From 711912cdfb71e7faaaede003c984452a8d061500 Mon Sep 17 00:00:00 2001 From: Levi Neuwirth Date: Sat, 16 May 2026 19:20:57 -0400 Subject: [PATCH] auto: 2026-05-16T23:20:57Z [skip ci] --- .../essays/index-period-normal-forms/index.md | 1963 +++++++++++++++++ .../essays/index-period-normal-forms/mark.svg | 70 + 2 files changed, 2033 insertions(+) create mode 100644 content/essays/index-period-normal-forms/index.md create mode 100644 content/essays/index-period-normal-forms/mark.svg diff --git a/content/essays/index-period-normal-forms/index.md b/content/essays/index-period-normal-forms/index.md new file mode 100644 index 0000000..46c3aa4 --- /dev/null +++ b/content/essays/index-period-normal-forms/index.md @@ -0,0 +1,1963 @@ +--- +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" +affiliation: + - "Brown University | https://www.brown.edu" +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 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: + +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. + +
+::: + +::: {.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](#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 $(0_M)_{P +\in D(\mathcal{R})}$. + +
+::: + +::: {.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* + +$$ +\gamma_b \in M^{D(\mathcal{R})} +$$ + +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. + +
+::: + +::: {.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} +:::: + +::: + +::: {.annotation .annotation--static #rem-right-object} +
+Remark 4.5 +Why this is the right object +
+
+ +The old bound used only the size of $M^{D(\mathcal{R})}$. That is enough to +prove eventual pumping, but it ignores the actual contribution elements +$\gamma_b$. The exact bound for a child type $b$ is controlled by the cyclic +submonoid generated by $\gamma_b$, not by the whole product monoid. + +
+::: + +## 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](#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. 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](#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. + +
+::: + +::: {.annotation .annotation--static #ex-stars} +
+Example 8.4 +Stars in standard monoids +
+
+ +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, + +$$ +(U_a(b))_P = f_P(a, \alpha_P(b_P)). +$$ + +
+::: + +::: {.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](#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. + +
+::: + +::: {.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](#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 $b \in B_{\mathcal{R}}$. + +
+::: + +::: {.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](#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 $\preceq$ 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](#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. + +
+::: + +::: {.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](#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. + +::::: diff --git a/content/essays/index-period-normal-forms/mark.svg b/content/essays/index-period-normal-forms/mark.svg new file mode 100644 index 0000000..ddbf638 --- /dev/null +++ b/content/essays/index-period-normal-forms/mark.svg @@ -0,0 +1,70 @@ + + A small rooted tree above an index-period rho-orbit, joined by a dashed abstraction arrow + A frontispiece mark for "Index-Period Normal Forms for Monoid-Aggregated Recursive Summaries." The upper figure is a small rooted tree with one node marked: the child whose contribution is being analyzed. The lower figure is a rho-shape orbit — a pre-periodic tail of three points leading into a four-cycle — the classical picture of an element's index and period in a finite monoid. The dashed arrow between them is the paper's central move: abstract from the combinatorial tree to the algebraic invariant of one element's contribution. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +