Convex Contact And Mixing

Let $P$ be a set of primitive type labels. A formal type $\mathrm{\Gamma }$ records a finitely supported amount function ${\nu }_{\mathrm{\Gamma }}:P\to {\mathbb{R}}_{\ge 0}$, not identically zero. Primitive generators have their unit amount function. Positive scaling, separated product, and thermal-join constructors retain distinct formal tags and obey

${\nu }_{s\mathrm{\Gamma }}=s{\nu }_{\mathrm{\Gamma }},{\nu }_{\mathrm{\Gamma }\otimes {\mathrm{\Gamma }}^{\mathrm{\prime }}}={\nu }_{\mathrm{\Gamma }}+{\nu }_{{\mathrm{\Gamma }}^{\mathrm{\prime }}}(s>0)\mathrm{.}$

Separated and joined types are distinct constructors even when their amount functions agree. Permutation, reassociation, and scaled-splitting conventions preserve the amount function. Define the positive bookkeeping weight $w_{\mathrm{\Gamma }}:={\sum }_{p\in P}{\nu }_{\mathrm{\Gamma }}(p)$, a finite sum over support. Thus $w_{s\mathrm{\Gamma }}=s{w}_{\mathrm{\Gamma }}$ and every product or join has weight $w_{\mathrm{\Gamma }}+w_{{\mathrm{\Gamma }}^{\mathrm{\prime }}}$. The optional empty unit has zero amount function and weight zero; it is represented only by the empty compound, not as a simple chart. These are formal amounts, not masses in kilograms or an empirical conservation law. The functions and pointwise operations use The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$; their finite-support convention and finite sums are those of Tagged scalable compounds and Compound mass is well-defined and additive.

For Euclidean vectors use $a\cdot b={\sum }_j{a}_j{b}_j$ and $\mathrm{\parallel }a\mathrm{\parallel }=(a\cdot a{)}^{1\mathrm{/}2}$. Write $B(a,r)=\{b:\mathrm{\parallel }b-a\mathrm{\parallel }<r\}$. Open means each point has such a ball contained in the set; convex means $(1-t)a+tb$ belongs to the set for all its $a,b$ and $0\le t\le 1$. In a subspace or subset, use intersections of these balls for the relative topology. A map $P:D\to {\mathbb{R}}^n$ is locally Lipschitz if each $a\in D$ has a relative neighborhood $V$ and finite $L\ge 0$ with $\mathrm{\parallel }P(b)-P(c)\mathrm{\parallel }\le L\mathrm{\parallel }b-c\mathrm{\parallel }$ for all $b,c\in V$. A set $D$ is arcwise connected here if each two distinct points are joined by a continuous injective map $\gamma :[0,1]\to D$ with those endpoints; singleton and empty sets satisfy this condition vacuously. Continuity uses the indicated Euclidean relative topology. Let $I$ index the declared family of simple types ${\mathrm{\Gamma }}_i$. For each simple type $\mathrm{\Gamma }$ in that family, let $n_{\mathrm{\Gamma }}\ge 1$ and let $\mathrm{\Gamma }\subset {\mathbb{R}}^{n_{\mathrm{\Gamma }}+1}$ be nonempty, open and convex, with coordinates $x=(u,v)$. The notation for a type and its chart is shared, but each state has a distinct raw tag $(\mathrm{\Gamma },x)$. Scaled charts are distinct tagged copies $s\mathrm{\Gamma }=\{sx:x\in \mathrm{\Gamma }\}$ with coordinate transport $x\mapsto sx$; transported raw tags remain distinct from the original raw tags. Let $e_{(\mathrm{\Gamma },x)}$ be the unit point mass from Tagged scalable compounds. The represented state is

${\mathsf{J}}_{\mathrm{\Gamma }}(x):=w_{\mathrm{\Gamma }}{e}_{(\mathrm{\Gamma },x)},m({\mathsf{J}}_{\mathrm{\Gamma }}(x))=w_{\mathrm{\Gamma }}\mathrm{.}$

Write $\mathsf{J}(x):={\mathsf{J}}_{\mathrm{\Gamma }}(x)$ when $x$ already carries its type tag. All state notation in this framework's compound comparisons, including a bare $x$, denotes ${\mathsf{J}}_{\mathrm{\Gamma }}(x)$; a raw unit point mass is always written $e_{(\mathrm{\Gamma },x)}$. A separated pair is addition of represented states, not a unit product tag. Nonnegative multiples are pointwise and zero terms are omitted. For an encoded compound $c$, its aggregate composition is the finite amount function

$\nu (c):={\sum }_{(\mathrm{\Gamma },x)\in \mathrm{supp}(c)}\frac{c(\mathrm{\Gamma },x)}{w_{\mathrm{\Gamma }}}{\nu }_{\mathrm{\Gamma }};m(c)={\sum }_{p\in P}\nu (c)(p)\mathrm{.}$

Fix the formal interpretation $\mathrm{\Pi }(c)$ as the finite separated preparation whose factor at each nonzero raw tag $(\mathrm{\Gamma },x)$ is the amount $c(\mathrm{\Gamma },x)\mathrm{/}{w}_{\mathrm{\Gamma }}$ of that typed state, using the declared coordinate scaling transport. Let $\mathcal{P}:=\{\mathrm{\Pi }(c):c\text{ is a finite nonnegative encoded compound}\}$ be the set of these preparations, with an invariant bookkeeping mass $M:\mathcal{P}\to {\mathbb{R}}_{\ge 0}$ satisfying $M(\mathrm{\Pi }(c))=m(c)$. As additional model data choose a binary relation $Q\subseteq \{(p,q)\in {\mathcal{P}}^2:M(p)=M(q)\}$. Define its pullback on encoded compounds by $A\prec B$ if and only if $(\mathrm{\Pi }(A),\mathrm{\Pi }(B))\in Q$. The transport, reversibility and A1–A7 assumptions below constrain this chosen relation; they do not construct it from coordinates. Merging or splitting repeated copies of the same preparation is reversible; permutation and reassociation preserve the preparation. Thus $\mathrm{\Pi }({\mathsf{J}}_{s\mathrm{\Gamma }}(sx))$ and $\mathrm{\Pi }(s{\mathsf{J}}_{\mathrm{\Gamma }}(x))$ are the same declared scaled preparation and yield

${\mathsf{J}}_{s\mathrm{\Gamma }}(sx)\sim s{\mathsf{J}}_{\mathrm{\Gamma }}(x),$

although their raw point masses have different tags and are not equal. This reversible transport commutes with finite composition, nested type constructors, and scaled splitting. It is part of the precise formal scaling interpretation, not a deduction about physical accessibility from coordinate multiplication. A physical use must adopt the stated preparation maps and accessibility assumptions; fixed-composition models compare only preparations with the same $\nu$, while a declared channel model may additionally compare different compositions of equal $m$.

This Definition restricts comparisons $A\prec B$ to $m(A)=m(B)$; the coefficient-map supplier defines the mass, not the comparison relation. Write $A\sim B$ when both directions hold, and $A\prec \prec B$ when only the forward direction holds. On any fixed simple type, dividing every represented state by the same positive constant $w_{\mathrm{\Gamma }}$ gives its unit-point-mass comparison. Positive scaling and its inverse preserve and reflect order. An additive entropy density for this fixed type means a function $S:\mathrm{\Gamma }\to \mathbb{R}$ such that, for every two finite nonnegative unit-point-mass compounds $C={\sum }_j{a}_j{e}_{(\mathrm{\Gamma },x_j)}$ and $D={\sum }_k{b}_k{e}_{(\mathrm{\Gamma },y_k)}$ with ${\sum }_j{a}_j={\sum }_k{b}_k$, one has

$C\prec D⟺{\sum }_j{a}_{j}S(x_j)\le {\sum }_k{b}_{k}S(y_k)\mathrm{.}$

Empty sums are zero. Given such an $S$, define $H_S(C):={\sum }_j{a}_{j}S(x_j)$ on compounds supported on this type. Its represented-state values are $H_S({\mathsf{J}}_{\mathrm{\Gamma }}(x))=w_{\mathrm{\Gamma }}S(x)$ and $H_S({\sum }_j{t}_j{\mathsf{J}}_{\mathrm{\Gamma }}(x_j))={\sum }_j{t}_j{w}_{\mathrm{\Gamma }}S(x_j)$ for finite nonnegative $t_j$. These identities follow by substituting the raw coefficients in the finite sum. Representing only the comparisons $e_{(\mathrm{\Gamma },x)}\prec e_{(\mathrm{\Gamma },y)}$ is insufficient for the displayed compound equivalence. Existence of an additive entropy density is not assumed here; a later representation theorem must establish it under its full compound-comparison hypotheses. No independent rescaling across different types is assumed without a calibrator.

The common relation satisfies A1–A6 on all equal-mass finite compounds: (A1) reflexivity; (A2) transitivity; (A3) if $A\prec B$, then $A+C\prec B+C$ for every compound $C$; (A4) if $A\prec B$, then $tA\prec tB$ for every $t>0$; (A5) $t{\mathsf{J}}_{\mathrm{\Gamma }}(x)$ is equivalent to $s{\mathsf{J}}_{\mathrm{\Gamma }}(x)+(t-s){\mathsf{J}}_{\mathrm{\Gamma }}(x)$ for $t>0$ and $0<s<t$, with the reversible scaled-copy transport above; (A6) for equal-mass $A,B$, fixed positive equal-mass spectators $Z_0,Z_1$, and every sequence ${ϵ}_n>0$ decreasing to zero, $A+{ϵ}_n{Z}_0\prec B+{ϵ}_n{Z}_1$ for all $n$ implies $A\prec B$. These are hypotheses, not conclusions. No comparison hypothesis is included here.

For each $i$ and $x\in {\mathrm{\Gamma }}_i$, define the forward sector $A_x:=\{y\in {\mathrm{\Gamma }}_i:{\mathsf{J}}_{{\mathrm{\Gamma }}_i}(x)\prec {\mathsf{J}}_{{\mathrm{\Gamma }}_i}(y)\}$. The simple-domain assumptions are: (A7) for $x,y\in {\mathrm{\Gamma }}_i$ and $0<t<1$, the encoded compound $t{\mathsf{J}}_{{\mathrm{\Gamma }}_i}(x)+(1-t){\mathsf{J}}_{{\mathrm{\Gamma }}_i}(y)$ compares to ${\mathsf{J}}_{{\mathrm{\Gamma }}_i}(tx+(1-t)y)$; both have bookkeeping mass $w_{{\mathrm{\Gamma }}_i}$, and convexity puts the recombined coordinate in the chart; (S1) each $x$ has a strict successor; (S2) $A_x$ has exactly one supporting hyperplane at $x$, expressible as $u-u_x+P_i(x)\cdot (v-v_x)=0$ with a finite-valued slope map $P_i:{\mathrm{\Gamma }}_i\to {\mathbb{R}}^{n_i}$, and $P_i$ is locally Lipschitz; (S3) the boundary of $A_x$ relative to ${\mathrm{\Gamma }}_i$ is arcwise connected. A supporting hyperplane means that $A_x$ lies in one of its closed half-spaces and meets the plane at $x$. The affine hull is the smallest affine subspace containing a set; relative interior is interior in that affine hull. Closure and boundary are taken in the indicated affine hull or in ${\mathrm{\Gamma }}_i$, as named; boundary is closure intersected with closure of the complement in that relative topology.

This definition imposes no physical interpretation, units, or empirical claim. A contact theorem that uses joins must separately assume that each join is a simple domain satisfying these same forward-sector hypotheses.

For two simple domains ${\mathrm{\Gamma }}_i\subset {\mathbb{R}}^{n_i+1}$ and ${\mathrm{\Gamma }}_j\subset {\mathbb{R}}^{n_j+1}$, write their coordinates $(u_i,v_i)$ and $(u_j,v_j)$. Their thermal-join domain is

${\mathrm{\Delta }}_{ij}:=\{(u,v_i,v_j):\mathrm{\exists }{u}_i,u_j,(u_i,v_i)\in {\mathrm{\Gamma }}_i,(u_j,v_j)\in {\mathrm{\Gamma }}_j,u=u_i+u_j\}\mathrm{.}$

The joined simple type is a distinct formal tag with amount function ${\nu }_{{\mathrm{\Delta }}_{ij}}={\nu }_{{\mathrm{\Gamma }}_i}+{\nu }_{{\mathrm{\Gamma }}_j}$ and weight $w_{{\mathrm{\Delta }}_{ij}}=w_{{\mathrm{\Gamma }}_i}+w_{{\mathrm{\Gamma }}_j}$, as defined in Forward-sector hypotheses for a simple state domain. Encode the separated pair by ${\mathsf{J}}_{{\mathrm{\Gamma }}_i}(x)+{\mathsf{J}}_{{\mathrm{\Gamma }}_j}(y)$ and a joined coordinate $Z$ by ${\mathsf{J}}_{{\mathrm{\Delta }}_{ij}}(Z)$. Their bookkeeping masses agree, and their aggregate composition functions agree. These encoded maps and their raw tags remain distinct. Positive coordinate-scaled copies use the reversible preparation transport of that definition; nested joins preserve their finite primitive amount function under reassociation and permutation. The empty type contributes the zero map.

Write $J_{ij}(x,y)$ for the join state with the summed energy coordinate and the unchanged work coordinates. A compound pair $(x,y)$ and $J_{ij}(x,y)$ have the same total energy and work coordinates but are distinct tagged objects. Define $x{\sim }_{T}y$ exactly when ${\mathsf{J}}_{{\mathrm{\Gamma }}_i}(x)+{\mathsf{J}}_{{\mathrm{\Gamma }}_j}(y)$ and ${\mathsf{J}}_{{\mathrm{\Delta }}_{ij}}(J_{ij}(x,y))$ are accessible in both directions. Nested joins use the declared reassociation and permutation identifications.

Assume every finite join is itself a simple domain satisfying Forward-sector hypotheses for a simple state domain. The contact hypotheses are mathematical assumptions on the common relation: (T1) $(x,y)\prec J_{ij}(x,y)$; (T2) every join state has at least one reversible split into component states at its stated coordinates, and the equal-state split of any state into positive scaled copies of itself is reversible; (T3) ${\sim }_T$ is transitive; (T4) for every simple state $x$ there are states $x_0,x_1$ in its domain with $x_0\prec \prec x\prec \prec x_1$ and $x_0{\sim }_T{x}_1$; (T5) for every ordered pair of simple systems, every $x$ in the first and every work coordinate $v$ in the second's projection $\rho ({\mathrm{\Gamma }}_j):=\{v:\mathrm{\exists }u,(u,v)\in {\mathrm{\Gamma }}_j\}$, there is a state $y$ with work coordinate $v$ and $x{\sim }_{T}y$.

For the completed calibrator branch, add (C1) join coherence: whenever $x{\sim }_{T}y$, the join state $J_{ij}(x,y)$ is thermally equivalent to each of $x$ and $y$ under the nested-join identifications. The coordinate sum in the definition alone does not assert C1. In later theorem statements, T1–T5 and C1 are explicitly listed as hypotheses; they are not treated as established physical laws.

For a nonempty set $C$, define $\mathrm{aff}C=\{{\sum }_{j=1}^N{a}_j{c}_j:N\ge 1,c_j\in C,a_j\in \mathbb{R},{\sum }_j{a}_j=1\}$. With a fixed $c_0\in C$, this is $c_0+\mathrm{span}\{c-c_0:c\in C\}$, where span consists of finite linear combinations. An affine Euclidean space is such a translate of a finite-dimensional linear subspace with its induced Euclidean distance. Relative balls use this distance; interior means containing a relative ball, and closure means every relative ball meets the set. Let $C$ be a nonempty convex subset of a finite-dimensional affine Euclidean space $H$. Its relative interior ${\mathrm{ri}}_{H}C$ is the interior in $H=\mathrm{aff}C$. Then ${\mathrm{ri}}_{H}C$ is nonempty. If $w\in {\mathrm{ri}}_{H}C$, $y\in {\bar{C}}^H$, and $0\le t<1$, then $(1-t)w+ty\in {\mathrm{ri}}_{H}C$.

For nonempty $C$, define $\mathrm{aff}C=\{{\sum }_{j=1}^N{a}_j{c}_j:N\ge 1,c_j\in C,a_j\in \mathbb{R},{\sum }_j{a}_j=1\}$. An affine Euclidean space is a translate of a finite-dimensional real linear subspace with the induced Euclidean metric. A supporting hyperplane at $z$ is $\{q:n\cdot (q-z)=0\}$ for a nonzero vector $n$, with $z$ in the set and all its points in one closed half-space. Let $C$ be a nonempty convex set in a finite-dimensional affine Euclidean space $H$ of dimension $k\ge 1$, let $\mathrm{aff}C=H$, and put $K={\bar{C}}^H$. If $z\in {\mathrm{\partial }}_{H}K$ (that is, $z\in K$ and every relative neighborhood of $z$ meets $H\setminus K$), then there is a nonzero linear functional $\mathrm{\ell }$ on the translation space of $H$ such that $\mathrm{\ell }(q-z)\le 0$ for all $q\in C$. If $C\subset {\mathbb{R}}^d$ has affine dimension at most $d-2$, then every $z\in C$ has at least two distinct ambient supporting hyperplanes containing $\mathrm{aff}C$.

Use the Euclidean distance $\mathrm{\parallel }q-z\mathrm{\parallel }$; $\mathrm{int}K$ consists of points containing a ball in $K$, closure consists of points whose every ball meets $K$, and $\mathrm{\partial }K=\bar{K}\setminus \mathrm{int}K$. Closed means equal to closure. A supporting hyperplane at $x\in K$ is $\{q:n\cdot (q-x)=0\}$ with $n\mathrm{\ne }0$ and $K$ contained in one closed half-space. Let $K\subset {\mathbb{R}}^d$ be closed and convex with nonempty interior, $d\ge 1$, and let $x\in \mathrm{\partial }K$. Suppose $K$ has exactly one supporting hyperplane at $x$. Choose its inward unit normal $n$, so $n\cdot (q-x)\ge 0$ for every $q\in K$. Then for every $v$ with $n\cdot v>0$, there is $\delta >0$ such that $x+tv\in \mathrm{int}K$ for every $0<t<\delta$.

Use the Euclidean norm $\mathrm{\parallel }z\mathrm{\parallel }=({\sum }_{j=1}^d{z}_j^2{)}^{1\mathrm{/}2}$, and balls $B(a,r)=\{z:\mathrm{\parallel }z-a\mathrm{\parallel }<r\}$. A set is open if it contains a ball about each of its points. Let $U\subset {\mathbb{R}}^d$ be open and convex, and let $f:U\to \mathbb{R}$ be finite and convex, meaning $f((1-t)x+ty)\le (1-t)f(x)+tf(y)$ for every $x,y\in U$ and $t\in [0,1]$. For every $x_0\in U$, there are a neighborhood $V$ of $x_0$ and $L<\mathrm{\infty }$ such that $\mathrm{\mid }f(x)-f(y)\mathrm{\mid }\le L\mathrm{\parallel }x-y\mathrm{\parallel }$ for all $x,y\in V$. The same conclusion holds for finite concave functions by applying the result to $-f$.

Use the Darboux integral on nondegenerate intervals and the convention ${\int }_a^{a}f=0$. Let $L\ge 0$, $T>0$, and let $f:[0,T]\to [0,\mathrm{\infty })$ be continuous. If $f(t)\le L{\int }_0^{t}f(s)ds$ for every $t\in [0,T]$, then $f(t)=0$ for every $t\in [0,T]$.

Under A1–A6 of Forward-sector hypotheses for a simple state domain, for compounds $A,B$ with $m(A)=m(B)$ and every finite spectator compound $C$, $A+C\prec B+C$ if and only if $A\prec B$. Consequently, for finite-support signed maps $u,v$ of equal total coefficient, define $u{⪯}_{s}v$ by $(v-u{)}^{-}\prec (v-u{)}^{+}$. This relation is well-defined, agrees with $\prec$ on nonnegative maps, is invariant under adding a common signed map, is transitive, and is closed under addition of two comparisons. No comparison hypothesis is used.

Let $u,v$ be finite-support real coefficient maps on the tagged state labels of Tagged scalable compounds with equal total coefficient. For a typed state $x$, compound notation uses the represented map $\mathsf{J}(x)$ from Forward-sector hypotheses for a simple state domain; its raw unit point mass is $e_{(\mathrm{\Gamma },x)}$. Signed maps still use the ordinary total of their raw coefficients, exactly the bookkeeping mass of the represented nonnegative maps. Put $d=v-u$, $d^{+}(x)=\max (d(x),0)$ and $d^{-}(x)=\max (-d(x),0)$. Define $u{⪯}_{s}v$ to mean $d^{-}\prec d^{+}$ under the compound relation of Forward-sector hypotheses for a simple state domain. Write $u{\sim }_{s}v$ when both $u{⪯}_{s}v$ and $v{⪯}_{s}u$. These are formal signed comparison maps, not physical negative amounts. The positive and negative parts are finite-support compounds of equal mass. If $u{\sim }_{s}v$, then $cu{\sim }_{s}cv$ for every real $c$: positive $c$ uses scaling, $c=0$ is reflexivity, and negative $c$ swaps the positive and negative parts while equivalence supplies both comparison directions. This Definition introduces the signed notation; ordinary-order compatibility and the signed-order laws are proved separately.

Under T1–T3 and the equal-state reversible split in Thermal joins and conditional contact hypotheses, if $x{\sim }_{T}y$ then $sx{\sim }_{T}ty$ for every $s,t>0$.

Let a simple domain and its finite thermal joins satisfy Forward-sector hypotheses for a simple state domain and Thermal joins and conditional contact hypotheses. Let $x_0\prec \prec x_1$ with $x_0{\sim }_T{x}_1$, and let $x$ satisfy $x_0\prec \prec x\prec \prec x_1$. For $0\le \lambda \le 1$, write $R_{\lambda }$ for the compound state $((1-\lambda )x_0,\lambda x_1)$, omitting a zero coefficient at the endpoints. Then there is a unique $\lambda \in (0,1)$ with $R_{\lambda }\sim x$ in the common accessibility relation.