# Canonical statement and exact scope

UnsolvedMath v1.6.0, commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e,
record30006529, OWR-14299904-011. Normalized statement SHA256:
f74365d6e3ccb7962f5a3dde5fa941b03c2fd785737d087fb9e7492f1612e61e.

Primary source: Guillaume Chapuy, Counting w-trees, OWR2/2026,
printed pp145-146, DOI10.4171/OWR/2026/2. Page146 was visually
checked in the original PDF. A w-tree has exactly ONE CYCLIC POINT,
not merely one cycle or one weak component. Composition follows
w=w1 o ... o wk, so application order is right to left.
The source expressly requests AB. The 2022 LPSM slides expressly
propose AAB. Both cases, and the whole singleton-letter family,
are completely counted by this manuscript.

The general word-tree question is NOT closed. Neither the TV
question nor Feray's adjacent permutation cycle conjectures,
accidentally bundled into this frozen record, is resolved here.
One source-explicit AB case and one auxiliary-explicit AAB case
are answered within ONE manuscript and ONE complete scoped result.
Zero whole frozen-record closures. The independent letters in the
generalization may exceed two; binary specialization is A^rB.

# Detailed mathematical self-audit

1. Weighted Cayley: sum rooted-tree weights at root i is p_i for
n>=2, so adding its loop yields p_i^2. Zero weights are allowed.
n=1 is checked separately. Pitman Eq(27) is exactly the same
classical identity; no novelty claim is attached to it.
2. For fixed g, independent uniform b values give iid pushforward
probabilities s_i/n. The conditional count is n^(n-2) sum s_i^2.
No independence is assumed between different values of g.
3. Simultaneous conjugation invariance makes every distinct
input pair have the same collision probability. Diagonal pairs
contribute n, off-diagonal pairs n(n-1)c_V. This proves the count.
4. The application sequence is the reversal of composition notation.
The checker predicted_count reverses the core, and extra nonpalindromic
colored-core checks prevent an unnoticed convention error.
5. Reveal all transitions from start0 then start1, preserving the
same partial table. Old input/color entries are never independently
resampled. Every actual map tuple has one canonical leaf.
6. Labeling a leaf uses (n-2)_(v-2) injective labels, without
automorphism divisors. Distinct input/color values are independent;
the prescribed event has probability n^-e. Unrevealed entries are
free. Summation neither omits nor double-counts paths.
7. The set-partition formulation independently enumerates all
position equalities, keeping starting blocks distinct and enforcing
functional consistency. It reproduces coefficients for15 cores
and total mass1 at9 distinct n values.
8. For AA, coincident first images contribute1/n; the two distinct
pairs (1,2),(2,1) contribute0; every other distinct pair contributes
conditional1/n. This includes n=2 and proves the displayed polynomial.
9. Cyclic rotation preserves the number of cyclic points by injective
intertwining on finite cyclic sets. This does NOT claim equality of
the whole pushforward distributions. Positive powers preserve the
cyclic set, not necessarily its permutation action.
10. AB bipartite encoding is a bijection with a spanning tree of
K_n,n with one marked edge. Unique cyclic point means unique2-cycle
in the bipartite graph, including connectivity of both copies.
11. Exact exhaustive diagnostics check full map tuples, not samples.
There are93 independently computed colored-collision cases, using
2054711 complete map-pair evaluations; the paper's main proof is
analytic/combinatorial, not justified by finite testing alone.
12. Laurent-polynomial conclusion follows from finite falling
factorials. Word-dependent state space can grow quickly; no
polynomial complexity in word length or arbitrary-word closure.
13. This is the authoring system's self-audit, not independent expert
review or proof-assistant verification. Full proof of the stated
scope is accepted; the wider dataset record stays partially solved.

# Bounded primary-source literature audit

Read OWR2/2026 pp145-146 and its definition of cyclic points.
Chapuy-Perarnau, arXiv2207.14108v2, Short Synchronizing Words for
Random Automata, introduces w-trees and gives asymptotic methods;
its introduction records the difficulty of exact enumeration.
The final journal DOI is10.1145/3736722. The inspected full text
is the public arXiv version, not a claimed line-by-line journal audit.
Chapuy's LPSM2022 lecture, PDF page48 (slide48), explicitly suggests
AAB. Web-accessible PDF text was read. A local HTTPS download failed
certificate-chain verification and the web screenshot was unavailable;
no certificate check was disabled and no visual inspection of this
slide is claimed. The independently visualized OWR AB statement is
the primary source target.

Martinsson, arXiv2306.09040v2, Section3, proof of Theorem1.2:
conditional on a, a^r o b has iid images with the fiber-size
distribution. This conditioning idea is PRIOR WORK and explicitly
credited; that paper studies synchronization bounds and p-mapping
statistics, not the exact singleton-letter tree formulas here.
Pitman, Random mappings, forests, and subsets associated with
Abel-Cayley-Hurwitz multinomial expansions, SLC46(2001),B46h,
Eq(27) in the inspected Berkeley report593: the exact cyclic-set
law gives P(one cyclic point)=sum p_i^2. This is CLASSICAL.
The manuscript also gives its weighted Prufer derivation.
The complete-bipartite spanning-tree formula is classical and
credited, with its matrix-tree derivation summarized.

Chapuy-Perarnau, Composition of random functions and word
reconstruction, arXiv2603.28936v1: recent word statistics,
length/exponent recovery and TV-separation. Theorems and introductory
comparison were checked; no full general enumeration resolution
or identical singleton-letter formulas were found there.

Bounded public searches on26OctoberUTC/27Octoberlocal2026 used:
w-trees exact enumeration Chapuy; aab trees Chapuy formula;
random functions 2n-1 tree; composition random mappings tree
enumeration; random mapping collision probability iterates exact.
No identical earlier exact AB/AAB result was found in this search.
This does NOT prove absolute priority. The arguments are elementary
consequences of known tools; they are presented as exact answers
to the expressly posed cases, not as new underlying mapping theory.
Third-party source PDFs are preserved locally and excluded from uploads.