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<a href="/grothendieckian-programme/">The Programme</a>
<a href="/#works">The works</a>
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<p class="kicker">Series 00–07 · WiseMotif Monograph of Mathematical Intelligence</p>
<h1 id="hero-title"><span class="nb">The Grothendieckian</span><br>Programme</h1>
<p class="subtitle">Recursive Mathematical Discovery in the Age of Machine Reasoning</p>
<p class="lead">The Grothendieckian Programme is a programme for machine reasoning about mathematics, set out in seven works of the WiseMotif Monograph of Mathematical Intelligence. Series 00 is their overview, and it adds no mathematical result to them.</p>
<div class="actions">
<a class="btn btn-primary" href="/grothendieckian-programme/">Read the overview</a>
<a class="btn" href="#works">The seven works</a>
<a class="btn" href="https://zenodo.org/communities/grothendieckian-programme">Zenodo community</a>
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</div>
<figure class="honeycomb">
<div class="hc">
<img src="/static/img/honeycomb.webp" width="1400" height="1400" fetchpriority="high"
alt="The seven works as a seven-cell honeycomb: GAI-MUA at the centre, Topos-Native above it, and Riemann · GRH, BSD, Hodge, abc and Langlands around them">
<svg viewBox="0 0 1400 1400" role="group" aria-label="The Grothendieckian Programme">
<a href="/grothendieckian-programme/grothendieckian-machine/" aria-label="Series 01 · GAI-MUA: The Grothendieckian Machine"><title>Series 01 · GAI-MUA: The Grothendieckian Machine</title><polygon points="963.3,700.0 831.7,928.1 568.3,928.1 436.7,700.0 568.3,471.9 831.7,471.9" class="cell cell-cyan"/></a>
<a href="/grothendieckian-programme/topos-native-intelligence/" aria-label="Series 02 · GM-Omega-Infinity: Topos-Native Intelligence"><title>Series 02 · GM-Omega-Infinity: Topos-Native Intelligence</title><polygon points="963.3,245.0 831.7,473.1 568.3,473.1 436.7,245.0 568.3,16.9 831.7,16.9" class="cell cell-cyan"/></a>
<a href="/grothendieckian-programme/arithmetic-spectral-descent/" aria-label="Series 03 · Arithmetic Spectral Descent"><title>Series 03 · Arithmetic Spectral Descent</title><polygon points="1357.4,472.5 1225.7,700.6 962.4,700.6 830.7,472.5 962.4,244.4 1225.7,244.4" class="cell cell-blue"/></a>
<a href="/grothendieckian-programme/fractured-adelic-geometry/" aria-label="Series 04 · Fractured Adelic Geometry"><title>Series 04 · Fractured Adelic Geometry</title><polygon points="1357.4,927.5 1225.7,1155.6 962.4,1155.6 830.7,927.5 962.4,699.4 1225.7,699.4" class="cell cell-violet"/></a>
<a href="/grothendieckian-programme/the-archimedean-prism/" aria-label="Series 05 · The Archimedean Prism"><title>Series 05 · The Archimedean Prism</title><polygon points="963.3,1155.0 831.7,1383.1 568.3,1383.1 436.7,1155.0 568.3,926.9 831.7,926.9" class="cell cell-blue"/></a>
<a href="/grothendieckian-programme/strained-prismatic-geometry/" aria-label="Series 06 · Strained Prismatic Geometry"><title>Series 06 · Strained Prismatic Geometry</title><polygon points="569.3,927.5 437.6,1155.6 174.3,1155.6 42.6,927.5 174.3,699.4 437.6,699.4" class="cell cell-violet"/></a>
<a href="/grothendieckian-programme/the-arithmetic-topos/" aria-label="Series 07 · The Arithmetic Topos"><title>Series 07 · The Arithmetic Topos</title><polygon points="569.3,472.5 437.6,700.6 174.3,700.6 42.6,472.5 174.3,244.4 437.6,244.4" class="cell cell-blue"/></a>
</svg>
</div>
<figcaption>The seven works as seven cells of one honeycomb. Select a cell to open its work.</figcaption>
</figure>
</div>
</section>
<section class="numbers" aria-label="The Programme in numbers">
<div class="wrap">
<ul class="strip">
<li><span class="n">7</span><span class="l">works</span></li>
<li><span class="n">34</span><span class="l">volumes</span></li>
<li><span class="n">26,021</span><span class="l">pages</span></li>
<li><span class="n">995</span><span class="l">chapters</span></li>
<li><span class="n">349,000+</span><span class="l">formulas</span></li>
<li><span class="n">10,500+</span><span class="l">numbered statements</span></li>
<li><span class="n">714</span><span class="l">algorithms</span></li>
<li><span class="n">1,434</span><span class="l">figures and plates</span></li>
</ul>
<p class="note">Each count was taken by two independent methods from the sealed editions; + marks a lower bound. The numbers measure extent, not achievement. <a href="https://github.com/AlienMuse/grothendieckian-programme/blob/main/STATISTICS.md">Definitions and both counts</a></p>
</div>
</section>
<section class="wrap three" aria-label="The question, the method, the boundary">
<article class="panel"><h2>The question</h2><p>A system trained to predict the next token can answer a remarkable range of questions. The Programme is concerned with a different capacity. Grothendieck's way with a hard problem was seldom to attack it where it stood: he built the setting in which the problem became natural, and let the solution come to it. The Programme asks whether a machine can take part in mathematics of that kind — whether it can construct a world in which a problem becomes natural, test that world against everything already known, say precisely where it fails, and pose the next question there — and what such a machine must be, internally, if anything it concludes is to be trusted.</p></article>
<article class="panel"><h2>The method</h2><p>The Programme calls its method recursive mathematical discovery. A structure is proposed; it is tested against the cases in which the answer is known; what survives is abstracted into a new setting, and the question is posed again there. What does not survive is not discarded. It becomes a conjecture, a gap or a falsification test, with a statement, a status and a place in a ledger, and it is carried forward as part of the result.</p></article>
<article class="panel panel-boundary"><h2>The boundary</h2><p><strong>None of the seven works claims a proof of the problems it studies.</strong> The Riemann Hypothesis, the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture and the abc conjecture remain open, and so does the Langlands correspondence over number fields beyond the cases the literature has proved; each work says so in its first pages, in a sentence of its own. What each offers is the state of the question after the method has been applied without concession: what has been proved, what has been reduced to stated hypotheses, and where each route stops.</p></article>
</section>
<section class="wrap works" id="works" aria-labelledby="works-title">
<header class="section-head">
<h2 id="works-title">Series 00–07</h2>
<p>The overview and the seven works, in series order. Each has its own repository, its own DOI and its own boundary.</p>
</header>
<ol class="cards">
<li class="card card-gold card-overview">
<a class="card-link" href="/grothendieckian-programme/">
<img class="cover cover-card" src="/static/img/covers/series-00.webp" width="480" height="674" alt="Cover of The Grothendieckian Programme" loading="lazy" decoding="async">
<span class="card-body">
<span class="card-series">Series 00</span>
<span class="card-title">The Grothendieckian Programme</span>
<span class="card-sub">Recursive Mathematical Discovery in the Age of Machine Reasoning</span>
<span class="card-line">The Programme's overview: it stands before the seven works and is not one of them; it sets out the idea of the Programme, its architecture and what the seven share, and adds no mathematical result to them.</span>
<span class="card-extent">one book · 53 pages</span>
</span>
</a>
<a class="card-doi" href="https://doi.org/10.5281/zenodo.23669107">doi:10.5281/zenodo.23669107</a>
</li>
<li class="card card-cyan">
<a class="card-link" href="/grothendieckian-programme/grothendieckian-machine/">
<img class="cover cover-card" src="/static/img/covers/series-01.webp" width="480" height="674" alt="Cover of GAI-MUA: The Grothendieckian Machine" loading="lazy" decoding="async">
<span class="card-body">
<span class="card-series">Series 01</span>
<span class="card-title">GAI-MUA: The Grothendieckian Machine</span>
<span class="card-sub">A Mathematical Universe Architecture for Verifiable and Self-Evolving Artificial Intelligence</span>
<span class="card-line">The core of the Programme: it sets out how mathematical worlds are constructed, compared and carried from one setting to another, and how a machine may hold them as typed, versioned and verifiable objects.</span>
<span class="card-extent">6 volumes · 14,905 pages</span>
</span>
</a>
<a class="card-doi" href="https://doi.org/10.5281/zenodo.23480696">doi:10.5281/zenodo.23480696</a>
</li>
<li class="card card-cyan">
<a class="card-link" href="/grothendieckian-programme/topos-native-intelligence/">
<img class="cover cover-card" src="/static/img/covers/series-02.webp" width="480" height="674" alt="Cover of GM-Omega-Infinity: Topos-Native Intelligence" loading="lazy" decoding="async">
<span class="card-body">
<span class="card-series">Series 02</span>
<span class="card-title">GM-Omega-Infinity: Topos-Native Intelligence</span>
<span class="card-sub">A Sheaf-Theoretic Architecture for Verifiable Machine Intelligence</span>
<span class="card-line">The meta-reasoning kernel: it states what an intelligence must be, internally, if it is to reason inside such worlds: reasoning native to a topos, in which a conclusion exists only where the local evidence glues.</span>
<span class="card-extent">5 volumes · 4,020 pages</span>
</span>
</a>
<a class="card-doi" href="https://doi.org/10.5281/zenodo.23538544">doi:10.5281/zenodo.23538544</a>
</li>
<li class="card card-blue">
<a class="card-link" href="/grothendieckian-programme/arithmetic-spectral-descent/">
<img class="cover cover-card" src="/static/img/covers/series-03.webp" width="480" height="674" alt="Cover of Arithmetic Spectral Descent" loading="lazy" decoding="async">
<span class="card-body">
<span class="card-series">Series 03</span>
<span class="card-title">Arithmetic Spectral Descent</span>
<span class="card-sub">Obstruction Theory, Hodge–Krein Realization, and a Falsifiable Architecture for the Riemann Hypothesis</span>
<span class="card-line">Research subject 1: the Riemann Hypothesis. The single substantive choice of the work is the replacement of a positivity condition by an index condition.</span>
<span class="card-extent">4 volumes · 1,549 pages</span>
</span>
</a>
<a class="card-doi" href="https://doi.org/10.5281/zenodo.23564898">doi:10.5281/zenodo.23564898</a>
</li>
<li class="card card-violet">
<a class="card-link" href="/grothendieckian-programme/fractured-adelic-geometry/">
<img class="cover cover-card" src="/static/img/covers/series-04.webp" width="480" height="674" alt="Cover of Fractured Adelic Geometry" loading="lazy" decoding="async">
<span class="card-body">
<span class="card-series">Series 04</span>
<span class="card-title">Fractured Adelic Geometry</span>
<span class="card-sub">A Foundational Framework for the Birch and Swinnerton-Dyer Conjecture</span>
<span class="card-line">Research subject 2: the conjecture of Birch and Swinnerton-Dyer. The governing idea is one object with two descriptions.</span>
<span class="card-extent">4 volumes · 571 pages</span>
</span>
</a>
<a class="card-doi" href="https://doi.org/10.5281/zenodo.23585887">doi:10.5281/zenodo.23585887</a>
</li>
<li class="card card-blue">
<a class="card-link" href="/grothendieckian-programme/the-archimedean-prism/">
<img class="cover cover-card" src="/static/img/covers/series-05.webp" width="480" height="674" alt="Cover of The Archimedean Prism" loading="lazy" decoding="async">
<span class="card-body">
<span class="card-series">Series 05</span>
<span class="card-title">The Archimedean Prism</span>
<span class="card-sub">Tame Period Geometry and the Hodge Conjecture</span>
<span class="card-line">Research subject 3: the Hodge conjecture. Algebraicity is not a property that a cohomology class possesses; it is a descent datum that a class carries along a prism.</span>
<span class="card-extent">5 volumes · 1,309 pages</span>
</span>
</a>
<a class="card-doi" href="https://doi.org/10.5281/zenodo.23597954">doi:10.5281/zenodo.23597954</a>
</li>
<li class="card card-violet">
<a class="card-link" href="/grothendieckian-programme/strained-prismatic-geometry/">
<img class="cover cover-card" src="/static/img/covers/series-06.webp" width="480" height="674" alt="Cover of Strained Prismatic Geometry" loading="lazy" decoding="async">
<span class="card-body">
<span class="card-series">Series 06</span>
<span class="card-title">Strained Prismatic Geometry</span>
<span class="card-sub">Deformation over the Absolute Base and the abc Conjecture</span>
<span class="card-line">Research subject 4: the abc conjecture. Over ℤ there is no base; the work asks for a substrate over which Spec ℤ acquires a relative cotangent direction.</span>
<span class="card-extent">5 volumes · 1,666 pages</span>
</span>
</a>
<a class="card-doi" href="https://doi.org/10.5281/zenodo.23610034">doi:10.5281/zenodo.23610034</a>
</li>
<li class="card card-blue">
<a class="card-link" href="/grothendieckian-programme/the-arithmetic-topos/">
<img class="cover cover-card" src="/static/img/covers/series-07.webp" width="480" height="674" alt="Cover of The Arithmetic Topos" loading="lazy" decoding="async">
<span class="card-body">
<span class="card-series">Series 07</span>
<span class="card-title">The Arithmetic Topos</span>
<span class="card-sub">Foundations of a Trace-Theoretic Approach to the Langlands Programme</span>
<span class="card-line">Research subject 5: the Langlands programme. The Langlands correspondence is not a bijection; it is the value, at π<sub>0</sub>, of a trace taken on a self-dual six-functor geometry over a single global base.</span>
<span class="card-extent">5 volumes · 2,001 pages</span>
</span>
</a>
<a class="card-doi" href="https://doi.org/10.5281/zenodo.23629681">doi:10.5281/zenodo.23629681</a>
</li>
</ol>
</section>
<section class="wrap architecture" aria-labelledby="arch-title">
<div class="arch-text">
<h2 id="arch-title">The architecture</h2>
<p>Seven works carry the Programme, drawn as seven cells of one honeycomb. <strong>At the centre</strong> stands <em>The Grothendieckian Machine</em> (Series 01), which sets out how mathematical worlds are constructed, compared and carried from one setting to another, and how a machine may hold them as typed, versioned and verifiable objects. <strong>Above it</strong>, on a shared edge, stands <em>Topos-Native Intelligence</em> (Series 02), which states what an intelligence must be, internally, if it is to reason inside such worlds: reasoning native to a topos, in which a conclusion exists only where the local evidence glues. <strong>Around the two</strong> stand five research subjects — the Riemann Hypothesis and its generalisation, the conjecture of Birch and Swinnerton-Dyer, the Hodge conjecture, the abc conjecture and the Langlands programme — on each of which the method is carried through in full, with every step recorded in a form that a mathematician can check and a machine can execute.</p><p>The two meet at the site. A world of the first work presents its evidence as a site; the kernel of the second work places a sheaf of local information over that site and computes what glues. A bridge of the first work becomes a morphism of sites, and an obligation becomes an open cell of a contract, printed as open until it is discharged. What returns from the periphery to the core is method rather than result: a scoped archive of negative results that a search can consult before it proposes; the lesson that the measure of AI-assisted mathematics is theorems, and that an exact account of where they are missing is part of the mathematics; a record of each failure at the place where it occurred, so that the next recursion can begin from it.</p> <p><a class="more" href="/grothendieckian-programme/#architecture">Figure 1 and the architecture in full →</a></p>
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<figure class="arch-figure">
<a href="/grothendieckian-programme/#architecture"><img src="/static/img/figure-1-960.webp" width="960" height="1280" loading="lazy" decoding="async"
alt="Figure 1 of the book: the seven works as a seven-cell honeycomb, with the core decomposed below"></a>
<figcaption>Figure 1 of the overview</figcaption>
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</section>
<section class="wrap collections" aria-labelledby="coll-title">
<h2 id="coll-title">Collections</h2>
<ul>
<li><a href="/grothendieckian-programme/">The Grothendieckian Programme</a> · Series 00–07</li>
<li><a href="https://grothendieckmath.com/">Lives of Mathematical Thought</a> · 1: <em>Alexander Grothendieck: A Life in Three Universes</em></li> </ul>
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<p class="foot-name">Vincent Liu</p>
<p>WiseMotif.ai</p>
<p><a href="https://orcid.org/0009-0000-8718-222X">ORCID 0009-0000-8718-222X</a> · <a href="https://github.com/AlienMuse">GitHub</a> · <a href="https://zenodo.org/communities/grothendieckian-programme">Zenodo</a></p>
<p><a href="https://alienmuse.space/">Alien Muse · alienmuse.space</a></p>
<p><a href="https://grothendiecklab.com/">Grothendieck Lab · grothendiecklab.com</a></p>
<p><a href="https://grothendieckmath.com/">Lives of Mathematical Thought · grothendieckmath.com</a></p> </div>
<nav class="foot-works" aria-label="The Grothendieckian Programme">
<p class="foot-head"><a href="/grothendieckian-programme/">The Grothendieckian Programme</a></p>
<ul>
<li><a href="/grothendieckian-programme/"><span class="sn">00</span> The Grothendieckian Programme</a></li>
<li><a href="/grothendieckian-programme/grothendieckian-machine/"><span class="sn">01</span> GAI-MUA: The Grothendieckian Machine</a></li>
<li><a href="/grothendieckian-programme/topos-native-intelligence/"><span class="sn">02</span> GM-Omega-Infinity: Topos-Native Intelligence</a></li>
<li><a href="/grothendieckian-programme/arithmetic-spectral-descent/"><span class="sn">03</span> Arithmetic Spectral Descent</a></li>
<li><a href="/grothendieckian-programme/fractured-adelic-geometry/"><span class="sn">04</span> Fractured Adelic Geometry</a></li>
<li><a href="/grothendieckian-programme/the-archimedean-prism/"><span class="sn">05</span> The Archimedean Prism</a></li>
<li><a href="/grothendieckian-programme/strained-prismatic-geometry/"><span class="sn">06</span> Strained Prismatic Geometry</a></li>
<li><a href="/grothendieckian-programme/the-arithmetic-topos/"><span class="sn">07</span> The Arithmetic Topos</a></li>
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<p>Texts and images of the works: CC BY-NC-ND 4.0, as each work's copyright page states.</p>
<p>© 2026 Vincent Liu</p>
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