The error a reduced model has, and the bound computed before it existed

At its defaults it draws the error a reduced model has, and the bound computed before it existed. ‖H − Hᵣ‖∞ for balanced truncation of a 20-state model, against 2(σᵣ₊₁ + … + σₙ) — a number available from two Lyapunov solves before any reduced model is formed — and against σᵣ₊₁, which is a lower bound no reduced model of that order can beat. The measured error and the upper bound are the same curve: the ratio is 1.0000 to 1.0000 across the sweep, spread 1.0000. The bound is attained rather than approached, because on a spectrum falling this fast the tail sum is its own first term and removing one state costs exactly twice it. The measured error sits at 2.03 times the lower bound, so the whole question of what order to truncate at is answered by a curve nobody has to compute a model to see.

truncation-bound is one function in lib/figures/gramian.js — the bound known before the computation — hankel singular values, and the product that halves them. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

‖H − Hᵣ‖∞ for balanced truncation of a 20-state model, against 2(σᵣ₊₁ + … + σₙ) — a number available from two Lyapunov solves before any reduced model is formed — and against σᵣ₊₁, which is a lower bound no reduced model of that order can beat. The measured error and the upper bound are the same curve: the ratio is 1.0000 to 1.0000 across the sweep, spread 1.0000. The bound is attained rather than approached, because on a spectrum falling this fast the tail sum is its own first term and removing one state costs exactly twice it. The measured error sits at 2.03 times the lower bound, so the whole question of what order to truncate at is answered by a curve nobody has to compute a model to see.

Twelve states — six modes with damping ratio 0.03 and frequencies spread geometrically from 1 to 30 — reduced by balanced truncation to every order from 1 to 11; for each, twice the sum of the discarded Hankel singular values divided by the measured largest gain of the error over frequency. The ratios are 3.15, 4.04, 3.06, 4.84, 4.22, 3.26, 2.22, 2.90, 1.89, 2.02, 1.00. The dashed line is one, where the bound is attained, as it is at every order on the heat model.

Twelve states — six modes with damping ratio 0.1 and frequencies spread geometrically from 1 to 30 — reduced by balanced truncation to every order from 1 to 11; for each, twice the sum of the discarded Hankel singular values divided by the measured largest gain of the error over frequency. The ratios are 3.23, 4.53, 3.52, 4.47, 4.55, 1.75, 2.43, 3.16, 2.01, 2.13, 1.00. The dashed line is one, where the bound is attained, as it is at every order on the heat model.

Hankel singular values in decreasing order on a logarithmic axis. At ζ = 0.7: 3.56, 0.331, 0.225, 0.0869, 0.0252, 0.0187, 0.00583, 0.00241, 0.00193, 0.00161, 4.55·10⁻⁴, 1.75·10⁻⁴. At ζ = 0.3: 6.19, 2.29, 0.925, 0.813, 0.794, 0.288, 0.223, 0.158, 0.122, 0.106, 0.0713, 0.0545. At ζ = 0.1: 13.2, 10.4, 6.15, 5.35, 2.78, 1.95, 1.94, 1.79, 0.927, 0.817, 0.406, 0.355. At ζ = 0.03: 40.8, 38.4, 21.5, 20.4, 8.44, 7.72, 7.63, 7.38, 3.38, 3.2, 1.44, 1.36. At ζ = 0.01: 120, 118, 63.9, 62.7, 24.4, 23.9, 23.2, 22.9, 10.1, 9.88, 4.28, 4.19. As the damping falls they arrive in pairs — one pair per mode — whose members agree to about the damping.

Bound over measured error when balanced truncation removes the two smallest Hankel singular values — the last mode — and when it removes the single smallest, against the damping ratio on a logarithmic axis. Last mode: 1.398 at ζ = 0.7, 2.019 at ζ = 0.3, 2.131 at ζ = 0.1, 2.019 at ζ = 0.03, 2.002 at ζ = 0.01. Last state: 1.0000, 1.0000, 1.0000, 1.0000, 1.0000.

For each order kept, the frequency at which the gain of the error system peaks, on a logarithmic axis, with the six modal frequencies 1.00, 1.97, 3.90, 7.70, 15.19, 30.00 dashed. The peaks sit at 1.00, 1.97, 1.97, 3.88, 3.88, 3.88, 4.28, 15.17, 15.17, 29.98, 29.98.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

23 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

It draws a decomposition and prints its residual. It calls hankelSingularValues, balancedTruncation, reductionError, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.