What does Gameness actually look like?

Rules, State, Possibility Space, Trajectory, Role, Agency. The words are already there. This piece assigns each of them a mathematical object and keeps the assignments fixed.

The construction proceeds as

Existing name	Formal object
Rule presentation	$\mathfrak R$
Configured form	$G$
Candidate extraction	$\operatorname{str}:\mathfrak C\to\mathcal G$
State space	$S_G=X/{\sim_G}$
Possibility Space	$\mathcal P_G$
Trajectory	$\operatorname{trace}_G(I)$
Continuation map at a State	$\kappa_s$
Structural Agency	$\operatorname{Ag}(G)$
Gameness-support profile	$\operatorname{GProf}(G)$

Table of contents

• Rule presentation
• Inspections and State
• Possibility Space and operating instances
• Continuation and Agency
• The Game bridge
• Witness geometry
• Observable quotient and structural invariance
• Comparison and realization
• Structural gameness-support profile
• Epistemic extension

Rule presentation

Take a carrier of raw configurations $X$, a set of interventions $A$, and a finite set of intervention roles $R$. For each role $r$, let

be its authorized contribution alphabet. For $U\subseteq R$, write

For each role $r$, let $\operatorname{obs}(r)$ be a family of total readings $o:X\to V_o$.

Definition R1. Rule factorization. Let $Z$ be a set of typed configuration ports, with non-empty value carrier $V_z$ for each $z\in Z$. For $J\subseteq Z$, define

Each port has a total coordinate reading $\pi_z:X\to V_z$. Require the joint map

to be injective. Write $\pi_J$ for the corresponding tuple of coordinate readings.

Let $C$ be a finite set of Rule-cell labels.

Rule formation discipline. Content identities may occur only as values in typed carriers. They may not index or name elements of $Z$, $C$, or $R$, and they may not name a disclosure map.

Let $\mathcal V$ be the typed carrier vocabulary with its declared predicates and authority after content-identity labels have been forgotten. Take

Each $\sigma\in\Sigma$ acts by bijections on the port carriers, disclosure-result carriers, and $A$, fixes the port and role labels, and satisfies

A Rule presentation is

where each cell $c\in C$ has finite input scope $I_c\subseteq Z$, intervention-role scope $U_c\subseteq R$, output scope $O_c\subseteq Z$, an application domain

and one local relation

For each role $r$, $\operatorname{read}_c(r)$ is a finite family of typed total disclosure maps

One cell is one scoped clause. A bound input $(i,u)\in D_c$ activates it. A candidate output $v$ is accepted when $(i,u,v)\in L_c$; if no such $v$ exists, the active cell refuses the candidate transition. Outside $D_c$, the cell is idle. Cells with $U_c=\varnothing$ are intervention-free; $I_c=\varnothing$ gives a generative cell; $O_c=\varnothing$ gives a constraint cell.

The logical directions are

for read-only, updated, and produced ports respectively.

Lift each local relation to the global carriers by

For $x\in X$ and $p\in\operatorname{Prof}$, define

Rule axiom R2. Equivariance. Within the Rule formation discipline, for every $\sigma\in\Sigma$,

and

Every $d\in\operatorname{read}_c(r)$ is equivariant:

Rule axiom R3. Local composition. The admitted global transition relation is exactly

The intersection is the natural join of the lifted local relations. The final equality is the frame clause. Overlapping outputs agree because they become one configuration.

Two cells have a directed Rule intersection when

The shared port is time-polarized by Rule axiom R3: $c$ constrains its value in $x'$, and $d$ may consume it when $x'$ becomes the source of a later transition.

Definition R4. Local Rule dependence. Regard $(i,u)\in\operatorname{Val}(I_c)\times\operatorname{Prof}(U_c)$ as one valuation on $I_c\sqcup U_c$. For $W\subseteq O_c$, define

When $a=(i,u)$, write $\operatorname{Out}_c(a;W)$.

A non-empty input block $B\subseteq I_c\sqcup U_c$ activates $c$ when two input valuations agreeing outside $B$ differ in membership in $D_c$. The block feeds $W$ through an applied $c$ when two such valuations both lie in $D_c$ and their $\operatorname{Out}_c(-;W)$ sets differ.

With $W=\varnothing$, the output set distinguishes active refusal from positive support. For non-empty $W$, it retains nondeterministic support and tuple correlation.

For $d\in\operatorname{read}_c(r)$ and non-empty $B\subseteq I_c$, the block feeds the disclosure map when there are $i,i'\in\operatorname{Val}(I_c)$ agreeing outside $B$ with $d(i)\neq d(i')$.

Definition R4.1. Globally extendable composite dependence. Let

Its globally extendable application traces are

For

write

For a valuation on a finite typed coordinate set $K$ and a block $B\subseteq K$, write

when $a$ and $a'$ agree on every input coordinate outside $B$.

For an initial valuation

let

A non-empty block $B\subseteq I_{c_0}\sqcup U_{c_0}$ changes the global extendability of $\mathbf c$ when there are $a\equiv_{-B}a'$ such that exactly one of

is empty. This is composite activation sensitivity. It is separate from transmission through an applied composite.

For $0\leq j<n$, put

Let $\varnothing\neq B\subseteq I_{c_0}\sqcup U_{c_0}$ and $\varnothing\neq W\subseteq O_{c_n}$. The block $B$ feeds $W$ through $\mathbf c$ when there are

in $\operatorname{ExtTrace}_{\mathfrak R}(\mathbf c)$ such that

and, for every $0\leq j<n$,

At the final cell, require

Both sides are complete admitted application traces. At each cell, the compared local inputs agree outside the incoming block, the local output support depends on that block, and the globally admitted outputs carry the distinction into the next shared block.

If $c_n\leadsto e$ and $d\in\operatorname{read}_e(r)$, put

The block $B$ feeds the disclosure $d$ through $\mathbf c$ when two traces $\xi,\xi'$ satisfy the initial condition and every intermediate condition above, and also

and

The fixed-other-input conditions rule out attribution through an unselected parallel input. Parallel routes may still coexist; this predicate establishes dependence through the selected composite without claiming unique provenance.

Rule axiom R5. Scoped information. The declared readings are exactly the Rule-disclosed maps:

$\operatorname{read}_c(r)$ is a standing disclosure interface. Its input may be rooted in $X_0$, retained by the frame clause, or supplied through directed intersections across successive applications.

For a transition claim $\varphi\subseteq X\times\operatorname{Prof}\times X$, define

An output occurrence is extensionally correct for $\varphi$ when

This inclusion establishes extensional correctness, not validation. Validation additionally requires a declared derivation from the Rule inputs supporting $\varphi$ to the asserting output. That derivation must be supplied by further Rule relations. A certificate remains a port value whose preservation and disclosure require further Rule relations.

Define the admitted profiles at a configuration by

Let $\varnothing\neq X_0\subseteq X$ be the admitted beginnings. The configured form is

Write $\mathfrak R\models G$ when Rule axioms R2, R3, and R5 hold over this configured vocabulary. From here, $G$ ranges over configured forms presented by at least one such $\mathfrak R$.

$\mathfrak R$ retains scope, intersection, dependence, and declared disclosure structure. $G$ retains the configured transition behavior from which State, continuation, and structural Agency will be derived.

The configured form is possibilistic: $\longrightarrow$ retains transition support. Every construction below that is derived from fixed $G$ alone is invariant under changes of outcome weights that preserve that support.

Axiom 0. Structural scope. Structural Agency support is determined by $G$ and no further data. An instance-indexed Agency claim may additionally use an operating instance only to identify a reached State, a transition window, and the roles bound there.

This scopes the calculus, not gamehood. Material, authorship, intent, reception, or another unrepresented fact does not enter $\operatorname{Ag}(G)$. The bridge to candidate games remains a separate assumption below.

Definition I1. Structural isomorphism. Let

A configured-form isomorphism $f:G_1\cong G_2$ consists of bijections

They induce

and preserve beginnings and authority:

For every role $r$, a bijection

pairs each reading $o:X_1\to V_o$ with $o':=f_{\operatorname{obs},r}(o):X_2\to V_{o'}$. A carrier bijection $f_{V_o}:V_o\to V_{o'}$ satisfies

Finally,

Now let $\mathfrak R_i\models G_i$. A Rule-presentation isomorphism

extends a configured-form isomorphism by bijections

carrier bijections $f_{V_z}:V_z\to V_{f_Z(z)}$, and a group isomorphism $f_\Sigma:\Sigma_1\to\Sigma_2$. For every $\sigma\in\Sigma_1$,

with the analogous equation on $A$. The coordinate maps satisfy

For every cell $c$, the isomorphism preserves its scopes:

The induced product maps preserve and reflect the application domain:

and the local relation:

For every $c$ and $r$, there is a bijection

For $d'=f_{\operatorname{read},c,r}(d)$, a result-carrier bijection satisfies

and, for every $\sigma\in\Sigma_1$,

The induced pairing of $d\circ\pi_{I_c}$ agrees with $f_{\operatorname{obs},r}$. Thus configured-form isomorphism preserves the full global denotation, while Rule-presentation isomorphism additionally preserves its declared factorization.

Inspections and State

Let $\mathsf{Insp}_G$ be the carrier of inspections.

Axiom 1. Self-disclosure. The atomic basis is exactly

Its result carriers and readings are

and, for every $(r,o)\in\{r\}\times\operatorname{obs}(r)$,

Every $b\in\mathcal B_G$ supplies one atomic inspection $\operatorname{atom}(b)\in\mathsf{Insp}_G$, and there are no other atomic inspections.

Axiom 2. Free formation. For every $1\leq n<\infty$ and every $p\in\operatorname{Prof}$ there are constructors

The map

obtained from $\operatorname{atom}$, all $\operatorname{tuple}_n$, and all $\operatorname{prefix}_p$ is injective. Write $\operatorname{tuple}_n(\iota_1,\ldots,\iota_n)$ as $(\iota_1,\ldots,\iota_n)$ and $\operatorname{prefix}_p(\iota)$ as $p\triangleright\iota$.

Axiom 3. Branching observation. Every inspection $\iota$ has a result carrier $V_\iota$ and a total result map $\operatorname{res}_\iota:X\to V_\iota$. These satisfy

and

Tupling retains correlation between residual probes. Prefixing returns the set of results over every admitted outcome of the fixed profile.

The following pair is separated by the branching inspection while the stated flattened profile-word result sets identify it. In a one-role form, take a total reading $o$ with one common value on every non-terminal configuration and values $0,1$ on two terminals. Let

Every middle configuration admits $b$ and $c$. From $u$, both profiles may reach terminal readings $0$ or $1$. From $v_0$, both reach only $0$; from $v_1$, both reach only $1$. All terminals admit no profile. Every linear sequence $ab$ or $ac$ therefore obtains $\{0,1\}$ at both roots. The correlated inspection

returns

at $x$, and

at $y$.

Axiom 4. Structural induction. For every $P\in\mathcal P(\mathsf{Insp}_G)$,

Definition 1. Inspection depth. Define

Corollary 4.1. Every inspection has finite depth.

Structural induction preserves finiteness under finite maxima and prefixing.

Definition 2. State. Define

The State space is

Proposition 1. $\sim_G$ is an equivalence relation and the coarsest equivalence relation respecting every inspection.

It is the kernel of the family $(\operatorname{res}_\iota)_{\iota\in\mathsf{Insp}_G}$.

Proposition 2. Configurations of one State admit the same profiles.

The atomic inspection $\operatorname{atom}(\mathbf{adm})$ returns $\operatorname{Adm}_G$. Therefore, for $s\in S_G$, define

Proposition 3. Every residual inspection of a branching continuation is an inspection of its root after prefixing the intervening profile.

For every $\iota$, its result after $p$ is $\operatorname{res}_{p\triangleright\iota}$ at the root.

Remark 4. Inspection depends on the current configuration only through its State. Any inspectable historical distinction therefore belongs to the current State.

Proposition 5. Strictly coarsening $\sim_G$ cannot preserve every inspection. Strictly refining it adds no inspectable behavior.

The first operation merges a pair separated by some result map. The second separates a pair outside the kernel without adding a result map.

Possibility Space and operating instances

Definition 3. Possibility Space. The Possibility Space $\mathcal P_G$ is the raw substructure reachable from $X_0$, including its configurations, admitted profiles, and full transition relation, with each configuration carrying its State as a label.

Write

for its raw configuration carrier and $S_G^{\mathrm{reach}}$ for the State labels occurring in it. Paths remain paths through raw configurations.

For $\mathfrak R\models G$, define the optional factorized Possibility Space by

where $\mathfrak R^{\mathrm{reach}}$ restricts each $\pi_z$ to $X_G^{\mathrm{reach}}$, and

$\mathcal P_G$ records configured continuations. $\widehat{\mathcal P}_{G,\mathfrak R}$ additionally records the accepting local clauses on each admitted edge.

Definition I2. Infrastructure. An infrastructure is the material, computational, or procedural arrangement through which $G$, and possibly its Rule presentation, is represented or operated.

Definition I3. Realization. An infrastructure behaviorally realizes $G$ when its represented layer is configured-form isomorphic to $G$ under Definition I1. It Rule-realizes $(G,\mathfrak R)$ when that isomorphism extends to the Rule-presentation isomorphism of Definition I1.

Proposition I4. Realization-invariance. Every construction defined in this article from $G$ alone is preserved under behavioral realization; numerical values agree. Rule realization additionally preserves application domains, cell and port incidence, local dependence, disclosure maps, and the Rule-mediated incidence defined below.

This follows from the relevant structural isomorphism. Concrete bindings and committed Trajectories require the corresponding instance data.

Let $\operatorname{Path}(G)$ be the set of finite or countably infinite sequences, including zero-edge paths,

with $x_0\in X_0$ and an admitted transition at every transition index. Let $\mathsf{Inst}_G$ be the carrier of declared operating instances and $\mathsf{Src}$ the carrier of sources.

Axiom 5. Operating-instance structure. There is a total trace map

Define

There is also a total binding assignment

Definition 4. Operating instance and Trajectory. An operating instance is an element $I\in\mathsf{Inst}_G$. Its committed Trajectory is $\operatorname{trace}_G(I)$.

Definition 4.1. Session and gameplay Run. Let $\mathsf{BInt}_G$ be the carrier of bounded trace intervals

A Session assignment is operating-context data

Its members are the bounded intervals opened and closed by the surrounding operation.

For $(I,i,j)\in\mathsf{BInt}_G$, define its finite State-profile word by

with the zero-edge word $(s_i^I)$ when $i=j$. Let $\mathsf{BWord}_G$ be the carrier of all finite State-profile words realized by bounded finite admitted path segments in $\mathcal P_G$. Thus

for every $(I,i,j)\in\mathsf{BInt}_G$, while $\mathsf{BWord}_G$ also contains configured path segments no operating instance has actualized.

A gameplay Run recognition structure is the optional configured extension $(G,\mathcal L_G^{\mathrm{run}})$ with a declared language

It is structural-isomorphism-invariant: every $f:G\cong H$ under Definition I1 induces the coordinatewise word bijection $f_{\mathrm{word}}$ and satisfies

The interval $(I,i,j)$ is a gameplay Run exactly when

Bare $G$ supplies no canonical $\mathcal L_G^{\mathrm{run}}$; declaring one selects a boundary relation over retained State and profile distinctions. The Session assignment remains operating-context data, and no axiom aligns it with $\mathcal L_G^{\mathrm{run}}$. A Session may therefore contain several gameplay Runs, and one gameplay Run may span several Sessions.

Definition 5. Transition window. $\Delta t_n$ is the logical window joining $s_n$ to $s_{n+1}$ in the order supplied by Axiom 5:

Definition 6. Role and binding. For $\alpha\in\mathsf{Src}$, define the role block occupied at $\Delta t_n$ by

Authority supplies contribution values. Binding attributes role coordinates to a source. Structural Agency will compare the continuations opened by those coordinates.

Definition 7. Role-indistinguishability. For a role $r$, let

Let $\mathsf{Insp}_{G,r}$ be the least subfamily of $\mathsf{Insp}_G$ containing $\operatorname{atom}(b)$ for every $b\in\mathcal B_{G,r}$ and closed under every tuple and prefix constructor of Axiom 2. Define

The admissibility atom is excluded from $\mathcal B_{G,r}$; profile prefixes still range over $\operatorname{Prof}$.

Proposition 6. $\approx_r$ is coarser than $\sim_G$. It is strictly coarser exactly when some pair of configurations is separated by a $G$-inspection but by no $r$-inspection.

Proposition 7. $\approx_r$ is fixed by $G$ and independent of the operating instance.

Definition 8. Available contributions. For a non-empty role set $U\subseteq R$ and reachable State $s$, define

For $\gamma\in\Gamma_G(U,s)$, define its completions by

Here $\gamma\oplus\delta$ is the unique profile agreeing with $\gamma$ on $U$ and with $\delta$ on $R\setminus U$.

Continuation and Agency

Definition 9. Raw branching continuation. Define the raw continuation carrier

Write

For every inspection $\iota$, its residual result on this continuation is

Nested prefix inspections carry every finite admitted continuation after the immediate outcomes. The token selects no particular successor.

Definition 10. Depth-$k$ indistinguishability. For configurations, define

For $\mathcal T_1,\mathcal T_2\in\mathsf{RCont}_G$, define

and define the equivalence relation on $\mathsf{RCont}_G$

Lemma 10.1. State-rooted continuation. If $x,y$ belong to one State and $p$ is admitted there, then

for every finite $k$.

Otherwise, prefixing a separating residual inspection by $p$ would separate $x$ from $y$. Therefore define

The induced finite-depth relation on continuation classes is

Proposition 8.

and

The inclusions follow because depth $k+1$ adds inspections. The intersections follow from Corollary 4.1.

Proposition 9. If two continuation classes differ, a least separating residual depth exists.

The set of separating depths is non-empty and upward closed in $\mathbb N_0$.

For a reachable State $s$, set

and define the continuation map

For $p,q\in D_s$, write

Definition 11. Agency. A non-empty role set $U\subseteq R$ bears structural Agency at $s$ when

Equivalently, there are $\gamma_1\neq\gamma_2$ in $\Gamma_G(U,s)$ and one shared completion

such that

At a concrete window with $U_I(\alpha,\Delta t_n)\neq\varnothing$, Agency is available through $\alpha$’s binding exactly when

The binding and State are actual. The witness pair is counterfactual.

Definition 11.1. Rule-mediated Agency incidence. For a witness

define its direct role incidences by

For $x\in s\cap X_G^{\mathrm{reach}}$, write

The cell is activation-sensitive when exactly one of these valuations lies in $D_c$. When both lie in $D_c$, it is locally output-sensitive toward $W\subseteq O_c$ when

$\operatorname{DirectInc}_{\mathfrak R}$ locates the first active Rule scope receiving the witness difference. $\kappa_s$ decides the complete continuation class.

Definition 12. Witnesses and latency. Let

For each witness pair, define

For a role block $U$, define

with value $\infty$ when the set is empty. For an occupied block,

Proposition 10. The following are equivalent:

and, within every shared completion, all contributions through $U$ lie in one continuation class.

Proposition 11. For $U_I(\alpha,\Delta t_n)\neq\varnothing$, Agency is present through the instance binding exactly when its latency is finite.

Proposition 12. Internal nondeterminism may select one outcome of an unchanged profile without producing an Agency witness for any binding.

Proposition 13. Role uncertainty does not veto local Agency.

$\Gamma_G(U,s)$ is structural availability at the reached State. Information and policy remain separate layers.

Remark A1. Rule mutation. Inside one fixed $\mathfrak R$, a mutable Rule parameter is data in $X$ read by cells that interpret it. An admitted intervention changes that data through Rule axiom R3. If the value changes a later inspection result, the affected configurations occupy different States.

Changing a cell, scope, local relation, or disclosure interface outside represented meta-Rules produces a new presentation $\mathfrak R'$. It may present the same $G$ or a different $G'$.

Definition 13. Agent. Where instance-indexed Agency is available, the source occupying the evaluated roles is an Agent at that binding and window.

Proposition 14. Locality. Agenthood is local to $(G,I,\alpha,s_n,\Delta t_n)$ through the reached State and occupied role set.

Definition 14. Structural grades. For $1\leq m\leq|R|$, define

and

Proposition 15. Global continuation constancy.

exactly when, at every reachable State, every two admitted profiles open the same continuation class.

Remark 15.1. $\operatorname{Ag}_1$ and $\operatorname{Ag}_*$ can differ.

Take two roles, each authorized for $0$ and $1$, and admit only $(0,0)$ and $(1,1)$ at one State, with the profiles opening different continuation classes. No singleton role block has a shared completion, while the two-role block does.

Proposition 15.2. Difference-set normal form. For every reachable $s$ and non-empty $U\subseteq R$,

Proposition 16. Role-block monotonicity. If $U\subseteq V$ and $(s,U)\in\operatorname{Ag}(G)$, then $(s,V)\in\operatorname{Ag}(G)$.

The same witness difference set lies inside both blocks.

Corollary 16.1. $\operatorname{Ag}(G)=\varnothing$ exactly when $(s,R)\notin\operatorname{Ag}(G)$ for every reachable State $s$.

Definition 15. Minimum Agency arity. Define

and $m_G^*(s):=\infty$ when $\mathcal W_G(s)$ is empty.

Define the reachable Agency-support locus by

Definition 16. Continuation fibres. The fibres of $\kappa_s$ partition $D_s$ by continuation class. Call $\kappa_s$ non-constant when

equivalently when $|\operatorname{im}\kappa_s|\geq2$.

Proposition 17. Agency is non-constancy. At a reachable State $s$, some non-empty role block bears Agency exactly when

For the reverse direction, choose profiles in different fibres and take $U=\operatorname{Diff}(p,q)$.

Definition 17. Structural playability. A configured form is structurally playable when some reachable continuation map has at least two values:

Therefore

The Game bridge

Let $\mathcal G$ be the class of configured forms admitted by the structural signature above, and let $\mathfrak C$ be a domain of candidate objects carrying whatever further data an account of gamehood may require.

Definition B1. Structural extraction. A declared extraction is a total map

The extraction is fixed before Agency is evaluated. Variation exposed by the candidate’s Rules as a contribution port becomes a role coordinate. Variation remaining after every represented contribution has been fixed remains inside $\longrightarrow$. Recasting one internal branch as a fictitious role changes $\operatorname{str}(z)$; it does not discover Agency in the same configured form.

No canonical extraction is asserted. Every bridge claim below is relative to the declared pair $(\mathfrak C,\operatorname{str})$. Let $\operatorname{Game}$ be a primitive predicate on $\mathfrak C$.

For $z\in\mathfrak C$, write

for the constructions above evaluated on its extraction.

Bridge axiom B2. Representation. Every game has a structurally playable extraction:

Equivalently,

Under the bridge,

Since the extracted role carrier is finite, $\operatorname{Diff}(p,q)$ is a finite role block.

The converse is absent. The bridge is refuted by one accepted candidate $z$ for which

Every structural result about Agency, latency, arity, and reconvergence survives that refutation. Only the bridge to the ordinary word game fails.

The bridge is support-level because $\operatorname{str}(z)$ retains transition support rather than outcome weights. An accepted game whose only contribution-sensitive differences alter probability over identical supports refutes B2 under that extraction. A distribution-sensitive bridge requires a richer configured form; it is not obtained by reading weights into $G$ after the fact.

Assumption bill and a concrete model

Assumption	What it fixes	Remove it and
Axiom 0	structural Agency reads only $G$	unrepresented material, intent, or reception may enter the Agency test
Rule formation discipline	content identities remain carrier values	Rule, port, role, or disclosure labels may hardcode proper nouns
R2	Rule clauses respect the declared carrier symmetries	structurally interchangeable values may be treated differently
R3	active cells jointly determine one framed transition	the Rule presentation no longer determines $\longrightarrow$
R5	every declared reading has a scoped Rule source	information may bypass Rule incidence
Axiom 1	admissibility and disclosures are atomic inspections	State-defined availability need not follow
Axiom 2	inspection constructors are free and unambiguous	inspection syntax may identify unrelated probes
Axiom 3	observation retains every admitted outcome and its correlation	a linear observation language induces another continuation relation
Axiom 4	every inspection is finitely generated	finite depth need not exhaust State sameness
Axiom 5	operation supplies traces, windows, and bindings	concrete Trajectory and Agenthood lose their instance index
B2	candidate gamehood requires non-empty structural support	the calculus no longer rules candidates out as games

The assumptions are jointly satisfiable. Take

$$R=\{r\}, \qquad A=\{0,a,b\}, \qquad \operatorname{auth}(r)=A,$$

$$X=\{x_0,x_1,x_2\}, \qquad X_0=\{x_0\},$$

and one configuration port $q$ with $V_q=\{0,1,2\}$ and $\pi_q(x_i)=i$. Choose the typed vocabulary so that its automorphism group is trivial, and take one cell $c$ with

$$I_c=O_c=\{q\}, \qquad U_c=\{r\},$$

$$D_c = \{(0,0),(0,a),(0,b),(1,a),(2,0)\},$$

$$L_c = \{ ((0,0),0), ((0,a),1), ((0,b),2), ((1,a),1), ((2,0),2) \}.$$

Let

$$\operatorname{read}_c(r)=\{\operatorname{id}_{V_q}\}, \qquad \operatorname{obs}(r)=\{\pi_q\}.$$

R2, R3, and R5 then hold, and the presentation admits exactly

$$x_0\xrightarrow{0}x_0, \quad x_0\xrightarrow{a}x_1, \quad x_0\xrightarrow{b}x_2, \quad x_1\xrightarrow{a}x_1, \quad x_2\xrightarrow{0}x_2.$$

Take the free inspection algebra of Axioms 1-4, let operating instances be admitted finite or countable paths with total bindings, set $\mathfrak C=\{z_*\}$, and declare

$$\operatorname{Game}(z_*), \qquad \operatorname{str}(z_*)=G.$$

The reading of $q$ separates $x_1$ from $x_2$, so $a$ and $b$ at $x_0$ open different continuation classes and $\operatorname{SPlay}(G)$ holds. This discharges joint satisfiability; it supplies no empirical evidence for B2.

Witness geometry

Definition W1. Raw reach and common recovery. For $p\in D_s$, define

For raw configurations, let

and define

For an admitted pair $p,q$, define

Proposition W2. Once $\operatorname{Recover}_{\leq j}^G(s;p,q)$ is non-empty, it remains non-empty at every greater depth.

Each raw reach set grows monotonically with $j$.

Definition W3. Possible reconvergence. For $\{p,q\}\in\mathcal W_G(s)$, define

and set $j_G(s;p,q):=\infty$ when no finite bound exists.

A finite value gives one uniform bound within which every immediate outcome retains a route to one common State. It asserts neither actual return nor control of that route. For a witness pair, $j_G(s;p,q)\neq0$.

Remark W4. Consequence persistence. Reconvergence depth is not the lifetime of a distinction. A persistence quantity would have to rebase the comparison at later States and declare how counterfactual paths, and under nondeterminism their outcomes, are paired. The signature supplies no uniform canonical pairing across arbitrary nondeterministic forms, although a particular form may supply one. The present calculus therefore retains latency and possible reconvergence without assigning a general persistence number.

Observable quotient and structural invariance

When State also carries transitions

Extension axiom $F_O$. Finite observable outcome images. For every configuration $x$ and profile $p$, define

$$\operatorname{SPost}_G(x,p) := \{\,[x']_{\sim_G}\mid x\xrightarrow{p}x'\,\},$$

and suppose

$$|\operatorname{SPost}_G(x,p)|<\infty.$$

Proposition O1 ($F_O$-transfer). Under $F_O$, $\sim_G$ is a reading-preserving bisimulation for the profile-labelled transition relation.

Suppose $x\sim_G y$ and $x\xrightarrow{p}x'$. Proposition 2 gives $p\in\operatorname{Adm}_G(y)$. If no $p$-successor of $y$ shared the State of $x'$, choose one representative for each of the finitely many successor State classes of $y$, choose an inspection separating each class from $x'$, tuple those inspections, and prefix the tuple by $p$. The resulting inspection would separate $x$ from $y$. The converse direction is symmetric.

Therefore the quotient admits the transition relation

$$s\xRightarrow{p}_G t \Longleftrightarrow \exists x\in s,\,y\in t: x\xrightarrow{p}y.$$

Proposition I5. Structural isomorphism-invariance. Let $f:G\cong G'$ be a configured-form isomorphism under Definition I1. Let

be the induced State bijection, and let $f_{\operatorname{Prof}}$ be the induced profile bijection. Then

and

Role bijections preserve difference-set cardinality. Configured-form isomorphisms preserve reachability, raw outcomes, inspections, and inspection depth.

Comparison and realization

Definition C1. Endogenous comparison scheme. Let $\mathcal D$ be a class of configured forms. A comparison scheme is a pair $(\Pi,q)$ with an assignment

and, for every $G\in\mathcal D$, a surjective bounded structural quotient

Write $H:=\Pi(G)$. The quotient certificate contains surjections

and

such that

Every source transition has its image,

and every target transition lifts from every representative:

Observations factor through the quotient. A partial surjection

pairs every target reading with at least one source reading. It is role-compatible on its domain:

Whenever

there is a surjection $q_{V_o}:V_o\twoheadrightarrow V_{\bar o}$ satisfying

The scheme respects structural renaming: every isomorphism $f:G\cong G'$ induces $\Pi(f):\Pi(G)\cong\Pi(G')$ with

Finally, $q_{\Pi(G)}$ is an isomorphism. One application fixes the granularity; a second has nothing further to discard.

Definition C2. Configured sameness relative to a comparison scheme. Two forms have the same configured denotation relative to $\Pi$ when

under the configured-form isomorphism of Definition I1. That isomorphism preserves every inspection result and its depth by Axioms 1-4 and Definition 1.

Definition C3. Rule-presentation sameness. Let $\mathfrak R_i\models G_i$. The two presented forms have the same Rule presentation when

under the Rule-presentation isomorphism of Definition I1. Definition C1 acts on configured forms and supplies no quotient of Rule presentations.

Remark C4. Configured sameness relative to $\Pi$ is distinct from Rule-presentation isomorphism. The first compares retained configured behavior. The second also preserves its declared factorization.

Structural gameness-support profile

Definition G1. Structural gameness-support profile. Define

so

Define the reachable witness population by

The witness-shape map is

Set

Corollary G2. Every game has positive structural gameness support:

If $S_G^{\mathrm{reach}}$ is finite, define

Then

Cross-form uses of $\rho$ use Definition C1.

For one comparison scheme $(\Pi,q)$, take

and, for $z,w\in\mathfrak C_{\Pi,\mathrm{fin}}$, define

Write $z\prec_{\Pi,\rho}w$ for strict inequality. This preorder compares candidate games by the density of structural gameness support retained after one declared normalization.

Finite continuation capacity

Extension axiom $F_P$. Finite local profile domains. Every reachable State admits finitely many profiles:

$$\forall s\in S_G^{\mathrm{reach}}: |D_s|<\infty.$$

Definition P1. Finite continuation capacity. Under $F_P$, define

$$\nu_G(s):=|\operatorname{im}\kappa_s|, \qquad a_G(s):=\max\{\nu_G(s)-1,0\},$$

and

$$\operatorname{Cap}(G) := \sup_{s\in S_G^{\mathrm{reach}}}a_G(s) \in \mathbb N_0\cup\{\infty\}.$$

Corollary P2 ($F_P$). Under $F_P$,

$$\operatorname{Cap}(G)>0 \Longleftrightarrow \operatorname{SPlay}(G).$$

Epistemic extension

Saturation

Definition S1. Saturation. Fix a comparison scheme $(\Pi,q)$ on $\mathcal D$. Define the carrier of retained Possibility Spaces by

$$\mathsf{RPS}_\Pi := \left\{ \mathcal P_H \;\middle|\; H\in\mathcal D, \ q_H:H\to\Pi(H) \text{ is a configured-form isomorphism under Definition I1} \right\}.$$

For $\mathcal P_H,\mathcal P_{H'}\in\mathsf{RPS}_\Pi$, write

$$\mathcal P_H\cong_\Pi^{\mathrm{reach}}\mathcal P_{H'}$$

when there are bijections satisfying the configured-form isomorphism clauses of Definition I1 after the raw configuration carriers are restricted to $X_H^{\mathrm{reach}}$ and $X_{H'}^{\mathrm{reach}}$, every reading is restricted to the corresponding reachable carrier, and the induced State labels are preserved. This is an isomorphism of reachable structures; it makes no claim about unreachable configurations.

Definition C1 gives

$$\mathcal P_{\Pi(G)}\in\mathsf{RPS}_\Pi$$

because $q_{\Pi(G)}$ is an isomorphism.

A factive hypothesis assignment $K$ gives each source $\alpha$ and retained Possibility Space $\mathcal P_{\Pi(G)}$ a non-empty subclass

$$\mathcal H_{\alpha}^{\Pi,K} \bigl(\mathcal P_{\Pi(G)}\bigr) \subseteq \mathsf{RPS}_\Pi$$

with

$$\mathcal P_{\Pi(G)} \in \mathcal H_{\alpha}^{\Pi,K} \bigl(\mathcal P_{\Pi(G)}\bigr).$$

$K$ is invariant under $\cong_\Pi^{\mathrm{reach}}$: an isomorphism between retained arguments transports one hypothesis class bijectively to the other.

Define

$$\operatorname{Sat}_{\Pi,K}(\alpha,G) \Longleftrightarrow \forall P\in \mathcal H_{\alpha}^{\Pi,K} \bigl(\mathcal P_{\Pi(G)}\bigr): P\cong_\Pi^{\mathrm{reach}}\mathcal P_{\Pi(G)}.$$

Saturation collapses the configured hypothesis class to one structural isomorphism type at the selected granularity. Rule presentation remains a separate hypothesis type. $\operatorname{Ag}(G)$ remains a function of $G$.

The resulting objects answer different questions:

Question	Structural object	It does not encode
Where can an admitted profile difference alter continuation?	$S_A(G)$ and $\mathcal W_G(s)$	quality, importance, or visitation frequency
How many role coordinates does the nearest witness change?	$m_G^*(s)$	number of people or interaction strength
How deep before one witness becomes distinguishable?	$\ell_G(s;p,q)$	elapsed time or consequence magnitude
How soon can every immediate outcome still reach one common State?	$j_G(s;p,q)$	actual return, control, or persistence
Has one source exhausted structural discovery at the selected granularity?	$\operatorname{Sat}_{\Pi,K}(\alpha,G)$	solving, current-State knowledge, or loss of Agency
How long does a distinction remain operative along paired later paths?	no canonical object in the present signature	Latency or Reconvergence by another name

Every symbol above is supposed to pay rent by keeping one of those arrows typed. Anything else goes.