Machine Societies

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1 \clearpage
2 \section{Theorem and proposition index}\label{app:statement-index}
3 This index links to all 62 theorem and proposition statements. The page numbers refer to this edition. Each result retains its stated hypotheses and prose proof. A final review of the central synthesis and accountability results is described in Appendix~\ref{app:release-review}.
4 \begingroup\small\setlength{\tabcolsep}{4pt}\renewcommand{\arraystretch}{1.15}
5 \begin{longtable}{@{}P{.19\linewidth}P{.69\linewidth}r@{}}
6 \toprule Statement & Result & Page\\\midrule\endfirsthead
7 \toprule Statement & Result & Page\\\midrule\endhead
8 \bottomrule\endfoot
9 \hyperref[prop:det]{Proposition~\ref*{prop:det}} & \hyperref[prop:det]{Deterministic quotient criterion} & \pageref{prop:det}\\
10 \hyperref[prop:lump]{Proposition~\ref*{prop:lump}} & \hyperref[prop:lump]{Controlled lumpability} & \pageref{prop:lump}\\
11 \hyperref[prop:time]{Proposition~\ref*{prop:time}} & \hyperref[prop:time]{Approximate closure over time} & \pageref{prop:time}\\
12 \hyperref[prop:lift]{Proposition~\ref*{prop:lift}} & \hyperref[prop:lift]{Implementation dependence} & \pageref{prop:lift}\\
13 \hyperref[prop:identifiability]{Proposition~\ref*{prop:identifiability}} & \hyperref[prop:identifiability]{Behavior alone does not identify implementation} & \pageref{prop:identifiability}\\
14 \hyperref[prop:memory]{Proposition~\ref*{prop:memory}} & \hyperref[prop:memory]{Exact finite-memory lower bound} & \pageref{prop:memory}\\
15 \hyperref[prop:erasure]{Proposition~\ref*{prop:erasure}} & \hyperref[prop:erasure]{Exact reconstruction after erasure} & \pageref{prop:erasure}\\
16 \hyperref[prop:repair]{Proposition~\ref*{prop:repair}} & \hyperref[prop:repair]{Repair budget} & \pageref{prop:repair}\\
17 \hyperref[prop:composition]{Proposition~\ref*{prop:composition}} & \hyperref[prop:composition]{Compositional one-step closure} & \pageref{prop:composition}\\
18 \hyperref[prop:flatten]{Proposition~\ref*{prop:flatten}} & \hyperref[prop:flatten]{Simulation by an unrestricted controller} & \pageref{prop:flatten}\\
19 \hyperref[prop:symmetry]{Proposition~\ref*{prop:symmetry}} & \hyperref[prop:symmetry]{Symmetry lock for identical deterministic agents} & \pageref{prop:symmetry}\\
20 \hyperref[prop:forkweight]{Proposition~\ref*{prop:forkweight}} & \hyperref[prop:forkweight]{Weight-conserving refinement of controlled votes} & \pageref{prop:forkweight}\\
21 \hyperref[prop:handover]{Proposition~\ref*{prop:handover}} & \hyperref[prop:handover]{Intersecting witnesses exclude conflicting certificates} & \pageref{prop:handover}\\
22 \hyperref[thm:govquot]{Theorem~\ref*{thm:govquot}} & \hyperref[thm:govquot]{Exact governance quotient} & \pageref{thm:govquot}\\
23 \hyperref[prop:govrefine]{Proposition~\ref*{prop:govrefine}} & \hyperref[prop:govrefine]{Coarsest finite governance summary} & \pageref{prop:govrefine}\\
24 \hyperref[prop:languagetrace]{Proposition~\ref*{prop:languagetrace}} & \hyperref[prop:languagetrace]{Preservation of finite institutional conversations} & \pageref{prop:languagetrace}\\
25 \hyperref[prop:amort]{Proposition~\ref*{prop:amort}} & \hyperref[prop:amort]{Institutional amortization} & \pageref{prop:amort}\\
26 \hyperref[prop:veto]{Proposition~\ref*{prop:veto}} & \hyperref[prop:veto]{Aggregate profitability without voluntary adoption} & \pageref{prop:veto}\\
27 \hyperref[prop:potential]{Proposition~\ref*{prop:potential}} & \hyperref[prop:potential]{Finite termination of strict institutional improvement} & \pageref{prop:potential}\\
28 \hyperref[prop:barrier]{Proposition~\ref*{prop:barrier}} & \hyperref[prop:barrier]{Institutional formation barrier} & \pageref{prop:barrier}\\
29 \hyperref[prop:seed]{Proposition~\ref*{prop:seed}} & \hyperref[prop:seed]{Minimum profitable coordinated seed} & \pageref{prop:seed}\\
30 \hyperref[prop:deterrence-tv]{Proposition~\ref*{prop:deterrence-tv}} & \hyperref[prop:deterrence-tv]{The exact bounded-sanction discrimination limit} & \pageref{prop:deterrence-tv}\\
31 \hyperref[prop:enforcement-garbling]{Proposition~\ref*{prop:enforcement-garbling}} & \hyperref[prop:enforcement-garbling]{Garbling does not increase sanctioning capacity} & \pageref{prop:enforcement-garbling}\\
32 \hyperref[thm:joint-enforcement]{Theorem~\ref*{thm:joint-enforcement}} & \hyperref[thm:joint-enforcement]{Simultaneous enforcement and mixture obstruction} & \pageref{thm:joint-enforcement}\\
33 \hyperref[prop:joint-robust]{Proposition~\ref*{prop:joint-robust}} & \hyperref[prop:joint-robust]{A robust simultaneous enforcement certificate} & \pageref{prop:joint-robust}\\
34 \hyperref[prop:safety]{Proposition~\ref*{prop:safety}} & \hyperref[prop:safety]{Constitutional inductive safety} & \pageref{prop:safety}\\
35 \hyperref[prop:viability]{Proposition~\ref*{prop:viability}} & \hyperref[prop:viability]{Finite-horizon almost-sure institutional viability} & \pageref{prop:viability}\\
36 \hyperref[thm:confluence]{Theorem~\ref*{thm:confluence}} & \hyperref[thm:confluence]{Order-independent amendment normal forms} & \pageref{thm:confluence}\\
37 \hyperref[prop:merge-convergence]{Proposition~\ref*{prop:merge-convergence}} & \hyperref[prop:merge-convergence]{Quiescent convergence and invariant preservation} & \pageref{prop:merge-convergence}\\
38 \hyperref[prop:exclusive-service]{Proposition~\ref*{prop:exclusive-service}} & \hyperref[prop:exclusive-service]{Exclusive service cannot be unconditionally local} & \pageref{prop:exclusive-service}\\
39 \hyperref[thm:undecidable]{Theorem~\ref*{thm:undecidable}} & \hyperref[thm:undecidable]{No universal constitutional safety decider} & \pageref{thm:undecidable}\\
40 \hyperref[thm:decsafe]{Theorem~\ref*{thm:decsafe}} & \hyperref[thm:decsafe]{Indistinguishability obstruction to joint safety} & \pageref{thm:decsafe}\\
41 \hyperref[prop:obligationtransport]{Proposition~\ref*{prop:obligationtransport}} & \hyperref[prop:obligationtransport]{Obligation-preserving succession and translation} & \pageref{prop:obligationtransport}\\
42 \hyperref[prop:transport]{Proposition~\ref*{prop:transport}} & \hyperref[prop:transport]{Accumulation of cultural transport defects} & \pageref{prop:transport}\\
43 \hyperref[prop:culturegrowth]{Proposition~\ref*{prop:culturegrowth}} & \hyperref[prop:culturegrowth]{Conditional cumulative culture} & \pageref{prop:culturegrowth}\\
44 \hyperref[prop:agendaclosure]{Proposition~\ref*{prop:agendaclosure}} & \hyperref[prop:agendaclosure]{Least closure of a finite problem--method system} & \pageref{prop:agendaclosure}\\
45 \hyperref[prop:lifetime]{Proposition~\ref*{prop:lifetime}} & \hyperref[prop:lifetime]{Survival and expected institutional lifetime} & \pageref{prop:lifetime}\\
46 \hyperref[thm:block-averaging]{Theorem~\ref*{thm:block-averaging}} & \hyperref[thm:block-averaging]{Finite-block institutional averaging} & \pageref{thm:block-averaging}\\
47 \hyperref[prop:robust-deterrence]{Proposition~\ref*{prop:robust-deterrence}} & \hyperref[prop:robust-deterrence]{A robust margin for reduced-model enforcement} & \pageref{prop:robust-deterrence}\\
48 \hyperref[prop:cycle]{Proposition~\ref*{prop:cycle}} & \hyperref[prop:cycle]{Cycle criterion for binary institutional compatibility} & \pageref{prop:cycle}\\
49 \hyperref[thm:holonomy]{Theorem~\ref*{thm:holonomy}} & \hyperref[thm:holonomy]{Exact frame synchronization} & \pageref{thm:holonomy}\\
50 \hyperref[prop:holonomysection]{Proposition~\ref*{prop:holonomysection}} & \hyperref[prop:holonomysection]{Compatible states and holonomy fixed points} & \pageref{prop:holonomysection}\\
51 \hyperref[prop:holonomybound]{Proposition~\ref*{prop:holonomybound}} & \hyperref[prop:holonomybound]{Loop defect bounds unavoidable edge error} & \pageref{prop:holonomybound}\\
52 \hyperref[thm:residual]{Theorem~\ref*{thm:residual}} & \hyperref[thm:residual]{Composition of typed institutional residuals} & \pageref{thm:residual}\\
53 \hyperref[prop:cyclerepair]{Proposition~\ref*{prop:cyclerepair}} & \hyperref[prop:cyclerepair]{Cycle residual and minimum authorized repair} & \pageref{prop:cyclerepair}\\
54 \hyperref[prop:cascade-lock]{Proposition~\ref*{prop:cascade-lock}} & \hyperref[prop:cascade-lock]{Almost-sure locking with a positive error probability} & \pageref{prop:cascade-lock}\\
55 \hyperref[prop:signal-majority]{Proposition~\ref*{prop:signal-majority}} & \hyperref[prop:signal-majority]{Conditional independence permits continuing accuracy gains} & \pageref{prop:signal-majority}\\
56 \hyperref[prop:commonreach]{Proposition~\ref*{prop:commonreach}} & \hyperref[prop:commonreach]{Common knowledge as a reachability condition} & \pageref{prop:commonreach}\\
57 \hyperref[prop:epistemictransport]{Proposition~\ref*{prop:epistemictransport}} & \hyperref[prop:epistemictransport]{Epistemic preservation and reflection} & \pageref{prop:epistemictransport}\\
58 \hyperref[prop:blocker]{Proposition~\ref*{prop:blocker}} & \hyperref[prop:blocker]{Blocking coalitions as hitting sets} & \pageref{prop:blocker}\\
59 \hyperref[prop:founding]{Proposition~\ref*{prop:founding}} & \hyperref[prop:founding]{When a founding fact is determined by internal records} & \pageref{prop:founding}\\
60 \hyperref[thm:robust-staffing]{Theorem~\ref*{thm:robust-staffing}} & \hyperref[thm:robust-staffing]{Authorized staffing after any bounded agent erasure} & \pageref{thm:robust-staffing}\\
61 \hyperref[thm:joint-viability]{Theorem~\ref*{thm:joint-viability}} & \hyperref[thm:joint-viability]{Joint continuation under a common observation history} & \pageref{thm:joint-viability}\\
62 \hyperref[prop:joint-progress]{Proposition~\ref*{prop:joint-progress}} & \hyperref[prop:joint-progress]{Safe service with a finite progress rank} & \pageref{prop:joint-progress}\\
63 \hyperref[prop:continuation-certificate]{Proposition~\ref*{prop:continuation-certificate}} & \hyperref[prop:continuation-certificate]{Soundness of a finite continuation certificate} & \pageref{prop:continuation-certificate}\\
64 \hyperref[thm:synthesis-transport]{Theorem~\ref*{thm:synthesis-transport}} & \hyperref[thm:synthesis-transport]{Governance abstraction preserves joint realizability} & \pageref{thm:synthesis-transport}\\
65 \hyperref[prop:strategic-inspector]{Proposition~\ref*{prop:strategic-inspector}} & \hyperref[prop:strategic-inspector]{Inspection without commitment} & \pageref{prop:strategic-inspector}\\
66 \hyperref[thm:privacy-envelope]{Theorem~\ref*{thm:privacy-envelope}} & \hyperref[thm:privacy-envelope]{Sharp pairwise privacy envelope for a bounded sanction} & \pageref{thm:privacy-envelope}\\
67 \hyperref[prop:privacy-transcript]{Proposition~\ref*{prop:privacy-transcript}} & \hyperref[prop:privacy-transcript]{Conditional disclosure accounting} & \pageref{prop:privacy-transcript}\\
68 \hyperref[prop:appeal-capacity]{Proposition~\ref*{prop:appeal-capacity}} & \hyperref[prop:appeal-capacity]{Capacity for a finite appeal contract} & \pageref{prop:appeal-capacity}\\
69 \hyperref[thm:stoppable-operation]{Theorem~\ref*{thm:stoppable-operation}} & \hyperref[thm:stoppable-operation]{Continued operation with an available authorized ending} & \pageref{thm:stoppable-operation}\\
70 \hyperref[thm:guarded-certificates]{Theorem~\ref*{thm:guarded-certificates}} & \hyperref[thm:guarded-certificates]{Guarded certificate use and authorized handover} & \pageref{thm:guarded-certificates}\\
71 \end{longtable}
72 \endgroup

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