8 Shared work and maintained obligations
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Joint information can support a task without creating a single decision maker.
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One agent may have a display record and another a calibration record. Neither need know the physical target bit alone. Together their records can determine it. This is an informational relation before it is any claim about collective agency.
8.1 What several accessible records can reconstruct
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Let \(V\) be a finite-dimensional vector space over a field. Participant \(i\) has a linear record \(A_i v\). Stack the records into \(Av=(A_1v,\ldots,A_mv)\), and let \(Qv\) be the protected linear target.
Proposition 8.1: Target-relative joint reconstruction
There is a linear decoder \(D\) with \(Q=DA\) if and only if \(\ker A\subseteq\ker Q\). Full-state reconstruction is possible exactly when \(A\) is injective. The intersection identity \(\ker A=\bigcap_i\ker A_i\) determines which ambiguities sharing removes.
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Proof
If \(Q=DA\), a vector annihilated by \(A\) is also annihilated by \(Q\). Conversely, define \(D(Av)=Qv\) on \({\operatorname{im}} A\). If \(Av=Aw\), then \(v-w\in\ker A\subseteq\ker Q\), so this definition is independent of the representative. It is linear and extends to the full record space by extending a basis. For \(Q=I_V\) the kernel condition is injectivity. A stacked record vanishes exactly when each component record vanishes, giving the intersection identity.
Source relationship. Target-relative specialization of the reconstruction and factorization machinery in Sufficient Resolution [SR] and Stratified Relational Workspace Theory [SRWT]. The proof here does not import their complete noise, dynamics, or identity theories.
Over \(\mathbb F_2\), Agent A observes \(A_A(b,s)=b+s\) and Agent B observes \(A_B(b,s)=s\). The joint matrix is invertible. Either agent alone lacks a direction needed to recover \(b\). The nuisance bit may remain unknown because the target map ignores it.
The theorem concerns accessible records. It does not send them. A communication protocol, truthful interpretation, permitted disclosure, and the recipient's executable decoder remain separate requirements. Chapter 4 states the corresponding fixed-recipient criterion.
8.2 Permissions are part of an operational state
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Now let a finite institutional state \(x\) have a protected label \(\ell(x)\), an enabled-action set \(D(x)\) within a fixed finite alphabet, and a deterministic successor \(F(x,a)\) for enabled actions. Labels may include resource balances, outstanding obligations, and offices, but need not retain every historical byte.
Theorem 8.2: A quotient that preserves legal operations
For a surjection \(\pi:X\to Z\), quotient labels, enabled sets, and successors satisfying
exist exactly when equal fibres have equal labels, equal enabled sets, and equal projected successors for every enabled action. The quotient data are unique and preserve every legal finite action and label sequence.
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Proof
Factoring each quantity proves necessity. For sufficiency choose a representative of each fibre to define its label, enabled set, and successor. The three equalities make those definitions independent of the representative. Surjectivity gives uniqueness. Equal enabled sets admit the same first action, and successor equality preserves membership in one fibre. Induction on the sequence length preserves legality and all labels.
Source relationship. [MS], “Exact governance quotient.” This extends the deterministic behavioral construction by explicitly protecting action availability. It does not derive moral or external legal authority from the state description.
A summary that preserves only the next sensor response can merge states in which a repair is authorized with states in which it is forbidden. It may remain a good predictor while becoming an invalid action interface. The necessary correction is to preserve the enabled-set distinction, not merely to attach a reassuring name to the old code.
8.3 Continuing should not erase the ability to finish
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Use a finite observed configuration space \(Z\). Under partial observation, the configurations would instead have to be justified public possibility states. At \(z\) there are finite catalogues of running actions and ending actions, each with a nonempty successor set. A declared safe set is \(S\), and an ending target is \(G\subseteq S\). A safe hold action is available at \(G\).
A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define
Let \(T_*\) be the stable set. Then define
and let \(L_*\) be its stable value. Both sequences stabilize because \(Z\) is finite.
Theorem 8.3: Safe continuation that retains an ending strategy
A starting configuration belongs to \(L_*\) exactly when one policy can run safely indefinitely without a stop command and, after a stop command at any decision epoch, reach \(G\) on every possible successor path using only the admitted ending actions. Before the command a policy preserving \(L_*\) suffices. After the command it can reach \(G\) within the first-entry rank of the current state in the \(T_k\) sequence.
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Proof
Induction shows that membership in \(T_k\) is equivalent to an ending-only policy reaching \(G\) within \(k\) transitions while staying safe. The chosen action moves every successor to a smaller rank. Outside \(T_*\), every ending action has a successor outside \(T_*\) or outside safety. Choosing such a successor at every step defeats any purported surely safe ending. Thus \(T_*\) is exactly the sure-ending region.
At \(L_*\) a running action keeps every successor inside \(L_*\). Repeatedly using such actions is safe and retains membership in \(T_*\). A command can therefore switch to the decreasing-rank ending policy at any epoch.
Conversely, if a policy has the stated guarantee, every pre-command reachable configuration must admit an ending immediately and hence lie in \(T_*\). For any finite number of further running steps, it must keep all paths in that set. Induction in the decreasing construction places the initial state in every \(L_k\), hence in \(L_*\). No action is chosen after secretly observing its successor.
Source relationship. Fully observed finite specialization of [MS], “Continued operation with an available authorized ending.” The more general public-possibility construction is not silently assumed in this version.
Counterexample 8.4: Safe inactivity without an ending
A state \(z\) has a safe idle self-loop and no ending action reaching \(G\). It is safe forever but outside \(T_*\). In a second state \(u\), an ending is available, while the only running action spends its reserved resource and moves to \(z\). Then \(u\) belongs to \(T_*\) but not \(L_*\). Being able to stop now does not establish that ordinary operation preserves that ability.
These results do not establish society formation, strategic compliance, or a just constitution. They establish information, action, and continuation properties of a specified finite institution. Their place at the end of the book is earned by the preceding account of accessible evidence, common artifacts, feasible controllers, and dependency-preserving revision.