Prerequisite Mathematical Definitions

1 Transition Independence

Setting.

We consider functions $f\colon S_{1}\times\cdots\times S_{n}\to T$ , where the factor sets $S_{i}$ and the codomain $T$ carry no assumed structure unless otherwise noted. We write $\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}}$ for the tuple identical to $\boldsymbol{a}$ except that the $i$ -th component is replaced by $a_{i}^{\prime}$ , and write

\mathcal{D}_{i}\coloneqq\{(f(\boldsymbol{a}),\,a_{i}):\boldsymbol{a}\in S_{1}% \times\cdots\times S_{n}\}\subseteq T\times S_{i}

for the joint range of $(f,\mathrm{pr}_{i})$ — the set of $(v,a_{i})$ pairs realized by some $\boldsymbol{a}$ . The auxiliary functions introduced in the conditions below are defined on $\mathcal{D}_{i}$ (or $\mathcal{D}_{i}\times S_{i}$ ).

We introduce three formulations of transition independence forming a hierarchy of decreasing generality: the transition-map condition requires no structure on $T$ ; the finite-difference condition requires that $(T,+)$ be a group, so that differences are defined, writing $x-y$ for $x+(-y)$ ; and the derivative condition requires that $S_{i}$ be an open interval of $\mathbb{R}$ , that $T$ be a normed vector space, and that $f$ be differentiable in $s_{i}$ , with the regularity supporting its equivalences stated in the summary below. Each formulation is stated at its natural level of generality.

Transition map.

The variable $s_{i}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ with respect to $f$ iff there exists a transition map $H_{i}\colon\mathcal{D}_{i}\times S_{i}\to T$ (necessarily unique on its declared domain) such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n},\;\;\forall\,a_{i}^{% \prime}\in S_{i}\quad f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})=H_{i}\bigl{(}f% (\boldsymbol{a}),\,a_{i},\,a_{i}^{\prime}\bigr{)}.

(1)

The non-trivial content of the condition is that the remaining coordinates $\boldsymbol{a}_{j\neq i}$ are dropped in favor of the output $f(\boldsymbol{a})$ . No structure on $T$ is required: the condition is meaningful for functions valued in discrete spaces, ordered sets, manifolds, or any other codomain. For $H_{i}$ to be well-defined as a function on $\mathcal{D}_{i}\times S_{i}$ , the following consistency condition is necessary and sufficient.

Remark 1 (Output-determinacy).

For every $\boldsymbol{a},\boldsymbol{b}\in S_{1}\times\cdots\times S_{n}$ , if $f(\boldsymbol{a})=f(\boldsymbol{b})$ and $a_{i}=b_{i}$ , then $f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})=f(\boldsymbol{b}|_{b_{i}=a_{i}^{% \prime}})$ for every $a_{i}^{\prime}\in S_{i}$ .

Proof of the equivalence. ( $\Rightarrow$ ) If $H_{i}$ satisfies (1) and $f(\boldsymbol{a})=f(\boldsymbol{b})$ with $a_{i}=b_{i}$ , then $f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})=H_{i}(f(\boldsymbol{a}),a_{i},a_{i}^% {\prime})=H_{i}(f(\boldsymbol{b}),b_{i},a_{i}^{\prime})=f(\boldsymbol{b}|_{b_{% i}=a_{i}^{\prime}})$ . ( $\Leftarrow$ ) Given output-determinacy, define $H_{i}(v,s,a_{i}^{\prime}):=f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})$ for any $\boldsymbol{a}$ with $f(\boldsymbol{a})=v$ and $a_{i}=s$ ; output-determinacy makes the choice of $\boldsymbol{a}$ immaterial, so $H_{i}$ is well-defined and (1) holds by construction.

Geometrically, output-determinacy says that the level sets of $f$ do not “twist” along the $s_{i}$ -direction: two points sharing both the same output and the same $i$ -th coordinate must land on the same level set as each other — in general a different one from where they started — when $s_{i}$ is varied. In full generality, the condition asserts that the pair $(f(\boldsymbol{a}),a_{i})$ is a sufficient representation for predicting how $f$ responds to movement of $s_{i}$ .

Finite-difference formulation.

When $(T,+)$ is a group, the transition-map condition has an equivalent algebraic formulation: there exists $G_{i}\colon\mathcal{D}_{i}\times S_{i}\to T$ such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n},\;\;\forall\,a_{i}^{% \prime}\in S_{i}\quad f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})-f(\boldsymbol{% a})=G_{i}\bigl{(}f(\boldsymbol{a}),\,a_{i},\,a_{i}^{\prime}\bigr{)}.

(2)

The two formulations are related by $G_{i}(v,a_{i},a_{i}^{\prime})=H_{i}(v,a_{i},a_{i}^{\prime})-v$ and $H_{i}(v,a_{i},a_{i}^{\prime})=G_{i}(v,a_{i},a_{i}^{\prime})+v$ .

Derivative formulation.

Suppose $S_{i}$ is an open interval of $\mathbb{R}$ and $T$ is a normed vector space. When $f$ is differentiable in $s_{i}$ , the derivative condition is: there exists $F_{i}\colon\mathcal{D}_{i}\to T$ such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n}\quad\frac{\partial f(% \boldsymbol{s})}{\partial s_{i}}\bigg{|}_{\boldsymbol{s}=\boldsymbol{a}}=F_{i}% \bigl{(}f(\boldsymbol{a}),\,a_{i}\bigr{)}.

(3)

When (3) holds, the map $t\mapsto f(\boldsymbol{a}|_{a_{i}=t})$ satisfies the initial value problem

\frac{dy}{dt}=F_{i}(y,t),\qquad y(a_{i})=f(\boldsymbol{a}),

(4)

whose solution determines $f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})$ entirely from $(f(\boldsymbol{a}),a_{i})$ provided the solution is unique.¹¹1Solution and uniqueness are taken, here and below, in the class specified in Appendix A. The ODE (4) is the bridge between the derivative condition and the transition map.

Summary of equivalences.

The relationships among (1), (2), and (3) can be summarized as follows:

(a)

Condition (1) is equivalent to output-determinacy (Remark 1). No structure on $T$ is required.
(b)

If $(T,+)$ is a group, conditions (1) and (2) are equivalent via $H_{i}(v,a_{i},a_{i}^{\prime})=G_{i}(v,a_{i},a_{i}^{\prime})+v$ and $G_{i}(v,a_{i},a_{i}^{\prime})=H_{i}(v,a_{i},a_{i}^{\prime})-v$ .
(c)

Suppose $S_{i}$ is an open interval of $\mathbb{R}$ , $T$ is a normed vector space, and $f$ is differentiable in $s_{i}$ throughout. Then (1) $\Rightarrow$ (3) holds, via $F_{i}(v,a_{i}):=\partial_{a_{i}^{\prime}}H_{i}(v,a_{i},a_{i}^{\prime})\big{|}_% {a_{i}^{\prime}=a_{i}}$ . The reverse direction (3) $\Rightarrow$ (1) holds provided the IVP (4) admits a unique solution from each initial datum $(v,a_{i})\in\mathcal{D}_{i}$ . Under these hypotheses the first two formulations are equivalent and imply the third; when in addition the IVPs (4) arising from (3) are uniquely solvable in the sense of Appendix A, the third returns the first, and the three formulations coincide. The trajectory $t\mapsto f(\boldsymbol{a}|_{a_{i}=t})$ solves the initial value problem

$\frac{dy}{dt}=F_{i}(y,t),\qquad y(a_{i})=f(\boldsymbol{a}),$

and, provided the integrand below is integrable — for instance when $f$ is continuously differentiable in $s_{i}$ — the finite-difference and derivative formulations are related by

$G_{i}\bigl{(}f(\boldsymbol{a}),a_{i},a_{i}^{\prime}\bigr{)}=\int_{a_{i}}^{a_{i% }^{\prime}}F_{i}\bigl{(}f(\boldsymbol{a}|_{a_{i}=t}),\,t\bigr{)}\,dt.$ (5)

By default we will mean (1) when referring to ‘transition independence.’

2 Transition Context

Setting.

Transition independence may be partial: $s_{i}$ might be transition-independent of some variables, with the others retained as its context, without being fully independent. We formalize this by partitioning $\{1,\ldots,n\}$ as $\{i\}\cup J\cup K$ , where $J$ is the context and $K=\{1,\ldots,n\}\setminus(\{i\}\cup J)$ contains the remaining variables. Each of the three formulations of Section 1 extends to this conditional setting, the auxiliary maps $H_{i,J}$ , $G_{i,J}$ , $F_{i,J}$ now allowed to depend on the context values $\boldsymbol{a}_{J}$ .

Transition-map formulation (no structure on $T$ ).

The variable $s_{i}$ is transition-independent of $\boldsymbol{s}_{K}$ given context $\boldsymbol{s}_{J}$ iff there exists a map $H_{i,J}\colon\mathcal{D}_{i}^{J}\times S_{i}\to T$ (denoted $H_{i,J}(f(\boldsymbol{a}),a_{i},a_{i}^{\prime},\boldsymbol{a}_{J})$ , with the target $a_{i}^{\prime}$ written before the context $\boldsymbol{a}_{J}$ ) such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n},\;\;\forall\,a_{i}^{% \prime}\in S_{i}\quad f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})=H_{i,J}\bigl{(% }f(\boldsymbol{a}),\,a_{i},\,a_{i}^{\prime},\,\boldsymbol{a}_{J}\bigr{)},

(6)

where $\mathcal{D}_{i}^{J}\coloneqq\{(f(\boldsymbol{a}),\,a_{i},\,\boldsymbol{a}_{J})% :\boldsymbol{a}\in S_{1}\times\cdots\times S_{n}\}$ . Setting $J=\emptyset$ recovers (1); setting $J=\{1,\ldots,n\}\setminus\{i\}$ holds trivially.

Remark 2 (Output-determinacy on context $J$ ).

For every $\boldsymbol{a},\boldsymbol{b}\in S_{1}\times\cdots\times S_{n}$ , if $f(\boldsymbol{a})=f(\boldsymbol{b})$ , $a_{i}=b_{i}$ , and $\boldsymbol{a}_{J}=\boldsymbol{b}_{J}$ , then $f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})=f(\boldsymbol{b}|_{b_{i}=a_{i}^{% \prime}})$ for every $a_{i}^{\prime}\in S_{i}$ .

As in the full case, output-determinacy on context $J$ is necessary and sufficient for $H_{i,J}$ to be well-defined on $\mathcal{D}_{i}^{J}\times S_{i}$ . Setting $J=\emptyset$ recovers Remark 1.

Geometrically, this is still essentially the no-twist condition of Section 1, now imposed within each context slice: with $\boldsymbol{s}_{J}$ fixed, the level sets of $f$ do not “twist” along the $s_{i}$ -direction.

Remark 3 (Partial-function form).

For every $\boldsymbol{c}_{J}\in\prod_{j\in J}S_{j}$ , the variable $s_{i}$ is fully transition-independent on the partial function $f|_{\boldsymbol{s}_{J}=\boldsymbol{c}_{J}}\colon\prod_{m\in\{i\}\cup K}S_{m}\to T$ , in the sense of Section 1.²²2Here and throughout, for $E\subseteq\{1,\ldots,n\}$ we write $\boldsymbol{a}_{E}\coloneqq(a_{j})_{j\in E}$ for the corresponding subtuple, abbreviated $\boldsymbol{a}_{j\neq i}$ when $E=\{1,\ldots,n\}\setminus\{i\}$ , and $\mathrm{pr}_{i}$ for the $i$ -th projection $\boldsymbol{a}\mapsto a_{i}$ . Products indexed by other finite sets, and products of products, are read through the evident bijections: every condition of Section 1 distinguishes the substituted coordinate from the remaining coordinates and never consults the indexing or grouping of the factors, so the framework transfers verbatim.

Remark 3 is equivalent to $s_{i}$ being transition-independent of $\boldsymbol{s}_{K}$ given context $\boldsymbol{s}_{J}$ : since $\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}}$ fixes $\boldsymbol{a}_{J}$ , the map $H_{i,J}(\,\cdot\,,\,\cdot\,,\,\cdot\,,\boldsymbol{c}_{J})$ is the transition map of $f|_{\boldsymbol{s}_{J}=\boldsymbol{c}_{J}}$ .

In other words, partial transition independence is full transition independence on sets of partial functions. Context is what’s fixed.

Finite-difference formulation (group $T$ ).

When $(T,+)$ is a group, the condition has the equivalent algebraic form: there exists $G_{i,J}\colon\mathcal{D}_{i}^{J}\times S_{i}\to T$ such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n},\;\;\forall\,a_{i}^{% \prime}\in S_{i}\quad f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})-f(\boldsymbol{% a})=G_{i,J}\bigl{(}f(\boldsymbol{a}),\,a_{i},\,a_{i}^{\prime},\,\boldsymbol{a}% _{J}\bigr{)}.

(7)

The two formulations are related by $G_{i,J}(v,a_{i},a_{i}^{\prime},\boldsymbol{a}_{J})=H_{i,J}(v,a_{i},a_{i}^{% \prime},\boldsymbol{a}_{J})-v$ and $H_{i,J}(v,a_{i},a_{i}^{\prime},\boldsymbol{a}_{J})=G_{i,J}(v,a_{i},a_{i}^{% \prime},\boldsymbol{a}_{J})+v$ .

Derivative formulation ( $S_{i}$ real, $T$ normed).

Suppose $S_{i}$ is an open interval of $\mathbb{R}$ and $T$ is a normed vector space. When $f$ is differentiable in $s_{i}$ , a weaker condition is: there exists $F_{i,J}\colon\mathcal{D}_{i}^{J}\to T$ such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n}\quad\frac{\partial f(% \boldsymbol{s})}{\partial s_{i}}\bigg{|}_{\boldsymbol{s}=\boldsymbol{a}}=F_{i,% J}\bigl{(}f(\boldsymbol{a}),\,a_{i},\,\boldsymbol{a}_{J}\bigr{)}.

(8)

When (8) holds, the map $t\mapsto f(\boldsymbol{a}|_{a_{i}=t})$ satisfies the initial value problem

\frac{dy}{dt}=F_{i,J}(y,t,\boldsymbol{a}_{J}),\qquad y(a_{i})=f(\boldsymbol{a}),

(9)

whose solution determines $f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})$ entirely from $(f(\boldsymbol{a}),a_{i},\boldsymbol{a}_{J})$ provided the solution is unique.³³3In the class of Appendix A, with $(y(t),t,\boldsymbol{a}_{J})\in\mathcal{D}_{i}^{J}$ in place of $(y(t),t)\in\mathcal{D}_{i}$ . The ODE (9) is the bridge between the derivative condition and the transition map. By default we will mean (6) when referring to ‘transition context’, ’partial transition independence’, or ’transition independence given …’.

Summary of equivalences.

The relationships among (6), (7), and (8) parallel those of Section 1, with the context tuple $\boldsymbol{a}_{J}$ entering each auxiliary map as a passenger argument:

(a)

Condition (6) is equivalent to output-determinacy on context $J$ (Remark 2). No structure on $T$ is required.
(b)

If $(T,+)$ is a group, conditions (6) and (7) are equivalent via $H_{i,J}(v,a_{i},a_{i}^{\prime},\boldsymbol{a}_{J})=G_{i,J}(v,a_{i},a_{i}^{% \prime},\boldsymbol{a}_{J})+v$ and $G_{i,J}(v,a_{i},a_{i}^{\prime},\boldsymbol{a}_{J})=H_{i,J}(v,a_{i},a_{i}^{% \prime},\boldsymbol{a}_{J})-v$ .
(c)

Suppose $S_{i}$ is an open interval of $\mathbb{R}$ , $T$ is a normed vector space, and $f$ is differentiable in $s_{i}$ throughout. Then (6) $\Rightarrow$ (8) holds. The reverse direction (8) $\Rightarrow$ (6) holds provided the IVP (9) admits a unique solution from each initial datum. Under these hypotheses the first two formulations are equivalent and imply the third; when in addition the IVPs (9) arising from (8) are uniquely solvable in the sense of Appendix A, the third returns the first, and the three formulations coincide. The trajectory $t\mapsto f(\boldsymbol{a}|_{a_{i}=t})$ solves the initial value problem

$\frac{dy}{dt}=F_{i,J}(y,t,\boldsymbol{a}_{J}),\qquad y(a_{i})=f(\boldsymbol{a}),$

and, provided the integrand below is integrable — for instance when $f$ is continuously differentiable in $s_{i}$ — the finite-difference and derivative formulations are related by

$G_{i,J}\bigl{(}f(\boldsymbol{a}),a_{i},a_{i}^{\prime},\boldsymbol{a}_{J}\bigr{% )}=\int_{a_{i}}^{a_{i}^{\prime}}F_{i,J}\bigl{(}f(\boldsymbol{a}|_{a_{i}=t}),\,% t,\,\boldsymbol{a}_{J}\bigr{)}\,dt.$ (10)

From now on we work with the transition-map version (6).

3 Transform Independence

Setting.

Sections 1 and 2 describe the change in a variable by its initial and final values. This section describes it by a transform carrying the initial value to the final one. Fix a coordinate $i$ , and let $\Delta_{i}$ be a family of transforms of $S_{i}$ : each $\delta_{i}\in\Delta_{i}$ is a map $\delta_{i}\colon S_{i}\to S_{i}$ , and we write $\delta_{i}\ast a_{i}$ for $\delta_{i}(a_{i})$ , so that the substitution of $a_{i}^{\prime}$ for $a_{i}$ is represented by a transform $\delta_{i}$ if $a_{i}^{\prime}=\delta_{i}\ast a_{i}$ . No structure on the family is assumed; in particular $\Delta_{i}$ need not be closed under composition. We write

\mathcal{E}_{i}\coloneqq\{(f(\boldsymbol{a}),\,\delta_{i}):\boldsymbol{a}\in S% _{1}\times\cdots\times S_{n},\;\delta_{i}\in\Delta_{i}\}=f(S_{1}\times\cdots% \times S_{n})\times\Delta_{i}

for the product of the range of $f$ with the family — the value and the transform do not constrain each other, since $\delta_{i}$ is chosen independently of $\boldsymbol{a}$ . The auxiliary functions introduced in the conditions below are defined on $\mathcal{E}_{i}$ : they consult the transform applied, not the initial value $a_{i}$ or the final value $a_{i}^{\prime}$ .

Transform-map formulation (no structure on $T$ ).

The variable $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and $\Delta_{i}$ iff there exists a transform map $H_{i}^{\Delta}\colon\mathcal{E}_{i}\to T$ (necessarily unique on its declared domain) such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n},\;\;\forall\,\delta_{% i}\in\Delta_{i}\quad f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})=H_{i}^{% \Delta}\bigl{(}f(\boldsymbol{a}),\,\delta_{i}\bigr{)}.

(11)

The non-trivial content of the condition is that the remaining coordinates $\boldsymbol{a}_{j\neq i}$ and the initial value $a_{i}$ are both dropped in favor of the output $f(\boldsymbol{a})$ and the transform $\delta_{i}$ alone. For $H_{i}^{\Delta}$ to be well-defined as a function on $\mathcal{E}_{i}$ , the following consistency condition is necessary and sufficient.

Remark 4 (Transform output-determinacy).

For every $\boldsymbol{a},\boldsymbol{b}\in S_{1}\times\cdots\times S_{n}$ , if $f(\boldsymbol{a})=f(\boldsymbol{b})$ , then $f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})=f(\boldsymbol{b}|_{b_{i}=% \delta_{i}\ast b_{i}})$ for every $\delta_{i}\in\Delta_{i}$ .

Proof of the equivalence. ( $\Rightarrow$ ) If $H_{i}^{\Delta}$ satisfies (11) and $f(\boldsymbol{a})=f(\boldsymbol{b})$ , then $f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})=H_{i}^{\Delta}(f(\boldsymbol{a% }),\delta_{i})=H_{i}^{\Delta}(f(\boldsymbol{b}),\delta_{i})=f(\boldsymbol{b}|_% {b_{i}=\delta_{i}\ast b_{i}})$ for every $\delta_{i}\in\Delta_{i}$ . ( $\Leftarrow$ ) Given transform output-determinacy, define $H_{i}^{\Delta}(v,\delta_{i}):=f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})$ for any $\boldsymbol{a}$ with $f(\boldsymbol{a})=v$ ; the condition makes the choice of $\boldsymbol{a}$ immaterial, so $H_{i}^{\Delta}$ is well-defined and (11) holds by construction.

Transform output-determinacy says that two tuples sharing the same output must respond identically to each other under every transform in $\Delta_{i}$ , whatever their initial values. Over the substitutions that $\Delta_{i}$ represents, ordinary output-determinacy (Remark 1) makes that same demand, but only of tuples that also share an initial value: for those, the transform applied and the final value describe the same substitution. Geometrically, the condition says that applying a transform $\delta_{i}$ to the $i$ -th coordinate carries every level set of $f$ into a single level set — in general a different one — and so descends along $f$ to the induced map $v\mapsto H_{i}^{\Delta}(v,\delta_{i})$ on the range of $f$ . In full generality, the condition asserts that the output $f(\boldsymbol{a})$ alone is a sufficient representation for predicting how $f$ responds to each transform in $\Delta_{i}$ .

Finite-difference formulation (group $T$ ).

When $(T,+)$ is a group, the transform-map condition has an equivalent algebraic formulation: there exists $G_{i}^{\Delta}\colon\mathcal{E}_{i}\to T$ such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n},\;\;\forall\,\delta_{% i}\in\Delta_{i}\quad f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})-f(% \boldsymbol{a})=G_{i}^{\Delta}\bigl{(}f(\boldsymbol{a}),\,\delta_{i}\bigr{)}.

(12)

The two formulations are related by $G_{i}^{\Delta}(v,\delta_{i})=H_{i}^{\Delta}(v,\delta_{i})-v$ and $H_{i}^{\Delta}(v,\delta_{i})=G_{i}^{\Delta}(v,\delta_{i})+v$ . No derivative (instantaneous) formulation is included; the two formulations above are the only ones we use.

Summary of equivalences.

The relationships between (11) and (12) parallel items (a) and (b) of Section 1, with the pair $(f(\boldsymbol{a}),\,\delta_{i})$ replacing $(f(\boldsymbol{a}),\,a_{i},\,a_{i}^{\prime})$ throughout:

(a)

Condition (11) is equivalent to transform output-determinacy (Remark 4). No structure on $T$ is required.
(b)

If $(T,+)$ is a group, conditions (11) and (12) are equivalent via $H_{i}^{\Delta}(v,\delta_{i})=G_{i}^{\Delta}(v,\delta_{i})+v$ and $G_{i}^{\Delta}(v,\delta_{i})=H_{i}^{\Delta}(v,\delta_{i})-v$ .

Remark 5 (Keeping the initial value).

Passing from the transition-map condition (1) to the transform-map condition (11) replaces the pair $(a_{i},a_{i}^{\prime})$ with the transform $\delta_{i}$ alone. If we replaced it instead with the pair $(a_{i},\delta_{i})$ — the initial value kept — the condition would read: there exists $H\colon\mathcal{D}_{i}\times\Delta_{i}\to T$ (necessarily unique on its declared domain) such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n},\;\;\forall\,\delta_{% i}\in\Delta_{i}\quad f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})=H\bigl{(}% f(\boldsymbol{a}),\,a_{i},\,\delta_{i}\bigr{)}.

(13)

Condition (13) is equivalent to condition (1) quantified over the substitutions that $\Delta_{i}$ represents: there exists $H_{i}$ such that $f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})=H_{i}(f(\boldsymbol{a}),a_{i},a_{i}^% {\prime})$ for every $\boldsymbol{a}\in S_{1}\times\cdots\times S_{n}$ and every target $a_{i}^{\prime}=\delta_{i}\ast a_{i}$ with $\delta_{i}\in\Delta_{i}$ .

Proof of the equivalence. ( $\Rightarrow$ ) Given $H$ satisfying (13), define $H_{i}(v,a_{i},a_{i}^{\prime}):=H(v,a_{i},\delta_{i})$ for any $\delta_{i}\in\Delta_{i}$ with $\delta_{i}\ast a_{i}=a_{i}^{\prime}$ . The choice is immaterial: for any $\boldsymbol{a}$ realizing $(v,a_{i})$ , every representing $\delta_{i}$ gives $H(v,a_{i},\delta_{i})=f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})=f(% \boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})$ — the substituted tuple depends only on the final value, not on the transform that represents the substitution — so $H_{i}$ is well-defined and the restricted condition holds by construction. ( $\Leftarrow$ ) Given $H_{i}$ , set $H(v,a_{i},\delta_{i}):=H_{i}(v,a_{i},\delta_{i}\ast a_{i})$ ; the target $\delta_{i}\ast a_{i}$ is represented by $\delta_{i}$ itself, so (13) holds.

When every substitution of $a_{i}^{\prime}$ for $a_{i}$ is represented by some transform in $\Delta_{i}$ , the restriction disappears: condition (13) is then equivalent to condition (1) itself.

Remark 6 (Reach and the closure).

Deleting the initial value from the pair $(a_{i},\delta_{i})$ of the preceding remark removes information and nothing else: any transform map serves, through the preceding remark, as a transition map on the substitutions that $\Delta_{i}$ represents. Transform output-determinacy (Remark 4) therefore implies output-determinacy (Remark 1) over the substitutions that $\Delta_{i}$ represents. The implication reaches further.

We write $\overline{\Delta}_{i}$ for the family of maps of $S_{i}$ carried out by one or more transforms in $\Delta_{i}$ applied in succession, each to the value the previous one produced. Each map in $\overline{\Delta}_{i}$ is itself a transform of $S_{i}$ , so the transform-map condition (11) applies with $\overline{\Delta}_{i}$ in the role of $\Delta_{i}$ .

Lemma 7 (Passage to the closure).

If $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and $\Delta_{i}$ , then it is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and $\overline{\Delta}_{i}$ : there exists a transform map $H_{i}^{\overline{\Delta}}\colon f(S_{1}\times\cdots\times S_{n})\times% \overline{\Delta}_{i}\to T$ (necessarily unique on its declared domain) such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n},\;\;\forall\,% \overline{\delta}_{i}\in\overline{\Delta}_{i}\quad f(\boldsymbol{a}|_{a_{i}=% \overline{\delta}_{i}\ast a_{i}})=H_{i}^{\overline{\Delta}}\bigl{(}f(% \boldsymbol{a}),\,\overline{\delta}_{i}\bigr{)}.

(14)

Proof. Suppose $H_{i}^{\Delta}$ satisfies (11). For a tuple $\boldsymbol{a}$ and a succession $\delta_{i,1},\ldots,\delta_{i,k}\in\Delta_{i}$ , write $a_{i}^{0}:=a_{i}$ and $a_{i}^{j}:=\delta_{i,j}\ast a_{i}^{j-1}$ for the values the succession visits, so that a succession carrying out $\overline{\delta}_{i}\in\overline{\Delta}_{i}$ ends at $a_{i}^{k}=\overline{\delta}_{i}\ast a_{i}$ . The tuple $\boldsymbol{a}|_{a_{i}=a_{i}^{j-1}}$ lies in $S_{1}\times\cdots\times S_{n}$ and carries initial value $a_{i}^{j-1}$ , so (11) applied to it with $\delta_{i,j}$ gives $f(\boldsymbol{a}|_{a_{i}=a_{i}^{j}})=H_{i}^{\Delta}\bigl{(}f(\boldsymbol{a}|_{% a_{i}=a_{i}^{j-1}}),\,\delta_{i,j}\bigr{)}$ : each response is again an output of $f$ and feeds the next application, and induction on $j$ gives $f(\boldsymbol{a}|_{a_{i}=a_{i}^{k}})=H_{i}^{\Delta}(\cdots H_{i}^{\Delta}(f(% \boldsymbol{a}),\,\delta_{i,1})\cdots,\,\delta_{i,k})$ — a value determined by the output $f(\boldsymbol{a})$ and the succession alone. Now define $H_{i}^{\overline{\Delta}}(v,\overline{\delta}_{i})$ by chaining $H_{i}^{\Delta}$ from $v$ along any succession carrying out $\overline{\delta}_{i}$ . The choice is immaterial: every $v$ in the declared domain is realized as $v=f(\boldsymbol{a})$ for some $\boldsymbol{a}$ , and two successions carrying out the same map substitute the same value into $\boldsymbol{a}$ , so both chains compute the output of the same substituted tuple, $f(\boldsymbol{a}|_{a_{i}=\overline{\delta}_{i}\ast a_{i}})$ . Hence $H_{i}^{\overline{\Delta}}$ is well-defined on its declared domain, and (14) holds by construction.

Proposition 8 (Reach).

Suppose $\overline{\Delta}_{i}$ represents every substitution of one value for another: for every $a_{i},a_{i}^{\prime}\in S_{i}$ with $a_{i}^{\prime}\neq a_{i}$ , some $\overline{\delta}_{i}\in\overline{\Delta}_{i}$ has $a_{i}^{\prime}=\overline{\delta}_{i}\ast a_{i}$ . If $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and $\Delta_{i}$ , then $s_{i}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ with respect to $f$ .

Proof. Lemma 7 yields $H_{i}^{\overline{\Delta}}$ satisfying (14). Through Remark 5, with $\overline{\Delta}_{i}$ in the role of $\Delta_{i}$ , it serves as a transition map on the substitutions that $\overline{\Delta}_{i}$ represents: there exists $H_{i}$ with $f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})=H_{i}\bigl{(}f(\boldsymbol{a}),\,a_{% i},\,a_{i}^{\prime}\bigr{)}$ for every $\boldsymbol{a}$ and every target $a_{i}^{\prime}=\overline{\delta}_{i}\ast a_{i}$ , $\overline{\delta}_{i}\in\overline{\Delta}_{i}$ . By hypothesis every target $a_{i}^{\prime}\neq a_{i}$ is of that form; the identity target is free: the substituted tuple is $\boldsymbol{a}$ itself, so setting $H_{i}(v,a_{i},a_{i}):=v$ satisfies (1) there and agrees with any value already forced. Thus $H_{i}$ is defined on all of $\mathcal{D}_{i}\times S_{i}$ and (1) holds: $s_{i}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ with respect to $f$ , and output-determinacy (Remark 1) follows by item (a) of Section 1.

Corollary 9 (Represented substitutions).

If $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and $\Delta_{i}$ , then there exists $H_{i}$ such that $f(\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}})=H_{i}(f(\boldsymbol{a}),a_{i},a_{i}^% {\prime})$ for every $\boldsymbol{a}\in S_{1}\times\cdots\times S_{n}$ and every target $a_{i}^{\prime}=\overline{\delta}_{i}\ast a_{i}$ with $\overline{\delta}_{i}\in\overline{\Delta}_{i}$ — condition (1) quantified over the substitutions that $\overline{\Delta}_{i}$ represents.

Transform independence alone yields the map: no hypothesis on the family enters.

Definition 10 (Achievability).

A value $a_{i}^{\prime}\in S_{i}$ is achievable from $a_{i}\in S_{i}$ with respect to $\Delta_{i}$ iff $a_{i}^{\prime}=a_{i}$ or some $\overline{\delta}_{i}\in\overline{\Delta}_{i}$ has $a_{i}^{\prime}=\overline{\delta}_{i}\ast a_{i}$ : either finitely many transforms in $\Delta_{i}$ applied in succession carry $a_{i}$ to $a_{i}^{\prime}$ , or $a_{i}^{\prime}$ is $a_{i}$ itself. The achievable set of $a_{i}$ is the set of values achievable from $a_{i}$ .

Achievable sets are closed under the transforms: appending one further transform extends a succession. A subset $R\subseteq S_{i}$ is the achievable set of each of its values iff $R$ is closed under the transforms and every value of $R$ is achievable from every other. The achievable set of a single value meets the first requirement but need not meet the second: a succession may carry $a_{i}$ to $a_{i}^{\prime}$ while no succession carries $a_{i}^{\prime}$ back to $a_{i}$ . In this vocabulary, the targets of Corollary 9, together with $a_{i}$ itself, are exactly the values achievable from $a_{i}$ .

Corollary 11 (Achievable sets).

Suppose $R\subseteq S_{i}$ is the achievable set of each of its values. If $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and $\Delta_{i}$ , then $s_{i}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ with respect to the partial function $f|_{s_{i}\in R}\colon S_{1}\times\cdots\times S_{i-1}\times R\times S_{i+1}% \times\cdots\times S_{n}\to T$ with the $i$ -th factor confined to $R$ , in the sense of Section 1.

Proof. The equivalence above unpacks the hypothesis: $R$ is closed under the transforms, and every value of $R$ is achievable from every other. Each $\delta_{i}\in\Delta_{i}$ carries $R$ into $R$ ; write $\delta_{i}|_{R}$ for the transform of $R$ it induces, and $\Delta_{i}|_{R}$ for the family of these. The partial function agrees with $f$ at every tuple of its domain; by (11), any two such tuples sharing an output respond identically to each other under every $\delta_{i}\in\Delta_{i}$ , and each response is again a value of the partial function: the substituted value $\delta_{i}|_{R}\ast a_{i}=\delta_{i}\ast a_{i}$ lies in $R$ . Hence transform output-determinacy (Remark 4) holds for $f|_{s_{i}\in R}$ and $\Delta_{i}|_{R}$ , and by item (a) of Section 3 $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f|_{s_{i}\in R}$ and $\Delta_{i}|_{R}$ . A succession in $\Delta_{i}$ starting at a value of $R$ visits only values of $R$ , and the induced succession in $\Delta_{i}|_{R}$ carries out the same map of $R$ ; since every value of $R$ is achievable from every other, $\overline{\Delta_{i}|_{R}}$ represents every substitution of one value of $R$ for another. Proposition 8, with $f|_{s_{i}\in R}$ in the role of $f$ , $R$ in the role of $S_{i}$ , and $\Delta_{i}|_{R}$ in the role of $\Delta_{i}$ , yields the claim: $s_{i}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ with respect to $f|_{s_{i}\in R}$ .

Setting $R=S_{i}$ recovers Proposition 8: closedness holds trivially, and $\overline{\Delta}_{i}$ represents every substitution of one value for another iff every value of $S_{i}$ is achievable from every other.

Remark 12 (Safe transforms).

Transform independence with respect to a family whose closure represents every substitution of one value for another gives, through Proposition 8, transition independence. Whether some family of transforms of $S_{i}$ satisfies both of those hypotheses is decided by $f$ alone. Call a transform $\delta_{i}$ of $S_{i}$ safe for $f$ when every two tuples sharing the same output but not the same initial value respond identically to each other under $\delta_{i}$ : if $f(\boldsymbol{a})=f(\boldsymbol{b})$ and $a_{i}\neq b_{i}$ , then $f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})=f(\boldsymbol{b}|_{b_{i}=% \delta_{i}\ast b_{i}})$ . We write $\Delta_{i}^{f}$ for the family of all safe transforms of $S_{i}$ . The identity transform of $S_{i}$ is safe for every $f$ : the substituted tuples are $\boldsymbol{a}$ and $\boldsymbol{b}$ themselves.

Lemma 13 (Closure of the safe family).

Suppose $s_{i}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ with respect to $f$ . Then every map of $S_{i}$ carried out by finitely many safe transforms applied in succession, each to the value the previous one produced, is itself safe for $f$ . With $\Delta_{i}^{f}$ in the role of $\Delta_{i}$ in Remark 6, the closure $\overline{\Delta}_{i}$ is $\Delta_{i}^{f}$ itself.

Proof. For $\boldsymbol{a},\boldsymbol{b}$ with $f(\boldsymbol{a})=f(\boldsymbol{b})$ and $a_{i}\neq b_{i}$ , and a succession $\delta_{i,1},\ldots,\delta_{i,k}$ of safe transforms carrying out $\overline{\delta}_{i}$ , write $a_{i}^{0}:=a_{i}$ and $a_{i}^{j}:=\delta_{i,j}\ast a_{i}^{j-1}$ for the values the succession visits from $a_{i}$ , and likewise $b_{i}^{j}$ from $b_{i}$ , so that $a_{i}^{k}=\overline{\delta}_{i}\ast a_{i}$ and $b_{i}^{k}=\overline{\delta}_{i}\ast b_{i}$ . The substituted tuples $\boldsymbol{a}|_{a_{i}=a_{i}^{j}}$ and $\boldsymbol{b}|_{b_{i}=b_{i}^{j}}$ lie in $S_{1}\times\cdots\times S_{n}$ , carry initial values $a_{i}^{j}$ and $b_{i}^{j}$ , and share an output at $j=0$ ; sharing survives each step. If $a_{i}^{j-1}=b_{i}^{j-1}$ , the tuples at $j-1$ share their initial value, and output-determinacy (Remark 1), which holds by item (a) of Section 1, keeps the outputs equal at the common target $a_{i}^{j}=b_{i}^{j}$ . If $a_{i}^{j-1}\neq b_{i}^{j-1}$ , the tuples at $j-1$ share the same output but not the same initial value, and $\delta_{i,j}$ is safe. Induction on $j$ gives $f(\boldsymbol{a}|_{a_{i}=\overline{\delta}_{i}\ast a_{i}})=f(\boldsymbol{b}|_{% b_{i}=\overline{\delta}_{i}\ast b_{i}})$ : the map the succession carries out is safe. The closure clause follows: every map in $\overline{\Delta}_{i}$ is carried out by a succession of safe transforms, and every safe transform lies in $\overline{\Delta}_{i}$ as a succession of length one.

Lemma 14 (Families of safe transforms).

Proof. ( $\Rightarrow$ ) If $H_{i}^{\Delta}$ satisfies (11), then for $f(\boldsymbol{a})=f(\boldsymbol{b})$ with $a_{i}\neq b_{i}$ and any $\delta_{i}\in\Delta_{i}$ , $f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})=H_{i}^{\Delta}(f(\boldsymbol{a% }),\delta_{i})=H_{i}^{\Delta}(f(\boldsymbol{b}),\delta_{i})=f(\boldsymbol{b}|_% {b_{i}=\delta_{i}\ast b_{i}})$ : every member is safe. ( $\Leftarrow$ ) Suppose every member of $\Delta_{i}$ is safe, and let $f(\boldsymbol{a})=f(\boldsymbol{b})$ with $\delta_{i}\in\Delta_{i}$ . If $a_{i}=b_{i}$ , then $\delta_{i}\ast a_{i}=\delta_{i}\ast b_{i}$ , and output-determinacy (Remark 1), which holds by item (a) of Section 1, keeps the outputs equal at that common target; if $a_{i}\neq b_{i}$ , $\delta_{i}$ is safe. Transform output-determinacy (Remark 4) therefore holds, and $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and $\Delta_{i}$ by item (a) of Section 3.

Proposition 15 (Reach of the safe family).

Suppose $s_{i}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ with respect to $f$ . Then $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and some family $\Delta_{i}$ of transforms of $S_{i}$ satisfying the hypothesis of Proposition 8 — its closure $\overline{\Delta}_{i}$ represents every substitution of one value for another — iff the safe family $\Delta_{i}^{f}$ itself represents every substitution of one value for another. In that case $\Delta_{i}^{f}$ is such a family.

Proof. ( $\Rightarrow$ ) Let $\Delta_{i}$ be such a family: every member is safe (Lemma 14), and every map in $\overline{\Delta}_{i}$ is carried out by finitely many members applied in succession, hence is itself safe (Lemma 13). Every substitution of one value for another is represented by some map in $\overline{\Delta}_{i}$ , hence by a safe transform: $\Delta_{i}^{f}$ represents every substitution of one value for another. ( $\Leftarrow$ ) Take $\Delta_{i}^{f}$ in the role of $\Delta_{i}$ — this also proves the closing assertion. Every member of $\Delta_{i}^{f}$ is safe, so $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and $\Delta_{i}^{f}$ (Lemma 14); the closure of $\Delta_{i}^{f}$ is $\Delta_{i}^{f}$ itself (Lemma 13), and it represents every substitution of one value for another by hypothesis.

Corollary 16 (Strictness).

Suppose $s_{i}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ with respect to $f$ , and let $\Delta_{i}$ be any family of transforms of $S_{i}$ . Then $s_{i}$ is not transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and $\Delta_{i}$ iff some member of $\Delta_{i}$ is not safe for $f$ . When $\Delta_{i}$ satisfies the hypothesis of Proposition 8, this is exactly the statement that the implication of Proposition 8 is strict at $f$ and $\Delta_{i}$ .

Proof. Lemma 14: transform independence with respect to $f$ and $\Delta_{i}$ holds iff every member of $\Delta_{i}$ is safe for $f$ ; strictness at $f$ and $\Delta_{i}$ is exactly its failure.

Remark 17 (Assignments).

For each value $c\in S_{i}$ , the assignment to $c$ is the transform of $S_{i}$ carrying every value to $c$ . The assignment family represents every substitution of one value for another (the substitution of $a_{i}^{\prime}$ for $a_{i}$ is represented by the assignment to $a_{i}^{\prime}$ ), and a succession of assignments carries out its last member, so the family is its own closure. Suppose $s_{i}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ with respect to $f$ . By Lemma 14, $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and the assignment family iff every assignment is safe for $f$ — iff $f(\boldsymbol{a}|_{a_{i}=c})$ is determined by $(f(\boldsymbol{a}),\,c)$ alone. If every assignment is safe, the assignment family lies inside $\Delta_{i}^{f}$ , and $\Delta_{i}^{f}$ then represents every substitution of one value for another; the reverse implication need not hold.

Remark 18 (Determination of the initial value).

Suppose $s_{i}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ with respect to $f$ , and suppose the output determines the initial value: $f(\boldsymbol{a})=f(\boldsymbol{b})$ only when $a_{i}=b_{i}$ . Then two tuples sharing the same output share the same initial value, every transform of $S_{i}$ is safe for $f$ , and $s_{i}$ is transform-independent of the remaining variables $\boldsymbol{s}_{j\neq i}$ and of its own initial value with respect to $f$ and every family $\Delta_{i}$ of transforms of $S_{i}$ (Lemma 14), whether or not its closure represents every substitution of one value for another.

Remark 19 (Equivariance form).

For $\delta_{i}\in\Delta_{i}$ , we write $\hat{\delta}_{i}$ for the map $\boldsymbol{a}\mapsto\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}}$ on $S_{1}\times\cdots\times S_{n}$ , which applies $\delta_{i}$ to the $i$ -th coordinate and leaves the remaining coordinates unchanged. Condition (11) is then the intertwining identity

f\circ\hat{\delta}_{i}=\rho(\delta_{i})\circ f,\qquad\rho(\delta_{i})\coloneqq H% _{i}^{\Delta}(\,\cdot\,,\delta_{i}).

Equivariance in the usual sense supplies actions on both sides and asks $f$ to intertwine them; here only the domain side is given, and the codomain family $\rho$ is quantified existentially. Transform output-determinacy (Remark 4) says exactly that the level sets of $f$ form a congruence for each $\hat{\delta}_{i}$ : each level set is carried into a single level set, in general a different one. Each $\hat{\delta}_{i}$ therefore descends along $f$ to $\rho(\delta_{i})$ , a self-map of the range of $f$ , unique there because its values are forced. Composites descend as well. Since $\widehat{\delta_{i}^{\prime}\circ\delta_{i}}=\hat{\delta}_{i}^{\prime}\circ% \hat{\delta}_{i}$ and the induced maps are forced, $\rho(\delta_{i}^{\prime}\circ\delta_{i})=\rho(\delta_{i}^{\prime})\circ\rho(% \delta_{i})$ , whether or not $\delta_{i}^{\prime}\circ\delta_{i}$ lies in $\Delta_{i}$ . So $\rho$ extends to an action of the closure $\overline{\Delta}_{i}$ on the range of $f$ : equivariance on the generators is equivariance on the generated semigroup — the induction behind Lemma 7. Under standard names, $f$ is a semiconjugacy, or factor map, onto its image, and the descent along $f$ is the deterministic lumpability of the level-set partition, also known as exact aggregation.

4 Transform Context

Setting.

Transform independence may be partial in the same way: $s_{i}$ might be transform-independent of some variables and of its own initial value, with the others retained as its context. We partition $\{1,\ldots,n\}$ as $\{i\}\cup J\cup K$ as in Section 2, with $J$ the context and $K$ the remaining variables. The variable $s_{i}$ is transform-independent of $\boldsymbol{s}_{K}$ and of its own initial value given context $\boldsymbol{s}_{J}$ , with respect to $f$ and $\Delta_{i}$ , if the conditions of Section 3 hold when the auxiliary maps are allowed to depend on the context values $\boldsymbol{a}_{J}$ . The family is unchanged: only the maps consult the context.

Transform-map formulation (no structure on $T$ ).

The variable $s_{i}$ is transform-independent of $\boldsymbol{s}_{K}$ and of its own initial value given context $\boldsymbol{s}_{J}$ , with respect to $f$ and $\Delta_{i}$ , iff there exists a map $H_{i,J}^{\Delta}\colon\mathcal{E}_{i}^{J}\to T$ (denoted $H_{i,J}^{\Delta}(f(\boldsymbol{a}),\delta_{i},\boldsymbol{a}_{J})$ , the transform written before the context) such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n},\;\;\forall\,\delta_{% i}\in\Delta_{i}\quad f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})=H_{i,J}^{% \Delta}\bigl{(}f(\boldsymbol{a}),\,\delta_{i},\,\boldsymbol{a}_{J}\bigr{)},

(15)

where $\mathcal{E}_{i}^{J}\coloneqq\{(f(\boldsymbol{a}),\,\boldsymbol{a}_{J}):% \boldsymbol{a}\in S_{1}\times\cdots\times S_{n}\}\times\Delta_{i}$ — the transform still constrains nothing, being chosen independently of $\boldsymbol{a}$ , while the output and the context may constrain each other. Setting $J=\emptyset$ recovers (11); setting $J=\{1,\ldots,n\}\setminus\{i\}$ does not hold trivially: the map consults the transform and every remaining coordinate, still not the initial value the transform acts on.

Remark 20 (Transform output-determinacy on context $J$ ).

For every $\boldsymbol{a},\boldsymbol{b}\in S_{1}\times\cdots\times S_{n}$ , if $f(\boldsymbol{a})=f(\boldsymbol{b})$ and $\boldsymbol{a}_{J}=\boldsymbol{b}_{J}$ , then $f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})=f(\boldsymbol{b}|_{b_{i}=% \delta_{i}\ast b_{i}})$ for every $\delta_{i}\in\Delta_{i}$ .

As in the full case, transform output-determinacy on context $J$ is necessary and sufficient for $H_{i,J}^{\Delta}$ to be well-defined on $\mathcal{E}_{i}^{J}$ . Setting $J=\emptyset$ recovers Remark 4. Geometrically, this is still the descent of Section 3, now within each context slice: with $\boldsymbol{s}_{J}$ fixed, applying a transform carries every level set of the partial function into a single level set — in general a different one.

Remark 21 (Partial-function form).

For every $\boldsymbol{c}_{J}\in\prod_{j\in J}S_{j}$ , the variable $s_{i}$ is fully transform-independent of the remaining variables and of its own initial value on the partial function $f|_{\boldsymbol{s}_{J}=\boldsymbol{c}_{J}}\colon\prod_{m\in\{i\}\cup K}S_{m}\to T$ , with respect to the same family $\Delta_{i}$ , in the sense of Section 3.

Remark 21 is equivalent to $s_{i}$ being transform-independent of $\boldsymbol{s}_{K}$ and of its own initial value given context $\boldsymbol{s}_{J}$ : since $\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}}$ fixes $\boldsymbol{a}_{J}$ , the map $H_{i,J}^{\Delta}(\,\cdot\,,\,\cdot\,,\,\boldsymbol{c}_{J})$ is the transform map of $f|_{\boldsymbol{s}_{J}=\boldsymbol{c}_{J}}$ . Partial transform independence is full transform independence on sets of partial functions, the family shared across them; context is what’s fixed.

Finite-difference formulation (group $T$ ).

When $(T,+)$ is a group, the condition has the equivalent algebraic form: there exists $G_{i,J}^{\Delta}\colon\mathcal{E}_{i}^{J}\to T$ such that

\forall\,\boldsymbol{a}\in S_{1}\times\cdots\times S_{n},\;\;\forall\,\delta_{% i}\in\Delta_{i}\quad f(\boldsymbol{a}|_{a_{i}=\delta_{i}\ast a_{i}})-f(% \boldsymbol{a})=G_{i,J}^{\Delta}\bigl{(}f(\boldsymbol{a}),\,\delta_{i},\,% \boldsymbol{a}_{J}\bigr{)}.

(16)

The relationships between (15) and (16) parallel items (a) and (b) of Section 3, with the context tuple $\boldsymbol{a}_{J}$ entering each map as a passenger argument: the formulations are related by $G_{i,J}^{\Delta}(v,\delta_{i},\boldsymbol{a}_{J})=H_{i,J}^{\Delta}(v,\delta_{i% },\boldsymbol{a}_{J})-v$ and $H_{i,J}^{\Delta}(v,\delta_{i},\boldsymbol{a}_{J})=G_{i,J}^{\Delta}(v,\delta_{i% },\boldsymbol{a}_{J})+v$ , and no derivative formulation is included.

The apparatus under context.

Every result of Section 3 applies within each context slice through the partial-function form, with $f|_{\boldsymbol{s}_{J}=\boldsymbol{c}_{J}}$ in the role of $f$ : each is proved for a function on a total product of factor sets, and each partial function is one. Remark 5 converts the slice’s transform map into a transition map on the substitutions the family represents. The closure $\overline{\Delta}_{i}$ of Remark 6, Lemma 7, and Proposition 8 apply per slice with one and the same closure: the closure and the reach hypothesis consult $S_{i}$ and $\Delta_{i}$ alone, so a single hypothesis serves every slice at once. Read back through Remark 3, the slice conclusions reassemble into context statements: under the hypothesis of Proposition 8, if $s_{i}$ is transform-independent of $\boldsymbol{s}_{K}$ and of its own initial value given context $\boldsymbol{s}_{J}$ with respect to $f$ and $\Delta_{i}$ , then $s_{i}$ is transition-independent of $\boldsymbol{s}_{K}$ given context $\boldsymbol{s}_{J}$ with respect to $f$ . Corollary 9, Definition 10, and Corollary 11 apply per slice verbatim, achievability being likewise independent of the context. The safe transforms of Remark 12 are read per slice — a transform safe for one partial function need not be safe for another — and Lemma 13, Lemma 14, Proposition 15, Corollary 16, and Remarks 17 and 18 hold in each slice so read. Remark 19 reads likewise, with $\rho$ acting on the range of the partial function.

5 Context-Related Definitions

Sufficient contexts and monotonicity.

Fix a coordinate $i$ and a context $J\subseteq\{1,\ldots,n\}\setminus\{i\}$ . The context $J$ is sufficient for $s_{i}$ (with respect to $f$ ) iff $s_{i}$ is transition-independent of $\boldsymbol{s}_{K}$ given context $\boldsymbol{s}_{J}$ , with $K=\{1,\ldots,n\}\setminus(\{i\}\cup J)$ , in the sense of Section 2, formulation (6). Sufficiency is monotone in the context: if $J\subseteq J^{\prime}$ and $J$ is sufficient for $s_{i}$ , then $J^{\prime}$ is, since tuples agreeing on $J^{\prime}$ agree on $J$ , so the condition on context $J^{\prime}$ (Remark 2) constrains a subset of the pairs the condition on $J$ does. The implication is strict in general, and the two boundary contexts of Section 2 — $J=\emptyset$ and $J=\{1,\ldots,n\}\setminus\{i\}$ — are its endpoints. The same monotonicity holds verbatim for the transform-context condition of Section 4, where the full residual need not be sufficient: the context set may be empty there, and the basis below has no automatic analogue.

Context sets and context bases.

Collect the sufficient contexts for $s_{i}$ into the context set

C_{i}(f)\coloneqq\{J\subseteq\{1,\ldots,n\}\setminus\{i\}:J\text{ is % sufficient for }s_{i}\},

and let the context basis $CB_{i}(f)\coloneqq\{J\in C_{i}(f):\nexists\,J^{\prime}\in C_{i}(f)\text{ with % }J^{\prime}\subsetneq J\}$ be its set of minimal elements. The two carry the same information: $C_{i}(f)$ is the up-set of $CB_{i}(f)$ . This rests on three facts. Non-emptiness: the full residual $\{1,\ldots,n\}\setminus\{i\}$ is always sufficient (Section 2 records it as trivial), and $C_{i}(f)$ lies in the finite power set of $\{1,\ldots,n\}\setminus\{i\}$ (finite even when the $S_{j}$ are not), so $CB_{i}(f)\neq\emptyset$ and every sufficient context contains a minimal one. Upward closure: this is the monotonicity above. Generation: hence $J\in C_{i}(f)$ iff $J\supseteq J_{0}$ for some $J_{0}\in CB_{i}(f)$ . The empty context is sufficient exactly when $s_{i}$ is transition-independent of all remaining variables in the sense of Section 1 — that is, $CB_{i}(f)=\{\emptyset\}$ . The basis need not be a singleton: with $S_{1}=S_{2}=S_{3}=\{0,1\}$ , $f(0,0,0)=f(0,1,1)$ , and the other six values distinct and fresh, both $\{2\}$ and $\{3\}$ are sufficient for $s_{1}$ while $\emptyset$ is not, so $CB_{1}(f)=\{\{2\},\{3\}\}$ . In general, whenever the basis holds two incomparable contexts their intersection lies properly below each, so its sufficiency would deny their minimality: $C_{i}(f)$ is not in general closed under intersection.

Trichotomy of variable roles.

Each variable $s_{j}$ , $j\neq i$ , is irrelevant, optional, or essential for $s_{i}$ according as $j$ lies in no element of $CB_{i}(f)$ , in some but not all, or in every one — mutually exclusive and exhaustive. The relation need not be symmetric: for $f(x,y)=x\,e^{y}$ on $\mathbb{R}^{2}$ , $y$ is essential for $x$ (the empty context fails at $x=0$ , so $CB_{x}(f)=\{\{y\}\}$ ) while $x$ is irrelevant for $y$ (from $f(x,y^{\prime})=f(x,y)\,e^{\,y^{\prime}-y}$ the empty context suffices, so $CB_{y}(f)=\{\emptyset\}$ ).

Setting.

The inputs $s_{1},\ldots,s_{n}$ are contextually ordered with respect to $f$ iff for every $i$ , the variable $s_{i}$ is transition-independent of $\boldsymbol{s}_{\{1,\ldots,i-1\}}$ given context $\boldsymbol{s}_{\{i+1,\ldots,n\}}$ , in the sense of Section 2: the tuple $(f(\boldsymbol{a}),\,a_{i},\,\boldsymbol{a}_{\{i+1,\ldots,n\}})$ is a sufficient representation for predicting how $f$ responds to movement of $s_{i}$ . The contexts nest and shrink along the order, and enlarging a context only weakens the requirement (the monotonicity above). At $i=n$ the context is empty and the condition is transition independence in the sense of Section 1: $s_{n}$ is transition-independent of the remaining variables $\boldsymbol{s}_{j\neq n}$ with respect to $f$ . At $i=1$ the condition holds trivially. The property is of the ordering as given; when some reordering of the inputs is contextually ordered, the inputs are contextually orderable.

Remark 22 (Adjacent grouping).

Grouping adjacent variables preserves contextual order: partition the indices into consecutive blocks and read $f$ on the product of the block products, each block tuple a single variable — the block tuples are contextually ordered with respect to $f$ so read. Substituting a block from its last variable to its first, each step’s context — the remainder of its own block, already substituted, and the later blocks — stands at values shared by any two tuples the substitution must reconcile, so output-determinacy on the context (Remark 2) carries equality of outputs through every step, and the composite response is determined by the output, the block’s values, and the later blocks’ values alone. In particular, split in two: for $\{s_{1},\ldots,s_{m}\}$ and $\{s_{m+1},\ldots,s_{n}\}$ , the latter tuple $\boldsymbol{s}_{\{m+1,\ldots,n\}}$ is transition-independent of $\boldsymbol{s}_{\{1,\ldots,m\}}$ with respect to $f$ so read. The preservation is one-way: the grouped order returns each variable’s condition only with the earlier variables of its own block as added context, and so returns the original condition exactly at the first variable of each block; the unconditional independence of the last variable is recovered precisely when the last block is the singleton $\{n\}$ .

Appendix A Solutions of the Bridging Initial Value Problems

By a solution of (4) we mean a differentiable $y\colon S_{i}\to T$ with $(y(t),t)\in\mathcal{D}_{i}$ for every $t\in S_{i}$ and $y^{\prime}(t)=F_{i}(y(t),t)$ ; a solution of (9) is defined likewise, with $(y(t),t,\boldsymbol{a}_{J})\in\mathcal{D}_{i}^{J}$ for every $t\in S_{i}$ and $y^{\prime}(t)=F_{i,J}(y(t),t,\boldsymbol{a}_{J})$ . Uniqueness of solutions, wherever it is invoked, is uniqueness in this class from the given datum: (4) is uniquely solvable when, for each datum $(v,t_{0})\in\mathcal{D}_{i}$ , at most one solution in this class takes the value $v$ at $t_{0}$ , and likewise for (9) at data $(v,t_{0},\boldsymbol{a}_{J})\in\mathcal{D}_{i}^{J}$ .

Local uniqueness at each datum suffices, where local uniqueness at a datum $(v,t_{0})\in\mathcal{D}_{i}$ means that any two solutions in this class taking the value $v$ at $t_{0}$ agree on a neighbourhood of $t_{0}$ in $S_{i}$ : the set on which two solutions agree is closed by continuity and open by local uniqueness, and it contains $t_{0}$ when the solutions share the datum, so it is all of the interval $S_{i}$ . The same reduction applies to (9), with data $(v,t_{0},\boldsymbol{a}_{J})\in\mathcal{D}_{i}^{J}$ and the context tuple held fixed.

Appendix B Notation

$s_{1},\ldots,s_{n}$ — the input variables of $f$ , with $s_{i}$ the $i$ -th.

$\boldsymbol{a}=(a_{1},\ldots,a_{n})$ — a tuple of input values, $a_{i}\in S_{i}$ ; $a_{i}^{\prime}$ denotes a further value of $S_{i}$ .

$\boldsymbol{a}|_{a_{i}=a_{i}^{\prime}}$ — the tuple identical to $\boldsymbol{a}$ except that the $i$ -th component is $a_{i}^{\prime}$ .

$\boldsymbol{a}_{E}\coloneqq(a_{j})_{j\in E}$ — the subtuple of $\boldsymbol{a}$ on $E\subseteq\{1,\ldots,n\}$ .

$\boldsymbol{a}_{j\neq i}$ — the subtuple on $E=\{1,\ldots,n\}\setminus\{i\}$ .

$\boldsymbol{a}_{J}$ — the subtuple on a context $J$ , where $J$ and $K=\{1,\ldots,n\}\setminus(\{i\}\cup J)$ partition $\{1,\ldots,n\}\setminus\{i\}$ .

$\boldsymbol{s}_{E}$ — the tuple of variables indexed by $E$ , of which $\boldsymbol{a}_{E}$ is a tuple of values; likewise $\boldsymbol{s}_{j\neq i}$ , $\boldsymbol{s}_{J}$ , $\boldsymbol{s}_{K}$ .

$\boldsymbol{c}_{J}$ — a fixed value of $\boldsymbol{s}_{J}$ , that is, an element of $\prod_{j\in J}S_{j}$ .

$\mathrm{pr}_{i}$ — the $i$ -th projection $\boldsymbol{a}\mapsto a_{i}$ .

Prerequisite Mathematical Definitions

1 Transition Independence

Setting.

Transition map.

Remark 1 (Output-determinacy).

Finite-difference formulation.

Derivative formulation.

Summary of equivalences.

2 Transition Context

Setting.

Transition-map formulation (no structure on T).

Remark 2 (Output-determinacy on context J).

Remark 3 (Partial-function form).

Finite-difference formulation (group T).

Derivative formulation (Si real, T normed).

Summary of equivalences.

3 Transform Independence

Setting.

Transform-map formulation (no structure on T).

Remark 4 (Transform output-determinacy).

Finite-difference formulation (group T).

Summary of equivalences.

Remark 5 (Keeping the initial value).

Remark 6 (Reach and the closure).

Lemma 7 (Passage to the closure).

Proposition 8 (Reach).

Corollary 9 (Represented substitutions).

Definition 10 (Achievability).

Corollary 11 (Achievable sets).

Remark 12 (Safe transforms).

Lemma 13 (Closure of the safe family).

Lemma 14 (Families of safe transforms).

Proposition 15 (Reach of the safe family).

Corollary 16 (Strictness).

Remark 17 (Assignments).

Remark 18 (Determination of the initial value).

Remark 19 (Equivariance form).

4 Transform Context

Setting.

Transform-map formulation (no structure on T).

Remark 20 (Transform output-determinacy on context J).

Remark 21 (Partial-function form).

Finite-difference formulation (group T).

The apparatus under context.

5 Context-Related Definitions

Sufficient contexts and monotonicity.

Context sets and context bases.

Trichotomy of variable roles.

Setting.

Remark 22 (Adjacent grouping).

Appendix A Solutions of the Bridging Initial Value Problems

Appendix B Notation

Transition-map formulation (no structure on $T$ ).

Remark 2 (Output-determinacy on context $J$ ).

Finite-difference formulation (group $T$ ).

Derivative formulation ( $S_{i}$ real, $T$ normed).

Transform-map formulation (no structure on $T$ ).

Finite-difference formulation (group $T$ ).

Transform-map formulation (no structure on $T$ ).

Remark 20 (Transform output-determinacy on context $J$ ).

Finite-difference formulation (group $T$ ).