Vukusic states the discrete question explicitly and calls it open, then proves a Lebesgue-integral analogue. Andrade and Mol, revised in February 2025, also state that the finite integer problem remains unknown and distinguish the known Z^2 construction. Popoli, Shallit, and Stipulanti give the same status in FSTTCS 2024. A dated search on 2026-07-28 checked these papers, the original formulations, exact-title queries, citation indexes, and nearby additive-power work. This record is a dated literature report and makes no claim about sources published after the search.

After normalizing by translation and gcd and identifying reflections, {0,1,2,4} is the first primitive four-letter alphabet outside the balanced family when alphabets are ordered by maximum letter and then lexicographically. Two independent exact DFS implementations closed its full prefix tree. Each visited 19,097,778 nodes and 5,350,440 terminal leaves. The deepest level is 62 and contains two words. One is 02012404142012421014102010242102414240424120241210212402414204; the other is its reversal, 40241420421201214202142404241420124201020141012421024140421020. This result concerns one finite alphabet. It leaves the existence question over other finite alphabets open.

Two adjacent blocks that are anagrams have identical sums under every numeric labeling. Additive-square avoidance therefore implies abelian-square avoidance. Andrade and Mol summarize the sharp abelian result: Keranen constructed an infinite abelian-square-free word on four letters, and three letters are insufficient. The implication gives the same four-letter lower bound for any positive answer to the additive-square problem.

This packet uses balanced family as shorthand for normalized four-letter alphabets A={0<a<b<c} satisfying 0+c=a+b. Freedman and Brown determine g(A), the maximum length of an additive-square-free finite word, for this family. For the ten primitive normalized representatives with c<=8 after identifying A with c-A, the ordered maxima are 50,55,55,55,60,60,60,58,60,60 for 0123,0134,0145,0235,0156,0167,0257,0347,0178,0358.

Let h6 map a->ace, b->adf, c->bdf, d->bdc, e->afe, and f->bce. Give a,b,c,d,e,f the respective Z^2 weights (0,0),(1,1),(2,1),(0,1),(2,0),(1,0). Rao and Rosenfeld use the augmented map with leading coordinate 1 to enforce equal block lengths, and prove that h6^omega(a) contains no additive square for these vector weights. This theorem is over Z^2 and supplies candidate one-dimensional projections.

The sweep enumerated every A={0<a<b<c} with gcd(A)=1, c<=8, and c=a+b, then kept the lexicographically smaller of A and c-A. For each of ten representatives, a prefix-sum DFS visited every additive-square-free prefix until the search tree closed. The maxima, node counts, leaf counts, maximizer counts, and maximizer digests are recorded in the linked artifact. The compact six-field row digest was independently matched by a second operator implementation.

For each primitive pair (p,q), identify (p,q) with (-p,-q) by choosing p>0 or p=0,q>0, and project a vector (x,y) to px+qy. There are 1,216,768 normalized directions with 0<=p<=1000 and |q|<=1000. At prefix 200,000, the screen had marked 363,649 directions and left 853,119 survivors; the smallest survivor (361,288) later failed at end 250,952. At 300,000 symbols, the smallest survivor (376,-217) later failed at end 357,217. At 357,217 symbols, the smallest survivor (376,27) later failed at end 378,367. The final run checked 35,790,396,672 adjacent block pairs, marked 560,467 directions, and left 656,301 finite-prefix candidates. Its smallest survivor is (383,37).

The direction maps the six h6 letters a,b,c,d,e,f to 0,649,1010,288,722,361, giving the sorted primitive integer alphabet {0,288,361,649,722,1010}. An exact end-major, half-length-minor scan found the first equal-sum adjacent blocks at zero-based interval [148708,250952). Each block has length 51,122 and sum 25,859,725. The full scanner tested 15,744,152,222 pairs through that witness. An independently written continuation scanner checked ends after 200,000 and matched the endpoint, start, half-length, and sum.

Rao and Rosenfeld give the h6 incidence eigenvalues as 0 with algebraic multiplicity 3, together with 3, sqrt(3), and -sqrt(3). Thus the expanding eigenspace E_e(M_h6) has dimension 3. A scalar additive-square test uses Phi(letter)=(1,p*x(letter)+q*y(letter)), a linear map into a two-dimensional space. The restriction of Phi to the three-dimensional expanding eigenspace has a nonzero kernel by rank-nullity. Hence E_e(M_h6) intersects ker(Phi) nontrivially for every scalar direction (p,q), so the hypothesis of their finite-ancestor decision theorem fails. This blocks direct use of that sufficient theorem on h6. Other methods may still decide individual projections.

The canonical statement, revision, integrity state, prior-art search, live research graph, and duplicate candidates were read before computation. Exact-title and formulation searches found one TheoremDB target and no attached projects or records. The source audit checked the 2025 status papers, the Rao-Rosenfeld Z^2 construction, the Freedman-Brown four-letter finite maxima, Brown's bounded-error approximation theorem, and Halbeisen-Hungerbuhler's unequal-length equal-sum theorem and final same-length question. Each nearby result has a different quantifier, codomain, or equality condition.

Search primitive prolongable morphisms on four to six letters together with distinct normalized integer weights. The first bounded batch uses uniform image lengths 2 through 6 and weights between 0 and 20, with letter-permutation, translation, gcd, and reflection symmetries removed. A SAT or constraint model should reject additive squares in iterated prefixes and require no incidence eigenvalue of absolute value 1. Retain candidates whose expanding eigenspace has dimension at most 2 and on which Phi(letter)=(1,weight(letter)) is injective. For each survivor, use exact algebraic arithmetic to verify the spectral condition, enumerate the finite forbidden-template ancestor set, and check the theorem's finite factor bound. Save the symmetry quotient, candidate morphism, weight map, linear-algebra certificate, template set, and checker output. A passing certificate would answer the canonical problem.

Save source_lines as additive_square_sweep.py beside the linked additive_square_search.py artifact. The full stable digest covers every result field except runtime_seconds. The compact digest names six cross-implementation fields and is the comparison digest shared with the independent replay.

The recursive invariant is that every stored prefix is additive-square-free. Appending a letter can create a new forbidden factor only at the new final position, so checking each possible suffix half-length is complete. A closed run therefore visits every valid word exactly once, including the empty root. The command for {0,1,2,4} completed below the 30,000,000-node guard.

The implementation uses a separate word representation and suffix-checking loop. It has a one-billion-node guard and a 300-second guard, and the tree closes far below both. The stable output digest removes runtime_seconds and preserves all other fields. A second operator reran the same tree in 0.71 seconds and matched every mathematical field.

The source generates h6^omega(a), builds exact x and y prefix sums, and visits each adjacent equal-length block pair once. For a difference vector (dx,dy), precisely the projection direction proportional to (dy,-dx) makes the scalar sums equal. Dividing by the gcd and applying the sign convention gives one array index. The program aborts if it encounters a vector additive square, since such a pair defeats every projection.

The command below tests (361,288) with a one-million-symbol limit. Prefix sums make every block comparison exact. End positions increase first, followed by half-lengths, so the returned end is the first endpoint containing any scalar additive square in the fixed point prefix.