html, body, .reveal, .reveal .slides, .reveal .slides > section {
  height: 100%;
  width: 100%;
}

.reveal {
    --slide-line-height: 1.4;
    --slide-body-gap: 0.2em;
    --slide-paragraph-spacing: 0.45em;
    --slide-heading-spacing-top: 0.35em;
    --slide-heading-spacing-bottom: 0.1em;
    font-family: 'DM Sans', sans-serif;
    font-size: 22px;
    line-height: var(--slide-line-height);
    color: #222;
    background-color: #fff;
}

.reveal .slides section {
    text-align: left;
}

.reveal .slides section.deck-slide {
    display: flex;
    flex-direction: column;
    align-items: flex-start;
    justify-content: flex-start;
}

.reveal .slides section.deck-slide.present {
    display: flex !important;
}

.reveal .slides .slide-body {
    margin-top: 120px;
    margin-bottom: 40px;
    margin-left: 65px;
    margin-right: 65px;
    width: calc(100% - 130px);
    box-sizing: border-box;
}

.reveal .slides p {
    margin-top: 0px;
    margin-bottom: var(--slide-paragraph-spacing);
}

.reveal .slides section.vertical {
  top: 0 !important;
  transform: none !important;
}

.presentation-title {
    color: white;
    background-color: #A81C3A;
    padding: 24px;
    text-align: center;
    margin: 40px 0px 40px 0px;
}

.author {
    text-align: center;
}

.affiliation {
    display: flex;
    flex-direction: row;
    align-items: center;
    justify-content: center;
    gap: 40px;
}

.slide-header {
  position: absolute;
  top: 0;
  left: 0;
  right: 0;
  height: 80px;                 /* adjust as you like */
  background-color: #A81C3A;    /* header bar color */
  color: white;
  display: flex;
  align-items: center;          /* vertically center text */
  padding-left: 3em;            /* align with body text */
  box-sizing: border-box;
}


.slide-header h2 {
  margin: 0;
  font-size: 1.2em;
}

.slide-body {
  margin-top: 100px;            /* space below the header */
  display: flex;
  flex-direction: column;
  align-items: flex-start;
  justify-content: flex-start;
  gap: var(--slide-body-gap);
}

.slide-body h3 {
  margin: var(--slide-heading-spacing-top) 0 var(--slide-heading-spacing-bottom);
  font-size: 1.05em;
}

.slide-body ul,
.slide-body ol {
  margin-top: 0;
  margin-bottom: 0.5em;
}

.slide-body > p,
.slide-body > div,
.slide-body > canvas,
.slide-body > label {
  width: 100%;
}

.slide-body p:has(> img) {
  text-align: center;
  margin: 0.25em 0 0.6em;
}

.slide-body img {
  display: block;
  height: auto;
  margin: 0 auto;
  max-width: 70%;
}

.slide-body .image-small img,
.slide-body img.image-small {
  max-width: 50%;
}

.slide-body .image-medium img,
.slide-body img.image-medium {
  max-width: 60%;
}

.slide-body .image-large img,
.slide-body img.image-large {
  max-width: 80%;
}

.slide-body .r-stack {
  min-height: 190px;
}

.slide-body .r-stack p {
  width: 100%;
}

.spacer-small {
  height: 0.5em;
}

.spacer {
  height: 1em;
}

.spacer-large {
  height: 2em;
}

.reveal .katex-display {
  margin: 0.35em 0;
}

.reveal canvas {
  display: block;
  max-width: 100%;
  margin: 0.25em auto;
}
.reveal .slides section.title-slide {
  align-items: center;
  justify-content: flex-start;
  padding: 0 40px;
  box-sizing: border-box;
  text-align: center;
}

.reveal .slides section.title-slide h1 {
  width: 100%;
  box-sizing: border-box;
  color: white;
  background-color: #A81C3A;
  padding: 24px;
  margin: 40px 0;
  text-align: center;
  font-size: 1.75em;
}

.reveal .slides section.title-slide h3 {
  display: flex;
  align-items: center;
  justify-content: center;
  gap: 40px;
  width: 100%;
  margin: 0.35em 0;
  font-size: 1em;
}

.reveal .slides section.title-slide h3 img {
  height: 120px;
  width: auto;
  max-width: 240px;
  object-fit: contain;
}

.reveal .slides section.title-slide p {
  width: min(920px, 100%);
  margin: 0.2em auto;
  text-align: left;
  font-size: 18px;
}

.reveal .slides section.closing-slide {
  justify-content: center;
}

.vslider {
  display: flex;
  flex-direction: column;
  align-items: center;
  gap: 6px;
}

/* Chrome / Edge / Safari */
.vslider input[type=range] {
  -webkit-appearance: slider-vertical;
  writing-mode: bt-lr; /* IE/Edge legacy */
  width: 10px;
  height: 140px;
  padding: 0;
}

/* optional: make the labels smaller */
.vslider .label, .vslider .value {
  font: 22px system-ui, sans-serif;
  color: #444;
}

.slider-value {
  display: inline-block;
  width: 50px;     /* fixed width → stops shifting */
  text-align: right;
}

.sliders {
  display: flex;
  gap: 40px;
  justify-content: center;
  align-items: flex-end;
  margin-top: 10px;
}

.two-formulae-row {
  display: flex;
  width: 100%;
  align-items: flex-start;
  justify-content: space-between;
  gap: 0em;
}

.left-formula {
  text-align: left;
}

.right-formula {
  margin-left: auto;
  text-align: right;
}

.slide-body img.nobel-medal {
  display: inline-block;
  width: 1.15em;
  height: 1.15em;
  max-width: none;
  margin: 0 0.2em;
  vertical-align: -0.2em;
}

.lattice-plot,
.vortex-plot,
.no-interfaces-plot,
.puncturingPlot,
.compactnessVorticesPlot,
.liminfVorticesPlot {
  width: 100%;
  text-align: center;
  margin-top: 0em;
}

.gap-plot-row {
  display: flex;
  gap: 2rem;
  align-items: center;
  justify-content: center;
}
.gap-plot-row > div { flex: 1; min-width: 0; text-align: center; }
.gap-plot-label { margin-top: 0.35rem; text-align: center; }
.gap-plot-row canvas { width: 100%; cursor: grab; }
.gap-plot-3d canvas { background: #fff; opacity: 1; }
.gap-plot-row canvas:active { cursor: grabbing; }

.structure-theorem-plot { width: 100%; margin-top: .25rem; text-align: center; }
.structure-plot-row { display:flex; gap:1rem; align-items:center; justify-content:center; }
.structure-plot-row > div { flex:0 1 240px; min-width:0; }
.structure-plot-row canvas { width:100%; max-width:240px; cursor:default; }
.structure-plot-row > div:last-child canvas { cursor:grab; transform:scale(1.12); transform-origin:center; background:#fff; opacity:1; }
.structure-plot-row > div:last-child canvas:active { cursor:grabbing; }
.structure-toggles { display:flex; justify-content:center; gap:1.4rem; margin-top:.35rem; margin-bottom:.25rem; }
.structure-toggle { cursor:pointer; }
.structure-toggle input { display:none; }
.structure-toggle span { display:inline-flex; align-items:center; justify-content:center; width:1.65em; height:1.65em; border:2px solid #555; border-radius:50%; font-weight:700; font-style:italic; color:#555; transition:.15s; }
.structure-toggle input:checked + span { background:#A81C3A; border-color:#A81C3A; color:white; }

.piecewiseGraphRow {
  display: flex;
  align-items: center;
  justify-content: center;
  gap: 2rem;
  width: 100%;
  margin-top: .75rem;
}
.piecewiseGraphRow > div {
  flex: 0 1 370px;
  min-width: 0;
  display: flex;
  justify-content: center;
}
.piecewiseGraphRow canvas {
  display: block;
  max-width: 100%;
  background: #fff;
  opacity: 1;
  touch-action: none;
}
.piecewiseGraphControl canvas { cursor: crosshair; }
.piecewiseGraph3d canvas { cursor: grab; }
.piecewiseGraph3d canvas:active { cursor: grabbing; }

.vortex-plot-row {
  display: flex;
  align-items: center;
  justify-content: center;
  width: 100%;
  gap: 1.5em;
}

.vortex-plot-row .vortex-plot,
.vortex-plot-row .limit-plot {
  position: relative;
  flex: 1 1 0;
  min-width: 0;
  display: flex;
  align-items: center;
  justify-content: center;
}

.vortex-plot-row canvas {
  max-width: 100%;
}

.limit-label {
  position: absolute;
  top: 20%;
  left: 12%;
}

/* retain the shared plot defaults */
.vortex-plot-row + .spacer { margin-top: 0; }

.puncturing-plot-layout,
.compactness-vortices-plot-layout,
.liminf-vortices-plot-layout {
  display: flex;
  align-items: center;
  justify-content: center;
  gap: .4rem;
  margin-top: -2rem;
}
.puncturingPlot canvas,
.compactnessVorticesPlot canvas,
.liminfVorticesPlot canvas {
  display: block;
  width: 100%;
  max-width: 440px;
  cursor: crosshair;
}
.puncturing-degree-toggle,
.compactness-vortices-degree-toggle,
.liminf-vortices-degree-toggle {
  display: flex;
  flex-direction: column;
  gap: .8rem;
  align-items: flex-start;
  white-space: nowrap;
  transform: translateX(-150px);
}
.puncturing-degree-toggle label,
.compactness-vortices-degree-toggle label,
.liminf-vortices-degree-toggle label {
  cursor: pointer;
  color: #666;
}
.puncturing-degree-toggle input,
.compactness-vortices-degree-toggle input,
.liminf-vortices-degree-toggle input { accent-color: currentColor; }
.puncturing-degree-toggle label:first-child:has(input:checked),
.compactness-vortices-degree-toggle label:first-child:has(input:checked),
.liminf-vortices-degree-toggle label:first-child:has(input:checked) {
  color: #d7191c;
  font-weight: 700;
}
.puncturing-degree-toggle label:last-child:has(input:checked),
.compactness-vortices-degree-toggle label:last-child:has(input:checked),
.liminf-vortices-degree-toggle label:last-child:has(input:checked) {
  color: #2166ac;
  font-weight: 700;
}

.ballConstructionPlot {
  width: 100%;
  display: flex;
  flex-direction: column;
  align-items: center;
  gap: .35em;
  margin-top: -.25em;
}

.ballConstructionPlot canvas {
  display: block;
  width: min(760px, 100%);
  max-width: 100%;
  border: 2px solid #222;
  box-sizing: border-box;
}

.ballConstructionPlot input[type="range"] {
  display: block;
  width: min(680px, 88%);
  margin: 0;
  accent-color: #A81C3A;
}

.dyadicPlot,
.limsupPlot {
  width: 100%;
  display: flex;
  flex-direction: column;
  align-items: center;
  gap: .25em;
  margin-top: -.15em;
}

.dyadic-plot-layout {
  display: flex;
  align-items: center;
  justify-content: center;
  gap: .7em;
  width: 100%;
}

.dyadicPlot canvas {
  display: block;
  width: min(280px, calc(100% - 42px));
  max-width: 100%;
  margin: 0;
  background: #fff;
  opacity: 1;
}

.dyadicPlot input[type="range"] {
  width: 18px;
  height: 240px;
  margin: 0;
  padding: 0;
  writing-mode: vertical-lr;
  direction: rtl;
  accent-color: #A81C3A;
}

.limsupPlot canvas {
  display: block;
  width: min(720px, 100%);
  max-width: 100%;
  margin: 0 auto;
}

.limsup-slider {
  position: relative;
  width: min(680px, 86%);
  height: 48px;
  margin: 0 auto;
}

.limsup-slider-rail {
  position: absolute;
  z-index: 0;
  top: 9px;
  left: 10px;
  right: 10px;
  height: 4px;
  border-radius: 2px;
  background: #b9b9b9;
  pointer-events: none;
}

.limsup-slider-stops {
  position: absolute;
  z-index: 1;
  top: 3px;
  left: 4px;
  right: 4px;
  display: flex;
  justify-content: space-between;
  pointer-events: none;
}

.limsup-slider-stops span {
  display: flex;
  flex-direction: column;
  align-items: center;
  gap: 5px;
  color: #444;
  font-size: 16px;
  line-height: 1;
}

.limsup-slider-stops i {
  width: 15px;
  height: 15px;
  box-sizing: border-box;
  border: 2px solid #777;
  border-radius: 50%;
  background: #fff;
}

.limsup-slider-stops b {
  font-weight: 700;
}

.limsup-slider input[type="range"] {
  position: absolute;
  z-index: 2;
  top: 0;
  left: 0;
  width: 100%;
  height: 22px;
  margin: 0;
  padding: 0;
  appearance: none;
  -webkit-appearance: none;
  background: transparent;
  cursor: pointer;
}

.limsup-slider input[type="range"]::-webkit-slider-runnable-track {
  height: 4px;
  background: transparent;
}

.limsup-slider input[type="range"]::-moz-range-track {
  height: 4px;
  background: transparent;
}

.limsup-slider input[type="range"]::-webkit-slider-thumb {
  width: 22px;
  height: 22px;
  margin-top: -9px;
  appearance: none;
  -webkit-appearance: none;
  border: 3px solid #fff;
  border-radius: 50%;
  background: #A81C3A;
  box-shadow: 0 0 0 2px #A81C3A;
}

.limsup-slider input[type="range"]::-moz-range-thumb {
  width: 16px;
  height: 16px;
  border: 3px solid #fff;
  border-radius: 50%;
  background: #A81C3A;
  box-shadow: 0 0 0 2px #A81C3A;
}

.limsup-slider input[type="range"]:focus-visible {
  outline: 2px solid #A81C3A;
  outline-offset: 4px;
  border-radius: 4px;
}

.notSubadditivePlot {
  width: 100%;
  margin-top: -0.7em;
  text-align: center;
}

.notSubadditivePlot canvas {
  display: block;
  width: min(310px, 100%);
  margin: 0 auto;
}

.not-subadditive-controls {
  display: flex;
  align-items: center;
  justify-content: center;
  gap: 2em;
  margin: -0.2em auto 0.35em;
}

.no-interfaces-plot {
  width: 100%;
  text-align: center;
  margin-top: 0em;
}

.no-interfaces-plot {
  display: flex;
  align-items: center;
  justify-content: center;
  gap: 1em;
}

.no-interfaces-plot .interface-label {
  flex: 0 0 auto;
  font-size: 1.15em;
}

.no-interfaces-plot canvas {
  flex: 1 1 auto;
  min-width: 0;
}

.transition-angle-control {
  position: relative;
  width: 100%;
  height: 220px;
  display: flex;
  justify-content: center;
  margin-top: -1em;
  margin-bottom: -4em;
}

.transition-angle-control canvas {
  cursor: grab;
}

.transition-angle-control canvas:active {
  cursor: grabbing;
}

.transition-angle-control .angle-label {
  position: absolute;
  pointer-events: none;
  font-size: 1.1em;
}

.lattice-controls {
  display: flex;
  width: 100%;
  align-items: center;
  justify-content: center;
  gap: 1em;
}

.color-toggle,
.ground-state-toggle {
  color: #222;
  white-space: nowrap;
}

.color-toggle,
.ground-state-toggle {
  display: inline-flex;
  align-items: center;
  gap: 0.45em;
  cursor: pointer;
  font-weight: 700;
}

.color-toggle input,
.ground-state-toggle input {
  appearance: none;
  width: 2.35em;
  height: 1.25em;
  margin: 0;
  border-radius: 999px;
  background: #787878;
  position: relative;
  cursor: pointer;
  transition: background 0.2s ease;
}

.color-toggle input::after,
.ground-state-toggle input::after {
  content: "";
  position: absolute;
  top: 0.15em;
  left: 0.15em;
  width: 0.95em;
  height: 0.95em;
  border-radius: 50%;
  background: white;
  transition: transform 0.2s ease;
}

.color-toggle input:checked,
.ground-state-toggle input:checked {
  background: #A81C3A;
}

.color-toggle input:checked::after,
.ground-state-toggle input:checked::after {
  transform: translateX(1.1em);
}

.color-toggle input:focus-visible,
.ground-state-toggle input:focus-visible {
  outline: 2px solid #A81C3A;
  outline-offset: 2px;
}

.MHIsing-plot,
.MHXY-plot {
  width: 100%;
  margin-top: 1em; /* controls the gap below the Energy paragraph */
  text-align: center;
}

.ising-plot-formula-row {
  display: flex;
  align-items: center;
  justify-content: center;
  gap: 0em;
  width: 100%;
}

.Ising-interface-plot {
  width: min(520px, 60%);
  text-align: center;
  margin-top: 0em;
}

#IsingInterfaceCanvas {
  display: block;
  margin: 0 auto;
  cursor: pointer;
  touch-action: none;
}

.ising-limit-formula {
  max-width: 38%;
  text-align: center;
}

.ising-paint-controls {
  display: flex;
  align-items: center;
  justify-content: center;
  gap: 0.7em;
  margin-top: 0.25em;
}

.ising-swatch {
  width: 1.5em;
  height: 1.5em;
  position: relative;
  border: 1px solid #222;
}
.ising-swatch.down { background: hsl(90 75% 68%); }
.ising-swatch.up { background: hsl(270 75% 68%); }
.ising-swatch::after {
  content: "";
  position: absolute;
  left: 50%;
  top: 50%;
  width: 2px;
  height: 1.1em;
  background: #222;
  transform: translate(-50%, -50%);
}
.ising-swatch::before {
  content: "";
  position: absolute;
  left: 50%;
  width: 0;
  height: 0;
  border-left: 0.18em solid transparent;
  border-right: 0.18em solid transparent;
}
.ising-swatch.down::before { bottom: 0.25em; border-top: 0.22em solid #222; transform: translateX(-50%); }
.ising-swatch.up::before { top: 0.25em; border-bottom: 0.22em solid #222; transform: translateX(-50%); }
.ising-toggle input { display: none; }
.ising-toggle span {
  display: block; width: 1.7em; height: 0.85em; border-radius: 1em;
  background: #787878; position: relative; cursor: pointer;
}
.ising-toggle span::after {
  content: ""; position: absolute; width: 0.7em; height: 0.7em;
  top: 0.075em; left: 0.075em; border-radius: 50%; background: white;
  transition: transform 0.2s ease;
}
.ising-toggle input:checked + span { background: #A81C3A; }
.ising-toggle input:checked + span::after { transform: translateX(0.85em); }

.ising-interface-controls {
  display: flex;
  justify-content: center;
  align-items: center;
  gap: 1em;
  color: #555;
  font-size: 0.55em;
}

.N-fixed-proof-row {
  display: flex;
  align-items: center;
  justify-content: center;
  gap: 2em;
  width: 100%;
  margin-top: -1.5em;
  margin-bottom: 0.5em;
}

.N-fixed-proof-plot {
  position: relative;
  flex: 1 1 52%;
  width: auto;
  text-align: center;
  margin: -1em 0;
}

.N-fixed-proof-sinus-plot {
  flex: 1 1 48%;
  text-align: center;
}

.N-fixed-proof-sinus-plot { position: relative; }
.sinus-label { position: absolute; transform: translate(-50%, -50%); white-space: nowrap; pointer-events: none; }
.sinus-label-2 { transform: translate(0, -50%); }

.N-fixed-proof-sinus-plot canvas {
  display: block;
  width: min(100%, 360px);
  height: auto;
  margin: 0 auto;
}

.N-fixed-label {
  position: absolute;
  transform: translate(-50%, -50%);
  pointer-events: none;
  white-space: nowrap;
}

.n-fixed-label-i { color: #222; }
.n-fixed-label-j { color: #222; }

.N-fixed-proof-plot canvas {
  display: block;
  width: min(100%, 560px) !important;
  height: auto !important;
  aspect-ratio: 1.3889;
  margin: 0 auto;
  cursor: crosshair;
  touch-action: none;
}

.SN-plot {
  width: 100%;
  margin-top: -2em; /* controls the gap below the Energy paragraph */
  text-align: center;
}

.SN-plot canvas {
  display: block; /* removes the inline-canvas baseline gap */
  margin: 0 auto;
}

#SNCanvas {
  cursor: grab;
  touch-action: none;
}

#SNCanvas:active {
  cursor: grabbing;
}

.SN-controls {
  display: flex;
  justify-content: center;
  align-items: center;
  gap: 0.6em;
  width: min(560px, 100%);
  margin: -2em auto;
}

.SN-slider-wrap {
  flex: 1;
  display: flex;
  align-items: center;
}

.SN-controls input[type="range"] {
  width: 100%;
  accent-color: #A81C3A;
}

.SN-control {
  font-weight: 700;
}

.SN-controls output {
  min-width: 1.5em;
  font-weight: 700;
  text-align: right;
}

.MHIsing-controls,
.MHXY-controls {
  display: flex;
  justify-content: center;
  align-items: center;
  gap: 0.6em;
  width: 100%;
  margin-top: 0.4em;
}

.beta-slider-wrap {
  position: relative;
  flex: 1;
  display: flex;
  align-items: center;
}

.MHIsing-controls input[type="range"],
.MHXY-controls input[type="range"] {
  width: 100%;
  accent-color: #A81C3A;
}

.beta-transition-marker {
  position: absolute;
  left: 13.25%;
  bottom: -5px;
  width: 3px;
  height: 14px;
  background: #222;
  pointer-events: none;
}

.beta-control {
  font-weight: 700;
}

.generate-button {
  display: block;
  margin: 0.4em 0 0;
  padding: 0.45em 1.2em;
  border: none;
  border-radius: 8px;
  background: #A81C3A;
  box-shadow: 0px 8px #787878;
  color: white;
  font-weight: 700;
  font-size: 1em;
  cursor: pointer;
}

.generate-button:hover {
    background: #b8546a;
}

.generate-button:active {
   box-shadow: 0px 4px #383838;
   transform: translateY(4px);
}

.N-eps-row { width: 100%; display: grid; grid-template-columns: 1fr 1fr; gap: 28px; align-items: center; margin-bottom: 2em;}
.lattice-to-zero-plot, .N-to-infty-plot { display: flex; align-items: center; justify-content: center; min-height: 270px; }
.N-eps-row canvas { margin: 0; max-width: 100%; }
.eps-to-zero-controls-row { grid-column: 1 / -1; display: grid; grid-template-columns: 1fr 1fr; gap: 28px; align-items: flex-end; margin-top: 8px; }
.eps-to-zero-slider, .rate-slider { justify-self: center; }
.eps-to-zero-slider, .rate-slider { display: flex; flex-direction: column; align-items: center; gap: 6px; font-size: 22px; }
.eps-to-zero-slider input { width: 220px; }
.rate-endpoints { display: flex; align-items: center; gap: 10px; font-size: 17px; }
.rate-endpoints input { width: 180px; }

/* Spectrum of possible rates, from the fast to the slow regime. */
.spectrum-incomplete-plot {
  width: min(760px, 82%);
  margin: 1.4em auto 0;
  color: #222;
  margin-top: 0em;
}
.spectrum-labels {
  display: grid;
  grid-template-columns: 1fr 1fr 1fr;
  align-items: center;
  text-align: center;
  font-size: 1.05em;
}
.spectrum-labels-top { margin-bottom: 0.28em; }
.spectrum-labels-bottom { margin-top: 0.35em; }
.spectrum-labels-bottom span:first-child { text-align: left; }
.spectrum-labels-bottom span:last-child { text-align: right; }
.spectrum-line {
  display: flex;
  width: 100%;
  height: 4px;
  align-items: center;
  position: relative;
}
.spectrum-incomplete-BV .spectrum-threshold-label {
  position: absolute;
  left: 66.6667%;
  top: 0.55em;
  transform: translateX(-50%);
  white-space: nowrap;
  font-size: 0.9em;
}
.spectrum-almost-complete .spectrum-left-threshold-label {
  left: 33.3333%;
}
.spectrum-incomplete-BV .spectrum-threshold-label::before {
  content: "";
  position: absolute;
  left: 50%;
  bottom: calc(100%);
  width: 12px;
  height: 12px;
  border-radius: 50%;
  background: white;
  border: 2px solid #222;
  transform: translateX(-50%);
}
.spectrum-segment {
  display: block;
  height: 2px;
  flex: 1;
}
.spectrum-segment.solid { background: #222; }
.spectrum-segment.dashed {
  background: repeating-linear-gradient(
    to right, #222 0, #222 10px, transparent 10px, transparent 18px
  );
}