Let s(w) count the distinct square words occurring among the rotations of a word w of length n, equivalently the distinct square factors of ww whose total length is at most n. Li and Song's Main Theorem gives s(w) <= 5n/3 for every finite word w. Since s(w) is integral, this also gives s(w) <= floor(5n/3). The result leaves the three-halves target open.

Every length-n word is globally equivalent under alphabet renaming to one restricted-growth string, so the Bell-number layer covers arbitrary finite alphabets without fixing an alphabet size. Two independent C++17 programs enumerate every restricted-growth string through n=14. One encodes square factors in 64-bit keys; the other compares factor text directly. Their normalized outputs agree byte for byte.

For n=1,...,14, the exact maxima are [0,1,1,2,3,4,4,6,6,9,8,10,11,13]. The corresponding maxima restricted to primitive words are [0,0,1,2,3,3,4,6,6,7,8,10,11,11]. For nonprimitive words they are undefined at n=1 and then [1,1,2,2,4,3,6,4,9,5,9,6,13]. A separate Python enumerator canonicalized alphabet renaming, rotation, and reversal through n=12 and returned the same total maxima. Every unrestricted maximum through n=14 has a binary witness. These finite values lie below both the floor and ceiling three-halves targets and do not settle the universal question.

Fixing the first bit to 0 identifies each binary word with its complement and still covers every binary word. Two independent programs exhaust this space through n=26. The total maxima for n=1,...,26 are [0,1,1,2,3,4,4,6,6,9,8,10,11,13,13,16,15,18,17,19,21,22,22,25,24,28].

The maxima restricted to primitive words are [0,0,1,2,3,3,4,6,6,7,8,10,11,11,13,15,15,17,17,19,21,20,22,24,24,26]. For nonprimitive words they are undefined at n=1 and then [1,1,2,2,4,3,6,4,9,5,9,6,13,9,16,8,18,9,19,13,22,11,25,14,28].

The maxima restricted to square factors of total length at most n/2 are [0,0,0,2,2,2,2,4,4,5,5,9,8,10,9,12,12,13,13,16,16,17,18,19,20,20]. The complementary long-square maxima are [0,1,1,2,2,3,3,4,4,6,7,6,7,9,11,10,11,14,13,13,17,16,17,20,19,20].

At n=16 and n=24, the exact total maxima 16 and 25 are attained by rotations of the k=0 and k=1 words in Amit and Gawrychowski's 2017 family. Their Lemma 1 gives 10k+16-(k mod 2) squares at length 8k+16. The region beyond n=14 is binary-only and makes no claim about larger alphabets.

Charalampopoulos, Mohamed, Radoszewski, Rytter, Waleń, and Zuba construct words from Fibonacci-word blocks and prove that CS(n) >= 1.5n - O(sqrt(n)) for infinitely many n. Here CS(n) is the maximum number of distinct squares of total length at most n in a circular word of length n. This is the strongest checked lower construction and supplies the source for the conjectured coefficient.

The live statement at revision tdbcr1:fd17647632161b23903c494cbd3f7f8404aec50a36197b28c3f7fc74e31db5fd asks s(w) <= ceil(3n/2). The CPM 2026 Final Remarks state the conjecture as CS(n) <= 1.5n. Because CS(n) is integer-valued, that source statement is equivalent to CS(n) <= floor(3n/2). The two versions differ by one when n is odd. Li and Song also describe the conjectured sharp upper bound as 1.5n and prove 5n/3.

The factor convention agrees across the sources and the live problem: a circular square is a square factor of a rotation, equivalently a square factor of ww of total length at most n. The dated search checked the CPM DOI page, PDF, and experimental HTML; arXiv:2605.12215 and its submission history; arXiv:1708.00639; exact-title and formula searches; DataCite metadata; the live TheoremDB directory; orient; and research search. TheoremDB contained one canonical target and zero attached research records. No later revision or later strengthening beyond 5n/3 appeared in the checked sources.

This mismatch blocks a universal resolution claim against the imported provenance until review aligns the canonical convention. The bounded computations in this packet state their literal finite regions and are valid under either convention.

A direct induction on word length would seek a deletion that loses at most the one- or two-unit increment in ceil(3n/2). This route fails on an infinite family.

Let u=a_0...a_{k-1}, where the k letters are pairwise distinct, and let w=u^2. In the circular word w, equality of letters h positions apart forces k to divide h. A square factor has half-length h<=k, so h=k. The k rotations of u^2 are therefore the complete square list and s(w)=k.

Delete either occurrence of a_i and rotate the remaining circular word into x a_i x, where x is u with a_i removed. A square containing a_i would contain that sole circular occurrence once, which is impossible because its two halves have equal letter counts. A square avoiding a_i lies in the block xx across the circular seam. The distinct letters of x force its half-length to be k-1, so xx is the only square. Every deletion leaves square count one. The best deletion loses k-1 squares, an unbounded quantity.

The exhaustive structural sweep first violates the parity-sensitive ceiling induction at w=012012, where s(w)=3 and every deletion has count one. It first violates every two-loss deletion rule at w=01230123, where s(w)=4 and every deletion has count one.

A tempting split would bound squares of total length at most n/2 by ceil(n/2), then charge the remaining long squares separately. Exact binary enumeration refutes the short part at n=12. The first counterexample in mask order is w=010100100000, with seven short squares: 00, 0000, 0101, 1010, 000000, 010010, and 100100.

The exact binary maximum at n=12 is nine, attained by w=011011010010. Its complete short-square list is 00, 11, 1010, 001001, 010010, 011011, 100100, 101101, and 110110. Each listed word has total length at most six. The structural sweep over arbitrary alphabets confirms nine as the all-alphabet maximum for this short region at n=12.

First align the live floor or ceiling convention with the cited conjecture. Then focus the proof search on primitive words. Li and Song's Lemma 9 already gives the three-halves bound for nonprimitive words, and their Lemma 10 gives it when every high-order Rauzy-graph basis contains a circuit of size at least n/2. The five-thirds loss occurs in the split branch handled by Lemma 13, where a primitive circuit root p has n/4 <= |p| <= n/3.

A concrete next experiment should build Gamma_m(ww) for the exact primitive maximizers in this packet and for targeted binary words through a larger bounded range. For every split, record n, m, |p|, |Class_p(ww)|, |O_p(ww)|, the cyclomatic number, and the large circuits forced into every basis. Test whether a second circuit class or unused odd-power count supplies an additional n/6 charge in the Lemma 13 branch. Stop with either a smallest profile refuting each proposed charge or a finite list of inequalities that covers every observed split type. Every basis claim needs a certificate or an independently replayable graph computation.

The program recursively generates restricted-growth strings. This gives one representative for every word under global alphabet renaming. It tests each circular factor at every start and even length, stores each square word in a length-tagged 64-bit encoding, and separates primitive words using exact period divisibility. The largest layer contains Bell(14)=190,899,322 strings. The run uses exact comparisons and no random choices.

This replay generates the same complete restricted-growth space, then stores square factors by start and length and checks duplicates through direct character comparison. It shares no packed-factor representation with the first program. After field-name normalization, the two 14-line outputs have the same SHA-256 digest.

The program fixes the first bit to 0, which selects one member of each complement pair. It checks every circular start and every even factor length. Distinct factors use a length-tagged 64-bit key. Exact period tests separate primitive and nonprimitive words. At n=26 it examines 33,554,432 words. It also emits the first counterexample to the proposed short-square half-length charge.

This program represents each word as a string, checks square halves with string comparison, and deduplicates factors by direct substring comparison. Exact string periods separate primitive and nonprimitive words. It independently reproduces the total, short, long, primitive, and nonprimitive maxima. Its raw 26-line output equals the normalized rows of the packed replay.