Visualizing the search for abelian square-free words. Over a three-letter alphabet {a, b, c}, every word of length 8 or greater contains an abelian square, making an infinite abelian square-free word impossible. In 1992, Veikko Keränen proved that an infinite abelian square-free word exists over a four-letter alphabet {a, b, c, d}.
Letter colors: a b c d. * Tab color categories: a=Learn, b=Explore, c=Research, d=How we know it
Every claim shown here traces to a row in MATH_CLAIMS.md —
view the live research ledger and source code
· The Bridge Story
An ordinary square is a word that repeats consecutively, like hotdoghotdog (hotdog + hotdog) or abcabc (abc + abc).
An Abelian square is a looser concept: it is a word consisting of two consecutive parts that have exactly the same letters, but possibly in a different order.
The left half (abc) and the right half (cab) both contain one a, one b, and one c. This is an Abelian square!
If you only have three letters (a, b, and c), the task turns out to be impossible.
How long of a word can you form so that no part of it contains an Abelian square?
No matter which letter you add (a, b, or c) after this, an Abelian square inevitably forms. Try adding 'a', 'b', or 'c' and see what happens in the "1. Tree Search (3 letters)" tab!
Since 3 letters are not enough, in 1992 Veikko Keränen proved that with four letters (a, b, c, d) an infinite word can be built without a single Abelian square.
This was done with a morphism. A morphism is a rule that replaces each single letter with a long string. Keränen's rule (g85) replaces each letter with exactly an 85-letter word.
You can explore this complex structure and current discoveries (such as the g3 morphism) on the other tabs!
Backtracking over {a, b, c}. Each branch tries one letter; branches marked red contain an abelian square and are pruned.
abcbacb). This was noted by Dekking (1979) and independently by Keränen. The tree search below hits a dead end every time it reaches depth 7.After each appended letter, all suffix splits are scanned. A match in Parikh vectors means an abelian square — branch is pruned.
Exhaustive finite search over {a,b,c}. The laboratory keeps every prefix that avoids abelian squares, then shows the exact collision that prevents extension to length 8.
Listen to the structure of the g₈₅ word. Slicing into chunks of 85 corresponds to exactly one morphism image per track. Because g(b), g(c), g(d) are cyclic permutations of g(a), each track shares the same set of pitches — only their order changes.
Build a word letter by letter. The app checks for abelian squares in real time. With 3 letters, you will hit a wall by length 8 — try it!
How many letters does it take to build an infinite abelian square-free word? This question took over three decades to answer — click the events to explore.
Certain short words (“unfavorable factors”) cannot be extended far in either direction without creating an abelian square. Enter a word or pick an example, then see how far left and right the word can grow before it gets stuck. (Concept from Keränen’s 2010 Mathematica Journal article.)
Two abelian square-free morphisms by Veikko Keränen. Both map each of the four letters to a longer word such that the image of b, c, d are cyclic permutations of the image of a. Iterating either morphism from the seed word “a” produces an infinite abelian square-free word.
Published at ICALP 1992. First proof that 4 letters suffice. Length of each image: 85.
Parikh vector of g₈₅(a):
g₈₅(a) →
Published in Theoretical Computer Science (2009). Described as “a powerful abelian square-free substitution.” Length of each image: ?.
Parikh vector of g₁₀₉(a):
g₁₀₉(a) →
Interactive concept map. Drag nodes to explore relationships. Click on a node to read details.
Click any node on the graph to see definitions, history, and relevance to the abelian square problem.
This glossary connects the formal mathematical terminology to the concepts in "The Bridge Story".
Design your own uniform morphism over {a,b,c,d}. The program will evaluate its Parikh vectors, check for cyclic permutations, and iteratively search for abelian squares.
Scanning the 14 valid bigrams of h₆ω(a) across 126 seam offsets for K=5 collisions. Evaluating single and dual boundary mutations to eliminate K=5 while preserving K>5 avoidance.
Each cell (x, y) shows whether an abelian square exists starting at position x with half-length y. ■ Red = abelian square found, ■ Yellow = near-miss (Parikh diff = 1 per letter), ■ Gray = no square. The g₈₅ word is abelian square-free, so the red cells come only from a random word for comparison.
Generating a unique puzzle...
You have successfully constructed an Abelian Square-Free sequence. The math holds strong.
Use the Start Extension button to start the background search. The algorithm automatically appends letters and backtracks when hitting dead ends. Try loading a 40-letter seed and run the search!
This is the program's most powerful "X-Ray Vision". First, Pause the search. You will see the letters displayed in the main view. Click any letter in the word! The program sends the sequence up to that point to the background engine, which looks 10 steps ahead. It will then tell you in the Predictive Lookahead panel whether the path is viable or a guaranteed dead end.
You can now manually build the sequence based on the Predictive Lookahead results! Use the ✍️ Manual Construction buttons (a, b, c) under the lookahead panel to append letters. If a letter leads to an immediate abelian square or forbidden factor, it is instantly rejected and the reason is explained.
While the search runs and encounters dead ends, the Search Pruning Heuristics tracker on the right side logs them. If it notices that a specific suffix (e.g., bcca) causes an abnormally high number of dead ends, it automatically suggests it as a Structural Trap heuristic rule and provides a statistical confidence interval.
Don't lose your findings! If the Heuristics tracker finds a trap, click Save to Notebook. If you manually discover a stuck path using Predictive Lookahead, click the 📓 Save Path button in the top bar. These will be permanently saved to your browser's memory in the bottom right corner.
aa, bb, cc), but strictly prohibit all other Abelian squares of half-length ≥ 2.baac, caab, abbc, cbba, accb, bcca.The app loads a valid 40-letter AA2FR word where exactly one of a, b, or c can be appended on the right. Guess the letter; wrong answers open the obstruction panel below.
Choose the next letter to extend the sequence on the right:
| Run | Seed | Max Length | Steps | Time (ms) |
|---|
| Motif | N | Dead % | Surv % | Avg Depth | Wilson CI Lower |
|---|---|---|---|---|---|
| No data yet. Run searches to aggregate statistics. | |||||
The mathematics of abelian square-free words touches many areas of computer science and beyond. Click a card to explore how combinatorics on words connects to real-world problems.
This dedicated laboratory isolates formal mathematical verification from exploratory heuristics.
Here we execute exact reproduction tests for published literature on Mäkelä's Conjecture and bounded audits of empirical morphisms,
while verifying core mathematical invariants like group symmetries.
Epistemological Note: Bounded computational auditing does not replace formal infinite proofs or verify theorems for all N; it confirms published results and heuristic behavior strictly within specified finite prefix lengths (N) and half-length inspection windows (K).
Every external mathematical claim and internal empirical invariant relied upon by this project is cataloged below with its precise citation, verification status, and scope. This ledger serves as the single source of truth for all tools in this laboratory.
| # | Claim | Source / Citation | Status | Notes / Verification Scope |
|---|---|---|---|---|
| 1 | Over a ternary ({a,b,c}) alphabet, abelian squares cannot be avoided indefinitely (full version, all half-lengths). | Folklore / Fici & Puzynina (2022), Abelian Combinatorics on Words, Prop. 17 | Computed | Exhaustive search: longest ternary abelian square-free word has length 7 (e.g., cbcacbc). All length-8 words contain an abelian square. |
| 2 | Pleasants (1970) proved the existence of an infinite abelian square-free word over a 5-letter alphabet. | P.A.B. Pleasants, Non-repetitive sequences, Proc. Cambridge Phil. Soc. 68 (1970) | Indirect | Distinct from Claim 1; not the source for the 3-letter length ≥ 8 impossibility theorem. |
| 3 | Keränen (1992) proved the existence of an infinite abelian square-free word over a 4-letter alphabet via morphism g₈₅. | V. Keränen, ICALP 1992, LNCS 623 | Indirect | Original proof verified in literature; implemented in Tab 3 of this laboratory. |
| 4 | Mäkelä's conjecture: there is an infinite ternary word whose only abelian squares are 00, 11, 22 (the AA2F setting). | Fici & Puzynina (2023), arXiv:2207.09937, Conjecture 20; same question as Rao & Rosenfeld, arXiv:1511.05875, Problem 1 | Open 🟢 Level 2: Verified Source | Quoted verbatim (Fici & Puzynina, Conj. 20): "There exists an infinite ternary word whose only abelian squares are 00, 11, 22." Rao & Rosenfeld, Problem 1: "Can you avoid abelian squares of the form uv where |u| ≥ 2 over three letters?" Both read from the arXiv sources on 2026-07-28. Open for half-lengths 2 ≤ K ≤ 5. Note Rao & Rosenfeld add: "Computer experiments show that you can avoid these patterns at least in words of length 450." — the words known to this project and to Keränen (25,379 letters) are far longer. See MATH_CLAIMS.md row 4. |
| 5 | h₆ uniform morphism (6 letters a–f): the fixed point h₆ˆω(a) is entirely abelian square-free. | M. Rao & M. Rosenfeld, arXiv:1511.05875, "Avoiding two consecutive blocks of same size and same sum over Z2", Theorem 4 | 🟢 Level 2: Verified Source | Quoted verbatim: "h₆ω(a) is abelian-square-free." Citation retracted and re-corrected 2026-07-28: the 2026-07-26 note claiming arXiv:1507.02581 was itself wrong. That record is "Avoidability of long k-abelian repetitions", a different result; the title "On Mäkelä's Conjectures" matches no arXiv record at all. Theorem number corrected from "Theorem 5" — the paper's actual Theorem 5 concerns a different morphism h₈. h₆'s images audited character-by-character against §5.4 and against Fici & Puzynina's digit relabelling; asserted by node test.js test 14. Independently re-derived: decide-realizability.js runs the paper's own decision procedure and reaches the same conclusion (MATH_CLAIMS.md rows 5, 32). |
| 6 | g₃(h₆ˆω(a)) (3 letters) contains zero abelian squares of half-length K > 5. | M. Rao & M. Rosenfeld, arXiv:1511.05875, Theorem 9 (§5.4 "Mäkelä's Problem 1") | 🟢 Level 2: Verified Source | Quoted verbatim: "The word obtained by applying g₃ to the fixed point of h₆, that is g₃(h₆ω(a)), does not contain any square of period more than 5." Theorem number corrected 2026-07-28 from "Theorem 11" — the paper's actual Theorem 11 concerns 2-abelian squares of period more than 60, a different statement. arXiv ID re-corrected as in row 5. g₃'s images audited character-by-character against §5.4. See MATH_CLAIMS.md row 6a. |
| 7 | There exists an infinite ternary word avoiding abelian squares of period > 5. | M. Rao & M. Rosenfeld, arXiv:1511.05875, Theorem 10; same result as Fici & Puzynina (2023) Theorem 19, part 2 | 🟢 Level 2: Verified Source | Quoted verbatim: "There is an infinite word over 3 letters avoiding abelian squares of period more than 5." Fici & Puzynina state the same as "There exists an infinite ternary word with no abelian square of length 12 or greater" (length 2K ≥ 12 ⇔ K ≥ 6). The authors write of their own result: "We show that the answer is positive if we replace 2 by 6" — so K = 2…5 is exactly the gap that remains open (row 4). See MATH_CLAIMS.md row 7. |
| 8 | h₆ morphism is derived from the Hall–Janko group. | — | Rejected | No literature support found. Removed from code and UI. Do not re-add without precise page citation. |
| 9 | FORBID4 set {baac, caab, abbc, cbba, accb, bcca} is closed under S₃ letter permutations and word reversal. |
Project Empirical Finding | Computed | S₃ orbit verified by Module A; reversal symmetry verified in Worker. Zero intersection with K = 2 abelian squares. |
| 10 | Morphism Verification Badge Levels: Level 1 (⚙️ Checksum OK) vs Level 2 (🟢 Verified from Source [DOI]). | Project Epistemological Standard | 🟢 Level 2 Standard | Level 1 proves only internal codebase integrity (`djb2` hash unchanged). Level 2 requires an explicit DOI/arXiv link and human character-by-character verification against printed literature. |
The figures below are exact. They are derived algebraically with rational and Q(√3) arithmetic — no floating point in any result path, and no prefix scanning. Where they concern the infinite word they are statements about the infinite word, not about a window.
These are not computed in your browser. They come from standalone Node scripts in the repository, listed in the last column. This panel reports their output; it does not reproduce it. Run the command to check any row yourself.
| Quantity | Exact value | Status | Reproduce with |
|---|---|---|---|
| Letter frequencies in h6ω(a) | exactly 1⁄6 each | ⚙️ Level 1 (exact derivation) | node perron-frobenius.js |
| Ternary densities in g3(h6ω(a)) | a=1⁄3, b=17⁄60, c=23⁄60 | ⚙️ Level 1 (exact derivation) | |
| Spectrum of Mh6, char. poly x³(x−3)(x²−3) | { 3, ±√3, 0, 0, 0 } | ⚙️ Level 1 (exact derivation) | |
| Abelian square densities ρK, K=1…5 | 109⁄180, 13⁄36, 41⁄180, 29⁄180, 2⁄45 | ⚙️ Level 1 (exact derivation) | node factor-frequencies.js --maxk 40 |
| Distinct abelian squares in g3(h6ω(a)) | exactly 34, longest of length 10 | ⚙️ Level 1 (complete factor sets) | |
| K = 5 collisions that span an image boundary | exactly 100% (K = 4: 79.3%, K = 3: 48.8%, K = 2: 35.4%) | ⚙️ Level 1 (complete factor sets) | |
| Jordan form of Mh6 — the matrix is defective | diag(3,√3,-√3)⊕ J2(0)⊕ J1(0) | ⚙️ Level 1 (exact derivation) | node jordan-decomposition.js |
| Index of im(Mg3) in Z3 | 10 — forced by g3 being 10-uniform | ⚙️ Level 1 (exact derivation) | node smith-normal-form.js |
| Factor complexity of g3(h6ω(a)) | linear; p(45) = 372, first differences in [6, 8] | ⚙️ Level 1 (complete factor sets) | node factor-complexity.js |
| Rao & Rosenfeld decision procedure, run end to end | bound s = 34; 0 factors realize the abelian-square template ⇒ h6ω(a) is abelian-square-free | ⚙️ Level 1 (re-derivation) | node decide-realizability.js |
What changed, and why it matters. The ρK values, the count of 34 and the boundary split were previously reported here as observations over a finite prefix. A prefix scan can only ever show that at least 34 distinct squares occur; it cannot rule out a 35th further out. These are now computed from the complete factor set of the infinite word at each length, so the count is exact in both directions. The older empirical figures agreed to within 0.1 percentage points, which validates the earlier engine rather than replacing it.
What the last row is and is not. Running the decision procedure to completion
re-derives Rao & Rosenfeld's Theorem 4 through their own machinery. It is a replication, not an
independent proof: it rests on their Propositions 5–8, which are Level 2, and on this
implementation being correct, which is Level 1. See MATH_CLAIMS.md rows 17–32 for
the full record, including the interpretive convention about empty blocks that the source leaves implicit.
Verifies that our 6 forbidden factors (baac, caab, abbc, cbba, accb, bcca) form exactly one closed transitive orbit under the symmetric group S₃ (all 6 letter permutations of {a, b, c}). Also launches an empirical Worker verification across all 6 relabeled alphabetical trees, generating a canonical traceHash to prove identical search behavior ("6/6 matched").
Generates the 59,049-letter prefix (N) of the uniform morphism h₆ fixed point (10 iterations from 'a') in a dedicated Web Worker and executes an O(1) prefix-sum audit across half-lengths K = 1 to 400.
Expected: Zero abelian squares observed in this prefix (N=59,049) and window (K ≤ 400). Note: The geometric map below highlights our exact inspected right-triangle domain { (i, K) : i + 2K ≤ N }. This is an empirical computational verification of string integrity (Level 1 Checksum OK), not an unconditional infinite theorem proof.
Applies the g₃ uniform morphism (length 10) to the h₆ word in a Web Worker, auditing the first 50,000 characters across half-length window K = 1..500.
Boundary Scan (K=1..5): Empirically observes 0 squares for K > 5 (N ≤ 196,830). Where squares occur at K ≤ 5 is tracked by our finite boundary scanner. The tool computes the Density Fade Ratio (Drop % vs. K-1), revealing how the morphism combinatorially narrows occurrences before reaching zero at K=6. The Audit Area Map below visualizes square density across the audited domain, with clickable samples that open directly in the Parikh Lens!
Executes strictly controlled, side-by-side empirical benchmarks in the Web Worker of AA2F (Mäkelä 2002, avoid K ≥ 2) versus AA2FR (Extension Lab, avoid K ≥ 2 plus FORBID4).
L12 Test Suite: Compares models across 10 frozen, independent AA2FR-valid initial seeds (L=12), reporting realized candidate nodes, depth curves sampled every 1,000 nodes, and a detailed stacked rejection breakdown (FORBID4 only, square only, both, plus minSquareK).
Projects the 6-letter cumulative Parikh trajectories of the h₆ fixed point onto its secondary eigenspaces (±√3). Because |±√3| < 3, the projection folds into a bounded, self-similar Rauzy Fractal (± 170 range over 177,147 letters).
2.5D Isometric Spiral Mode: Re-introduces the dominant linear growth eigenvalue (3) along the Z-axis, revealing how the fractal climbs as a continuous genomic helix.
Executes exhaustive DFS tree evaluation and visualizes search topology as an interactive radial Sunburst / Icicle map. In this layout, radius represents search depth (L) and angle represents descendant volume (360^∘ / N).
Welcome to the Morphic Art Gallery—a curated exhibition space designed to provide intuitive visual and intellectual nourishment for researchers and mathematicians. In accordance with our strict Mathematical Claims Protocol, every visual artwork displayed here is anchored by an immutable provenance badge detailing its exact computational derivation, algorithmic parameters, and epistemological status.
When the 6-letter uniform morphism h6 from the Rao–Rosenfeld construction (Theorem 4, arXiv:1511.05875) is iterated infinite times, its cumulative Parikh vector grows linearly along the dominant eigenvalue (3). By projecting the vector trajectory onto the 2D subspace spanned by the secondary eigenvalues (+√3 and -√3), linear drift is cancelled out, folding the trajectory into a bounded, self-similar geometric domain.
[Eigenspace Projection | N = 177,147 letters (3ˆ11) | Secondary Eigenvalues ±√3 | Bounded Area ±170]A side-by-side topological inspection of exhaustive ternary depth-first search up to length L=10. In this radial sunburst representation, radius denotes word length while angular slice width denotes the descendant budget (360^∘ / N). Dead-end terminations are colored by their exact mathematical rejection cause.
[Exhaustive DFS Tree | Depth L ≤ 10 | Exact Node Counts: 11,950 vs 4,498 | Dead-end ratio: 0.402% vs 0.800%] ⚙️ Exhaustive Checksum OKIn the Rao–Rosenfeld construction, the long abelian square avoidance of g3(h6ω(a)) is formally established by Theorem 9 (arXiv:1511.05875, §5.4). Our empirical boundary scanner reveals a striking regularity in the finite prefix N=200,000: exactly 100% of half-length 5 abelian square collisions occur across block image boundaries (i ≢ 0 (mod 10)). Because primitive morphic words exhibit uniform recurrence, short squares stabilize at positive stationary densities (ρ5 ≈ 4.3%), serving as a permanent combinatorial safety belt that blocks K > 5 squares.
[Empirical Boundary Scanner | Observed in N=200,000 prefix at half-length K=5 | 100% Boundary-Spanning (0% Internal)]An interactive experimental discovery engine that unifies macro-scale Parikh imbalance landscapes, meso-scale polyomino boundary seams, and micro-scale genealogical substitution atlases. When a collision or near-miss is selected, all observatory instruments synchronize instantly, tracing the exact cause-and-effect chain from macroscopic Parikh topography down to its ancestral morphic root.
[Integrated Observatory | N = 7,290 letters | Synchronized Topography & Genealogy]Runs bounded searches for short connecting words between two fixed morphism blocks, and keeps what has been checked separate from what has not. Nothing on this page is called a proof unless a standalone script, run independently of this page, has confirmed it.
[CERTIFIED] Provable asymptotic stability replicated for a computation it never ran — caught by a human reading the code, not by any automated check. It is fixed now, and the compute-budget counter below is real, browser-local telemetry, not decoration — but this module has produced no mathematical result recorded anywhere in this project's history other than the report on its own bug. Read the full account in The Graveyard →
In combinatorics on words, verifying abelian-square-freedom over infinite morphisms often stalls at block boundaries (seams). When two abelian-square-free blocks U and V are concatenated, Parikh vector collisions frequently emerge across the seam U · V.
Module 18 acts as an automated seam surgery laboratory. Instead of generating brute-force words from scratch, it executes bounded Depth-First Search (DFS) in Web Workers to discover minimal connecting words (bridges) that eliminate boundary collisions without introducing internal squares.
To achieve high-throughput scanning (hundreds of thousands of nodes/sec in browser memory), the engine relies on three algorithmic optimizations:
===).Follow a disciplined 3-step epistemological protocol:
Confirms the search finds zero collisions at period 6 and above before it is trusted with anything new.
Looks for a short word that removes boundary collisions between two fixed blocks, for periods 2 through 4.
Exhaustively checks that no ternary word longer than 7 avoids all abelian squares. Confirms the search reports zero when zero is correct.
Connect the search engine to open problems attributed to Matthieu Rosenfeld's PhD dissertation. When verification runs complete, the bounds shown here update automatically.
Provenance warning. Nobody in this project has opened the dissertation, so the problem statements below are unverified and their numbering may not match any published source. The SIAM J. Discrete Math. 32(4):2381–2397 (2018) reference formerly cited here has been rejected: the arXiv record for arXiv:1511.05875 shows Journal ref: (none), and no one has opened the publisher's page (MATH_CLAIMS.md row 23). Nothing else in this application depends on that reference — every verified claim rests on the arXiv preprint alone. Treat the problems below as prompts for investigation, not as citable statements.
Is there an abelian-square-free morphic word on 4 or 5 letters whose morphism has only 3 eigenvalues ≥ 1?
Status: OPEN (Spectral Matrix Search)What is the exact growth rate g(2) of ternary abelian-square-free words? Known bound: 5 ≤ g(2) ≤ 734.
Status: OPEN (ParikhFenwick Tree Target)What is the smallest alphabet size in Z2 for avoiding 2D additive squares?
Status: OPEN (2D Lattice Extension)When exploration concludes, results are separated into certified candidates and pruned branch trajectories. Click Try to break this on any verified candidate to launch an independent adversarial stress-test (10 randomized seed rounds).
[Seam Research Laboratory | Event-Driven Pulse | ParikhFenwickTree / Oracle Enabled]Browser sandboxes impose strict memory constraints (~2GB-4GB RAM) and throttle background threads when tabs are inactive or minimized. For intensive mathematical research, multi-hour parameter evaluations, or deep seam bridge searches across high-core server clusters (e.g., 32+ cores), download our self-contained, zero-dependency command-line runner.
nodejs.org).seam-hpc-cli.js and run-seam-search.bat into the same folder (e.g., Downloads or Desktop). Double-click run-seam-search.bat to open an interactive terminal menu without typing commands!node seam-hpc-cli.js --mode=weld --workers=8.In combinatorics, knowing what not to do saves millions of CPU hours. This laboratory visualizes the cognitive traps and algorithmic dead ends encountered during the search for abelian square-free words. These are not abstract theories, but empirical measurements that shattered our initial intuition.
Myth: "If we keep testing larger uniform morphisms (k = 7, 8, 9 …), the survival length grows, so we are getting closer to a solution!"
Reality: The growth is a statistical artifact of testing exponentially more morphisms. The maximum survival length perfectly follows y ≈ 2.29 · ln N. It is a wall.
Myth: "Algebraic morphisms are highly synchronized, so they must maintain a very tight Parikh imbalance (e.g. difference between max and min letter counts < 10)."
Reality: Exact measurements at length 25,379 prove the opposite. The expanding eigenvalue (|λ| = √3 > 1) forces the imbalance of g3(h6ω(a)) to diverge as √N.
Myth: "Factors like baac immediately lead to dead ends in all searches and must be hard-banned globally (FORBID4)."
Reality: They are only fatal in tight constraint boundaries. The 25,379-letter record word contains 2,820 occurrences of these so-called 'death factors'. Banning them makes finding the 25k word mathematically impossible.
Myth: "If we feed the 25k word to an AI or algorithm, we can extract the generating morphism and continue the word to infinity."
Reality: We measured the subword complexity p(k). A true morphism is highly restricted (e.g., p(15) = 144). The 25k word is maximum entropy (p(6) = 360): it explores every possible valid combination like a random walk. You cannot extract algebraic structure from maximum entropy noise.
Myth: "Rao & Rosenfeld already solved abelian squares for lengths K ≥ 6. Therefore, our scanner only needs to check K = 2…5."
Reality: Rao & Rosenfeld proved existence of a specific construction that avoids K ≥ 6. They did not prove that all candidate morphisms magically avoid them. By skipping the check, our scanner falsely reported trivial periodic words (like (aaabaac)ⁿ) as thousands of "successful" solutions.
(aaabaac)ⁿ with K ≤ 5: SUCCESS(aaabaac)ⁿ with K ≥ 6: FAILEDabaaca = aabaac)
Myth: "I can hand-craft a short 40-character example string that is abelian square-free just by making it look random."
Reality: The original version of this project shipped with a hand-crafted 40-character "aa2fr" example string. It was completely invalid. A rigorous scan revealed it contained 3 universal death factors and 3 pure abelian squares.
Myth: "The de Bruijn container relaxation that gave frequency bounds and SCC structure for abelian squares should also work as an elimination test for additive squares — keep growing the window until the container dies."
Reality: Container cost grows as |A|2k−1. It never died on any tested 4-letter alphabet at a reachable window size, while an exhaustive search on the true language finished in seconds for the same alphabets. Elimination turned out to be a search question, not a container question.
Myth: "A sound pruning oracle that removes 84–89% of search nodes when eliminating an alphabet should also help find longer record words on the alphabets that remain open."
Reality: Measured on two open alphabets at two budgets, pruned and unpruned search returned the exact same longest word every time. Branch-and-bound only cuts a branch that provably cannot beat the current best — and when the language never runs out, the best keeps growing, so nothing is ever provably beaten.
Myth: "A pruning table worth roughly 85× on search nodes should make any single search dramatically faster."
Reality: Building the table walks essentially the same tree as the unpruned search itself — 751,156 nodes to search one alphabet directly versus 725,960 nodes to build its table. Net gain for one run: 1.00×. The entire value is in reusing the table later, never in the run that built it.
Myth: "To maximize independence when a second model verifies a result, forbid it from using graphs, automata, or dynamic programming — that guarantees a structurally different implementation."
Reality: The ban forced exhaustive generation, whose cost grows as |A|N. It could check words up to roughly N = 10 while the results needing verification sit at length 50–62. It worked flawlessly on everything within reach — and could never reach the result it existed to check. Independence should vary the algorithmic idea at the same performance class, not cripple the performance class itself.
Myth: "A search-engine summary of an unopened paper is a usable lead, as long as it is labelled 'untraced' so nobody cites it directly."
Reality: The label prevented citation but not influence. A fabricated author name and figure from a summary sat on the critical path of two working sessions before the actual source was opened and the name turned out not to appear in it anywhere. A summary is noise until it is located inside a document someone has actually read.
Myth: "An alternating search-order technique reported to work spectacularly for additive cubes should transfer directly to this project's own record hunt."
Reality: Tested first in the project's own search, the technique looked useless. A controlled test in the setting it was actually reported for reversed that completely — and the technique still lost when honestly re-tried on the project's real target. A borrowed method has to be tested first where it came from, or the transfer gets measured instead of the method.
Myth: "When a withdrawn citation is found in an independent bibliography with matching volume, issue, pages, and year, it is corroborated and the retraction can be lifted."
Reality: The DOI — the one field never actually compared — turned out not to exist at all. Four correct fields felt like confirmation; the unchecked fifth field was the wrong one. Corroboration only covers the fields actually compared, never the record as a whole.
Myth: "Watching how the best result found so far grows across increasing computation budgets — flattening versus still climbing — should predict which of several open cases is closest to being resolved, so compute can be spent there first."
Reality: Run across 20 open cases at three budget scales, the growth shape was pure noise: no visible relationship to any other property of the cases. A three-point curve cannot separate real structure from sampling variance. The diagnostic was set aside in favour of a method that can actually settle a question either way, rather than one that only describes how a search is currently struggling.
Myth: "An AI agent can generate genuinely novel mathematical breakthroughs (like applying MCTS or Roth's theorem) if you just ask it for fresh ideas."
Reality: AI training data is built on linguistic and semantic links, not mathematical invariants. When asked for unconstrained new ideas, it produces flawed analogies (e.g., applying Roth's theorem on sets to prefix sums which are sequences) and circles the drain by renaming the same flawed finite-search approach with new terms (Toeplitz mining, MCTS, blind DFS) without understanding the mathematical kill condition is identical for all of them. The solution is to use AI strictly as an execution engine for finite tests where every idea has an explicit 5-minute kill condition.
Myth: "Keränen's g85 is perfectly abelian-square-free on 4 letters. Since the 3-letter problem allows trivial squares (K=1), we can just project g85 down to 3 letters (by merging two letters) and the resulting squares might only be the allowed K=1 trivial ones."
Reality: All 36 possible surjections from 4 letters to 3 were generated and tested up to the second iteration (length 7225). Every single one failed immediately at K=2. Keränen's structure fundamentally relies on cross-leveraging all 4 degrees of freedom. It cannot be compressed into 3 dimensions while maintaining balance, proving this projection route is a complete dead end.
Myth: "The original g3 morphism fails on K ∈ [2,5] but works perfectly for K ≥ 6. We can just use a CEGIS-style search to find a new, pure 10-uniform morphism that fixes the small squares."
Reality: A backtracking DFS with on-the-fly pruning was executed to search the 360 space of 10-uniform morphisms, relaxing the constraint to only avoid K ∈ [2,5]. The search bounded-exhausted at 500 million nodes, reaching depth 59/60 but perpetually backtracking without a single survivor. The space is overwhelmingly hostile. The original g3's success for large K is a mathematical anomaly that cannot be found by clean-slate uniform searching.
This trap is about Module 18 itself — the tab you are reading this on. seam-hpc-cli.js, the standalone script this module tells visitors to download, once printed a banner reading [CERTIFIED] Provable asymptotic stability replicated on every run, unconditionally. An audit found three independent bugs at once: (1) the certifying mode never loaded the actual morphisms, so it audited nothing and printed a hardcoded zero; (2) the weld mode's own default input (g3's first block) contains a small abelian square by construction, making the default run structurally unable to test what it claimed to test; (3) the search itself pruned nothing, making it slower than the tested function it was supposed to replace.
Why this belongs here rather than being quietly fixed and forgotten: the claims ledger (MATH_CLAIMS.md) audits documents, not running programs — a script's own stdout answers to nobody by default. This is the one time in the project's history that gap was actually exploited, even if accidentally rather than maliciously. It was caught by a human reading the code, not by any automated check, and two permanent drift checks were added afterward so no program in this repository may print the words certified / provable / proven again. The bug is fixed and the checks now pass — the "Provenance Chain" and compute-budget counter above are real, browser-local telemetry, not decoration. But this module has produced no other mathematical result in the project's history, which is itself worth knowing before trusting a "Certified" badge anywhere, on this page or any other. Full account: NEGATIVE_RESULTS.md §21, MATH_CLAIMS.md row 26.