tropical-prediction

At its defaults it draws a prediction from three numbers, exact at one end of the spectrum and wrong by n² at the other. The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 0.5K. The large root is within 5.7 per cent of the largest modulus at every size. The small root is out by 5.36, 17.1, 60.8 and 229.6 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.

tropical-prediction is one function in lib/figures/qepscale.js — the units a polynomial is written in — two lines of scaling, a prediction that is half right, and a format's edge. 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.

The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 0.5K. The large root is within 5.7 per cent of the largest modulus at every size. The small root is out by 5.36, 17.1, 60.8 and 229.6 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.

The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 0.5K. The large root is within 5.7 per cent of the largest modulus at every size. The small root is out by 5.36, 17.1, 60.8 and 229.6 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.

The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 0.2K. The large root is within 4.4 per cent of the largest modulus at every size. The small root is out by 5.76, 18.6, 66.7 and 252 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.

The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 4K. The large root is within 31 per cent of the largest modulus at every size. The small root is out by 3.09, 8.59, 29 and 107.8 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.

The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 1K. The large root is within 15 per cent of the largest modulus at every size. The small root is out by 4.8, 15, 52.9 and 199.1 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.

The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 2K. The large root is within 24 per cent of the largest modulus at every size. The small root is out by 4, 12, 41.6 and 156 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions 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.

7 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.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 173 of 325 generators — 158 print a residual and 15 are exempt with a published reason; 152 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.