Synthesis — Mathematics

The evolving thesis. Spun out of the hub _inbox on 2026-08-02 from three mathematics texts parked under three different tags (mathematics, signal-processing, and the arriving probability course) across five days. A fourth, keisler-elementary-calculus, arrived the same afternoon and opened the corpus's first tension — four sources in, the spoke already has something to argue about.

Current thesis

A spoke was missing, and the _inbox diagnosed it before the split happened. The fourier-analysis-for-beginners record, parked 2026-07-28, ended with the observation that Fourier analysis is "infrastructure beneath at least four spokes and the subject of none" and set the spin-out trigger at three. The vector-calculus-corral record the next day made the identical argument about calculus, tallied two, and named the shapes a third could take — "applied maths, linear algebra, probability/statistics, information theory, numerical methods." mcmullen-probability-theory arrived on 2026-08-02 as the second item on that list. The cluster was predicted, not discovered.

That matters for what this spoke is. It was not created because mathematics is interesting; it was created because thirty applied spokes kept declining the same class of source for the same reason, and a repeated correct decline is what a missing spoke looks like from the outside.

The boundary is the thesis

Each founding source had a spoke that nearly took it, and each was refused on the same line: the maths was infrastructure under that spoke, not its subject.

• corral-vector-calculus against ../optimization-algorithms-wiki — two of its thirty sections minimize a function, but that spoke owns metaheuristic global optimizers, the family that exists because the calculus route needs derivatives it doesn't always have.
• thibos-fourier-analysis against ../psychology-wiki — the fit is through the audience (vision science) and not the content (about 85% general Fourier analysis).
• mcmullen-probability-theory against ../quant-trading-wiki and ../machine-learning-wiki — both teach probability as a prerequisite to their own subject; neither is teaching probability.

Stated as a rule: the maths is the subject here, the application is the subject there. None of the mathematics here makes a claim about the world, which is what distinguishes this spoke from every other one in the hub and why the usual freshness question barely applies — none of this material dates. The claims that can be wrong are the pedagogical ones sitting on top of it, and those are exactly where the corpus has now split.

What the corpus can support, and what it can't

Three of the four subjects still rest on exactly one source. The fourth ingest changed that for vector-calculus, which now has two accounts that agree on every theorem and disagree about the foundation.

Updated 2026-08-06. probability-theory is the second doubled subject, and it doubled in a different way. stirzaker-elementary-probability (CUP, 2nd ed. 2003, 538pp) covers McMullen's syllabus and keeps going into martingales, Markov chains and Brownian motion; the two texts agree about everything they share. Where the calculus pair disagrees about foundations, the probability pair differs only in how much the reader must supply — 98 terse pages with Feller assumed, against a self-contained course with worked examples, problems and solutions. Both kinds of second source are worth having, and only one of them creates a tension. In a spoke about mathematics, single-sourcing is less alarming than elsewhere — a theorem does not need a second vendor to confirm it — but the pedagogy claims are where it bites, and those are where the sources have started to differ.

Updated 2026-08-10. osborne-rubinstein-course-in-game-theory adds a twelfth text and a new subject, game-theory, and it arrives saying something none of the others do: the authors state that they treat their subject "not as a branch of mathematics but as a social science", and that a book with the same content could in principle be written without mathematics at all. Every other text here is written by someone claiming the subject as mathematics. This one proves theorems for 373 pages while disclaiming the label. It does not unsettle the boundary rule — a text teaching a body of theory from first principles routes here regardless of the field its examples come from — but it is the first time the rule has had to override the author's own description of the work, rather than merely the field of the examples.

It also imports an argument the spoke had no page for: the rationality assumptions the whole subject rests on are, in the authors' own words, "under perpetual attack by experimental psychologists", and they decline to defend them. That attack is the heuristics-and-biases tradition held in ../psychology-wiki. Nothing in this spoke's corpus can adjudicate it; recorded as a cross-spoke tension, not an open question here.

The rigour spectrum, stated up front

The founding three sit at visibly different points, and the corpus is more interesting for it than a uniform set would be:

• McMullen writes for people who will read Feller alongside, glosses binomial coefficients with the transitive action of a symmetric group, and derives P(A′) = 1 − P(A) from the axioms rather than from intuition.
• Corral proves selectively and says so: 5 out of 10, on the argument that too much rigour "can impede the flow of learning for the vast majority of the audience at this level."
• Thibos starts at vector arithmetic and complex numbers, and spends three chapters on hypothesis testing that a mathematics text would not contain.

The through-line held for the founding three: the same mathematics gets taught at whatever rigour the audience will bear, and knowing which version you are reading matters more than the material dating.

keisler-elementary-calculus breaks that frame, and it is the most interesting thing in the corpus so far. Keisler does not move along the rigour axis at all. He changes the foundation — infinitesimals instead of limits — and claims this buys accessibility and rigour together rather than trading one for the other. His premise is that the century of ε-δ teaching cost students the original intuition for no gain once Robinson made infinitesimals precise in 1960. If that is right, the spectrum the other three sit on is not a necessary trade-off but an artifact of one historical choice.

A second author defends a foundation choice, and defends it the other way (2026-08-06). stirzaker-elementary-probability adopts the formal definition of probability "not because of high principles, but merely because the alternative intuitive approach seems to lead more students into errors." That is Keisler's own kind of argument — a claim about what happens to students — reaching the opposite conclusion in a different subject. Neither claim is measured, and Stirzaker's is the weaker of the two: an experienced teacher's impression, offered as an impression. What the pair establishes is that the rigour question in this corpus is argued on pedagogy rather than on principle by both authors who bother to argue it at all.

He is also the only author here who cites evidence for a pedagogical claim: a controlled experiment across five schools, run by Sister Kathleen Sullivan in 1972–74, reported in a Wisconsin PhD thesis and the American Mathematical Monthly. Weigh it honestly: it is the author summarizing a study of his own method in his own preface, and this hub holds neither the thesis nor the article. It is still the only such citation in the corpus.

The floor of the rigour spectrum, and what sits on it (2026-08-12)

illowsky-introductory-statistics extends the spectrum above downward past every text held here, and does it deliberately rather than by omission. No calculus, no proofs, intermediate algebra assumed, a graphing calculator standing in for the theory — and it still reaches confidence intervals, two-sample hypothesis testing, chi-square, regression and one-way ANOVA in 847 pages.

The spectrum is now wide enough to be read as a curriculum rather than a list of styles. From larsen-marx-mathematical-statistics deriving the generalized likelihood ratio, through herzog-understanding-statistics applying inference without derivation, to Illowsky & Dean teaching the same tests with a keystroke sequence, the corpus holds three altitudes of one subject. The mathematics is common; what varies is whether the reader is ever shown why it works, and mathematical-statistics now carries the comparison.

The tool a text assumes turns out to track the rigour, inversely. carroll-geometric-function-theory assumes no computational tool at all and proves everything. judson-abstract-algebra assumes Sage and proves everything. This one assumes a TI-83/84 and proves nothing — the calculator is doing the work the derivation would otherwise do, supplying numbers whose provenance the student is not expected to follow. Three points is not a law, but the spoke has been recording the assumed tool since carroll-geometric-function-theory and this is the first time it has paid.

A publishing arrangement the corpus had not seen. Every earlier open text here was released by its author. openstax is an institution that commissions, reviews and revises at scale, and the consequences are visible in the book: four practice exams, instructor guides behind verified accounts, an errata process, and a revision that preserves problem numbering so online homework providers can follow the edition change. A service-course textbook is infrastructure with downstream integrations, and this is the corpus's first source that shows the pressure. The licence also undercuts the "free textbook" framing slightly: CC BY-NC-SA, not the unrestricted CC BY of mari-measurement-across-sciences.

The canon has an edge, and it is the end of the first course

Every text in this corpus until now teaches a first course, and the rigour spectrum above is what you see when authors disagree inside a syllabus they all inherited. Keisler changes the foundation, Stirzaker defends the formal definition, Corral proves selectively at 5 out of 10 — three arguments about how to teach material none of them chose.

carroll-geometric-function-theory is the corpus's first second course, and its preface names the thing the rest of the shelf could not show. On the first course there is "general agreement on the list of main topics" — Cauchy, Liouville, Maximum Modulus, Residues, Rouché — and "the general syllabus of a first course in complex analysis is well-established." Then: "This consensus evaporates when it comes to a second course."

So the settledness the other texts enjoy is not a property of mathematics. It is a property of the first year of a subject, and it stops. That reframes the rigour spectrum: those authors could argue about pedagogy precisely because content was not in dispute, and Carroll cannot.

What an author does when nothing is settled. He writes §1.4, "What Is Not in This Book" — a numbered section listing the omissions and naming where each is properly treated: no curvature (Ahlfors, Krantz), no conformal invariants beyond the hyperbolic metric, no Riemann surfaces past the sphere, nothing on Denjoy–Wolff, which he says "would fit naturally in this book." And he concedes the selection is contestable: "many other topics could lay equal claim to being included."

No other text here does this. The nearest comparison is in another spoke and it is instructive: ../engineering-education-wiki's coding-interview-university also announces exclusions (frontend, full-stack, SQL) but its omissions are dictated by an exam. Carroll has no exam. His omissions are argued, and he supplies the forwarding address — which is the behaviour a reader actually needs from a text at the edge of a canon, and which this corpus can now say is rare.

A third apparatus finding. The book assumes no computational tool — no Sage (judson-abstract-algebra), no Matlab (thibos-fourier-analysis) — and carries 54 pages of complete worked solutions, where corral-vector-calculus's 420 exercises answer only a subset. Both follow from the SUMS specification, which is the first case here of a publisher rather than an author setting the pedagogy: one-to-two semesters, fully-worked solutions, self-study viable. The corpus has been reading these decisions as authorial; at least sometimes they are contractual.

The hole this opens is larger than the one it closes. complex-analysis enters the corpus from the top: the spoke now holds the geometric second course and none of the settled first one. It is also an edge nobody had listed — the shelf ran calculus → analysis → algebra and simply skipped the complex plane. That it went unnamed through eight growth-edge revisions is worth recording about the edge-listing method itself, which catches subjects the existing sources gesture at and misses the ones they never mention.

The hole opened three days ago is mostly filled, and by a book that denies it was a hole (2026-08-13)

carroll-geometric-function-theory opened the corpus's largest gap on 2026-08-10 — a second course in complex analysis with none of the first course beneath it. baggett-analysis-single-variable arrived on 2026-08-13 with most of that first course in its final chapter: Cauchy's Theorem and Integral Formula, Taylor and Laurent, Liouville, the Fundamental Theorem of Algebra, Maximum Modulus, Open Mapping, the Residue Theorem. Missing from Carroll's own list: the Argument Principle and Rouché's Theorem. Missing in the other direction: conformal mapping, where Carroll starts. The seam is thin but real, and complex-analysis now records both ends instead of one.

Baggett does not think he wrote a complex analysis book. He refuses the split: separating real from complex "does not at all jive with their development historically." What he cuts instead is dimension — several variables are "a very different kettle of fish." So this corpus, which had been running calculus → analysis → complex analysis as a sequence of subjects, now holds a text organised on a perpendicular axis: one variable, both fields, against trench-real-analysis's many variables, one field.

The two subject pages this spoke maintains, real-analysis and complex-analysis, are exactly the cut Baggett declines to make. Keeping them is a decision the spoke should be able to defend rather than inherit, and for now the defence is practical: two of the three texts are organised that way and one reader arriving at either page is better served than by a merged one.

A second thing this book does that no other here does: it is organised around results. Seven named truths from the preface — √2 irrational, A = πr², C = 2πr, e^(iπ) = −1, and the fundamental theorems of calculus and of algebra — with the whole definitional order arranged so that six of them can be proved. π is defined as the smallest positive zero of the sine function; sine is defined by a power series so that arc length need not come first; area is defined only for regions between graphs of continuous functions so that measure theory can be skipped and the disk still covered. That is a curriculum built backwards from what the student should be able to explain at the end.

Set against the corpus's other pedagogical arguments, it is a third kind. Keisler argues about the foundation, Corral and Illowsky about the rigour the audience will bear; Baggett argues about the order of definitions, and it is the only one of the three whose cost is visible in the book — he cannot prove e^(iπ) = −1 in Chapter III where the series already give exp(iz) = cos z + i sin z, because cos π = −1 is not yet available, and he says so instead of rearranging.

And it makes growth edge 3 worse again. The one text here that constructs the reals from Dedekind cuts and proves the complete ordered field unique never writes the word infinitesimal. Three assumers now, one arguer against, still nobody arguing for. Recorded under the foundational fork below.

A subject arrives without a book (2026-08-15)

Every other subject here came in with a text: a source page, a tier, a licence, a raw/ copy or a URL. Information and coding theory did not. The curator asked for the coding half of MacKay's book to be built out in this spoke, and the book is in ../machine-learning-wiki — routed there on 2026-08-02 on the machine-learning half of its substance, hours before this spoke existed, with the router's note recording at the time that no spoke owned the other half.

So the spoke now teaches a subject it holds no copy of. Eight pages cite information-theory-inference-learning-algorithms across a spoke boundary, and the Book section of index does not list it. Three things follow, and they are not all comfortable.

It vindicates the boundary rule rather than bending it. The rule says the maths is the subject here and the application is the subject there. Both halves of that sentence are now true of the same book at once: the ML half is the subject in that spoke, the coding half is the subject in this one, and the book was not moved or duplicated to make it work. A source can be the subject of two spokes in different parts, which the corpus had asserted before (thibos-fourier-analysis against ../psychology-wiki) but never acted on this literally.

It is the only subject here whose depth is capped by another spoke's decision. If that spoke ever re-scopes, deletes or narrows its holdings, eight pages here lose their footing. Nothing prevents that and no mechanism warns of it. The mitigation is the ordinary one — a text of this spoke's own — and growth edge 6 is left open for exactly that reason.

The material argues, for once, in the same direction the corpus already leans. Information theory is the corpus's first body of theory whose central results are negative — the source coding theorem says what no compressor can beat, the noisy-channel coding theorem says what no code can exceed. It is also the corpus's clearest case of a proof that establishes existence and withholds construction: Shannon's 1948 argument shows good codes exist by averaging over all of them, and fifty years passed before anyone built one that came close (low-density-parity-check-code was invented in 1962 and ignored until 1995). The rigour spectrum this synthesis tracks is about how much a text proves. This subject adds a second axis: what a proof gives you, and what it conspicuously does not.

Three entrances, one summit (stochastic processes, 2026-08-16)

The curator asked for research on stochastic processes, "basically the main field of mathematical finance." Three sources landed: lawler-stochastic-calculus (T1, free from the author's page at university-of-chicago) and two MIT courses published by mit-opencourseware, gamarnik-advanced-stochastic-processes (15.070J) and wu-introduction-stochastic-processes (18.445). Five pages: stochastic-processes, markov-chain, martingale, brownian-motion, stochastic-calculus.

The framing is right about the destination and wrong about the field. Two of the three sources end at option pricing and one never approaches it — 18.445 has no Brownian motion, no stochastic integral and no finance in twenty-four lectures. Mathematical finance is one exit from this subject, and the exit the curator named, but the subject is not organised around it.

The finding is about the entrances. These three teach the same material in three incompatible orders: finite Markov chains (Wu), discrete-time martingales (Lawler), metric spaces and large deviations (Gamarnik). Markov chains are thirteen of twenty-four lectures in one course and not a lecture topic in the other, at the same institution, two years apart. Brownian motion is chapter 2, lecture 6, and absent. What they converge on is Itô calculus. That is the same shape carroll-geometric-function-theory found in complex analysis — "consensus evaporates when it comes to a second course" — arriving now from the other side: consensus about the summit, none about the approach. The corpus has two independent instances of it, which makes it a pattern rather than one author's complaint.

Two things the sources add to axes this synthesis already tracks.

Rigour, handled typographically. lawler-stochastic-calculus asks for post-calculus probability, says a deep understanding needs measure theory, and refuses to choose: the measure-theoretic discussion is indented in a different font to be skipped or read. The spoke has seen split audiences (lal-algebra-1 addressing undergraduates and graduates in one volume) and never seen one solved on the page like this.

The tool assumed. Lawler expects the reader to program — C++, Matlab or Mathematica — and his exercises ask for a simulated Brownian path at Δt = 0.01 checked against the reflection principle. On the axis running from carroll-geometric-function-theory (no tool) through illowsky-introductory-statistics (a TI-83), this is the far end: the tool is not an aid to arithmetic, it is how you see the object.

A fourth source, 2026-09-05, and it breaks the section's own title. dynkin-yushkevich-markov-processes (Dynkin & Yushkevich, Moscow University, Plenum 1969) arrived by Telegram from the sources directory of the Harvard course page that gave us mcmullen-probability-theory. It is not a course, and it does not climb to the same summit.

It runs the other way. The three courses build analytic machinery and aim it at probability. This book proves the Dirichlet problem for the Laplace equation has a solution by probabilistic argument, and derives boundary conditions for differential equations from the paths of a birth-and-death process. No Itô calculus as a destination, no finance anywhere. So "one summit" was a claim about three teaching sources, not about the field: probability is also a tool for analysis, and was one long before the general theory of Markov processes existed — the book's own footnote dates it to Phillips and Wiener in 1923.

The direction has a name and a precedent. Doob derived the Dirichlet solution from Wiener paths in 1954, and the authors say plainly he ran it the opposite way, "for the derivation of the properties of paths from theorems of analysis." Two directions between the same two subjects, distinguished in print by the people doing it, which is a better statement of what this subject is than any of the four syllabi.

And it disagrees with Feller in method. Feller is credited as "the pioneer in the application of the probabilistic approach to boundary problems", with the objection that he "still introduces all his constructions in purely analytical form." This spoke now holds a text that follows Feller (mcmullen-probability-theory is explicitly a companion to him) and a text that names what it declines to take from him.

Organisation as a fourth axis. The three courses differ in order; this one differs in unit. It is arranged by problem area, and its problem sets carry results rather than exercises — the Martin boundary for a denumerable chain lives in a Chapter III problem. The analytical arguments are exiled to an appendix so as "not to impede the main flow of the probabilistic discussions", which is Lawler's font-switch solved by relocation instead of typography. Two sources in this spoke have now independently invented a way to keep one audience from blocking another on the page.

And the corpus's oldest hole gets louder. Each source handles the measure-theory question differently and none of them closes it: Lawler says a deep understanding needs it and indents that layer as optional; 15.070J teaches "measure theoretic probability" itself, opening from metric spaces and topology; 18.445 sidesteps it, asking only for basic probability and matrix operations. The spoke still holds no measure-theory text, and the gap is now under a subject it teaches rather than beside one it already had.

A prerequisite list that decides a syllabus (graph theory, 2026-08-17)

grinberg-graph-theory arrived from Telegram and opened graph-theory, and it is the first source here that sits entirely off the corpus's spine. Every other subject in this spoke descends from calculus or from probability, and several from both. This one states its prerequisites as determinants, polynomials and finite sums, and then adds a sentence no other text here could write:

No analysis (or even calculus) is required anywhere in this text.

That choice has a price, and the book pays it in public. Planarity and the four-colour theorem are not in it — the most famous corner of introductory graph theory, mentioned twice and handed to another textbook both times. Grinberg gives the reason at the first mention: any proof that K₅ cannot be drawn without crossings "must use some analysis or topology, since the result relies on the notion of a (continuous) curve in the plane."

So the syllabus was cut by the prerequisite list. That is a third kind of pedagogical decision, and the corpus now holds one of each:

• about rigour — keisler-elementary-calculus vs the ε-δ development, the foundational fork;
• about content selection — judson-abstract-algebra placing cryptography and coding theory a third of the way in, because students meeting proof need motivation;
• about entry cost — this book, deciding what the reader must already know and letting the syllabus follow from it.

The third is the only one of the three that is falsifiable from inside the text. Keisler's and Judson's arguments are assertions in a preface; Grinberg's shows up as a missing chapter, and the implication runs the way he says it does. Whether the trade was worth making is still an opinion, but for once the corpus can see both sides of it.

A second point about consensus, extending the carroll-geometric-function-theory finding. That book reported that the canon agrees on a first course in complex analysis and dissolves at the second. Grinberg lists roughly twenty-five graph theory textbooks and says that beyond the absolute basics they "often cover different ground" — at the first course. His own is "self-contained and does not follow any existing book," weighted toward the discrete and algebraic side, and bounded, he says, by the quarter and by "the limits of my own knowledge." Two subjects, two different places where agreement runs out, and the likeliest explanation is age: analysis had two centuries to settle a first course, graph theory has had one, and most of the material dates from after 1950. §6.8 makes the point by itself — a conjecture from 1971, proved in 2019, with the general case still open, printed in an introductory text.

The licence floor moved, and it is now the floor. CC0 1.0 is a public-domain dedication: no attribution, no share-alike, no non-commercial clause. Everything else free here asks for something (mari-measurement-across-sciences CC BY 4.0 was the previous most-open, and the GFDL and NC texts ask for more). Nothing more permissive exists, so the access axis described under Paid first, free second below now has both ends occupied — Springer at $89.99 in hardcover, and a graduate text with the copyright given away, arriving nine days apart.

The syllabus cut gets a control (2026-08-30)

Thirteen days later a second graph theory text arrived from Telegram — harju-graph-theory, 98 pages of lecture notes from the University of Turku — and it is the closest thing this spoke has had to a controlled comparison. Same subject, same first-course level, and its Chapter 5 is the chapter Grinberg refuses to write: planar graphs, Kuratowski, Euler's formula, the four- and five-colour theorems, and embeddings on surfaces of higher genus.

Harju does not contest the prerequisite argument. He agrees with it and pays the bill in a different currency, saying so in one sentence at the top of the chapter:

The geometry of the plane will be treated intuitively.

He then defines region and continuous curve by hand from open sets and proceeds, omitting the proof of the genus form of Euler's formula and citing Mohar & Thomassen for the rest.

So the corpus can now separate two things it could previously only assert together. The mathematical claim — a rigorous planarity chapter needs topology the reader does not have — is confirmed by the book that keeps the chapter, which is much stronger evidence than the book that dropped it saying so. What is not confirmed is that the syllabus had to shrink. There were two exits, both honest, and Grinberg took the one that protects rigour while Harju took the one that protects coverage.

That upgrades the section above rather than overturning it. Entry cost remains "the only one of the three falsifiable from inside the text," and it has now been tested from outside as well.

The exercise contrast points the same way. Grinberg prints 116 numbered exercises and no solutions; Harju prints no exercise sets at all — ten inline handoffs across 98 pages, always "Proof. Exercise." under a theorem, plus a few proofs omitted with a citation. A graduate course text builds the reader's work into a structure; a lecturer's notes leave it as gaps to be filled in the room. Both are pedagogical decisions about exercise design, which this spoke's domain names and had until now only observed in the one style.

A third position on the access axis. harju-graph-theory carries no copyright notice and no licence at all — searched in full. It is a PDF on a university web server: free to obtain, legally silent. Every other text here states its terms, whether commercial (lal-algebra-1, larsen-marx-mathematical-statistics) or open (grinberg-graph-theory CC0, hefferon-linear-algebra, trench-real-analysis, the NC and GFDL texts). Lecture notes on a personal university page are a large fraction of freely available mathematics and this is the corpus's first specimen of the condition; see Paid first, free second.

The generalising move, made twice (general algebra, 2026-08-22)

bergman-invitation-general-algebra adds a rung the spoke did not have. abstract-algebra already carries the corpus's cleanest argument for abstraction: prove a theorem once about groups and it holds for integers, polynomials, matrices, permutations and symmetries at no extra cost. General algebra applies that argument to itself. Fix a signature and a set of identities, prove the theorem about the resulting variety, and the group case, the ring case, the lattice case and the Boolean-algebra case fall out together. Same move, one level up — and having both rungs in the spoke makes the move visible as a method rather than as a fact about groups.

Three subjects arrive as tools rather than as topics, because the book needs them: order and lattice theory (semilattices, closure operators, Galois connections), set theory past naive use (ordinals, well-ordering, Zorn's Lemma), and category-theory — about 180 pages of it, from categories and functors through Yoneda and adjoints to Freyd's criterion. Only the last is paged, and it is paged with its limits stated: the spoke has no category-theory text, so it holds the apparatus a working algebraist needs and none of monoidal categories, topoi or the homotopy-theoretic side. That is a growth edge, not a covered subject.

A subject that disputes its own name. §1.5 is titled "The name of the game" and concedes that "Universal Algebra… has long been called" that, but that many in the field now dislike the term because "it promises too much." george-bergman uses General Algebra in the title and immediately grants that the replacement "may not be understood as referring to a specific area." Nothing else in the corpus opens by telling the reader its subject's name is contested — the closest is carroll-geometric-function-theory on where the canon stops agreeing, and that is about syllabus, not nomenclature.

A fourth kind of pedagogical argument. The spoke already holds three: keisler-elementary-calculus arguing about rigour foundations (with an experiment behind it), judson-abstract-algebra arguing about content selection (applications early, because students meeting rigorous proof need motivation), and grinberg-graph-theory where a prerequisite list cut the syllabus. Bergman argues about none of those. He argues about what class time is for: "Why, for an hour at a time, should an instructor write notes on a blackboard and students copy them into their notebooks – often too busy with the copying to pay attention to the content – when this work could be done just as well by a photocopying machine?" The notes exist to absorb transcription so that the room can be spent on questions and exercises, and the reading is enforced by a mechanism — a question a day, tagged urgent / important / unimportant / pro forma, with pro forma questions required to arrive with their own answers. He kept seven pages of course administration in the published book on the grounds that the arrangement is nonstandard enough to interest other instructors. That is the first argument here about the format of teaching rather than its content, and it is the only one that would survive translation to a subject other than mathematics.

It is also the only text here that states the risk its own style creates: "I am worried that the amount of handwaving in these notes may lead some students to think handwaving is an acceptable substitute for proof." Set beside the corpus's habit of authors rating their own rigour (corral-vector-calculus at 5 out of 10) this is a different act — not a calibration for the reader, but a warning about a failure mode the book might cause.

Two smaller firsts. It publishes its own difficulty map — "the hard material begins with Chapter 8", refined to a section list, plus a note that the last two sections of Chapters 7, 8 and 9 are sketchy and should be read for ideas. And its prerequisite is a test rather than a course list: "anyone conversant with fewer than three ways of proving the existence of free groups has something to learn from Chapters 2-3" — a bar the book then clears for the reader in Chapter 3, which supplies exactly three proofs. Against grinberg-graph-theory, where the prerequisite list decided what got cut, here the prerequisite is a self-assessment the text immediately answers.

On the age axis it is the extreme. The notes began around 1971 and were revised on every teaching for forty-four years before Springer published them. Nothing else in the spoke has that provenance, and it shows in the acknowledgements: a locally enhanced troff before the LaTeX conversion, and students thanked by name for corrections across decades. Access sits with larsen-marx-mathematical-statistics and carroll-geometric-function-theory — reachable, not openly licensed — but here the arrangement is stated by the author, who says free distribution "will not be feasible for the final published version" while committing to keep the submitted version on his Berkeley page.

Boundary note. The spoke header names seven subjects and lists neither abstract algebra (paged 2026-08-08) nor general algebra and category theory (today). The header has fallen behind the spoke twice now; whether to widen it again is left to the curator, and flagged in log.md.

Someone tries to move the edge (matrix calculus, 2026-08-22)

The spoke's standing finding is that the canon has an edge and it is the end of the first course — carroll-geometric-function-theory found agreement on a first course in complex analysis and none on the second; grinberg-graph-theory found the disagreement arriving at the first course, which the age of the subject explains. edelman-johnson-matrix-calculus is the first source here that treats that edge as a defect to be fixed rather than a fact to be reported.

Its argument is made against MIT's own catalog. After 18.01 and 18.02, "it seems as though this sequence of material is being cut off arbitrarily" — and the missing term is named: Scalar → Vector → Matrices → Higher-Order Arrays, with the observation that "this is how the sequence is portrayed in many computer programming languages, including Julia! Why should calculus stop with vectors?" The justification is external to mathematics: linear algebra "has gradually taken over a much larger part of today's tools" in machine learning, statistics and engineering, so the calculus has to follow it up the type hierarchy. Two MIT professors ran a 16-hour IAP short course to fill the hole and then posted the notes.

So the corpus now has three positions on the same question. Carroll: the canon dissolves after the first course, and says so. Grinberg: it dissolves at the first course, because the subject is young. Bright, Edelman and Johnson: the sequence was cut in the wrong place and here is the next term. Whether the gap is real mathematics or an artifact of one institution's course numbering is not settleable from one source — but the claim is explicit, dated, and comes with a syllabus.

The technical move is the pedagogical one. To make calculus survive matrix and function inputs, the notes stop dividing: df = f(x + dx) − f(x) = f′(x)[dx], where f′(x) is a linear operator and dx is a direction it acts on — "we avoid dividing by dx, because soon we will allow x (and hence dx) to be something other than a number — if dx is a vector, we won't be able to divide by it!" That reframing is what the whole course rests on, and d/dA A² ≠ 2A is its one-line advertisement. Banach spaces then appear in Chapter 5 as machinery to define the discarded o(δx), not as a subject — an inversion of the usual order, where the analysis comes first and the applications after.

It settles the spoke's oldest running tension in a single remark. Five texts here assume ε-δ and never name the alternative; keisler-elementary-calculus is the one that builds on infinitesimals and argues for them with an experiment. These notes use differential notation on every page and then say, in Remark 6: "Here, δx is not an infinitesimal but rather a small number." That is the first time a text in this corpus reaches for d and explicitly disclaims the infinitesimal reading rather than leaving the default unspoken. It does not adjudicate the argument. It does show the default being chosen, which is what the rigour-spectrum section has been unable to observe until now.

A subject that disowns one of its own chapters. "Modern automatic differentiation is more computer science than calculus (it is neither symbolic formulas nor finite differences)" — and Chapter 8 is built to demonstrate it, with dual numbers, computational graphs, and naive symbolic differentiation shown failing. Nothing else here argues that part of itself belongs to a different discipline. It is also where this spoke touches ../machine-learning-wiki: backpropagation is the same object from the training side there, and the mechanism is paged here. The arXiv classification says the same thing from outside — primary math.HO, cross-listed to cs.LG, math.NA and stat.ML.

Two smaller notes for the standing axes. The computational tool assumed is Julia, with running numerical examples and an install guide — but Chapter 14 gives the JAX equivalents (jax.custom_jvp / jax.custom_vjp against Julia's ChainRules frule/rrule), so the tool assumption is stated and portable rather than silent. And the notes name their own omissions in a closing chapter — custom Jacobian–vector products, complex/Wirtinger calculus, the wider matrix-factorization literature — which until now only carroll-geometric-function-theory's §1.4 "What Is Not in This Book" did.

The application picks the mathematics (Essential Math for AI, 2026-09-09)

[[nelson-essential-math-for-ai|Essential Math for AI]] (Hala Nelson, O'Reilly 2023) is the first text here organized by an application rather than by a mathematical subject, and it sharpens two of the spoke's standing threads without a proof in it.

It confirms the boundary from the direction that looks most like a violation. Every founding refusal turned on the same line: the maths is the subject here, the application is the subject there. Nelson's book is titled for AI, motivated entirely by AI, and pitched "rather than dense academic theory" — the profile of a book that belongs in ../machine-learning-wiki. It routes here anyway, because it teaches the mathematics (regression through differential equations) and makes no claim about a model. It is the thibos-fourier-analysis case at full breadth: the field the examples come from is not the subject the book teaches. The runner-up was logged, and the spoke's MacKay note applies — a maths text with an AI spine lands here, it does not pull ML's holdings over.

It is a new point on the organization axis, not just the rigour one. Every other source is filed under a mathematical structure and taught in the order the mathematics suggests — complex analysis, graph-theory, stochastic-processes. Nelson inverts that: the chapters are "the math for convolution," "the math for RL," and the sequencing is set by what a system needs, not by what a proof requires. edelman-johnson-matrix-calculus made this move inside one subject and argued the canon's edge should move to follow ML's tools; Nelson makes it across the whole of undergraduate applied maths at once. Two sources now say the shape of the mathematics is being pulled on by AI — one at survey breadth, one at the level of a single re-derived operator.

And it extends the rigour floor rather than resting on it. "Rather than dense academic theory" puts it beside illowsky-introductory-statistics (whole syllabus, no proof) and below corral-vector-calculus (author-rated 5/10) — but wider than either, touching a dozen subjects a proof-complete text would give a volume each. That width is the standing caveat: body not read, T2, and a survey this broad must be thin somewhere the description does not admit. What a later reading has to settle is whether it derives or merely states.

The counter-example arrives the next day (Gallier & Quaintance, 2026-09-10)

gallier-quaintance-math-deep is 2,204 pages by jean-gallier and jocelyn-quaintance at Penn, titled for Computer Science and Machine Learning and ending in support vector machines, ridge regression and Lasso. Routed one day after nelson-essential-math-for-ai, it is the same destination reached by the opposite road, and that makes the pair the cleanest test the spoke has of the organization axis.

Same audience, contradictory pedagogy. Nelson organizes by application — chapters named for convolution, for reinforcement learning — and reaches the mathematics a system needs, thinly, in ~1,000 pages. Gallier and Quaintance organize by subject and prove everything: groups, rings and fields first, then 740 pages of linear algebra, then affine and projective geometry, bilinear forms, PIDs and tensor algebras, topology, differential calculus, and only then extrema, convexity, linear programming and duality. Kernels appear on p. 1913, once Hilbert spaces and Lagrangian duality are already theorems. Both books answer "what mathematics does machine learning need"; one answers with a syllabus of applications, the other with a nineteenth-century algebra course that eventually gets there. The corpus can now state the disagreement but cannot adjudicate it — neither text reports what its readers learned, which is the same evidential hole herzog-understanding-statistics names elsewhere.

It breaks the one-book-one-subject pattern the spoke was built on. Every founding routing decision assumed a source teaches one body of theory; the corpus map in mathematics is a table of exactly that. This single file supplies linear-algebra, abstract-algebra (with the corpus's first modules and tensor algebra), topology, graph-theory's spectral side, differential calculus, and the whole of convex and nonlinear optimization. The dependency the spoke keeps recording as a gap between books — "linear algebra sits under every text here and is the subject of none" — is here a chapter boundary inside one, and nothing about the routing rule had to change to hold it.

Two subjects arrive that the domain header does not name. topology is new, and optimization theory is halfway in the door: Parts VI–VIII are convex sets, linear programming, the simplex algorithm, LP duality, subgradients, ADMM. ../optimization-algorithms-wiki owns optimization algorithms and already holds convex-optimization and linear-programming pages; the boundary that decided this route is the spoke's own — a body of theory taught from first principles routes here — and the dominant substance of a 2,204-page proof-complete text is the theory, with ~700 pages of it optimization. That runner-up is logged, and the seam is a real one to watch: a second optimization-heavy text would be worth re-arguing rather than assuming this precedent.

A fourth position on access. Free to download, forbidden to mirror. The corpus's other free texts are copyleft (hefferon-linear-algebra, judson-abstract-algebra), CC0 (grinberg-graph-theory) or open-access with a non-commercial clause (herzog-understanding-statistics); this one grants educational copying at cost and reserves redistribution entirely, so the author's page is the only address the book has. Free as in price, reserved as in rights — and, being a manuscript in progress with no edition, a moving target under a fixed URL. The landing page still advertises "2192 pages (2023)" over a file whose own cover says April 2025 and which counts 2,204; the page is signed 2017. What the corpus cites here is dated by the file, not by its advertisement.

The multivariable course gets its coordinate-free ending (Bowen & Wang, 2026-10-03)

bowen-wang-vectors-tensors-vol-2 is the geometry half of a two-volume text by an engineer (ray-bowen) and a mathematician (c-c-wang), and it answers a question the spoke had not asked: what does the multivariable course look like when it is pushed past R³ and past Cartesian coordinates? The answer is tensor-analysis on a euclidean-manifold: Christoffel symbols for the changing basis, differential-forms for the curl, one Stokes theorem for the three classical ones.

It extends the calculus edge in a third direction. The spoke already had two: down, to the foundations (keisler-elementary-calculus, real-analysis), and across, to matrix and function inputs (matrix-calculus). This is the first source that goes on — the authors write that Chapter 9 is "the first where the algebraic concepts developed thus far are combined with ideas from analysis" — and it does so by dropping the 3-D bias vector-calculus (Corral) is organised around. It shares matrix-calculus's move of treating the derivative as a coordinate-free object, and differs in what it differentiates: fields on curved coordinates, not matrix arguments.

The scope is a stated choice. The preface declines general differentiable manifolds and keeps to Euclidean ones plus hypersurfaces in them. That echoes the "stated avoidance" pattern the spoke has logged elsewhere (measure theory dodged in four sources), but here it is the authors' announced design, not an unspoken gap. topology is the neighbour that would supply the general case, and the spoke still holds only gallier-quaintance-math-deep Part V for it.

A caveat on the reading. Only the prose and stated theorems were read; the extracted equations are unreliable. Nothing in this section rests on a formula.

Half a text. The first volume (linear and multilinear algebra) is the prerequisite the authors cite constantly and is not held. A linear-algebra account that includes tensor and exterior algebra is now the obvious companion; gallier-quaintance-math-deep has one in Part IV, at a different altitude.

Arithmetic before rings (Baker, 2026-10-04)

baker-algebra-number-theory gives the algebra-and-number-theory growth edge a third course design, and the first where the order is reversed. lal-algebra-1 and judson-abstract-algebra both reach arithmetic through rings. Baker teaches number-theory first, by hand, with the Euclidean algorithm and congruences, and then uses group-theory to prove what chapter 1 only stated: the units modulo a prime form a cyclic group. The three texts agree on the mathematics (Fermat, Euclid's lemma, Lagrange, primitive roots) and differ on which of the two theories is the foundation. Neither ordering is flagged as wrong.

What it adds that the others lack: continued fractions with the full solution of Pell's equation, Möbius inversion, Burnside's counting formula, and a chapter on cardinality-and-countability. What it leaves out: rings, Sylow, quadratic reciprocity, and any application. It never mentions cryptography, so the "number theory is cryptography's toolkit" link in abstract-algebra stays resting on Judson alone.

Gap-relevance: no ranked edge closes. Edge 7 (algebra and number theory) was already closed; this supplies number theory as a standalone subject, which that edge's closure had folded into algebra. The author is a topologist teaching the course, so the quality of the number theory does not rest on a specialist's authority; the proofs are the evidence.

Open questions

1. Is this spoke mathematics or quantitative methods? Flagged in the parked record before the split and called mathematics on two-of-three. keisler-elementary-calculus makes it three of four, and does so emphatically — an argument about the foundations of a subject is not a quantitative method by any reading. Leaning settled; leaving it open one more source. 2026-08-06 pulls both ways in one day. stirzaker-elementary-probability is unambiguously mathematics — CUP, an Oxford mathematician, theorems and exercises. mari-measurement-across-sciences, routed hours earlier, is unambiguously not. The name question is now less about drift than about whether the spoke tolerates one deliberate outlier (see question 7).

2. Did the ε-δ definition win on merit? The sharpest question the corpus has raised and the one it cannot answer. Keisler's case is that limits took the curriculum in the 1870s because infinitesimals had no rigorous account, that Robinson removed that reason in 1960, and that the curriculum never revisited the decision. Corral, writing in 2008, simply assumes the limit-based sequence as prerequisite and does not mention the alternative — which is either evidence for Keisler's thesis about inertia, or evidence that the profession considered and declined it. The corpus holds no source that argues the standard side deliberately.

3. Where does information and coding theory live? ../machine-learning-wiki holds information-theory-inference-learning-algorithms, routed there hours before this spoke existed, half of which is compression and coding theory that spoke does not claim. The routing stands, but the subject belongs here. The next such source should land here rather than pulling the first one over. Answered halfway on 2026-08-15, and by neither branch the question anticipated. The curator asked for the coding half to be built out here, so the subject now lives here in eight pages (information-theory, coding-theory, entropy, source-coding-theorem, noisy-channel-coding-theorem, error-correcting-code, low-density-parity-check-code, fountain-code) while the book stays where it was routed. No second source arrived and none was needed. What the question did not foresee is the arrangement that resulted: a subject held here, cited across a spoke boundary, with the raw document in another spoke's raw/. See the section above for why that is worth watching rather than celebrating.

4. Does a linear-algebra text arrive, and from where? Answered 2026-08-04, and from neither of the two places the question guessed. hefferon-linear-algebra arrived via the research pass rather than a routed source, from a state university rather than Strang/MIT OCW, and it brought trench-real-analysis with it — so the analysis half is now filled from the standard side as well as Keisler's hyperreal one. See linear-algebra and real-analysis.

5. The research-methods fork just got live — and the deciding source came down on the fork's side, without settling it. Flagged in CLAUDE.md before the split and dormant since: if the corpus fills with experimental design, hypothesis testing and study statistics, that is arguably a different and better spoke. larsen-marx-mathematical-statistics brought ANOVA, randomized block designs, contrasts and multiple comparisons — that is experimental design — but it was read as strengthening the mathematics side, because it derives its distributions, proves its theorems, and teaches design as a consequence of the F test rather than as a protocol. A research-methods text would do the reverse. herzog-understanding-statistics (2026-08-06) does the reverse. It carries no derivations at all, teaches at the level a working experimentalist needs, and devotes a third of its length to meta-analysis, replication and the Test for Excess Success — statistics about research practice, which is the fork's own description. By the test this question set, that is one vote each way. It was routed here anyway and the reasoning is worth recording rather than hiding: the spoke's boundary rule files a text that teaches a body of theory from first principles here even when its examples come from one field, and this book does teach the subject from the ground up. But it is the first source whose centre of gravity sits outside the mathematics reading, and a second one like it should trigger the fork rather than another routing note. The condition to watch: a text that assumes the machinery and argues about how to use it, rather than building it. Re-read 2026-08-17 when the curator re-sent the book as a direct PDF, and the fork tilts a little further. Chapter 11 turns out to carry an audit, not just worked examples: of 133 psychology/education articles in Science 2005–2012, the 18 with enough detail were TES-tested and 15 (83%) look too good to be true. It also supplies two diagnostics usable without any computation — effect size should not correlate with sample size (valid set r = 0.25; problematic sets r = −0.86, −0.83; the precognition studies r = −0.89), and valid experiments produce very small p-values rather than a shelf of them just under 0.05. That is a research-methods instrument, not a theorem, and it is the part of the book most likely to be used by anyone. The routing still stands on the first-principles rule; the centre of gravity has moved again, in the same direction. A third source, 2026-09-27, and it moves the same direction from a different genre. ramos-controlling-variables is not a textbook or a monograph at all — a short news explainer, T2, no derivations, no citation trail of its own. Judged purely on the first-principles rule it is the weakest of the three fork candidates. But its content is the fork's clearest case yet: confounding, the difference between experimental and statistical control, stratification, and the collider failure mode are pure study-design vocabulary, with no probability or inference theory attached at all — Ramos never derives a distribution or states a test. It was still routed here, on the same rule as the other two: it teaches the general distinction (not a claim about GLP-1 drugs specifically), and the drug-trial material is the illustration, not the subject. What it adds to the fork question: the three candidates now span genre as well as centre of gravity — a graduate statistics text, an open-access research-methods monograph, and a piece of public science journalism — and all three were kept by the identical rule. That is either a sign the rule is doing real work regardless of genre, or a sign the rule is elastic enough to keep anything this spoke is handed. A fourth research-methods-flavoured source should be read for which.

6. Does the origin-predicts-quality reading survive five texts? The corpus recorded early that where a text came from predicts its tier and almost nothing about its usefulness (schoolcraft-college). Five texts in, the pattern is holding but the reason has shifted: the two new T1s earn the grade from a professional-society award and a prior commercial edition respectively — neither of which is about the institution. Worth one more source before it is stated as a finding.

7. Should a book that is not mathematics be in the mathematics spoke? Live from 2026-08-06, when mari-measurement-across-sciences was routed in from Telegram. It is measurement science and philosophy of measurement — metrology, the ontology of properties, scale types, psychometric validity — with no theorem in it that the spoke would otherwise hold. Three arguments carried the routing: no spoke in the hub owns measurement as a subject; at least three borrow it (statistics here, psychometrics in ../psychology-wiki, capability benchmarking in ../agentic-tooling-wiki and ../ai-governance-wiki); and the spoke's own boundary rule files a text that builds a body of theory from first principles here even when its examples come from one field. The argument against is simply that the rule says mathematics as the subject, and this book's subject is not mathematics. ../psychology-wiki was the runner-up on the validity and psychometrics half; ../philosophy-wiki was considered for chapter 4 and declined, since that spoke owns philosophers and their systems rather than philosophy of science. This bears on question 1 more than on question 5. The fork question 5 watches is mathematics vs research methods, and this book is neither — it sits under both. What it does supply is the strongest evidence yet that the spoke's real subject is closer to quantitative methods than to mathematics proper, which is exactly what question 1 asked and had provisionally answered the other way. Two of the last three arrivals now sit outside a strict reading of the name. The condition to watch: a second measurement-science or philosophy-of-measurement source. Two would make a cluster the hub could spin out rather than a stretch this spoke absorbs.

Paid first, free second

lal-algebra-1 closes the last untouched subject on the growth-edge list and breaks the corpus's licence pattern doing it: Springer, $49.99 for the eBook, $89.99 in hardcover, no open licence.

That has now happened twice in the same shape. Statistics arrived as larsen-marx-mathematical-statistics (Pearson, paid), and an open text — herzog-understanding-statistics, CC BY-NC — followed two days later to close the licence half separately. Algebra has arrived paid, and the open half is filed as its own edge rather than pretended away.

The observation worth keeping is about this wiki's own collection policy: a corpus that only accepted open licences would have neither statistics nor algebra today. The free texts here are excellent and several are better than their commercial peers (hefferon-linear-algebra took a professional society's award), but they arrive when someone chooses to write one, not when a subject needs covering. Access and quality are independent axes, and so are access and coverage. springer makes the same point from the publisher side: the same house issued the corpus's paywalled algebra text and its open-access statistics text.

Updated hours later: the free text was pulled rather than waited for. At the curator's instruction to go deeper, judson-abstract-algebra (GFDL, maintained since 1997) was brought in — closing edge 8 the same day it opened, and giving the subject a text this wiki has actually read rather than one it knows from a publisher's page. So the pattern above holds in shape and collapses in time: paid first, free second, four hours apart, because the second one was fetched on purpose.

That changes what the corpus can claim. abstract-algebra now carries read definitions — the group axioms, and the ring → commutative ring → integral domain → division ring → field hierarchy with the examples that separate them (ℤ is a domain and not a field; ℤ₁₂ has zero divisors; the quaternions are a noncommutative division ring). group-theory is paged as the subject's deepest corner, which it is in both books: Lagrange and its number-theoretic corollaries, the isomorphism theorems, Sylow, the classification of finite abelian groups.

And the subject arrived with a disagreement already in it, which no other subject here has on day one. The two texts differ about what a first course is for: Lal opens with two chapters of logic and set theory (Zermelo–Fraenkel with choice, motivated by Russell's and Cantor's paradoxes) and spends his later chapters on classification — Schreier, Jordan–Hölder, Remak–Krull–Schmidt, solvable and nilpotent groups. Judson opens with a short note on proofs, spends two full chapters on cryptography and coding theory a third of the way in, and reaches Galois theory in the same volume. Judson states his reason: applications exist to motivate students meeting rigorous proof for the first time. That is this corpus's recurring move — a pedagogical argument, offered without evidence, in a preface — arriving now about content selection rather than about rigour or foundations.

The asymmetry to keep in view: Judson's body was read, Lal's was not. Where the two are described as differing, that is a comparison of stated scope, not of exposition.

A third condition, and it is neither paid nor licensed (2026-08-30). harju-graph-theory states no terms whatsoever — no copyright line, no licence, nothing about reuse. The axis this section describes runs from "$89.99 hardcover" to "CC0 public-domain dedication," and both ends are stated positions. Lecture notes posted to a university web server are a third thing: freely downloadable, legally undefined, and probably the largest category of free mathematics by volume. Worth noting what it costs this wiki specifically — a text whose terms are unstated cannot be redistributed here with confidence, only cited and linked, which is a practical difference from the CC0 text sitting beside it in raw/.

The foundation finally has a defender, and he is arguing about meaning (2026-09-14)

Growth edge 3 asked for a text that chose ε-δ over infinitesimals in print, after five assumers and one dissenter (keisler-elementary-calculus). bishop-review-keisler is that text, and it changes what the edge was about. Bishop does not defend the standard definition as more rigorous — he grants the technical side is "of minor importance" — he defends it as the one a student can recognise as a statement about approximation, where Keisler's axioms V* and VI* deliver a hyperreal line by decree. So the corpus's ε-δ-vs-infinitesimal question is not rigour against intuition, as the assumers' silence made it look; it is two accounts of where the meaning of a limit lives. Davis's reply — that Bishop never states Keisler's aims or the constructivism his objection rests on — and Sullivan's 1976 classroom result in Keisler's favour are recorded on the page and keep it from settling anything: the defence exists, it is philosophical, and it lost the exchange on the evidence its opponents brought. The textbooks still owe an argument.

Tensions

The foundational fork (opened 2026-08-02, on the fourth ingest)

The corpus's first genuine disagreement, and it needs stating precisely because it is easy to overstate.

• corral-vector-calculus names the standard limit-based Calculus I and II as its prerequisite and builds multivariable calculus on top of them.
• keisler-elementary-calculus rebuilds exactly that prerequisite on the hyperreals, gets derivative, continuity and integral by chapter 4, holds limits back to chapter 5, and then covers Corral's own material again in chapters 10–13 — gradients, line integrals, Green's, Stokes, Gauss.

This is not a contradiction about mathematics. The theorems match, the notation matches, the methods for solving problems match; Keisler says so himself and keeps the traditional limit definition in the book rather than replacing it. The disagreement is about what a student should meet first, and behind that, about whether the ε-δ definition earned its place on merit or won by default during the century when infinitesimals had no rigorous account.

Recorded as a live tension rather than resolved, because the corpus cannot resolve it: one side is an assumption Corral never argues for, and the other is an argument Keisler makes at length with one second-hand study behind it. What the hub can say is that the two agree on every result they both cover, which is the strongest corroboration this spoke has of anything.

Second assumer, 2026-08-04. trench-real-analysis is the rigorous course where the ε-δ choice is made explicit for the first time in a curriculum — chapter 1 builds the reals from completeness and the whole development follows — and it does not mention infinitesimals either. That is the place an argument would live if anyone were making one, so the silence is more informative here than Corral's was. It still is not evidence against Keisler; it is a second data point for the inertia reading and no data point for the merit reading. The corpus now covers multivariable calculus three times over (corral-vector-calculus, Keisler ch. 10–13, Trench ch. 5–7) and the three agree on the results while two of them decline to discuss the foundation. Restated as growth edge 3.

Third assumer, 2026-08-13, and the best-placed one. baggett-analysis-single-variable does not contain the word infinitesimal — not in the text, and not in the appendix where he constructs ℝ from Dedekind cuts and proves a complete ordered field unique up to isomorphism. Trench's silence was informative because analysis is where the foundational choice is first made explicit. Baggett's is sharper: he is the only author here who works below the completeness axiom, writing in 2006, and he does not mention the alternative that had been rigorous since 1960. Three silences still do not add up to an argument for ε–δ, and one of them now comes from the book with the most room to have made it.

Provenance versus usefulness

The other standing tension is between the corpus's own provenance and its usefulness. The most authoritative source (mcmullen-probability-theory, T1, Harvard, primary) is also the least self-sufficient — terse notes that assume a 1950 textbook the hub does not hold. The least authoritative (corral-vector-calculus, T2, community-college adjunct, self-published) is the most complete and the only one under a free licence. Provenance tier and reader value point in opposite directions here, which is worth remembering the next time a source is graded.

Weakened 2026-08-04, deliberately not withdrawn. hefferon-linear-algebra is T1 and also the most self-sufficient text in the corpus — full proofs, one calculus course assumed, solutions manual, video lectures. It is the first source here where tier and reader value point the same way, which shows the tension was never a law, only the shape of the founding four. The reasoning behind each grade is now written on the page rather than left to the letter (trench-real-analysis earns T1 from a prior Pearson edition, Hefferon from an MAA award), because a bare tier letter carried none of what the sentence above is actually about. Corral's case stands unchanged.

Growth edges

Ranked; each names the kind of source that would close it (see ../QUALITY.md → Growth edges).

1. Linear algebra — closed 2026-08-04 by hefferon-linear-algebra (T1, research pass).

2. Analysis — closed 2026-08-04 by trench-real-analysis (T1, research pass).

3. A text that argues for the ε-δ foundation — now the spoke's sharpest hole, and promoted here because the two closures above made it worse rather than better. The corpus holds four texts that assume the standard development and one that argues against it; nothing defends it. 2026-08-13: one more assumer, and the worst case yet for the merit reading — baggett-analysis-single-variable constructs the reals from Dedekind cuts in an appendix and still never names the alternative. 2026-08-18: and now a worse one. klazar-elementary-real-analysis re-derives all five numeric domains from ZFC, proves induction instead of assuming it, adds a characterization theorem for the extended reals its author calls "completely new", writes an appendix section headed "How do we know that a theorem is true?", and opens by listing sixteen rival texts he intends to improve on. Searched in full: no Keisler, no Robinson, no hyperreals, no "non-standard analysis"; the word infinitesimal appears three times, twice as the thing an Archimedean field excludes by definition. An author auditing the foundations from the bottom, across fourteen public versions, who does not mention the alternative foundation exists. The edge is no longer plausibly explained by inattention. — needs: a T1 text or paper that considered non-standard analysis and declined it in print (a foundations-of-mathematics-education article, or a preface that engages Keisler directly). CLOSED 2026-09-14 (research pass) by bishop-review-keisler — Errett Bishop's review of Keisler in the Bulletin of the AMS (83(2), 1977): the ε-δ definition defended as "common sense … central to the important practical problems of approximation and estimation," against a text that "can be used to confirm their experience of mathematics as an esoteric and meaningless exercise in technique." It is a constructivist's defence, and Davis and Keisler said so in the same year, which the page records. Read in quotation only — the review is paywalled. Successor (3a): the defence is now one philosopher's four pages from 1977; the textbooks that assume ε-δ still do not argue for it — needs: a modern text or education paper that engages Keisler or Sullivan (1976) and keeps the standard route on stated grounds.

4. Statistics and inference — closed 2026-08-04 by larsen-marx-mathematical-statistics (T1, Pearson 5th ed.), which brings estimation, testing, regression, ANOVA, blocked designs and nonparametrics in one text. See mathematical-statistics.

5. A freely-licensed statistics text. CLOSED 2026-08-06 by herzog-understanding-statistics (Herzog, Francis & Clarke, Springer 2019, open access), routed in from Telegram. It restores the corpus's free-by-licence pattern that Larsen & Marx broke — with one qualification worth keeping: the licence is CC BY-NC, so it is free to read and share but carries a non-commercial restriction that hefferon-linear-algebra and trench-real-analysis do not. The edge said "open-licensed" without saying how open, and this is the weaker kind.

6. Information and coding theory as a subject. Subject opened here 2026-08-15 at the curator's instruction — eight pages built from MacKay's ITILA, which still sits in ../machine-learning-wiki. The edge's own requirement is not met: it asked for a source landing here, and no source did. This spoke teaches the subject and holds no text for it, which makes the residue sharper than the original edge rather than softer. — needs: a T1 information-theory text held in this spoke — Cover & Thomas is the obvious candidate (MacKay cites it for the proofs he skips), or an open-licensed alternative. A coding-theory text proper would also close the half MacKay treats at survey depth: BCH, Reed–Solomon and Goppa codes get named here and derived nowhere.

7. Abstract algebra and number theory — subject closed 2026-08-08 by lal-algebra-1 (Ramji Lal, Springer 2017, T1): groups, rings, fields and arithmetic in one volume, with number theory woven through it rather than separated out (abstract-algebra). The edge's own qualifier is not met — it asked for an open textbook and this one is commercial ($49.99 / $89.99). So the subject is covered and the licence hole is new: an open-licensed algebra text is still wanted, and that is the residue of this edge rather than the whole of it.

8. An open-licensed algebra text — added and closed the same day (2026-08-08) by judson-abstract-algebra (GFDL, T1), pulled deliberately once lal-algebra-1 closed the subject and left the licence half open. It also carries Galois theory to the fundamental theorem, which Lal defers to a later volume. Still open within algebra: modules and representation theory, in neither book. — needs: a graduate-level text or a survey.

9. Measure theory, now load-bearing. Promoted from a standing absence to a ranked edge on 2026-08-16, when stochastic-processes opened on three sources that each dodge it a different way (indented as optional, taught in passing from metric spaces, or ruled out of the prerequisites). Filtrations, σ-algebras and the probability space are assumed by every page in that subject and built by none. — needs: a T1 measure-theory or measure-theoretic-probability text, open-licensed if possible. 2026-09-05: a fourth avoidance, and the only one stated as a decision. dynkin-yushkevich-markov-processes writes it into the foreword — "We have consciously avoided references in the main text to measure theory and measurability. The reader who has these concepts at his command will have no difficulty in grasping the presentation at a more rigorous set-theoretic level." Four sources, four ways around it, and this one is an announced editorial choice rather than an omission or a prerequisite line. The edge is not closer; it is better evidenced, which is a real result about how the subject is written: nobody in this spoke's stochastic cluster builds the probability space, and at least one of them decided not to on purpose.

Coverage edges (added 2026-08-08, at the curator's request for a wider backlog). These widen what the spoke covers instead of answering an open question above; one ordinary solid source closes any.

9. Differential equations. The applied subject with the widest reach across every other spoke, and the shelf here goes from calculus straight to analysis without it. — needs: an open-licensed ODE text (Trench wrote one, matching trench-real-analysis).
10. Discrete mathematics and combinatorics. The half of the subject computing readers need most — counting, recurrences, graphs — is unheld. Graphs closed 2026-08-17 by grinberg-graph-theory (T1, CC0, routed in from Telegram), which over-closes that third for a first course: 454 pages, a graduate quarter, and the matrix-tree-theorem taken further than most introductions go. Counting and recurrences are untouched, and the source makes the residue easy to name — Grinberg repeatedly forwards the reader to enumerative combinatorics for proofs he does not give (Stanley, Galvin, Aigner & Ziegler on Cayley's formula alone). — needs: an enumerative-combinatorics text — generating functions, recurrences, inclusion-exclusion. Within graphs, the standing holes are planarity (excluded by this book's prerequisites, see the 2026-08-17 section above) and random / extremal / spectral graph theory.
11. Logic and set theory. abstract-algebra and real-analysis both sit on foundations no page states, and the proof assistants in ../research-wiki assume the same material. — needs: an open-licensed logic or set-theory text.
12. Decision theory and utility theory. Opened 2026-08-10 by osborne-rubinstein-course-in-game-theory, which assumes von Neumann–Morgenstern expected utility and Savage's subjective probability as prerequisites and explicitly declines to defend them ("we do not discuss the assumptions that underlie the theory of a rational decision-maker"). game-theory therefore sits on foundations no page here states. — needs: a T1 text on decision theory under uncertainty, or von Neumann & Morgenstern itself.
13. Mechanism design, and experimental / behavioral game theory. The two halves the source names as its own gaps — the authors say they would add chapters on experimental game theory and on learning and evolution if starting over, and implementation theory (ch. 10) is only the pure end of mechanism design. — needs: a modern game-theory text, or an auction/mechanism-design text.
14. The first course in complex analysis. Opened 2026-08-10 by carroll-geometric-function-theory and mostly closed 2026-08-13 by baggett-analysis-single-variable, whose Chapter VII carries Cauchy, Liouville, the Fundamental Theorem of Algebra, Maximum Modulus, Open Mapping, Laurent and the Residue Theorem. The residue of the edge: the Argument Principle and Rouché's Theorem (the last two items on Carroll's own list), and conformal mapping, which no source here holds and which is where the second course starts. — needs: a standard first-course text after all (Howie, in the same series, is the one Carroll names as sufficient; Stein & Shakarchi the one he recommends alongside).
15. Curvature and Riemann surfaces. Named by carroll-geometric-function-theory §1.4 as deliberate omissions, with the forwarding addresses given: Ahlfors's Conformal Invariants and Krantz's Complex Analysis: The Geometric Viewpoint. — needs: either of those.
16. Numerical methods. Floating point, conditioning and quadrature are where the mathematics in this corpus meets the machines in the rest of it. — needs: a standard text, or Goldberg's floating-point paper for the arithmetic half.

Index — Mathematics Wiki

Catalog of every page, grouped by schema.org @type. Spine: synthesis (thesis), log.md (history), this file (catalog). Spun out of the hub _inbox on 2026-08-02 (3 sources, three tags); a fourth landed the same day. Most subjects here rest on a single source. Five are doubled: vector-calculus (three accounts, disagreeing about foundations), probability-theory and mathematical-statistics (two each, agreeing about the mathematics and differing in altitude), and — since 2026-08-13 — real-analysis and complex-analysis, doubled by one book that argues they were never two subjects. Record the level a text is pitched at alongside its tier — the texts sit at visibly different rigour. One subject here holds no source of its own: information and coding theory (8 pages, added 2026-08-15) is built from information-theory-inference-learning-algorithms, which lives in ../machine-learning-wiki and is cited across the spoke boundary. Nothing in the Book section below covers it.

DefinedTerm (concepts)

• mathematics — the root page: why a mathematics spoke exists at all, the four texts, and the boundary rule (subject here, application there) · root
• probability-theory — sample spaces, counting, distributions, the law of large numbers; specifying the space is the whole difficulty · field
• stochastic-processes — probability with time in it: what a sequence of random variables does. Three sources, three incompatible entrances (finite Markov chains / discrete martingales / large deviations) and one agreed destination, Itô calculus · field
• markov-chain — the memoryless process: transition matrices, stationary distributions, mixing and hitting times. A finite chain is a matrix, which is where linear-algebra becomes probability's working tool — and no applied spoke in the hub uses it by name yet · concept
• martingale — the fair game, formalised: optional stopping, convergence, and the change of measure that turns a discounted price into one. Built first and in discrete time by two of the three sources · concept
• brownian-motion — the random walk's continuous limit, constructed rather than assumed; continuous martingale, Markov, Gaussian and self-similar at once. Quadratic variation = t is the fact that breaks ordinary calculus · concept
• stochastic-calculus — calculus along a random path: the integral defined first because the derivative cannot be, Itô's formula and its surviving second-order term, and the SDE. Ends in the Black-Scholes derivation, where the stock's drift disappears · concept
• matrix-calculus — calculus where the inputs and outputs are matrices, operators, functions or random variables: stop dividing by dx and define the derivative as a linear operator, df = f′(x)[dx]. Kronecker products, adjoint/reverse-mode differentiation (backpropagation), forward and reverse AD, second derivatives as bilinear maps, derivatives of determinants, inverses, eigenproblems, ODE solutions, functionals and random functions. d/dA A² ≠ 2A is the advertisement · field
• tensor-analysis — calculus for vector- and tensor-valued fields in curved coordinates: chart transformation rules, holonomic vs anholonomic vs physical components, Christoffel symbols and the covariant derivative, Lie derivatives. One source, the geometry volume of a two-volume text · field
• euclidean-manifold — N-dimensional space as a set of points with an inner-product translation space (an affine space without the inner product); charts as diffeomorphisms to Rᴺ, and hypersurfaces inside it. The book declines general manifolds on purpose · concept
• differential-forms — skew covariant fields and the exterior derivative d: the Poincaré lemma (closed is exact on a star-shaped domain), the two forms of Frobenius' theorem, and the generalized Stokes theorem with the classical ones as cases · concept
• topology — continuity without a metric: open sets, compactness, connectedness, taught here as the run-up to defining the derivative as a linear map. Not one of the spoke's five named subject families — it arrived because a text that develops it from first principles did (2026-09-10), and rests on that one source · field
• vector-calculus — several-variable and vector-field calculus, ending at Green's/Stokes'/Divergence; exports the gradient to half the hub · field
• fourier-analysis — time and frequency as two descriptions of one signal; orthogonality, sampling, linear systems · field
• infinitesimal-calculus — calculus built on hyperreals instead of limits; Robinson's 1960 result, standard parts, and the claim that ε-δ won by default · foundation
• linear-algebra — vector spaces and the maps between them; the subject every other one here consumes and none supplied · field
• real-analysis — calculus proved rather than computed, from completeness to metric spaces; where the ε-δ choice becomes explicit, and where two texts now disagree about what the next course is (several variables, or the complex plane) · field
• mathematical-statistics — probability pointed backwards: estimation (MLE, Cramér–Rao, Bayesian) and hypothesis testing (the generalized likelihood ratio); the first subject here that sits downstream of three others · field
• universal-algebra — general algebra: structures as such, one rung above abstract-algebra — every set carrying operations of a given signature subject to given identities, so a theorem proved about varieties yields the group, ring and lattice cases at once. Signature/arity, terms, free algebras, derived operations, clones, and Lawvere's Structure and Semantics. The one subject here whose own name is disputed by its practitioners ('Universal Algebra' vs 'General Algebra') · field
• category-theory — objects and the maps between them, where the maps carry the content: functors, natural transformations, Yoneda, adjoint functors, limits and colimits, ending at Freyd's adjoint-functor criterion. Held only as ~180pp inside a general-algebra book, so it is the apparatus an algebraist needs rather than a survey — no dedicated text, and monoidal categories, topoi and the homotopy side are absent · field
• abstract-algebra — sets carrying operations that satisfy axioms, with the ring → domain → division ring → field hierarchy and the examples that separate them (ℤ is a domain, not a field; ℤ₁₂ has zero divisors); the structure under linear-algebra rather than beside it. The corpus's first subject to arrive with two texts that disagree about what a first course is for · field
• group-theory — the subject's deepest corner and the bulk of both algebra texts: Lagrange (and Fermat/Euler as corollaries), the isomorphism theorems, Sylow, finite abelian classification, group actions; symmetry and permutation/matrix groups as the two sources · field
• number-theory — arithmetic of the integers taught before any ring: Euclid, congruences, Fermat and Wilson, continued fractions, Pell's equation, Möbius inversion. The third ordering of arithmetic and algebra the spoke holds · field
• cardinality-and-countability — bijection as the definition of size: pigeonhole, power sets, Cantor's diagonal argument for ℝ · concept
• game-theory — what happens when decision-makers whose outcomes depend on each other choose: strategic vs extensive games, perfect vs imperfect information, and the solution concepts (Nash, sequential, the core, the Shapley value) that are most of the subject. Arrives with its own authors denying it is mathematics · field
• complex-analysis — calculus of the plane, and where analysis meets geometry through conformal mapping; entered from a second course, with the first course arriving three days later (baggett-analysis-single-variable) — what is still missing is narrow: the Argument Principle, Rouché, and conformal mapping itself · field
• information-theory — the subject: Shannon information content, entropy, and the two limits — how far a source compresses and how fast a noisy channel carries. Dual-lens with ../machine-learning-wiki, which holds the learning-facing page under the same slug; the canonical owner of a bare [[information-theory]] link is this spoke · field
• coding-theory — the constructive half, organised around a fifty-year gap: Shannon proved good codes exist in 1948 by averaging over random codes, and nothing both good and runnable existed until the mid-1990s. Very good / good / bad / practical, and why minimum distance was the wrong obsession · field
• entropy — H(X) = Σ pᵢ log₂(1/pᵢ), why the logarithm is forced rather than chosen, mutual information, and typicality as the law of large numbers applied to log 1/P(x) · concept
• source-coding-theorem — compression's floor: NH bits and no fewer. Kraft, Huffman's one-bit slack, arithmetic coding, and MacKay's verdict on Lempel–Ziv's universality · concept
• noisy-channel-coding-theorem — capacity C = max I(X;Y), the three parts, and a proof that weighs all the babies at once and points at none of them · concept
• error-correcting-code — repetition and Hamming (7,4) as the two baselines, linear codes, why decoding is the wall (NP-complete), concatenation and the Voyager/Galileo state of the art · concept
• low-density-parity-check-code — Gallager 1962, forgotten for 33 years, rediscovered 1995; sparse random parity checks decoded by belief propagation, and the thresholds against the Shannon limit · concept
• fountain-code — rateless erasure coding: any K′ ≈ 1.05K drops recover the file, no feedback and no rate chosen in advance · concept
• graph-theory — finite sets with a relation on their pairs: simple graphs, multigraphs, digraphs and tournaments, then Euler vs Hamilton, trees, colorings, matchings, flows and Menger. The corpus's first subject that needs no calculus and no analysis at all, and the first whose syllabus is visibly cut by that prerequisite (planarity needs continuity, so it is absent) · field
• matrix-tree-theorem — spanning arborescences counted by the determinant of a Laplacian minor; Cayley's n^(n−2) (Borchardt, 1860) and the BEST theorem falling out of it, and de Bruijn sequences out of that. Where linear-algebra stops being a prerequisite and does the work · concept
• measurement-theory — what makes a procedure a measurement, and what its number means: four necessary conditions, model-dependent realism, scale types, and the error / uncertainty / validity split between the physical and human sciences. Not mathematics — the spoke's odd one out, recorded as open question 7 · field
• confounding — a third variable driving both exposure and outcome; the failure sitting in front of every inference machinery this spoke holds. Experimental vs statistical control, stratification as blocking by another name, and the collider case where more controls make an estimate worse · concept

Course (sources)

• mcmullen-probability-theory — Curtis McMullen, Harvard 2011 (rev. 2021), 98pp: terse notes accompanying Feller, with the mailman paradox, Jovian birthdays and Benford's law as its teaching set · source · T1 · people.math.harvard.edu

• thibos-fourier-analysis — Larry Thibos, Indiana University School of Optometry, 6th ed. 2014, 201pp: Fourier analysis as an experimentalist's method, including three chapters on the statistics of measuring a coefficient · source · T2 · PDF only, no URL

• gamarnik-advanced-stochastic-processes — MIT 15.070J, Fall 2013, david-gamarnik, CC BY-NC-SA, 22 lectures: measure-theoretic probability, five lectures of large deviations before Brownian motion appears, martingales and stopping, four on Itô calculus, then financial economics and reflected Brownian motion. Read as a syllabus — the lecture list and description, not the notes · source · T1 · ocw.mit.edu

• wu-introduction-stochastic-processes — MIT 18.445, Spring 2015, Dr. Hao Wu, CC BY-NC-SA, 24 lectures: thirteen of them Markov chains and random walks (mixing times get five), then martingales, Galton-Watson, the Poisson process and continuous-time chains. No Brownian motion, no stochastic integral, no finance; prerequisites are conditional expectation and matrix operations. Read as a syllabus · source · T1 · ocw.mit.edu

• baker-algebra-number-theory — andrew-baker, university-of-glasgow, third-year course notes (© 2009, 73pp, held in raw/): number theory, groups and actions, arithmetic functions, countability; arithmetic taught before rings · source · T1 · maths.gla.ac.uk

Review (sources)

• bishop-review-keisler — Errett Bishop, Bull. AMS 83(2), March 1977, pp. 205–208: the review of Keisler that is the corpus's only defence of ε-δ in print — "common sense … central to … approximation and estimation" against a text that confirms mathematics as "an esoteric and meaningless exercise in technique." Constructivist, and answered by Davis and Keisler the same year; read in quotation (paywalled). Closes edge 3 · source · T1 · projecteuclid.org

Book (sources)

• gallier-quaintance-math-deep — jean-gallier & jocelyn-quaintance (Penn CIS), Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning, book in progress, 2,204pp, cover-dated 14 April 2025 (the landing page still says "approx. 2192 pages (2023)" and is signed 2017). Eight parts: linear algebra through SVD, graph Laplacians and iterative solvers · affine and projective geometry · bilinear forms and Witt's theorem · PIDs, UFDs, tensor and exterior algebras, modules, canonical forms · topology and differential calculus · extrema, Newton's method, quadratic optimization · convex sets, linear programming, the simplex algorithm, LP duality · Hilbert spaces, nonlinear optimization, subgradients, ADMM, kernels, SVMs, ridge/Lasso. The corpus's largest text by four times and the only one that supplies several of its subjects at once. Free to download, redistribution forbidden — the author's page is the book's only address · source · T1 · cis.upenn.edu

• bowen-wang-vectors-tensors-vol-2 — ray-bowen (texas-am-university, mechanical engineering) & c-c-wang (rice-university, mathematical sciences), Introduction to Vectors and Tensors, Vol. 2: Vector and Tensor Analysis, ISBN 0-306-37509-5, 246pp, no year on the title page (web: Plenum 1976, Dover 2009). Five chapters: Euclidean manifolds and covariant differentiation · Lie derivatives, Frobenius, differential forms and Poincaré · hypersurfaces with Gauss-Codazzi · classical continuous groups and Cartan parallelism · integration to the generalized Stokes theorem. Declines general manifolds in the preface; Volume 1 (algebra) not held. Equations garble in extraction; read from prose and theorem statements · source · T2 · pdf (Texas A&M repository)

• dynkin-yushkevich-markov-processes — Evgenii Dynkin & Aleksandr Yushkevich (Moscow University), Markov Processes: Theorems and Problems, Plenum 1969, tr. James S. Wood, foreword dated 22 January 1966; 245pp scanned with no text layer, held in raw/, read as page images. Out of Dynkin's 1962–63 Moscow lectures, reorganised by problem area rather than curriculum: recurrence via harmonic functions, potential and capacity on a random walk (I); the Dirichlet and Poisson problems solved probabilistically (II); optimal stopping, from the secretary problem to the Wiener process (III); boundary conditions for differential equations read off a birth-and-death process (IV). The spoke's fourth stochastic source and the one that runs the other way — probability as a tool for analysis, no Itô calculus, no finance. Names Feller as the pioneer and objects to his "purely analytical form"; quotes Doob (1954) as running the same link in the converse direction. Consciously avoids measure theory, and says so — the fourth avoidance in four sources, and the only one stated as a decision. Problem sets carry results, not exercises (the Martin boundary is in one) · source · T1 · people.math.harvard.edu

• klazar-elementary-real-analysis — martin-klazar, Elementary Real Analysis (arxiv:2508.19405, v14 of 16 July 2026, 400pp, CC BY 4.0): lecture notes for Mathematical Analysis 1 in MFF UK's School of Computer Science, grown into a book and openly unfinished — chapters 11–14 (Newton, Riemann and Henstock–Kurzweil integrals, applications) are title pages only, and the contents mark which sections are "relatively complete". Chapter 1 rebuilds five numeric domains from ZFC with a characterization theorem each (induction is a theorem, not an axiom; the ℝ* result is "completely new"). A combinatorialist's examples throughout, and counterexamples to a theorem of Cauchy's Cours d'analyse. The fifth text here to assume ε-δ and never name the alternative — and the least explicable, given what else it audits · source · T1 · arxiv.org

• keisler-elementary-calculus — H. Jerome Keisler, 1976/1986 (Dover 3rd ed. 2012), 992pp scanned, CC BY-NC-SA: the whole calculus sequence on infinitesimals, limits deferred to ch. 5; the corpus's only pedagogical claim with an experiment behind it · source · T1 · people.math.wisc.edu

• corral-vector-calculus — Michael Corral, 2008, 222pp, GNU FDL: a complete Calculus III course with 420 graded exercises, whose author rates his own rigour 5 out of 10 · source · T2 · mecmath.net

• lawler-stochastic-calculus — Gregory Lawler, Stochastic Calculus: An Introduction with Applications (© 2014, revised Feb 2023, 260pp, free from the author's university-of-chicago page): discrete martingales → Brownian motion, constructed → the stochastic integral and Itô's formula → Girsanov, Black-Scholes, martingale pricing and the martingale representation theorem → Lévy and jump processes, fractional Brownian motion, harmonic functions. Two audiences in one file — the measure-theoretic layer is indented in a different font to be skipped — and it expects the reader to program (C++/Matlab/Mathematica). States its own thesis before any mathematics: "if the assumptions are bad, one cannot expect a good answer" · source · T1 · math.uchicago.edu

• hefferon-linear-algebra — Jim Hefferon, 4th ed. 2020, GFDL/CC BY-SA: proves everything, assumes one semester of calculus, MAA Solow Award 2020 · source · T1 · hefferon.net

• larsen-marx-mathematical-statistics — Larsen & Marx, Prentice Hall/Pearson, 5th ed. 2012 (1st 1981), ~753pp: the two-semester mathematical-statistics sequence — estimation through Cramér–Rao and Bayes, testing through the generalized likelihood ratio, then regression, ANOVA, blocked designs and nonparametrics. A "Taking a Second Look at Statistics" essay per chapter on how the methods mislead. The corpus's first text that is accessible without being freely licensed · source · T1 · © Pearson, read from an Emory course-page PDF

• trench-real-analysis — William F. Trench, free ed. 2013, CC BY-NC-SA: two-term real analysis, reissued after a Pearson edition; the corpus's first rights-reverted text · source · T1 · digitalcommons.trinity.edu

• stirzaker-elementary-probability — David Stirzaker, Elementary Probability (CUP, 2nd ed. 2003, 538pp): a self-contained undergraduate course in three parts — basic probability, discrete random variables, continuous random variables — plus counting, generating functions and Markov chains, with worked examples, problems and solutions. The 2nd edition added martingales, optional stopping and the Wiener process (Black–Scholes as the stated motivation). The spoke's second probability text, and its second in-copyright one · source · T1 · ctanujit.org PDF, © CUP

• herzog-understanding-statistics — Herzog, Francis & Clarke, Understanding Statistics and Experimental Design: How to Not Lie with Statistics (Springer, open access CC BY-NC, 2019, 142pp): applied inference in 8 chapters, then 4 on meta-analysis, replication and the Test for Excess Success — sum the power across a set of studies and compare with how often they actually succeeded (precognition: 6.27 expected vs 9 reported, p=0.058; bystander effect: 10.77 vs 10, p=0.76). Argues replication cannot arbitrate at moderate power. Closed the free-statistics-text edge; no derivations, so it adds no theory · source · T1 · link.springer.com

• lal-algebra-1 — Ramji Lal, Algebra 1: Groups, Rings, Fields and Arithmetic (Springer 2017, Infosys Science Foundation Series, 11 chapters / ~433pp): two chapters of logic and set theory before any algebra, then number systems, groups, rings and fields, with arithmetic returning twice once rings can state it. Volume one of three, undergraduate and graduate audience. Closes the abstract-algebra edge and breaks the free-licence pattern doing it ($49.99 eBook / $89.99 hardcover). Body not read — this page rests on the publisher's chapter list and description · source · T1 · link.springer.com

• edelman-johnson-matrix-calculus — paige-bright, alan-edelman & steven-g-johnson, Matrix Calculus (for Machine Learning and Beyond) (arxiv:2501.14787, 7 Jan 2025, 101pp, CC BY-NC-SA 4.0, held in raw/): the write-up of MIT 18.063, a 16-hour IAP short course. Rebuilds the derivative as a linear operator so it survives matrix, function and stochastic inputs, then runs Kronecker products → finite differences → Banach spaces → adjoint differentiation → AD via dual numbers and computational graphs → ODE sensitivity → Euler–Lagrange → the reparameterization trick → Hessians as bilinear maps → eigenproblem derivatives. Filed math.HO and claimed by three other fields (cs.LG, math.NA, stat.ML). Its curriculum claim is the spoke's first attempt to move the canon's edge rather than describe it — the 18.01 → 18.02 sequence "is being cut off arbitrarily". Assumes Julia; names its own omissions in a closing chapter; and settles one running tension in a line — "δx is not an infinitesimal but rather a small number" · source · T1 · arxiv.org

• bergman-invitation-general-algebra — george-bergman, An Invitation to General Algebra and Universal Constructions (springer 2015, 551pp, held in raw/): the Math 245 graduate course at uc-berkeley, begun as mimeographed outlines around 1971 and revised on every teaching for 44 years — the longest-maintained text here. 100 pages of worked examples before any general theory (free groups proved to exist three separate ways), then order/lattices/Galois connections, ~180pp of category-theory, varieties and clones, ending at Freyd's criterion, which he calls "the climax of the course". 629 exercises, no solutions, each with a telegraphic description in a back-matter index. Publishes its own difficulty map ("the hard material begins with Chapter 8") and its own risk ("I am worried that the amount of handwaving in these notes may lead some students to think handwaving is an acceptable substitute for proof"). Its pedagogy is a fourth kind: not rigour, content or prerequisites, but what class time is for — notes replace blackboard copying ("this work could be done just as well by a photocopying machine") and a question a day enforces the reading · source · T1 · math.berkeley.edu, © Springer

• judson-abstract-algebra — Thomas Judson, Abstract Algebra: Theory and Applications (GNU FDL, 1997–, 23 chapters): groups → rings → fields, reaching Galois theory in one volume, with two full chapters of applications (cryptography, algebraic coding theory) placed early because students meeting rigorous proof need motivation — the corpus's pedagogical argument arriving about content selection. Sage material by Rob Beezer; maintained for nearly three decades. Closed the open-licensed-algebra edge the day it opened · source · T1 · abstract.ups.edu

• mari-measurement-across-sciences — Mari, Wilson & Maul, Measurement Across the Sciences (Springer, open access CC BY 4.0, 2nd ed. 2023, 307pp): a shared concept system for measurement covering thermometers and reading-comprehension tests alike — four necessary conditions, model-dependent realism, an ontology of properties and scale types, the Hexagon Framework for measurand identification, and the comparison of the physical sciences' uncertainty approach with the human sciences' validity approach. A conceptual monograph, not a textbook; the corpus's first unrestricted open licence, and its first source that is not mathematics · source · T1 · link.springer.com

• carroll-geometric-function-theory — Tom Carroll, Geometric Function Theory: A Second Course in Complex Analysis (springer 2024, SUMS, 358pp): the Riemann sphere and hyperbolic disk, Schwarz–Pick in three widening versions, normal families to the Great Picard Theorem, the Riemann Mapping Theorem, the Koebe ¼-Theorem, and uniformisation of planar domains — plus 54 pages of full solutions and no computational tool assumed. The corpus's sharpest curriculum finding: "consensus evaporates when it comes to a second course", and the only text here with an explicit §1.4 "What Is Not in This Book" · source · T1 · PDF held in raw/, © Springer

• baggett-analysis-single-variable — Lawrence Baggett, Analysis of Functions of a Single Variable: A Detailed Development (University of Colorado, 29 Oct 2006, 229pp, held in raw/): one variable at a time, real and complex together — the author calls separating them historically false — from the least upper bound to the Residue Theorem, with Green's Theorem proved in Chapter VI so that Cauchy's Theorem can be deduced from it in Chapter VII, and an appendix constructing ℝ from Dedekind cuts. Organised backwards from seven named truths (√2 irrational, A = πr², C = 2πr, e^(iπ) = −1, FTC, FTA), which forces the definitional order: π is the smallest positive zero of sine, sine is a power series, area exists only for regions between graphs. Integrable means uniform limit of step functions — deliberately fewer functions than Riemann. Closed most of the first-course complex-analysis hole; never writes the word infinitesimal · source · T1 · spot.colorado.edu (CC BY 3.0 via the OpenStax CNX edition)

• osborne-rubinstein-course-in-game-theory — Osborne & Rubinstein, A Course in Game Theory (mit-press 1994, 372pp scanned): the graduate one-semester course — strategic games and Nash equilibrium, extensive games through bargaining, repeated games and four folk theorems, sequential equilibrium, then coalitional games to the core and the Shapley value. Proof-complete and deliberately pure-theory ("the art of applying an abstract model to a real-life situation should be the subject of another tome"). The corpus's first text whose authors call their subject a social science, not a branch of mathematics · source · T1 · PDF from an Internet Archive scan, © MIT Press

• grinberg-graph-theory — darij-grinberg, An introduction to graph theory (arxiv:2308.04512, v3 of 8 June 2025, 454pp, held in raw/): the text for Math 530 at drexel-university, Spring 2022 — a quarter-long graduate course in ten chapters, with trees and arborescences taking a third of it and the matrix-tree-theorem complex alone taking 65 pages. 116 exercises, no solutions. Prerequisites are determinants, polynomials and finite sums, and "no analysis (or even calculus) is required anywhere" — which is also why planarity and the four-colour theorem are absent, handed to another book because their proofs need continuity. Names ~25 rival textbooks and says that past the basics they cover different ground. Carries a 2019 theorem and an unsolved conjecture in §6.8, and cites MathOverflow. The corpus's first CC0 source, its first from arXiv, and the most permissive licence it holds · source · T1 · arxiv.org

• harju-graph-theory — tero-harju, Lecture Notes on Graph Theory (Turku, 98pp, title page "1994 – 2012", PDF generated 2014-04-24, held in raw/): the corpus's second graph theory text, and it carries the chapter the first one refuses — planar graphs, Kuratowski, Euler's formula, the four- and five-colour theorems, and genus, plus Ramsey theory and network flows. Handles the topology Grinberg cites as his reason for dropping the subject by stating "the geometry of the plane will be treated intuitively" and building region and continuous curve by hand. ~215 theorems / 88 lemmas / 54 examples, and no exercise sets at all — ten inline "Proof. Exercise." handoffs. No copyright notice and no licence anywhere in it: free to obtain, legally silent, the corpus's first such text · source · T1 · users.utu.fi

• illowsky-introductory-statistics — Illowsky & Dean, Introductory Statistics 2e (openstax/Rice University, ©2026, CC BY-NC-SA 4.0, 847pp, held in raw/): the standard one-semester American service course in 13 chapters — sampling → description → probability → CLT → confidence intervals → one- and two-sample tests → chi-square → regression → one-way ANOVA — reaching all of it without calculus and without a proof, for "students majoring in fields other than math or engineering". Assumes intermediate algebra and a TI-83/84 calculator; carries the authors' own statistics labs, four practice exams, and a 2e revision whose stated themes are "currency, relevance, accuracy, and belonging". The corpus's rigour floor, and its first institutionally-published open text · source · T1 · openstax.org

• nelson-essential-math-for-ai — Hala Nelson, Essential Math for AI (O'Reilly 2023, ISBN 978-1-098-10763-5): a survey of the mathematics AI needs — regression, neural nets/convolution, optimization, probability, graphs/random walks, Markov processes, ODEs — organized by application (CV, NLP, generative, RL, operations research) "rather than dense academic theory." The corpus's first text sequenced by what an application needs rather than by mathematical structure, and near its rigour floor. Body not read — publisher description + OpenLibrary · source · T2 · amazon.com

AnalysisNewsArticle (sources)

• ramos-controlling-variables — mark-louie-ramos (Penn State) in The Conversation, 25 Sept 2026: the two meanings of "control" (experimental assignment vs statistical adjustment), confounding, stratification, and the too-few/too-many-controls failure modes (colliders). No derivation, no citation trail of its own — the spoke's first popular-science source and its first explicitly about study design rather than inference machinery · source · T2 · theconversation.com

Person (entities)

• alan-edelman — MIT mathematics + CSAIL, runs the Julia lab; co-author of edelman-johnson-matrix-calculus · author

• steven-g-johnson — MIT mathematics + physics; co-author of edelman-johnson-matrix-calculus · author

• paige-bright — wrote up the MIT 18.063 lectures as edelman-johnson-matrix-calculus · thin node

• george-bergman — uc-berkeley algebraist; author of bergman-invitation-general-algebra, maintained across 44 years of teaching Math 245 · author

• martin-klazar — MFF UK combinatorialist writing klazar-elementary-real-analysis in the open, fourteen arXiv versions deep · author

• jiri-matousek — (1963–2015) Charles University mathematician and computer scientist; the analysis course he was preparing when illness stopped him is why klazar-elementary-real-analysis exists · dedicatee

• claude-shannon — the founder of information-theory and the author of both limits this spoke holds; paged for his method (prove existence by averaging over all codes) as much as his results. The only entity here reached through a source held in another spoke · originator

• darij-grinberg — drexel-university algebraic combinatorialist; the only author here who gave up the copyright (CC0) rather than choosing a licence that asks for something back, and whose web address has outlived three affiliations · author

• tero-harju — university-of-turku professor emeritus and research director; a combinatorics-on-words specialist rather than a graph theorist, which shows in notes that tour the standard results and hand the algebraic and algorithmic sides to other books. Still teaching the course in 2018, six years past the last date on its title page · author

• lawrence-baggett — Colorado professor emeritus; harmonic analysis and wavelets by research, an undergraduate analysis course by authorship. Blind since the age of five, and the author of a 2012 MAA memoir about it · author

• michael-herzog — Brain Mind Institute, epfl; lead author of herzog-understanding-statistics (not michael-corral — the index near-matches the names)

• gregory-francis — Purdue; co-author, and the researcher behind the Test for Excess Success

• martin-j-osborne — economist, mcmaster-university at the time of writing; taught game theory at Stanford by Aumann, Hart, Kurz and Wilson · author

• ariel-rubinstein — tel-aviv-university and Princeton; the alternating-offers bargaining model and the bounded-rationality thread that shapes three chapters of the book · author

• tom-carroll — university-college-cork analyst, Hayman and Rippon student; the first author here who had to argue about what the syllabus is rather than how to teach an agreed one · author

• gregory-lawler — university-of-chicago probabilist; random walks and their continuum limits, and a free book he has revised for nine years · author

• david-gamarnik — MIT Sloan, applied probability and operations research; instructor of the graduate stochastic-processes course · instructor

• hao-wu — probabilist; taught MIT 18.445 in 2015. The OCW listing gives a title and a department and nothing else, and the page says so · instructor

• curtis-mcmullen — Harvard mathematician; author of the probability notes · author

• david-stirzaker — Oxford mathematician (St John's; Grimmett & Stirzaker); defends formal definitions on student-error grounds, the mirror image of h-jerome-keisler's argument · author

• luca-mari — LIUC industrial engineering; the metrology half of mari-measurement-across-sciences · author

• mark-wilson — Berkeley School of Education; the psychometrics half · author

• andrew-maul — UCSB Gevirtz School of Education; the philosophy-of-measurement half · author

• thomas-judson — stephen-f-austin-state-university; kept his algebra text free under the GFDL since 1997 and still maintains it — the sustained-maintenance pattern he shares with jim-hefferon, h-jerome-keisler and william-trench · author

• ramji-lal — HRI Allahabad mathematician; author of the three-volume algebra series whose first volume is here · author

• barbara-illowsky — De Anza College; co-author of illowsky-introductory-statistics and of the Collaborative Statistics it grew from; the spoke's first community-college author · author

• susan-dean — De Anza College; co-author of the same text and co-creator of its statistics labs · author

• ray-bowen — mechanical engineer, texas-am-university; co-author of bowen-wang-vectors-tensors-vol-2; same-as Wikipedia + Wikidata Q26236282 (both checked) · author

• c-c-wang — rice-university mathematical sciences; co-author of bowen-wang-vectors-tensors-vol-2; initials only in the source, no same-as · thin node

• michael-corral — Schoolcraft College adjunct; writes free-licensed mathematics textbooks · author

• larry-thibos — Indiana vision scientist; six editions of the same coursenotes over 25 years · author

• h-jerome-keisler — Wisconsin logician; kept his textbook free for 24 years, including through a Dover edition · author

• jim-hefferon — Vermont mathematician; CTAN maintainer and TUG officer, three open textbooks · author

• richard-larsen — Vanderbilt statistician; five editions over 31 years, the corpus's longest sustained authorship · author

• morris-marx — University of West Florida statistician; co-author of the same · author

• william-trench — Trinity professor emeritus; reissued a Pearson title free once the rights reverted · author

• jean-gallier — Penn CIS and mathematics, full professor since 1990; Horn-satisfiability in linear time with Dowling, then geometry and its applications from 1993. Serves six books from departmental web space, redistribution reserved · author

• jocelyn-quaintance — co-author of gallier-quaintance-math-deep at Penn CIS · author · thin

• hala-nelson — author of Essential Math for AI; writes the mathematics of AI for an applied audience · author · thin

• mark-louie-ramos — Penn State statistician, Health Policy and Administration; author of ramos-controlling-variables · author · thin

• andrew-baker — Glasgow algebraic topologist (honorary senior research fellow); author of baker-algebra-number-theory; homepage identifier, no Wikipedia entry · author

Organization (entities)

• charles-university — Prague; its Faculty of Mathematics and Physics (MFF UK) is where klazar-elementary-real-analysis was taught — to computer-science students, not mathematics majors
• university-of-colorado-boulder — CollegeOrUniversity: lawrence-baggett's department, and the corpus's least institutional hosting — a tilde URL on staff web space, no licence field, no record page, unrevised since 2006 · institution
• stephen-f-austin-state-university — CollegeOrUniversity: thomas-judson's institution; hosts the book in its ScholarWorks repository · institution
• arxiv — Project: the Cornell-run e-print archive; the spoke's third publishing route, and the only one that keeps dated earlier versions addressable (the graph theory text has three). Deposit is not review, and the page says so · publisher
• drexel-university — CollegeOrUniversity: darij-grinberg's department, and the corpus's first text written against a quarter rather than a semester — ten weeks is why the graph theory book stops where it does · institution
• university-of-turku — CollegeOrUniversity: founded 1920, tero-harju's department, and harju-graph-theory's only publisher — its own web server, which is why that text has no imprint and no licence. The spoke's first non-American institution · institution
• mit-opencourseware — Project: MIT's course-publishing programme, CC BY-NC-SA. Supplies what a textbook cannot — a syllabus with a date and an instructor on it, read here for sequence rather than theorems · publisher
• university-of-chicago — CollegeOrUniversity: its mathematics department serves lawler-stochastic-calculus from the author's own page · institution
• mit-press — academic publisher; the spoke's second publisher node, holding osborne-rubinstein-course-in-game-theory, which the authors typeset themselves in TeX · publisher
• mcmaster-university — CollegeOrUniversity: martin-j-osborne's department in 1994 · institution
• tel-aviv-university — CollegeOrUniversity: ariel-rubinstein's home department · institution
• openstax — nonprofit publisher at Rice University; free peer-reviewed textbooks under CC BY-NC-SA, produced institutionally rather than by their authors — revisions keep problem numbering stable so online homework providers can follow an edition change · publisher
• de-anza-college — Cupertino community college; home of barbara-illowsky and susan-dean, and the roster's second two-year college · institution
• springer — academic publisher; the spoke's first publisher node holding two of its books, at opposite ends of the access axis (paywalled lal-algebra-1, open-access herzog-understanding-statistics) · publisher
• springer-undergraduate-mathematics-series — CreativeWorkSeries: SUMS, springer's undergraduate line; a publisher-imposed pedagogical specification — one-to-two semesters, fully-worked solutions, self-study viable — visible in the one volume held · series
• oreilly-media — technical-book publisher and learning platform; the spoke's first trade publisher (vs the academic springer/mit-press and the course-page/arXiv routes), holding nelson-essential-math-for-ai · publisher
• university-college-cork — CollegeOrUniversity: tom-carroll's school, and the corpus's one case of a department's disciplinary taste being given as the reason a book exists · institution
• infosys-science-foundation-series — CreativeWorkSeries: the Springer series (ISFS/ISFM) carrying lal-algebra-1; a hook for volumes two and three · series
• harish-chandra-research-institute — ResearchOrganization: mathematics and theoretical-physics institute in Allahabad; ramji-lal's affiliation · institution
• epfl — École Polytechnique Fédérale de Lausanne; the Brain Mind Institute is michael-herzog's home
• texas-am-university — CollegeOrUniversity: ray-bowen's affiliation; the PDF is from its OAKTrust repository · institution
• rice-university — CollegeOrUniversity: c-c-wang's affiliation · institution
• purdue-university — psychological sciences; gregory-francis's home
• university-of-pennsylvania — CollegeOrUniversity: its computer-science department (CIS, not mathematics) is where the corpus's largest mathematics text was written, which shows in what the book optimizes toward · institution
• harvard-university — CollegeOrUniversity: hosts the notes on faculty web space, no publisher in between · institution
• indiana-university — CollegeOrUniversity: its School of Optometry is why the Fourier text has statistics chapters · institution
• schoolcraft-college — CollegeOrUniversity: community college, and the origin of the corpus's only GFDL text · institution
• university-of-wisconsin-madison — CollegeOrUniversity: hosts the Keisler text, and produced the 1974 thesis that tested its method · institution
• university-of-vermont — CollegeOrUniversity: the fifth institution in five texts, and the one that breaks the origin-predicts-quality pattern · institution
• vanderbilt-university — CollegeOrUniversity: where the institution and the publisher finally come apart — the first text here that reached readers through Pearson rather than its authors' servers · institution
• university-of-west-florida — CollegeOrUniversity: a regional public university co-authoring with a private research one, indistinguishably, inside one text · institution
• university-of-oxford — CollegeOrUniversity: david-stirzaker's Mathematical Institute and St John's College; with harvard-university, the second elite-department origin, and both produced probability texts · institution
• pennsylvania-state-university — CollegeOrUniversity: mark-louie-ramos's department, Health Policy and Administration · institution
• the-conversation — NewsMediaOrganization: publisher of ramos-controlling-variables; the spoke's first news-media publisher, distinct from its academic, course-page and trade routes · publisher
• liuc — CollegeOrUniversity: Università Cattaneo, Castellanza — the corpus's first non-US institution and first engineering school · institution
• uc-berkeley — CollegeOrUniversity: Berkeley School of Education, mark-wilson's home · institution
• uc-santa-barbara — CollegeOrUniversity: Gevirtz Graduate School of Education, andrew-maul's home · institution
• university-of-glasgow — CollegeOrUniversity: andrew-baker's school; hosts the notes on staff web space · institution
• trinity-university — CollegeOrUniversity: its Digital Commons repository hosts the analysis text — an institutional archive rather than a personal domain · institution