One earthquake pinned my error bound forever

TL;DR — The deterministic error bound carries a global magnitude factor T★ — a single worst-case constant over the entire stream. One large spike pins it at a uselessly large value for every tick that follows, even though the actual error stays tiny. The fix is a windowed factor that heals once the spike ages out: on the printed synthetic run the post-exit ratio between the two envelopes averages ~3.2×10⁵. The capstone scoreboard covers the main empirical legs of both papers: zero violations.

The last article ended on a warning: a certificate is only as good as the data you throw at it. So here is the data. The bound's magnitude factor T★ is a single global constant — the largest product magnitude anywhere in the entire stream. One large spike anywhere fixes it for the whole bound. Feed the kernel that spike early, and T★ sits at the spike's scale for every remaining tick. The error itself stays tiny; the envelope around it stays enormous. The theorem is true, and on exactly the high-dynamic-range data that motivates streaming in the first place — seismic data — its global form tells you almost nothing.

Global pins, local heals

My answer is locality. The first patch I reached for — a prefix maximum T_t that grows as the stream does — repaired nothing: it is non-decreasing, so after the spike it equals T★ at every later tick.

The prefix-max patch also forfeits the bound's defining property — uniformity in stream length — which is the whole point of the re-anchoring topology.
The patch that works: window the magnitude factor on the re-anchoring topology itself. Under a Periodic{k} reseed, only a bounded window of recent ticks can still influence the carry, so the magnitude factor can be taken over that window alone. Call it T_local(t). Once the spike exits the window, T_local collapses back to the pre-spike level — on the printed run, the global-only envelope I would otherwise have shipped runs five orders of magnitude looser for the entire post-exit stream. The global bound remembers forever; the local one is allowed to forget.

The printed synthetic run makes the gap concrete. Stream of 20,000 ticks, window 50, reseed every 200, fp32, worst-case carry lag 199. A spike of scale 10³ enters at tick 300 and leaves the data 21 ticks later. The global envelope jumps from about 0.23 to about 8.1×10⁴ and never returns. The local envelope rides at the same inflated level only while the spike sits inside its window — then, at tick ≈621, it heals back to roughly where it started. (In the chart below the drop lands at the t = 600 re-anchor boundary, where the envelope is next re-evaluated, just ahead of the printed window-exit tick of ≈621.) Post-exit ratio between the two: a mean of ~3.2×10⁵. Same theorem, same stream, five orders of magnitude of difference in what the bound is worth.

![Log-scale forward error and bounds versus stream tick for the printed synthetic healing run (N=20,000, w=50, k=200, fp32): a spike of scale 1e3 at ticks 300,321) pins the global T-star envelope at ~8.1e4 for the rest of the stream, while the windowed local envelope drops back to ~0.2-0.35 at the t=600 re-anchor boundary, just before the printed window-exit tick of ~621; the measured error stays between ~4.5e-11 and ~5.5e-4 throughout, below both envelopes at every tick.

Global versus windowed envelope on the synthetic healing run: one spike pins the global bound forever; the local bound heals at tick ≈621, post-exit ratio mean ≈3.2×10⁵. Measured error never exceeds either. Rebuilt from the data behind Figure 3 of the preprint (doi:10.5281/zenodo.20599315).

The earthquake

The synthetic spike is a planted toy, so I ran the same comparison on the real Parkfield broadband seismogram. The toy's pathology reproduced exactly: the global factor stays pinned at the quake's peak for the entire post-event stream, while T_local recovers to four to five orders of magnitude smaller across the benign tail. The measured drift never exceeded either envelope at any point of the post-event stream. The bound held on data that was designed, by nature rather than by me, to embarrass it.

The figure below traces the whole record — every series across the full stream, not a window around the event. The mainshock arrives partway through. The global envelope steps up to the quake's peak and stays there flat to the end. The local envelope rides up with it through the event, then bleeds back down across the long benign tail that follows. The measured drift sits a dozen-plus orders of magnitude below both the whole way across, with no excursion toward either line at the mainshock or anywhere after it.

The full long-record run: the global envelope pins at the mainshock

Every number in this article matches the public preprint text; nothing here is quoted from private runs.

What "holds" means

Here is the empirical record of both papers, with the failure count printed. This table doubles as the series index.

empirical leg	scale	result	owner
unbroken-chain drift sweep	N = 2¹³…2²⁴, 7 corpora × 30 seeds, f64 + f32	traces the predicted carry-chain growth (numbers in Part 1)	Part 1
anchored Periodic{k} grid	18 (precision, accumulator, k) cells	0 envelope violations	Part 1
regime-boundary sweep	full regime-boundary grid	0 failures; every in-regime cell contained	Part 1
stochastic-rounding ensemble	20,000 seeds × 4 precision tiers	coverage held at every tested failure probability	Part 2
centred cross-covariance grid	100 (precision, window, distance, band) cells	0 envelope violations	Part 3
Parkfield seismic record	mainshock broadband record	measured drift never exceeded either envelope	this article

A worst-case bound that survives all of this is a falsifiable prediction that was given every chance to fail — planted adversaries, 20,000 random seeds, a real earthquake — and did not. That is the standard the gate-design article argues for: these checks could have failed, visibly, in any of those cells.

One narrow family of kernels

These are theorems about one family of streaming inner-product kernels under IEEE-754 arithmetic, machine-checked because the domain is narrow enough that the spec-faithfulness question can be mechanized — the trust stack works because the territory is small. Both manuscripts are public preprints (doi:10.5281/zenodo.20599315, doi:10.5281/zenodo.20599478) and currently under review. The work is part of an ongoing research program, and that is all this series will say about what comes next.