levineuwirth.org/content/essays/index-period-normal-forms/index.md at 37665f67db48e4d6818341dfd2fbc6f34fd09f22

63 KiB

Raw Blame History

title

subtitle

date

abstract

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 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 is a semilattice of abelian groups. That statement holds for special regular/Clifford-type commutative monoids, not for arbitrary finite commutative monoids. Threshold monoids already contain aperiodic 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.

:::

::: {.annotation .annotation--static #def-context}

Definition 2.2 Rooted one-hole context

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.

:::

::: {.annotation .annotation--static #def-resource-datum}

Definition 2.3 Finite resource datum

A finite resource datum is a tuple


\mathcal{R} = (A, \mu, S, M, \oplus, 0_M)

where:

A is a finite multiplicity alphabet;
\mu : \mathbb{N} \to A is a fixed multiplicity observation map;
S is a finite state set;
(M, \oplus, 0_M) is a finite commutative monoid.

:::

::: {.annotation .annotation--static #warn-actual-resources}

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}.

:::

::: {.annotation .annotation--static #def-summary}

Definition 2.5 Monoid-aggregated summary

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).

:::

::: {.annotation .annotation--static #def-fixed-class}

Definition 2.6 The fixed-resource class

Let D(\mathcal{R}) be the finite class of all monoid-aggregated summaries over \mathcal{R}.

:::

::: {.annotation .annotation--static #lem-cardinality}

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.

:::

::: {.annotation .annotation--static #def-context-equiv}

Definition 3.2 Fixed-resource context equivalence

For rooted cop-labeled trees X, Y, define


X \sim_{\mathcal{R}} Y

if for every rooted one-hole context K[\square] and every summary $P \in D(\mathcal{R})$,


P(K[X]) = P(K[Y]).

:::

::: {.annotation .annotation--static #thm-behavior-vector}

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.

:::

::: {.annotation .annotation--static #cor-congruence}

Corollary 3.5 Fixed-resource congruence

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, 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 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 $(0_M)_{P \in D(\mathcal{R})}$.

:::

::: {.annotation .annotation--static #def-contribution}

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})}


(\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.

:::

::: {.annotation .annotation--static #rem-notation}

Remark 4.3 Notation checkpoint

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.

:::

::: {.annotation .annotation--static #lem-aggregate}

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.

:::

::: {.annotation .annotation--static #lem-existence}

Lemma 5.2 Existence of index and period

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.

:::

::: {.annotation .annotation--static #lem-unary-pumping}

Lemma 5.4 Exact unary pumping

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 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).

:::

::: {.annotation .annotation--static #thm-sibling-pumping}

Theorem 6.2 Exact sibling pumping at one node

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, 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, 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 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. Hence every behavior type has bound 0 if its contribution is zero and bound 1 otherwise.

:::

Finite cyclic groups

::: {.annotation .annotation--static #prop-cyclic}

Proposition 7.3 Cyclic group pumping

Let M = \mathbb{Z}/q\mathbb{Z} under addition. For g \in M,


\operatorname{ind}(g) = 0, \quad \operatorname{per}(g) = \operatorname{ord}(g) = \frac{q}{\gcd(q, g)},

with the convention that \operatorname{ord}(0) = 1. Thus


N(g) = \operatorname{ord}(g) - 1.

:::

::: {.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 of g in the 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.

:::

Threshold monoids

::: {.annotation .annotation--static #def-threshold}

Definition 7.5 Threshold monoid

For T \geq 0, let


\Theta_T := \{0, 1, \ldots, T\}

with operation


x \oplus y := \min(T, x + y)

and identity 0.

:::

::: {.annotation .annotation--static #prop-threshold}

Proposition 7.6 Threshold pumping

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.

:::

Threshold-times-cyclic products

::: {.annotation .annotation--static #prop-hybrid}

Proposition 7.8 Hybrid threshold-residue pumping

Let


M = \Theta_T \times \mathbb{Z}/q\mathbb{Z}

with coordinatewise operation, and let g = (g_{\text{thr}}, g_{\text{cyc}}). Then


\operatorname{ind}(g) = \begin{cases} 0, & g_{\text{thr}} = 0, \\ \lceil T/g_{\text{thr}} \rceil, & g_{\text{thr}} > 0, \end{cases}

and


\operatorname{per}(g) = \operatorname{ord}(g_{\text{cyc}}).

Consequently


N(g) = \operatorname{ind}(g) + \operatorname{per}(g) - 1.

:::

::: {.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 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.

:::

::: {.annotation .annotation--static #ex-stars}

Example 8.4 Stars in standard monoids

Assume unit contribution \gamma_b = g.

Boolean support: all positive n are equivalent; n = 0 is separate from n > 0 if g \neq 0.
Cyclic \mathbb{Z}/q\mathbb{Z}: n and m are equivalent exactly modulo \operatorname{ord}(g).
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)).

:::

::: {.annotation .annotation--static #prop-unary-periodic}

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.

:::

::: {.annotation .annotation--static #rem-variable-labels}

Remark 8.8 Variable labels

If labels vary along a unary path, one obtains the finite 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 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 and Proposition 8.9, not by aggregate inequality alone. This is precisely why split-versus-concentrated examples are diagnostically interesting rather than trivial.

:::

::: {.annotation .annotation--static #rem-family-matters}

Remark 8.11 Why this family matters

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).

:::

::: {.annotation .annotation--static #lem-label-normalization}

Lemma 9.2 Label normalization preserves behavior

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.

:::

::: {.annotation .annotation--static #lem-minimal-exists}

Lemma 9.4 Size-minimal representatives exist

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 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, 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, 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, Corollary 3.5 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:

every root-to-leaf path has at most |B_{\mathcal{R}}| vertices;
at every node, behavior type b occurs among the children at most N_{\mathcal{R}}(b) times, for every b \in B_{\mathcal{R}}.

:::

::: {.annotation .annotation--static #rem-sharper-universe}

Remark 9.8 Sharper realizable normal universe

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.

:::

::: {.annotation .annotation--static #thm-normal-universe}

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, choose a normalized size-minimal representative T realizing b. By Theorem 9.5, $T$ satisfies the exact sibling bounds. By Theorem 9.6, 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 \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.

:::

::: {.annotation .annotation--static #def-canonical-rep}

Definition 10.2 Canonical representative

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.

:::

::: {.annotation .annotation--static #thm-canonical}

Theorem 10.3 Canonical representative theorem

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, 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. 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, 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. [□]{.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:

support-only resources with bounded |S|;
threshold resources with threshold T \leq t and |S| \leq s;
cyclic resources with modulus q \leq Q and |S| \leq s;
hybrid threshold-residue resources with bounded thresholds, moduli, and state counts.

:::

::: {.annotation .annotation--static #def-separation}

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}.

:::

::: {.annotation .annotation--static #def-separation-cost}

Definition B.3 Separation cost, schematic

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.

:::

::: {.annotation .annotation--static #rem-games-deferred}

Remark B.4 Why games are deferred

For fixed \mathcal{R}, a game characterization 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.
T2. Canonical family bounds. Boolean, cyclic, threshold, and hybrid monoids have explicit pumping signatures. Proved in Section 7.
T3. Finite normal representatives. Every fixed-resource class has a representative in a finite normal universe. Proved in Theorem 9.9.
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.

:::::

63 KiB Raw Blame History Unescape Escape