auto: 2026-05-16T23:42:26Z [skip ci]

content/essays/index-period-normal-forms/index.md

@@ -39,7 +39,8 @@ history:
 
 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
+[pigeonhole](https://en.wikipedia.org/wiki/Pigeonhole_principle) bound in a
+huge product monoid. The correct next move is to
 analyze, for each behavior type, the cyclic submonoid generated by its child
 contribution. This gives an exact index-period pumping rule.
 
@@ -51,10 +52,13 @@ 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
+every [finite commutative monoid](https://en.wikipedia.org/wiki/Monoid) is a
+[semilattice](https://en.wikipedia.org/wiki/Semilattice) of [abelian
+groups](https://en.wikipedia.org/wiki/Abelian_group). That
 statement holds for special regular/Clifford-type commutative monoids, not
 for arbitrary finite commutative monoids. Threshold monoids already contain
-aperiodic saturation behavior that is not group-like. The universal
+[aperiodic](https://en.wikipedia.org/wiki/Aperiodic_semigroup) saturation
+behavior that is not group-like. The universal
 finite-monoid fact needed here is simpler: for each element $g$ of a finite
 monoid, the sequence
 
@@ -622,7 +626,8 @@ $$
 N(g) := \operatorname{ind}(g) + \operatorname{per}(g) - 1.
 $$
 
-The exact pumping lemma says every coefficient of $g$ can be reduced to at
+The exact [pumping lemma](https://en.wikipedia.org/wiki/Pumping_lemma) says
+every coefficient of $g$ can be reduced to at
 most $N(g)$ without changing the monoid value.
 
 </div>
@@ -808,7 +813,8 @@ $0$ with period $1$. If $g = 1$, then $0g = 0$ and $ng = 1$ for all $n \geq
 <div class="annotation-body">
 
 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
+1\}, \vee, 0)$, every element is
+[idempotent](https://en.wikipedia.org/wiki/Idempotence). Hence every behavior type has
 bound $0$ if its contribution is zero and bound $1$ otherwise.
 
 </div>
@@ -842,7 +848,8 @@ $$
 
 :::: 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
+is the [additive order](https://en.wikipedia.org/wiki/Order_(group_theory))
+of $g$ in the [cyclic group](https://en.wikipedia.org/wiki/Cyclic_group). The displayed formula for
 the order in $\mathbb{Z}/q\mathbb{Z}$ is standard. [□]{.proof-qed}
 ::::
 
@@ -1262,8 +1269,9 @@ bound is $|B_{\mathcal{R}}| + 1$ vertices for a repeated behavior vector.
 </div>
 <div class="annotation-body">
 
-If labels vary along a unary path, one obtains the finite transformation
+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,
+semigroup](https://en.wikipedia.org/wiki/Transformation_semigroup)
+generated by the maps $U_a$ for $a \in A$: this is the subsemigroup,
 under composition, of the finite monoid of all self-maps of $B_{\mathcal{R}}$
 generated by the maps $U_a$. Long labeled unary words can be pumped using
 repetitions in this finite transformation semigroup, but the
@@ -1651,7 +1659,7 @@ are intrinsically unique, which is generally false.
 </div>
 <div class="annotation-body">
 
-Fix a total order $\preceq$ on the finite universe $U_{\mathcal{R}}$. For
+Fix a [total order](https://en.wikipedia.org/wiki/Total_order) $\preceq$ on the finite universe $U_{\mathcal{R}}$. For
 example, order first by number of vertices, then by height, then recursively
 by sorted child lists and vertex labels.
 
@@ -1905,7 +1913,9 @@ separating $X$ and $Y$, and set $\operatorname{sep}_{\mathcal{F}}(X, Y) =
 </div>
 <div class="annotation-body">
 
-For fixed $\mathcal{R}$, a game characterization is nearly tautological:
+For fixed $\mathcal{R}$, a
+[game characterization](https://en.wikipedia.org/wiki/Ehrenfeucht%E2%80%93Fra%C3%AFss%C3%A9_game)
+is nearly tautological:
 Spoiler can choose a differing coordinate $P \in D(\mathcal{R})$ if one
 exists, and otherwise Duplicator wins because behavior vectors agree. A
 nontrivial game should therefore characterize resource growth, restricted