Accessibility Foundation

Fix a material constitution $c$ and its equilibrium state domain $D_c$ from Equilibrium state domains. For each admitted record $P\in {\mathcal{P}}_s$, use the nonnegative, nondecreasing cumulative incoming and outgoing thermal-energy maps $H_P^{\mathrm{i}\mathrm{n}},H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}:I_P\to [0,\mathrm{\infty })$ of Thermodynamic process records and transfer conventions, with numerical values in joules and initial values zero. For $x,y\in D_c$, write $x{\prec }_{c}y$, and say that $y$ is adiabatically accessible from $x$, exactly when the declared process model admits at least one record $P\in {\mathcal{P}}_s$ with $x_P^{-}=x$, $x_P^{+}=y$, $H_P^{\mathrm{i}\mathrm{n}}(t)=H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}(t)=0$ for every $t\in I_P$, and the model’s stated auxiliary and work-device endpoint conditions satisfied. The material constitution is the same at both endpoints. The process record must account for the selected outer boundary and every auxiliary included by its protocol; transfers inside that boundary are internal. These maps are supplied on all admitted records, but no energy-balance law outside the core closed-record class follows.

The sign convention is $Q>0$ for net heat into the system and $W_{\mathrm{o}\mathrm{n}}>0$ for work done on it, as fixed in Thermodynamic process records and transfer conventions. An adiabatic record has $Q[P]=0$, but zero net heat alone is insufficient: equal nonzero incoming and outgoing transfers cancel in $Q$. Adiabatic means absence of thermal transfer across the selected boundary throughout the record; it imposes neither zero work nor zero energy change, isolation, quasistatic evolution, or reversibility. The relation concerns equilibrium endpoints only; intermediate configurations may be nonequilibrium. Existence of a witness record is model input, not a consequence of this definition.

When the constitution is clear, write $x\prec y$. Define $x\sim y$ by $x\prec y$ and $y\prec x$, and $x\prec \prec y$ by $x\prec y$ and not $y\prec x$. These are respectively accessibility equivalence and strict one-way accessibility; neither is defined by equality of energy alone.

The order-theoretic relation of Lieb and Yngvason is an abstract model of this operation. Goddard’s Carathéodory formulation uses a distinct local-inaccessibility route; it is cited as context and does not supply the product comparison results used later.

Fix a family $I$ of primitive material labels. A tagged type $c\in I$ retains its identity even if another type has the same coordinate domain. For the accessibility/contact branch, select a fully scalable model family: for every declared type $c$ and every dimensionless $\lambda >0$, explicitly assume the tag $c^{(\lambda )}$ exists, $B_{c,\lambda }=D_c$, and $D_{c^{(\lambda )}}={\delta }_{\lambda }(D_c)$, with ${\delta }_{\lambda }:D_c\to D_{c^{(\lambda )}}$ the coordinate-multiplication bijection. The inverse divides coordinates by $\lambda$. These are additional model conditions; the general, partial-domain postulate Additive separated systems and tagged bulk scaling alone does not supply them. On a declared energy/work chart $X_c$ the same condition means $X_{c^{(\lambda )}}=\lambda X_c$, scaling every numerical extensive coordinate and the fixed material amounts. An ordered finite separated type $T=(c_1^{({\lambda }_1)},\dots ,c_k^{({\lambda }_k)})$, ${\lambda }_i>0$, has the full state domain $X_T={\prod }_{i=1}^k{D}_{c_i^{({\lambda }_i)}}$. A separated state retains its tagged constituent tuple. Permuting factors together with their states and reassociating products give the declared identifications; a compound state means the resulting equivalence class of these representatives. The empty type has the singleton state domain $\{()\}$. The aggregate extensive quantities are the sums specified by Additive separated systems and tagged bulk scaling wherever that postulate admits composition; those sums do not replace the constituent tuple or identify different decompositions.

Supply a process model on these endpoint domains as follows. For each ordered pair $T,T^{\mathrm{\prime }}$ with the same formal composition defined below, declare a configuration set $Z_{T,T^{\mathrm{\prime }}}$ containing the system, auxiliaries and work devices, equilibrium endpoint subsets $Z_{T,T^{\mathrm{\prime }}}^{-},Z_{T,T^{\mathrm{\prime }}}^{+}$, and endpoint maps ${\xi }_{T,T^{\mathrm{\prime }}}^{-}:Z_{T,T^{\mathrm{\prime }}}^{-}\to X_T$, ${\xi }_{T,T^{\mathrm{\prime }}}^{+}:Z_{T,T^{\mathrm{\prime }}}^{+}\to X_{T^{\mathrm{\prime }}}$. Also declare control/contact and transfer-record sets ${\mathcal{K}}_{T,T^{\mathrm{\prime }}},{\mathcal{R}}_{T,T^{\mathrm{\prime }}}$ and a selected outer boundary. The typed records are tuples $P=(I_P,A^{-},A^{+},z_P,{\kappa }_P,r_P)$ with a finite time interval $I_P=[t_0,t_1]$, $t_0<t_1$, in seconds, $A^{-}\in X_T$, $A^{+}\in X_{T^{\mathrm{\prime }}}$, $z_P:I_P\to Z_{T,T^{\mathrm{\prime }}}$, ${\kappa }_P\in {\mathcal{K}}_{T,T^{\mathrm{\prime }}}$, $r_P\in {\mathcal{R}}_{T,T^{\mathrm{\prime }}}$, and $z_P(t_0)\in Z_{T,T^{\mathrm{\prime }}}^{-}$, $z_P(t_1)\in Z_{T,T^{\mathrm{\prime }}}^{+}$ mapping to $A^{-},A^{+}$. No regularity of $z_P$ is presumed. The model specifies an admitted subset ${\mathcal{P}}_{T,T^{\mathrm{\prime }}}$ of these records, including its auxiliary and work-device endpoint conditions and any adopted conservation constraints. Each admitted transfer record supplies the nonnegative, nondecreasing cumulative maps $H_P^{\mathrm{i}\mathrm{n}},H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}:I_P\to [0,\mathrm{\infty })$, initially zero and with numerical values in joules, with exactly the outer-boundary convention of Thermodynamic process records and transfer conventions; internal thermal exchanges between constituents are excluded from these totals. No admission or balance law follows from the tuple types.

For $A\in X_T$, $B\in X_{T^{\mathrm{\prime }}}$ of equal composition, define $A\prec B$ exactly when there is $P\in {\mathcal{P}}_{T,T^{\mathrm{\prime }}}$ with $A^{-}=A$, $A^{+}=B$ and $H_P^{\mathrm{i}\mathrm{n}}(t)=H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}(t)=0$ for every $t\in I_P$. Require the declared permutation and reassociation maps to transport admitted records bijectively, preserving endpoint maps, the selected boundary, thermal transfers and auxiliary/work-device conditions. Hence existence of an adiabatic witness is independent of representatives. For one-factor endpoints $D_c$, require this record model to reproduce the records and endpoint conditions of Adiabatic accessibility, so its restriction is ${\prec }_c$. On fixed-amount charts use the constituent coordinate embeddings into these domains. A later contact model may add joined endpoint domains using this same typed-record scheme and must agree with this relation on existing endpoints. Product accessibility is additional process data, not the coordinate-sum pullback and not necessarily the componentwise relation. As for single systems, write $A\sim B$ when both directions hold and $A\prec \prec B$ when only the forward direction holds.

Scaling a separated compound scales each factor. Adopt coherent preparation transport: $1x$, $x$, $s(tx)$ and $(st)x$ denote the same preparation after their declared tag transport, and scaling commutes with permutation, reassociation and finite separated product. Their raw tags remain distinct; the model declares these transports reversible under $\prec$, and transports of later joined types commute with the same operations. This is an explicit accessibility assumption, not an inference from coordinate multiplication. Merging or splitting copies of one preparation is reversible when A5 is adopted.

The formal composition vector of a compound records, for each primitive label $c$, the sum of its positive scale coefficients. Two compounds have the same composition exactly when these vectors agree. The distinct formal mass $m(A)$ is the sum of all scale coefficients in $A$, a dimensionless bookkeeping quantity; it is not physical mass in kilograms, mole amount, energy, or volume. Same-composition compounds have equal formal mass. In particular the relation just defined includes states of the same compound type and may compare different decompositions of the same composition when the model admits a witness.

An equal-composition comparison is a comparison between states of types with the same formal composition vector. The later reaction-network definition may separately admit transitions between distinct composition types, but the mathematical supplier still requires equal formal mass at both endpoints. A physical application must state how its channel preserves the chosen formal mass and separately record its dimensioned conservation balances. The empty unit has one state and formal mass zero. No identity is imposed between a primitive type and a scaled tag, or between two different decompositions of a compound, beyond the explicitly declared reassociation, permutation, and bulk-composition maps.

Use the fully scalable model family of Tagged scaled systems and separated products: every positive scale map is defined on the full state domain, finite separated state spaces are full products, and nested scaling has the declared reversible preparation transport. These additional existence and transport assumptions are not supplied by the general partial-domain bulk postulate. Adopt the following axioms for $\prec$ on every declared pair of finite compounds with equal formal mass in Tagged scaled systems and separated products. Initially this covers equal-composition adiabatic accessibility. If a later channel model admits comparisons between distinct composition types, the same axioms apply to those comparisons and all their finite scaled products; no cross-type transition is admitted by this postulate alone. They are assumptions of the operational model, not consequences of the process definition or of energy additivity.

A1 (reflexivity). $A\prec A$ for every state $A$.

A2 (transitivity). If $A\prec B$ and $B\prec C$, then $A\prec C$.

A3 (consistency). If $A\prec B$, then $A+C\prec B+C$ for every finite spectator compound $C$ for which both endpoint compounds are admitted and the resulting formal masses agree.

A4 (scaling). If $A\prec B$, then $\lambda A\prec \lambda B$ for every $\lambda >0$, using the full-domain tagged scaling maps expressly adopted in Tagged scaled systems and separated products.

A5 (splitting). For each state $x$, positive $s<t$, and tagged scaled copies of its material, the state $tx$ is accessible in both directions to the separated pair $(sx,(t-s)x)$. This states reversible splitting of an equilibrium preparation into scaled copies, not additivity of arbitrary state functions.

A6 (stability). Let $A,B$ have equal formal mass. Let $Z_0,Z_1$ be fixed positive spectator compounds of equal formal mass, and let ${ϵ}_n>0$ decrease to zero. If $A+{ϵ}_n{Z}_0\prec B+{ϵ}_n{Z}_1$ for every $n$, then $A\prec B$.

Equality of formal masses in A6 is required so each comparison is defined; equality of composition vectors is not required when a declared model includes cross-type transitions. A6 is a limiting closure assumption on the relation; it does not assert continuity of any physical process or the existence of a limiting apparatus.

For the simple-compressible physical branch, a primitive label is a pair $i=(c_i,n^i)$ of constitution and fixed positive mole amounts. Its numerical state domain is exactly ${\mathrm{\Gamma }}_i={\mathrm{\Gamma }}_{c_i,n^i}$ of Equilibrium state domains, with embedding $b_i(u,v)=(u,v,{\widehat{n}}^i)\in D_{c_i}$. The unit map sends $(u,v)$ to $(u\mathrm{J},v{\mathrm{m}}^3)$; thus $u$ is the numerical value of the fixed internal energy, not a newly oriented energy coordinate. The underlying accessibility is the pullback of the admitted physical endpoint relation along $b_i$. Positive bulk scaling uses the declared scaling of Additive separated systems and tagged bulk scaling, carries the mole vector to $\lambda n^i$, and retains the distinct scaled tag. More general work coordinates or formal thermometer examples require their own explicitly declared domain, coordinate unit map and endpoint relation; no identification with $D_c$ is supplied by this definition.

For two named simple-system types $i,j$ with energy/work coordinates $x=(u_i,v_i)$ and $y=(u_j,v_j)$, a thermal join is the model-declared compound state $J_{ij}(x,y)$ whose energy coordinate is $u_i+u_j$ and whose work coordinates retain the ordered pair $(v_i,v_j)$. Its join domain consists exactly of those tuples $(u,v_i,v_j)$ for which there are $u_i,u_j$ with $(u_i,v_i)$ and $(u_j,v_j)$ in the component domains and $u=u_i+u_j$. This is a tagged join state, distinct from the separated pair $(x,y)$, even when their total coordinates agree. The joined domain retains both work coordinates and has aggregate mole composition $n^i+n^j$ (in a common declared species list); it is not thereby a simple-compressible slice of one $D_c$. The contact model supplies its joined equilibrium endpoint space, endpoint coordinate map and admitted process relation. The contact model declares the pair encoding and join encoding to have the same aggregate composition and extends its accessibility relation to these two encodings; the contact axioms specify which such comparisons are admitted.

Write $x{\sim }_{T}y$ exactly when the pair $(x,y)$ and its join state $J_{ij}(x,y)$ are accessible in both directions under the declared contact operation. Here ${\sim }_T$ is a relation between states of the two named systems; it is not numerical equality of temperature. Nested joins use only the model’s declared reassociation and permutation identifications. The coordinate sum alone supplies no accessibility comparison.

A cross-type mixing and reaction network consists of a declared set $\mathcal{G}$ of tagged composition types and a state set $X_{\mathrm{\Gamma }}$ for each type. Each state set comes with a fixed formal composition ${\nu }_{\mathrm{\Gamma }}$, and its states satisfy $\nu (x)={\nu }_{\mathrm{\Gamma }}$ as in Decorated finite-chain costs with catalysts. These mathematical coefficients and their positive type weights are distinct from dimensioned material amounts. For physical channels the model additionally supplies endpoint quantity maps $U_{\mathrm{\Gamma }}:X_{\mathrm{\Gamma }}\to \mathbb{R}\mathrm{J}$ and $n_{\mathrm{\Gamma }}:X_{\mathrm{\Gamma }}\to {\mathbb{R}}^r\mathrm{m}\mathrm{o}\mathrm{l}$ in a declared finite species list, together with any separately tracked mechanical or external-field endpoint quantities. No identification of two distinct scaled tags is made.

For each channel $\alpha$ from $\mathrm{\Gamma }$ to ${\mathrm{\Gamma }}^{\mathrm{\prime }}$, the model supplies sets $Z_{\alpha }$ of configurations, ${\mathcal{K}}_{\alpha }$ of protocol records, ${\mathcal{B}}_{\alpha }$ of outer-boundary records, and ${\mathcal{R}}_{\alpha }$ of transfer records, and equilibrium subsets and endpoint maps ${\xi }_{\alpha }^{-}:Z_{\alpha }^{-}\to X_{\mathrm{\Gamma }}$, ${\xi }_{\alpha }^{+}:Z_{\alpha }^{+}\to X_{{\mathrm{\Gamma }}^{\mathrm{\prime }}}$. A typed record is $P=(I,x^{-},x^{+},z,\kappa ,b,r),$ where $I=[t_0,t_1]$, $t_0<t_1$, is a finite time interval in seconds, $x^{-}\in X_{\mathrm{\Gamma }}$, $x^{+}\in X_{{\mathrm{\Gamma }}^{\mathrm{\prime }}}$, $z:I\to Z_{\alpha }$, $z(t_0)\in Z_{\alpha }^{-}$, $z(t_1)\in Z_{\alpha }^{+}$, ${\xi }_{\alpha }^{-}(z(t_0))=x^{-}$, ${\xi }_{\alpha }^{+}(z(t_1))=x^{+}$, $\kappa \in {\mathcal{K}}_{\alpha }$, $b\in {\mathcal{B}}_{\alpha }$, and $r\in {\mathcal{R}}_{\alpha }$. Write ${\mathcal{T}}_{\alpha }$ for this typed record set. Transfer maps $Q_{\alpha },W_{\mathrm{o}\mathrm{n},\alpha }:{\mathcal{T}}_{\alpha }\to \mathbb{R}\mathrm{J}$ and $N_{\alpha }:{\mathcal{T}}_{\alpha }\to {\mathbb{R}}^r\mathrm{m}\mathrm{o}\mathrm{l}$ record heat, work and material transfer with declared signs; any tracked $\mathrm{\Delta }{K}_{\mathrm{c}\mathrm{m}}$ and $\mathrm{\Delta }{U}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ are also joule-valued record maps. The transfer maps may depend on the whole record; factorization through the field $r$ requires an additional declared condition. The channel specifies a balance predicate on ${\mathcal{T}}_{\alpha }$, or equivalently a balance map into a declared product of quantity spaces whose required value is zero, and a model-admitted subset satisfying that predicate. The predicate states the actual energy and material constraints of that channel; no balance law, physical admission or endpoint realizability follows from the tuple alone. Other channels require their own adopted constraints. Define $(\mathrm{\Gamma },x)\mathcal{R}({\mathrm{\Gamma }}^{\mathrm{\prime }},y)$ exactly when an admitted channel record has those endpoints. No channel is inferred from separated composition.

For a fixed-constitution core closed channel of system $s$, use $X_{\mathrm{\Gamma }}=X_{{\mathrm{\Gamma }}^{\mathrm{\prime }}}=D_c$ and all six original fields, configuration/equilibrium sets and endpoint maps of Thermodynamic process records and transfer conventions. Fix the outer-boundary record $b_s$ of that system and take ${\mathcal{B}}_{\alpha }=\{b_s\}$. The embedding

${\iota }_s(I,x^{-},x^{+},z,\kappa ,r)=(I,x^{-},x^{+},z,\kappa ,b_s,r)$

is a bijection from ${\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)$ onto the core channel's admitted subset; its inverse deletes the fixed field $b_s$. Thus the subsets are identified by this map, not by equality of six- and seven-tuples. On the image assign $Q_{\alpha }({\iota }_s(P))=Q[P](1\mathrm{J})$ and $W_{\mathrm{o}\mathrm{n},\alpha }({\iota }_s(P))=W_{\mathrm{o}\mathrm{n}}[P](1\mathrm{J})$, where the supplier's real values are numerical joule values. Set $N_{\alpha }=0$ there and retain the supplier's stronger no-matter-crossing condition throughout the interval, together with its fixed-chart mass, endpoint velocities and zero kinetic/external changes. These assignments need not factor through $r$ alone. Values on other typed records are separate protocol data and do not enlarge the admitted subset or assert an energy-balance law.

For any permitted equal-formal-mass compound states $A,B$ of the network, define $A\prec B$ exactly when at least one admitted channel record has initial state $A$ and final state $B$; for named single types this is the relation $\mathcal{R}$ above. This is the exact relation supplied to Forward-sector hypotheses for a simple state domain, not a separate abstract order that enlarges or restricts the channel edges. Its extension to finite tagged products and common spectators is governed by the adopted A1–A6 conditions below. Thus every edge in a decorated chain is precisely an admitted channel transition. This identification is a declared model condition.

The network is finite-factor closed when its types admit the declared finite composition and positive scaling operations and each allowed channel respects the model’s stated conservation constraints. In this network class, explicitly adopt the represented state-family conditions of Decorated finite-chain costs with catalysts. Each $X_T$ is a nonempty set of finite coefficient-map states with its declared fixed ${\nu }_T$. The separated-product map

${\pi }_{T,T\mathrm{\prime }}:X_T\times X_{T\mathrm{\prime }}\to X_{T\times T\mathrm{\prime }},{\pi }_{T,T\mathrm{\prime }}(a,b)=a+b$

is surjective, so each admitted separated-product state has a factorization, which need not be unique. For every $t>0$, the positive coefficient-scaling map $s_{t,T}:X_T\to X_{tT}$, $s_{t,T}(a)=ta$, is a bijection with inverse scaling by $1\mathrm{/}t$. These maps commute with the declared reassociation, permutation and scaling conventions and preserve the exact amount-function and formal-mass identities. The empty unit has $X_1=\{0\}$ with the empty coefficient map $0$. A chosen finite real baseline entropy family used for the chain costs must obey

${\sigma }_{T\times T\mathrm{\prime }}(a+b)={\sigma }_T(a)+{\sigma }_{T\mathrm{\prime }}(b),{\sigma }_{tT}(ta)=t{\sigma }_T(a),{\sigma }_1(0)=0,$

independently of a nonunique factorization. These are explicit admissibility and calibration conditions of the declared model, not consequences of a channel record or claims that every physical network satisfies them. Separated products remain sums of represented states; the distinct raw tags of scaled and joined types are retained, with reversible preparation transport as specified in Forward-sector hypotheses for a simple state domain, rather than identifying those tags. The dimensioned endpoint-quantity maps and channel balance predicates remain the separately supplied physical data. In addition, each channel must preserve the dimensionless formal mass $m$ of Tagged scaled systems and separated products; this is the equal-coefficient-mass convention required by the mathematical chain-cost supplier, not a statement that formal mass is kilograms or moles. The common relation on all equal-formal-mass compounds satisfies A1–A6 of Accessibility axioms A1–A6, including consistency with spectators and stability. A decorated finite chain is exactly a finite sequence of such allowed transitions with a common catalyst and the real baseline-entropy edge costs in Decorated finite-chain costs with catalysts.

For $\mathrm{\Gamma },{\mathrm{\Gamma }}^{\mathrm{\prime }}$, let $F(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\mathrm{\prime }})\in \mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$ be the infimum of those finite decorated-chain costs, with $+\mathrm{\infty }$ if no witness exists and $-\mathrm{\infty }$ if costs are unbounded below. The network has no sinks when, for every ordered type pair,

$F(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\mathrm{\prime }})<+\mathrm{\infty }⟺F({\mathrm{\Gamma }}^{\mathrm{\prime }},\mathrm{\Gamma })<+\mathrm{\infty }\mathrm{.}$

No-sinks is an explicit property required of a network used in the global-offset theorem; it is not proved by this definition, and it is not asserted for every physical mixing or reaction. Raw channel data may have an empty channel set, but such data do not satisfy the finite-factor-closed A1–A6 subclass above: each nonempty $X_T$ requires an admitted identity-endpoint witness by A1. That subclass may have no channels between distinct composition types while retaining those reflexive witnesses. In every case no-sinks must be checked, not inferred from the absence of cross-composition channels.

For each simple-system chart ${\mathrm{\Gamma }}_i\subset {\mathbb{R}}^{n_i+1}$, $n_i\ge 1$, let $x=(u,v)$, with one energy coordinate $u$ and $n_i$ work coordinates $v$. The chart is a nonempty open convex domain. The simple-compressible physical case is the fixed-composition slice and embedding of Thermal joins and thermal equivalence, with $n_i=1$, $u=U\mathrm{/}(1\mathrm{J})$ and $v=V\mathrm{/}(1{\mathrm{m}}^3)$. A declared multi-work chart or joined chart instead supplies its own equilibrium endpoint domain and diagonal unit map: $U=u(1\mathrm{J})$ and $W_a=v_a{w}_a$ for fixed positive work-coordinate units $w_a$. The numerical domain is identified with these physical endpoints by that supplied map, and accessibility is pulled back from the admitted endpoint relation. These identifications preserve the internal-energy meaning and fixed composition; extra work coordinates do not follow from the simple-compressible state definition. Adopt these assumptions for every state $x$ and for every thermal join treated as a simple system:

A7 (convex-combination accessibility). For $0<t<1$, the separated scaled pair $(tx,(1-t)y)$ is accessible to the chart state $tx+(1-t)y$, whenever these states are in the declared domains.

For each $x$, define its forward sector $A_x=\{y\in {\mathrm{\Gamma }}_i:x\prec y\}$. Adopt:

S1 (strict successor). Some $y\in {\mathrm{\Gamma }}_i$ satisfies $x\prec \prec y$.

S2 (unique finite-slope support). At $x=(u_x,v_x)$, $A_x$ has exactly one supporting hyperplane through $x$, and it can be written

with finite $P_i(x)\in {\mathbb{R}}^{n_i}$. The map $P_i:{\mathrm{\Gamma }}_i\to {\mathbb{R}}^{n_i}$ is locally Lipschitz: every point has a neighborhood on which one finite Lipschitz constant bounds $\mathrm{\parallel }{P}_i(x)-P_i(y)\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }x-y\mathrm{\parallel }$. A supporting hyperplane means $A_x$ lies in one of its closed half-spaces and meets it at $x$.

S3 (connected relative boundary). The boundary ${\mathrm{\partial }}_{{\mathrm{\Gamma }}_i}{A}_x$, taken in the topology relative to ${\mathrm{\Gamma }}_i$, is arcwise connected.

These are explicit adopted conditions. The slope $P_i$ is a geometric support parameter here; no equation identifying it with measured pressure is asserted. In particular, S1–S3 and A7 do not assert comparison hypothesis for products.

Use thermal joins and ${\sim }_T$ exactly as defined in Thermal joins and thermal equivalence. For every ordered pair of simple systems and every admitted finite join, adopt:

T1 (joining). $(x,y)\prec J_{ij}(x,y)$.

T2 (reversible splitting). Every join state has at least one split into component states at its stated total energy and work coordinates that is reversible to the join state. Also, for every simple-system state $x=(u,v)$ and every $0<t<1$, adopt

$J_{ti,(1-t)i}(tx,(1-t)x)=(u,tv,(1-t)v)\sim (tx,(1-t)x)\mathrm{.}$

The displayed equality specifies coordinates in the tagged join of the two scaled types; it does not identify that join with the original one-work state $x$. A5 of Accessibility axioms A1–A6 separately gives $(tx,(1-t)x)\sim x$, hence transitivity also gives equivalence of the join and $x$.

T3 (thermal-equilibrium transitivity). If $x{\sim }_{T}y$ and $y{\sim }_{T}z$ in the model’s declared contact family, then $x{\sim }_{T}z$. This requires the model to provide the relevant joined types and identifications.

T4 (transversality). For each simple-system state $x$, there are $x_0,x_1$ in that system with $x_0\prec \prec x\prec \prec x_1$ and $x_0{\sim }_T{x}_1$.

The axioms apply to admitted types and their joins only. T4 is an assumption about the contact relation; it is not a consequence of an equation of state or of a thermometer reading. The stronger range condition T5 is stated separately in Universal thermal-equilibrium range T5, and join coherence C1 is separately stated in Equilibrium-join coherence C1.

Adopt T5 (universal thermal-equilibrium range): for every ordered pair of simple systems $i,j$, each state $x$ of $i$, and every work-coordinate point $v_j$ in the projection of the state domain of $j$, there is some state $y=(u_j,v_j)$ of $j$ with $x{\sim }_{T}y$, using Thermal joins and thermal equivalence.

This is a quantified existence assumption for a thermal-equilibrium partner at each permitted work coordinate. It does not define a numerical temperature, assert that all systems possess the same temperature range as a real apparatus, or supply a calibration map and its uncertainty.

Adopt C1 (equilibrium-join coherence): whenever $x{\sim }_{T}y$, the joined state $J_{ij}(x,y)$ is thermally equivalent to each constituent, $J_{ij}(x,y){\sim }_{T}x$ and $J_{ij}(x,y){\sim }_{T}y$, using the model’s declared nested-join identifications.

C1 is an additional postulate of this completed contact branch. The source’s Theorem 4.8 is stated with T1–T4; the local conditional calibrator route uses the constituent-to-join bridge in addition to the stronger T5 range assumption. Neither fact licenses deriving C1 from $u=u_i+u_j$, from T1–T4, or from transitivity alone. The displayed source supports the contact context, not the claim that its bare printed axioms already contain C1.

For every declared family obtained as the closure of finitely many simple-system generators under positive scaling and finite thermal joins of Thermal joins and thermal equivalence, assume the accessibility axioms A1–A6, the simple-system assumptions A7 and S1–S3, the contact assumptions T1–T4, the universal thermal-equilibrium range T5, and equilibrium-join coherence C1, with the exact formulations in Accessibility axioms A1–A6, Simple-system geometric axioms, Thermal-contact axioms T1–T4, Universal thermal-equilibrium range T5, and Equilibrium-join coherence C1. The closure itself may be infinite; each join and product expression is finite. Then for any two simple types $i,j$, there are $x_0\prec \prec x_1$ in $i$ and $y_0\prec \prec y_1$ in $j$ such that

$(x_0,y_1)\sim (x_1,y_0)\mathrm{.}$

Moreover, for this fixed pair $i,j$, every finite separated product made only from positive scaled copies of $i$ and $j$ satisfies the comparison hypothesis: any two equal-formal-mass states $A,B$ of that product type satisfy $A\prec B$ or $B\prec A$. No comparison conclusion for products involving a third primitive type is contained in this pairwise result. These are conditional model deductions. In particular, the conclusion is not claimed from the printed Theorem 4.8 interface with T1–T4 alone; the local mathematical supplier explicitly requires the additional T5 and C1 branch.

Lieb–Yngvason’s printed Theorem 4.8 states T1–T4, while its surrounding contact development includes T5 separately. The current local mathematical calibrator carrier makes its actual input stronger and explicit: it also uses C1, the equilibrium-join coherence bridge. Until that carrier and its upstream batch 6 prerequisites are accepted, this physics theorem remains conditional and escalated.

Let $I$ be a nonempty, possibly infinite set of primitive system labels. Assume the declared family includes every finite separated product and every finite thermal join of positive scaled copies. For every nonempty finite subset $F\subseteq I$, assume its closure under positive scaling and finite thermal joining satisfies the full hypotheses A1–A7, S1–S3, T1–T5 and C1 of Contact calibrators under explicit join coherence, with the exact represented-state and preparation conventions of Forward-sector hypotheses for a simple state domain and Thermal joins and conditional contact hypotheses. The closure may be infinite; “finite” refers to the generators and to each individual product or join expression. These hypotheses concern one common declared physical endpoint relation, with its chart and join interpretations, not independently chosen orders on overlapping families. Also assume the explicit finite-factor representation and fully scalable preparation transport of Tagged scaled systems and separated products and the Axiom of Choice. Then there is a single dimensionless function $\tilde{S}$ on all states of the declared finite-factor family that is additive under separation, extensive under every admitted positive scaling, and characterizes every equal-composition comparison:

$A\prec B⟺\tilde{S}(A)\le \tilde{S}(B)\mathrm{.}$

For each simple chart, A7 also makes $\tilde{S}$ concave on every convex region where the stated recombination operation is admitted. The result does not set the entropy unit to $\mathrm{J}{\mathrm{K}}^{-1}$, determine additive constants across different reaction compositions, or establish empirical validity of the contact assumptions.

The mathematical calibration carrier is a current authored draft with a supplied proof; independent review is still required. The transfer here is conditional on its stated mathematical hypotheses and on the adopted physical accessibility/contact axioms, and is not evidence that those axioms hold in Nature. The calibrated meter in the mathematical construction fixes a common dimensionless unit. Conversion to the physical entropy unit $\mathrm{J}{\mathrm{K}}^{-1}$ requires an additional scale calibration. Additive type offsets for cross-composition transitions are handled separately in Conditional weak entropy principle across mixing channels.

Let $\mathcal{G}\mathrm{\ne }\mathrm{\varnothing }$ be a finite-factor-closed network of Cross-type mixing and reaction network. Assume the contact hypotheses of Entropy representation for finite separated products provide a finite real baseline entropy ${\sigma }_{\mathrm{\Gamma }}$ on each type. Also assume that network satisfies A1–A6, its exact decorated-chain cost $F$ has the no-sinks property defined there, and the Axiom of Choice. Then there is a real offset $B(\mathrm{\Gamma })$ for each type, additive under finite composition and extensive under every positive scaling, such that the corrected entropy

$S_{\mathrm{\Gamma }}(x):={\sigma }_{\mathrm{\Gamma }}(x)+B(\mathrm{\Gamma })$

is nondecreasing across every admitted channel transition: if $(\mathrm{\Gamma },x)\mathcal{R}({\mathrm{\Gamma }}^{\mathrm{\prime }},y)$, then $S_{\mathrm{\Gamma }}(x)\le S_{{\mathrm{\Gamma }}^{\mathrm{\prime }}}(y)$. The conclusion is weak monotonicity only. It does not assert a no-gap identity, uniqueness of $B$, strict increase, or a global entropy principle for networks that fail the stated hypotheses.

The cross-type cost, quotient and no-sinks sandwich carriers belong to the mathematical supplier page and are currently authored draft suppliers on that page; this conditional use records no independent acceptance. This item records their exact conditional use; it does not certify those proofs or the physical network assumptions. The stronger no-gap branch in the research notes is an additional hypothesis and is not part of this theorem.

The order relation in Adiabatic accessibility is a model-defined relation between equilibrium endpoints; it is not a measured trajectory law. A1–A6, A7, S1–S3, T1–T5 and C1 are adopted physical assumptions. The contact-product conclusion is conditional on the full list. In the inspected Lieb–Yngvason source, Theorem 4.8 is stated under T1–T4; the local supplier’s proof branch explicitly also needs the universal range T5 and the separate join-coherence bridge C1. The bare printed source interface does not discharge those stronger inputs.

The product representation $\tilde{S}$ is a dimensionless order parameter normalized by a chosen strict meter. A physical entropy reported in $\mathrm{J}{\mathrm{K}}^{-1}$ requires an external scale calibration and stated uncertainty. The additive constants for composition types are not fixed by the no-mixing theorem; the mixing theorem fixes only a weak monotonicity selection under its extra hypotheses.

The arbitrary-family calibration supplier assumes an explicit finite-factor representation and the Axiom of Choice for simultaneous reference/calibrator selections. The mixing supplier further requires an actual cross-type transition network, finite decorated-chain costs, A1–A6 on those transitions, and the no-sinks condition. It does not establish that arbitrary mixing or reaction channels satisfy these conditions. No-gap equality, uniqueness of reaction offsets, or strict entropy increase requires additional hypotheses and is not asserted here.

The Carathéodory treatment of Goddard is a distinct route based on local inaccessibility. It supports the conceptual distinction between accessibility assumptions and derived consequences, but it is not a second proof of the convex-contact product theorem.