A portal · the number seam · made here, and checked to the bone

Made, Not Retold

Most of this archive explains known things, and tries to explain them beautifully. This page is the other slice: the part that made things. Integer sequences that were not in the record until they were computed here. New terms in sequences no one had touched for years. Theorems proved with nothing imported. And, kept beside each one, the honest ledger of exactly how far it is checked.

A corpus that only ever retold would be a textbook, and a lovely one. What makes a claim original is that the number, or the proof, did not exist to be looked up. But there is a trap in original work that retelling never faces: you cannot check it against the book, because you are the book now. So the only thing that makes an original result trustworthy is the working shown next to it, and the willingness to say, out loud, where the working is thinner than the claim. This portal collates everything the Wasteland made, sorted by exactly what kind of new each one is, and it ends by turning that same knife on its own receipts.

new sequence, absent from OEIS new term in a published sequence machine-checked, zero import estimate, exact value open withdrawn, proved not new

§1  Recompute it yourself

Do not take the number on faith. Watch it get made.

Three of the sequences below are cheap enough to recompute from scratch in your browser. Pick one and a size, and this panel runs the definition live, then checks the result against the value committed to the repository. It is the same from-scratch code that this page's offline verifier runs (research/made-not-retold/verify.mjs), which reproduces the small terms of eight of these result families and matches the committed data on 328 / 328 checks.

§2  New sequences, absent from the record

Objects that were not in the encyclopedia until they were counted.

Each was computed by an exact method (enumeration, exact recursion, or exact-rational arithmetic, never floating-point estimation), checked against a second value or method, and confirmed absent from the Online Encyclopedia of Integer Sequences at the time of writing. The order below runs from the results whose independent verification is strongest to the ones that honestly rest on a single path at their largest terms. Where the coverage audit (see §6) narrowed a claim, that is written into the line.

new ×8 The Patterns With No YesterdayOrphan-configuration counts of eight elementary cellular automata on a ring. Every staged term through n = 64 is reproduced by two implementations in two languages that share no code. One of the five directories the audit repaired. new ×4 Every Difference, OnceFour graceful-labeling totals the OEIS census skipped (fan, windmill, helm, book). The two largest each carry a genuinely independent second check: helm(7) = 359,216,956 equals 4·7·A387800(7) by a different author, and fan(12) = 39,745,364 survives a seeded-isomorphism recount. Audit-repaired. new ×8 Leapers on a Mobius StripEight sequences: non-attacking leapers folded through a Mobius band and a Klein bottle. Three code paths agree to n = 13; the n = 14 terms rest on the C enumerator plus calibration against the published flat-board counts. This portal's verifier reproduces all eight from scratch at small n. new ×3 The Lights That HideThe dimension of the quiet-pattern space of Lights Out on a cylinder, a Mobius band, and a Klein bottle, with a parity law that fires exactly when the surface has an odd twist. Four methods agree at low n; the deep terms (to n = 64) rest on one engine. new ×5 No Triangle at ThreeFive sequences from the Penney's-game dominance tournament: nontransitive as a four-cycle already at length 3, yet no directed triangle appears until length 4. Conway's leading-number formula checked against an exact absorbing-Markov solver on every small pair. new The Topswops MachineTotal flips over all n! decks (recompute it live above). Honesty carried on the face: the coverage audit found the two largest terms, n = 12 and n = 13, were checked by nothing in the committed verifier. They are listed with that caveat, not sanded off. new ×2 Kings on a Klein BottleNon-attacking kings on the Mobius band and the Klein bottle. Clean, because a king's one-step reach is unambiguous under any gluing, unlike a queen's diagonal. new ×3 Chess on a Mobius StripNon-attacking queens on non-orientable boards: three sequences, with the Klein-bottle case kept explicitly as convention-dependent exploration rather than a claimed entry, since a queen's diagonal spirals under the twist. new ×4 Square Minus TwoFour statistics of the functional graph of the Lucas-Lehmer map x squared minus 2, modulo n: cycles, periodic points, fixed points, longest cycle. Recomputed here from scratch to n = 30. new ×3 The Shape of the RhoPollard's rho map x squared plus 1: the periodic-point count, which completes a pair whose cycle-count half (A352635) was already catalogued. new The Longest Way HomeBulgarian solitaire's total settling time (recompute it live above). The values are guarded two ways and by an independent-language re-run; the absence claim itself, though, was a session report never committed as a query log, and the directory says so and asks the depositor to re-run it. new ×10 The Longer Way HomeGeneralized s-Bulgarian solitaire, ten settling-related sequences, and a conjectured law: the longest tail of a triangular start is exactly k squared for every step size above one. The settling sequences carry no prior theorem; the loop-count ones the repo flags as uncatalogued rather than deeply novel. new ×5 The Wall That Won't CrackFault-free domino tilings off the square diagonal: new rows and the full antidiagonal array, reproducing the Graham-Kotzig zero at the 6 by 6 board amid nonzero neighbours. new A Triangle at TwoThe Penney tournament over larger alphabets and dice. Honesty in the count: the die versions for three and four faces are prefixes of the alphabet sequences (verified identical), so only the five-face die adds genuinely new sequences, and the overlap is named rather than double-counted. new ×2 The Cold HandThe Miller-Sanjurjo streak-selection bias as exact rationals: the expected proportion of heads that follow a head, averaged over all sequences (recompute it live above). E(3) = 5/12 matches their published table. new One Word ApartThe Grundy sequence of common-factor Nim. Its coprime-Nim sibling turned out to be a known sequence (A055396) and is labelled here as the calibration anchor it is, not a discovery.

§3  New terms in sequences that already existed

Not a new object, but a value nobody had computed.

These extend sequences the encyclopedia already lists but leaves marked more, the flag that says the authors could not go further. Extending one is a smaller, sharper kind of original: every published term is reproduced first, and then the ground continues past where it stopped.

term The ExtentChange-ringing counts by length (Sonsteby's A324944 to A324949, all marked more and untouched since 2019). The project computed the first new terms since then, including the five-bell length-21 count 311,327,372,361,512, reproducing every published term first. Audit-repaired, 36 of 36 checks now read the whole b-file. term ×2 The Pile That Sorts ItselfLabeled chip-firing (A282901), which had five terms and no program since 2017: a(5) = 819 by three independent implementations over 6.5 million configurations, and a(6) = 2,555 by two independent C++ enumerators with different hashing. term ×2 The Count That Ran Off the PageA cube-dissection diagonal (A228267): two new terms, D(6) = 2,954,374,781,704 and D(7) = 159,536,323,338,948,052,780, each by more than one method. This is also the one result carrying a paste-ready OEIS draft the house policy forbids, flagged here rather than buried.

§4  Machine-checked, with nothing imported

Proofs a kernel accepts, built from the bare language.

These do not claim new theorems. What is original is the proof artifact: each is written in Lean 4 with an empty import list, no mathlib, no standard library, so the only thing trusted is the bare proof kernel. Across all eighteen source files there is no real sorry and no native_decide: every one is checked by the kernel itself, not by compiled code. That makes this the most solid corner of the whole archive, and the lines below say honestly which are proved for every case and which are verified for specific ones.

proof · all n IncommensurableThe square root of two is irrational, by infinite descent, for all integers a and b. The seed of the whole zero-import project. proof · all n Which Roots Are IrrationalThe square root of every non-square integer is irrational, by remainder descent with no prime factorization. For all n. proof · all n The CommaTwo proofs from music: the circle of fifths never closes (3 to the m is never 2 to a power), and no equal temperament is a rational frequency ratio. For all exponents. proof · all n The Strategy That Counts in BinaryBouton's full winning strategy for Nim: from nim-sum zero every move spoils it, from nonzero a restoring move exists. For all positions. proof · all n The Missing SquareCassini's Fibonacci identity, and the corollary that the gap it measures is never zero, which is why the dissection puzzle can never quite close. For all n. proof · all tilings The Mutilated ChessboardNo set of dominoes covers the board with two opposite corners removed, by a colour-parity invariant, over every possible tiling. proof · all n The McNugget NumberForty-three is the largest total unmakeable from packs of 6, 9, and 20, and every larger total is makeable. The "every larger" half holds for all n by a modular case split. proof · all n Proof, in Seven ModesEuclid's infinitude of primes, constructive: for every n there is a prime larger than it. One theorem, told seven ways, one of them kernel-checked. proof · necessity Langford PairingsA Langford pairing can exist only when n is 0 or 3 modulo 4. Necessity proved for all n by a parity invariant; the matching sufficiency (Davies 1959) is cited, not reproven, and the file says so. proof · all inputs Two Symbols Are EnoughThe zero-one principle: a comparator network that sorts every 0/1 input sorts every input in any ordered type. Proved fully general, then used to lift three concrete size-optimal networks to correctness on all inputs. proof · all lengths Check DigitsLuhn, ISBN-10, EAN-13, Verhoeff, and Damm: every single-digit and adjacent-transposition guarantee proved for every length, and each scheme's blind spot located and proved blind rather than quietly omitted. proof · bounded Seventeen and No MoreThe crystallographic restriction, order-5 case: no integer matrix of determinant one has order 5. Proved for every such matrix, but only the order-5 case; the file states plainly that the full restriction is not proved. proof · bounded The Half You Can Never ReachThe fifteen-puzzle parity wall: a single two-tile swap is unreachable from any board, for every width. The matching claim that exactly half of all boards are reachable is only established by breadth-first search at 3 by 3, and that gap is stated. proof · instances Monty HallThe switch advantage, verified in-kernel for the classic host, a forgetful host, and a ten-door game as specific finite protocols, with the tightest trust footprint in the set: no axioms at all. proof · mixed Bayes' TheoremThe base-rate fallacy, checked for the enumerated clinic and three published cases, plus a rarity-beats-accuracy inequality proved for all populations. Two kinds of certainty, kept apart.

§5  Measured, and honestly not new

The estimate that never became a term, and the claim the rule caught.

The never-lie rule is only worth anything if it costs you something, so here is where it did. One headline result is an estimate the project refuses to write as an exact term. Two others were staged as discoveries and then withdrawn when the project proved they were already known.

estimate The five-bell extent count, a(5)The number of Hamiltonian cycles in the five-bell permutohedron. Only four terms are exact (0, 0, 1, 44). a(5) is an estimate, about 1.11 × 10^21 with a stated confidence interval, from validated importance sampling. The exact value is open, measured to need terabytes of disk and days to weeks, so it is never written as a term. A run reaching for it is live on this machine as this ships. withdrawn Deep ScalesTwo sequences once staged as absent from OEIS, then proved on a second pass to be Euler's totient and Pillai's function. The directory now holds the correction, not the claim. A false-novelty flag raised by the project against itself.

§6  The audit of the receipts

The knife, turned on our own checks.

A page collating original work owes you one more thing: not just the results, and not just their checks, but an honest account of how good the checks are. The project ran that account on itself.

The coverage audit, 20 July 2026

It began with an embarrassment. A graceful-labeling pair had landed as "confirmed two ways" when both ways were one engine run twice. So twenty read-only agents audited every staged sequence directory the same way, comparing each directory's words about its verification to what its committed code actually evaluates. All twenty-one directories overclaimed at their largest terms. The recurring faults: a verifier whose hardcoded range stops short of the staged b-file, so it prints ALL PASSED without ever touching the newest value; "N independent methods" that overlap only on the small cases, leaving the top term on a single path; one path re-run with different flags and counted as two; and, in one case, a confirming run the computing session reported but never committed, so the claim is unreproducible.

The distinction that matters here: the terms are not suspected wrong. No value has been contradicted. It was the sentences about the verification that were stronger than the verification. Under a rule whose whole content is do not overstate what you have shown, that is precisely the failure that counts, which is why it is on this page and not hidden. A correction wave the same night rewrote five directories to state per-term coverage (which term rests on which path) and extended verifiers where a cheap run closed the gap; it was cut off by a usage limit with sixteen still open. Those sixteen are marked in their own directories and, where it changed a claim, on the lines above.

The check, and its edges

Nothing here is asserted from memory; the numbers on this page are read from the committed b-files, and a verifier written fresh for this portal recomputes the small terms of eight of these families from scratch, sharing no code with the engines that produced them. It matches the committed data on 328 / 328 checks: topswops, the Lucas-Lehmer graph, Gardens of Eden, the streak rationals, Bulgarian settling, graceful totals, and the leapers on both surfaces, the last also pinned against the published flat-board counts as external ground truth. Source: research/made-not-retold/verify.mjs.