/* ============================================================
   This page on top of the kpress design system.

   Everything typographic -- the reading measure, the type ramp, headings,
   lists, code, the color roles and both themes -- comes from the kpress
   stylesheets above. What follows is only what kpress has no component for:
   the certificate's figures and the two interactive panels. Every value here
   is a kpress token, so the figures move with the system rather than beside
   it.
   ============================================================ */

/* This standalone article scrolls the document, so the browser preserves its
   reading position on reload and history traversal. KPress also supports host
   panes, but its pane reset would leave the document at scroll position zero.
   Keep main's width-query container without creating a second scroll context. */
html,
body {
  height: auto;
  overflow: visible;
}
.kpress-page-main.kpress-viewport {
  block-size: auto;
  overflow: visible;
}

/* One accent, everywhere. kpress ships a separate link blue and sets it on
   `:root`, on `.kpress`, and again per theme and palette on attribute selectors,
   all of which outrank `body`; the popovers it appends carry `.kpress-tooltip`
   and their own resolved-theme attribute. Redeclare the token in the same
   places kpress does, so links and controls inside those popovers resolve the
   same accent as the surrounding page. */
:root,
.kpress,
.kpress-page-main,
.kpress-tooltip,
:is(.kpress, .kpress-page-main, .kpress-tooltip)[data-kpress-resolved-theme],
:is(.kpress, .kpress-page-main, .kpress-tooltip)[data-kpress-palette][data-kpress-resolved-theme],
body {
  --kpress-doc-link: var(--kpress-doc-accent);
  --color-primary-light: var(--kpress-doc-accent);
}
/* kpress declares `--kpress-doc-accent` once, for both themes, and gives
   `--kpress-doc-link` a value per theme. Aliasing the link to the accent above
   therefore took the light teal into dark mode as well, where it measured 3.37:1 on
   the page ground against the 4.5:1 AA needs -- every link on the page, plus the
   footnote references and the pressed chooser label at 2.99:1 and 3.07:1 on their
   tinted grounds. The accent is what carries the alias, so the accent is what gains a
   dark value: the same hue and chroma lifted to the lightness kpress's own dark link
   uses, which brings everything that reads the accent up together rather than fixing
   links and leaving the borders and the range control behind. Scoped exactly as kpress
   scopes its own dark tokens, on the resolver's output attribute, so an explicit
   choice and the system default both land here. */
:root:where([data-kpress-resolved-theme="dark"]),
:where([data-kpress-resolved-theme="dark"]) :is(.kpress, .kpress-page-main, .kpress-tooltip),
:is(.kpress, .kpress-page-main, .kpress-tooltip)[data-kpress-resolved-theme="dark"] {
  --kpress-doc-accent: oklch(76.68% 0.0861 186.4);
}
/* The reading measure (forty of the base) and the h2 scale (1.2) are in paper-type.css,
   shared with every page of the site. */
.cert-page {
  padding-block-end: calc(var(--kpress-font-size-base) * 6);
}
/* Paper typography profile; see paper-design.md beside this template.
   Keep the values and their scope together for review upstream to KPress:
   a 19px sans base against 18px prose, regular 410, medium 550, bold 680.
   Captions and footnotes share a slightly smaller size than figure labels.
   Print keeps the same size ratio.
   Scope the bold token to sans components because KPress also uses it for serif
   emphasis. Diagram labels with explicit SVG weights join the same scale.

   Three weights and no others, which is the rule rather than the tally: one bold
   everywhere in the sans, one medium everywhere, one light for supporting text.
   `sans_instances --weights` lists what the page asks for in both media with the
   declaration behind each request, and it is how a fourth weight is found. Two used
   to be there: the caption label sat at the medium while the title credits were bold,
   and the footnote controls at KPress's literal 600, which is neither token. */
/* The values, and the `--paper-*` role scales, are in paper-type.css, shared with every
   page of the site; this page reads them under its own names. */
:root {
  /* The tints a paper paints with, each a share of a token colour in the rule that uses
     it, so the dark theme's colours mix in where they are set
     (`devtools.check_colour_tokens`): the accent's under a tag, a passed verdict and a
     pressed button; the failing colour's under a failed verdict; the text's under a
     hovered button; the probe's under a hovered handle. */
  --paper-accent-tint: 12%;
  --paper-below-tint: 16%;
  --paper-hover-tint: 10%;
  --paper-probe-tint: 12%;
  --cert-font-scale-sans: var(--paper-font-scale-sans);
  --cert-font-weight-sans-light: var(--kpress-font-weight-sans-regular);
  --cert-font-weight-sans-medium: var(--paper-font-weight-sans-medium);
  --cert-font-weight-sans-bold: var(--paper-font-weight-sans-bold);
}
/* Resolve the sans base in the prose scope so nested sans components all use
   the same size instead of multiplying the scale at each level. */
:root,
.cert-page {
  --cert-font-size-sans-base: calc(var(--kpress-font-size-base) * var(--cert-font-scale-sans));
  --paper-font-size-support: calc(var(--cert-font-size-sans-base) * var(--paper-support-scale));
  --paper-font-size-note: calc(var(--cert-font-size-sans-base) * var(--paper-note-scale));
  --paper-support-color: var(--kpress-doc-muted);
}
.cert-page
  :is(
    .credits,
    .kpress-figcaption,
    .kpress-footnotes,
    .colophon,
    .mass-line,
    .panel,
    .tip-panel,
    .caps,
    .tag,
    button,
    .doc-links .chip,
    .hero h1,
    .hero .subtitle
  ),
.kpress-tooltip {
  --kpress-font-size-base: var(--cert-font-size-sans-base);
  --kpress-font-size-normal: var(--kpress-font-size-base);
  --kpress-font-size-large: calc(var(--kpress-font-size-base) * 1.2);
  --kpress-font-size-small: calc(var(--kpress-font-size-base) * 0.95);
  --kpress-font-size-smaller: calc(var(--kpress-font-size-base) * 0.9);
  --kpress-font-size-tiny: calc(var(--kpress-font-size-base) * 0.85);
  --kpress-caps-label-size: calc(
    var(--kpress-font-size-smaller) *
    var(--kpress-caps-heading-size-multiplier)
  );
}
.cert-page .panel {
  font-size: var(--kpress-font-size-normal);
}
.cert-page .hero h1 {
  font-size: calc(var(--cert-font-size-sans-base) * var(--paper-title-scale));
}
.cert-page,
.kpress-tooltip {
  --kpress-font-weight-sans-light: var(--cert-font-weight-sans-light);
  /* The medium is mapped at this scope rather than at the sans components' because it
     is the same 550 KPress already uses, so nothing the page does not own can change
     under it; naming it here is what lets every medium on the page read as a token. */
  --kpress-font-weight-sans-medium: var(--cert-font-weight-sans-medium);
}
.cert-page
  :is(
    .credits,
    .kpress-figcaption,
    .kpress-footnotes,
    .colophon,
    .mass-line,
    .panel,
    .tip-panel,
    .line-fig,
    .chart,
    button
  ),
.kpress-tooltip {
  --kpress-font-weight-sans-bold: var(--cert-font-weight-sans-bold);
  font-weight: var(--cert-font-weight-sans-light);
}
/* Diagram labels carry their weight as an SVG presentation attribute, because that is
   what an `<svg>` written by `render_n11_lower_bounds_explainer` can say about one `<text>`. The two
   numbers it emits are markers, not weights: they select the scale's medium and its
   bold here, so a token moves them with everything else. */
.cert-page :is(.line-fig, .chart) [font-weight="550"] {
  font-weight: var(--kpress-font-weight-sans-medium);
}
.cert-page :is(.line-fig, .chart) [font-weight="650"] {
  font-weight: var(--cert-font-weight-sans-bold);
}
.cert-page a,
.cert-page .kpress-footnote-ref a,
.cert-page a.kpress-footnote-backref,
.cert-page a:hover,
.kpress a.kpress-footnote-backref:hover,
.kpress-tooltip a,
.kpress-tooltip a:hover,
.kpress-tooltip .kpress-footnote-nav-link {
  color: var(--kpress-doc-accent);
}
.cert-page a,
.kpress-tooltip a {
  text-decoration: none;
}

/* Supporting text shares its family, weight and color in both media.
   KPress's print text-role reset below turns the inherited gray into black. */
.cert-page
:is(.kpress-figcaption, .kpress-footnotes, .mass-line, .line-fig, .chart, .panel, .tip-panel),
.cert-page
  figure
  :is(.caps, .hint, .legend, .kv, .val, .mass-val, .mass-dec, .verdict, .tag, button) {
  font-family: var(--kpress-font-sans);
  font-size: var(--paper-font-size-support);
  font-weight: var(--cert-font-weight-sans-light);
  line-height: var(--paper-support-leading);
  color: var(--paper-support-color);
}
.cert-page :is(.kpress-figcaption, .kpress-footnotes) {
  font-size: var(--paper-font-size-note);
  padding-inline: var(--paper-note-inset);
}
.cert-page :is(.kpress-figcaption, .kpress-footnotes) strong {
  color: inherit;
}
/* Sources, not controls: the ↑ after each footnote is kpress's footnote control and
   keeps the control styling (accent here, through the link alias above). */
.cert-page :is(figure, .kpress-footnotes) a:not(.kpress-footnote-backref),
.cert-page :is(figure, .kpress-footnotes) a:not(.kpress-footnote-backref):hover {
  color: var(--paper-support-color);
  text-decoration: none;
}
.cert-page :is(.kpress-figcaption, .kpress-footnotes, .mass-line, .panel) .katex {
  color: inherit;
}
.cert-page .kpress-footnotes code {
  color: inherit;
}
.cert-page :is(.line-fig, .chart) svg text {
  font-family: var(--kpress-font-sans);
  font-size: var(--paper-diagram-font-size, var(--paper-font-size-support));
  fill: var(--paper-support-color);
}
.cert-page figure .figure-note {
  font: inherit;
  color: inherit;
  margin-block: 0.25rem 0;
}
.cert-page figure .verdict.ok {
  color: var(--kpress-doc-accent);
}
.cert-page figure .verdict.no {
  color: var(--cert-below);
}
/* Diagram geometry scales with its viewport; the shared script compensates
   label sizes. Narrow screens scroll the diagram instead of shrinking its text. */
.cert-page :is(.line-fig, .chart) {
  overflow-x: auto;
  margin-block-end: 0;
}
.cert-page :is(.line-fig, .chart) svg {
  min-width: calc(var(--paper-font-size-support) * 34);
}
@media print {
  .cert-page a,
  .cert-page a:hover {
    text-decoration: none;
  }
  .cert-page :is(.line-fig, .chart) {
    overflow: visible;
  }
  /* Fix the drawing width before pagination so label compensation sees the
     same viewport as the PDF: Letter text column minus figure side padding. */
  .cert-page :is(.line-fig, .chart) svg {
    min-width: 0;
    width: calc(8.5in - 2 * var(--kpress-print-page-margin) - 2.2rem);
  }
}

/* The footnote controls -- the reference in the body, the arrow back from each source,
   the tooltip's next and previous -- at the scale's medium. KPress sets all three from
   one literal 600, which is neither of this page's tokens and was the page's fourth sans
   weight: two instanced faces in the PDF for thirty superscript figures. 550 is 50 units
   below it and still reads as a mark rather than as body text at the 0.75em these run
   at; if a control ever needs more presence than that, the answer is the bold token and
   not a number between the two.

   The tooltip's link is named at KPress's own specificity and comes later in the
   cascade, which is how it wins; the two in the page win on `.cert-page`. */
.cert-page .kpress-footnote-ref a,
.cert-page a.kpress-footnote-backref,
.kpress-tooltip .kpress-footnote-nav-link {
  font-weight: var(--kpress-font-weight-sans-medium);
}

/* One transition for every hover and every replaced readout: the site's one hover
   transition (`--site-hover-transition`, site-nav.css, paper-design.md Motion), fast
   and the same everywhere. The scripts set `.swap` on a readout whose text they
   replace, so it fades in rather than flicking. */
.cert-page a,
.cert-page button,
.cert-page .verdict,
.cert-page .mass-dec,
.cert-page .kpress-footnote-ref a,
.cert-page a.kpress-footnote-backref {
  transition: var(--site-hover-transition);
}
@keyframes cert-swap {
  from {
    opacity: 0;
  }
  to {
    opacity: 1;
  }
}
.cert-page .swap {
  animation: cert-swap 160ms ease-out;
}
/* The two footnote anchors are named again rather than left to `.cert-page a`: a
   media query adds no specificity, so the transition above at (0,2,1) would outrank
   this rule's (0,1,1) and keep running for a reader who asked for no motion. It is
   also a render hazard -- a transition caught in flight is one of the three things
   that make the PDF export draw the page differently twice (D-490). */
@media (prefers-reduced-motion: reduce) {
  .cert-page a,
  .cert-page button,
  .cert-page .verdict,
  .cert-page .mass-dec,
  .cert-page .swap,
  .cert-page .kpress-footnote-ref a,
  .cert-page a.kpress-footnote-backref {
    animation: none;
    transition: none;
  }
}

/* Prose keeps the kpress measure; figures are allowed a wider track. The
   Markdown body has no column wrappers, so every direct child of the page and
   of an article takes the measure unless it is a figure. kpress indents a list
   with a margin, which the measure resets; the indent comes back as padding
   inside the measured box, so a list's text is narrower by the same amount. */
.cert-page > :is(ul, ol) {
  box-sizing: border-box;
  padding-inline-start: 1.8rem;
}
.cert-page > :not(figure, .cert-figure, .kpress-figure),
.col {
  margin-inline: auto;
  max-width: calc(var(--kpress-measure) + 2 * var(--kpress-column-inset));
}
.cert-page figure {
  margin-inline: auto;
  margin-block: calc(var(--kpress-font-size-base) * 1.8) 0;
  max-width: calc(var(--kpress-measure) + 14rem);
  border: 1px solid var(--kpress-doc-border);
}

/* A section label, in the kpress caps idiom. */
.caps {
  color: var(--kpress-doc-muted);
  font-family: var(--kpress-font-sans);
  font-size: var(--kpress-caps-label-size);
  font-weight: var(--kpress-font-weight-sans-medium);
  letter-spacing: var(--kpress-caps-spacing);
  text-transform: var(--kpress-caps-transform);
}
/* A section starts with room above its heading, so it reads as a new step. */
.section-head {
  margin-block: calc(var(--kpress-font-size-base) * 2.6) 0.9rem;
  text-align: center;
}
/* A section starts with room above its heading and a breath below it. */
.cert-page h2 {
  margin-block: var(--paper-section-space) var(--paper-section-space-below);
  text-align: center;
}
/* A box holds discrete statements rather than continuous argument, so its
   paragraphs stand a little further apart than running prose, which kpress sets at
   0.75rem. One rule serves both boxes on the page, the five conditions and the
   contradiction, so they cannot drift into two rhythms.

   The conditions were a bulleted list and are now paragraphs. The bold lead already
   numbers and names each one, so a marker beside it said the same thing twice, and
   losing the list's hanging indent lets each condition start on the measure with
   everything else. `.conditions ol` and `.conditions li` styled an ordered list this
   page had stopped emitting; they are gone with it. */
.cert-page .boxed-text {
  margin-block: calc(1rem + 0.5lh);
}
.cert-page .boxed-text p {
  margin-block: 0.95rem;
}
.intro .section-head {
  margin-block-start: 0;
}

.hero {
  padding-block-start: calc(var(--kpress-font-size-base) * 3);
}
/* The heading is set in caps by kpress; a formula in a title is mathematics and keeps
   both mathematical italic and lowercase rather than inheriting the heading transform,
   which would print Part I's `n = 11` as `N = 11` and Part II's `s(11)` as `S(11)`,
   different symbols. It also stays on one line: KaTeX allows a break after a relation,
   and at phone width the title ended one line on `n =` and began the next with `11`.
   Part I's prepared math is a `.tex` run; Parts II and III write `$...$`, which KPress
   sets as `.kpress-math`. One rule holds both. */
.hero h1 .tex,
.hero h1 .kpress-math {
  letter-spacing: 0;
  text-transform: none;
  white-space: nowrap;
}
/* Prepared math reserves its full measured ascent and descent. In the title, a math
   run's 1.27-1.30em profile reservations (measured on `s(11)`, which the title once
   carried; `n = 11` reserves 1.20-1.27em) are taller than the heading's 1.05
   line-height and can push Figure 1 onto the next printed page. Both profiles keep a
   0.9123em ascent, so this limits only the invisible inline box while leaving the 1em
   glyphs in the same painted position: 1.05 - 0.9123 = 0.1377em below the baseline. */
.hero h1 .squares-math-box {
  /* biome-ignore lint/complexity/noImportantStyles: prepared math reserves a measured inline box and KPress sets this dimension inline. */
  height: 1.05em !important;
  /* biome-ignore lint/complexity/noImportantStyles: the paired baseline correction must override the prepared inline value. */
  vertical-align: -0.138em !important;
}
/* Mathematical relations set in a sans line: `≥` in the title and the certificate
   buttons, `≈` in two captions and in the chart labels, `→` in Figure 7's label.
   Source Sans 3 carries none of the three, so the browser used to resolve them to
   whatever family on the reader's machine had them.

   That was drawn as a weight bug first. The hero is set at 550 and CSS weight matching
   searches upward from anything above 500, so the fallback family, offering 400 and
   700, answered with Bold: one bold character in a semibold line, 18% heavier in stroke
   and up to twice the ink of the digits beside it, from DejaVu Sans Bold in the printed
   file while every other glyph on the line came from Source Sans 3. A `.rel { 400 }`
   class fixed the weight and left the family to the host, and the three treatments
   compared then -- the host sans at 400, the page's own KaTeX_Main, and the whole line
   set as mathematics -- picked the host sans, because it is drawn like the digits beside
   it and a Computer Modern relation is a hairline against Source Sans.

   The rule that the page resolves every run to a face it ships retires that choice: a
   glyph from the reader's machine is a different glyph for every reader, and on macOS
   the PDF writes it as outline paths. Setting the whole title as mathematics would
   hand the relation and bound to formula-specific glyph selection, spacing, and weight
   rather than the established title typography, so the relation comes from KaTeX_Main,
   subset to those three code points and declared on the sans families themselves
   (`relation_face_css`) rather than named by a class. The title's `s(11)` is separately
   set as mathematics, where the italic `s` carries meaning, while the relation and
   value remain part of the sans title line. The relation face also reaches the chart
   labels and the caption prose, which no class did.

   KaTeX_Main-Bold, not Regular, and the comparison is the reason. Measured with
   `compare_math_fonts metrics`, in units of a thousandth of an em, the rule thickness of
   the minus -- which is the bar every relation here is drawn on -- is 40 in
   KaTeX_Main-Regular and 60 in its Bold, against Source Sans 3's own 62 at the 410 the
   captions run at and 78 at the 550 of the title. Bold lands within 3% of the sans at
   410 and at 77% of it at 550; Regular is at 65% and 51%, which is the hairline the
   earlier comparison saw. One face for all of them, declared across the whole weight
   range, so the relation cannot change weight between the title and a caption the way
   the host's fallback did -- the same rule as one bold in the sans.

   Scaled to 70% by `size-adjust`, because a mathematics face draws its relations for a
   mathematics line and this one's `≈` is 1.7 times the size of the arithmetic beside it:
   `RELATION_SIZE_ADJUST` carries the measurements and what the scale costs. Scaling takes
   the stroke down with the width, so the sign is lighter than the sans's own `=` at every
   size. That is the price of the rule, and it is stated rather than hidden: the route it
   replaces gave a different relation to every reader.

   `.rel` is gone with the class-based fix: it marked three of the sites and the face
   reaches all of them. */
/* Under the title: the description in sans caps, a step below the title's
   size, then the deck in the prose face, italic. */
.cert-page .hero .subtitle {
  color: var(--kpress-doc-text);
  font-family: var(--kpress-font-sans);
  font-feature-settings: var(--kpress-font-features-sans);
  font-size: calc(var(--cert-font-size-sans-base) * var(--paper-subtitle-scale));
  font-style: normal;
  font-variant-caps: var(--kpress-caps-variant);
  font-weight: var(--kpress-font-weight-sans-medium);
  letter-spacing: var(--kpress-caps-spacing);
  line-height: 1.2;
  margin-block: 0.5rem 0;
  text-transform: var(--kpress-caps-transform);
}
/* `.centred` is the page's one word for "this block is centred, in every medium".
   It exists because the print block left-aligns prose wholesale, and the elements that
   must escape that need to be nameable in one place: the rule there carves them out by
   `:not(.centred)` and `check_print_layout` holds them to it. The hero's `h1` is named
   separately because a Markdown heading carries no class of its own.
   The hero is centred -- title and credits; the deck below reads as prose. */
.centred,
.cert-page .hero h1 {
  text-align: center;
}
/* The credits are one component on both papers (`devtools.paper_front`): one span a
   line, names in bold, addresses as plain links, the version plain. The grid is one
   column the width of the page, and an address in it may break anywhere: as a grid's
   automatic column the credits were as wide as the optimality paper's source address,
   wider than a phone's column, so every credit centred on a column wider than the page
   and was cut at its edge. */
.credits {
  display: grid;
  font-family: var(--kpress-font-sans);
  font-feature-settings: var(--kpress-font-features-sans);
  font-size: var(--kpress-font-size-normal);
  gap: 0.2rem;
  grid-template-columns: minmax(0, 1fr);
  line-height: 1.45;
  margin-block: 2rem 2.2rem;
}
.credits a {
  overflow-wrap: anywhere;
}
/* biome-ignore lint/style/noDescendingSpecificity: credit emphasis inherits the shared bold role after the broader support-text rules. */
.credits strong {
  font-weight: var(--kpress-font-weight-sans-bold);
}
/* A paper that explains someone else's work credits its source first, by name and
   address; a line's space then sets the paper's own credits apart from it, and another
   sets the dates apart below. A paper with no source begins at its own credits, with no
   space above them. A paper in a series ends with the series strip, which part it is
   and the other parts by title, a line's space under the dates. */
.credits .credits-source + .credits-own,
.credits .publication-date,
.credits .publication-date + .series {
  margin-block-start: 1lh;
}
.cert-page .hero .deck {
  font-family: var(--kpress-font-prose);
  font-style: italic;
  margin-block: 0;
}
/* Where this page came from and how to take it away, at the top corner and out of
   the way. Chips rather than prose: they are tools, not part of the argument. */
.doc-links {
  display: flex;
  float: inline-end;
  gap: 0.4rem;
  margin-block-start: 0.15rem;
  margin-inline-start: 1rem;
}
.doc-links .chip {
  align-items: center;
  border: 1px solid var(--kpress-doc-border);
  border-radius: var(--kpress-radius-sm);
  color: var(--kpress-doc-muted);
  display: inline-flex;
  font-family: var(--kpress-font-sans);
  font-size: var(--kpress-font-size-tiny);
  font-weight: var(--kpress-font-weight-sans-medium);
  gap: 0.3rem;
  letter-spacing: 0.02em;
  /* `normal`, not `1`, and the padding is what it is because of that.

     `line-height: 1` makes the line box 15.3px while the sans content area at that size
     is 17.28px. The half-leading is then negative, and negative leading is not shared
     evenly the way positive leading is -- ascent and descent are not symmetric about the
     em box -- so the text sat 0.65px above the chip's centre: 2.11px of space over it and
     3.41px under. Small, and a reader saw it.

     A line box at least as tall as the content area centres exactly, so the height comes
     back from the padding instead: `normal` gives a 23.98px content area, and 0.11rem
     either side puts the chip at 27.50px against the 27.52px it had. Measured after:
     2.75px above, 2.75px below, offset 0. `check_print_layout` holds it there. */
  line-height: normal;
  padding: 0.11rem 0.5rem;
  text-decoration: none;
}
.doc-links .chip:hover {
  background: var(--kpress-doc-surface-hover);
  border-color: var(--kpress-doc-accent);
  color: var(--kpress-doc-accent);
  /* kpress underlines a link on hover, which is right for a link in a sentence and
     wrong for a bordered chip: the rule lands inside the box and reads as a defect.
     The chip already answers the hover with its border, its ground and its colour,
     all three on the site's one hover timing. */
  text-decoration: none;
}
.doc-links .chip svg {
  display: block;
}
/* On screen the page's first block is the source chips' row, a little below the bar's
   rule, and the hero starts the site's shared `--site-page-top` below that row
   (site-nav.css), as every page's title does; print keeps the space above.
   These follow the hero's, the credits' and the chips' own rules above, which they
   adjust on screen. */
@media screen {
  .cert-page .doc-links {
    margin-block-start: 0.5rem;
  }
  .hero {
    clear: both;
    padding-block-start: var(--site-page-top);
  }
  .cert-page .hero h1 {
    margin-block-start: 0;
  }
  /* The credits start a little further under the title on screen than in print. */
  .cert-page .hero .credits {
    margin-block-start: 2.25rem;
  }
}

/* The deck is a div so its footnote renders; the paragraph inside keeps its edges. */
.cert-page .hero .deck > p {
  margin-block: 0;
}

/* The certificate switch sits at the foot of every figure it changes; the articles
   are whole documents, so they are hidden outright. */
/* Wrapping, because the two buttons are 148px and 176px and the page's scroller
   clips rather than scrolls: at 320px they spilled about five pixels off each edge and
   the labels were cut. They fit on one line from 360px up, which is where wrapping
   costs nothing. */
.cert-toggle {
  display: inline-flex;
  flex-wrap: wrap;
  gap: 0.3rem;
  justify-content: center;
}
.cert-toggle button {
  white-space: nowrap;
}
.intro {
  margin-block-start: 1.6rem;
}
/* The atlas rendering is inlined: its ground and strokes take the page's
   theme, and its own caption is replaced by the figure's. */
.trump a {
  display: block;
  width: min(100%, 24rem);
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.trump svg {
  display: block;
  height: auto;
  width: 100%;
}
.trump svg rect[fill="#ffffff"] {
  fill: var(--kpress-doc-bg);
}
.trump svg [stroke="#000000"] {
  stroke: var(--kpress-doc-text);
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.trump svg text {
  display: none;
}
.cert-figure[hidden] {
  display: none;
}
/* Separate the closing credit with whitespace. Qualify the selector so its
   margins take precedence over kpress's paragraph rules in both media. */
.cert-page .colophon {
  border-block-start: 0;
  color: var(--kpress-doc-muted);
  font-family: var(--kpress-font-sans);
  font-size: var(--kpress-font-size-tiny);
  margin-block: 4rem 0;
  padding-block-start: 0;
}

/* KPress paints a 0.226em square. Keep its actual height: the old glyph-era
   height:1lh stretched that square into a full-line vertical bar.

   The item's resolved line-height keeps the marker's lh tied to the first text
   line. Centre the square there, then lower its optical centre by 0.04 of the
   body base: 0.72px at the 18px screen size and 0.64px at the 16px print size.
   In the shipped PT Serif outlines, the A and E crossbars are centred 0.262em
   and 0.357em above the baseline. The adjusted prose marker sits between them,
   about 0.32em above the baseline in both media. */
.cert-page.kpress-prose ul > li {
  line-height: 1lh;
}
.cert-page.kpress-prose ul > li::before {
  height: 0.226em;
  top: calc(0.5lh - 0.113em + var(--kpress-font-size-base) * 0.04);
}

.line-fig {
  padding: 1.3rem 1.1rem 0.4rem;
}
/* `.kpress-diagram svg` sets display, height and centring; these figures also
   need full width and labels allowed outside the viewBox. */
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.line-fig svg,
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.chart svg {
  overflow: visible;
  width: 100%;
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.line-fig svg text,
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
  .chart
  svg
  text {
  fill: var(--kpress-doc-muted);
  font-family: var(--kpress-font-sans);
}

/* Figure chrome. The figure carries no title: the caption names it, and repeating
   the number above the frame said the same thing twice. What the head did carry that
   the caption cannot is the certificate chooser, which belongs with the diagram it
   changes, so it sits at its foot. */
.fig-choose {
  border-block-start: 1px solid var(--kpress-doc-border);
  display: flex;
  justify-content: center;
  padding: 0.7rem 1.1rem;
}
.tag {
  background: color-mix(in srgb, var(--kpress-doc-accent) var(--paper-accent-tint), transparent);
  border-radius: var(--kpress-radius-sm);
  color: var(--kpress-doc-accent);
  font-family: var(--kpress-font-sans);
  font-size: var(--kpress-font-size-tiny);
  font-weight: var(--kpress-font-weight-sans-medium);
  letter-spacing: var(--kpress-caps-spacing);
  padding: 0.05rem 0.4rem;
}
/* Pure charts and packing drawings need no outer box. Composite apparatus uses the
   same hairline as its internal panel so its diagram, readout and controls read as one
   figure.

   Caption padding keeps borderless figures inset from the measure, so a reader can see
   where the figure ends and the prose resumes. */
.cert-page .kpress-figure {
  border: 0;
}
.cert-page .kpress-figure.apparatus {
  border: 1px solid var(--kpress-doc-border);
}
.cert-page .kpress-figcaption {
  border-block-start: 0;
  margin-block-start: 0;
  padding-block: 0.85rem 1rem;
}
/* The "Figure N." prefix is the caption label. It uses the title-credit bold so a
   reader scanning for Figure 7 can find it; the condition and caption text stay at the
   caption's regular weight. */
.cert-page .kpress-figcaption strong {
  font-weight: var(--kpress-font-weight-sans-bold);
}

/* The mass budget. */
.mass-line {
  color: var(--kpress-doc-muted);
  display: flex;
  flex-wrap: wrap;
  font-family: var(--kpress-font-sans);
  font-size: var(--kpress-font-size-tiny);
  gap: 1rem;
  justify-content: space-between;
  margin-block-start: 0.6rem;
  padding: 0 1.1rem 0.9rem;
}
.mass-line .v {
  display: block;
  margin-block-start: 0.15rem;
}

/* The two interactive stages. */
.split {
  display: grid;
  grid-template-columns: minmax(0, 1.75fr) minmax(14rem, 1fr);
}
/* Figures sit on the page's own ground, and the canvases take the room the
   figure track gives them. */
.stage {
  align-items: center;
  background: var(--kpress-doc-bg);
  display: flex;
  justify-content: center;
  padding: 0.6rem;
  position: relative;
}
/* A canvas is 40% of a phone's viewport height and there are three of them, so what
   it does with a touch decides whether the page can be read on a phone at all. The
   blanket `touch-action: none` here took every gesture: a 250px upward swipe starting
   on a canvas scrolled the page 0px, against 338px for the same swipe two centimetres
   to the left, leaving 26px of gutter either side as the only way past three dead
   zones. Keep vertical scrolling on the canvases; only the small native rotation
   handles capture a two-axis touch gesture. The atom field, which only taps,
   restricts nothing. */
.stage canvas {
  display: block;
  height: auto;
  max-width: 42rem;
  width: 100%;
}
/* The film under the atlas. Sized like a canvas, and its box is settled before a byte of
   video arrives: the `width`/`height` attributes give the intrinsic 16:9, the poster is a
   frame of the film at that same ratio, and `aspect-ratio` holds it if either is missing.
   Without them a `preload="none"` player is drawn at the 300x150 default and then jumps to
   1920x1080 when someone presses play (the owner, 2026-09-22). */
.stage video.film {
  aspect-ratio: 16 / 9;
  background: var(--kpress-doc-text);
  display: block;
  height: auto;
  max-width: 42rem;
  width: 100%;
}
.stage canvas.draggable {
  cursor: grab;
  touch-action: pan-y;
}
.stage canvas.draggable:active {
  cursor: grabbing;
}
/* The atom readout: a fixed panel beside the canvas, in the sans face. */
.tip-panel {
  color: var(--kpress-doc-text);
  font-family: var(--kpress-font-sans);
  font-size: var(--kpress-font-size-small);
  line-height: 1.5;
}
.tip-panel .tip-w {
  color: var(--kpress-doc-muted);
  display: block;
  margin-block-start: 0.1rem;
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.panel {
  border-inline-start: 1px solid var(--kpress-doc-border);
  display: flex;
  flex-direction: column;
  font-family: var(--kpress-font-sans);
  gap: 1.1rem;
  padding: 1.1rem;
}
@media (max-width: 52rem) {
  .split {
    grid-template-columns: 1fr;
  }
  /* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
  .panel {
    border-block-start: 1px solid var(--kpress-doc-border);
    border-inline-start: 0;
  }
}

.readout {
  padding: 0.2rem 0 0.7rem;
}
.readout .caps {
  display: block;
  margin-block-end: 0.5rem;
}
.mass-val {
  align-items: center;
  display: flex;
  font-size: calc(var(--kpress-font-size-base) * 1.9);
  line-height: 1.1;
  min-height: 2.4em;
}
.mass-row {
  align-items: baseline;
  display: flex;
  flex-wrap: wrap;
  gap: 0.15rem 0.7rem;
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.mass-dec {
  color: var(--kpress-doc-text);
  font-size: var(--kpress-font-size-large);
}
/* Figure 5 needs room for a mass fraction and three separate direction values.
   Keep its controls below the drawing even on a wide screen. */
.prover .split {
  display: block;
}
.prover .stage canvas {
  max-width: 36rem;
}
.prover .panel {
  border-inline-start: 0;
  border-block-start: 1px solid var(--kpress-doc-border);
  text-align: left;
}
.prover .readout,
.prover .ctl {
  text-align: center;
}
.prover .mass-row {
  justify-content: center;
  align-items: center;
  gap: 0.6rem 1rem;
}
.cert-page .prover .mass-val {
  font-size: var(--kpress-font-size-large);
  min-height: 0;
}
.prover .mass-val .katex {
  white-space: nowrap;
}
.prover .ctl {
  align-items: center;
  gap: 0.8rem;
}
.prover input[type="range"] {
  max-width: 34rem;
}
.prover .direction-values {
  display: flex;
  flex-wrap: wrap;
  justify-content: center;
  align-items: center;
  gap: 0.8rem 1.6rem;
}
.prover :is(.math-item, .mass-dec) {
  display: inline-block;
  white-space: nowrap;
}
.prover .btns {
  justify-content: center;
}
.prover .shading-toggle {
  display: inline-flex;
  align-items: center;
  gap: 0.45rem;
  cursor: pointer;
}
.prover .shading-toggle input {
  accent-color: var(--kpress-doc-accent);
}
.prover .legend {
  flex-direction: row;
  flex-wrap: wrap;
  gap: 0.5rem 1.4rem;
}
.prover .control-status {
  border-inline-start: 2px solid var(--kpress-doc-accent);
  padding-inline-start: 0.8rem;
}
.prover .fig-choose {
  flex-direction: column;
  gap: 0.5rem;
}
.prover .fig-choose > .screen-only {
  display: block;
}
.prover [hidden] {
  display: none;
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.verdict {
  border-radius: var(--kpress-radius-sm);
  display: inline-block;
  font-size: var(--kpress-font-size-tiny);
  font-weight: var(--kpress-font-weight-sans-medium);
  letter-spacing: var(--kpress-caps-spacing);
  margin-block-start: 0.6rem;
  padding: 0.1rem 0.4rem;
  text-transform: var(--kpress-caps-transform);
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.verdict.ok {
  background: color-mix(in srgb, var(--kpress-doc-accent) var(--paper-accent-tint), transparent);
  color: var(--kpress-doc-accent);
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.verdict.no {
  background: color-mix(in srgb, var(--cert-below) var(--paper-below-tint), transparent);
  color: var(--cert-below);
}
.verdict.out {
  background: var(--kpress-doc-code-bg);
  color: var(--kpress-doc-muted);
}

/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.ctl {
  display: flex;
  flex-direction: column;
  gap: 0.4rem;
}
.ctl .val {
  color: var(--kpress-doc-text);
  font-size: var(--kpress-font-size-tiny);
  line-height: 1.6;
}
.cert-page input[type="range"] {
  accent-color: var(--kpress-doc-accent);
  margin: 0;
  width: 100%;
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.btns {
  display: flex;
  flex-wrap: wrap;
  gap: 0.4rem;
}
.cert-page button {
  background: var(--kpress-doc-bg);
  border: 1px solid var(--kpress-doc-border-hairline);
  border-radius: var(--kpress-radius-sm);
  color: var(--kpress-doc-text);
  cursor: pointer;
  font-family: var(--kpress-font-sans);
  font-size: var(--kpress-font-size-tiny);
  padding: 0.3rem 0.55rem;
}
/* The theme's dark surface-hover is mostly muted text, which washes out a
   button's pale label. Tint the document ground instead. */
.cert-page button:hover {
  background: color-mix(
    in srgb,
    var(--kpress-doc-text) var(--paper-hover-tint),
    var(--kpress-doc-bg)
  );
}
.cert-page button[aria-pressed="true"] {
  background: color-mix(in srgb, var(--kpress-doc-accent) var(--paper-accent-tint), transparent);
  border-color: var(--kpress-doc-accent);
  color: var(--kpress-doc-accent);
}
/* A visible native target works before hover and keeps touch rotation local.
   Its position follows the canvas, while its hit area stays finger-sized. */
.cert-page figure .rotation-handle {
  align-items: center;
  background: var(--kpress-doc-bg);
  border: 2px solid var(--cert-probe);
  border-radius: 50%;
  box-shadow: 0 0 0 3px var(--kpress-doc-bg);
  box-sizing: border-box;
  color: var(--cert-probe);
  cursor: grab;
  display: flex;
  font-size: 25px;
  height: 44px;
  justify-content: center;
  line-height: 1;
  min-height: 44px;
  min-width: 44px;
  padding: 0;
  position: absolute;
  touch-action: none;
  transform: translate(-50%, -50%);
  user-select: none;
  width: 44px;
}
.cert-page .rotation-handle:hover {
  background: color-mix(in srgb, var(--cert-probe) var(--paper-probe-tint), var(--kpress-doc-bg));
}
.cert-page .rotation-handle:focus-visible {
  outline: 2px solid var(--kpress-doc-accent);
  outline-offset: 3px;
}
.cert-page .rotation-handle[data-dragging] {
  cursor: grabbing;
}
.cert-page .rotation-handle[hidden] {
  display: none;
}
.hint {
  color: var(--kpress-doc-muted);
  font-size: var(--kpress-font-size-tiny);
  line-height: 1.55;
  margin: 0;
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.legend {
  color: var(--kpress-doc-muted);
  display: flex;
  flex-direction: column;
  font-size: var(--kpress-font-size-tiny);
  gap: 0.3rem;
}
.legend > span {
  align-items: center;
  display: flex;
  gap: 0.5rem;
}
.legend i {
  display: block;
  flex: none;
  height: 0.7rem;
  width: 0.9rem;
}

.kv {
  align-items: baseline;
  display: grid;
  font-size: var(--kpress-font-size-tiny);
  gap: 0.3rem 0.9rem;
  grid-template-columns: auto 1fr;
}
.kv dt {
  color: var(--kpress-doc-muted);
}
.kv dd {
  margin: 0;
  text-align: end;
}
.kv dd.hi {
  color: var(--cert-probe);
}

.chart {
  padding: 0.5rem 1.1rem 0.4rem;
}

/* One role kpress has no token for: the square the reader drags, which must
   read as a separate instrument from the mass it measures. */
:root {
  /* The type base, 18px, is paper-type.css's, shared with every page of the site. */
  --cert-probe: oklch(55% 0.145 52);
  --cert-below: #e26e82;
  --cert-near: #c9a13a;
  --cert-probe-wash: color-mix(in srgb, var(--cert-probe) 14%, transparent);
  --cert-accent-wash: color-mix(
    in srgb,
    var(--kpress-doc-accent) var(--paper-accent-tint),
    transparent
  );
}
:root:where([data-kpress-resolved-theme="dark"]) {
  --cert-probe: oklch(76% 0.13 60);
}

/* Keep an inline result, its punctuation and its source reference together. */
.math-reference {
  white-space: nowrap;
}

/* Every outer formula uses its surrounding text size, including displays and
   saved system-font settings. KaTeX still controls scripts and large operators
   inside that em. Override the stock face's legacy display-size compensation. */
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.cert-page .katex,
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.kpress-tooltip .katex {
  /* biome-ignore lint/complexity/noImportantStyles: this measured KPress or print override needs to beat a vendor or inline declaration. */
  font-size: inherit !important;
}

/* Published math reserves widths in em. Linux Chromium's hinted advances can
   round each glyph to a whole pixel, so a width measured there is wrong on
   another platform or at another size. Override KaTeX's explicit `auto` for
   both preparation and reading, including print. */
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.kpress .katex,
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.kpress-tooltip .katex {
  text-rendering: geometricPrecision;
}
/* On macOS, geometricPrecision also disables native hinting and visibly lightens
   the unchanged PT Serif and Source Sans glyphs. CoreText already gives linear
   advances under auto; the preparation guard verifies them at 16x as well. */
@media screen {
  html[data-squares-native-math-metrics] .kpress .katex,
  html[data-squares-native-math-metrics] .kpress-tooltip .katex {
    text-rendering: auto;
  }
}

/* Measured bases reserve ascent and descent too. Empty inline carriers must not
   add a font-dependent strut when a fallback face changes, especially in print's
   tighter lines. Preparation copies each base's measured height and depth onto
   its existing KaTeX strut, which also fixes the glyph baseline when print
   changes the font size. */
.kpress [data-kpress-math-prepared="true"] .katex,
.kpress [data-kpress-math-prepared="true"] .katex-html {
  /* biome-ignore lint/complexity/noImportantStyles: this measured KPress or print override needs to beat a vendor or inline declaration. */
  line-height: 0 !important;
}
.kpress [data-kpress-math-prepared="true"] .base {
  /* biome-ignore lint/complexity/noImportantStyles: this measured KPress or print override needs to beat a vendor or inline declaration. */
  line-height: 0 !important;
}

/* Math is held until KaTeX has typeset it, so no raw TeX flashes. */
/* Saved preferences are stamped in the head. Select their measured math before
   first paint; inactive variants occupy no space and are never hydrated. */
.squares-math-variant {
  display: none;
}
html:not([data-kpress-font-set="system"]):not([data-kpress-prose-font="sans"])
  .squares-math-variant[data-squares-math-contexts~="custom-serif"],
html:not([data-kpress-font-set="system"])[data-kpress-prose-font="sans"]
  .squares-math-variant[data-squares-math-contexts~="custom-sans"],
html[data-kpress-font-set="system"]:not([data-kpress-prose-font="sans"])
  .squares-math-variant[data-squares-math-contexts~="system-serif"],
html[data-kpress-font-set="system"][data-kpress-prose-font="sans"]
  .squares-math-variant[data-squares-math-contexts~="system-sans"] {
  display: inline;
}
/* The head bootstrap marks pending enhancement and clears it on failure. Without
   JavaScript, raw TeX remains legible alongside KPress's semantic MathML. */
html[data-kpress-math-pending]
:is(.tex, .tex-d, [data-kpress-math-prepared="true"]):not([data-squares-math-ready="true"]) {
  visibility: hidden;
}
/* Once startup has queued a wrapper, the root watchdog cannot expose it before
   its render runs. Include native MathML and descendants with explicit visibility. */
.kpress [data-squares-math-queued],
.kpress [data-squares-math-queued] * {
  /* biome-ignore lint/complexity/noImportantStyles: this measured KPress or print override needs to beat a vendor or inline declaration. */
  visibility: hidden !important;
}
/* Wide equations scroll sideways; the block keeps headroom for fractions, since
   overflow-x: auto also clips vertically and the display margin is zeroed. */
.tex-d {
  margin-block: 1.1rem;
  overflow-x: auto;
  overflow-y: hidden;
  padding-block: 0.45rem;
}
.tex-d .katex-display {
  margin: 0;
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.caps .katex,
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.fig-title .katex,
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.legend .katex,
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.hint .katex {
  text-transform: none;
}
/* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
.caps .katex {
  letter-spacing: 0;
}
/* ============================================================
   Print. kpress prints a document well: page margins, 11pt type,
   forced-light tokens, break rules for headings, figures, boxes and
   display math, hidden buttons and popovers. It cannot know this page
   carries canvases, control panels and a figure chooser, so what
   follows is only what kpress does not cover.
   ============================================================ */
@media print {
  /* Nothing on paper can be dragged, hovered or chosen. The buttons kpress
     already hides; these are the rest of the apparatus, including the empty
     rule the chooser would otherwise leave under every figure. */
  /* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
  .panel,
  .fig-choose,
  .cert-page input[type="range"],
  /* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
  .screen-only,
  .cert-page figure .rotation-handle {
    display: none;
  }

  /* Paper keeps inline code's Planetaire metrics and text role without drawing the
     screen chip behind it. Only decoration changes, so family, color, size, weight,
     baseline and wrapping continue to come from kpress and the surrounding role. */
  .cert-page code:not(pre code) {
    background: transparent;
    border: 0;
  }

  /* Stack the printable figure body and cap canvases so their captions fit. */
  .split {
    display: block;
  }
  .stage canvas {
    max-height: 3.2in;
    width: auto;
    max-width: 100%;
    margin-inline: auto;
  }
  .trump a {
    width: min(100%, 3in);
  }
  .cert-page figure img {
    max-height: 6.5in;
    width: auto;
    max-width: 100%;
  }

  /* Keep the complete hundred-packing atlas and caption on one Letter page
     (think-yabp). The 2260x4023 portrait drawing overflows at the six-inch text
     width. A 6.75in height leaves 2.75in of the 9.5in content box for the caption,
     stage padding and figure margins. Auto width/height retain its true aspect
     ratio; the linked standalone PDF keeps the full-size printable drawing. */
  .cert-page .stage:has(img[src$="known-best-1-100.svg"]) {
    padding-inline: 0;
  }
  .cert-page .stage > a:has(img[src$="known-best-1-100.svg"]) {
    width: 100%;
  }
  .cert-page figure img[src$="known-best-1-100.svg"] {
    max-height: 6.75in;
    width: auto;
    height: auto;
  }

  /* Link targets are deliberately not spelled out after each link: a URL set in
     the middle of a sentence breaks the line badly and reads worse than the
     sentence it interrupts. The footnotes carry the sources, and the colophon
     carries the repository. */

  /* kpress forces its light tokens on `:root` and `.kpress`, which leaves the
     page's own main element and this page's one extra token on their dark
     values. A dark-mode reader would otherwise print prose on a dark ground
     and lose the Figure 7 threshold line entirely. */
  .kpress-page-main {
    --kpress-doc-bg: oklch(100% 0 0);
    --kpress-doc-border: oklch(78.26% 0 0);
    background: var(--kpress-doc-bg);
    color-scheme: light;
  }
  /* Print prose, captions, labels and the colophon in solid black. Reset both
     text roles at every KPress scope so screen theme tokens cannot shadow them. */
  :root,
  .kpress,
  .kpress-page-main,
  .kpress-tooltip,
  :is(.kpress, .kpress-page-main, .kpress-tooltip)[data-kpress-resolved-theme],
  :is(
    .kpress,
    .kpress-page-main,
    .kpress-tooltip
  )[data-kpress-palette][data-kpress-resolved-theme] {
    --kpress-doc-muted: #000;
    --kpress-doc-text: #000;
    color: var(--kpress-doc-text);
  }
  :root {
    --cert-probe: oklch(55% 0.145 52);
  }

  /* The accent, which the block above forgot. The dark rule near the top of this sheet
     lifts `--kpress-doc-accent` to L=76.68% so links clear 4.5:1 on a dark ground; the
     print reset puts the ground back to white and left the accent where it was, so a
     dark-mode reader printed #6ec5bc links on paper at 2.02:1 against the 5.47:1 the
     light accent gives. Restored to kpress's own light value.

     Declared on the same elements the dark rule names, including the attribute form at
     (0,2,0). Putting it on `.kpress-page-main` alone would be shadowed by the dark
     rule's own declaration on `.kpress` -- exactly how the print type size went missing
     two rules down. */
  :root,
  .kpress,
  .kpress-page-main,
  .kpress-tooltip,
  :is(.kpress, .kpress-page-main, .kpress-tooltip)[data-kpress-resolved-theme] {
    --kpress-doc-accent: oklch(51.09% 0.0861 186.4);
  }

  /* Legend chips carry their meaning in colour alone, so they keep it even
     when the reader prints without background graphics. */
  .legend i {
    border: 1px solid currentColor;
    print-color-adjust: exact;
  }

  /* The title block is a title page. */
  .intro {
    break-before: page;
  }

  /* Inline math is hidden until KaTeX finishes; on paper it should fall back
     to its source rather than to nothing. */
  /* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
  .tex,
  .tex-d {
    visibility: visible;
  }

  /* Unrounded advances. This is what made the serif text look loose and, worse,
     uneven. Chromium's default `text-rendering: auto` rounds every glyph advance
     to a whole CSS pixel at layout time and then writes that rounded position
     into the PDF: measured here, 98.44% of the 11,177 glyph moves in the printed
     file sat on integers. At the print size the rounding is not small. PT Serif
     at 11pt is 14.6667px, where the word space gains 12.3% and `i` and `l` gain
     13.3% while the comma and the full stop lose about 10% -- a spread of 1.395px,
     or 9.5% of an em, between adjacent gaps. The true metrics are identical to the
     screen's; a sweep across 9pt to 14pt puts 11pt at the worst of the range.
     `geometricPrecision` is the only value that turns linear metrics on, it
     inherits, so one declaration covers prose, captions, tables, code and KaTeX,
     and being print-only it leaves the screen's hinted rendering alone. */
  .kpress {
    text-rendering: geometricPrecision;
  }

  /* Quotation marks used to be overridden here. kpress borrowed Georgia's into the PT
     Serif stack through a `LocalPunct` @font-face whose `src` was `local("Georgia")`,
     so on the Mac and Windows machines most readers print from, every apostrophe and
     quotation mark came from a face the reader happened to own -- at its own metrics,
     and impossible to embed in the PDF. This scope replaced the prose stack for print
     to keep that out of the printed page.

     kpress ships the marks now (`KPress Quotes`, six glyphs of Source Serif 4 at about
     700 bytes, leading the prose stack through `--kpress-font-punctuation`), so the
     override would do the opposite of what it was for: naming the stack again would
     drop the shipped face and send print back to PT Serif's own marks. It is gone, and
     `render_n11_lower_bounds_explainer_pdf --fonts` is what holds the result -- no Georgia in the
     export, and `KPressQuotes` embedded. */

  /* Ragged right. kpress justifies printed prose and asks for automatic
     hyphenation to keep the lines even, but hyphenation needs dictionaries the
     printing engine may not carry: headless Chromium has none, so `hyphens: auto`
     is inert there and justification has nothing to stretch but the word spaces.
     Even word spacing is worth more here than a straight right edge.

     `!important` and the wider selector are deliberate. kpress's rule and this one
     have the same specificity, so this one wins on nothing but the order the two
     stylesheets are inlined in; a kpress release that adds `!important` or narrows
     its selector to `.kpress-prose p` would silently restore justification. The
     extra elements are not justified by kpress today and cost nothing to name.

     What is centred by design carries `.centred` and is carved out of the selector
     rather than restated after it. Restating it is what shipped the bug: the exemption
     was `.colophon { text-align: center !important }` at (0,1,0) against this rule's
     (0,2,0), both `!important` in one stylesheet, so the more specific one won and the
     colophon printed flush left. An exemption that has to out-shout the rule it exempts
     is a race the rule can win; an exemption the rule does not match is not a race at
     all, which is why this one is written with `:not()`. `:is()` takes the specificity
     of its most specific argument -- all type selectors here -- so the selector is
     (0,2,1) and the `!important` still stands against a kpress bump.

     `check_print_layout` holds `.centred` to its word in both media. */
  .kpress :is(p, li, blockquote, dd, dt, figcaption, td, th, caption):not(.centred) {
    /* biome-ignore lint/complexity/noImportantStyles: this measured KPress or print override needs to beat a vendor or inline declaration. */
    text-align: left !important;
  }

  /* Carving `.centred` out of the rule above is only half of it: kpress justifies
     printed prose itself, so a block this page excuses from the left-align rule lands
     on the vendor's `justify` instead of on its own centring, which is (0,1,0) and
     loses. This is the restatement the rule above refuses to be, and the difference is
     that it is not exempting anything from a rule beside it -- the two selectors cannot
     match the same element -- it is beating a stylesheet we do not control, which is
     what the ragged-right rule above already uses `!important` for. */
  .kpress .centred {
    /* biome-ignore lint/complexity/noImportantStyles: this measured KPress or print override needs to beat a vendor or inline declaration. */
    text-align: center !important;
  }

  /* kpress breaks words anywhere inside an ordered list so a long token cannot
     overflow. With no hyphen to mark the break it reads as a typo. `overflow-wrap`
     stays, so a URL too long for the measure still wraps -- at a boundary. */
  /* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
  .kpress ol > li > * {
    word-break: normal;
  }

  /* The measure, which is what "the margins look wrong" turned out to mean.

     Measured across the document rather than estimated, median characters per full body
     line, over the paragraphs that carry no inline math -- a KaTeX span sits at its own
     top, so counting distinct line tops in a paragraph containing one counts more lines
     than the paragraph has, and an earlier pass that did not exclude them read 45 where
     the answer was 64:

       0.70in / 11pt   6.94in column   102 chars   11 pages
       1.00in / 11pt   6.30in column    94 chars   12 pages   <- was here
       1.25in / 11pt   5.84in column    80 chars
       1.25in / 12pt   5.84in column    64 chars   13 pages   <- is here

     The margin alone cannot fix this. Going 0.7in to 1in bought eight characters and
     cost a page, and 94 is still far outside the 45 to 75 that reads comfortably. It is
     a type size decision wearing a margin's clothes, so both move.

     The two rows that matter are the last two, and they are the same margin: the type
     size is what carries the measure from 80 into the band. That is worth stating
     plainly because the first version of this change did not deliver it. The margin
     took effect and the type size did not (see the rule below), so the document shipped
     at 1.25in and 11pt -- the 80-character row -- while the row underneath it, the one
     this comment argued for, was labelled 12pt and had never been rendered. The 80
     figure was real; only its label was wrong.

     `size` is declared because the exporter passes `preferCSSPageSize`, which makes
     this rule the one that decides the paper. Declaring `size` WITHOUT that flag is the
     trap: Chromium then centres the CSS page box on the API's paper and leaves dead
     bands around it. The two belong together and are commented at both ends. */
  @page {
    size: Letter;
    margin-top: 0.75in;
    margin-bottom: 0.75in;
  }
  .cert-page {
    padding-block-end: 0;
  }
  .cert-page .hero {
    padding-block-start: 0.25in;
  }
  .cert-page h2 {
    margin-block: calc(var(--kpress-font-size-base) * 1.5) 0.75rem;
  }
  .cert-page .colophon {
    margin-block-start: 0.5rem;
  }
  /* Keep reading-guide groups together and set their short entries compactly. */
  #further-reading + ul {
    font-size: var(--kpress-font-size-small);
  }
  #further-reading + ul > li {
    break-inside: avoid;
  }
  #further-reading + ul ul {
    margin-block-end: 0;
  }
  #further-reading + ul ul > li {
    margin-block-start: 0.3rem;
  }
  /* Keep an equation with the prose introducing it, and short lists intact. */
  /* biome-ignore lint/style/noDescendingSpecificity: historical selector order preserves the established KPress publication cascade. */
  .cert-page p {
    break-inside: avoid;
  }
  .cert-page p:has(+ .kpress-math-display, + ul) {
    break-after: avoid;
  }
  .cert-page p + .kpress-math-display {
    break-before: avoid;
  }
  .cert-page > ul:not(:has(ul)) {
    break-inside: avoid;
  }
  /* On `.kpress` as well as `:root`, and that is the whole of why the type size moved.
     kpress declares both tokens on `:root, .kpress, .kpress-page-main, .kpress-tooltip`
     and then reads the size back on `.kpress` -- `font-size: var(--kpress-print-font-size)`
     -- so a declaration only at the root is shadowed by kpress's own on the element that
     consumes it, and inheritance never gets a turn. The margin escaped this because
     `@page { margin: var(--kpress-print-page-margin) }` resolves at the root, where
     nothing shadows it. So the first version of this rule shipped 1.25in margins around
     11pt type, half of a change whose whole argument was that the two move together, and
     set the folio -- also resolved at the root -- one step larger than the body it
     numbers. Measured in the PDF before the fix: body 11.00pt, folio 12pt. */
  :root,
  .kpress {
    --kpress-print-page-margin: 1.25in;
    --kpress-print-font-size: 12pt;
  }

  /* The running foot carries the folio and nothing else.

     kpress puts its own credit in `@bottom-left` on every page -- `--kpress-print-footer`
     is the string "Formatted by KPress" -- and the page number in `@bottom-right`. The
     credit is right for a page kpress publishes and wrong for a paper: it is the same
     eighteen characters under all thirteen pages, saying at the foot of the argument what
     the colophon already says once at the end of it. The folio stays, because a paper's
     pages need numbering.

     Chromium 141 supports margin boxes and `@page :first` -- verified, not assumed; the
     advice that Blink ignores them is out of date. */
  @page {
    @bottom-left {
      content: none;
    }
    @bottom-left {
      color: #000;
    }
    @bottom-right {
      color: #000;
      vertical-align: top;
      padding-top: 0.15in;
    }
  }

  /* No folio on the title page either: it already carries the publication dates and the
     repository, and a page number under a title block reads as a body page. */
  @page :first {
    @bottom-right {
      content: none;
    }
  }

  /* The footnotes are the paper's sources, not an aside, so they are set in the text
     colour rather than the muted one kpress gives them. */
  .cert-page .kpress-footnotes {
    color: var(--kpress-doc-text);
    break-after: avoid-page;
    margin-block-start: 1rem;
    padding-block-start: 0.5rem;
  }
  /* Replace both print list margins so the source list leaves room for its colophon. */
  .cert-page .kpress-footnotes > ol {
    margin-block: 0.5rem 0;
  }
  .cert-page .kpress-footnotes ol > li {
    margin-block: 0.125rem 0;
  }

  /* Their numbers in the sans the entries are set in. kpress replaces the list marker in
     print with `content: counter(...)` on `li::before` and sets that in
     `--kpress-font-prose`, which is right for a numbered list inside serif prose and
     wrong here: the footnote block is sans throughout, so its own numbers were the one
     serif thing in it. Scoped to the footnotes for that reason -- an ordered list in the
     body keeps serif figures, because the text beside them is serif. */
  .cert-page .kpress-footnotes ol > li::before {
    font-family: var(--kpress-font-sans);
    font-feature-settings: var(--kpress-font-features-sans);
  }

  /* The footnote references in the body, in the sans they wear on screen. kpress prints
     them in the prose face as small-cap figures at 400 -- a quiet treatment for prose
     that cites rarely. This paper cites in most paragraphs, and each reference is a
     signpost to the sources block, which is sans throughout; the prose-face figure sat
     a little too close to the digits of the text around it. The weight is the screen's,
     and the screen's is now the scale's medium, set once for both media in the rule
     above rather than repeated here as a number. No chip in print; kpress's print rule
     already removes it.

     kpress prints every superscript at 110% of the text, a compensation for the
     small-cap figure it sets the reference in, which is short. A sans figure at that
     size and weight stood taller than the capitals beside it; a superscript's own
     size, 0.75em, puts it back where a reference mark belongs. */
  .cert-page sup.kpress-footnote-ref {
    font-size: 0.75em;
  }
  .cert-page .kpress-footnote-ref a {
    font-family: var(--kpress-font-sans);
    font-feature-settings: var(--kpress-font-features-sans);
    font-variant-caps: normal;
  }
}