gram-schmidt-order

At its defaults it draws classical and modified gram–schmidt: the same subtraction, in a different order. Two panels of pseudocode differing in one argument, with the resulting pairwise dot products of the computed columns listed beneath each.

gram-schmidt-order is one function in lib/figures/ortho.js — orthogonality — ‖qᵀq − i‖ as a measurement rather than an adjective. 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.

Two panels of pseudocode differing in one argument, with the resulting pairwise dot products of the computed columns listed beneath each.

Two panels of pseudocode differing in one argument, with the resulting pairwise dot products of the computed columns listed beneath each.

Two panels of pseudocode differing in one argument, with the resulting pairwise dot products of the computed columns listed beneath each.

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.

13 distinct claims across 3 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 calls a factoriser without drawing a factorisation (gramSchmidt), so the rule is written down as not applying, with the reason: prints the pairwise dot products, which is ‖QᵀQ − I‖ entry by entry

The exemption list is the interesting half of the rule rather than an escape hatch — it is where a decision about a figure had to be argued in one line. residualcheck refuses an exemption that is not doing work, and rejected ten of the fifteen written for the expansion's figures on exactly that ground: a figure whose vertical axis is a residual satisfies the rule by construction, and touching a factoriser does not by itself require an entry.

Across the library: the rule bites on 52 of 99 generators — 37 print a residual and 15 are exempt with a published reason; 47 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.