cascade. Consequence: metal fill must be tied, not floating. This should be a requirement on the design, not an option; a design with floating fill has no valid locality argument at all.

Driven signal nets screen only partially. A driven net is low-impedance through its driver's output resistance, which rises with frequency; digital edges carry content well above where the drivers are stiff. So "grounded" is an idealisation and the honest hypothesis carries Z_max over the band, which is why the conjecture states it that way.

What follows if it holds

• The coupling graph has support bounded by the truncation radius — each net has O(10) neighbours instead of O(61,000). That sparsity is what makes L1's crosstalk treatment and 09's per-window solves possible at all.
• X3 stops being an axiom. It is currently the assumption standing in for this theorem.
• Truncation radius becomes a derived quantity: choose d so that K·α^{d/p} fits the margin, rather than choosing 2–3 tracks by convention.

What is actually at stake

Read this carefully, because the naive reading is wrong. The property — the aggregate coupling converges — is not in doubt: working silicon establishes it, and every industrial extractor already truncates at a fixed window and matches measured chips, which is the empirical certificate that the far field is negligible (it is the content of axiom E3, the calibrated rule deck). So screening is the mechanism and it is present — the divergence of the unscreened sum is elementary, so something must supply convergence, and a grounded mesh is the only candidate. What M2 buys is not the property but its a-priori proof: without it, the far-field truncation stays an empirically calibrated step (E3 trusted, not derived) and only the industrial capacitance numbers — calibrated, not proved — are available; with it, the truncation radius becomes a derived quantity with a certified error bound, and E3's far-field part discharges into a theorem. The open problem is the rate that licenses cheap local extraction over the empirical deck, never whether the coupling is bounded.

Obligations

1. The one-cell harmonic-measure bound — the technical core, and the piece I would treat as genuine analysis rather than engineering.
2. The cascade lemma: iterated maximum principle across successive barriers.
3. Potential bound → capacitance bound (surface integral; routine given 1–2).
4. A checkable shielding-coverage predicate on the layout: the design-side hypothesis that there really is grid metal interposed. Combinatorial, and it belongs with 12.

First experiments

• Numerically test the cascade before committing. A 2D mesh of grounded strips with a source on one side: measure attenuation per cell, check it is geometric in the number of cells, and extract an empirical α(a/p). Cheap, and it either supports the exponential form or kills it early.
• Compute the shielding coverage actually present in the flow's layout between representative net pairs — is there grid metal interposed, or are there sparsely-gridded regions where the hypothesis fails locally?
• Check whether the flow's metal fill is tied or floating. If floating, the locality argument needs the harder version.

Effort

Unknown, and the widest error bars in the project — obligation 1 could be six months or a thesis. It is also the highest-leverage single result in L1, since everything in local extraction depends on it.

Reading

Pólya & Szegő for capacity comparison methods. Standard potential-theory texts for harmonic measure and the maximum principle; Garnett & Marshall, Harmonic Measure, for the estimates Step 2 needs.