BranchesTags

Repository files navigation

Proof of the Lyons-White Conjecture

This is a Lean formalization of the rate-monotonicity of even-exponent distances for continuous-time random walks, and of its failure at every other exponent above 1.

Main Results

  • For all positive integers n and m, raising the jump rates of a symmetric walk on the dihedral group D_n cannot increase the ℓ^{2m} distance of its time-t distribution from uniform.
  • The same holds over every inversion extension of a finite abelian group by an involution, a family including the generalized dihedral, dicyclic and generalized quaternion groups.
  • For every real p > 1 that is not an even integer, there is an n for which the pair (D_n, p) is not rate-monotonic.

See §Formal Challenge for a formal certificate.

Dependencies

This depends on Mathlib.

Formal Challenge

A formal challenge file certifying that this repository does formalize the results claimed above is located at Challenge/Basic.lean. This file only depends on the dependency above. It contains formal statements of §Main Results with sorry as proof.

This repository can be verified against the formal challenge with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:

lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.

About

No description, website, or topics provided.

Resources

Readme
Activity
Custom properties

Stars

0 stars

Watchers

0 watching

Forks

0 forks
Report repository

Releases

Packages

Contributors

Languages

, 'i'); if (__m === '*' || __re.test(location.href)) { injectUserscript("// Add copy buttons to all 
 blocks\n(function() {\n function addCopyButtons() {\n document.querySelectorAll('pre code').forEach(function(codeBlock) {\n if (codeBlock.parentElement.hasAttribute('data-copy-added')) return;\n codeBlock.parentElement.setAttribute('data-copy-added', 'true');\n \n var btn = document.createElement('button');\n btn.textContent = 'Copy';\n btn.style.cssText = 'position:absolute;top:4px;right:4px;padding:2px 8px;font-size:11px;background:#4ecdc4;border:none;border-radius:4px;color:#1a1a2e;cursor:pointer;opacity:0.7;transition:opacity 0.2s;';\n btn.onmouseover = function() { this.style.opacity = '1'; };\n btn.onmouseout = function() { this.style.opacity = '0.7'; };\n btn.onclick = function() {\n navigator.clipboard.writeText(codeBlock.textContent).then(function() {\n btn.textContent = 'Copied!';\n setTimeout(function() { btn.textContent = 'Copy'; }, 1500);\n });\n };\n codeBlock.parentElement.style.position = 'relative';\n codeBlock.parentElement.appendChild(btn);\n });\n }\n \n addCopyButtons();\n \n // Re-run on dynamic content\n var observer = new MutationObserver(addCopyButtons);\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Add Copy Buttons to Code Blocks");
}
} catch(__e) { console.warn('[Userscript:Add Copy Buttons to Code Blocks]', __e); }
})();
(function(){
try {
var __m = "github.com";
var __re = new RegExp('^' + "github\\.com" + '

Skip to content
BranchesTags
Repository files navigation
Proof of the Lyons-White Conjecture
This is a Lean formalization of the rate-monotonicity of even-exponent distances for
continuous-time random walks, and of its failure at every other exponent above 1.
Main Results
  • For all positive integers n and m, raising the jump rates of a symmetric walk on the
dihedral group D_n cannot increase the ℓ^{2m} distance of its time-t distribution
from uniform.
  • The same holds over every inversion extension of a finite abelian group by an involution,
a family including the generalized dihedral, dicyclic and generalized quaternion groups.
  • For every real p > 1 that is not an even integer, there is an n for which the pair
(D_n, p) is not rate-monotonic.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results
claimed above is located at Challenge/Basic.lean. This file only
depends on the dependency above. It contains formal statements of
§Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean
comparator on a Linux machine. First, follow the instructions in
https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
Resources
Readme
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Force GitHub README to respect dark mode\n(function() {\n var style = document.createElement('style');\n style.textContent = '\n .markdown-body {\n color-scheme: dark light;\n }\n .markdown-body pre { background: #161b22 !important; }\n .markdown-body code { background: rgba(110, 118, 129, 0.4) !important; }\n .markdown-body table th, .markdown-body table td { border-color: #30363d !important; }\n .markdown-body img { background: #0d1117; }\n .markdown-body blockquote { border-left-color: #8b949e; }\n .markdown-body hr { border-color: #30363d; }\n ';\n document.head.appendChild(style);\n})();", "GitHub Dark Mode README Fix");
}
} catch(__e) { console.warn('[Userscript:GitHub Dark Mode README Fix]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
BranchesTags
Repository files navigation
Proof of the Lyons-White Conjecture
This is a Lean formalization of the rate-monotonicity of even-exponent distances for
continuous-time random walks, and of its failure at every other exponent above 1.
Main Results
  • For all positive integers n and m, raising the jump rates of a symmetric walk on the
dihedral group D_n cannot increase the ℓ^{2m} distance of its time-t distribution
from uniform.
  • The same holds over every inversion extension of a finite abelian group by an involution,
a family including the generalized dihedral, dicyclic and generalized quaternion groups.
  • For every real p > 1 that is not an even integer, there is an n for which the pair
(D_n, p) is not rate-monotonic.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results
claimed above is located at Challenge/Basic.lean. This file only
depends on the dependency above. It contains formal statements of
§Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean
comparator on a Linux machine. First, follow the instructions in
https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
Resources
Readme
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Highlight search terms from Google/DuckDuckGo/Bing referrer\n(function() {\n var ref = document.referrer;\n var terms = [];\n \n if (ref.includes('google.com') || ref.includes('duckduckgo.com') || ref.includes('bing.com')) {\n var url = new URL(ref);\n var q = url.searchParams.get('q') || url.searchParams.get('p');\n if (q) {\n terms = q.split(/\\s+/).filter(function(t) { return t.length > 2; });\n }\n }\n \n if (terms.length === 0) return;\n \n var style = document.createElement('style');\n style.textContent = '.userscript-highlight { background: #fbbf24; color: #1a1a2e; padding: 1px 3px; border-radius: 2px; }';\n document.head.appendChild(style);\n \n function highlight(node) {\n if (node.nodeType === 3) { // text node\n var text = node.textContent;\n var found = false;\n terms.forEach(function(term) {\n var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\\]\\\\]/g, '\\\\') + ')', 'gi');\n if (regex.test(text)) {\n found = true;\n var frag = document.createDocumentFragment();\n var parts = text.split(regex);\n parts.forEach(function(part, i) {\n if (i % 2 === 0) {\n frag.appendChild(document.createTextNode(part));\n } else {\n var span = document.createElement('span');\n span.className = 'userscript-highlight';\n span.textContent = part;\n frag.appendChild(span);\n }\n });\n node.parentNode.replaceChild(frag, node);\n }\n });\n } else if (node.nodeType === 1 && node.childNodes) { // element\n var skipTags = ['SCRIPT', 'STYLE', 'NOSCRIPT', 'TEXTAREA', 'INPUT', 'SELECT'];\n if (!skipTags.includes(node.tagName)) {\n Array.from(node.childNodes).forEach(highlight);\n }\n }\n }\n \n highlight(document.body);\n \n // Re-highlight on dynamic content\n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1 || node.nodeType === 3) highlight(node);\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Highlight Search Terms");
}
} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
BranchesTags
Repository files navigation
Proof of the Lyons-White Conjecture
This is a Lean formalization of the rate-monotonicity of even-exponent distances for
continuous-time random walks, and of its failure at every other exponent above 1.
Main Results
  • For all positive integers n and m, raising the jump rates of a symmetric walk on the
dihedral group D_n cannot increase the ℓ^{2m} distance of its time-t distribution
from uniform.
  • The same holds over every inversion extension of a finite abelian group by an involution,
a family including the generalized dihedral, dicyclic and generalized quaternion groups.
  • For every real p > 1 that is not an even integer, there is an n for which the pair
(D_n, p) is not rate-monotonic.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results
claimed above is located at Challenge/Basic.lean. This file only
depends on the dependency above. It contains formal statements of
§Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean
comparator on a Linux machine. First, follow the instructions in
https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
Resources
Readme
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Strip utm_, fbclid, gclid, etc. from all links on page\n(function() {\n var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',\n 'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',\n 'ref', 'ref_src', 'source', 'medium', 'campaign'];\n \n function cleanUrl(url) {\n try {\n var u = new URL(url, window.location.origin);\n var changed = false;\n trackingParams.forEach(function(p) {\n if (u.searchParams.has(p)) {\n u.searchParams.delete(p);\n changed = true;\n }\n });\n return changed ? u.toString() : url;\n } catch (e) {\n return url;\n }\n }\n \n function cleanLinks() {\n document.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n \n cleanLinks();\n \n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1) {\n if (node.tagName === 'A') cleanLinks();\n node.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Remove Tracking Parameters from Links");
}
} catch(__e) { console.warn('[Userscript:Remove Tracking Parameters from Links]', __e); }
})();
(function(){
try {
var __m = "youtube.com";
var __re = new RegExp('^' + "youtube\\.com" + '

Skip to content
BranchesTags
Repository files navigation
Proof of the Lyons-White Conjecture
This is a Lean formalization of the rate-monotonicity of even-exponent distances for
continuous-time random walks, and of its failure at every other exponent above 1.
Main Results
  • For all positive integers n and m, raising the jump rates of a symmetric walk on the
dihedral group D_n cannot increase the ℓ^{2m} distance of its time-t distribution
from uniform.
  • The same holds over every inversion extension of a finite abelian group by an involution,
a family including the generalized dihedral, dicyclic and generalized quaternion groups.
  • For every real p > 1 that is not an even integer, there is an n for which the pair
(D_n, p) is not rate-monotonic.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results
claimed above is located at Challenge/Basic.lean. This file only
depends on the dependency above. It contains formal statements of
§Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean
comparator on a Linux machine. First, follow the instructions in
https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
Resources
Readme
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Auto-enable theater mode on YouTube\n(function() {\n function tryTheater() {\n var btn = document.querySelector('button[aria-label=\"Theater mode\"], ytd-player #player button[title=\"Theater mode\"]');\n if (btn && !btn.classList.contains('activated')) {\n btn.click();\n }\n }\n \n // Try immediately\n tryTheater();\n \n // Try after navigation (SPA)\n var lastUrl = location.href;\n setInterval(function() {\n if (location.href !== lastUrl) {\n lastUrl = location.href;\n setTimeout(tryTheater, 500);\n }\n }, 1000);\n \n // Also try on player load\n var observer = new MutationObserver(tryTheater);\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "YouTube Theater Mode Default");
}
} catch(__e) { console.warn('[Userscript:YouTube Theater Mode Default]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
BranchesTags
Repository files navigation
Proof of the Lyons-White Conjecture
This is a Lean formalization of the rate-monotonicity of even-exponent distances for
continuous-time random walks, and of its failure at every other exponent above 1.
Main Results
  • For all positive integers n and m, raising the jump rates of a symmetric walk on the
dihedral group D_n cannot increase the ℓ^{2m} distance of its time-t distribution
from uniform.
  • The same holds over every inversion extension of a finite abelian group by an involution,
a family including the generalized dihedral, dicyclic and generalized quaternion groups.
  • For every real p > 1 that is not an even integer, there is an n for which the pair
(D_n, p) is not rate-monotonic.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results
claimed above is located at Challenge/Basic.lean. This file only
depends on the dependency above. It contains formal statements of
§Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean
comparator on a Linux machine. First, follow the instructions in
https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
Resources
Readme
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Remove or un-stick sticky/fixed headers that block content\n(function() {\n function unstick() {\n document.querySelectorAll('header, nav, [role=\"banner\"], .header, .navbar, .sticky, .fixed-top, [style*=\"position: fixed\"], [style*=\"position:sticky\"]').forEach(function(el) {\n if (el.style.position === 'fixed' || el.style.position === 'sticky' || \n getComputedStyle(el).position === 'fixed' || getComputedStyle(el).position === 'sticky') {\n el.style.position = 'static';\n el.style.top = 'auto';\n el.style.zIndex = 'auto';\n }\n });\n }\n \n unstick();\n \n var observer = new MutationObserver(unstick);\n observer.observe(document.body, { childList: true, subtree: true, attributes: true, attributeFilter: ['style', 'class'] });\n})();", "Kill Sticky Headers");
}
} catch(__e) { console.warn('[Userscript:Kill Sticky Headers]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
BranchesTags
Repository files navigation
Proof of the Lyons-White Conjecture
This is a Lean formalization of the rate-monotonicity of even-exponent distances for
continuous-time random walks, and of its failure at every other exponent above 1.
Main Results
  • For all positive integers n and m, raising the jump rates of a symmetric walk on the
dihedral group D_n cannot increase the ℓ^{2m} distance of its time-t distribution
from uniform.
  • The same holds over every inversion extension of a finite abelian group by an involution,
a family including the generalized dihedral, dicyclic and generalized quaternion groups.
  • For every real p > 1 that is not an even integer, there is an n for which the pair
(D_n, p) is not rate-monotonic.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results
claimed above is located at Challenge/Basic.lean. This file only
depends on the dependency above. It contains formal statements of
§Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean
comparator on a Linux machine. First, follow the instructions in
https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
Resources
Readme
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Universal Dark Mode - works on any site\n(function() {\n var enabled = true;\n \n function applyDarkMode() {\n if (!enabled) return;\n \n // Create style element if it doesn't exist\n var style = document.getElementById('universal-dark-mode-style');\n if (!style) {\n style = document.createElement('style');\n style.id = 'universal-dark-mode-style';\n document.head.appendChild(style);\n }\n \n // Dark mode CSS - inverts colors but preserves images/video\n style.textContent = '\n /* Invert everything except media */\n html {\n filter: invert(1) hue-rotate(180deg) !important;\n background: #1a1a2e !important;\n }\n \n /* Restore images, videos, iframes, canvas */\n img, video, iframe, canvas, svg, picture, [style*=\"background-image\"] {\n filter: invert(1) hue-rotate(180deg) !important;\n }\n \n /* Preserve specific elements that should not be inverted */\n .no-dark-mode, .no-dark-mode *,\n [data-theme=\"light\"], [data-theme=\"light\"],\n .ace_editor, .ace_editor *,\n .CodeMirror, .CodeMirror *,\n .monaco-editor, .monaco-editor *,\n .markdown-body pre, .markdown-body pre *,\n .highlight, .highlight *,\n pre code, pre code * {\n filter: none !important;\n }\n \n /* Fix common UI elements */\n .modal, .popup, .dropdown-menu, .tooltip, .popover {\n filter: invert(1) hue-rotate(180deg) !important;\n background: #2d2d44 !important;\n border-color: #444 !important;\n }\n \n /* Scrollbars */\n ::-webkit-scrollbar { background: #1a1a2e !important; }\n ::-webkit-scrollbar-thumb { background: #444 !important; }\n ::-webkit-scrollbar-thumb:hover { background: #555 !important; }\n \n /* Selection */\n ::selection { background: #4ecdc4 !important; color: #1a1a2e !important; }\n ::-moz-selection { background: #4ecdc4 !important; color: #1a1a2e !important; }\n ';\n }\n \n function removeDarkMode() {\n var style = document.getElementById('universal-dark-mode-style');\n if (style) style.remove();\n }\n \n // Toggle with Alt+Shift+D\n document.addEventListener('keydown', function(e) {\n if (e.altKey && e.shiftKey && e.key === 'D') {\n e.preventDefault();\n enabled = !enabled;\n if (enabled) {\n applyDarkMode();\n console.log('[Universal Dark Mode] Enabled');\n } else {\n removeDarkMode();\n console.log('[Universal Dark Mode] Disabled');\n }\n }\n });\n \n // Apply on load\n applyDarkMode();\n \n // Re-apply on dynamic content\n var observer = new MutationObserver(function(mutations) {\n if (enabled && !document.getElementById('universal-dark-mode-style')) {\n applyDarkMode();\n }\n });\n observer.observe(document.head, { childList: true });\n \n console.log('[Universal Dark Mode] Loaded - Press Alt+Shift+D to toggle');\n})();", "Universal Dark Mode");
}
} catch(__e) { console.warn('[Userscript:Universal Dark Mode]', __e); }
})();
})();

Skip to content
BranchesTags
Repository files navigation
Proof of the Lyons-White Conjecture
This is a Lean formalization of the rate-monotonicity of even-exponent distances for
continuous-time random walks, and of its failure at every other exponent above 1.
Main Results
  • For all positive integers n and m, raising the jump rates of a symmetric walk on the
dihedral group D_n cannot increase the ℓ^{2m} distance of its time-t distribution
from uniform.
  • The same holds over every inversion extension of a finite abelian group by an involution,
a family including the generalized dihedral, dicyclic and generalized quaternion groups.
  • For every real p > 1 that is not an even integer, there is an n for which the pair
(D_n, p) is not rate-monotonic.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results
claimed above is located at Challenge/Basic.lean. This file only
depends on the dependency above. It contains formal statements of
§Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean
comparator on a Linux machine. First, follow the instructions in
https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
Resources
Readme
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
Releases
Packages
Contributors
Languages