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----
-title: "Index-Period Normal Forms for Monoid-Aggregated Recursive Summaries"
-subtitle: "Exact Pumping, Canonical Representatives, and Computable Test Families"
-date: 2026-05-16
-abstract: >
- A monoid-aggregated summary evaluates a finite rooted cop-labeled tree
- bottom-up through a finite state set and a finite commutative
- child-aggregation monoid. Once the multiplicity observation map and the
- monoid are fixed, context equivalence has finite index and is exactly
- equality of a finite behavior vector. This note sharpens the resulting
- pumping and normal-form theory: the crude pigeonhole bound in the product
- monoid is replaced by an exact index–period bound on each behavior type's
- child contribution, isolating support, modular, and saturation counting in
- the Boolean, cyclic, and threshold families. Combining exact sibling
- pumping with a size-minimality argument — no behavior vector may repeat
- along a root-to-leaf path — yields a finite universe of normal
- representatives, and an external tie-break selects one canonical
- representative per class. Worked computations for one-node trees, stars,
- unary chains, and split-versus-concentrated examples make the bounds
- concrete.
-tags:
- - research
- - research/mathematics
- - research/algebra
- - research/graph-theory
-authors:
- - "Levi Neuwirth | /me.html"
-no-collapse: true
-status: "Working model"
-confidence: 80
-evidence: 4
-peer-status: unreviewed
-result-shape: positive
-history:
- - date: 2026-05-16
----
-
-## Purpose and executive diagnosis
-
-The fixed-resource monoid-aggregated model gives a genuine finite-index
-theory, but the first normal-form bound is far too coarse if stated only as a
-[pigeonhole](https://en.wikipedia.org/wiki/Pigeonhole_principle) bound in a
-huge product monoid. The correct next move is to
-analyze, for each behavior type, the cyclic submonoid generated by its child
-contribution. This gives an exact index-period pumping rule.
-
-The result is a more useful theory. Sibling multiplicities reduce by
-explicit index–period normal forms; the Boolean, cyclic, and threshold
-monoids acquire transparent pumping signatures; fixed-resource equivalence
-classes gain finite normal representatives; canonical representatives exist
-after a harmless external tie-break; and the example computations become
-concrete rather than schematic.
-
-There is also an important algebraic correction. One should not assume that
-every [finite commutative monoid](https://en.wikipedia.org/wiki/Monoid) is a
-[semilattice](https://en.wikipedia.org/wiki/Semilattice) of [abelian
-groups](https://en.wikipedia.org/wiki/Abelian_group). That
-statement holds for special regular/Clifford-type commutative monoids, not
-for arbitrary finite commutative monoids. Threshold monoids already contain
-[aperiodic](https://en.wikipedia.org/wiki/Aperiodic_semigroup) saturation
-behavior that is not group-like. The universal
-finite-monoid fact needed here is simpler: for each element $g$ of a finite
-monoid, the sequence
-
-$$
-0,\; g,\; 2g,\; 3g,\; \ldots
-$$
-
-is ultimately periodic.
-
-**Main principle.** For fixed resources, the relevant algebra is not a global
-decomposition of the whole monoid. It is the index-period decomposition of
-the cyclic submonoid generated by each realized child-contribution element.
-
-## The fixed-resource model, recalled
-
-This section repeats the definitions needed for the present note. The
-conventions are unchanged from the finite-resource foundations note.
-
-::: {.annotation .annotation--static #def-rooted-tree}
-
-Definition 2.1
-Rooted cop-labeled tree
-
-
-
-A *rooted cop-labeled tree* is a finite rooted unordered tree $T$ with root
-$\rho_T$ together with a multiplicity function
-
-$$
-m_T : V(T) \to \mathbb{N}.
-$$
-
-Sibling order is not part of the structure.
-
-
-
-A *rooted one-hole context* $K[\square]$ is a finite rooted cop-labeled tree
-with one distinguished subtree slot. If $X$ is a rooted cop-labeled tree, then
-$K[X]$ is obtained by plugging $X$ into the slot. Contexts compose, and the
-empty context is $E[\square] = \square$.
-
-
-
-A *finite resource datum* is a tuple
-
-$$
-\mathcal{R} = (A, \mu, S, M, \oplus, 0_M)
-$$
-
-where:
-
-1. $A$ is a finite multiplicity alphabet;
-2. $\mu : \mathbb{N} \to A$ is a fixed multiplicity observation map;
-3. $S$ is a finite state set;
-4. $(M, \oplus, 0_M)$ is a finite commutative monoid.
-
-
-Warning 2.4
-Actual resources, not just cardinalities
-
-
-
-For the clean fixed-resource theory, $\mu$ and $(M, \oplus, 0_M)$ are part of
-the resource datum. Fixing only $|A|$ would allow infinitely many exact
-multiplicity tests by varying $\mu$. Fixing only $|M|$ still leaves only
-finitely many monoid structures on a fixed finite set, but the pumping
-constants depend on the actual operation. Therefore all sharp statements
-below are parametrized by the actual resource datum $\mathcal{R}$.
-
-
-
-A *monoid-aggregated summary* over $\mathcal{R}$ is a pair
-
-$$
-P = (\alpha_P, f_P)
-$$
-
-with
-
-$$
-\alpha_P : S \to M, \qquad f_P : A \times M \to S.
-$$
-
-It evaluates a rooted tree bottom-up by
-
-$$
-P(T_v) = f_P\!\left( \mu(m_T(v)),\; \bigoplus_{u \text{ child of } v} \alpha_P(P(T_u)) \right),
-$$
-
-where the empty sum is $0_M$. The root value is denoted $P(T)$.
-
-
-Lemma 2.7
-Crude cardinality of the summary class
-
-
-
-The number of syntactic summaries over $\mathcal{R}$ is
-
-$$
-|D(\mathcal{R})| = |M|^{|S|} \cdot |S|^{|A||M|},
-$$
-
-where equality means syntactic equality of pairs $(\alpha, f)$. The number of
-extensionally distinct summaries is at most this quantity.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Crude cardinality of the summary class" data-exhibit-type="proof" data-exhibit-caption="Count the choices of α : S → M and f : A × M → S independently."}
-
-:::: exhibit-body
-There are $|M|^{|S|}$ choices of $\alpha : S \to M$ and $|S|^{|A||M|}$ choices
-of $f : A \times M \to S$. [□]{.proof-qed}
-::::
-
-:::
-
-## Behavior vectors and fixed-resource equivalence
-
-::: {.annotation .annotation--static #def-behavior-vector}
-
-Definition 3.1
-Behavior vector
-
-
-
-The *$\mathcal{R}$-behavior vector* of a tree $T$ is
-
-$$
-\beta_{\mathcal{R}}(T) = (P(T))_{P \in D(\mathcal{R})} \in S^{D(\mathcal{R})}.
-$$
-
-We write
-
-$$
-B_{\mathcal{R}} := S^{D(\mathcal{R})}
-$$
-
-for the finite set of *formal* behavior vectors. A vector $b \in
-B_{\mathcal{R}}$ is *realizable* if $b = \beta_{\mathcal{R}}(T)$ for some tree
-$T$.
-
-
-Theorem 3.3
-Fixed-resource equivalence is behavior-vector equality
-
-
-
-For all rooted cop-labeled trees $X, Y$,
-
-$$
-X \sim_{\mathcal{R}} Y \iff \beta_{\mathcal{R}}(X) = \beta_{\mathcal{R}}(Y).
-$$
-
-Consequently $\sim_{\mathcal{R}}$ has finite index, with at most
-$|S|^{|D(\mathcal{R})|}$ classes.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Fixed-resource equivalence is behavior-vector equality" data-exhibit-type="proof" data-exhibit-caption="Single-summary context equivalence is root-state equality; intersect over all summaries."}
-
-:::: exhibit-body
-For a single fixed summary $P$, context equivalence is exactly equality of
-root state: if two inserted trees have the same root state, the computation
-above the hole is identical; conversely, the empty context detects root-state
-inequality. Intersecting over all $P \in D(\mathcal{R})$ gives precisely
-equality of all coordinates of $\beta_{\mathcal{R}}$. Since $B_{\mathcal{R}} =
-S^{D(\mathcal{R})}$ is finite, the finite-index bound follows.
-[□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #rem-behavior-type}
-
-Remark 3.4
-Behavior type
-
-
-
-In this note a *behavior type* means an element of $B_{\mathcal{R}}$, usually
-a realizable one. Two trees have the same behavior type exactly when they are
-$\sim_{\mathcal{R}}$-equivalent.
-
-
-
-If $X \sim_{\mathcal{R}} Y$, then for every rooted one-hole context
-$K[\square]$,
-
-$$
-K[X] \sim_{\mathcal{R}} K[Y].
-$$
-
-Equivalently, replacing a subtree by another subtree with the same
-$\mathcal{R}$-behavior vector preserves the $\mathcal{R}$-behavior vector of
-the whole tree.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Fixed-resource congruence" data-exhibit-type="proof" data-exhibit-caption="The inserted subtree is seen above the hole only through its single root state, which agrees for every summary."}
-
-:::: exhibit-body
-By [Theorem 3.3](#thm-behavior-vector), $X \sim_{\mathcal{R}} Y$ means
-$\beta_{\mathcal{R}}(X) = \beta_{\mathcal{R}}(Y)$. In the bottom-up evaluation
-of any summary $P \in D(\mathcal{R})$ on $K[X]$ or $K[Y]$, the inserted
-subtree is seen above the hole only through the single state $P(X)$ or
-$P(Y)$. These states agree for every $P$, so the computation above the hole
-agrees for every $P$. Applying [Theorem 3.3](#thm-behavior-vector) again gives
-$K[X] \sim_{\mathcal{R}} K[Y]$. [□]{.proof-qed}
-::::
-
-:::
-
-## The product contribution monoid
-
-Sibling pumping is most naturally stated in a product monoid that tracks all
-summaries simultaneously.
-
-::: {.annotation .annotation--static #def-product-monoid}
-
-Definition 4.1
-Product monoid
-
-
-
-Let $M^{D(\mathcal{R})}$ denote the product monoid of $D(\mathcal{R})$ copies
-of $M$ — equivalently, the set of functions $D(\mathcal{R}) \to M$ — with
-coordinatewise operation, also denoted $\oplus$, and zero element $(0_M)_{P
-\in D(\mathcal{R})}$.
-
-
-Definition 4.2
-Contribution element of a behavior type
-
-
-
-For a formal behavior vector
-
-$$
-b = (b_P)_{P \in D(\mathcal{R})} \in B_{\mathcal{R}},
-$$
-
-define its *product contribution element*
-
-$$
-\gamma_b \in M^{D(\mathcal{R})}
-$$
-
-by
-
-$$
-(\gamma_b)_P := \alpha_P(b_P).
-$$
-
-Thus $\gamma_b$ is the simultaneous child contribution made by a child
-subtree of behavior type $b$ to every summary $P \in D(\mathcal{R})$. This
-definition also makes sense for formal, non-realizable behavior vectors; only
-realizable vectors occur as actual child types in trees.
-
-
-
-The symbols used below are as follows: $B_{\mathcal{R}} = S^{D(\mathcal{R})}$
-is the set of formal behavior vectors; $M^{D(\mathcal{R})}$ is the product
-contribution monoid; $\gamma_b \in M^{D(\mathcal{R})}$ is the contribution
-element of a behavior type $b$; $\operatorname{ind}(\gamma_b)$ and
-$\operatorname{per}(\gamma_b)$ are computed inside $M^{D(\mathcal{R})}$; and
-$N_{\mathcal{R}}(b) = \operatorname{ind}(\gamma_b) +
-\operatorname{per}(\gamma_b) - 1$ is the exact per-type sibling bound.
-
-
-Lemma 4.4
-Sibling aggregate as a product-monoid sum
-
-
-
-Let a node have child behavior-type multiplicities
-
-$$
-(n_b)_{b \in B_{\mathcal{R}}},
-$$
-
-with all but finitely many $n_b$ zero. Then the simultaneous child aggregate
-seen by all summaries is
-
-$$
-\Gamma := \bigoplus_{b \in B_{\mathcal{R}}} n_b \gamma_b \in M^{D(\mathcal{R})}.
-$$
-
-The $P$-coordinate of $\Gamma$ is exactly
-
-$$
-\bigoplus_{u \text{ child}} \alpha_P(P(T_u)),
-$$
-
-the aggregate used by $P$ at the parent.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Sibling aggregate as a product-monoid sum" data-exhibit-type="proof" data-exhibit-caption="Group children by behavior vector; each contributes α_P(b_P) in coordinate P."}
-
-:::: exhibit-body
-Group the children according to their behavior vector $b$. For each child $u$
-of type $b$, the $P$-coordinate contribution is $\alpha_P(b_P)$. Summing over
-all children and all behavior types gives the stated product-monoid
-expression. Coordinate $P$ is exactly the ordinary child aggregate for the
-summary $P$. [□]{.proof-qed}
-::::
-
-:::
-
-## Index-period decomposition in a finite monoid
-
-We now isolate the elementary finite-monoid fact used throughout the note.
-Additive notation means repeated use of the monoid operation: $ng = g \oplus
-\cdots \oplus g$ with $n$ copies, and $0g = 0_N$.
-
-::: {.annotation .annotation--static #def-index-period}
-
-Definition 5.1
-Index and period of an element
-
-
-
-Let $(N, +, 0_N)$ be a finite monoid and let $g \in N$. The sequence
-
-$$
-0g,\; 1g,\; 2g,\; 3g,\; \ldots
-$$
-
-is eventually periodic. Define $\operatorname{ind}_N(g)$ to be the least $i
-\geq 0$ for which there exists a $p \geq 1$ such that
-
-$$
-(n+p)g = ng \quad \text{for all } n \geq i.
-$$
-
-Given this least index, define $\operatorname{per}_N(g)$ to be the least such
-positive period $p$. When $N$ is clear, write simply $\operatorname{ind}(g)$
-and $\operatorname{per}(g)$. This is the least-index-then-least-period
-convention; other equivalent conventions are possible, but this one is fixed
-throughout the note.
-
-
-
-For every element $g$ of a finite monoid $N$, $\operatorname{ind}(g)$ and
-$\operatorname{per}(g)$ exist. Moreover
-
-$$
-\operatorname{ind}(g) + \operatorname{per}(g) \leq |N|.
-$$
-
-Equivalently, the exact contribution bound satisfies
-$\operatorname{ind}(g) + \operatorname{per}(g) - 1 \leq |N| - 1$.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Existence of index and period" data-exhibit-type="proof" data-exhibit-caption="Pigeonhole on the |N|+1 elements 0g,…,|N|g, then associativity gives eventual periodicity."}
-
-:::: exhibit-body
-Among the $|N|+1$ elements
-
-$$
-0g,\; 1g,\; \ldots,\; |N|g
-$$
-
-two are equal, say $ig = jg$ with $0 \leq i < j \leq |N|$. Let $p = j - i$.
-Then for every $n \geq i$, write $n = i + r$. Associativity gives
-
-$$
-(n+p)g = (i + r + p)g = (j + r)g = (i + r)g = ng.
-$$
-
-Thus eventual periodicity holds with $i + p = j \leq |N|$. The
-least-index-then-least-period pair can only improve this sum, so
-$\operatorname{ind}(g) + \operatorname{per}(g) \leq |N|$. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #def-canon-reduction}
-
-Definition 5.3
-Canonical reduction of a coefficient
-
-
-
-Let $g \in N$, and put
-
-$$
-i = \operatorname{ind}(g), \quad p = \operatorname{per}(g).
-$$
-
-Define
-
-$$
-\operatorname{red}_g(n) = \begin{cases} n, & n < i, \\ i + ((n-i) \bmod p), & n \geq i. \end{cases}
-$$
-
-Then $0 \leq \operatorname{red}_g(n) \leq i + p - 1$.
-
-
-
-For every $n \geq 0$,
-
-$$
-ng = \operatorname{red}_g(n)\, g.
-$$
-
-Moreover $\operatorname{red}_g(n) \leq n$, and if $n > \operatorname{ind}(g) +
-\operatorname{per}(g) - 1$, then $\operatorname{red}_g(n) < n$.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Exact unary pumping" data-exhibit-type="proof" data-exhibit-caption="Reduce n modulo the period beyond the index; a strict drop occurs once n exceeds ind+per−1."}
-
-:::: exhibit-body
-If $n < i$, the claim is immediate. If $n \geq i$, write
-
-$$
-n = i + qp + r
-$$
-
-with $q \geq 0$ and $0 \leq r < p$. By eventual periodicity in steps of $p$
-beyond $i$,
-
-$$
-ng = (i + qp + r)g = (i + r)g = \operatorname{red}_g(n)\, g.
-$$
-
-The inequality $\operatorname{red}_g(n) \leq n$ is clear from the formula. If
-$n > i + p - 1$, then $q \geq 1$, hence $\operatorname{red}_g(n) = i + r < n$.
-[□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #def-contribution-bound}
-
-Definition 5.5
-Contribution bound
-
-
-
-For $g \in N$, define
-
-$$
-N(g) := \operatorname{ind}(g) + \operatorname{per}(g) - 1.
-$$
-
-The exact [pumping lemma](https://en.wikipedia.org/wiki/Pumping_lemma) says
-every coefficient of $g$ can be reduced to at
-most $N(g)$ without changing the monoid value.
-
-
-:::
-
-## Exact sibling pumping
-
-We now apply the index-period decomposition to behavior-type contributions.
-
-::: {.annotation .annotation--static #def-sibling-signature}
-
-Definition 6.1
-Sibling signature
-
-
-
-For a behavior type $b \in B_{\mathcal{R}}$, its *sibling signature* is
-
-$$
-\sigma_{\mathcal{R}}(b) := \bigl(\operatorname{ind}(\gamma_b), \operatorname{per}(\gamma_b)\bigr),
-$$
-
-computed inside the product monoid $M^{D(\mathcal{R})}$. Its *exact sibling
-bound* is
-
-$$
-N_{\mathcal{R}}(b) := \operatorname{ind}(\gamma_b) + \operatorname{per}(\gamma_b) - 1.
-$$
-
-A uniform exact sibling bound is
-
-$$
-N^{\max}_{\mathcal{R}} := \max_{b \in B_{\mathcal{R}}} N_{\mathcal{R}}(b).
-$$
-
-
-
-Let a node have child behavior-type multiplicities $(n_b)_{b \in
-B_{\mathcal{R}}}$. For each $b$, set
-
-$$
-n'_b := \operatorname{red}_{\gamma_b}(n_b).
-$$
-
-Replace the child multiset by one having exactly $n'_b$ children of behavior
-type $b$ for every $b$, using any available representatives of those behavior
-types. Then the simultaneous child aggregate in $M^{D(\mathcal{R})}$ is
-unchanged. Consequently, if the node's observed multiplicity label is
-unchanged, then its parent behavior vector is unchanged.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Exact sibling pumping at one node" data-exhibit-type="proof" data-exhibit-caption="Per-type unary pumping leaves each n_b γ_b unchanged, hence the whole product aggregate."}
-
-:::: exhibit-body
-By [Lemma 4.4](#lem-aggregate), the original simultaneous child aggregate is
-
-$$
-\Gamma = \bigoplus_b n_b \gamma_b.
-$$
-
-The new aggregate is
-
-$$
-\Gamma' = \bigoplus_b n'_b \gamma_b.
-$$
-
-By [Lemma 5.4](#lem-unary-pumping), $n_b \gamma_b = n'_b \gamma_b$ for each
-$b$. Therefore $\Gamma = \Gamma'$. Coordinatewise, every summary $P \in
-D(\mathcal{R})$ receives the same child aggregate at the node. Since the
-observed multiplicity label is also unchanged, every $P$ assigns the same
-parent state as before. Hence the whole behavior vector at the node is
-unchanged. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #cor-sibling-normal-form}
-
-Corollary 6.3
-Exact sibling normal form
-
-
-
-Every sibling multiset is equivalent, as seen by all summaries in
-$D(\mathcal{R})$, to one in which each behavior type $b$ occurs at most
-
-$$
-N_{\mathcal{R}}(b) = \operatorname{ind}(\gamma_b) + \operatorname{per}(\gamma_b) - 1
-$$
-
-times. In particular, the total number of children after exact sibling
-normalization is at most
-
-$$
-C_{\mathcal{R}} := \sum_{b \in B_{\mathcal{R}}} N_{\mathcal{R}}(b) \leq |B_{\mathcal{R}}| \cdot N^{\max}_{\mathcal{R}}.
-$$
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Exact sibling normal form" data-exhibit-type="proof" data-exhibit-caption="Apply the one-node pumping theorem per behavior type."}
-
-:::: exhibit-body
-Apply [Theorem 6.2](#thm-sibling-pumping) to each behavior type. The
-resulting count $n'_b$ satisfies $n'_b \leq N_{\mathcal{R}}(b)$.
-[□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #rem-realizable-formal}
-
-Remark 6.4
-Realizable versus formal behavior types
-
-
-
-The bounds may be sharpened by taking $b$ only over realizable behavior
-vectors. The present statement uses all formal $b \in B_{\mathcal{R}}$ to
-avoid introducing a separate realizability analysis. Note that realizability
-of behavior vectors is defined existentially over all trees and is not in
-general algorithmically transparent, so the formal-version bounds are also
-the practically computable ones.
-
-
-:::
-
-## Canonical monoid families
-
-The index-period form makes the standard monoid families transparent.
-
-### Boolean semilattices
-
-::: {.annotation .annotation--static #prop-boolean}
-
-Proposition 7.1
-Boolean support pumping
-
-
-
-Let $M = (\{0, 1\}, \vee, 0)$. Then for $g = 0$,
-
-$$
-\operatorname{ind}(g) = 0, \quad \operatorname{per}(g) = 1, \quad N(g) = 0,
-$$
-
-and for $g = 1$,
-
-$$
-\operatorname{ind}(g) = 1, \quad \operatorname{per}(g) = 1, \quad N(g) = 1.
-$$
-
-Thus a nonzero child contribution is remembered only by presence or absence.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Boolean support pumping" data-exhibit-type="proof" data-exhibit-caption="g = 0 is periodic from index 0; g = 1 stabilizes at 1 from index 1."}
-
-:::: exhibit-body
-If $g = 0$, then $ng = 0$ for all $n$, so the sequence is periodic from index
-$0$ with period $1$. If $g = 1$, then $0g = 0$ and $ng = 1$ for all $n \geq
-1$, so the sequence has index $1$ and period $1$. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #cor-boolean-product}
-
-Corollary 7.2
-Boolean product bound
-
-
-
-If $M$ is a finite Boolean semilattice, for example a finite power of $(\{0,
-1\}, \vee, 0)$, every element is
-[idempotent](https://en.wikipedia.org/wiki/Idempotence). Hence every behavior type has
-bound $0$ if its contribution is zero and bound $1$ otherwise.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Cyclic group pumping" data-exhibit-type="proof" data-exhibit-caption="ng is periodic from the start with least period the additive order of g."}
-
-:::: exhibit-body
-The sequence $ng$ is periodic from the beginning. Its least positive period
-is the [additive order](https://en.wikipedia.org/wiki/Order_(group_theory))
-of $g$ in the [cyclic group](https://en.wikipedia.org/wiki/Cyclic_group). The displayed formula for
-the order in $\mathbb{Z}/q\mathbb{Z}$ is standard. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #cor-cyclic-product}
-
-Corollary 7.4
-Product of cyclic groups
-
-
-
-If $M$ is a finite abelian group and $g \in M$, then
-
-$$
-\operatorname{ind}(g) = 0, \quad \operatorname{per}(g) = \operatorname{ord}(g), \quad N(g) = \operatorname{ord}(g) - 1.
-$$
-
-For a product element $g = (g_i)$, $\operatorname{ord}(g)$ is the least
-common multiple of the coordinate orders.
-
-
-
-Let $M = \Theta_T$. If $g = 0$, then $\operatorname{ind}(g) = 0$,
-$\operatorname{per}(g) = 1$, and $N(g) = 0$. If $1 \leq g \leq T$, then
-
-$$
-\operatorname{ind}(g) = \lceil T/g \rceil, \quad \operatorname{per}(g) = 1, \quad N(g) = \lceil T/g \rceil.
-$$
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Threshold pumping" data-exhibit-type="proof" data-exhibit-caption="The sequence climbs until it saturates at T after ⌈T/g⌉ steps, then is constant."}
-
-:::: exhibit-body
-For $g = 0$ the sequence is constantly zero. If $T = 0$, this is the only
-case. For $g > 0$,
-
-$$
-ng = \min(T, ng)
-$$
-
-in ordinary integer notation. The first $n$ for which $ng$ reaches $T$ is
-$\lceil T/g \rceil$. From that index onward the sequence is constantly $T$,
-hence the period is $1$. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #rem-aperiodic}
-
-Remark 7.7
-Aperiodic saturation
-
-
-
-Threshold monoids show why arbitrary finite commutative monoids cannot be
-treated as semilattices of abelian groups. In $\Theta_2$, the element $1$ has
-the sequence $0, 1, 2, 2, 2, \ldots$; this has a genuine preperiod and no
-group-like cycle before saturation.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Hybrid threshold-residue pumping" data-exhibit-type="proof" data-exhibit-caption="The threshold coordinate fixes the index; the cyclic coordinate fixes the period."}
-
-:::: exhibit-body
-If $g_{\text{thr}} = 0$, the threshold coordinate is constantly $0$ and the
-product period is exactly the cyclic order. If $g_{\text{thr}} > 0$, the
-threshold coordinate strictly changes until the first index
-
-$$
-i = \lceil T/g_{\text{thr}} \rceil,
-$$
-
-at which it reaches $T$ and remains constant. Thus no smaller index can work.
-From index $i$ onward, the threshold coordinate contributes period $1$, while
-the cyclic coordinate has least period $\operatorname{ord}(g_{\text{cyc}})$.
-Therefore the product has least period $\operatorname{ord}(g_{\text{cyc}})$
-from the least possible index $i$. [□]{.proof-qed}
-::::
-
-:::
-
-### Product bounds in general
-
-::: {.annotation .annotation--static #prop-coord-product}
-
-Proposition 7.9
-Coordinatewise product bound
-
-
-
-Let $N = N_1 \times \cdots \times N_r$ be a product of finite monoids and let
-$g = (g_1, \ldots, g_r)$. If $i_j = \operatorname{ind}(g_j)$ and $p_j =
-\operatorname{per}(g_j)$, then a valid index-period pair for $g$ is
-
-$$
-i = \max_j i_j, \quad p = \operatorname{lcm}_j p_j.
-$$
-
-Thus
-
-$$
-N(g) \leq \max_j i_j + \operatorname{lcm}_j p_j - 1.
-$$
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Coordinatewise product bound" data-exhibit-type="proof" data-exhibit-caption="Beyond the max index, adding the lcm of periods preserves every coordinate."}
-
-:::: exhibit-body
-For every coordinate $j$, the sequence $n g_j$ is periodic with period $p_j$
-from index $i_j$ onward. Once $n \geq \max_j i_j$, adding $p =
-\operatorname{lcm}_j p_j$ preserves every coordinate. Hence it preserves the
-product element. [□]{.proof-qed}
-::::
-
-:::
-
-## Examples: exact computations
-
-This section records concrete test families. These are not yet
-pursuit-evasion applications; they are calibration examples for the summary
-model.
-
-### One-node trees
-
-Let $A_n$ be the one-node tree whose root multiplicity is $n$.
-
-::: {.annotation .annotation--static #prop-one-node}
-
-Proposition 8.1
-One-node criterion
-
-
-
-For fixed $\mathcal{R}$, if
-
-$$
-\mu(n) = \mu(m),
-$$
-
-then
-
-$$
-A_n \sim_{\mathcal{R}} A_m.
-$$
-
-Conversely, if $\mu(n) \neq \mu(m)$ and $|S| \geq 2$, then $A_n$ and $A_m$
-are separated by some summary in $D(\mathcal{R})$.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="One-node criterion" data-exhibit-type="proof" data-exhibit-caption="A one-node tree has empty child aggregate, so its state depends only on μ(n)."}
-
-:::: exhibit-body
-A one-node tree has empty child aggregate. Hence for every $P = (\alpha_P,
-f_P)$,
-
-$$
-P(A_n) = f_P(\mu(n), 0_M).
-$$
-
-If $\mu(n) = \mu(m)$, these values are equal for all $P$, so [Theorem
-3.3](#thm-behavior-vector) gives equivalence.
-
-If $\mu(n) \neq \mu(m)$ and $|S| \geq 2$, choose two distinct states $s_0,
-s_1 \in S$. Define $f$ so that $f(\mu(n), 0_M) = s_0$ and $f(\mu(m), 0_M) =
-s_1$, extending $f$ arbitrarily elsewhere. Choose any $\alpha : S \to M$. The
-resulting summary separates $A_n$ and $A_m$. [□]{.proof-qed}
-::::
-
-:::
-
-### Stars
-
-Fix a rooted tree $Q$ with behavior vector $b = \beta_{\mathcal{R}}(Q)$. Let
-$\mathrm{Star}_n(a; Q)$ be the tree with root observed multiplicity label $a
-\in A$ and $n$ children, each isomorphic to $Q$. More precisely, choose any
-root multiplicity $r$ with $\mu(r) = a$.
-
-::: {.annotation .annotation--static #prop-stars}
-
-Proposition 8.2
-Star aggregate criterion
-
-
-
-For fixed $a$ and $Q$, if
-
-$$
-n \gamma_b = m \gamma_b
-$$
-
-in the product monoid $M^{D(\mathcal{R})}$, then
-
-$$
-\mathrm{Star}_n(a; Q) \sim_{\mathcal{R}} \mathrm{Star}_m(a; Q).
-$$
-
-More generally, the two stars have the same behavior vector exactly when, for
-every summary $P \in D(\mathcal{R})$,
-
-$$
-f_P(a, n \alpha_P(b_P)) = f_P(a, m \alpha_P(b_P)).
-$$
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Star aggregate criterion" data-exhibit-type="proof" data-exhibit-caption="Root state is f_P(a, n α_P(b_P)); product-aggregate equality forces coordinatewise equality."}
-
-:::: exhibit-body
-For every summary $P$, the root state is $f_P(a, n \alpha_P(b_P))$ for the
-first star and $f_P(a, m \alpha_P(b_P))$ for the second. The coordinatewise
-equality displayed in the proposition is therefore exactly behavior-vector
-equality. The product-monoid identity $n \gamma_b = m \gamma_b$ implies that
-equality, since its $P$-coordinate is precisely
-
-$$
-n \alpha_P(b_P) = m \alpha_P(b_P)
-$$
-
-for every $P$. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #rem-aggregate-sufficient}
-
-Remark 8.3
-Why aggregate equality is sufficient, not necessary
-
-
-
-The implication from product-aggregate equality to star equivalence is the
-one needed for pumping and normal forms. It is not generally necessary: two
-different aggregates may be identified by all root update maps at the
-observed label $a$ for the summaries under discussion. In particular examples
-one can often force separation by choosing a summary whose update map
-distinguishes the two aggregates, but the exact statement is the displayed
-coordinatewise criterion.
-
-
-
-Assume unit contribution $\gamma_b = g$.
-
-1. Boolean support: all positive $n$ are equivalent; $n = 0$ is separate from
- $n > 0$ if $g \neq 0$.
-2. Cyclic $\mathbb{Z}/q\mathbb{Z}$: $n$ and $m$ are equivalent exactly modulo
- $\operatorname{ord}(g)$.
-3. Threshold $\Theta_T$ with $g = 1$: $n$ and $m$ are equivalent iff either
- $n = m < T$ or both $n, m \geq T$.
-
-
-:::
-
-### Unary chains
-
-Unary-chain behavior is controlled by finite transformations on behavior
-vectors, not directly by the horizontal child-aggregation monoid.
-
-::: {.annotation .annotation--static #def-unary-map}
-
-Definition 8.5
-Unary extension map
-
-
-
-For each observed multiplicity label $a \in A$, define
-
-$$
-U_a : B_{\mathcal{R}} \to B_{\mathcal{R}}
-$$
-
-by declaring $U_a(b)$ to be the behavior vector of a new root with observed
-label $a$ and exactly one child of behavior type $b$. Coordinatewise,
-
-$$
-(U_a(b))_P = f_P(a, \alpha_P(b_P)).
-$$
-
-
-Proposition 8.6
-Unary chains are eventually periodic
-
-
-
-Fix $a \in A$ and $b \in B_{\mathcal{R}}$. The sequence
-
-$$
-b,\; U_a(b),\; U_a^2(b),\; U_a^3(b),\; \ldots
-$$
-
-is eventually periodic. In particular, among the first $|B_{\mathcal{R}}| +
-1$ terms two are equal.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Unary chains are eventually periodic" data-exhibit-type="proof" data-exhibit-caption="U_a is a self-map of the finite set B_R; every finite-set orbit is eventually periodic."}
-
-:::: exhibit-body
-The map $U_a$ is a self-map of the finite set $B_{\mathcal{R}}$. Every orbit
-of a self-map on a finite set is eventually periodic, and the pigeonhole
-principle gives a repetition among the first $|B_{\mathcal{R}}| + 1$ terms.
-[□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #cor-unary-pumping}
-
-Corollary 8.7
-Unary chain pumping
-
-
-
-Any sufficiently long constant-label unary chain contains a proper subchain
-whose deletion preserves the behavior vector at the top of the chain. A crude
-bound is $|B_{\mathcal{R}}| + 1$ vertices for a repeated behavior vector.
-
-
-
-If labels vary along a unary path, one obtains the finite [transformation
-semigroup](https://en.wikipedia.org/wiki/Transformation_semigroup)
-generated by the maps $U_a$ for $a \in A$: this is the subsemigroup,
-under composition, of the finite monoid of all self-maps of $B_{\mathcal{R}}$
-generated by the maps $U_a$. Long labeled unary words can be pumped using
-repetitions in this finite transformation semigroup, but the
-minimal-representative argument in [Section
-9](#global-normal-representatives) gives a simpler global height bound for
-entire trees.
-
-
-:::
-
-### Split versus concentrated examples
-
-Let $Q$ be a tree of behavior type $b$. A simple split tree has a root with
-two children of type $b$, hence horizontal contribution
-
-$$
-2 \gamma_b.
-$$
-
-A concentrated competitor has a root with one child $R$ of behavior type $c$,
-where the internal construction of $R$ may have encoded some information that
-resembles two copies of $b$ at a lower level.
-
-::: {.annotation .annotation--static #prop-split-concentrated}
-
-Proposition 8.9
-Diagnostic criterion
-
-
-
-Suppose two trees have the same observed root label $a$. One has child
-multiset consisting of two children of behavior type $b$, and the other has
-one child of behavior type $c$. Their root behavior vectors are equal exactly
-when, for every $P \in D(\mathcal{R})$,
-
-$$
-f_P(a, 2\alpha_P(b_P)) = f_P(a, \alpha_P(c_P)).
-$$
-
-Equivalently, equality follows from the stronger aggregate identity
-
-$$
-2 \gamma_b = \gamma_c
-$$
-
-in $M^{D(\mathcal{R})}$, but may also occur accidentally because all update
-maps in question identify the two aggregates at label $a$.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Diagnostic criterion" data-exhibit-type="proof" data-exhibit-caption="Immediate from coordinatewise root evaluation; aggregate equality is sufficient but not necessary."}
-
-:::: exhibit-body
-This is immediate from the coordinatewise evaluation formula at the root.
-Aggregate equality is sufficient. It is not necessary for an arbitrary fixed
-subfamily of updates because two different aggregates may be mapped to the
-same state by every relevant $f_P$ at the label $a$. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #rem-no-converse}
-
-Remark 8.10
-No automatic converse from aggregate inequality
-
-
-
-The stronger aggregate identity $2\gamma_b = \gamma_c$ is a clean sufficient
-condition for equality of the two root behavior vectors. Its failure is not,
-by itself, a clean separation theorem. The behavior coordinates $b_P, c_P$ of
-the child subtrees are fixed properties of those subtrees, indexed by every
-$P \in D(\mathcal{R})$. If a particular coordinate $P^*$ witnesses aggregate
-inequality in $M^{D(\mathcal{R})}$, the summary $P^*$ itself may still fail to
-separate the parents because $f_{P^*}$ may identify the two aggregates. One
-cannot remedy this by "switching to a different $f$" while holding the child
-contributions fixed: choosing a different summary $P'$ means looking at a
-different coordinate $P'$ of the child behavior vectors, with potentially
-different aggregate values. Thus separation should be checked by the exact
-coordinatewise criterion in [Proposition 8.2](#prop-stars) and [Proposition
-8.9](#prop-split-concentrated), not by aggregate inequality alone. This is
-precisely why split-versus-concentrated examples are diagnostically
-interesting rather than trivial.
-
-
-
-This is the first family where horizontal aggregation interacts with vertical
-recursion. It is a natural bridge to later pursuit-evasion questions about
-whether support is split across branches or concentrated inside one branch.
-
-
-:::
-
-## Global normal representatives
-
-Exact sibling pumping bounds branching. To obtain a finite universe of
-representatives, one also needs a height bound. The cleanest argument is not a
-unary-chain analysis; it is minimality.
-
-::: {.annotation .annotation--static #def-label-rep}
-
-Definition 9.1
-Observed-label representative
-
-
-
-Choose once and for all a representative integer $r(a) \in \mathbb{N}$ for
-each $a \in A$ with $\mu(r(a)) = a$, for every $a$ in the image of $\mu$. No
-representative is needed for $a \notin \operatorname{im}(\mu)$, since no
-vertex of any tree has observed label $a$. A tree is *label-normalized* if
-every vertex with observed label $a$ has actual multiplicity $r(a)$.
-
-
-
-Every tree $T$ is $\sim_{\mathcal{R}}$-equivalent to a label-normalized tree
-$T^{\ell}$ with the same underlying rooted unordered tree and the same
-observed labels.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Label normalization preserves behavior" data-exhibit-type="proof" data-exhibit-caption="Summaries use multiplicities only through μ, so replacing m by r(μ(m)) changes nothing."}
-
-:::: exhibit-body
-Replace each vertex multiplicity $m$ by $r(\mu(m))$. Every summary in
-$D(\mathcal{R})$ uses multiplicities only through $\mu$, so every bottom-up
-computation is unchanged. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #def-size-minimal}
-
-Definition 9.3
-Size-minimal representative
-
-
-
-A tree $T$ is *size-minimal* for its behavior vector if among all trees $T'$
-with $\beta_{\mathcal{R}}(T') = \beta_{\mathcal{R}}(T)$, the number of
-vertices of $T'$ is minimized. It is *normalized size-minimal* if it is also
-label-normalized.
-
-
-
-Every realizable behavior vector has a normalized size-minimal
-representative.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Size-minimal representatives exist" data-exhibit-type="proof" data-exhibit-caption="Pick a fewest-vertex realizer, then label-normalize it without changing vertex count."}
-
-:::: exhibit-body
-The behavior vector is realizable, so at least one tree realizes it. Among
-all realizing trees, choose one with the fewest vertices. Apply [Lemma
-9.2](#lem-label-normalization) to normalize labels without changing the
-number of vertices or the behavior vector. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #thm-minimal-sibling-bounded}
-
-Theorem 9.5
-Minimal representatives are sibling-bounded
-
-
-
-Let $T$ be a normalized size-minimal representative. At every node $v$ of
-$T$, each child behavior type $b$ occurs at most
-
-$$
-N_{\mathcal{R}}(b) = \operatorname{ind}(\gamma_b) + \operatorname{per}(\gamma_b) - 1
-$$
-
-times.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Minimal representatives are sibling-bounded" data-exhibit-type="proof" data-exhibit-caption="An over-full sibling type could be pumped down, contradicting size-minimality."}
-
-:::: exhibit-body
-Suppose some node $v$ has $n_b > N_{\mathcal{R}}(b)$ children of behavior
-type $b$. By [Lemma 5.4](#lem-unary-pumping), replacing $n_b$ by
-$\operatorname{red}_{\gamma_b}(n_b)$ preserves the contribution of type $b$,
-and by the same lemma this reduced number satisfies
-$\operatorname{red}_{\gamma_b}(n_b) < n_b$. Delete enough children of type
-$b$ to leave exactly $\operatorname{red}_{\gamma_b}(n_b)$ such children,
-leaving all other child types unchanged. The simultaneous child aggregate at
-$v$ is unchanged, so the behavior vector of the subtree rooted at $v$ is
-unchanged. By [Corollary 3.5](#cor-congruence), the behavior vector of the
-whole tree is unchanged. But the number of vertices strictly decreases,
-contradicting size-minimality. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #thm-no-repeat}
-
-Theorem 9.6
-Minimal representatives have no repeated behavior along a path
-
-
-
-Let $T$ be a normalized size-minimal representative. No root-to-leaf path of
-$T$ contains two distinct vertices $u$ and $v$, with $v$ a proper descendant
-of $u$, such that
-
-$$
-\beta_{\mathcal{R}}(T_u) = \beta_{\mathcal{R}}(T_v).
-$$
-
-Consequently every root-to-leaf path has at most $|B_{\mathcal{R}}|$
-vertices.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Minimal representatives have no repeated behavior along a path" data-exhibit-type="proof" data-exhibit-caption="A repeated behavior vector lets the upper subtree be replaced by the lower one, shrinking the tree."}
-
-:::: exhibit-body
-Suppose $v$ is a proper descendant of $u$ and the rooted subtrees $T_u$ and
-$T_v$ have the same behavior vector. Replace the subtree $T_u$ by the proper
-descendant subtree $T_v$. Since the two subtrees are
-$\sim_{\mathcal{R}}$-equivalent by [Theorem 3.3](#thm-behavior-vector),
-[Corollary 3.5](#cor-congruence) implies that the behavior vector of the
-whole tree is unchanged. The replacement strictly decreases the number of
-vertices, contradicting size-minimality. Therefore no behavior vector
-repeats along a path. Since an actual path encounters only realizable
-behavior vectors, every path has at most the number of realizable behavior
-vectors, and in particular at most $|B_{\mathcal{R}}|$ vertices.
-[□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #def-normal-universe}
-
-Definition 9.7
-Normal universe
-
-
-
-Let $U_{\mathcal{R}}$ be the finite set of all label-normalized rooted
-cop-labeled trees satisfying:
-
-1. every root-to-leaf path has at most $|B_{\mathcal{R}}|$ vertices;
-2. at every node, behavior type $b$ occurs among the children at most
- $N_{\mathcal{R}}(b)$ times, for every $b \in B_{\mathcal{R}}$.
-
-
-
-One may replace $|B_{\mathcal{R}}|$ and the sum over all formal $b \in
-B_{\mathcal{R}}$ by the corresponding quantities for realizable behavior
-vectors. The formal version is cruder but avoids a separate realizability
-computation. Since realizability is defined existentially over all trees and
-is not in general algorithmically transparent, the formal version is also the
-version most directly usable in computations.
-
-
-Theorem 9.9
-Finite global normal representatives
-
-
-
-Every $\sim_{\mathcal{R}}$-equivalence class has a representative in the
-finite universe $U_{\mathcal{R}}$.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Finite global normal representatives" data-exhibit-type="proof" data-exhibit-caption="A normalized size-minimal representative satisfies both universe constraints; the universe is finite."}
-
-:::: exhibit-body
-Let $b$ be a realizable behavior vector. By [Lemma
-9.4](#lem-minimal-exists), choose a normalized size-minimal representative
-$T$ realizing $b$. By [Theorem 9.5](#thm-minimal-sibling-bounded), $T$
-satisfies the exact sibling bounds. By [Theorem 9.6](#thm-no-repeat), its
-root-to-leaf paths have at most $|B_{\mathcal{R}}|$ vertices. Thus $T \in
-U_{\mathcal{R}}$.
-
-The set $U_{\mathcal{R}}$ is finite because labels come from the finite image
-of $\mu$, height is bounded, and at each node the number of children is
-bounded by
-
-$$
-C_{\mathcal{R}} = \sum_{b \in B_{\mathcal{R}}} N_{\mathcal{R}}(b).
-$$
-
-There are only finitely many finite unordered rooted trees with bounded
-height, bounded branching, and labels from a finite alphabet.
-[□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #cor-size-bound}
-
-Corollary 9.10
-Crude size bound
-
-
-
-Let
-
-$$
-H_{\mathcal{R}} := |B_{\mathcal{R}}|, \quad C_{\mathcal{R}} := \sum_{b \in B_{\mathcal{R}}} N_{\mathcal{R}}(b).
-$$
-
-Then every class has a representative with at most
-
-$$
-1 + C_{\mathcal{R}} + C_{\mathcal{R}}^2 + \cdots + C_{\mathcal{R}}^{H_{\mathcal{R}} - 1}
-$$
-
-vertices, with the usual interpretation as $H_{\mathcal{R}}$ when
-$C_{\mathcal{R}} = 1$.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Crude size bound" data-exhibit-type="proof" data-exhibit-caption="Sum the geometric series for ≤ H_R levels with branching ≤ C_R."}
-
-:::: exhibit-body
-Here $H_{\mathcal{R}}$ bounds the number of vertices on a root-to-leaf path,
-so the tree has at most $H_{\mathcal{R}}$ levels, indexed $0, 1, \ldots,
-H_{\mathcal{R}} - 1$. With branching at most $C_{\mathcal{R}}$, level $d$ has
-at most $C_{\mathcal{R}}^d$ vertices. Summing over the levels gives the
-displayed bound; when $C_{\mathcal{R}} = 1$ the geometric sum has
-$H_{\mathcal{R}}$ terms each equal to $1$, giving $H_{\mathcal{R}}$.
-[□]{.proof-qed}
-::::
-
-:::
-
-## Canonical representatives
-
-A finite normal universe gives canonical representatives once one imposes an
-external tie-break. This is safer than claiming that minimal representatives
-are intrinsically unique, which is generally false.
-
-::: {.annotation .annotation--static #def-external-order}
-
-Definition 10.1
-External ordering
-
-
-
-Fix a [total order](https://en.wikipedia.org/wiki/Total_order) $\preceq$ on the finite universe $U_{\mathcal{R}}$. For
-example, order first by number of vertices, then by height, then recursively
-by sorted child lists and vertex labels.
-
-
-
-For a realizable behavior vector $b$, define
-
-$$
-\operatorname{Can}_{\mathcal{R}}(b)
-$$
-
-to be the $\preceq$-least tree in $U_{\mathcal{R}}$ with behavior vector $b$.
-
-
-
-After choosing the external order $\preceq$, every
-$\sim_{\mathcal{R}}$-equivalence class has a unique selected canonical
-representative.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Canonical representative theorem" data-exhibit-type="proof" data-exhibit-caption="Each class meets the finite totally ordered universe in a nonempty set with a unique least element."}
-
-:::: exhibit-body
-By [Theorem 9.9](#thm-normal-universe), every class has at least one
-representative in $U_{\mathcal{R}}$. Since $U_{\mathcal{R}}$ is finite and
-totally ordered by $\preceq$, each nonempty subset has a unique least
-element. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #rem-no-uniqueness}
-
-Remark 10.4
-No intrinsic uniqueness claimed
-
-
-
-The theorem does not claim that each class has a unique minimal tree in any
-intrinsic sense. Different non-isomorphic trees of the same size may realize
-the same behavior vector. The uniqueness is selected uniqueness after a
-chosen tie-breaking order.
-
-
-:::
-
-## What has been proved and what has not
-
-- *Proved:* fixed-resource equivalence equals behavior-vector equality.
-- *Proved:* exact sibling pumping is governed by the index and period of each
- product contribution element $\gamma_b$.
-- *Proved:* Boolean, cyclic, threshold, and hybrid threshold-residue monoids
- have explicit pumping bounds.
-- *Proved:* every fixed-resource class has a finite normal representative,
- and an externally selected canonical representative.
-- *Not proved:* any global decomposition theorem for arbitrary finite
- commutative monoids.
-- *Not claimed:* intrinsic uniqueness of minimal representatives.
-- *Not proved:* comparison with order-$r$ profiles.
-- *Not proved:* nondefinability or definability of pursuit-evasion
- properties.
-
-## Conclusion
-
-The fixed-resource monoid-aggregated model now has a sharper structural core.
-The essential invariant for sibling multiplicities is the index-period pair
-of the product contribution element $\gamma_b$. This converts the original
-crude finite-product pumping lemma into an exact normal-form statement. It
-also clarifies the qualitative meanings of the standard monoid families:
-Boolean semilattices track support, cyclic groups track residue, threshold
-monoids track saturation, and hybrids combine these effects.
-
-The global canonical-form theory is also cleaner than expected. Sibling
-pumping bounds branching; size-minimality bounds height, because a repeated
-behavior vector along a path could be contracted. Therefore every
-fixed-resource equivalence class has a representative in a finite universe,
-and canonical representatives exist after external tie-breaking.
-
-The next mathematical frontier is no longer fixed-resource equivalence
-itself. That relation is fully finite and behavior-vector controlled. The
-hard questions concern how separation cost grows as one varies the allowed
-multiplicity observations, state sets, and monoids, and whether
-pursuit-evasion properties cut across the resulting bounded-resource
-theories.
-
-::::: aftermatter
-
-## Appendix A — Algorithmic consequences
-
-The theory is constructive, although often computationally enormous.
-
-::: {.annotation .annotation--static #prop-decidability}
-
-Proposition A.1
-Decidability for fixed resources
-
-
-
-For fixed finite resource data $\mathcal{R}$ and finite rooted cop-labeled
-trees $X, Y$, it is decidable whether
-
-$$
-X \sim_{\mathcal{R}} Y.
-$$
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Decidability for fixed resources" data-exhibit-type="proof" data-exhibit-caption="Enumerate the finite class D(R), evaluate both trees bottom-up, compare all coordinates."}
-
-:::: exhibit-body
-The class $D(\mathcal{R})$ is finite by [Lemma 2.7](#lem-cardinality). One
-may enumerate all summaries $P \in D(\mathcal{R})$, compute $P(X)$ and $P(Y)$
-bottom-up, and compare all coordinates. By [Theorem
-3.3](#thm-behavior-vector), the trees are equivalent iff all coordinates
-agree. [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #prop-computable-canonical}
-
-Proposition A.2
-Computable canonical representative, in principle
-
-
-
-For fixed $\mathcal{R}$, a chosen external order $\preceq$ on
-$U_{\mathcal{R}}$, and an input tree $T$, the canonical representative of the
-class of $T$ is computable by finite search.
-
-
-:::
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Computable canonical representative, in principle" data-exhibit-type="proof" data-exhibit-caption="Compute β_R(T), then scan U_R in ≼-order for the first tree with the same behavior vector."}
-
-:::: exhibit-body
-Compute $b = \beta_{\mathcal{R}}(T)$ by enumerating $D(\mathcal{R})$. Then
-enumerate the finite universe $U_{\mathcal{R}}$ in $\preceq$-order and return
-the first tree $U$ with $\beta_{\mathcal{R}}(U) = b$. Termination follows from
-[Theorem 9.9](#thm-normal-universe). [□]{.proof-qed}
-::::
-
-:::
-
-::: {.annotation .annotation--static #rem-practical}
-
-Remark A.3
-Practical versus theoretical computation
-
-
-
-The bounds involving $|D(\mathcal{R})|$ are generally huge. The point of the
-theorem is structural finiteness and conceptual normalization, not immediate
-efficient implementation. For special monoid families, the exact
-index-period bounds above are the first route toward usable computations.
-
-
-:::
-
-## Appendix B — Resource-growth separation complexity
-
-For fixed $\mathcal{R}$, equivalence is completely characterized by the
-behavior vector. The more interesting long-term invariant appears when
-resources vary.
-
-::: {.annotation .annotation--static #def-resource-family}
-
-Definition B.1
-Resource family
-
-
-
-A *resource family* $\mathcal{F}$ is a collection of finite resource data
-$\mathcal{R}$. Examples include:
-
-1. support-only resources with bounded $|S|$;
-2. threshold resources with threshold $T \leq t$ and $|S| \leq s$;
-3. cyclic resources with modulus $q \leq Q$ and $|S| \leq s$;
-4. hybrid threshold-residue resources with bounded thresholds, moduli, and
- state counts.
-
-
-Definition B.2
-Separation over a resource family
-
-
-
-Given a resource family $\mathcal{F}$, say that $X$ and $Y$ are *separated by
-$\mathcal{F}$* if there exists $\mathcal{R} \in \mathcal{F}$ and $P \in
-D(\mathcal{R})$ such that
-
-$$
-P(X) \neq P(Y).
-$$
-
-Equivalently, $\beta_{\mathcal{R}}(X) \neq \beta_{\mathcal{R}}(Y)$ for some
-$\mathcal{R} \in \mathcal{F}$.
-
-
-
-Let $\kappa(\mathcal{R})$ be a chosen cost of a resource datum, for example a
-tuple involving $|A|$, $|S|$, $|M|$, the threshold parameter, the modulus, or
-the description length of $\mu$ and $\oplus$. Define
-
-$$
-\operatorname{sep}_{\mathcal{F}}(X, Y)
-$$
-
-to be the least cost of a resource datum $\mathcal{R} \in \mathcal{F}$
-separating $X$ and $Y$, and set $\operatorname{sep}_{\mathcal{F}}(X, Y) =
-\infty$ if no $\mathcal{R} \in \mathcal{F}$ separates them.
-
-
-
-For fixed $\mathcal{R}$, a
-[game characterization](https://en.wikipedia.org/wiki/Ehrenfeucht%E2%80%93Fra%C3%AFss%C3%A9_game)
-is nearly tautological:
-Spoiler can choose a differing coordinate $P \in D(\mathcal{R})$ if one
-exists, and otherwise Duplicator wins because behavior vectors agree. A
-nontrivial game should therefore characterize resource growth, restricted
-resource families, or bounded access to coordinates, not merely the
-fixed-$\mathcal{R}$ relation. In particular, the interesting game will need to
-encode a budget on which coordinates $P$ Spoiler is permitted to access at
-each round, with cost tied to $\kappa$.
-
-
-:::
-
-## Appendix C — Near-term theorem targets
-
-The present note proves the first three targets below and sets up the rest.
-
-- **T1. Exact sibling pumping.** Child counts of behavior type $b$ reduce by
- the index-period pair of $\gamma_b$. Proved in [Theorem
- 6.2](#thm-sibling-pumping).
-- **T2. Canonical family bounds.** Boolean, cyclic, threshold, and hybrid
- monoids have explicit pumping signatures. Proved in [Section
- 7](#canonical-monoid-families).
-- **T3. Finite normal representatives.** Every fixed-resource class has a
- representative in a finite normal universe. Proved in [Theorem
- 9.9](#thm-normal-universe).
-- **T4. Efficient special-case canonicalization.** For concrete resource
- families, replace the huge all-summary behavior vector by smaller
- sufficient invariants.
-- **T5. Separation complexity examples.** Compute exact or asymptotic costs
- for one-node, star, unary-chain, and split/concentrated families under
- support, threshold, cyclic, and hybrid resources.
-- **T6. Resource-growth games.** Design a game that characterizes bounded
- resource families rather than fixed-$\mathcal{R}$ equality of behavior
- vectors.
-- **T7. Pursuit-evasion tests.** Only after the previous items, ask whether
- local pursuit properties are definable or separable in specific resource
- families.
-
-:::::
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