Notation by work
Machine Societies
All notation · Glossary · Source coverage
712 symbol spellings in 1619 mathematical passages across 4 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 712 entries are available below.
*Notation table · 51 passages
Search aliases: *
Notation table · Machine_Societies_v1.0.0.tex, line 112
W_*, R_* · Safe continuation kernel and certified-service attractor for one contract. The first need not imply the second.
Exact mathematical passage
W_*
=Notation table · 287 passages
Search aliases: =
Notation table · Machine_Societies_v1.0.0.tex, line 114
λ=e^ε · Pairwise disclosure factor in Section~[source reference]. Elsewhere λ may denote a mixture or intervention.
Exact mathematical passage
\lambda=e^\varepsilon
ANotation table · 60 passages
Search aliases: A
Notation table · Machine_Societies_v1.0.0.tex, line 109
I, A(s) · Institutional state and admissible protocol catalogue at state s. Admissibility requires its stipulated implementation.
Exact mathematical passage
A(s)
BNotation table · 119 passages
Search aliases: B
Notation table · Machine_Societies_v1.0.0.tex, line 110
B, B · A nonempty set of states consistent with a common history, and the space of such sets. These are not posterior distributions.
Exact mathematical passage
B
\mathcal BNotation table · 14 passages
Search aliases: B mathcal
Notation table · Machine_Societies_v1.0.0.tex, line 110
B, B · A nonempty set of states consistent with a common history, and the space of such sets. These are not posterior distributions.
Exact mathematical passage
\mathcal B
eNotation table · 59 passages
Search aliases: e
Notation table · Machine_Societies_v1.0.0.tex, line 114
λ=e^ε · Pairwise disclosure factor in Section~[source reference]. Elsewhere λ may denote a mixture or intervention.
Exact mathematical passage
\lambda=e^\varepsilon
e^\varepsilonNotation table · 2 passages
Search aliases: e^ε varepsilon
Notation table · Machine_Societies_v1.0.0.tex, line 114
λ=e^ε · Pairwise disclosure factor in Section~[source reference]. Elsewhere λ may denote a mixture or intervention.
Exact mathematical passage
\lambda=e^\varepsilon
FNotation table · 68 passages
Search aliases: F
Notation table · Machine_Societies_v1.0.0.tex, line 113
F, g, α · Authorized collectible penalty, deviation gain in the same utility units, and the protected false-positive limit.
Exact mathematical passage
F
gNotation table · 73 passages
Search aliases: g
Notation table · Machine_Societies_v1.0.0.tex, line 113
F, g, α · Authorized collectible penalty, deviation gain in the same utility units, and the protected false-positive limit.
Exact mathematical passage
g
INotation table · 33 passages
Search aliases: I
Notation table · Machine_Societies_v1.0.0.tex, line 109
I, A(s) · Institutional state and admissible protocol catalogue at state s. Admissibility requires its stipulated implementation.
Exact mathematical passage
I
KNotation table · 22 passages
Search aliases: K
Notation table · Machine_Societies_v1.0.0.tex, line 106
K_u, Q_u · Row-stochastic transition and report kernels. The model states when their joint law requires separate treatment.
Exact mathematical passage
K_u
K_uNotation table · 6 passages
Search aliases: K_u
Notation table · Machine_Societies_v1.0.0.tex, line 106
K_u, Q_u · Row-stochastic transition and report kernels. The model states when their joint law requires separate treatment.
Exact mathematical passage
K_u
LNotation table · 32 passages
Search aliases: L
Notation table · Machine_Societies_v1.0.0.tex, line 115
T_*, L_* · Ending attractor and the running kernel that preserves an available authorized ending.
Exact mathematical passage
L_*
L_*Notation table · 12 passages
Search aliases: L_*
Notation table · Machine_Societies_v1.0.0.tex, line 115
T_*, L_* · Ending attractor and the running kernel that preserves an available authorized ending.
Exact mathematical passage
L_*
PNotation table · 75 passages
Search aliases: P
Notation table · Machine_Societies_v1.0.0.tex, line 108
d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].
Exact mathematical passage
\TV(P,Q)
QNotation table · 16 passages
Search aliases: Q
Notation table · Machine_Societies_v1.0.0.tex, line 106
K_u, Q_u · Row-stochastic transition and report kernels. The model states when their joint law requires separate treatment.
Exact mathematical passage
Q_u
Q_uNotation table · 3 passages
Search aliases: Q_u
Notation table · Machine_Societies_v1.0.0.tex, line 106
K_u, Q_u · Row-stochastic transition and report kernels. The model states when their joint law requires separate treatment.
Exact mathematical passage
Q_u
RNotation table · 47 passages
Search aliases: R
Notation table · Machine_Societies_v1.0.0.tex, line 112
W_*, R_* · Safe continuation kernel and certified-service attractor for one contract. The first need not imply the second.
Exact mathematical passage
R_*
R_*Notation table · 5 passages
Search aliases: R_*
Notation table · Machine_Societies_v1.0.0.tex, line 112
W_*, R_* · Safe continuation kernel and certified-service attractor for one contract. The first need not imply the second.
Exact mathematical passage
R_*
SNotation table · 60 passages
Search aliases: S
Notation table · Machine_Societies_v1.0.0.tex, line 104
X_t, S, X · Complete modeled state or its state space. The space is finite unless an explicit extension says otherwise.
Exact mathematical passage
S
sNotation table · 119 passages
Search aliases: s
Notation table · Machine_Societies_v1.0.0.tex, line 109
I, A(s) · Institutional state and admissible protocol catalogue at state s. Admissibility requires its stipulated implementation.
Exact mathematical passage
A(s)
\operatorname{Succ}Notation table · 8 passages
Search aliases: Succ operatorname
Notation table · Machine_Societies_v1.0.0.tex, line 111
Δ, Succ · Complete-state successors and the induced successors of public possibility sets.
Exact mathematical passage
\operatorname{Succ}
\operatorname{supp}Notation table · 2 passages
Search aliases: supp operatorname
Notation table · Machine_Societies_v1.0.0.tex, line 107
suppK · The full supported successor set. A small positive probability is not discarded in a sure guarantee.
Exact mathematical passage
\operatorname{supp}K
TNotation table · 68 passages
Search aliases: T
Notation table · Machine_Societies_v1.0.0.tex, line 105
π, T · State abstraction or transport. Preservation of outputs does not automatically preserve actions, knowledge, or rights.
Exact mathematical passage
T
tNotation table · 113 passages
Search aliases: t
Notation table · Machine_Societies_v1.0.0.tex, line 104
X_t, S, X · Complete modeled state or its state space. The space is finite unless an explicit extension says otherwise.
Exact mathematical passage
X_t
T_*Notation table · 7 passages
Search aliases: T_*
Notation table · Machine_Societies_v1.0.0.tex, line 115
T_*, L_* · Ending attractor and the running kernel that preserves an available authorized ending.
Exact mathematical passage
T_*
\TVNotation table · 19 passages
Search aliases: TV d_TV mathrm
Notation table · Machine_Societies_v1.0.0.tex, line 108
d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].
Exact mathematical passage
\TV(P,Q)
uNotation table · 46 passages
Search aliases: u
Notation table · Machine_Societies_v1.0.0.tex, line 106
K_u, Q_u · Row-stochastic transition and report kernels. The model states when their joint law requires separate treatment.
Exact mathematical passage
K_u
WNotation table · 35 passages
Search aliases: W
Notation table · Machine_Societies_v1.0.0.tex, line 112
W_*, R_* · Safe continuation kernel and certified-service attractor for one contract. The first need not imply the second.
Exact mathematical passage
W_*
W_*Notation table · 14 passages
Search aliases: W_*
Notation table · Machine_Societies_v1.0.0.tex, line 112
W_*, R_* · Safe continuation kernel and certified-service attractor for one contract. The first need not imply the second.
Exact mathematical passage
W_*
XNotation table · 12 passages
Search aliases: X
Notation table · Machine_Societies_v1.0.0.tex, line 104
X_t, S, X · Complete modeled state or its state space. The space is finite unless an explicit extension says otherwise.
Exact mathematical passage
X_t
\mathcal XNotation table · 1 passage
Search aliases: X mathcal
Notation table · Machine_Societies_v1.0.0.tex, line 104
X_t, S, X · Complete modeled state or its state space. The space is finite unless an explicit extension says otherwise.
Exact mathematical passage
\mathcal X
xNotation table · 55 passages
Search aliases: x
Notation table · Machine_Societies_v1.0.0.tex, line 108
d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].
Exact mathematical passage
\tfrac12\sum_x|P(x)-Q(x)|
X_tNotation table · 5 passages
Search aliases: X_t
Notation table · Machine_Societies_v1.0.0.tex, line 104
X_t, S, X · Complete modeled state or its state space. The space is finite unless an explicit extension says otherwise.
Exact mathematical passage
X_t
|Notation table · 20 passages
Search aliases: |
Notation table · Machine_Societies_v1.0.0.tex, line 108
d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].
Exact mathematical passage
\tfrac12\sum_x|P(x)-Q(x)|
\alphaNotation table · 42 passages
Search aliases: α alpha
Notation table · Machine_Societies_v1.0.0.tex, line 113
F, g, α · Authorized collectible penalty, deviation gain in the same utility units, and the protected false-positive limit.
Exact mathematical passage
\alpha
\DeltaNotation table · 15 passages
Search aliases: Δ Delta
Notation table · Machine_Societies_v1.0.0.tex, line 111
Δ, Succ · Complete-state successors and the induced successors of public possibility sets.
Exact mathematical passage
\Delta
\varepsilonNotation table · 46 passages
Search aliases: ε varepsilon
Notation table · Machine_Societies_v1.0.0.tex, line 114
λ=e^ε · Pairwise disclosure factor in Section~[source reference]. Elsewhere λ may denote a mixture or intervention.
Exact mathematical passage
\lambda=e^\varepsilon
\lambdaNotation table · 53 passages
Search aliases: λ lambda
Notation table · Machine_Societies_v1.0.0.tex, line 114
λ=e^ε · Pairwise disclosure factor in Section~[source reference]. Elsewhere λ may denote a mixture or intervention.
Exact mathematical passage
\lambda=e^\varepsilon
\piNotation table · 40 passages
Search aliases: π pi
Notation table · Machine_Societies_v1.0.0.tex, line 105
π, T · State abstraction or transport. Preservation of outputs does not automatically preserve actions, knowledge, or rights.
Exact mathematical passage
\pi
\sumNotation table · 35 passages
Search aliases: ∑ sum
Notation table · Machine_Societies_v1.0.0.tex, line 108
d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].
Exact mathematical passage
\tfrac12\sum_x|P(x)-Q(x)|
\sum_xNotation table · 1 passage
Search aliases: ∑_x sum
Notation table · Machine_Societies_v1.0.0.tex, line 108
d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].
Exact mathematical passage
\tfrac12\sum_x|P(x)-Q(x)|
\mathcal ADefinition passage · 7 passages
Search aliases: A mathcal
Definition passage: Institutional realization · Machine_Societies_v1.0.0.tex, line 647
[Institutional realization] An institutional realization comprises an environment state x, an institutional state I, a set of local actions A_i(x,I) for each decision center, a transition kernel K_I on the complete state, and a family of admissible interventions on I and its implementation. A written rule has operative force only relative to changes in the feasible action sets, authorized transitions, incentives, or observable outcomes that it actually induces.
Exact mathematical passage
\mathcal A_i(x,I)
\mathcal A_iDefinition passage · 1 passage
Search aliases: A_i mathcal
Definition passage: Institutional realization · Machine_Societies_v1.0.0.tex, line 647
[Institutional realization] An institutional realization comprises an environment state x, an institutional state I, a set of local actions A_i(x,I) for each decision center, a transition kernel K_I on the complete state, and a family of admissible interventions on I and its implementation. A written rule has operative force only relative to changes in the feasible action sets, authorized transitions, incentives, or observable outcomes that it actually induces.
Exact mathematical passage
\mathcal A_i(x,I)
DDefinition passage · 75 passages
Search aliases: D
Definition passage: Endogenous institutional formation · Machine_Societies_v1.0.0.tex, line 615
[Endogenous institutional formation] Let D record the rules, affordances, and permissions fixed by the designer and let I_t be the institutional state. Formation is endogenous relative to that family when interactions among participating decision centers make a causal difference to the content, adoption, revision, or enforcement of I_t beyond what is fixed by D. Its strength may be measured by an explicitly defined counterfactual effect on those institutional observables, not by an assumed unive…
Exact mathematical passage
D
F^A_mDefinition passage · 1 passage
Search aliases: F^A_m
Definition passage: Operatively faithful language transport · Machine_Societies_v1.0.0.tex, line 795
[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities apply to admitted messages. Any required state observations, such as outstanding obligations, must also agree after translation.
Exact mathematical passage
F^A_m(s)\text{ is defined}&\quad\Longleftrightarrow\quad F^B_{\tau(s,m)}(T(s))\text{ is defined},\\ T(F^A_m(s))&=F^B_{\tau(s,m)}(T(s)),\\ \ell_A(s,m)&=\ell_B(T(s),\tau(s,m)),
F^B_{\tau(s,m)}Definition passage · 1 passage
Search aliases: F^B_τ(s,m) tau
Definition passage: Operatively faithful language transport · Machine_Societies_v1.0.0.tex, line 795
[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities apply to admitted messages. Any required state observations, such as outstanding obligations, must also agree after translation.
Exact mathematical passage
F^A_m(s)\text{ is defined}&\quad\Longleftrightarrow\quad F^B_{\tau(s,m)}(T(s))\text{ is defined},\\ T(F^A_m(s))&=F^B_{\tau(s,m)}(T(s)),\\ \ell_A(s,m)&=\ell_B(T(s),\tau(s,m)),
iDefinition passage · 119 passages
Search aliases: i
Definition passage: Declared decomposition and access model · Machine_Societies_v1.0.0.tex, line 240
[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.
Exact mathematical passage
O_i:\X\to\mathcal O_i
I_tDefinition passage · 7 passages
Search aliases: I_t
Definition passage: Endogenous institutional formation · Machine_Societies_v1.0.0.tex, line 615
[Endogenous institutional formation] Let D record the rules, affordances, and permissions fixed by the designer and let I_t be the institutional state. Formation is endogenous relative to that family when interactions among participating decision centers make a causal difference to the content, adoption, revision, or enforcement of I_t beyond what is fixed by D. Its strength may be measured by an explicitly defined counterfactual effect on those institutional observables, not by an assumed unive…
Exact mathematical passage
I_t
K_IDefinition passage · 2 passages
Search aliases: K_I
Definition passage: Institutional realization · Machine_Societies_v1.0.0.tex, line 647
[Institutional realization] An institutional realization comprises an environment state x, an institutional state I, a set of local actions A_i(x,I) for each decision center, a transition kernel K_I on the complete state, and a family of admissible interventions on I and its implementation. A written rule has operative force only relative to changes in the feasible action sets, authorized transitions, incentives, or observable outcomes that it actually induces.
Exact mathematical passage
K_I
\LongleftrightarrowDefinition passage · 7 passages
Search aliases: Longleftrightarrow
Definition passage: Operatively faithful language transport · Machine_Societies_v1.0.0.tex, line 795
[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities apply to admitted messages. Any required state observations, such as outstanding obligations, must also agree after translation.
Exact mathematical passage
F^A_m(s)\text{ is defined}&\quad\Longleftrightarrow\quad F^B_{\tau(s,m)}(T(s))\text{ is defined},\\ T(F^A_m(s))&=F^B_{\tau(s,m)}(T(s)),\\ \ell_A(s,m)&=\ell_B(T(s),\tau(s,m)),
mDefinition passage · 49 passages
Search aliases: m
Definition passage: Operatively faithful language transport · Machine_Societies_v1.0.0.tex, line 794
[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined,
Exact mathematical passage
m
ODefinition passage · 22 passages
Search aliases: O
Definition passage: Declared decomposition and access model · Machine_Societies_v1.0.0.tex, line 240
[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.
Exact mathematical passage
O_i:\X\to\mathcal O_i
\mathcal ODefinition passage · 3 passages
Search aliases: O mathcal
Definition passage: Declared decomposition and access model · Machine_Societies_v1.0.0.tex, line 240
[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.
Exact mathematical passage
O_i:\X\to\mathcal O_i
O_iDefinition passage · 1 passage
Search aliases: O_i
Definition passage: Declared decomposition and access model · Machine_Societies_v1.0.0.tex, line 240
[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.
Exact mathematical passage
O_i:\X\to\mathcal O_i
\mathcal O_iDefinition passage · 1 passage
Search aliases: O_i mathcal
Definition passage: Declared decomposition and access model · Machine_Societies_v1.0.0.tex, line 240
[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.
Exact mathematical passage
O_i:\X\to\mathcal O_i
O_SDefinition passage · 2 passages
Search aliases: O_S
Definition passage: Declared decomposition and access model · Machine_Societies_v1.0.0.tex, line 240
[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.
Exact mathematical passage
O_S(X)
\pushDefinition passage · 4 passages
Search aliases: push
Definition passage: Candidate macrostate · Machine_Societies_v1.0.0.tex, line 247
[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.
Exact mathematical passage
(\push\mu)(z)=\sum_{x:\pi(x)=z}\mu(x).
\XDefinition passage · 8 passages
Search aliases: X mathcal
Definition passage: Declared decomposition and access model · Machine_Societies_v1.0.0.tex, line 240
[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.
Exact mathematical passage
O_i:\X\to\mathcal O_i
\ZDefinition passage · 6 passages
Search aliases: Z mathcal
Definition passage: Candidate macrostate · Machine_Societies_v1.0.0.tex, line 246
[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.
Exact mathematical passage
\pi:\X\to\Z
zDefinition passage · 64 passages
Search aliases: z
Definition passage: Candidate macrostate · Machine_Societies_v1.0.0.tex, line 247
[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.
Exact mathematical passage
(\push\mu)(z)=\sum_{x:\pi(x)=z}\mu(x).
\muDefinition passage · 17 passages
Search aliases: μ mu
Definition passage: Candidate macrostate · Machine_Societies_v1.0.0.tex, line 246
[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.
Exact mathematical passage
\mu
\pi^{-1}Definition passage · 2 passages
Search aliases: π^-1 pi
Definition passage: Candidate macrostate · Machine_Societies_v1.0.0.tex, line 250
[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.
Exact mathematical passage
\pi^{-1}(z)
\tauDefinition passage · 12 passages
Search aliases: τ tau
Definition passage: Operatively faithful language transport · Machine_Societies_v1.0.0.tex, line 794
[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined,
Exact mathematical passage
(T,\tau)
\ellDefinition passage · 24 passages
Search aliases: ℓ ell
Definition passage: Operatively faithful language transport · Machine_Societies_v1.0.0.tex, line 795
[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities apply to admitted messages. Any required state observations, such as outstanding obligations, must also agree after translation.
Exact mathematical passage
F^A_m(s)\text{ is defined}&\quad\Longleftrightarrow\quad F^B_{\tau(s,m)}(T(s))\text{ is defined},\\ T(F^A_m(s))&=F^B_{\tau(s,m)}(T(s)),\\ \ell_A(s,m)&=\ell_B(T(s),\tau(s,m)),
\ell_ADefinition passage · 1 passage
Search aliases: ℓ_A ell
Definition passage: Operatively faithful language transport · Machine_Societies_v1.0.0.tex, line 795
[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities apply to admitted messages. Any required state observations, such as outstanding obligations, must also agree after translation.
Exact mathematical passage
F^A_m(s)\text{ is defined}&\quad\Longleftrightarrow\quad F^B_{\tau(s,m)}(T(s))\text{ is defined},\\ T(F^A_m(s))&=F^B_{\tau(s,m)}(T(s)),\\ \ell_A(s,m)&=\ell_B(T(s),\tau(s,m)),
\ell_BDefinition passage · 1 passage
Search aliases: ℓ_B ell
Definition passage: Operatively faithful language transport · Machine_Societies_v1.0.0.tex, line 795
[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities apply to admitted messages. Any required state observations, such as outstanding obligations, must also agree after translation.
Exact mathematical passage
F^A_m(s)\text{ is defined}&\quad\Longleftrightarrow\quad F^B_{\tau(s,m)}(T(s))\text{ is defined},\\ T(F^A_m(s))&=F^B_{\tau(s,m)}(T(s)),\\ \ell_A(s,m)&=\ell_B(T(s),\tau(s,m)),
\toDefinition passage · 40 passages
Search aliases: → to
Definition passage: Declared decomposition and access model · Machine_Societies_v1.0.0.tex, line 240
[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.
Exact mathematical passage
O_i:\X\to\mathcal O_i
\sum_{x:\pi(x)=z}Definition passage · 1 passage
Search aliases: ∑_x:π(x)=z sum pi
Definition passage: Candidate macrostate · Machine_Societies_v1.0.0.tex, line 247
[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.
Exact mathematical passage
(\push\mu)(z)=\sum_{x:\pi(x)=z}\mu(x).
+Introduction passage · 77 passages
Search aliases: +
Introduction passage · Machine_Societies_v1.0.0.tex, line 374
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
Y^+
aIntroduction passage · 81 passages
Search aliases: a
Introduction passage · Machine_Societies_v1.0.0.tex, line 2171
A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also when every available profile has a possible successor from which the remaining contract cannot be sustained.
Exact mathematical passage
W_0&=\{B\in\mathcal B:B\subseteq S_{\rm safe}\},\label{eq:belief-w0}\\ W_{k+1}&=\{B\in W_0:\exists u\in A(B),\ \operatorname{Succ}(B,u)\subseteq W_k\}.\label{eq:belief-wk}
CIntroduction passage · 106 passages
Search aliases: C
Introduction passage · Machine_Societies_v1.0.0.tex, line 947
[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C). For every 0≤ s≤1, ∑_o s(o)d(o)≤∑_d(o)>0d(o). Equality is attained by s(o)=1 on the positive terms and zero elsewhere. Substitution into~[source refe…
Exact mathematical passage
g\leq F\,\TV(P_D,P_C).
\mathcal CIntroduction passage · 17 passages
Search aliases: C mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 1426
Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t), Q_t+1=G( Q_t, C_t,I_t,u_t), where u_t records permitted external inputs and I_t governs admission, allocation, and revision. No common utility functi…
Exact mathematical passage
\mathcal C_t
cIntroduction passage · 84 passages
Search aliases: c
Introduction passage · Machine_Societies_v1.0.0.tex, line 685
Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assign…
Exact mathematical passage
c_t:V_t\to\mathcal D
\mathcal C_tIntroduction passage · 2 passages
Search aliases: C_t mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 1426
Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t), Q_t+1=G( Q_t, C_t,I_t,u_t), where u_t records permitted external inputs and I_t governs admission, allocation, and revision. No common utility functi…
Exact mathematical passage
\mathcal C_t
c_tIntroduction passage · 9 passages
Search aliases: c_t
Introduction passage · Machine_Societies_v1.0.0.tex, line 685
Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assign…
Exact mathematical passage
c_t:V_t\to\mathcal D
\mathcal C_{t+1}Introduction passage · 1 passage
Search aliases: C_t+1 mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 1427
Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t), Q_t+1=G( Q_t, C_t,I_t,u_t), where u_t records permitted external inputs and I_t governs admission, allocation, and revision. No common utility functi…
Exact mathematical passage
\mathcal C_{t+1}=F(\mathcal C_t,\mathcal Q_t,I_t,u_t),\qquad \mathcal Q_{t+1}=G(\mathcal Q_t,\mathcal C_t,I_t,u_t),
c_{t+1}Introduction passage · 1 passage
Search aliases: c_t+1
Introduction passage · Machine_Societies_v1.0.0.tex, line 2058
may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t( ·mid G_t,r_t,c_t,H_t), where H_t denotes the retained information used by the up…
Exact mathematical passage
(G_{t+1},r_{t+1},c_{t+1})\sim \Gamma_{I_t}(\,\cdot\mid G_t,r_t,c_t,H_t),
C_vIntroduction passage · 5 passages
Search aliases: C_v
Introduction passage · Machine_Societies_v1.0.0.tex, line 2547
v,z) with z∈ C_v, a selector may either use the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z) lies in the declared successor set D_v(z). The local certificate proves D_v(z)⊆ C_v. A handover from v to w must keep every intermediate state in C and end in C_w. Its command, evidence transfer, resou…
Exact mathematical passage
D_v(z)\subseteq C_v
C_wIntroduction passage · 1 passage
Search aliases: C_w
Introduction passage · Machine_Societies_v1.0.0.tex, line 2547
andover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z) lies in the declared successor set D_v(z). The local certificate proves D_v(z)⊆ C_v. A handover from v to w must keep every intermediate state in C and end in C_w. Its command, evidence transfer, resources, and delay belong to the transition model. Intermediate handover and checking states c…
Exact mathematical passage
C_w
\mathcal DIntroduction passage · 2 passages
Search aliases: D mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 685
Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assign…
Exact mathematical passage
\mathcal D
dIntroduction passage · 64 passages
Search aliases: d
Introduction passage · Machine_Societies_v1.0.0.tex, line 953
[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C). For every 0≤ s≤1, ∑_o s(o)d(o)≤∑_d(o)>0d(o). Equality is attained by s(o)=1 on the positive terms and zero elsewhere. Substitution into~[source refe…
Exact mathematical passage
d(o)=P_D(o)-P_C(o)
D_vIntroduction passage · 3 passages
Search aliases: D_v
Introduction passage · Machine_Societies_v1.0.0.tex, line 2547
requires indefinite preservation. At (v,z) with z∈ C_v, a selector may either use the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z) lies in the declared successor set D_v(z). The local certificate proves D_v(z)⊆ C_v. A handover from v to w must keep every intermediate state in C and end in C_w.…
Exact mathematical passage
D_v(z)
EIntroduction passage · 26 passages
Search aliases: E
Introduction passage · Machine_Societies_v1.0.0.tex, line 2340
Membership, decision rights, resource control, and exposure to consequences define different sets. Let M denote participating members, V positions with specified decision rights, H infrastructure controllers, and E affected parties outside the membership. These sets may overlap where the description allows it. A human resource owner can be in H without holding an internal vote. A person whose reserved service is delayed can be in E without ever communicating with the agents. Conversely, membersh…
Exact mathematical passage
E
\mathcal EIntroduction passage · 4 passages
Search aliases: E mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the alre…
Exact mathematical passage
\mathcal E
\mathsf{epoch}Introduction passage · 1 passage
Search aliases: epoch mathsf
Introduction passage · Machine_Societies_v1.0.0.tex, line 781
A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a moral verdict on the message. The interpreter, authentication assumptions, role registry, and relevant history must be included in s or fixed by the c…
Exact mathematical passage
m=(\mathsf{kind},\mathsf{issuer},\mathsf{epoch},\mathsf{payload},\mathsf{nonce}).
fIntroduction passage · 24 passages
Search aliases: f
Introduction passage · Machine_Societies_v1.0.0.tex, line 2171
A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also when every available profile has a possible successor from which the remaining contract cannot be sustained.
Exact mathematical passage
W_0&=\{B\in\mathcal B:B\subseteq S_{\rm safe}\},\label{eq:belief-w0}\\ W_{k+1}&=\{B\in W_0:\exists u\in A(B),\ \operatorname{Succ}(B,u)\subseteq W_k\}.\label{eq:belief-wk}
F_mIntroduction passage · 2 passages
Search aliases: F_m
Introduction passage · Machine_Societies_v1.0.0.tex, line 784
A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a moral verdict on the message. The interpreter, authentication assumptions, role registry, and relevant history must be included in s or fixed by the c…
Exact mathematical passage
F_m(s)
GIntroduction passage · 38 passages
Search aliases: G
Introduction passage · Machine_Societies_v1.0.0.tex, line 1427
Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t), Q_t+1=G( Q_t, C_t,I_t,u_t), where u_t records permitted external inputs and I_t governs admission, allocation, and revision. No common utility functi…
Exact mathematical passage
\mathcal C_{t+1}=F(\mathcal C_t,\mathcal Q_t,I_t,u_t),\qquad \mathcal Q_{t+1}=G(\mathcal Q_t,\mathcal C_t,I_t,u_t),
G_tIntroduction passage · 2 passages
Search aliases: G_t
Introduction passage · Machine_Societies_v1.0.0.tex, line 2057
A continuing office need not be one of a permanently supplied list. A population may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t(…
Exact mathematical passage
G_t
G_{t+1}Introduction passage · 1 passage
Search aliases: G_t+1
Introduction passage · Machine_Societies_v1.0.0.tex, line 2058
may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t( ·mid G_t,r_t,c_t,H_t), where H_t denotes the retained information used by the up…
Exact mathematical passage
(G_{t+1},r_{t+1},c_{t+1})\sim \Gamma_{I_t}(\,\cdot\mid G_t,r_t,c_t,H_t),
HIntroduction passage · 27 passages
Search aliases: H
Introduction passage · Machine_Societies_v1.0.0.tex, line 2058
may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t( ·mid G_t,r_t,c_t,H_t), where H_t denotes the retained information used by the up…
Exact mathematical passage
(G_{t+1},r_{t+1},c_{t+1})\sim \Gamma_{I_t}(\,\cdot\mid G_t,r_t,c_t,H_t),
hIntroduction passage · 52 passages
Search aliases: h
Introduction passage · Machine_Societies_v1.0.0.tex, line 2153
The state of possible institutional realizations. The relevant public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Writ…
Exact mathematical passage
\mathcal B=\{B\subseteq S:B\ne\varnothing,\ h\text{ is constant on }B\}.
h^{-1}Introduction passage · 2 passages
Search aliases: h^-1
Introduction passage · Machine_Societies_v1.0.0.tex, line 2161
of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Write Succ(B,u) for their collection. Induction on the public history shows that~[source re…
Exact mathematical passage
\label{eq:belief-update} \operatorname{Post}(B,u,y)= \left(\bigcup_{s\in B}\Delta(s,u)\right)\cap h^{-1}(y).
H_tIntroduction passage · 2 passages
Search aliases: H_t
Introduction passage · Machine_Societies_v1.0.0.tex, line 2058
may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t( ·mid G_t,r_t,c_t,H_t), where H_t denotes the retained information used by the up…
Exact mathematical passage
(G_{t+1},r_{t+1},c_{t+1})\sim \Gamma_{I_t}(\,\cdot\mid G_t,r_t,c_t,H_t),
\mathcal IIntroduction passage · 2 passages
Search aliases: I mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the alre…
Exact mathematical passage
\mathcal I
I_nIntroduction passage · 1 passage
Search aliases: I_n
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the alre…
Exact mathematical passage
I_n=i
\mathsf{issuer}Introduction passage · 1 passage
Search aliases: issuer mathsf
Introduction passage · Machine_Societies_v1.0.0.tex, line 781
A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a moral verdict on the message. The interpreter, authentication assumptions, role registry, and relevant history must be included in s or fixed by the c…
Exact mathematical passage
m=(\mathsf{kind},\mathsf{issuer},\mathsf{epoch},\mathsf{payload},\mathsf{nonce}).
jIntroduction passage · 94 passages
Search aliases: j
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
with a declared review schedule Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the already specified i…
Exact mathematical passage
R_i(\widehat X_n,j)
kIntroduction passage · 67 passages
Search aliases: k
Introduction passage · Machine_Societies_v1.0.0.tex, line 2171
A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also when every available profile has a possible successor from which the remaining contract cannot be sustained.
Exact mathematical passage
W_0&=\{B\in\mathcal B:B\subseteq S_{\rm safe}\},\label{eq:belief-w0}\\ W_{k+1}&=\{B\in W_0:\exists u\in A(B),\ \operatorname{Succ}(B,u)\subseteq W_k\}.\label{eq:belief-wk}
K_iIntroduction passage · 9 passages
Search aliases: K_i
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the alre…
Exact mathematical passage
K_i
\mathsf{kind}Introduction passage · 1 passage
Search aliases: kind mathsf
Introduction passage · Machine_Societies_v1.0.0.tex, line 781
A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a moral verdict on the message. The interpreter, authentication assumptions, role registry, and relevant history must be included in s or fixed by the c…
Exact mathematical passage
m=(\mathsf{kind},\mathsf{issuer},\mathsf{epoch},\mathsf{payload},\mathsf{nonce}).
MIntroduction passage · 20 passages
Search aliases: M
Introduction passage · Machine_Societies_v1.0.0.tex, line 685
Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assign…
Exact mathematical passage
M_t\subseteq\mathcal U
M_tIntroduction passage · 5 passages
Search aliases: M_t
Introduction passage · Machine_Societies_v1.0.0.tex, line 685
Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assign…
Exact mathematical passage
M_t\subseteq\mathcal U
\midIntroduction passage · 17 passages
Search aliases: mid
Introduction passage · Machine_Societies_v1.0.0.tex, line 1805
If a message M has the same conditional distribution under two equally possible hidden states θ=0,1, then Bayes' rule gives Pr(θ=1mid M)=Pr(θ=1) wherever the posterior is defined. Such a channel carries no information about the distinction, regardless of how fluent or authoritative its reports appear. In strategic environments, senders may have preferences that differ from those of receivers. The theory of strategic information transmission shows why informative communication can depend on prefe…
Exact mathematical passage
\Pr(\theta=1\mid M)=\Pr(\theta=1)
nIntroduction passage · 57 passages
Search aliases: n
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the alre…
Exact mathematical passage
n
\mathsf{nonce}Introduction passage · 1 passage
Search aliases: nonce mathsf
Introduction passage · Machine_Societies_v1.0.0.tex, line 781
A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a moral verdict on the message. The interpreter, authentication assumptions, role registry, and relevant history must be included in s or fixed by the c…
Exact mathematical passage
m=(\mathsf{kind},\mathsf{issuer},\mathsf{epoch},\mathsf{payload},\mathsf{nonce}).
oIntroduction passage · 23 passages
Search aliases: o
Introduction passage · Machine_Societies_v1.0.0.tex, line 953
[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C). For every 0≤ s≤1, ∑_o s(o)d(o)≤∑_d(o)>0d(o). Equality is attained by s(o)=1 on the positive terms and zero elsewhere. Substitution into~[source refe…
Exact mathematical passage
d(o)=P_D(o)-P_C(o)
\overline AIntroduction passage · 2 passages
Search aliases: overline A overline
Introduction passage · Machine_Societies_v1.0.0.tex, line 2245
Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longleftrightarrowπ s∈overline G, π[Δ(s,u)] · =overlineΔ(π s,u). · · The notation π[B] denotes the image of a set, not a probability pushforward. These are…
Exact mathematical passage
h(s)&=\overline h(\pi s), & A(s)&=\overline A(\pi s),\label{eq:synthesis-labels}\\ s\in S_{\rm safe}&\Longleftrightarrow\pi s\in\overline S_{\rm safe}, & s\in G&\Longleftrightarrow\pi s\in\overline G,\label{eq:synthesis-goals}\\ \pi[\Delta(s,u)]&=\overline\Delta(\pi s,u).&&\label{eq:synthesis-successors}
\overline GIntroduction passage · 2 passages
Search aliases: overline G overline
Introduction passage · Machine_Societies_v1.0.0.tex, line 2245
Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longleftrightarrowπ s∈overline G, π[Δ(s,u)] · =overlineΔ(π s,u). · · The notation π[B] denotes the image of a set, not a probability pushforward. These are…
Exact mathematical passage
h(s)&=\overline h(\pi s), & A(s)&=\overline A(\pi s),\label{eq:synthesis-labels}\\ s\in S_{\rm safe}&\Longleftrightarrow\pi s\in\overline S_{\rm safe}, & s\in G&\Longleftrightarrow\pi s\in\overline G,\label{eq:synthesis-goals}\\ \pi[\Delta(s,u)]&=\overline\Delta(\pi s,u).&&\label{eq:synthesis-successors}
\overline hIntroduction passage · 2 passages
Search aliases: overline h overline
Introduction passage · Machine_Societies_v1.0.0.tex, line 2245
Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longleftrightarrowπ s∈overline G, π[Δ(s,u)] · =overlineΔ(π s,u). · · The notation π[B] denotes the image of a set, not a probability pushforward. These are…
Exact mathematical passage
h(s)&=\overline h(\pi s), & A(s)&=\overline A(\pi s),\label{eq:synthesis-labels}\\ s\in S_{\rm safe}&\Longleftrightarrow\pi s\in\overline S_{\rm safe}, & s\in G&\Longleftrightarrow\pi s\in\overline G,\label{eq:synthesis-goals}\\ \pi[\Delta(s,u)]&=\overline\Delta(\pi s,u).&&\label{eq:synthesis-successors}
\overline SIntroduction passage · 3 passages
Search aliases: overline S overline
Introduction passage · Machine_Societies_v1.0.0.tex, line 2245
Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longleftrightarrowπ s∈overline G, π[Δ(s,u)] · =overlineΔ(π s,u). · · The notation π[B] denotes the image of a set, not a probability pushforward. These are…
Exact mathematical passage
h(s)&=\overline h(\pi s), & A(s)&=\overline A(\pi s),\label{eq:synthesis-labels}\\ s\in S_{\rm safe}&\Longleftrightarrow\pi s\in\overline S_{\rm safe}, & s\in G&\Longleftrightarrow\pi s\in\overline G,\label{eq:synthesis-goals}\\ \pi[\Delta(s,u)]&=\overline\Delta(\pi s,u).&&\label{eq:synthesis-successors}
\overline S_{\rm safe}Introduction passage · 2 passages
Search aliases: overline S_rm safe overline rm
Introduction passage · Machine_Societies_v1.0.0.tex, line 2245
Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longleftrightarrowπ s∈overline G, π[Δ(s,u)] · =overlineΔ(π s,u). · · The notation π[B] denotes the image of a set, not a probability pushforward. These are…
Exact mathematical passage
h(s)&=\overline h(\pi s), & A(s)&=\overline A(\pi s),\label{eq:synthesis-labels}\\ s\in S_{\rm safe}&\Longleftrightarrow\pi s\in\overline S_{\rm safe}, & s\in G&\Longleftrightarrow\pi s\in\overline G,\label{eq:synthesis-goals}\\ \pi[\Delta(s,u)]&=\overline\Delta(\pi s,u).&&\label{eq:synthesis-successors}
\overline\DeltaIntroduction passage · 2 passages
Search aliases: overlineΔ overline Delta
Introduction passage · Machine_Societies_v1.0.0.tex, line 2245
Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longleftrightarrowπ s∈overline G, π[Δ(s,u)] · =overlineΔ(π s,u). · · The notation π[B] denotes the image of a set, not a probability pushforward. These are…
Exact mathematical passage
h(s)&=\overline h(\pi s), & A(s)&=\overline A(\pi s),\label{eq:synthesis-labels}\\ s\in S_{\rm safe}&\Longleftrightarrow\pi s\in\overline S_{\rm safe}, & s\in G&\Longleftrightarrow\pi s\in\overline G,\label{eq:synthesis-goals}\\ \pi[\Delta(s,u)]&=\overline\Delta(\pi s,u).&&\label{eq:synthesis-successors}
P_CIntroduction passage · 38 passages
Search aliases: P_C
Introduction passage · Machine_Societies_v1.0.0.tex, line 947
[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C). For every 0≤ s≤1, ∑_o s(o)d(o)≤∑_d(o)>0d(o). Equality is attained by s(o)=1 on the positive terms and zero elsewhere. Substitution into~[source refe…
Exact mathematical passage
g\leq F\,\TV(P_D,P_C).
P_DIntroduction passage · 19 passages
Search aliases: P_D
Introduction passage · Machine_Societies_v1.0.0.tex, line 947
[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C). For every 0≤ s≤1, ∑_o s(o)d(o)≤∑_d(o)>0d(o). Equality is attained by s(o)=1 on the positive terms and zero elsewhere. Substitution into~[source refe…
Exact mathematical passage
g\leq F\,\TV(P_D,P_C).
P_\lambdaIntroduction passage · 9 passages
Search aliases: P_λ lambda
Introduction passage · Machine_Societies_v1.0.0.tex, line 374
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
P_\lambda
P_{\lambda'}Introduction passage · 1 passage
Search aliases: P_λ' lambda
Introduction passage · Machine_Societies_v1.0.0.tex, line 375
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
\eta(x,z')=\sup_{\lambda,\lambda'\in\Lambda(x,z')} \TV(P_\lambda,P_{\lambda'}).
\mathsf{payload}Introduction passage · 1 passage
Search aliases: payload mathsf
Introduction passage · Machine_Societies_v1.0.0.tex, line 781
A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a moral verdict on the message. The interpreter, authentication assumptions, role registry, and relevant history must be included in s or fixed by the c…
Exact mathematical passage
m=(\mathsf{kind},\mathsf{issuer},\mathsf{epoch},\mathsf{payload},\mathsf{nonce}).
\operatorname{Post}Introduction passage · 2 passages
Search aliases: Post operatorname
Introduction passage · Machine_Societies_v1.0.0.tex, line 2161
of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Write Succ(B,u) for their collection. Induction on the public history shows that~[source re…
Exact mathematical passage
\label{eq:belief-update} \operatorname{Post}(B,u,y)= \left(\bigcup_{s\in B}\Delta(s,u)\right)\cap h^{-1}(y).
\PrIntroduction passage · 1 passage
Search aliases: Pr
Introduction passage · Machine_Societies_v1.0.0.tex, line 1805
If a message M has the same conditional distribution under two equally possible hidden states θ=0,1, then Bayes' rule gives Pr(θ=1mid M)=Pr(θ=1) wherever the posterior is defined. Such a channel carries no information about the distinction, regardless of how fluent or authoritative its reports appear. In strategic environments, senders may have preferences that differ from those of receivers. The theory of strategic information transmission shows why informative communication can depend on prefe…
Exact mathematical passage
\Pr(\theta=1\mid M)=\Pr(\theta=1)
\mathcal QIntroduction passage · 11 passages
Search aliases: Q mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 1426
Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t), Q_t+1=G( Q_t, C_t,I_t,u_t), where u_t records permitted external inputs and I_t governs admission, allocation, and revision. No common utility functi…
Exact mathematical passage
\mathcal Q_t
\mathcal Q_tIntroduction passage · 2 passages
Search aliases: Q_t mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 1426
Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t), Q_t+1=G( Q_t, C_t,I_t,u_t), where u_t records permitted external inputs and I_t governs admission, allocation, and revision. No common utility functi…
Exact mathematical passage
\mathcal Q_t
\mathcal Q_{t+1}Introduction passage · 1 passage
Search aliases: Q_t+1 mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 1427
Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t), Q_t+1=G( Q_t, C_t,I_t,u_t), where u_t records permitted external inputs and I_t governs admission, allocation, and revision. No common utility functi…
Exact mathematical passage
\mathcal C_{t+1}=F(\mathcal C_t,\mathcal Q_t,I_t,u_t),\qquad \mathcal Q_{t+1}=G(\mathcal Q_t,\mathcal C_t,I_t,u_t),
rIntroduction passage · 84 passages
Search aliases: r
Introduction passage · Machine_Societies_v1.0.0.tex, line 2057
A continuing office need not be one of a permanently supplied list. A population may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t(…
Exact mathematical passage
r_t
R_iIntroduction passage · 3 passages
Search aliases: R_i
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
with a declared review schedule Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the already specified i…
Exact mathematical passage
R_i(\widehat X_n,j)
r_tIntroduction passage · 2 passages
Search aliases: r_t
Introduction passage · Machine_Societies_v1.0.0.tex, line 2057
A continuing office need not be one of a permanently supplied list. A population may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t(…
Exact mathematical passage
r_t
r_{t+1}Introduction passage · 1 passage
Search aliases: r_t+1
Introduction passage · Machine_Societies_v1.0.0.tex, line 2058
may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t( ·mid G_t,r_t,c_t,H_t), where H_t denotes the retained information used by the up…
Exact mathematical passage
(G_{t+1},r_{t+1},c_{t+1})\sim \Gamma_{I_t}(\,\cdot\mid G_t,r_t,c_t,H_t),
\rmIntroduction passage · 31 passages
Search aliases: rm
Introduction passage · Machine_Societies_v1.0.0.tex, line 2171
A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also when every available profile has a possible successor from which the remaining contract cannot be sustained.
Exact mathematical passage
W_0&=\{B\in\mathcal B:B\subseteq S_{\rm safe}\},\label{eq:belief-w0}\\ W_{k+1}&=\{B\in W_0:\exists u\in A(B),\ \operatorname{Succ}(B,u)\subseteq W_k\}.\label{eq:belief-wk}
S_{\rm safe}Introduction passage · 10 passages
Search aliases: S_rm safe rm
Introduction passage · Machine_Societies_v1.0.0.tex, line 2171
A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also when every available profile has a possible successor from which the remaining contract cannot be sustained.
Exact mathematical passage
W_0&=\{B\in\mathcal B:B\subseteq S_{\rm safe}\},\label{eq:belief-w0}\\ W_{k+1}&=\{B\in W_0:\exists u\in A(B),\ \operatorname{Succ}(B,u)\subseteq W_k\}.\label{eq:belief-wk}
\supIntroduction passage · 7 passages
Search aliases: sup
Introduction passage · Machine_Societies_v1.0.0.tex, line 375
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
\eta(x,z')=\sup_{\lambda,\lambda'\in\Lambda(x,z')} \TV(P_\lambda,P_{\lambda'}).
\sup_{\lambda,\lambda'\in\Lambda(x,z')}Introduction passage · 1 passage
Search aliases: sup_λ,λ'∈Λ(x,z') sup lambda lambda in Lambda
Introduction passage · Machine_Societies_v1.0.0.tex, line 375
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
\eta(x,z')=\sup_{\lambda,\lambda'\in\Lambda(x,z')} \TV(P_\lambda,P_{\lambda'}).
\mathcal TIntroduction passage · 5 passages
Search aliases: T mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 253
An organization is always evaluated relative to a contract: permitted controls, target outputs, horizon, environment family, resource limits, and error tolerances. Let T denote this contract. Claims such as ``the group is irreducible'' or ``the identity survived'' are incomplete without it. A state sufficient for tomorrow's scheduling decision may be inadequate for a later audit of who authorized that decision.
Exact mathematical passage
\mathcal T
\mathcal UIntroduction passage · 3 passages
Search aliases: U mathcal
Introduction passage · Machine_Societies_v1.0.0.tex, line 685
Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assign…
Exact mathematical passage
\mathcal U
u_tIntroduction passage · 2 passages
Search aliases: u_t
Introduction passage · Machine_Societies_v1.0.0.tex, line 1427
Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t), Q_t+1=G( Q_t, C_t,I_t,u_t), where u_t records permitted external inputs and I_t governs admission, allocation, and revision. No common utility functi…
Exact mathematical passage
\mathcal C_{t+1}=F(\mathcal C_t,\mathcal Q_t,I_t,u_t),\qquad \mathcal Q_{t+1}=G(\mathcal Q_t,\mathcal C_t,I_t,u_t),
VIntroduction passage · 42 passages
Search aliases: V
Introduction passage · Machine_Societies_v1.0.0.tex, line 685
Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assign…
Exact mathematical passage
V_t\subseteq M_t
vIntroduction passage · 39 passages
Search aliases: v
Introduction passage · Machine_Societies_v1.0.0.tex, line 2547
For a finite set of modes V, let C_v⊆ Z be a certified state domain, with each C_v contained in a common protected set C. A certified hold exists when a final state requires indefinite preservation. At (v,z) with z∈ C_v, a selector may either use the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z…
Exact mathematical passage
\kappa_v
V_tIntroduction passage · 3 passages
Search aliases: V_t
Introduction passage · Machine_Societies_v1.0.0.tex, line 685
Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assign…
Exact mathematical passage
V_t\subseteq M_t
wIntroduction passage · 38 passages
Search aliases: w
Introduction passage · Machine_Societies_v1.0.0.tex, line 2547
se the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z) lies in the declared successor set D_v(z). The local certificate proves D_v(z)⊆ C_v. A handover from v to w must keep every intermediate state in C and end in C_w. Its command, evidence transfer, resources, and delay belong to the transition …
Exact mathematical passage
w
W_0Introduction passage · 1 passage
Search aliases: W_0
Introduction passage · Machine_Societies_v1.0.0.tex, line 2171
A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also when every available profile has a possible successor from which the remaining contract cannot be sustained.
Exact mathematical passage
W_0&=\{B\in\mathcal B:B\subseteq S_{\rm safe}\},\label{eq:belief-w0}\\ W_{k+1}&=\{B\in W_0:\exists u\in A(B),\ \operatorname{Succ}(B,u)\subseteq W_k\}.\label{eq:belief-wk}
W_kIntroduction passage · 5 passages
Search aliases: W_k
Introduction passage · Machine_Societies_v1.0.0.tex, line 2171
A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also when every available profile has a possible successor from which the remaining contract cannot be sustained.
Exact mathematical passage
W_0&=\{B\in\mathcal B:B\subseteq S_{\rm safe}\},\label{eq:belief-w0}\\ W_{k+1}&=\{B\in W_0:\exists u\in A(B),\ \operatorname{Succ}(B,u)\subseteq W_k\}.\label{eq:belief-wk}
W_{k+1}Introduction passage · 1 passage
Search aliases: W_k+1
Introduction passage · Machine_Societies_v1.0.0.tex, line 2171
A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also when every available profile has a possible successor from which the remaining contract cannot be sustained.
Exact mathematical passage
W_0&=\{B\in\mathcal B:B\subseteq S_{\rm safe}\},\label{eq:belief-w0}\\ W_{k+1}&=\{B\in W_0:\exists u\in A(B),\ \operatorname{Succ}(B,u)\subseteq W_k\}.\label{eq:belief-wk}
\widehat XIntroduction passage · 2 passages
Search aliases: widehat X widehat
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the alre…
Exact mathematical passage
\widehat X_n
\widehat X_nIntroduction passage · 2 passages
Search aliases: widehat X_n widehat
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the alre…
Exact mathematical passage
\widehat X_n
X_nIntroduction passage · 1 passage
Search aliases: X_n
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the alre…
Exact mathematical passage
X_n=x
YIntroduction passage · 11 passages
Search aliases: Y
Introduction passage · Machine_Societies_v1.0.0.tex, line 374
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
Y^+
yIntroduction passage · 19 passages
Search aliases: y
Introduction passage · Machine_Societies_v1.0.0.tex, line 2160
nt public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Write Succ(B,u) for their collection. Induction on the public hi…
Exact mathematical passage
y
Y^+Introduction passage · 5 passages
Search aliases: Y^+
Introduction passage · Machine_Societies_v1.0.0.tex, line 374
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
Y^+
z'Introduction passage · 14 passages
Search aliases: z'
Introduction passage · Machine_Societies_v1.0.0.tex, line 375
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
\eta(x,z')=\sup_{\lambda,\lambda'\in\Lambda(x,z')} \TV(P_\lambda,P_{\lambda'}).
\cdotIntroduction passage · 16 passages
Search aliases: · cdot
Introduction passage · Machine_Societies_v1.0.0.tex, line 2058
may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t( ·mid G_t,r_t,c_t,H_t), where H_t denotes the retained information used by the up…
Exact mathematical passage
(G_{t+1},r_{t+1},c_{t+1})\sim \Gamma_{I_t}(\,\cdot\mid G_t,r_t,c_t,H_t),
\GammaIntroduction passage · 9 passages
Search aliases: Γ Gamma
Introduction passage · Machine_Societies_v1.0.0.tex, line 2058
may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t( ·mid G_t,r_t,c_t,H_t), where H_t denotes the retained information used by the up…
Exact mathematical passage
(G_{t+1},r_{t+1},c_{t+1})\sim \Gamma_{I_t}(\,\cdot\mid G_t,r_t,c_t,H_t),
\Gamma_{I_t}Introduction passage · 1 passage
Search aliases: Γ_I_t Gamma
Introduction passage · Machine_Societies_v1.0.0.tex, line 2058
may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t( ·mid G_t,r_t,c_t,H_t), where H_t denotes the retained information used by the up…
Exact mathematical passage
(G_{t+1},r_{t+1},c_{t+1})\sim \Gamma_{I_t}(\,\cdot\mid G_t,r_t,c_t,H_t),
\etaIntroduction passage · 16 passages
Search aliases: η eta
Introduction passage · Machine_Societies_v1.0.0.tex, line 375
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
\eta(x,z')=\sup_{\lambda,\lambda'\in\Lambda(x,z')} \TV(P_\lambda,P_{\lambda'}).
\thetaIntroduction passage · 21 passages
Search aliases: θ theta
Introduction passage · Machine_Societies_v1.0.0.tex, line 1805
If a message M has the same conditional distribution under two equally possible hidden states θ=0,1, then Bayes' rule gives Pr(θ=1mid M)=Pr(θ=1) wherever the posterior is defined. Such a channel carries no information about the distinction, regardless of how fluent or authoritative its reports appear. In strategic environments, senders may have preferences that differ from those of receivers. The theory of strategic information transmission shows why informative communication can depend on prefe…
Exact mathematical passage
\theta=0,1
\kappaIntroduction passage · 13 passages
Search aliases: κ kappa
Introduction passage · Machine_Societies_v1.0.0.tex, line 2547
For a finite set of modes V, let C_v⊆ Z be a certified state domain, with each C_v contained in a common protected set C. A certified hold exists when a final state requires indefinite preservation. At (v,z) with z∈ C_v, a selector may either use the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z…
Exact mathematical passage
\kappa_v
\kappa_vIntroduction passage · 2 passages
Search aliases: κ_v kappa
Introduction passage · Machine_Societies_v1.0.0.tex, line 2547
For a finite set of modes V, let C_v⊆ Z be a certified state domain, with each C_v contained in a common protected set C. A certified hold exists when a final state requires indefinite preservation. At (v,z) with z∈ C_v, a selector may either use the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z…
Exact mathematical passage
\kappa_v
\LambdaIntroduction passage · 3 passages
Search aliases: Λ Lambda
Introduction passage · Machine_Societies_v1.0.0.tex, line 375
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
\eta(x,z')=\sup_{\lambda,\lambda'\in\Lambda(x,z')} \TV(P_\lambda,P_{\lambda'}).
\lambda'Introduction passage · 1 passage
Search aliases: λ' lambda
Introduction passage · Machine_Societies_v1.0.0.tex, line 375
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
\eta(x,z')=\sup_{\lambda,\lambda'\in\Lambda(x,z')} \TV(P_\lambda,P_{\lambda'}).
\existsIntroduction passage · 7 passages
Search aliases: ∃ exists
Introduction passage · Machine_Societies_v1.0.0.tex, line 2171
A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also when every available profile has a possible successor from which the remaining contract cannot be sustained.
Exact mathematical passage
W_0&=\{B\in\mathcal B:B\subseteq S_{\rm safe}\},\label{eq:belief-w0}\\ W_{k+1}&=\{B\in W_0:\exists u\in A(B),\ \operatorname{Succ}(B,u)\subseteq W_k\}.\label{eq:belief-wk}
\varnothingIntroduction passage · 4 passages
Search aliases: ∅ varnothing
Introduction passage · Machine_Societies_v1.0.0.tex, line 2153
The state of possible institutional realizations. The relevant public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Writ…
Exact mathematical passage
\mathcal B=\{B\subseteq S:B\ne\varnothing,\ h\text{ is constant on }B\}.
\inIntroduction passage · 84 passages
Search aliases: ∈ in
Introduction passage · Machine_Societies_v1.0.0.tex, line 375
Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specified.
Exact mathematical passage
\eta(x,z')=\sup_{\lambda,\lambda'\in\Lambda(x,z')} \TV(P_\lambda,P_{\lambda'}).
\sum_{d(o)>0}Introduction passage · 1 passage
Search aliases: ∑_d(o)>0 sum
Introduction passage · Machine_Societies_v1.0.0.tex, line 954
p:deterrence-tv In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C). For every 0≤ s≤1, ∑_o s(o)d(o)≤∑_d(o)>0d(o). Equality is attained by s(o)=1 on the positive terms and zero elsewhere. Substitution into~[source reference] proves both assertions.
Exact mathematical passage
\sum_o s(o)d(o)\leq\sum_{d(o)>0}d(o).
\sum_oIntroduction passage · 3 passages
Search aliases: ∑_o sum
Introduction passage · Machine_Societies_v1.0.0.tex, line 953
[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C). For every 0≤ s≤1, ∑_o s(o)d(o)≤∑_d(o)>0d(o). Equality is attained by s(o)=1 on the positive terms and zero elsewhere. Substitution into~[source refe…
Exact mathematical passage
\sum_o d(o)=0
\capIntroduction passage · 2 passages
Search aliases: ∩ cap
Introduction passage · Machine_Societies_v1.0.0.tex, line 2161
of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Write Succ(B,u) for their collection. Induction on the public history shows that~[source re…
Exact mathematical passage
\label{eq:belief-update} \operatorname{Post}(B,u,y)= \left(\bigcup_{s\in B}\Delta(s,u)\right)\cap h^{-1}(y).
\simIntroduction passage · 7 passages
Search aliases: ∼ sim
Introduction passage · Machine_Societies_v1.0.0.tex, line 2058
may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their transition belongs to the realized state: (G_t+1,r_t+1,c_t+1)∼ Γ_I_t( ·mid G_t,r_t,c_t,H_t), where H_t denotes the retained information used by the up…
Exact mathematical passage
(G_{t+1},r_{t+1},c_{t+1})\sim \Gamma_{I_t}(\,\cdot\mid G_t,r_t,c_t,H_t),
\neIntroduction passage · 10 passages
Search aliases: ≠ ne
Introduction passage · Machine_Societies_v1.0.0.tex, line 2153
The state of possible institutional realizations. The relevant public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Writ…
Exact mathematical passage
\mathcal B=\{B\subseteq S:B\ne\varnothing,\ h\text{ is constant on }B\}.
\leqIntroduction passage · 56 passages
Search aliases: ≤ leq
Introduction passage · Machine_Societies_v1.0.0.tex, line 947
[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C). For every 0≤ s≤1, ∑_o s(o)d(o)≤∑_d(o)>0d(o). Equality is attained by s(o)=1 on the positive terms and zero elsewhere. Substitution into~[source refe…
Exact mathematical passage
g\leq F\,\TV(P_D,P_C).
\geqIntroduction passage · 25 passages
Search aliases: ≥ geq
Introduction passage · Machine_Societies_v1.0.0.tex, line 1522
A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision kernel R_i(widehat X_n,j). A subsequent fast-state reset may depend on the endpoint and adopted institution, but cannot revise the alre…
Exact mathematical passage
m\geq1
\subseteqIntroduction passage · 34 passages
Search aliases: ⊆ subseteq
Introduction passage · Machine_Societies_v1.0.0.tex, line 685
Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assign…
Exact mathematical passage
M_t\subseteq\mathcal U
\bigcapIntroduction passage · 1 passage
Search aliases: ⋂ bigcap
Introduction passage · Machine_Societies_v1.0.0.tex, line 2157
The state of possible institutional realizations. The relevant public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Writ…
Exact mathematical passage
A(B)=\bigcap_{s\in B}A(s).
\bigcap_{s\in B}Introduction passage · 1 passage
Search aliases: ⋂_s∈ B bigcap in
Introduction passage · Machine_Societies_v1.0.0.tex, line 2157
The state of possible institutional realizations. The relevant public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Writ…
Exact mathematical passage
A(B)=\bigcap_{s\in B}A(s).
\bigcupIntroduction passage · 2 passages
Search aliases: ⋃ bigcup
Introduction passage · Machine_Societies_v1.0.0.tex, line 2161
of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Write Succ(B,u) for their collection. Induction on the public history shows that~[source re…
Exact mathematical passage
\label{eq:belief-update} \operatorname{Post}(B,u,y)= \left(\bigcup_{s\in B}\Delta(s,u)\right)\cap h^{-1}(y).
\bigcup_{s\in B}Introduction passage · 1 passage
Search aliases: ⋃_s∈ B bigcup in
Introduction passage · Machine_Societies_v1.0.0.tex, line 2161
of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y). Only nonempty successor sets are possible observations. Write Succ(B,u) for their collection. Induction on the public history shows that~[source re…
Exact mathematical passage
\label{eq:belief-update} \operatorname{Post}(B,u,y)= \left(\bigcup_{s\in B}\Delta(s,u)\right)\cap h^{-1}(y).
\mathbf1Use in source · 7 passages
Search aliases: 1 mathbf
Use in source · Machine_Societies_v1.0.0.tex, line 1478
[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random variable gives Eτ=∑_t≥0 P(τ>t). Under ρ(Q)<1, the matrix geometric series converges to (Id-Q)^-1, giving the result.
Exact mathematical passage
\Prob_\mu(\tau>t)=\mu Q^t\mathbf1.
A_*Use in source · 2 passages
Search aliases: A_*
Use in source · Machine_Societies_v1.0.0.tex, line 2195
Let G⊆ S_rm safe be a stipulated completion set. The target is certified completion, meaning that the current possibility set lies wholly in G. Let G={B∈ W_*:B⊆ G}, A_*(B)={u∈ A(B):Succ(B,u)⊆ W_*}. Define R_0= G and R_k+1=R_k∪ {B∈ W_*:∃ u∈ A_*(B), Succ(B,u)⊆ R_k}. The increasing sequence stabilizes at R_*. For B∈ R_*, let r(B) be its first index of entry.
Exact mathematical passage
\mathcal G=\{B\in W_*:B\subseteq G\}, \quad A_*(B)=\{u\in A(B):\operatorname{Succ}(B,u)\subseteq W_*\}.
a_1Use in source · 3 passages
Search aliases: a_1
Use in source · Machine_Societies_v1.0.0.tex, line 1307
Why separate marginal ignorance is insufficient. [Local ignorance with a perfectly correct joint action] Let U be a fair bit. Agent 1 observes H_1=U and agent 2 observes H_2=U⊕θ. Each individual observation has the same law for both values of θ. Nevertheless, the joint action (a_1,a_2)=(H_1,H_2) always satisfies a_1⊕ a_2=θ. The joint observation laws differ and have disjoint supports. Equal marginal laws therefore do not imply the hypothesis of Theorem~[source reference].
Exact mathematical passage
(a_1,a_2)=(H_1,H_2)
a_2Use in source · 2 passages
Search aliases: a_2
Use in source · Machine_Societies_v1.0.0.tex, line 1307
Why separate marginal ignorance is insufficient. [Local ignorance with a perfectly correct joint action] Let U be a fair bit. Agent 1 observes H_1=U and agent 2 observes H_2=U⊕θ. Each individual observation has the same law for both values of θ. Nevertheless, the joint action (a_1,a_2)=(H_1,H_2) always satisfies a_1⊕ a_2=θ. The joint observation laws differ and have disjoint supports. Equal marginal laws therefore do not imply the hypothesis of Theorem~[source reference].
Exact mathematical passage
(a_1,a_2)=(H_1,H_2)
a_iUse in source · 2 passages
Search aliases: a_i
Use in source · Machine_Societies_v1.0.0.tex, line 886
A formation barrier despite mutual benefit. The stationary cost comparison does not describe a route to adoption. Consider n≥2 agents choosing a_i∈{0,1}. An adopter pays cost c>0 and receives benefit b>0 from each other adopter: u_i(a)=a_i(b∑_j≠ ia_j-c). Nonadopters receive zero. This is a finite network-coordination model, not a universal utility function for societies.
Exact mathematical passage
a_i\in\{0,1\}
a_jUse in source · 1 passage
Search aliases: a_j
Use in source · Machine_Societies_v1.0.0.tex, line 887
A formation barrier despite mutual benefit. The stationary cost comparison does not describe a route to adoption. Consider n≥2 agents choosing a_i∈{0,1}. An adopter pays cost c>0 and receives benefit b>0 from each other adopter: u_i(a)=a_i(b∑_j≠ ia_j-c). Nonadopters receive zero. This is a finite network-coordination model, not a universal utility function for societies.
Exact mathematical passage
u_i(a)=a_i\left(b\sum_{j\ne i}a_j-c\right).
A_{\rm end}Use in source · 2 passages
Search aliases: A_rm end rm
Use in source · Machine_Societies_v1.0.0.tex, line 2502
The finite construction of Section~[source reference] yields an exact characterization of continued operation with an available authorized ending. Use its possibility space B, successor operation, and protected set S_rm safe. For each possibility set B, distinguish implemented running profiles A_rm run(B) from implemented ending profiles A_rm end(B). Both must be admissible in every state in B. Ending profiles cannot accept new obligations outside the declared settlement protocol. Let G⊆ S_rm sa…
Exact mathematical passage
A_{\rm end}(B)
A_{\rm run}Use in source · 2 passages
Search aliases: A_rm run rm
Use in source · Machine_Societies_v1.0.0.tex, line 2502
The finite construction of Section~[source reference] yields an exact characterization of continued operation with an available authorized ending. Use its possibility space B, successor operation, and protected set S_rm safe. For each possibility set B, distinguish implemented running profiles A_rm run(B) from implemented ending profiles A_rm end(B). Both must be admissible in every state in B. Ending profiles cannot accept new obligations outside the declared settlement protocol. Let G⊆ S_rm sa…
Exact mathematical passage
A_{\rm run}(B)
A_tUse in source · 3 passages
Search aliases: A_t
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
a_tUse in source · 6 passages
Search aliases: a_t
Use in source · Machine_Societies_v1.0.0.tex, line 485
Repair needs a causal and temporal test. Let e_t≥0 measure deviation from an acceptable organization after accounting for legitimate changes in task state. Suppose a repair controller ensures e_t+1≤ a_t e_t+b_t, where a_t measures retained deviation and b_t covers new disturbance and model mismatch.
Exact mathematical passage
e_{t+1}\leq a_t e_t+b_t,
a_{t-1}Use in source · 1 passage
Search aliases: a_t-1
Use in source · Machine_Societies_v1.0.0.tex, line 1830
A decision-only public record. Let an unknown state θ∈{0,1} have equal prior probabilities. Participant t receives a private signal s_t∈{0,1} with P(s_t=θmidθ)=p, 1/2<p<1. Signals are independent conditional on θ, and their accuracy is common knowledge. In sequence, each participant observes previous actions a_1,…,a_t-1 and its own signal, then chooses a_t∈{0,1} to maximize its probability of matching θ. At exact posterior indifference it follows its private signal. The public record contains ac…
Exact mathematical passage
a_1,\ldots,a_{t-1}
\absUse in source · 1 passage
Search aliases: abs
Use in source · Machine_Societies_v1.0.0.tex, line 467
Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.
Exact mathematical passage
d_E=\min_{z\ne z'}\abs{\{i:E_i(z)\ne E_i(z')\}}.
\mathsf{Assumptions}Use in source · 1 passage
Search aliases: Assumptions mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2539
An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a component, or unavailable. Those statuses should not all become the Boolean value verified. A failed guard means that continued reliance is not justified by t…
Exact mathematical passage
\mathcal K=(\mathsf{Claim},\mathsf{Assumptions},\mathsf{Evidence}, \mathsf{Domain},\mathsf{Guard},\mathsf{Fallback},\mathsf{Revision}).
\mathsf{Auth}Use in source · 2 passages
Search aliases: Auth mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2343
A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. Standing specifies who may initiate a challenge and on which grounds. Remedy specifies the actions an authorized review can actually require. The cont…
Exact mathematical passage
\mathcal B=(Z_{\rm ext},O_{\rm ext},\mathsf{Auth},\mathsf{Effect}, \mathsf{Standing},\mathsf{Remedy}).
\mathsf{authority}Use in source · 1 passage
Search aliases: authority mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 1322
tibility, and knowledge Temporal obligations, contracts, and office succession. Permissions describe available actions; obligations constrain continuations. The debt created by an urgency exchange is not satisfied merely because the creditor remains authorized to ask for service. It requires a particular future event. A finite obligation record can be written as o=(id,i,j,φ,d,authority,status), where i and j are the debtor and beneficiary offices, φ specifies a fulfillment event, and d is a dead…
Exact mathematical passage
o=(\mathsf{id},i,j,\varphi,d,\mathsf{authority},\mathsf{status}),
bUse in source · 66 passages
Search aliases: b
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
B_0Use in source · 8 passages
Search aliases: B_0
Use in source · Machine_Societies_v1.0.0.tex, line 1288
[Indistinguishability obstruction to joint safety] Let B_0 and B_1 be disjoint subsets of a common finite joint-action alphabet, containing the actions that meet a declared requirement under conditions 0 and 1. If the laws of H are within η in total variation, then P_0(B_0)+P_1(B_1)≤1+η. Consequently at least one condition has failure probability at least (1-η)/2. If every admissible protocol has identical joint-history laws under the conditions, no admissible protocol can meet the requirement a…
Exact mathematical passage
B_0
\mathcal B_0Use in source · 3 passages
Search aliases: B_0 mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 761
The smallest exact summary under a fixed contract. The conditions yield a finite construction rather than only an existence criterion. Begin with the partition B_0 that identifies exactly the states sharing (ℓ(s),D(s)). At each refinement, identify s,t only when they were previously in the same block and every action in their common enabled set leads to the same previous block: s≡_k+1t Longleftrightarrow s≡_kt and F(s,a)≡_kF(t,a) for all a∈ D(s).
Exact mathematical passage
\mathcal B_0
B_1Use in source · 3 passages
Search aliases: B_1
Use in source · Machine_Societies_v1.0.0.tex, line 1288
[Indistinguishability obstruction to joint safety] Let B_0 and B_1 be disjoint subsets of a common finite joint-action alphabet, containing the actions that meet a declared requirement under conditions 0 and 1. If the laws of H are within η in total variation, then P_0(B_0)+P_1(B_1)≤1+η. Consequently at least one condition has failure probability at least (1-η)/2. If every admissible protocol has identical joint-history laws under the conditions, no admissible protocol can meet the requirement a…
Exact mathematical passage
B_1
b_{12}Use in source · 1 passage
Search aliases: b_12
Use in source · Machine_Societies_v1.0.0.tex, line 1624
This result is a standard parity fact on graphs, closely related to cycle-space algebra and discrete cocycles. Its value here is methodological. It shows why pairwise satisfiable cannot be equated with globally realizable. For a triangle with b_12=0, b_23=0, and b_31=1, every individual edge requirement has a solution, but the three cannot hold together. A translator between two agents could appear successful while a three-party federation is impossible under the combined rules.
Exact mathematical passage
b_{12}=0
b_{23}Use in source · 1 passage
Search aliases: b_23
Use in source · Machine_Societies_v1.0.0.tex, line 1624
This result is a standard parity fact on graphs, closely related to cycle-space algebra and discrete cocycles. Its value here is methodological. It shows why pairwise satisfiable cannot be equated with globally realizable. For a triangle with b_12=0, b_23=0, and b_31=1, every individual edge requirement has a solution, but the three cannot hold together. A translator between two agents could appear successful while a three-party federation is impossible under the combined rules.
Exact mathematical passage
b_{23}=0
b_{31}Use in source · 1 passage
Search aliases: b_31
Use in source · Machine_Societies_v1.0.0.tex, line 1624
This result is a standard parity fact on graphs, closely related to cycle-space algebra and discrete cocycles. Its value here is methodological. It shows why pairwise satisfiable cannot be equated with globally realizable. For a triangle with b_12=0, b_23=0, and b_31=1, every individual edge requirement has a solution, but the three cannot hold together. A translator between two agents could appear successful while a three-party federation is impossible under the combined rules.
Exact mathematical passage
b_{31}=1
b_iUse in source · 5 passages
Search aliases: b_i
Use in source · Machine_Societies_v1.0.0.tex, line 864
In an idealized transferable-utility setting there is a corresponding conditional observation. If the discounted net advantages b_i satisfy ∑_i b_i≥0, then there exist budget-balanced transfers t_i with ∑_i t_i=0 and b_i+t_i≥0 for all i: choose nonnegative w_i summing to ∑_i b_i and set t_i=w_i-b_i. This algebra does not supply liquidity, a transfer mechanism, consent, enforceability, or a right to redistribute. The difference between mathematical feasible compensation and institutionally permit…
Exact mathematical passage
b_i
b_{ij}Use in source · 3 passages
Search aliases: b_ij
Use in source · Machine_Societies_v1.0.0.tex, line 1606
Consider an undirected graph G=(V,E) whose vertices are institutional positions. Each position carries a binary coordinate x_i∈ F_2 and each edge {i,j} imposes the relative constraint x_i⊕ x_j=b_ij, b_ij∈ F_2. The coordinates might represent a toy authorization bit or a binary convention, but the mathematical model itself makes no claim about a realistic legal code.
Exact mathematical passage
\label{eq:edge-rule} x_i\oplus x_j=b_{ij},\qquad b_{ij}\in\mathbb F_2.
b_{ij}'Use in source · 1 passage
Search aliases: b_ij'
Use in source · Machine_Societies_v1.0.0.tex, line 1767
The binary jurisdiction model admits a sharper statement. A local convention change x_i'=x_i⊕ c_i transforms its edge labels to b_ij'=b_ij⊕ c_i⊕ c_j.
Exact mathematical passage
b_{ij}'=b_{ij}\oplus c_i\oplus c_j.
b_jUse in source · 1 passage
Search aliases: b_j
Use in source · Machine_Societies_v1.0.0.tex, line 497
[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.
Exact mathematical passage
e_T\leq e_0\prod_{t=0}^{T-1}a_t+ \sum_{j=0}^{T-1}b_j\prod_{t=j+1}^{T-1}a_t.
\mathcal B_kUse in source · 1 passage
Search aliases: B_k mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 772
mber of states among those exact quotients, up to relabeling. Every strict refinement increases the number of blocks, which cannot exceed |S|. At a fixed point, states in one block have the same labels and enabled sets and their enabled successors remain in one block, so the quotient conditions hold. Any compatible partition refines B_0. If it refines B_k, compatible successors show that it also refines B_k+1. Induction proves that it refines the stable partition. The latter is itself compatible…
Exact mathematical passage
\mathcal B_k
\mathcal B_{k+1}Use in source · 1 passage
Search aliases: B_k+1 mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 772
ndproposition Every strict refinement increases the number of blocks, which cannot exceed |S|. At a fixed point, states in one block have the same labels and enabled sets and their enabled successors remain in one block, so the quotient conditions hold. Any compatible partition refines B_0. If it refines B_k, compatible successors show that it also refines B_k+1. Induction proves that it refines the stable partition. The latter is itself compatible, giving the minimum block count.
Exact mathematical passage
\mathcal B_{k+1}
B_tUse in source · 1 passage
Search aliases: B_t
Use in source · Machine_Societies_v1.0.0.tex, line 686
hcal U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assigns nonnegative voting weights, q_t is an approval threshold, and e_t is a configuration ep…
Exact mathematical passage
B_t=(M_t,V_t,c_t,w_t,q_t,e_t),
b_tUse in source · 4 passages
Search aliases: b_t
Use in source · Machine_Societies_v1.0.0.tex, line 485
Repair needs a causal and temporal test. Let e_t≥0 measure deviation from an acceptable organization after accounting for legitimate changes in task state. Suppose a repair controller ensures e_t+1≤ a_t e_t+b_t, where a_t measures retained deviation and b_t covers new disturbance and model mismatch.
Exact mathematical passage
e_{t+1}\leq a_t e_t+b_t,
\bar aUse in source · 2 passages
Search aliases: bar a bar
Use in source · Machine_Societies_v1.0.0.tex, line 492
[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.
Exact mathematical passage
0\leq a_t\leq\bar a<1
\bar a^tUse in source · 1 passage
Search aliases: bar a^t bar
Use in source · Machine_Societies_v1.0.0.tex, line 493
[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.
Exact mathematical passage
e_t\leq \bar a^t e_0+\bar b\frac{1-\bar a^t}{1-\bar a}.
\bar bUse in source · 2 passages
Search aliases: bar b bar
Use in source · Machine_Societies_v1.0.0.tex, line 492
[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.
Exact mathematical passage
0\leq b_t\leq\bar b
\bar DUse in source · 2 passages
Search aliases: bar D bar
Use in source · Machine_Societies_v1.0.0.tex, line 738
[Exact governance quotient] Let S be finite and nonempty and A finite, and let π:S→ Z be surjective. There exist functions barℓ, bar D, and partial transitions bar F satisfying ℓ=barℓπ, D=bar Dπ, π F(s,a)=bar F(π s,a) (a∈ D(s)) if and only if whenever π s=π t, ℓ(s)=ℓ(t), D(s)=D(t), π F(s,a)=π F(t,a) for all a∈ D(s). The quotient data are unique. From any two states in the same fibre, the legal finite action sequences and their corresponding label sequences are identical. This includes sequences …
Exact mathematical passage
\bar D
\bar FUse in source · 2 passages
Search aliases: bar F bar
Use in source · Machine_Societies_v1.0.0.tex, line 738
[Exact governance quotient] Let S be finite and nonempty and A finite, and let π:S→ Z be surjective. There exist functions barℓ, bar D, and partial transitions bar F satisfying ℓ=barℓπ, D=bar Dπ, π F(s,a)=bar F(π s,a) (a∈ D(s)) if and only if whenever π s=π t, ℓ(s)=ℓ(t), D(s)=D(t), π F(s,a)=π F(t,a) for all a∈ D(s). The quotient data are unique. From any two states in the same fibre, the legal finite action sequences and their corresponding label sequences are identical. This includes sequences …
Exact mathematical passage
\bar F
\bar wUse in source · 2 passages
Search aliases: bar w bar
Use in source · Machine_Societies_v1.0.0.tex, line 2019
tor v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-step monotonic estimate followed by induction. Under uniformly bounded disturbance normw_t_v,∞≤bar w, the last term is at most bar w(1-κ^t)/(1-κ). This gives a conditional attenuation result, not a universal contagion threshold. A graph alone cannot supply the matrix B; the rates must come from a justified dynamical model. M…
Exact mathematical passage
\norm{w_t}_{v,\infty}\le\bar w
\bar\ellUse in source · 2 passages
Search aliases: barℓ bar ell
Use in source · Machine_Societies_v1.0.0.tex, line 738
[Exact governance quotient] Let S be finite and nonempty and A finite, and let π:S→ Z be surjective. There exist functions barℓ, bar D, and partial transitions bar F satisfying ℓ=barℓπ, D=bar Dπ, π F(s,a)=bar F(π s,a) (a∈ D(s)) if and only if whenever π s=π t, ℓ(s)=ℓ(t), D(s)=D(t), π F(s,a)=π F(t,a) for all a∈ D(s). The quotient data are unique. From any two states in the same fibre, the legal finite action sequences and their corresponding label sequences are identical. This includes sequences …
Exact mathematical passage
\bar\ell
\bigwedgeUse in source · 2 passages
Search aliases: bigwedge
Use in source · Machine_Societies_v1.0.0.tex, line 1888
For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling complementary private evidence can therefore reveal a fact that is not public to any participant.
Exact mathematical passage
E_J\varphi=\bigwedge_{i\in J}K_i\varphi
\bigwedge_{i\in J}Use in source · 1 passage
Search aliases: bigwedge_i∈ J bigwedge in
Use in source · Machine_Societies_v1.0.0.tex, line 1888
For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling complementary private evidence can therefore reveal a fact that is not public to any participant.
Exact mathematical passage
E_J\varphi=\bigwedge_{i\in J}K_i\varphi
\bigwedge_{n\ge0}Use in source · 1 passage
Search aliases: bigwedge_n≥0 bigwedge ge
Use in source · Machine_Societies_v1.0.0.tex, line 1889
For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling complementary private evidence can therefore reveal a fact that is not public to any participant.
Exact mathematical passage
C_J\varphi=\bigwedge_{n\ge0}E_J^n\varphi, \qquad E_J^0\varphi=\varphi.
\bottomruleUse in source · 2 passages
Search aliases: bottomrule
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\mathcal C_{\mathcal T}Use in source · 1 passage
Search aliases: C_ T mathcal mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 257
A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may be unidentified. They should not be replaced by zero or hidden inside a favorable weighted average.
Exact mathematical passage
\mathcal C_{\mathcal T}=(\varepsilon_{\rm pred},\varepsilon_{\rm dyn},\varepsilon_{\rm int}, \Delta_{\rm coal},R_{\rm repair},D_{\rm lineage},G_{\rm formation}).
\mathcal C_*Use in source · 2 passages
Search aliases: C_* mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 1434
A finite closure theorem, not an open-endedness theorem. Take finite universes C_* and Q_*. Consider a fixed collection of production rules of the form ``if these finitely many records are present, add this record.'' Prerequisites are positive: no rule requires the absence of a record. The content and correctness of those rules are assumptions. An added method is certified only if the specified certification procedure actually supports it.
Exact mathematical passage
\mathcal C_*
\mathcal C_0Use in source · 2 passages
Search aliases: C_0 mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 1438
Let Φ simultaneously add all conclusions whose prerequisites are satisfied, retaining the input records. Order pairs ( C, Q) by componentwise inclusion. [Least closure of a finite problem--method system] Starting from ( C_0, Q_0), iteration of Φ reaches the least closed pair containing that initial pair. There are at most | C_*∖ C_0|+| Q_*∖ Q_0| strict growth steps. A sequential rule schedule reaches the same closure if every rule that remains enabled with an absent conclusion is eventually appl…
Exact mathematical passage
(\mathcal C_0,\mathcal Q_0)
c_0Use in source · 2 passages
Search aliases: c_0
Use in source · Machine_Societies_v1.0.0.tex, line 836
Suppose a coordination protocol without institution has expected per-period cost c_0. A proposed institutional protocol has operating cost c_1, per-period maintenance m, and a one-time setup cost B. With discount factor 0≤δ<1, write g=c_0-c_1-m.
Exact mathematical passage
c_0
C_1Use in source · 1 passage
Search aliases: C_1
Use in source · Machine_Societies_v1.0.0.tex, line 2485
For sufficiency of the interval tests, fix U. Its union of integer windows decomposes into disjoint maximal integer intervals C_1,…,C_k. Each window is contained in one component because it is an interval. The number of appeals from U in C_i is no larger than the number of all appeals whose entire windows lie in C_i. By~[source reference], this is at most the total capacity in C_i. Summing over the disjoint components proves~[source reference], and hence the matching exists.
Exact mathematical passage
C_1,\ldots,C_k
c_1Use in source · 2 passages
Search aliases: c_1
Use in source · Machine_Societies_v1.0.0.tex, line 836
Suppose a coordination protocol without institution has expected per-period cost c_0. A proposed institutional protocol has operating cost c_1, per-period maintenance m, and a one-time setup cost B. With discount factor 0≤δ<1, write g=c_0-c_1-m.
Exact mathematical passage
c_1
c_aUse in source · 1 passage
Search aliases: c_a
Use in source · Machine_Societies_v1.0.0.tex, line 2598
The audit provider receives a separate collectible fee w for verified execution, with w larger than its stipulated cost c_a. A genuine-execution receipt is checked by a specified trusted mechanism. The fee is not a reward for accusing someone. It makes execution preferable to skipping in this narrow two-action provider model. Monetary units are tracked separately from execution units. The payment budget, receipt checker, and record-generation law remain explicit assumptions. The perfect detector…
Exact mathematical passage
c_a
c_gUse in source · 2 passages
Search aliases: c_g
Use in source · Machine_Societies_v1.0.0.tex, line 2067
Specialization has a cost and a reachability condition. Consider two services performed once per period. A generalist arrangement costs 2c_g per period. Two specialists cost 2c_s+b, where b is the required communication and integration cost, and training costs h for each specialist. Over T periods the specialist arrangement has lower total cost precisely when T[2(c_g-c_s)-b]>2h. This is an instance of institutional amortization, not a universal law of specialization. Verification overhead, repla…
Exact mathematical passage
2c_g
C_iUse in source · 3 passages
Search aliases: C_i
Use in source · Machine_Societies_v1.0.0.tex, line 2485
For sufficiency of the interval tests, fix U. Its union of integer windows decomposes into disjoint maximal integer intervals C_1,…,C_k. Each window is contained in one component because it is an interval. The number of appeals from U in C_i is no larger than the number of all appeals whose entire windows lie in C_i. By~[source reference], this is at most the total capacity in C_i. Summing over the disjoint components proves~[source reference], and hence the matching exists.
Exact mathematical passage
C_i
c_iUse in source · 3 passages
Search aliases: c_i
Use in source · Machine_Societies_v1.0.0.tex, line 1766
The binary jurisdiction model admits a sharper statement. A local convention change x_i'=x_i⊕ c_i transforms its edge labels to b_ij'=b_ij⊕ c_i⊕ c_j.
Exact mathematical passage
x_i'=x_i\oplus c_i
C_JUse in source · 4 passages
Search aliases: C_J
Use in source · Machine_Societies_v1.0.0.tex, line 1889
For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling complementary private evidence can therefore reveal a fact that is not public to any participant.
Exact mathematical passage
C_J\varphi=\bigwedge_{n\ge0}E_J^n\varphi, \qquad E_J^0\varphi=\varphi.
\mathcal C_jUse in source · 3 passages
Search aliases: C_j mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 1399
Let C_j⊆ Q be the queries for which generation j has at least one admitted executable method satisfying the contract. This is an attainable-capability set, not a measurement of what an arbitrary method-selection policy actually achieves.
Exact mathematical passage
\mathcal C_j\subseteq\mathcal Q
c_jUse in source · 1 passage
Search aliases: c_j
Use in source · Machine_Societies_v1.0.0.tex, line 1767
The binary jurisdiction model admits a sharper statement. A local convention change x_i'=x_i⊕ c_i transforms its edge labels to b_ij'=b_ij⊕ c_i⊕ c_j.
Exact mathematical passage
b_{ij}'=b_{ij}\oplus c_i\oplus c_j.
\mathcal C_{j+1}Use in source · 1 passage
Search aliases: C_j+1 mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 1403
[Conditional cumulative culture] Suppose every certified method available at generation j has an admitted realization at generation j+1 preserving its query semantics, correctness premises, required resources, and execution rights. Suppose newly added methods satisfy the same declared correctness contract. Then C_j⊆ C_j+1. The inclusion is strict exactly when at least one newly available method correctly covers a previously uncovered query. The conclusion remains compatible with correction or re…
Exact mathematical passage
\mathcal C_j\subseteq\mathcal C_{j+1}.
C_kUse in source · 1 passage
Search aliases: C_k
Use in source · Machine_Societies_v1.0.0.tex, line 2485
For sufficiency of the interval tests, fix U. Its union of integer windows decomposes into disjoint maximal integer intervals C_1,…,C_k. Each window is contained in one component because it is an interval. The number of appeals from U in C_i is no larger than the number of all appeals whose entire windows lie in C_i. By~[source reference], this is at most the total capacity in C_i. Summing over the disjoint components proves~[source reference], and hence the matching exists.
Exact mathematical passage
C_1,\ldots,C_k
c_sUse in source · 2 passages
Search aliases: c_s
Use in source · Machine_Societies_v1.0.0.tex, line 2067
Specialization has a cost and a reachability condition. Consider two services performed once per period. A generalist arrangement costs 2c_g per period. Two specialists cost 2c_s+b, where b is the required communication and integration cost, and training costs h for each specialist. Over T periods the specialist arrangement has lower total cost precisely when T[2(c_g-c_s)-b]>2h. This is an instance of institutional amortization, not a universal law of specialization. Verification overhead, repla…
Exact mathematical passage
2c_s+b
C_tUse in source · 1 passage
Search aliases: C_t
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
C_{v_0}Use in source · 1 passage
Search aliases: C_v_0
Use in source · Machine_Societies_v1.0.0.tex, line 2550
[Guarded certificate use and authorized handover] Assume that the selector, at every reachable decision epoch, either uses a local policy with a sound positive guard or performs a handover satisfying the preceding conditions. Starting from z_0∈ C_v_0, every decision state belongs to its current certified domain and every intermediate handover state remains in C.
Exact mathematical passage
z_0\in C_{v_0}
C_{\{A,B\}}Use in source · 1 passage
Search aliases: C_{A,B}
Use in source · Machine_Societies_v1.0.0.tex, line 1907
[Everyone knows, but it is not common knowledge] Let W={0,1,2} and let φ hold exactly at 0,1. Agent A has partition {0,1}mid{2}, and agent B has partition {0}mid{1,2}. At world 0, both agents know φ. However, the chain 0∼_A1∼_B2 reaches a world where it is false. Thus E_{A,B}φ holds at 0 but C_{A,B}φ does not.
Exact mathematical passage
C_{\{A,B\}}\varphi
\mathsf{Claim}Use in source · 1 passage
Search aliases: Claim mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2539
An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a component, or unavailable. Those statuses should not all become the Boolean value verified. A failed guard means that continued reliance is not justified by t…
Exact mathematical passage
\mathcal K=(\mathsf{Claim},\mathsf{Assumptions},\mathsf{Evidence}, \mathsf{Domain},\mathsf{Guard},\mathsf{Fallback},\mathsf{Revision}).
\mathsf{created}Use in source · 1 passage
Search aliases: created mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 1328
For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy follows from admitting discharge in this particular contract.
Exact mathematical passage
\mathbf G\bigl(\mathsf{created}(o)\Rightarrow \mathbf F_{\leq D}\mathsf{settled}(o)\bigr).
\mathsf{creditor}Use in source · 1 passage
Search aliases: creditor mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 192
ubsectionTwo rules and an inherited obligation Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The exchange creates a registered obligation o=(id,debtor=j,creditor=k, quantity=1,due=e+1). At the next epoch the creditor receives two slots and the debtor n…
Exact mathematical passage
o=(\mathsf{id},\mathsf{debtor}=j,\mathsf{creditor}=k, \mathsf{quantity}=1,\mathsf{due}=e+1).
\mathfrak DUse in source · 1 passage
Search aliases: D mathfrak
Use in source · Machine_Societies_v1.0.0.tex, line 1731
ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The symmetric permission residual measures both changes and is deliberately stronger.
Exact mathematical passage
\mathfrak D(T)= (\varepsilon_{\rm dyn},\varepsilon_P,\varepsilon_O, \varepsilon_R,\varepsilon_E).
d_{\mathcal T}Use in source · 1 passage
Search aliases: d_ T mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 415
For approximate comparison, use the pseudometric d_ T(x,x')=sup_τ∈ T d_TV( L_x(Y^+midτ), L_x'(Y^+midτ)). A finite test suite may make distinct implementations distance zero. That is indistinguishability under those tests, not universal sameness.
Exact mathematical passage
d_{\mathcal T}(x,x')=\sup_{\tau\in\mathcal T} \TV\bigl(\Law_x(Y^+\mid\tau),\Law_{x'}(Y^+\mid\tau)\bigr).
D_1Use in source · 3 passages
Search aliases: D_1
Use in source · Machine_Societies_v1.0.0.tex, line 977
When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gains from both deviations. A deployable certificate must retain the joint system of constraints, not select a different optimal monitor after learning wh…
Exact mathematical passage
D_1,\ldots,D_k
D_2Use in source · 2 passages
Search aliases: D_2
Use in source · Machine_Societies_v1.0.0.tex, line 977
When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gains from both deviations. A deployable certificate must retain the joint system of constraints, not select a different optimal monitor after learning wh…
Exact mathematical passage
P_{D_2}=(0,1)
d_EUse in source · 4 passages
Search aliases: d_E
Use in source · Machine_Societies_v1.0.0.tex, line 467
Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.
Exact mathematical passage
d_E=\min_{z\ne z'}\abs{\{i:E_i(z)\ne E_i(z')\}}.
D_jUse in source · 7 passages
Search aliases: D_j
Use in source · Machine_Societies_v1.0.0.tex, line 977
When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gains from both deviations. A deployable certificate must retain the joint system of constraints, not select a different optimal monitor after learning wh…
Exact mathematical passage
\E_{P_{D_j}}s-\E_{P_C}s
d_jUse in source · 7 passages
Search aliases: d_j
Use in source · Machine_Societies_v1.0.0.tex, line 1373
Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε_j=sup_z∈ Z_jd_j+1(T_jF_j(z),F_j+1T_j(z)). An exact zero defect means the representation change intertwines the declared operation. It does not mean a…
Exact mathematical passage
(Z_j,d_j)
d_{j+1}Use in source · 1 passage
Search aliases: d_j+1
Use in source · Machine_Societies_v1.0.0.tex, line 1374
Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε_j=sup_z∈ Z_jd_j+1(T_jF_j(z),F_j+1T_j(z)). An exact zero defect means the representation change intertwines the declared operation. It does not mean a…
Exact mathematical passage
\label{eq:transport-defect} \varepsilon_j=\sup_{z\in Z_j}d_{j+1}\bigl(T_jF_j(z),F_{j+1}T_j(z)\bigr).
D_kUse in source · 1 passage
Search aliases: D_k
Use in source · Machine_Societies_v1.0.0.tex, line 977
When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gains from both deviations. A deployable certificate must retain the joint system of constraints, not select a different optimal monitor after learning wh…
Exact mathematical passage
D_1,\ldots,D_k
D_mUse in source · 1 passage
Search aliases: D_m
Use in source · Machine_Societies_v1.0.0.tex, line 1390
dproposition For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second by ε_m-1. The recurrence D_m≤ L_m-1D_m-1+ε_m-1 unfolds to~[source reference].
Exact mathematical passage
D_m\le L_{m-1}D_{m-1}+\varepsilon_{m-1}
d_mUse in source · 1 passage
Search aliases: d_m
Use in source · Machine_Societies_v1.0.0.tex, line 1381
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z)) · ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source r…
Exact mathematical passage
\label{eq:cumulative-transport} \begin{split} &d_m\bigl(T_{0:m}F_0(z),F_mT_{0:m}(z)\bigr)\\ &\qquad\le\sum_{j=0}^{m-1}\varepsilon_j\prod_{k=j+1}^{m-1}L_k, \end{split}
D_{m-1}Use in source · 1 passage
Search aliases: D_m-1
Use in source · Machine_Societies_v1.0.0.tex, line 1390
dproposition For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second by ε_m-1. The recurrence D_m≤ L_m-1D_m-1+ε_m-1 unfolds to~[source reference].
Exact mathematical passage
D_m\le L_{m-1}D_{m-1}+\varepsilon_{m-1}
D_{\rm lineage}Use in source · 1 passage
Search aliases: D_rm lineage rm
Use in source · Machine_Societies_v1.0.0.tex, line 257
A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may be unidentified. They should not be replaced by zero or hidden inside a favorable weighted average.
Exact mathematical passage
\mathcal C_{\mathcal T}=(\varepsilon_{\rm pred},\varepsilon_{\rm dyn},\varepsilon_{\rm int}, \Delta_{\rm coal},R_{\rm repair},D_{\rm lineage},G_{\rm formation}).
D_\alphaUse in source · 6 passages
Search aliases: D_α alpha
Use in source · Machine_Societies_v1.0.0.tex, line 970
False-positive limits and multiple deviations. A constitutional protection may impose E_P_Cs≤α, with 0≤α≤1. The relevant discrimination capacity is then the finite linear program D_α(P_D,P_C)= max_substack0≤ s≤1 E_P_Cs≤α ( E_P_Ds- E_P_Cs). The feasible set is nonempty and compact, so a maximum exists. The preceding proof with the feasible set restricted shows that deterrence is attainable exactly when g≤ F D_α. For the example above and α=1/10, the optimum is s=(0,2/5), giving D_α=1/5, not 1/2. …
Exact mathematical passage
\label{eq:protected-discrimination} D_\alpha(P_D,P_C)= \max_{\substack{0\leq s\leq1\\\E_{P_C}s\leq\alpha}} \bigl(\E_{P_D}s-\E_{P_C}s\bigr).
\mathsf{debtor}Use in source · 1 passage
Search aliases: debtor mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 192
ubsectionTwo rules and an inherited obligation Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The exchange creates a registered obligation o=(id,debtor=j,creditor=k, quantity=1,due=e+1). At the next epoch the creditor receives two slots and the debtor n…
Exact mathematical passage
o=(\mathsf{id},\mathsf{debtor}=j,\mathsf{creditor}=k, \mathsf{quantity}=1,\mathsf{due}=e+1).
\mathsf{deliver}Use in source · 1 passage
Search aliases: deliver mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2238
[A safe amendment can destroy every safe continuation] There are four publicly distinguishable states r,ℓ,g,b. The safe set is {r,ℓ,g}. At r, profile deliver moves to g, while profile freeze moves to ℓ. At g, wait loops safely. At ℓ, the only admissible profile advances an outstanding deadline and moves to breach state b. The amendment freeze preserves the immediate safe-state predicate but not W_*. No certificate proving unbounded continuation from r can choose it. The relevant defect is loss o…
Exact mathematical passage
\mathsf{deliver}
\mathsf{Domain}Use in source · 1 passage
Search aliases: Domain mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2539
An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a component, or unavailable. Those statuses should not all become the Boolean value verified. A failed guard means that continued reliance is not justified by t…
Exact mathematical passage
\mathcal K=(\mathsf{Claim},\mathsf{Assumptions},\mathsf{Evidence}, \mathsf{Domain},\mathsf{Guard},\mathsf{Fallback},\mathsf{Revision}).
\mathsf{due}Use in source · 1 passage
Search aliases: due mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 192
ubsectionTwo rules and an inherited obligation Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The exchange creates a registered obligation o=(id,debtor=j,creditor=k, quantity=1,due=e+1). At the next epoch the creditor receives two slots and the debtor n…
Exact mathematical passage
o=(\mathsf{id},\mathsf{debtor}=j,\mathsf{creditor}=k, \mathsf{quantity}=1,\mathsf{due}=e+1).
\EUse in source · 13 passages
Search aliases: E mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 273
A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distribution, observation maps, and coalition family. It is not equivalent to any one general-purpose partial-information-decomposition statistic~[source re…
Exact mathematical passage
\mathcal R_\ell(O)=\inf_f\E\bigl[\ell(f(O),Y)\bigr].
e'Use in source · 1 passage
Search aliases: e'
Use in source · Machine_Societies_v1.0.0.tex, line 2732
[.5em] What state must persist? · The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens. [.5em] Can a due debt be erased? · The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are required to preserve pending commitments. [.5em] Who can block change? · Two registered offices block a three-of-four rule under independent approval …
Exact mathematical passage
.5em] What state must persist? & The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens.\\[.5em] Can a due debt be erased? & The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are required to preserve pending commitments.\\[.5em] Who can block change? & Two registered offices block a three-of-four rule under independent approval rights. A hidden administrator changes this calculation.\\[.5em] Can every request be safely served? & Resource conservation can hold even when private information or conflicting deadlines prevent timely service. Viability with full information is not decentralized implementability.\\[.5em] What crosses a replacement? & An authorized office handover preserves debt and credential semantics. Copying a model does not duplicate an office's budget or approval right.\\[.5em] Can two jurisdictions merge? & Their translator must preserve the agreed action semantics, permissions, obligations, resources, and evidence coordinates. Nonzero residuals remain visible until an authorized repair is found.\\[.5em] How long does the organization last? & Lifetime is first exit from a declared operational contract under a specified transition law. It is neither occupant longevity nor mere archive retention.\\ \bottomrule \end{longtable} The common event mechanism and constitutional envelope are supplied. Within that envelope, offices can choose rules, incur obligations, revise decisions, and replace occupants. This demonstrates how the theory's objects fit together. It does not prove that the envelope itself arose endogenously or that these rules are uniquely desirable. \subsection{The additional boundary, language, and knowledge contract} An admission changes the membership configuration, not merely the occupant list. A message is interpreted using its type, issuer, and epoch. A resource approval creates the specified ledger event only after its admissibility check. A debtor replacement transports the obligation monitor but does not manufacture new approval rights. A notification history records which office has received which adoption record; an available ledger entry is not automatically common knowledge. For concurrent amendments, the contract states whether ordering is significant, whether a joinable normal form is required, and which external effects remain visible. For federation, the role-label transformations must be checked around ordered loops. A compatible allocation on one invariant subset does not certify synchronization of every role. For inquiry, the commons records unresolved scheduling questions independently of which questions it chooses to fund. For founding authority, it preserves the resource owner's stipulated grant separately from its own subsequent votes. Together these clauses specify what it would mean to preserve this analytical institution across translations and replacements. They do not claim that every useful society needs this exact architecture or that the four-office example exhibits autonomous institutional formation. \subsection{Distributed realization, enforcement, and adaptive staffing} In the atomic reference commons, a unique slot token prevents two successful grants of the same slot. In its replicated variant, two local approvals can each look valid while their merged grant set violates exclusivity. The institution must therefore distinguish a provisional reservation from a committed grant and identify the primitive that prevents double execution. A convergent archive is not that primitive by itself. Accurate urgency reporting is a different problem. A committed monitor can discourage a specified profitable misreport only within the discrimination and penalty budget of Section~\ref{sec:enforcement}. Its false-positive limit, evidence access, and remedy authority remain separate. When an averaged model supplies the observation laws, the margin must include its approximation error. A fixed review schedule and a data-selected review schedule cannot share the same mixing certificate without further work. The commons also distinguishes an urgency vote from the signal supporting it. Decision-only publication can reproduce a mistaken cascade; preservation of independent evidence changes the information available to later reviewers, not the legal effect of an already committed vote. Finally, creating a specialist verifier changes the capability graph and the distribution of authority. A staffing certificate covers all protected roles jointly, while the handover procedure must still deliver credentials, pending debts, and evidence before service deadlines. These clauses supplement the reference semantics rather than claim that the society has generated its own infrastructure without assumptions. \subsection{The joint continuation contract} The bounded audit-and-service variant in Section~\ref{sec:joint-synthesis} combines three protections that cannot be checked in isolation: the common monitor must deter every listed deviation, its false-positive rate must meet the cap, and its resource use must leave enough capacity for all due work. The current public possibility set must also admit the same protocol in every compatible complete state. A continuation certificate records this common choice and all possible successors. A progress rank distinguishes eventual certified service from indefinite safe waiting. An amendment is acceptable under this particular continuation contract only when its current authority is valid and its resulting states retain certified continuations. A transport certificate additionally checks public observations, enabled profiles, safe and goal labels, and successor sets. Replacing an office, changing an audit channel, or adding a slot changes specified parts of that contract. Such a change may repair an obstruction, but does not retroactively fulfill an earlier unmet promise. \FloatBarrier \subsection{The accountability extension} Section~\ref{sec:accountable-commons} deliberately changes the audit catalogue to satisfy its pairwise privacy constraint. It separately includes an outside claim, an admitted appeal, a genuine-execution fee, and a funded final record with revocation. The resulting eight-unit settlement requirement must not be confused with the five-unit service-and-audit variant. The supplied reference trace checks one declared five-profile settlement, while the stopping theorem characterizes an entire finite ending-preserving running kernel. Both are model-relative results, not certificates for an unspecified deployment. \section{A finite algebraic example of reusable state}\label{sec:worked} The following analytical example makes the preceding distinctions concrete. Four components initially hold independent fair bits $(a,b,c,d)$. Two local workspaces expose \[ p=a\oplus b,\qquad q=c\oplus d,
e^{\varepsilon_t}Use in source · 1 passage
Search aliases: e^ε_t varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 2461
Writing λ_t=e^ε_t gives the additive parameter ∑_tε_t. The proposition requires a fixed finite number of releases and uniform conditional bounds. A data-selected stopping rule must itself be represented in the transcript or controlled by an appropriate conditional argument. Privileged side information not covered by the declared record is not protected by a certificate for that record alone. The preceding counterexample fails the conditional premise at the second release, even though both uncond…
Exact mathematical passage
\lambda_t=e^{\varepsilon_t}
e_0Use in source · 7 passages
Search aliases: e_0
Use in source · Machine_Societies_v1.0.0.tex, line 346
[Approximate closure over time] Under~[source reference], let the actual and reduced processes use the same predeclared open-loop control sequence. If their initial macro-laws are within e_0 in total variation, then at time t they are within e_t≤min{1,e_0+tε}. If each overline K_u contracts total variation by at most κ<1, then e_t≤κ^t e_0+εfrac1-κ^t1-κ. With matched macro-history policies, the weaker path-law bound min{1,e_0+tε} also holds, but the contraction claim needs a separate closed-loop …
Exact mathematical passage
e_0
E_1Use in source · 2 passages
Search aliases: E_1
Use in source · Machine_Societies_v1.0.0.tex, line 466
Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.
Exact mathematical passage
E(z)=(E_1(z),\ldots,E_n(z))
E_cUse in source · 2 passages
Search aliases: E_c
Use in source · Machine_Societies_v1.0.0.tex, line 1713
Common semantic coordinates. For each component type c∈{P,O,R,E}, let g_j^c:Z_j⟶(E_c,δ_c) be a contract observable in a common metric codomain. P records permissions over the common action alphabet; O records obligation-monitor state; R records resources in common physical or accounting units; and E records the required evidentiary and provenance structure. A change of units, action names, or obligation identifiers requires an explicitly specified alignment before these comparisons can be made.
Exact mathematical passage
g_j^c:Z_j\longrightarrow(E_c,\delta_c)
e_GUse in source · 9 passages
Search aliases: e_G
Use in source · Machine_Societies_v1.0.0.tex, line 1651
[Exact frame synchronization] Global frames exist if and only if H_γ=e_G for every closed walk. When they exist, they are unique up to common right multiplication h_i↦ h_i k. It suffices to check the rooted fundamental loops associated with one spanning tree. If g_ij=h_jh_i^-1, adjacent factors cancel in every path product, giving H_γ=h_v_kh_v_0^-1 and hence identity on a closed walk. Conversely, choose a root r, set h_r=e_G, and define h_i as the product on the unique tree path from r to i. Tre…
Exact mathematical passage
H_\gamma=e_G
\E_HUse in source · 1 passage
Search aliases: E_H E mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 1499
r · 1-r-d, 0<f≤1, r≥0, d>0, r+d≤1. Every nonfailure state has a positive-probability path to failure, so the finite substochastic chain is transient. Solving the two first-step equations gives E_Hτ=fracf+r+dfd. For f=1/10, r=2/5, and d=1/10, the expected first exit is 60 epochs. With r=0 and the other parameters fixed it is 20 epochs. These are exact values of the stated chain, not estimates of real machine populations.
Exact mathematical passage
\E_H\tau=\frac{f+r+d}{fd}.
E_iUse in source · 2 passages
Search aliases: E_i
Use in source · Machine_Societies_v1.0.0.tex, line 467
Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.
Exact mathematical passage
d_E=\min_{z\ne z'}\abs{\{i:E_i(z)\ne E_i(z')\}}.
E_JUse in source · 1 passage
Search aliases: E_J
Use in source · Machine_Societies_v1.0.0.tex, line 1888
For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling complementary private evidence can therefore reveal a fact that is not public to any participant.
Exact mathematical passage
E_J\varphi=\bigwedge_{i\in J}K_i\varphi
E_J^0Use in source · 1 passage
Search aliases: E_J^0
Use in source · Machine_Societies_v1.0.0.tex, line 1889
For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling complementary private evidence can therefore reveal a fact that is not public to any participant.
Exact mathematical passage
C_J\varphi=\bigwedge_{n\ge0}E_J^n\varphi, \qquad E_J^0\varphi=\varphi.
E_J^nUse in source · 2 passages
Search aliases: E_J^n
Use in source · Machine_Societies_v1.0.0.tex, line 1889
For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling complementary private evidence can therefore reveal a fact that is not public to any participant.
Exact mathematical passage
C_J\varphi=\bigwedge_{n\ge0}E_J^n\varphi, \qquad E_J^0\varphi=\varphi.
E_nUse in source · 2 passages
Search aliases: E_n
Use in source · Machine_Societies_v1.0.0.tex, line 466
Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.
Exact mathematical passage
E(z)=(E_1(z),\ldots,E_n(z))
\E_{\overline P_C}Use in source · 1 passage
Search aliases: E_overline P_C E overline mathbb overline
Use in source · Machine_Societies_v1.0.0.tex, line 1584
rline P_D(o)>overline P_C(o) and put overline d=d_TV(overline P_D,overline P_C). In the model of Section~[source reference], this fixed policy makes compliance weakly preferable whenever g≤ F(overline d-ε_C-ε_D). Conversely, if g>Fmin{1,overline d+ε_C+ε_D}, no observation-based sanction with penalty F deters the deviation. Also, E_P_Cs^⋆≤ E_overline P_Cs^⋆+ε_C. The expectation of a [0,1]-valued function changes by at most total variation. Hence the true difference of sanction probabilities is at…
Exact mathematical passage
\E_{P_C}s^\star\leq\E_{\overline P_C}s^\star+\varepsilon_C.
\E_{P_C}Use in source · 8 passages
Search aliases: E_P_C E mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 940
Under these assumptions the deviation's net expected advantage is g-F( E_P_Ds- E_P_Cs). Compliance is weakly preferable exactly when this quantity is nonpositive. Equality permits indifference and does not predict which action will be selected.
Exact mathematical passage
\label{eq:deterrence-gap} g-F\bigl(\E_{P_D}s-\E_{P_C}s\bigr).
\E_{P_D}Use in source · 5 passages
Search aliases: E_P_D E mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 940
Under these assumptions the deviation's net expected advantage is g-F( E_P_Ds- E_P_Cs). Compliance is weakly preferable exactly when this quantity is nonpositive. Equality permits indifference and does not predict which action will be selected.
Exact mathematical passage
\label{eq:deterrence-gap} g-F\bigl(\E_{P_D}s-\E_{P_C}s\bigr).
\E_{P_{D_j}}Use in source · 1 passage
Search aliases: E_P_D_j E mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 977
When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gains from both deviations. A deployable certificate must retain the joint system of constraints, not select a different optimal monitor after learning wh…
Exact mathematical passage
\E_{P_{D_j}}s-\E_{P_C}s
E_tUse in source · 2 passages
Search aliases: E_t
Use in source · Machine_Societies_v1.0.0.tex, line 641
To formalize an operative institution, augment the microstate of Section~[source reference] with a distinguished component I_t. In a finite setting we may describe it as I_t=(N_t,P_t,E_t,A_t,L_t), where N specifies norms and role obligations, P permissions and capabilities, E enforcement and conflict procedures, A admissible amendments, and L records of decisions and their provenance. The components need not be physically stored together, and their representation is not unique. What matters is t…
Exact mathematical passage
\label{eq:institutional-state} I_t=(N_t,P_t,E_t,A_t,L_t),
e_TUse in source · 1 passage
Search aliases: e_T
Use in source · Machine_Societies_v1.0.0.tex, line 497
[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.
Exact mathematical passage
e_T\leq e_0\prod_{t=0}^{T-1}a_t+ \sum_{j=0}^{T-1}b_j\prod_{t=j+1}^{T-1}a_t.
e_tUse in source · 9 passages
Search aliases: e_t
Use in source · Machine_Societies_v1.0.0.tex, line 347
[Approximate closure over time] Under~[source reference], let the actual and reduced processes use the same predeclared open-loop control sequence. If their initial macro-laws are within e_0 in total variation, then at time t they are within e_t≤min{1,e_0+tε}. If each overline K_u contracts total variation by at most κ<1, then e_t≤κ^t e_0+εfrac1-κ^t1-κ. With matched macro-history policies, the weaker path-law bound min{1,e_0+tε} also holds, but the contraction claim needs a separate closed-loop …
Exact mathematical passage
e_t\leq\min\{1,e_0+t\varepsilon\}.
e_{t+1}Use in source · 3 passages
Search aliases: e_t+1
Use in source · Machine_Societies_v1.0.0.tex, line 357
o-history policies, the weaker path-law bound min{1,e_0+tε} also holds, but the contraction claim needs a separate closed-loop contraction assumption. Averaging~[source reference] over the actual microstate law shows that its next projected law is within ε of the current projected law evolved by overline K_u. Kernel nonexpansiveness gives e_t+1≤ε+e_t. Under contraction it gives e_t+1≤ε+κ e_t; iterate. For history policies, couple the histories while they agree. The controls can then be coupled t…
Exact mathematical passage
e_{t+1}\leq\varepsilon+e_t
E_{\{A,B\}}Use in source · 1 passage
Search aliases: E_{A,B}
Use in source · Machine_Societies_v1.0.0.tex, line 1907
[Everyone knows, but it is not common knowledge] Let W={0,1,2} and let φ hold exactly at 0,1. Agent A has partition {0,1}mid{2}, and agent B has partition {0}mid{1,2}. At world 0, both agents know φ. However, the chain 0∼_A1∼_B2 reaches a world where it is false. Thus E_{A,B}φ holds at 0 but C_{A,B}φ does not.
Exact mathematical passage
E_{\{A,B\}}\varphi
\E_\muUse in source · 1 passage
Search aliases: E_μ E mu mathbb mu
Use in source · Machine_Societies_v1.0.0.tex, line 1482
[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random variable gives Eτ=∑_t≥0 P(τ>t). Under ρ(Q)<1, the matrix geometric series converges to (Id-Q)^-1, giving the result.
Exact mathematical passage
\E_\mu[\tau]=\mu(\mathrm{Id}-Q)^{-1}\mathbf1.
\mathrm{eff}Use in source · 1 passage
Search aliases: eff mathrm
Use in source · Machine_Societies_v1.0.0.tex, line 1798
ginequation Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equal-variance estimators, the variance-equivalent effective sample size is n_eff=fracn1+(n-1)ρ. If the errors are perfectly correlated, adding copies does not reduce variance. This equation does not give a general probability that a majority is right, and real error structures n…
Exact mathematical passage
n_{\mathrm{eff}}=\frac{n}{1+(n-1)\rho}.
\mathsf{Effect}Use in source · 2 passages
Search aliases: Effect mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2343
A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. Standing specifies who may initiate a challenge and on which grounds. Remedy specifies the actions an authorized review can actually require. The cont…
Exact mathematical passage
\mathcal B=(Z_{\rm ext},O_{\rm ext},\mathsf{Auth},\mathsf{Effect}, \mathsf{Standing},\mathsf{Remedy}).
\mathsf{Evidence}Use in source · 1 passage
Search aliases: Evidence mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2539
An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a component, or unavailable. Those statuses should not all become the Boolean value verified. A failed guard means that continued reliance is not justified by t…
Exact mathematical passage
\mathcal K=(\mathsf{Claim},\mathsf{Assumptions},\mathsf{Evidence}, \mathsf{Domain},\mathsf{Guard},\mathsf{Fallback},\mathsf{Revision}).
\mathbb FUse in source · 2 passages
Search aliases: F mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 1605
Consider an undirected graph G=(V,E) whose vertices are institutional positions. Each position carries a binary coordinate x_i∈ F_2 and each edge {i,j} imposes the relative constraint x_i⊕ x_j=b_ij, b_ij∈ F_2. The coordinates might represent a toy authorization bit or a binary convention, but the mathematical model itself makes no claim about a realistic legal code.
Exact mathematical passage
x_i\in\mathbb F_2
\mathbf FUse in source · 1 passage
Search aliases: F mathbf
Use in source · Machine_Societies_v1.0.0.tex, line 1328
For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy follows from admitting discharge in this particular contract.
Exact mathematical passage
\mathbf G\bigl(\mathsf{created}(o)\Rightarrow \mathbf F_{\leq D}\mathsf{settled}(o)\bigr).
F^AUse in source · 1 passage
Search aliases: F^A
Use in source · Machine_Societies_v1.0.0.tex, line 787
Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated office names, configuration epochs, and reference records. The translation of a message sequence is constructed step by step from the evolving source st…
Exact mathematical passage
F^A,F^B
F^BUse in source · 1 passage
Search aliases: F^B
Use in source · Machine_Societies_v1.0.0.tex, line 787
Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated office names, configuration epochs, and reference records. The translation of a message sequence is constructed step by step from the evolving source st…
Exact mathematical passage
F^A,F^B
F_0Use in source · 1 passage
Search aliases: F_0
Use in source · Machine_Societies_v1.0.0.tex, line 1381
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z)) · ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source r…
Exact mathematical passage
\label{eq:cumulative-transport} \begin{split} &d_m\bigl(T_{0:m}F_0(z),F_mT_{0:m}(z)\bigr)\\ &\qquad\le\sum_{j=0}^{m-1}\varepsilon_j\prod_{k=j+1}^{m-1}L_k, \end{split}
F_0^aUse in source · 1 passage
Search aliases: F_0^a
Use in source · Machine_Societies_v1.0.0.tex, line 1751
-Lipschitz. Then ε_rm dyn(UT) ≤ Lε_rm dyn(T)+ε_rm dyn(U), and, separately for every c∈{P,O,R,E}, ε_c(UT)≤ε_c(T)+ε_c(U). Consequently translations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continuity of U, bound the two terms by Lε_rm dyn(T) and ε_rm dyn(U). For an observable type c, insert g_1^cT(z) between g_2^cUT(z) and g_0^c(z). Both compari…
Exact mathematical passage
UTF_0^a(z)
F_1^aUse in source · 1 passage
Search aliases: F_1^a
Use in source · Machine_Societies_v1.0.0.tex, line 1751
es. Suppose U is L-Lipschitz. Then ε_rm dyn(UT) ≤ Lε_rm dyn(T)+ε_rm dyn(U), and, separately for every c∈{P,O,R,E}, ε_c(UT)≤ε_c(T)+ε_c(U). Consequently translations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continuity of U, bound the two terms by Lε_rm dyn(T) and ε_rm dyn(U). For an observable type c, insert g_1^cT(z) between g_2^cUT(z) and g_0^…
Exact mathematical passage
UF_1^aT(z)
\mathbb F_2Use in source · 2 passages
Search aliases: F_2 mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 1605
Consider an undirected graph G=(V,E) whose vertices are institutional positions. Each position carries a binary coordinate x_i∈ F_2 and each edge {i,j} imposes the relative constraint x_i⊕ x_j=b_ij, b_ij∈ F_2. The coordinates might represent a toy authorization bit or a binary convention, but the mathematical model itself makes no claim about a realistic legal code.
Exact mathematical passage
x_i\in\mathbb F_2
F_2^aUse in source · 1 passage
Search aliases: F_2^a
Use in source · Machine_Societies_v1.0.0.tex, line 1751
ε_rm dyn(UT) ≤ Lε_rm dyn(T)+ε_rm dyn(U), and, separately for every c∈{P,O,R,E}, ε_c(UT)≤ε_c(T)+ε_c(U). Consequently translations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continuity of U, bound the two terms by Lε_rm dyn(T) and ε_rm dyn(U). For an observable type c, insert g_1^cT(z) between g_2^cUT(z) and g_0^c(z). Both comparisons occur in the…
Exact mathematical passage
F_2^aUT(z)
F_iUse in source · 1 passage
Search aliases: F_i
Use in source · Machine_Societies_v1.0.0.tex, line 1630
Even when a global static state exists, a federation may fail dynamically. Local rule applications could commute in one context and conflict in another. A translator T_ij:Z_i→ Z_j should preserve the operations relevant to the interaction, approximately satisfying T_ijF_i≃ F_jT_ij. A federation requires both representational compatibility and admissible operational coordination. Static agreement alone does not define an executable constitution. Nor does faithful gluing prove that the institution…
Exact mathematical passage
T_{ij}F_i\simeq F_jT_{ij}
F_i^aUse in source · 1 passage
Search aliases: F_i^a
Use in source · Machine_Societies_v1.0.0.tex, line 1721
For a translation T:Z_i→ Z_j, define ε_rm dyn(T) · =sup_z,ad_j(TF_i^a(z),F_j^aT(z)), ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The symmetric permission residual measures both changes and is delib…
Exact mathematical passage
\begin{split} \varepsilon_{\rm dyn}(T) &=\sup_{z,a}d_j\bigl(TF_i^a(z),F_j^aT(z)\bigr),\\ \varepsilon_c(T) &=\sup_z\delta_c\bigl(g_j^cT(z),g_i^c(z)\bigr) \quad(c=P,O,R,E). \end{split}
F_jUse in source · 3 passages
Search aliases: F_j
Use in source · Machine_Societies_v1.0.0.tex, line 1373
Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε_j=sup_z∈ Z_jd_j+1(T_jF_j(z),F_j+1T_j(z)). An exact zero defect means the representation change intertwines the declared operation. It does not mean a…
Exact mathematical passage
F_j:Z_j\to Z_j
F_{j+1}Use in source · 1 passage
Search aliases: F_j+1
Use in source · Machine_Societies_v1.0.0.tex, line 1374
Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε_j=sup_z∈ Z_jd_j+1(T_jF_j(z),F_j+1T_j(z)). An exact zero defect means the representation change intertwines the declared operation. It does not mean a…
Exact mathematical passage
\label{eq:transport-defect} \varepsilon_j=\sup_{z\in Z_j}d_{j+1}\bigl(T_jF_j(z),F_{j+1}T_j(z)\bigr).
F_j^aUse in source · 2 passages
Search aliases: F_j^a
Use in source · Machine_Societies_v1.0.0.tex, line 1709
We formalize a restricted compositional calculus, rather than assuming that every mismatch is measured in one natural unit. Three realizations j=0,1,2 have metric state spaces (Z_j,d_j) and a common finite action alphabet A. For algebraic comparison, each action has a total update F_j^a: a disabled action goes to an explicit rejection state. Translations map rejection to rejection. A separate permission predicate records which actions are genuinely admitted. Totalization is a bookkeeping device …
Exact mathematical passage
F_j^a
F_{m-1}Use in source · 1 passage
Search aliases: F_m-1
Use in source · Machine_Societies_v1.0.0.tex, line 1390
· ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second by ε_m-1. The recurrence D_m≤ L_m-1D_m-1+ε_m-1 unfolds to~[source reference].
Exact mathematical passage
T_{m-1}F_{m-1}T_{0:m-1}(z)
F_TUse in source · 3 passages
Search aliases: F_T
Use in source · Machine_Societies_v1.0.0.tex, line 2806
A deterministic microscopic realization. For any permitted macro-operation T, write (p',q')=T(p,q) and define F_T(a,b,c,d)=(a, a⊕ p', c, c⊕ q'). Then the two local XORs of F_T(a,b,c,d) are exactly (p',q'). Consequently π F_T=Tπ for every microstate, where π(a,b,c,d)=(p,q).
Exact mathematical passage
F_T(a,b,c,d)=(a,\ a\oplus p',\ c,\ c\oplus q').
F_uUse in source · 5 passages
Search aliases: F_u
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\mathbf F_{\leq D}Use in source · 1 passage
Search aliases: F_≤ D mathbf leq
Use in source · Machine_Societies_v1.0.0.tex, line 1328
For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy follows from admitting discharge in this particular contract.
Exact mathematical passage
\mathbf G\bigl(\mathsf{created}(o)\Rightarrow \mathbf F_{\leq D}\mathsf{settled}(o)\bigr).
\mathsf{Fallback}Use in source · 1 passage
Search aliases: Fallback mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2539
An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a component, or unavailable. Those statuses should not all become the Boolean value verified. A failed guard means that continued reliance is not justified by t…
Exact mathematical passage
\mathcal K=(\mathsf{Claim},\mathsf{Assumptions},\mathsf{Evidence}, \mathsf{Domain},\mathsf{Guard},\mathsf{Fallback},\mathsf{Revision}).
\FloatBarrierUse in source · 1 passage
Search aliases: FloatBarrier
Use in source · Machine_Societies_v1.0.0.tex, line 2732
[.5em] What state must persist? · The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens. [.5em] Can a due debt be erased? · The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are required to preserve pending commitments. [.5em] Who can block change? · Two registered offices block a three-of-four rule under independent approval …
Exact mathematical passage
.5em] What state must persist? & The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens.\\[.5em] Can a due debt be erased? & The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are required to preserve pending commitments.\\[.5em] Who can block change? & Two registered offices block a three-of-four rule under independent approval rights. A hidden administrator changes this calculation.\\[.5em] Can every request be safely served? & Resource conservation can hold even when private information or conflicting deadlines prevent timely service. Viability with full information is not decentralized implementability.\\[.5em] What crosses a replacement? & An authorized office handover preserves debt and credential semantics. Copying a model does not duplicate an office's budget or approval right.\\[.5em] Can two jurisdictions merge? & Their translator must preserve the agreed action semantics, permissions, obligations, resources, and evidence coordinates. Nonzero residuals remain visible until an authorized repair is found.\\[.5em] How long does the organization last? & Lifetime is first exit from a declared operational contract under a specified transition law. It is neither occupant longevity nor mere archive retention.\\ \bottomrule \end{longtable} The common event mechanism and constitutional envelope are supplied. Within that envelope, offices can choose rules, incur obligations, revise decisions, and replace occupants. This demonstrates how the theory's objects fit together. It does not prove that the envelope itself arose endogenously or that these rules are uniquely desirable. \subsection{The additional boundary, language, and knowledge contract} An admission changes the membership configuration, not merely the occupant list. A message is interpreted using its type, issuer, and epoch. A resource approval creates the specified ledger event only after its admissibility check. A debtor replacement transports the obligation monitor but does not manufacture new approval rights. A notification history records which office has received which adoption record; an available ledger entry is not automatically common knowledge. For concurrent amendments, the contract states whether ordering is significant, whether a joinable normal form is required, and which external effects remain visible. For federation, the role-label transformations must be checked around ordered loops. A compatible allocation on one invariant subset does not certify synchronization of every role. For inquiry, the commons records unresolved scheduling questions independently of which questions it chooses to fund. For founding authority, it preserves the resource owner's stipulated grant separately from its own subsequent votes. Together these clauses specify what it would mean to preserve this analytical institution across translations and replacements. They do not claim that every useful society needs this exact architecture or that the four-office example exhibits autonomous institutional formation. \subsection{Distributed realization, enforcement, and adaptive staffing} In the atomic reference commons, a unique slot token prevents two successful grants of the same slot. In its replicated variant, two local approvals can each look valid while their merged grant set violates exclusivity. The institution must therefore distinguish a provisional reservation from a committed grant and identify the primitive that prevents double execution. A convergent archive is not that primitive by itself. Accurate urgency reporting is a different problem. A committed monitor can discourage a specified profitable misreport only within the discrimination and penalty budget of Section~\ref{sec:enforcement}. Its false-positive limit, evidence access, and remedy authority remain separate. When an averaged model supplies the observation laws, the margin must include its approximation error. A fixed review schedule and a data-selected review schedule cannot share the same mixing certificate without further work. The commons also distinguishes an urgency vote from the signal supporting it. Decision-only publication can reproduce a mistaken cascade; preservation of independent evidence changes the information available to later reviewers, not the legal effect of an already committed vote. Finally, creating a specialist verifier changes the capability graph and the distribution of authority. A staffing certificate covers all protected roles jointly, while the handover procedure must still deliver credentials, pending debts, and evidence before service deadlines. These clauses supplement the reference semantics rather than claim that the society has generated its own infrastructure without assumptions. \subsection{The joint continuation contract} The bounded audit-and-service variant in Section~\ref{sec:joint-synthesis} combines three protections that cannot be checked in isolation: the common monitor must deter every listed deviation, its false-positive rate must meet the cap, and its resource use must leave enough capacity for all due work. The current public possibility set must also admit the same protocol in every compatible complete state. A continuation certificate records this common choice and all possible successors. A progress rank distinguishes eventual certified service from indefinite safe waiting. An amendment is acceptable under this particular continuation contract only when its current authority is valid and its resulting states retain certified continuations. A transport certificate additionally checks public observations, enabled profiles, safe and goal labels, and successor sets. Replacing an office, changing an audit channel, or adding a slot changes specified parts of that contract. Such a change may repair an obstruction, but does not retroactively fulfill an earlier unmet promise. \FloatBarrier \subsection{The accountability extension} Section~\ref{sec:accountable-commons} deliberately changes the audit catalogue to satisfy its pairwise privacy constraint. It separately includes an outside claim, an admitted appeal, a genuine-execution fee, and a funded final record with revocation. The resulting eight-unit settlement requirement must not be confused with the five-unit service-and-audit variant. The supplied reference trace checks one declared five-profile settlement, while the stopping theorem characterizes an entire finite ending-preserving running kernel. Both are model-relative results, not certificates for an unspecified deployment. \section{A finite algebraic example of reusable state}\label{sec:worked} The following analytical example makes the preceding distinctions concrete. Four components initially hold independent fair bits $(a,b,c,d)$. Two local workspaces expose \[ p=a\oplus b,\qquad q=c\oplus d,
\mathsf{freeze}Use in source · 2 passages
Search aliases: freeze mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2238
[A safe amendment can destroy every safe continuation] There are four publicly distinguishable states r,ℓ,g,b. The safe set is {r,ℓ,g}. At r, profile deliver moves to g, while profile freeze moves to ℓ. At g, wait loops safely. At ℓ, the only admissible profile advances an outstanding deadline and moves to breach state b. The amendment freeze preserves the immediate safe-state predicate but not W_*. No certificate proving unbounded continuation from r can choose it. The relevant defect is loss o…
Exact mathematical passage
\mathsf{freeze}
\FtwoUse in source · 1 passage
Search aliases: Ftwo F_2 mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 479
For z=(p,q)∈ F_2^2, the encoding (p,q,p⊕ q) has distance two. Any two components reconstruct the state, and no one component reconstructs all of it. It tolerates one known erasure. It does not correct an arbitrary one-component corruption at an unknown location. Confusing these fault models overstates robustness.
Exact mathematical passage
z=(p,q)\in\Ftwo^2
\Ftwo^2Use in source · 1 passage
Search aliases: Ftwo^2 F_2^2 Ftwo mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 479
For z=(p,q)∈ F_2^2, the encoding (p,q,p⊕ q) has distance two. Any two components reconstruct the state, and no one component reconstructs all of it. It tolerates one known erasure. It does not correct an arbitrary one-component corruption at an unknown location. Confusing these fault models overstates robustness.
Exact mathematical passage
z=(p,q)\in\Ftwo^2
\mathbf GUse in source · 1 passage
Search aliases: G mathbf
Use in source · Machine_Societies_v1.0.0.tex, line 1328
For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy follows from admitting discharge in this particular contract.
Exact mathematical passage
\mathbf G\bigl(\mathsf{created}(o)\Rightarrow \mathbf F_{\leq D}\mathsf{settled}(o)\bigr).
\mathcal GUse in source · 2 passages
Search aliases: G mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 2195
Let G⊆ S_rm safe be a stipulated completion set. The target is certified completion, meaning that the current possibility set lies wholly in G. Let G={B∈ W_*:B⊆ G}, A_*(B)={u∈ A(B):Succ(B,u)⊆ W_*}. Define R_0= G and R_k+1=R_k∪ {B∈ W_*:∃ u∈ A_*(B), Succ(B,u)⊆ R_k}. The increasing sequence stabilizes at R_*. For B∈ R_*, let r(B) be its first index of entry.
Exact mathematical passage
\mathcal G=\{B\in W_*:B\subseteq G\}, \quad A_*(B)=\{u\in A(B):\operatorname{Succ}(B,u)\subseteq W_*\}.
g_0^cUse in source · 1 passage
Search aliases: g_0^c
Use in source · Machine_Societies_v1.0.0.tex, line 1751
zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continuity of U, bound the two terms by Lε_rm dyn(T) and ε_rm dyn(U). For an observable type c, insert g_1^cT(z) between g_2^cUT(z) and g_0^c(z). Both comparisons occur in the same metric space E_c, so their bounds add directly. Taking suprema proves the claims.
Exact mathematical passage
g_0^c(z)
g_1Use in source · 3 passages
Search aliases: g_1
Use in source · Machine_Societies_v1.0.0.tex, line 859
dination environments in which a durable institution has strictly positive aggregate discounted benefit but is rejected by a rational agent who unilaterally bears setup cost and has veto power, assuming quasilinear payoffs, no transfers, and no additional benefits to that agent. Let the setup cost be B=10, borne entirely by agent 1, and let per-period gains be g_1=0, g_2=2 with δ=.9. The aggregate advantage is -10+2/(1-.9)=10>0, whereas agent 1's payoff difference is -10. If agent 1 can reject a…
Exact mathematical passage
g_1=0
g_1^cUse in source · 1 passage
Search aliases: g_1^c
Use in source · Machine_Societies_v1.0.0.tex, line 1751
). Consequently translations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continuity of U, bound the two terms by Lε_rm dyn(T) and ε_rm dyn(U). For an observable type c, insert g_1^cT(z) between g_2^cUT(z) and g_0^c(z). Both comparisons occur in the same metric space E_c, so their bounds add directly. Taking suprema proves the claims.
Exact mathematical passage
g_1^cT(z)
g_2Use in source · 3 passages
Search aliases: g_2
Use in source · Machine_Societies_v1.0.0.tex, line 859
environments in which a durable institution has strictly positive aggregate discounted benefit but is rejected by a rational agent who unilaterally bears setup cost and has veto power, assuming quasilinear payoffs, no transfers, and no additional benefits to that agent. Let the setup cost be B=10, borne entirely by agent 1, and let per-period gains be g_1=0, g_2=2 with δ=.9. The aggregate advantage is -10+2/(1-.9)=10>0, whereas agent 1's payoff difference is -10. If agent 1 can reject and has no…
Exact mathematical passage
g_2=2
g_2^cUse in source · 1 passage
Search aliases: g_2^c
Use in source · Machine_Societies_v1.0.0.tex, line 1751
ranslations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continuity of U, bound the two terms by Lε_rm dyn(T) and ε_rm dyn(U). For an observable type c, insert g_1^cT(z) between g_2^cUT(z) and g_0^c(z). Both comparisons occur in the same metric space E_c, so their bounds add directly. Taking suprema proves the claims.
Exact mathematical passage
g_2^cUT(z)
g_{AB}Use in source · 1 passage
Search aliases: g_AB
Use in source · Machine_Societies_v1.0.0.tex, line 1661
A three-jurisdiction example. Take G=S_3, acting on three role labels. Let the translations on the oriented triangle A→ B→ C→ A be g_AB=(12), g_BC=(23), g_CA=e_G. Every edge is a valid bijection. With the specified composition convention, the round trip is (23)(12)≠ e_G. Thus no single assignment of complete role frames realizes all three interfaces. Replacing the final edge by ((23)(12))^-1 repairs this loop, but whether that edit is authorized is a separate institutional question.
Exact mathematical passage
g_{AB}=(12),\qquad g_{BC}=(23),\qquad g_{CA}=e_G.
g_{BC}Use in source · 1 passage
Search aliases: g_BC
Use in source · Machine_Societies_v1.0.0.tex, line 1661
A three-jurisdiction example. Take G=S_3, acting on three role labels. Let the translations on the oriented triangle A→ B→ C→ A be g_AB=(12), g_BC=(23), g_CA=e_G. Every edge is a valid bijection. With the specified composition convention, the round trip is (23)(12)≠ e_G. Thus no single assignment of complete role frames realizes all three interfaces. Replacing the final edge by ((23)(12))^-1 repairs this loop, but whether that edit is authorized is a separate institutional question.
Exact mathematical passage
g_{AB}=(12),\qquad g_{BC}=(23),\qquad g_{CA}=e_G.
g_{CA}Use in source · 1 passage
Search aliases: g_CA
Use in source · Machine_Societies_v1.0.0.tex, line 1661
A three-jurisdiction example. Take G=S_3, acting on three role labels. Let the translations on the oriented triangle A→ B→ C→ A be g_AB=(12), g_BC=(23), g_CA=e_G. Every edge is a valid bijection. With the specified composition convention, the round trip is (23)(12)≠ e_G. Thus no single assignment of complete role frames realizes all three interfaces. Replacing the final edge by ((23)(12))^-1 repairs this loop, but whether that edit is authorized is a separate institutional question.
Exact mathematical passage
g_{AB}=(12),\qquad g_{BC}=(23),\qquad g_{CA}=e_G.
g_i^cUse in source · 1 passage
Search aliases: g_i^c
Use in source · Machine_Societies_v1.0.0.tex, line 1721
For a translation T:Z_i→ Z_j, define ε_rm dyn(T) · =sup_z,ad_j(TF_i^a(z),F_j^aT(z)), ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The symmetric permission residual measures both changes and is delib…
Exact mathematical passage
\begin{split} \varepsilon_{\rm dyn}(T) &=\sup_{z,a}d_j\bigl(TF_i^a(z),F_j^aT(z)\bigr),\\ \varepsilon_c(T) &=\sup_z\delta_c\bigl(g_j^cT(z),g_i^c(z)\bigr) \quad(c=P,O,R,E). \end{split}
g_{ij}Use in source · 12 passages
Search aliases: g_ij
Use in source · Machine_Societies_v1.0.0.tex, line 1641
Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
g_{ij}\in G
g_{ij}^{-1}Use in source · 1 passage
Search aliases: g_ij^-1
Use in source · Machine_Societies_v1.0.0.tex, line 1641
Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
g_{ji}=g_{ij}^{-1}
g_jUse in source · 5 passages
Search aliases: g_j
Use in source · Machine_Societies_v1.0.0.tex, line 1007
The common-monitor problem. Let O be a finite nonempty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous…
Exact mathematical passage
g_j\geq0
g_j^cUse in source · 2 passages
Search aliases: g_j^c
Use in source · Machine_Societies_v1.0.0.tex, line 1713
Common semantic coordinates. For each component type c∈{P,O,R,E}, let g_j^c:Z_j⟶(E_c,δ_c) be a contract observable in a common metric codomain. P records permissions over the common action alphabet; O records obligation-monitor state; R records resources in common physical or accounting units; and E records the required evidentiary and provenance structure. A change of units, action names, or obligation identifiers requires an explicitly specified alignment before these comparisons can be made.
Exact mathematical passage
g_j^c:Z_j\longrightarrow(E_c,\delta_c)
g_{ji}Use in source · 1 passage
Search aliases: g_ji
Use in source · Machine_Societies_v1.0.0.tex, line 1641
Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
g_{ji}=g_{ij}^{-1}
G_{\rm formation}Use in source · 1 passage
Search aliases: G_rm formation rm
Use in source · Machine_Societies_v1.0.0.tex, line 257
A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may be unidentified. They should not be replaced by zero or hidden inside a favorable weighted average.
Exact mathematical passage
\mathcal C_{\mathcal T}=(\varepsilon_{\rm pred},\varepsilon_{\rm dyn},\varepsilon_{\rm int}, \Delta_{\rm coal},R_{\rm repair},D_{\rm lineage},G_{\rm formation}).
G_{\rm settle}Use in source · 1 passage
Search aliases: G_rm settle rm
Use in source · Machine_Societies_v1.0.0.tex, line 2498
Let G_rm suspend require that the declared external capabilities are inactive and that the pending-claim record remains available. Let G_rm settle additionally require that every covered claim has been fulfilled, discharged under the stipulated authority, or transferred to an authorized successor that has accepted it, and that the required final record has been committed. A retained unresolved claim can be compatible with suspension but not with this stronger settlement target. Neither target by…
Exact mathematical passage
G_{\rm settle}
G_{\rm suspend}Use in source · 1 passage
Search aliases: G_rm suspend rm
Use in source · Machine_Societies_v1.0.0.tex, line 2498
Let G_rm suspend require that the declared external capabilities are inactive and that the pending-claim record remains available. Let G_rm settle additionally require that every covered claim has been fulfilled, discharged under the stipulated authority, or transferred to an authorized successor that has accepted it, and that the required final record has been committed. A retained unresolved claim can be compatible with suspension but not with this stronger settlement target. Neither target by…
Exact mathematical passage
G_{\rm suspend}
g_{v_0v_1}Use in source · 1 passage
Search aliases: g_v_0v_1
Use in source · Machine_Societies_v1.0.0.tex, line 1642
Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
H_\gamma=g_{v_{k-1}v_k}\cdots g_{v_0v_1}.
g_{v_{k-1}v_k}Use in source · 1 passage
Search aliases: g_v_k-1v_k
Use in source · Machine_Societies_v1.0.0.tex, line 1642
Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
H_\gamma=g_{v_{k-1}v_k}\cdots g_{v_0v_1}.
g_\lambdaUse in source · 6 passages
Search aliases: g_λ lambda
Use in source · Machine_Societies_v1.0.0.tex, line 1019
For a probability vector λ∈Δ_m, set P_λ=∑_jλ_jP_j and g_λ=∑_jλ_jg_j. Write V_α(P,P_C)=max_s∈ M_α⟨ P-P_C,s⟩. This is the same constrained observation advantage used in the single-deviation analysis, now evaluated on mixtures of deviation laws.
Exact mathematical passage
g_\lambda=\sum_j\lambda_jg_j
\mathsf{Guard}Use in source · 1 passage
Search aliases: Guard mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2539
An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a component, or unavailable. Those statuses should not all become the Boolean value verified. A failed guard means that continued reliance is not justified by t…
Exact mathematical passage
\mathcal K=(\mathsf{Claim},\mathsf{Assumptions},\mathsf{Evidence}, \mathsf{Domain},\mathsf{Guard},\mathsf{Fallback},\mathsf{Revision}).
\mathcal HUse in source · 4 passages
Search aliases: H mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 1975
The question can be stated without selecting a universal theory of legitimacy. Let H be a set of possible founding histories, let a: H→{0,1} be an externally stipulated predicate such as ``the resource owner authorized this founding event,'' and let R: H→ R be the record available to the institution. The predicate models one particular authorization criterion, not moral legitimacy in its entirety.
Exact mathematical passage
\mathcal H
h'Use in source · 1 passage
Search aliases: h'
Use in source · Machine_Societies_v1.0.0.tex, line 1981
nd only if a is constant on every nonempty fibre of R. This is a statement of informational determination, not necessarily of computability. If two histories with different labels induce identical distributions of every record available to an evaluator, no evaluator restricted to those records distinguishes them with accuracy above 1/2 under equal priors. If R(h)=R(h'), any function of the record gives the same answer on both histories, proving necessity. If the label is constant on every nonemp…
Exact mathematical passage
R(h)=R(h')
h_0Use in source · 1 passage
Search aliases: h_0
Use in source · Machine_Societies_v1.0.0.tex, line 2352
[Exact internal accounting with an omitted external claim] Four execution slots exist. The internal state contains four member requests and a rule granting one slot to each. In realization h_0, all four slots are available to that institution. In realization h_1, an external owner has reserved one slot for a nonmember, and has authorized the institution to use only the other three. Suppose the reservation is absent from every internal record. The same four grants satisfy the internal predicate ∑…
Exact mathematical passage
h_0
H_1Use in source · 2 passages
Search aliases: H_1
Use in source · Machine_Societies_v1.0.0.tex, line 1307
Why separate marginal ignorance is insufficient. [Local ignorance with a perfectly correct joint action] Let U be a fair bit. Agent 1 observes H_1=U and agent 2 observes H_2=U⊕θ. Each individual observation has the same law for both values of θ. Nevertheless, the joint action (a_1,a_2)=(H_1,H_2) always satisfies a_1⊕ a_2=θ. The joint observation laws differ and have disjoint supports. Equal marginal laws therefore do not imply the hypothesis of Theorem~[source reference].
Exact mathematical passage
H_1=U
h_1Use in source · 2 passages
Search aliases: h_1
Use in source · Machine_Societies_v1.0.0.tex, line 2352
[Exact internal accounting with an omitted external claim] Four execution slots exist. The internal state contains four member requests and a rule granting one slot to each. In realization h_0, all four slots are available to that institution. In realization h_1, an external owner has reserved one slot for a nonmember, and has authorized the institution to use only the other three. Suppose the reservation is absent from every internal record. The same four grants satisfy the internal predicate ∑…
Exact mathematical passage
h_1
H_2Use in source · 2 passages
Search aliases: H_2
Use in source · Machine_Societies_v1.0.0.tex, line 1307
Why separate marginal ignorance is insufficient. [Local ignorance with a perfectly correct joint action] Let U be a fair bit. Agent 1 observes H_1=U and agent 2 observes H_2=U⊕θ. Each individual observation has the same law for both values of θ. Nevertheless, the joint action (a_1,a_2)=(H_1,H_2) always satisfies a_1⊕ a_2=θ. The joint observation laws differ and have disjoint supports. Equal marginal laws therefore do not imply the hypothesis of Theorem~[source reference].
Exact mathematical passage
H_2=U\oplus\theta
h_iUse in source · 7 passages
Search aliases: h_i
Use in source · Machine_Societies_v1.0.0.tex, line 1645
ph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
h_i\in G
h_i^{-1}Use in source · 4 passages
Search aliases: h_i^-1
Use in source · Machine_Societies_v1.0.0.tex, line 1646
i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
g_{ij}=h_jh_i^{-1}\qquad\text{on every edge}.
h_jUse in source · 4 passages
Search aliases: h_j
Use in source · Machine_Societies_v1.0.0.tex, line 1646
i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
g_{ij}=h_jh_i^{-1}\qquad\text{on every edge}.
h_j^{-1}Use in source · 2 passages
Search aliases: h_j^-1
Use in source · Machine_Societies_v1.0.0.tex, line 1654
cancel in every path product, giving H_γ=h_v_kh_v_0^-1 and hence identity on a closed walk. Conversely, choose a root r, set h_r=e_G, and define h_i as the product on the unique tree path from r to i. Tree edges satisfy the required identity. For any non-tree edge i→ j, the rooted loop consisting of the path to i, that edge, and the reversed path from j has product h_j^-1g_ijh_i. Its being identity gives g_ij=h_jh_i^-1. Thus the fundamental-loop tests suffice. Any other solution obeys the same p…
Exact mathematical passage
h_j^{-1}g_{ij}h_i
h_rUse in source · 2 passages
Search aliases: h_r
Use in source · Machine_Societies_v1.0.0.tex, line 1654
ery closed walk. When they exist, they are unique up to common right multiplication h_i↦ h_i k. It suffices to check the rooted fundamental loops associated with one spanning tree. If g_ij=h_jh_i^-1, adjacent factors cancel in every path product, giving H_γ=h_v_kh_v_0^-1 and hence identity on a closed walk. Conversely, choose a root r, set h_r=e_G, and define h_i as the product on the unique tree path from r to i. Tree edges satisfy the required identity. For any non-tree edge i→ j, the rooted l…
Exact mathematical passage
h_r=e_G
h_r^{-1}Use in source · 1 passage
Search aliases: h_r^-1
Use in source · Machine_Societies_v1.0.0.tex, line 1701
th k≥1 traversals, d(H_γ,e_G)≤∑_(i,j) traversed by γδ_ij. In particular, some traversed edge has error at least d(H_γ,e_G)/k for every candidate frame family. Gauge-transform each edge to widetilde g_ij=h_j^-1g_ijh_i. Bi-invariance gives d(widetilde g_ij,e_G)=δ_ij. The transformed loop product is h_r^-1H_γ h_r, at the same distance from identity as H_γ. Repeatedly applying the triangle inequality and invariance to a product bounds that distance by the sum of its factors' distances from identity.…
Exact mathematical passage
h_r^{-1}H_\gamma h_r
h_{v_0}^{-1}Use in source · 1 passage
Search aliases: h_v_0^-1
Use in source · Machine_Societies_v1.0.0.tex, line 1654
act frame synchronization] Global frames exist if and only if H_γ=e_G for every closed walk. When they exist, they are unique up to common right multiplication h_i↦ h_i k. It suffices to check the rooted fundamental loops associated with one spanning tree. If g_ij=h_jh_i^-1, adjacent factors cancel in every path product, giving H_γ=h_v_kh_v_0^-1 and hence identity on a closed walk. Conversely, choose a root r, set h_r=e_G, and define h_i as the product on the unique tree path from r to i. Tree e…
Exact mathematical passage
H_\gamma=h_{v_k}h_{v_0}^{-1}
h_{v_k}Use in source · 1 passage
Search aliases: h_v_k
Use in source · Machine_Societies_v1.0.0.tex, line 1654
act frame synchronization] Global frames exist if and only if H_γ=e_G for every closed walk. When they exist, they are unique up to common right multiplication h_i↦ h_i k. It suffices to check the rooted fundamental loops associated with one spanning tree. If g_ij=h_jh_i^-1, adjacent factors cancel in every path product, giving H_γ=h_v_kh_v_0^-1 and hence identity on a closed walk. Conversely, choose a root r, set h_r=e_G, and define h_i as the product on the unique tree path from r to i. Tree e…
Exact mathematical passage
H_\gamma=h_{v_k}h_{v_0}^{-1}
H_\gammaUse in source · 10 passages
Search aliases: H_γ gamma
Use in source · Machine_Societies_v1.0.0.tex, line 1642
Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
H_\gamma=g_{v_{k-1}v_k}\cdots g_{v_0v_1}.
\HistUse in source · 1 passage
Search aliases: Hist H mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 364
Predictive closure has a target and a distribution. For a fixed future target Y^+ and data-generating policy, a predictive statistic satisfies I(Y^+; H_tmid Z_t)=0. Approximate versions bound this residual information or, more directly, the excess predictive loss on held-out data. This is related to predictive-state constructions~[source reference]. It is not equivalent to uniform controlled lumpability: it may hold for one observational distribution while failing after intervention or under a n…
Exact mathematical passage
I(Y^+;\Hist_t\mid Z_t)=0.
\Hist_tUse in source · 1 passage
Search aliases: Hist_t H_t Hist mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 364
Predictive closure has a target and a distribution. For a fixed future target Y^+ and data-generating policy, a predictive statistic satisfies I(Y^+; H_tmid Z_t)=0. Approximate versions bound this residual information or, more directly, the excess predictive loss on held-out data. This is related to predictive-state constructions~[source reference]. It is not equivalent to uniform controlled lumpability: it may hold for one observational distribution while failing after intervention or under a n…
Exact mathematical passage
I(Y^+;\Hist_t\mid Z_t)=0.
I'Use in source · 2 passages
Search aliases: I'
Use in source · Machine_Societies_v1.0.0.tex, line 652
For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared family, not universal inertness. An enforceable institution can also alter which actions are institutionally permitted while producing the same observed …
Exact mathematical passage
\lambda(I),\lambda(I')
I_0Use in source · 1 passage
Search aliases: I_0
Use in source · Machine_Societies_v1.0.0.tex, line 1537
[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1-(1-ε)^T≤min{1,Tε}, where 0≤ε≤1 may be assumed and the empty product at T=0 is 1. No independent initialization of the fast state at successive review…
Exact mathematical passage
\label{eq:institutional-path-error} \TV\bigl(\Law(I_0,\ldots,I_T),\Law(J_0,\ldots,J_T)\bigr) \leq 1-(1-\varepsilon)^T\leq\min\{1,T\varepsilon\},
I_TUse in source · 1 passage
Search aliases: I_T
Use in source · Machine_Societies_v1.0.0.tex, line 1537
[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1-(1-ε)^T≤min{1,Tε}, where 0≤ε≤1 may be assumed and the empty product at T=0 is 1. No independent initialization of the fast state at successive review…
Exact mathematical passage
\label{eq:institutional-path-error} \TV\bigl(\Law(I_0,\ldots,I_T),\Law(J_0,\ldots,J_T)\bigr) \leq 1-(1-\varepsilon)^T\leq\min\{1,T\varepsilon\},
\mathrm{Id}Use in source · 4 passages
Search aliases: Id mathrm
Use in source · Machine_Societies_v1.0.0.tex, line 1482
[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random variable gives Eτ=∑_t≥0 P(τ>t). Under ρ(Q)<1, the matrix geometric series converges to (Id-Q)^-1, giving the result.
Exact mathematical passage
\E_\mu[\tau]=\mu(\mathrm{Id}-Q)^{-1}\mathbf1.
\mathsf{id}Use in source · 2 passages
Search aliases: id mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 192
ubsectionTwo rules and an inherited obligation Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The exchange creates a registered obligation o=(id,debtor=j,creditor=k, quantity=1,due=e+1). At the next epoch the creditor receives two slots and the debtor n…
Exact mathematical passage
o=(\mathsf{id},\mathsf{debtor}=j,\mathsf{creditor}=k, \mathsf{quantity}=1,\mathsf{due}=e+1).
\infUse in source · 2 passages
Search aliases: inf
Use in source · Machine_Societies_v1.0.0.tex, line 273
A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distribution, observation maps, and coalition family. It is not equivalent to any one general-purpose partial-information-decomposition statistic~[source re…
Exact mathematical passage
\mathcal R_\ell(O)=\inf_f\E\bigl[\ell(f(O),Y)\bigr].
\inf_fUse in source · 1 passage
Search aliases: inf_f inf
Use in source · Machine_Societies_v1.0.0.tex, line 273
A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distribution, observation maps, and coalition family. It is not equivalent to any one general-purpose partial-information-decomposition statistic~[source re…
Exact mathematical passage
\mathcal R_\ell(O)=\inf_f\E\bigl[\ell(f(O),Y)\bigr].
\mathrm{inst}Use in source · 1 passage
Search aliases: inst mathrm
Use in source · Machine_Societies_v1.0.0.tex, line 653
For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared family, not universal inertness. An enforceable institution can also alter which actions are institutionally permitted while producing the same observed …
Exact mathematical passage
\label{eq:inst-effect} \Delta_{\mathrm{inst}}^{\mathcal E,T}(I,I') =\sup_{e\in\mathcal E}\TV\bigl(\Law_e(Y_{0:T}\mid\lambda(I)),\Law_e(Y_{0:T}\mid\lambda(I'))\bigr).
\mathsf{Int}Use in source · 1 passage
Search aliases: Int mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 516
Hierarchy, gluing, and the limits of flattening. A lower-level organization can serve as a component at the next level when it exports a sufficient interface. That interface must describe not only a state but also the operations that are permitted on it, their outcome semantics, and their implementation error. A useful interface certificate is Int=( Z, U,overline K,overline Q,Λ, ε_rm dyn,ε_rm int,permissions). This tuple is a design proposal. It prevents a higher-level controller from treating a…
Exact mathematical passage
\mathsf{Int}=(\Z,\U,\overline K,\overline Q,\Lambda, \varepsilon_{\rm dyn},\varepsilon_{\rm int},\mathsf{permissions}).
JUse in source · 18 passages
Search aliases: J
Use in source · Machine_Societies_v1.0.0.tex, line 1536
[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1-(1-ε)^T≤min{1,Tε}, where 0≤ε≤1 may be assumed and the empty product at T=0 is 1. No independent initialization of the fast state at successive review…
Exact mathematical passage
(J_n)
J_0Use in source · 1 passage
Search aliases: J_0
Use in source · Machine_Societies_v1.0.0.tex, line 1537
[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1-(1-ε)^T≤min{1,Tε}, where 0≤ε≤1 may be assumed and the empty product at T=0 is 1. No independent initialization of the fast state at successive review…
Exact mathematical passage
\label{eq:institutional-path-error} \TV\bigl(\Law(I_0,\ldots,I_T),\Law(J_0,\ldots,J_T)\bigr) \leq 1-(1-\varepsilon)^T\leq\min\{1,T\varepsilon\},
J_nUse in source · 1 passage
Search aliases: J_n
Use in source · Machine_Societies_v1.0.0.tex, line 1536
[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1-(1-ε)^T≤min{1,Tε}, where 0≤ε≤1 may be assumed and the empty product at T=0 is 1. No independent initialization of the fast state at successive review…
Exact mathematical passage
(J_n)
J_TUse in source · 1 passage
Search aliases: J_T
Use in source · Machine_Societies_v1.0.0.tex, line 1537
[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1-(1-ε)^T≤min{1,Tε}, where 0≤ε≤1 may be assumed and the empty product at T=0 is 1. No independent initialization of the fast state at successive review…
Exact mathematical passage
\label{eq:institutional-path-error} \TV\bigl(\Law(I_0,\ldots,I_T),\Law(J_0,\ldots,J_T)\bigr) \leq 1-(1-\varepsilon)^T\leq\min\{1,T\varepsilon\},
\mathcal KUse in source · 1 passage
Search aliases: K mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 2539
An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a component, or unavailable. Those statuses should not all become the Boolean value verified. A failed guard means that continued reliance is not justified by t…
Exact mathematical passage
\mathcal K=(\mathsf{Claim},\mathsf{Assumptions},\mathsf{Evidence}, \mathsf{Domain},\mathsf{Guard},\mathsf{Fallback},\mathsf{Revision}).
k_iUse in source · 1 passage
Search aliases: k_i
Use in source · Machine_Societies_v1.0.0.tex, line 1657
Changing local conventions by elements k_i replaces g_ij by k_jg_ijk_i^-1. A loop based at r then has holonomy k_rH_γ k_r^-1. In particular, identity holonomy is invariant under local changes of coordinates. An arbitrary binary cycle basis with unordered multiplication is not a valid replacement for rooted fundamental loops in a noncommutative group.
Exact mathematical passage
k_i
k_i^{-1}Use in source · 1 passage
Search aliases: k_i^-1
Use in source · Machine_Societies_v1.0.0.tex, line 1657
Changing local conventions by elements k_i replaces g_ij by k_jg_ijk_i^-1. A loop based at r then has holonomy k_rH_γ k_r^-1. In particular, identity holonomy is invariant under local changes of coordinates. An arbitrary binary cycle basis with unordered multiplication is not a valid replacement for rooted fundamental loops in a noncommutative group.
Exact mathematical passage
k_jg_{ij}k_i^{-1}
K_i^mUse in source · 2 passages
Search aliases: K_i^m
Use in source · Machine_Societies_v1.0.0.tex, line 1525
Assume K_i has a stationary law μ_i and sup_i,xd_TV(K_i^m(x,·),μ_i)≤ε. The actual one-review law from (i,x) and its stationary average are P(i,x;j) · =∑_y K_i^m(x,y)R_i(y,j),
Exact mathematical passage
\label{eq:uniform-fast-mixing} \sup_{i,x}\TV\bigl(K_i^m(x,\cdot),\mu_i\bigr)\leq\varepsilon.
k_jUse in source · 1 passage
Search aliases: k_j
Use in source · Machine_Societies_v1.0.0.tex, line 1657
Changing local conventions by elements k_i replaces g_ij by k_jg_ijk_i^-1. A loop based at r then has holonomy k_rH_γ k_r^-1. In particular, identity holonomy is invariant under local changes of coordinates. An arbitrary binary cycle basis with unordered multiplication is not a valid replacement for rooted fundamental loops in a noncommutative group.
Exact mathematical passage
k_jg_{ij}k_i^{-1}
k_rUse in source · 1 passage
Search aliases: k_r
Use in source · Machine_Societies_v1.0.0.tex, line 1657
Changing local conventions by elements k_i replaces g_ij by k_jg_ijk_i^-1. A loop based at r then has holonomy k_rH_γ k_r^-1. In particular, identity holonomy is invariant under local changes of coordinates. An arbitrary binary cycle basis with unordered multiplication is not a valid replacement for rooted fundamental loops in a noncommutative group.
Exact mathematical passage
k_rH_\gamma k_r^{-1}
k_r^{-1}Use in source · 1 passage
Search aliases: k_r^-1
Use in source · Machine_Societies_v1.0.0.tex, line 1657
Changing local conventions by elements k_i replaces g_ij by k_jg_ijk_i^-1. A loop based at r then has holonomy k_rH_γ k_r^-1. In particular, identity holonomy is invariant under local changes of coordinates. An arbitrary binary cycle basis with unordered multiplication is not a valid replacement for rooted fundamental loops in a noncommutative group.
Exact mathematical passage
k_rH_\gamma k_r^{-1}
lUse in source · 5 passages
Search aliases: l
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
L_0Use in source · 1 passage
Search aliases: L_0
Use in source · Machine_Societies_v1.0.0.tex, line 2512
T_k+1 · =T_k∪{B∈ B:B⊆ S_rm safe, ∃ u∈ A_rm end(B), Succ(B,u)⊆ T_k}. Let T_* be the stable set and r_rm end(B) the first entry index. Then define L_0=T_*, L_k+1={B∈ T_*:∃ u∈ A_rm run(B), Succ(B,u)⊆ L_k}, with stable set L_*. These are finite constructions with the same potentially exponential possibility space as before.
Exact mathematical passage
\label{eq:stoppable-kernel} L_0=T_*,\qquad L_{k+1}=\{B\in T_*:\exists u\in A_{\rm run}(B),\ \operatorname{Succ}(B,u)\subseteq L_k\},
L_jUse in source · 1 passage
Search aliases: L_j
Use in source · Machine_Societies_v1.0.0.tex, line 1380
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))
Exact mathematical passage
L_j
L_kUse in source · 3 passages
Search aliases: L_k
Use in source · Machine_Societies_v1.0.0.tex, line 1381
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z)) · ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source r…
Exact mathematical passage
\label{eq:cumulative-transport} \begin{split} &d_m\bigl(T_{0:m}F_0(z),F_mT_{0:m}(z)\bigr)\\ &\qquad\le\sum_{j=0}^{m-1}\varepsilon_j\prod_{k=j+1}^{m-1}L_k, \end{split}
L_{k+1}Use in source · 1 passage
Search aliases: L_k+1
Use in source · Machine_Societies_v1.0.0.tex, line 2512
T_k+1 · =T_k∪{B∈ B:B⊆ S_rm safe, ∃ u∈ A_rm end(B), Succ(B,u)⊆ T_k}. Let T_* be the stable set and r_rm end(B) the first entry index. Then define L_0=T_*, L_k+1={B∈ T_*:∃ u∈ A_rm run(B), Succ(B,u)⊆ L_k}, with stable set L_*. These are finite constructions with the same potentially exponential possibility space as before.
Exact mathematical passage
\label{eq:stoppable-kernel} L_0=T_*,\qquad L_{k+1}=\{B\in T_*:\exists u\in A_{\rm run}(B),\ \operatorname{Succ}(B,u)\subseteq L_k\},
L_{m-1}Use in source · 2 passages
Search aliases: L_m-1
Use in source · Machine_Societies_v1.0.0.tex, line 1390
qquad≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second by ε_m-1. The recurrence D_m≤ L_m-1D_m-1+ε_m-1 unfolds to~[source reference].
Exact mathematical passage
L_{m-1}
L_tUse in source · 1 passage
Search aliases: L_t
Use in source · Machine_Societies_v1.0.0.tex, line 641
To formalize an operative institution, augment the microstate of Section~[source reference] with a distinguished component I_t. In a finite setting we may describe it as I_t=(N_t,P_t,E_t,A_t,L_t), where N specifies norms and role obligations, P permissions and capabilities, E enforcement and conflict procedures, A admissible amendments, and L records of decisions and their provenance. The components need not be physically stored together, and their representation is not unique. What matters is t…
Exact mathematical passage
\label{eq:institutional-state} I_t=(N_t,P_t,E_t,A_t,L_t),
\LawUse in source · 4 passages
Search aliases: Law L mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 407
For a family T of finite continuation experiments, define exact behavioral equivalence by x≡_ Tx' Longleftrightarrow L_x(Y^+midτ)= L_x'(Y^+midτ) for every τ∈ T. Equality of these indexed laws is reflexive, symmetric, and transitive. It therefore defines a quotient. This equivalence is relative to the chosen experiments. It does not by itself equal bisimulation, and it says nothing about untested interventions or provenance.
Exact mathematical passage
x\equiv_{\mathcal T}x'\quad\Longleftrightarrow\quad \Law_x(Y^+\mid\tau)=\Law_{x'}(Y^+\mid\tau) \quad\text{for every }\tau\in\mathcal T.
\Law_eUse in source · 1 passage
Search aliases: Law_e L_e Law mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 653
For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared family, not universal inertness. An enforceable institution can also alter which actions are institutionally permitted while producing the same observed …
Exact mathematical passage
\label{eq:inst-effect} \Delta_{\mathrm{inst}}^{\mathcal E,T}(I,I') =\sup_{e\in\mathcal E}\TV\bigl(\Law_e(Y_{0:T}\mid\lambda(I)),\Law_e(Y_{0:T}\mid\lambda(I'))\bigr).
\Law_xUse in source · 2 passages
Search aliases: Law_x L_x Law mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 407
For a family T of finite continuation experiments, define exact behavioral equivalence by x≡_ Tx' Longleftrightarrow L_x(Y^+midτ)= L_x'(Y^+midτ) for every τ∈ T. Equality of these indexed laws is reflexive, symmetric, and transitive. It therefore defines a quotient. This equivalence is relative to the chosen experiments. It does not by itself equal bisimulation, and it says nothing about untested interventions or provenance.
Exact mathematical passage
x\equiv_{\mathcal T}x'\quad\Longleftrightarrow\quad \Law_x(Y^+\mid\tau)=\Law_{x'}(Y^+\mid\tau) \quad\text{for every }\tau\in\mathcal T.
\Law_{x'}Use in source · 2 passages
Search aliases: Law_x' L_x' Law mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 407
For a family T of finite continuation experiments, define exact behavioral equivalence by x≡_ Tx' Longleftrightarrow L_x(Y^+midτ)= L_x'(Y^+midτ) for every τ∈ T. Equality of these indexed laws is reflexive, symmetric, and transitive. It therefore defines a quotient. This equivalence is relative to the chosen experiments. It does not by itself equal bisimulation, and it says nothing about untested interventions or provenance.
Exact mathematical passage
x\equiv_{\mathcal T}x'\quad\Longleftrightarrow\quad \Law_x(Y^+\mid\tau)=\Law_{x'}(Y^+\mid\tau) \quad\text{for every }\tau\in\mathcal T.
\lceilUse in source · 2 passages
Search aliases: lceil
Use in source · Machine_Societies_v1.0.0.tex, line 457
[Exact finite-memory lower bound] Every deterministic summary that supports exact continuation on these histories requires at least N distinct states, hence at least lceillog_2Nrceil fixed-length bits. If the permitted continuation family is closed under prefixing and the history equivalence is a transition congruence, the quotient classes themselves provide an exact state representation. If a summary assigns the same state to two inequivalent histories, some permitted continuation requires diff…
Exact mathematical passage
\lceil\log_2N\rceil
\lfloorUse in source · 1 passage
Search aliases: lfloor
Use in source · Machine_Societies_v1.0.0.tex, line 903
[Minimum profitable coordinated seed] Starting from universal nonadoption, a simultaneously adopting coalition of size m makes each member strictly better off exactly when b(m-1)>c. When such a coalition exists, the smallest size is m_* = lfloorfrac cbrfloor+2≤ n. Once at least m_* adopters are present, every remaining nonadopter has a profitable unilateral entry. Under a schedule that eventually updates every remaining agent, strict best responses reach universal adoption without an adopter exi…
Exact mathematical passage
m_* = \left\lfloor\frac cb\right\rfloor+2\leq n.
\logUse in source · 2 passages
Search aliases: log
Use in source · Machine_Societies_v1.0.0.tex, line 457
[Exact finite-memory lower bound] Every deterministic summary that supports exact continuation on these histories requires at least N distinct states, hence at least lceillog_2Nrceil fixed-length bits. If the permitted continuation family is closed under prefixing and the history equivalence is a transition congruence, the quotient classes themselves provide an exact state representation. If a summary assigns the same state to two inequivalent histories, some permitted continuation requires diff…
Exact mathematical passage
\lceil\log_2N\rceil
\log_2Use in source · 2 passages
Search aliases: log_2 log
Use in source · Machine_Societies_v1.0.0.tex, line 457
[Exact finite-memory lower bound] Every deterministic summary that supports exact continuation on these histories requires at least N distinct states, hence at least lceillog_2Nrceil fixed-length bits. If the permitted continuation family is closed under prefixing and the history equivalence is a transition congruence, the quotient classes themselves provide an exact state representation. If a summary assigns the same state to two inequivalent histories, some permitted continuation requires diff…
Exact mathematical passage
\lceil\log_2N\rceil
\LongrightarrowUse in source · 4 passages
Search aliases: Longrightarrow
Use in source · Machine_Societies_v1.0.0.tex, line 299
[Deterministic quotient criterion] For maps F_u: X→ X and a surjection π: X→ Z, maps overline F_u: Z→ Z satisfying π F_u=overline F_uπ exist if and only if π(x)=π(x') Longrightarrow π(F_u(x))=π(F_u(x')) for every u,x,x'. When they exist, they are unique. Necessity follows by applying overline F_u to the common value π(x). For sufficiency, set overline F_u(z)=π(F_u(x)) for any x in the fibre of z. The condition makes this independent of the representative, and surjectivity gives uniqueness.
Exact mathematical passage
\pi(x)=\pi(x')\quad\Longrightarrow\quad \pi(F_u(x))=\pi(F_u(x'))
\mathcal MUse in source · 6 passages
Search aliases: M mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 787
Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated office names, configuration epochs, and reference records. The translation of a message sequence is constructed step by step from the evolving source st…
Exact mathematical passage
\mathcal M_A,\mathcal M_B
M^{\rm act}Use in source · 1 passage
Search aliases: M^rm act rm
Use in source · Machine_Societies_v1.0.0.tex, line 1363
Let a history H be summarized by an inheritance map C(H). A practically meaningful retained state can be divided into an active canon M^rm act for immediate use, an archive M^rm arc for recovery and criticism, and a provenance and uncertainty record P. These stores can overlap physically, but serve distinct decision contracts. The canon keeps interaction costs low. The archive makes forgotten exceptions, alternative explanations, and evidence recoverable. Provenance records why a claim should be…
Exact mathematical passage
M^{\rm act}
M^{\rm arc}Use in source · 1 passage
Search aliases: M^rm arc rm
Use in source · Machine_Societies_v1.0.0.tex, line 1363
Let a history H be summarized by an inheritance map C(H). A practically meaningful retained state can be divided into an active canon M^rm act for immediate use, an archive M^rm arc for recovery and criticism, and a provenance and uncertainty record P. These stores can overlap physically, but serve distinct decision contracts. The canon keeps interaction costs low. The archive makes forgotten exceptions, alternative explanations, and evidence recoverable. Provenance records why a claim should be…
Exact mathematical passage
M^{\rm arc}
m_*Use in source · 3 passages
Search aliases: m_*
Use in source · Machine_Societies_v1.0.0.tex, line 903
[Minimum profitable coordinated seed] Starting from universal nonadoption, a simultaneously adopting coalition of size m makes each member strictly better off exactly when b(m-1)>c. When such a coalition exists, the smallest size is m_* = lfloorfrac cbrfloor+2≤ n. Once at least m_* adopters are present, every remaining nonadopter has a profitable unilateral entry. Under a schedule that eventually updates every remaining agent, strict best responses reach universal adoption without an adopter exi…
Exact mathematical passage
m_* = \left\lfloor\frac cb\right\rfloor+2\leq n.
\mathcal M_AUse in source · 2 passages
Search aliases: M_A mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 787
Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated office names, configuration epochs, and reference records. The translation of a message sequence is constructed step by step from the evolving source st…
Exact mathematical passage
\mathcal M_A,\mathcal M_B
\mathcal M_BUse in source · 2 passages
Search aliases: M_B mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 787
Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated office names, configuration epochs, and reference records. The translation of a message sequence is constructed step by step from the evolving source st…
Exact mathematical passage
\mathcal M_A,\mathcal M_B
M_{t+1}Use in source · 1 passage
Search aliases: M_t+1
Use in source · Machine_Societies_v1.0.0.tex, line 690
nter controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assigns nonnegative voting weights, q_t is an approval threshold, and e_t is a configuration epoch. Admission and exit proposals determine M_t+1=(M_t∖ E_t)∪ A_t, E_t⊆ M_t, A_t⊆ U∖ M_t, but this equation alone specifies neither their authorization nor th…
Exact mathematical passage
M_{t+1}=(M_t\setminus E_t)\cup A_t, \qquad E_t\subseteq M_t,\quad A_t\subseteq\mathcal U\setminus M_t,
\mathcal M_\alphaUse in source · 4 passages
Search aliases: M_α mathcal alpha
Use in source · Machine_Societies_v1.0.0.tex, line 1008
The common-monitor problem. Let O be a finite nonempty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous…
Exact mathematical passage
\mathcal M_\alpha=\{s\in[0,1]^O:\langle P_C,s\rangle\leq\alpha\}.
\maxUse in source · 6 passages
Search aliases: max
Use in source · Machine_Societies_v1.0.0.tex, line 970
False-positive limits and multiple deviations. A constitutional protection may impose E_P_Cs≤α, with 0≤α≤1. The relevant discrimination capacity is then the finite linear program D_α(P_D,P_C)= max_substack0≤ s≤1 E_P_Cs≤α ( E_P_Ds- E_P_Cs). The feasible set is nonempty and compact, so a maximum exists. The preceding proof with the feasible set restricted shows that deterrence is attainable exactly when g≤ F D_α. For the example above and α=1/10, the optimum is s=(0,2/5), giving D_α=1/5, not 1/2. …
Exact mathematical passage
\label{eq:protected-discrimination} D_\alpha(P_D,P_C)= \max_{\substack{0\leq s\leq1\\\E_{P_C}s\leq\alpha}} \bigl(\E_{P_D}s-\E_{P_C}s\bigr).
\max_{0\le s\le1,\ \E_{P_C}s\le\alpha}Use in source · 1 passage
Search aliases: max_0≤ s≤1, E_P_Cs≤α max le le E le alpha max le le mathbb le alpha
Use in source · Machine_Societies_v1.0.0.tex, line 2403
Keep the earlier definition D_α(P_D,P_C)=max_0≤ s≤1, E_P_Cs≤α ( E_P_Ds- E_P_Cs), 0≤α≤1. The same randomized sanction s must obey the protected false-positive limit and supply the deviation margin.
Exact mathematical passage
D_\alpha(P_D,P_C)=\max_{0\le s\le1,\ \E_{P_C}s\le\alpha} (\E_{P_D}s-\E_{P_C}s),\qquad 0\le\alpha\le1.
\max_iUse in source · 1 passage
Search aliases: max_i max
Use in source · Machine_Societies_v1.0.0.tex, line 2013
nnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-step monotonic estimate followed by induction. Under uniformly bounded disturbance normw_t…
Exact mathematical passage
\norm{x}_{v,\infty}=\max_i x_i/v_i
\max_{\substack{0\leq s\leq1\\\E_{P_C}s\leq\alpha}}Use in source · 1 passage
Search aliases: max_substack0≤ s≤1 E_P_Cs≤α max substack leq leq E leq alpha max substack leq leq mathbb leq alpha
Use in source · Machine_Societies_v1.0.0.tex, line 970
False-positive limits and multiple deviations. A constitutional protection may impose E_P_Cs≤α, with 0≤α≤1. The relevant discrimination capacity is then the finite linear program D_α(P_D,P_C)= max_substack0≤ s≤1 E_P_Cs≤α ( E_P_Ds- E_P_Cs). The feasible set is nonempty and compact, so a maximum exists. The preceding proof with the feasible set restricted shows that deterrence is attainable exactly when g≤ F D_α. For the example above and α=1/10, the optimum is s=(0,2/5), giving D_α=1/5, not 1/2. …
Exact mathematical passage
\label{eq:protected-discrimination} D_\alpha(P_D,P_C)= \max_{\substack{0\leq s\leq1\\\E_{P_C}s\leq\alpha}} \bigl(\E_{P_D}s-\E_{P_C}s\bigr).
\max_{s\in\mathcal M_\alpha}Use in source · 2 passages
Search aliases: max_s∈ M_α max in mathcal alpha
Use in source · Machine_Societies_v1.0.0.tex, line 1012
th probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous margin by D_j(s)=F⟨ P_j-P_C,s⟩-g_j, Γ_α=max_s∈ M_αmin_1≤ j≤ mD_j(s). Weak deterrence of all listed deviations means Γ_α≥0. Strict deterrence means Γ_α>0. At equality a participant can remain indifferent. A rand…
Exact mathematical passage
\label{eq:joint-margin} D_j(s)=F\langle P_j-P_C,s\rangle-g_j, \qquad \Gamma_\alpha=\max_{s\in\mathcal M_\alpha}\min_{1\leq j\leq m}D_j(s).
\minUse in source · 17 passages
Search aliases: min
Use in source · Machine_Societies_v1.0.0.tex, line 277
A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distribution, observation maps, and coalition family. It is not equivalent to any one general-purpose partial-information-decomposition statistic~[source re…
Exact mathematical passage
\label{eq:coalition} \Delta_{\mathcal A}=\min_{S\in\mathcal A}\mathcal R_\ell(O_S) -\mathcal R_\ell(O_V)
\min_{1\leq j\leq m}Use in source · 1 passage
Search aliases: min_1≤ j≤ m min leq leq
Use in source · Machine_Societies_v1.0.0.tex, line 1012
th probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous margin by D_j(s)=F⟨ P_j-P_C,s⟩-g_j, Γ_α=max_s∈ M_αmin_1≤ j≤ mD_j(s). Weak deterrence of all listed deviations means Γ_α≥0. Strict deterrence means Γ_α>0. At equality a participant can remain indifferent. A rand…
Exact mathematical passage
\label{eq:joint-margin} D_j(s)=F\langle P_j-P_C,s\rangle-g_j, \qquad \Gamma_\alpha=\max_{s\in\mathcal M_\alpha}\min_{1\leq j\leq m}D_j(s).
\min_eUse in source · 1 passage
Search aliases: min_e min
Use in source · Machine_Societies_v1.0.0.tex, line 1772
esidual and minimum authorized repair] The XOR of edge labels around a cycle is unchanged by every local convention change above. On a single simple cycle with odd XOR, suppose an editable subset of edges can be flipped at nonnegative costs w_e, with other edges protected. An authorized repair exists exactly when the editable subset is nonempty. The minimum cost is then min_e w_e over editable edges. If all edges are protected, no change of local binary conventions can produce a compatible globa…
Exact mathematical passage
\min_e w_e
\min_{S\in\mathcal A}Use in source · 1 passage
Search aliases: min_S∈ A min in mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 277
A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distribution, observation maps, and coalition family. It is not equivalent to any one general-purpose partial-information-decomposition statistic~[source re…
Exact mathematical passage
\label{eq:coalition} \Delta_{\mathcal A}=\min_{S\in\mathcal A}\mathcal R_\ell(O_S) -\mathcal R_\ell(O_V)
\min_{z\ne z'}Use in source · 1 passage
Search aliases: min_z≠ z' min ne
Use in source · Machine_Societies_v1.0.0.tex, line 467
Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.
Exact mathematical passage
d_E=\min_{z\ne z'}\abs{\{i:E_i(z)\ne E_i(z')\}}.
\min_{\lambda\in\Delta_m}Use in source · 1 passage
Search aliases: min_λ∈Δ_m min lambda in Delta
Use in source · Machine_Societies_v1.0.0.tex, line 1027
[Simultaneous enforcement and mixture obstruction] Under the finite common-monitor model, Γ_α=min_λ∈Δ_m (FV_α(P_λ,P_C)-g_λ). Both extrema are attained. Consequently, one monitor weakly deters all listed deviations if and only if FV_α(P_λ,P_C)≥ g_λ for every λ∈Δ_m. Without an additional false-positive restriction, α=1 and V_1(P,P_C)=d_TV(P,P_C). For any fixed s, the minimum of the numbers D_j(s) is the minimum of ∑_jλ_jD_j(s) over Δ_m. This expression is bilinear in s and λ, and both domains are …
Exact mathematical passage
\label{eq:joint-dual} \Gamma_\alpha=\min_{\lambda\in\Delta_m} \bigl(FV_\alpha(P_\lambda,P_C)-g_\lambda\bigr).
\min_{\varnothing\ne S\subseteq J}Use in source · 1 passage
Search aliases: min_∅≠ S⊆ J min varnothing ne subseteq
Use in source · Machine_Societies_v1.0.0.tex, line 2097
The theorem permits a simple robustness index. If the initial graph admits a staffing, define σ=min_∅≠ S⊆ J(|N(S)|-|S|). Then exactly the erasure budgets k≤σ are guaranteed tolerable in this model. This is a static allocation certificate, not a claim that the surviving agents can discover or execute the reassignment before a deadline. That further question uses the communication and permission constraints of Sections~[source reference] and~[source reference].
Exact mathematical passage
\sigma=\min_{\varnothing\ne S\subseteq J}\bigl(|N(S)|-|S|\bigr).
\modelsUse in source · 2 passages
Search aliases: models
Use in source · Machine_Societies_v1.0.0.tex, line 1897
[Common knowledge as a reachability condition] Let Reach_J(w) be the worlds reachable from w by a finite chain of relations ∼_i with i∈ J, including the zero-length chain. Then wmodels C_Jφ Longleftrightarrow Reach_J(w)⊆{v:vmodelsφ}. Induction on n shows that E_J^nφ requires φ at the endpoint of every chain of n indistinguishability steps. Requiring this for every finite n is exactly the stated reachability inclusion.
Exact mathematical passage
w\models C_J\varphi\quad\Longleftrightarrow\quad \mathsf{Reach}_J(w)\subseteq\{v:v\models\varphi\}.
NUse in source · 26 passages
Search aliases: N
Use in source · Machine_Societies_v1.0.0.tex, line 453
Memory lower bounds from distinguishable histories. Fix a deterministic exact predictor for a family of future input continuations. Call two past histories equivalent when they require the same output sequence for every permitted continuation. Let N be the number of equivalence classes among the histories under consideration.
Exact mathematical passage
N
\mathbb NUse in source · 2 passages
Search aliases: N mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 2229
[Soundness of a finite continuation certificate] If the preceding checks hold, κ preserves the complete contract from B_0 for all transitions. If, in addition, there is a map r: C→ N such that r(B)=0 implies B⊆ G and every successor of a set with positive rank has strictly smaller rank, then a set contained in G is reached within at most r(B_0) transitions. Checking these finite conditions does not require proving C=W_*. Induction preserves membership in C, admissibility, and safety. Until rank …
Exact mathematical passage
r:\mathcal C\to\mathbb N
n_{\mathrm{eff}}Use in source · 1 passage
Search aliases: n_eff mathrm
Use in source · Machine_Societies_v1.0.0.tex, line 1798
ginequation Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equal-variance estimators, the variance-equivalent effective sample size is n_eff=fracn1+(n-1)ρ. If the errors are perfectly correlated, adding copies does not reduce variance. This equation does not give a general probability that a majority is right, and real error structures n…
Exact mathematical passage
n_{\mathrm{eff}}=\frac{n}{1+(n-1)\rho}.
N_FUse in source · 2 passages
Search aliases: N_F
Use in source · Machine_Societies_v1.0.0.tex, line 1462
Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not describe all cultural evolution. It establishes that selection for reproduction alone does not logically impose selection for accuracy. An institution nee…
Exact mathematical passage
N_T,N_F
N_iUse in source · 1 passage
Search aliases: N_i
Use in source · Machine_Societies_v1.0.0.tex, line 673
A norm can be presented operationally as a condition on histories and actions, N_i(h,a)∈{0,1}, interpreted as the system's declared permission for action a by position i after history h. This does not make the declared permission morally correct. Enforcement changes the action transition or cost structure; appeal changes what happens after an alleged violation; amendment changes the permissible norm itself.
Exact mathematical passage
N_i(h,a)\in\{0,1\}
N_TUse in source · 2 passages
Search aliases: N_T
Use in source · Machine_Societies_v1.0.0.tex, line 1462
Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not describe all cultural evolution. It establishes that selection for reproduction alone does not logically impose selection for accuracy. An institution nee…
Exact mathematical passage
N_T,N_F
N_tUse in source · 1 passage
Search aliases: N_t
Use in source · Machine_Societies_v1.0.0.tex, line 641
To formalize an operative institution, augment the microstate of Section~[source reference] with a distinguished component I_t. In a finite setting we may describe it as I_t=(N_t,P_t,E_t,A_t,L_t), where N specifies norms and role obligations, P permissions and capabilities, E enforcement and conflict procedures, A admissible amendments, and L records of decisions and their provenance. The components need not be physically stored together, and their representation is not unique. What matters is t…
Exact mathematical passage
\label{eq:institutional-state} I_t=(N_t,P_t,E_t,A_t,L_t),
\normUse in source · 3 passages
Search aliases: norm
Use in source · Machine_Societies_v1.0.0.tex, line 2013
nnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-step monotonic estimate followed by induction. Under uniformly bounded disturbance normw_t…
Exact mathematical passage
\norm{x}_{v,\infty}=\max_i x_i/v_i
O_1Use in source · 2 passages
Search aliases: O_1
Use in source · Machine_Societies_v1.0.0.tex, line 2445
[Two uninformative records with an informative joint history] Let θ∈{0,1} distinguish two protected cases, and let R be a fair independent bit. Release O_1=R and O_2=R⊕θ. Each marginal is a fair bit under both cases and satisfies~[source reference] with λ=1. But O_1⊕ O_2=θ. The joint laws have disjoint support and satisfy no finite multiplicative bound.
Exact mathematical passage
O_1=R
O_2Use in source · 2 passages
Search aliases: O_2
Use in source · Machine_Societies_v1.0.0.tex, line 2445
[Two uninformative records with an informative joint history] Let θ∈{0,1} distinguish two protected cases, and let R be a fair independent bit. Release O_1=R and O_2=R⊕θ. Each marginal is a fair bit under both cases and satisfies~[source reference] with λ=1. But O_1⊕ O_2=θ. The joint laws have disjoint support and satisfy no finite multiplicative bound.
Exact mathematical passage
O_2=R\oplus\theta
O_{\rm ext}Use in source · 2 passages
Search aliases: O_rm ext rm
Use in source · Machine_Societies_v1.0.0.tex, line 2343
A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. Standing specifies who may initiate a challenge and on which grounds. Remedy specifies the actions an authorized review can actually require. The cont…
Exact mathematical passage
\mathcal B=(Z_{\rm ext},O_{\rm ext},\mathsf{Auth},\mathsf{Effect}, \mathsf{Standing},\mathsf{Remedy}).
O_VUse in source · 1 passage
Search aliases: O_V
Use in source · Machine_Societies_v1.0.0.tex, line 277
A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distribution, observation maps, and coalition family. It is not equivalent to any one general-purpose partial-information-decomposition statistic~[source re…
Exact mathematical passage
\label{eq:coalition} \Delta_{\mathcal A}=\min_{S\in\mathcal A}\mathcal R_\ell(O_S) -\mathcal R_\ell(O_V)
\mathsf{open}Use in source · 4 passages
Search aliases: open mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 1131
[An unchecked change of the checker] Let the institutional state contain a validator mode v∈{strict,open} and a resource policy p∈{safe,unsafe}. Under v=strict, the current policy must remain safe, but the validator mode itself is unprotected. Thus (strict,safe)→(open,safe) is admitted. Under v=open, any policy transition is admitted, including (open,safe)→(open,unsafe). The original current-policy check does not imply that all reachable states remain safe. To prove that property, the inductive …
Exact mathematical passage
v\in\{\mathsf{strict},\mathsf{open}\}
\overline dUse in source · 5 passages
Search aliases: overline d overline
Use in source · Machine_Societies_v1.0.0.tex, line 1579
[A robust margin for reduced-model enforcement] Let s^⋆(o) sanction precisely where overline P_D(o)>overline P_C(o) and put overline d=d_TV(overline P_D,overline P_C). In the model of Section~[source reference], this fixed policy makes compliance weakly preferable whenever g≤ F(overline d-ε_C-ε_D). Conversely, if g>Fmin{1,overline d+ε_C+ε_D}, no observation-based sanction with penalty F deters the deviation. Also, E_P_Cs^⋆≤ E_overline P_Cs^⋆+ε_C. The expectation of a [0,1]-valued function change…
Exact mathematical passage
\overline d=\TV(\overline P_D,\overline P_C)
\overline FUse in source · 4 passages
Search aliases: overline F overline
Use in source · Machine_Societies_v1.0.0.tex, line 298
[Deterministic quotient criterion] For maps F_u: X→ X and a surjection π: X→ Z, maps overline F_u: Z→ Z satisfying π F_u=overline F_uπ exist if and only if π(x)=π(x') Longrightarrow π(F_u(x))=π(F_u(x')) for every u,x,x'. When they exist, they are unique. Necessity follows by applying overline F_u to the common value π(x). For sufficiency, set overline F_u(z)=π(F_u(x)) for any x in the fibre of z. The condition makes this independent of the representative, and surjectivity gives uniqueness.
Exact mathematical passage
\overline F_u:\Z\to\Z
\overline F_uUse in source · 4 passages
Search aliases: overline F_u overline
Use in source · Machine_Societies_v1.0.0.tex, line 298
[Deterministic quotient criterion] For maps F_u: X→ X and a surjection π: X→ Z, maps overline F_u: Z→ Z satisfying π F_u=overline F_uπ exist if and only if π(x)=π(x') Longrightarrow π(F_u(x))=π(F_u(x')) for every u,x,x'. When they exist, they are unique. Necessity follows by applying overline F_u to the common value π(x). For sufficiency, set overline F_u(z)=π(F_u(x)) for any x in the fibre of z. The condition makes this independent of the representative, and surjectivity gives uniqueness.
Exact mathematical passage
\overline F_u:\Z\to\Z
\overline KUse in source · 9 passages
Search aliases: overline K overline
Use in source · Machine_Societies_v1.0.0.tex, line 313
[Controlled lumpability] A common kernel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-model with initial law pushμ_0. For outputs to agree as well, require Q_u(x,·) to be constant on each fibre. The common fibre value in~[source reference] defines overline K…
Exact mathematical passage
\overline K_u
\overline K_uUse in source · 8 passages
Search aliases: overline K_u overline
Use in source · Machine_Societies_v1.0.0.tex, line 313
[Controlled lumpability] A common kernel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-model with initial law pushμ_0. For outputs to agree as well, require Q_u(x,·) to be constant on each fibre. The common fibre value in~[source reference] defines overline K…
Exact mathematical passage
\overline K_u
\overline PUse in source · 5 passages
Search aliases: overline P overline
Use in source · Machine_Societies_v1.0.0.tex, line 1572
Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bound and the same declared endpoint-to-evidence channel is used in the actual and averaged model.
Exact mathematical passage
\overline P_C,\overline P_D
\overline P_aUse in source · 1 passage
Search aliases: overline P_a overline
Use in source · Machine_Societies_v1.0.0.tex, line 1573
Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bound and the same declared endpoint-to-evidence channel is used in the actual and averaged model.
Exact mathematical passage
\TV(P_a,\overline P_a)\leq\varepsilon_a,\qquad a\in\{C,D\}.
\overline P_CUse in source · 4 passages
Search aliases: overline P_C overline
Use in source · Machine_Societies_v1.0.0.tex, line 1572
Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bound and the same declared endpoint-to-evidence channel is used in the actual and averaged model.
Exact mathematical passage
\overline P_C,\overline P_D
\overline P_DUse in source · 3 passages
Search aliases: overline P_D overline
Use in source · Machine_Societies_v1.0.0.tex, line 1572
Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bound and the same declared endpoint-to-evidence channel is used in the actual and averaged model.
Exact mathematical passage
\overline P_C,\overline P_D
\overline QUse in source · 1 passage
Search aliases: overline Q overline
Use in source · Machine_Societies_v1.0.0.tex, line 516
Hierarchy, gluing, and the limits of flattening. A lower-level organization can serve as a component at the next level when it exports a sufficient interface. That interface must describe not only a state but also the operations that are permitted on it, their outcome semantics, and their implementation error. A useful interface certificate is Int=( Z, U,overline K,overline Q,Λ, ε_rm dyn,ε_rm int,permissions). This tuple is a design proposal. It prevents a higher-level controller from treating a…
Exact mathematical passage
\mathsf{Int}=(\Z,\U,\overline K,\overline Q,\Lambda, \varepsilon_{\rm dyn},\varepsilon_{\rm int},\mathsf{permissions}).
\overline RUse in source · 4 passages
Search aliases: overline R overline
Use in source · Machine_Societies_v1.0.0.tex, line 1529
Assume K_i has a stationary law μ_i and sup_i,xd_TV(K_i^m(x,·),μ_i)≤ε. The actual one-review law from (i,x) and its stationary average are P(i,x;j) · =∑_y K_i^m(x,y)R_i(y,j), overline R(i,j) · =∑_yμ_i(y)R_i(y,j). The endpoint is the only fast input to this particular revision rule. A rule that uses the full trajectory, accumulated costs, or a data-dependent stopping time requires a different state or a different averaging argument.
Exact mathematical passage
P(i,x;j)&=\sum_y K_i^m(x,y)R_i(y,j),\label{eq:actual-review}\\ \overline R(i,j)&=\sum_y\mu_i(y)R_i(y,j).\label{eq:averaged-review}
\overline R_kUse in source · 1 passage
Search aliases: overline R_k overline
Use in source · Machine_Societies_v1.0.0.tex, line 2260
[Governance abstraction preserves joint realizability] Under~[source reference]--[source reference], for every B∈ B and every horizon k, B∈ W_kLongleftrightarrowπ[B]∈overline W_k, B∈ W_*Longleftrightarrowπ[B]∈overline W_*. The progress sets satisfy B∈ R_kLongleftrightarrowπ[B]∈overline R_k, so finite progress ranks agree. A reduced safe or ranked strategy pulls back by choosing at B the profile assigned to π[B]. The observation equality makes π[B] a valid reduced possibility set. Admissibility a…
Exact mathematical passage
B\in R_k\Longleftrightarrow\pi[B]\in\overline R_k
\overline WUse in source · 1 passage
Search aliases: overline W overline
Use in source · Machine_Societies_v1.0.0.tex, line 2255
[Governance abstraction preserves joint realizability] Under~[source reference]--[source reference], for every B∈ B and every horizon k, B∈ W_kLongleftrightarrowπ[B]∈overline W_k, B∈ W_*Longleftrightarrowπ[B]∈overline W_*. The progress sets satisfy B∈ R_kLongleftrightarrowπ[B]∈overline R_k, so finite progress ranks agree. A reduced safe or ranked strategy pulls back by choosing at B the profile assigned to π[B]. The observation equality makes π[B] a valid reduced possibility set. Admissibility a…
Exact mathematical passage
B\in W_k\Longleftrightarrow\pi[B]\in\overline W_k, \qquad B\in W_*\Longleftrightarrow\pi[B]\in\overline W_*.
\overline W_*Use in source · 1 passage
Search aliases: overline W_* overline
Use in source · Machine_Societies_v1.0.0.tex, line 2255
[Governance abstraction preserves joint realizability] Under~[source reference]--[source reference], for every B∈ B and every horizon k, B∈ W_kLongleftrightarrowπ[B]∈overline W_k, B∈ W_*Longleftrightarrowπ[B]∈overline W_*. The progress sets satisfy B∈ R_kLongleftrightarrowπ[B]∈overline R_k, so finite progress ranks agree. A reduced safe or ranked strategy pulls back by choosing at B the profile assigned to π[B]. The observation equality makes π[B] a valid reduced possibility set. Admissibility a…
Exact mathematical passage
B\in W_k\Longleftrightarrow\pi[B]\in\overline W_k, \qquad B\in W_*\Longleftrightarrow\pi[B]\in\overline W_*.
\overline W_kUse in source · 1 passage
Search aliases: overline W_k overline
Use in source · Machine_Societies_v1.0.0.tex, line 2255
[Governance abstraction preserves joint realizability] Under~[source reference]--[source reference], for every B∈ B and every horizon k, B∈ W_kLongleftrightarrowπ[B]∈overline W_k, B∈ W_*Longleftrightarrowπ[B]∈overline W_*. The progress sets satisfy B∈ R_kLongleftrightarrowπ[B]∈overline R_k, so finite progress ranks agree. A reduced safe or ranked strategy pulls back by choosing at B the profile assigned to π[B]. The observation equality makes π[B] a valid reduced possibility set. Admissibility a…
Exact mathematical passage
B\in W_k\Longleftrightarrow\pi[B]\in\overline W_k, \qquad B\in W_*\Longleftrightarrow\pi[B]\in\overline W_*.
\overline{\operatorname{Post}}Use in source · 1 passage
Search aliases: overlinePost overline operatorname
Use in source · Machine_Societies_v1.0.0.tex, line 2264
progress ranks agree. A reduced safe or ranked strategy pulls back by choosing at B the profile assigned to π[B]. The observation equality makes π[B] a valid reduced possibility set. Admissibility and safe-set membership are preserved and reflected. For every possible next observation y, successor equality gives the commuting update identity π[Post(B,u,y)] =overlinePost(π[B],u,y). Indeed, applying π to the union of complete successors gives the union of their reduced successors. The observation …
Exact mathematical passage
\label{eq:synthesis-post-commute} \pi[\operatorname{Post}(B,u,y)] =\overline{\operatorname{Post}}(\pi[B],u,y).
\mathcal PUse in source · 2 passages
Search aliases: P mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 727
Rule-changing computation and governance-preserving abstraction. A persistent society does not merely choose actions under a rule. Some authorized actions change which future actions are admitted. Write a complete state as s=(x,I, P,r,λ,c), where x is operational state, I institutional records, P the current population and office registry, r resources, λ representation or protocol configuration, and c the current checking and amendment mechanism. These coordinates may overlap in storage. They id…
Exact mathematical passage
s=(x,I,\mathcal P,r,\lambda,c),
pUse in source · 49 passages
Search aliases: p
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
p'Use in source · 3 passages
Search aliases: p'
Use in source · Machine_Societies_v1.0.0.tex, line 2805
A deterministic microscopic realization. For any permitted macro-operation T, write (p',q')=T(p,q) and define F_T(a,b,c,d)=(a, a⊕ p', c, c⊕ q'). Then the two local XORs of F_T(a,b,c,d) are exactly (p',q'). Consequently π F_T=Tπ for every microstate, where π(a,b,c,d)=(p,q).
Exact mathematical passage
(p',q')=T(p,q)
p^*Use in source · 1 passage
Search aliases: p^*
Use in source · Machine_Societies_v1.0.0.tex, line 2378
[Inspection without commitment] Under the preceding strict inequalities, the game has no pure Nash equilibrium and has a unique Nash equilibrium in independently mixed strategies. Its deviation and inspection probabilities are p^*=fraccb, q^*=fracgF. In particular, certain compliance is not sustained in this simultaneous game. If the auditor inspects with probability q, the participant's advantage from deviation is g-qF. If the participant deviates with probability p, the auditor's advantage fro…
Exact mathematical passage
p^*=\frac{c}{b},\qquad q^*=\frac{g}{F}.
p^2Use in source · 2 passages
Search aliases: p^2
Use in source · Machine_Societies_v1.0.0.tex, line 1836
[Almost-sure locking with a positive error probability] In this model a cascade begins almost surely after finitely many revealing decisions. Its probability of selecting the wrong state is η(p)=frac(1-p)^2p^2+(1-p)^2>0. The probability that no cascade has begun after 2r decisions is [2p(1-p)]^r. After locking, further actions contain no additional information about θ conditional on the preceding public history. The posterior argument above identifies the revealing region and the two absorbing a…
Exact mathematical passage
\label{eq:cascade-error} \eta(p)=\frac{(1-p)^2}{p^2+(1-p)^2}>0.
P^\star_{z'}Use in source · 1 passage
Search aliases: P^⋆_z' star
Use in source · Machine_Societies_v1.0.0.tex, line 383
[Implementation dependence] If every P_λ is within ε of a proposed macro-intervention law P^⋆_z', then η(x,z')≤2ε. Conversely, if two admissible implementations are distance η apart, no single law can approximate both within less than η/2. Both claims follow from the triangle inequality for total variation.
Exact mathematical passage
P^\star_{z'}
P_0Use in source · 4 passages
Search aliases: P_0
Use in source · Machine_Societies_v1.0.0.tex, line 1285
,1} and a class of admissible protocols. For one protocol, let H be the tuple of all local histories up to the decision, together with the private and shared random tapes used to choose actions. The tapes have their specified law and cannot encode extra information about θ. The joint action is a measurable function of H even though no participant need be able to read all of H. Let P_0 and P_1 be its laws under the two conditions.
Exact mathematical passage
P_0
P_1Use in source · 6 passages
Search aliases: P_1
Use in source · Machine_Societies_v1.0.0.tex, line 1040
[Individually enforceable deviations with no common monitor] Take O={0,1}, P_C=(1/2,1/2), P_1=(1,0), P_2=(0,1), F=2, g_1=g_2=1/2, and α=1/2. The monitor (1,0) strictly deters deviation 1, and (0,1) strictly deters deviation 2. Both respect the false-positive cap. For any common monitor, D_1(s)=s(0)-s(1)-tfrac12, D_2(s)=s(1)-s(0)-tfrac12. Their sum is -1, so both cannot be nonnegative. The best simultaneous margin is -1/2, attained when s(0)=s(1). The dual obstruction is λ=(1/2,1/2), for which P_…
Exact mathematical passage
P_1=(1,0)
P_2Use in source · 2 passages
Search aliases: P_2
Use in source · Machine_Societies_v1.0.0.tex, line 1040
[Individually enforceable deviations with no common monitor] Take O={0,1}, P_C=(1/2,1/2), P_1=(1,0), P_2=(0,1), F=2, g_1=g_2=1/2, and α=1/2. The monitor (1,0) strictly deters deviation 1, and (0,1) strictly deters deviation 2. Both respect the false-positive cap. For any common monitor, D_1(s)=s(0)-s(1)-tfrac12, D_2(s)=s(1)-s(0)-tfrac12. Their sum is -1, so both cannot be nonnegative. The best simultaneous margin is -1/2, attained when s(0)=s(1). The dual obstruction is λ=(1/2,1/2), for which P_…
Exact mathematical passage
P_2=(0,1)
P_aUse in source · 1 passage
Search aliases: P_a
Use in source · Machine_Societies_v1.0.0.tex, line 1573
Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bound and the same declared endpoint-to-evidence channel is used in the actual and averaged model.
Exact mathematical passage
\TV(P_a,\overline P_a)\leq\varepsilon_a,\qquad a\in\{C,D\}.
P_C^tUse in source · 1 passage
Search aliases: P_C^t
Use in source · Machine_Societies_v1.0.0.tex, line 2451
[Conditional disclosure accounting] Consider m releases on finite alphabets. At each common prefix h, suppose their conditional next-release laws P_C^t(·mid h) and P_D^t(·mid h) satisfy both pointwise inequalities with factor λ_t≥1, uniformly over prefixes that can occur. Then the joint transcript laws satisfy~[source reference] with λ=∏_t=1^mλ_t. The bound also holds after applying the same stochastic postprocessing to the transcript under both cases. In particular, any sanction computed only f…
Exact mathematical passage
P_C^t(\cdot\mid h)
P_D^tUse in source · 1 passage
Search aliases: P_D^t
Use in source · Machine_Societies_v1.0.0.tex, line 2451
[Conditional disclosure accounting] Consider m releases on finite alphabets. At each common prefix h, suppose their conditional next-release laws P_C^t(·mid h) and P_D^t(·mid h) satisfy both pointwise inequalities with factor λ_t≥1, uniformly over prefixes that can occur. Then the joint transcript laws satisfy~[source reference] with λ=∏_t=1^mλ_t. The bound also holds after applying the same stochastic postprocessing to the transcript under both cases. In particular, any sanction computed only f…
Exact mathematical passage
P_D^t(\cdot\mid h)
P_{D_1}Use in source · 1 passage
Search aliases: P_D_1
Use in source · Machine_Societies_v1.0.0.tex, line 977
When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gains from both deviations. A deployable certificate must retain the joint system of constraints, not select a different optimal monitor after learning wh…
Exact mathematical passage
P_{D_1}=(1,0)
P_{D_2}Use in source · 1 passage
Search aliases: P_D_2
Use in source · Machine_Societies_v1.0.0.tex, line 977
When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gains from both deviations. A deployable certificate must retain the joint system of constraints, not select a different optimal monitor after learning wh…
Exact mathematical passage
P_{D_2}=(0,1)
P_{D_j}Use in source · 1 passage
Search aliases: P_D_j
Use in source · Machine_Societies_v1.0.0.tex, line 977
When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gains from both deviations. A deployable certificate must retain the joint system of constraints, not select a different optimal monitor after learning wh…
Exact mathematical passage
\E_{P_{D_j}}s-\E_{P_C}s
P_jUse in source · 4 passages
Search aliases: P_j
Use in source · Machine_Societies_v1.0.0.tex, line 1007
The common-monitor problem. Let O be a finite nonempty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous…
Exact mathematical passage
P_j
P_{M,w}Use in source · 1 passage
Search aliases: P_M,w
Use in source · Machine_Societies_v1.0.0.tex, line 1265
[No universal constitutional safety decider] There is no total algorithm that decides safety for every program in this class. In the same class, occurrence of the forbidden event is semidecidable, whereas safety is not semidecidable in general. Given a Turing machine M and input w, construct a program P_M,w that simulates M(w) and produces the forbidden event exactly when the simulation halts. Each simulation step is computable and terminates. The constructed program is safe exactly when M(w) do…
Exact mathematical passage
P_{M,w}
P_tUse in source · 1 passage
Search aliases: P_t
Use in source · Machine_Societies_v1.0.0.tex, line 641
To formalize an operative institution, augment the microstate of Section~[source reference] with a distinguished component I_t. In a finite setting we may describe it as I_t=(N_t,P_t,E_t,A_t,L_t), where N specifies norms and role obligations, P permissions and capabilities, E enforcement and conflict procedures, A admissible amendments, and L records of decisions and their provenance. The components need not be physically stored together, and their representation is not unique. What matters is t…
Exact mathematical passage
\label{eq:institutional-state} I_t=(N_t,P_t,E_t,A_t,L_t),
\mathsf{permissions}Use in source · 1 passage
Search aliases: permissions mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 516
Hierarchy, gluing, and the limits of flattening. A lower-level organization can serve as a component at the next level when it exports a sufficient interface. That interface must describe not only a state but also the operations that are permitted on it, their outcome semantics, and their implementation error. A useful interface certificate is Int=( Z, U,overline K,overline Q,Λ, ε_rm dyn,ε_rm int,permissions). This tuple is a design proposal. It prevents a higher-level controller from treating a…
Exact mathematical passage
\mathsf{Int}=(\Z,\U,\overline K,\overline Q,\Lambda, \varepsilon_{\rm dyn},\varepsilon_{\rm int},\mathsf{permissions}).
\pmodUse in source · 1 passage
Search aliases: pmod
Use in source · Machine_Societies_v1.0.0.tex, line 327
[Deterministic microstate, non-Markov macrostate] Let X_t+1=X_t+1pmod 3 with uniform initial distribution. Set π(0)=π(1)=a and π(2)=b. Stationarity gives P(Z_t+1=bmid Z_t=a)=1/2. Yet P(Z_t+1=bmid Z_t=a,Z_t-1=b)=0, whereas the same probability conditional on Z_t=a,Z_t-1=a is 1. Thus the apparent randomness comes with memory: the one-step marginal does not define a valid autonomous first-order Markov model.
Exact mathematical passage
X_{t+1}=X_t+1\pmod 3
\ProbUse in source · 7 passages
Search aliases: Prob P mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\Prob_\muUse in source · 1 passage
Search aliases: Prob_μ P_μ Prob mu mathbb mu
Use in source · Machine_Societies_v1.0.0.tex, line 1478
[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random variable gives Eτ=∑_t≥0 P(τ>t). Under ρ(Q)<1, the matrix geometric series converges to (Id-Q)^-1, giving the result.
Exact mathematical passage
\Prob_\mu(\tau>t)=\mu Q^t\mathbf1.
qUse in source · 35 passages
Search aliases: q
Use in source · Machine_Societies_v1.0.0.tex, line 185
public record contains the current allocation rule, valid credentials, accepted urgency certificates, outstanding service obligations, and the amendment history. Slots have unique names (e,k), where e is the epoch and k∈{1,2,3,4}. A grant consumes one available slot token. Copying a message that describes a grant does not create an additional token. The resource invariant is therefore ∑_i∈{A,B,C,D}q_i(e)≤4, with no slot name assigned to two offices. In the worked trace all four offices demand th…
Exact mathematical passage
\sum_{i\in\{A,B,C,D\}}q_i(e)\leq4,
q'Use in source · 3 passages
Search aliases: q'
Use in source · Machine_Societies_v1.0.0.tex, line 2805
A deterministic microscopic realization. For any permitted macro-operation T, write (p',q')=T(p,q) and define F_T(a,b,c,d)=(a, a⊕ p', c, c⊕ q'). Then the two local XORs of F_T(a,b,c,d) are exactly (p',q'). Consequently π F_T=Tπ for every microstate, where π(a,b,c,d)=(p,q).
Exact mathematical passage
(p',q')=T(p,q)
q^*Use in source · 1 passage
Search aliases: q^*
Use in source · Machine_Societies_v1.0.0.tex, line 2378
[Inspection without commitment] Under the preceding strict inequalities, the game has no pure Nash equilibrium and has a unique Nash equilibrium in independently mixed strategies. Its deviation and inspection probabilities are p^*=fraccb, q^*=fracgF. In particular, certain compliance is not sustained in this simultaneous game. If the auditor inspects with probability q, the participant's advantage from deviation is g-qF. If the participant deviates with probability p, the auditor's advantage fro…
Exact mathematical passage
p^*=\frac{c}{b},\qquad q^*=\frac{g}{F}.
Q^tUse in source · 3 passages
Search aliases: Q^t
Use in source · Machine_Societies_v1.0.0.tex, line 1478
[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random variable gives Eτ=∑_t≥0 P(τ>t). Under ρ(Q)<1, the matrix geometric series converges to (Id-Q)^-1, giving the result.
Exact mathematical passage
\Prob_\mu(\tau>t)=\mu Q^t\mathbf1.
\mathcal Q_*Use in source · 2 passages
Search aliases: Q_* mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 1434
A finite closure theorem, not an open-endedness theorem. Take finite universes C_* and Q_*. Consider a fixed collection of production rules of the form ``if these finitely many records are present, add this record.'' Prerequisites are positive: no rule requires the absence of a record. The content and correctness of those rules are assumptions. An added method is certified only if the specified certification procedure actually supports it.
Exact mathematical passage
\mathcal Q_*
\mathcal Q_0Use in source · 2 passages
Search aliases: Q_0 mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 1438
Let Φ simultaneously add all conclusions whose prerequisites are satisfied, retaining the input records. Order pairs ( C, Q) by componentwise inclusion. [Least closure of a finite problem--method system] Starting from ( C_0, Q_0), iteration of Φ reaches the least closed pair containing that initial pair. There are at most | C_*∖ C_0|+| Q_*∖ Q_0| strict growth steps. A sequential rule schedule reaches the same closure if every rule that remains enabled with an absent conclusion is eventually appl…
Exact mathematical passage
(\mathcal C_0,\mathcal Q_0)
q_iUse in source · 2 passages
Search aliases: q_i
Use in source · Machine_Societies_v1.0.0.tex, line 185
public record contains the current allocation rule, valid credentials, accepted urgency certificates, outstanding service obligations, and the amendment history. Slots have unique names (e,k), where e is the epoch and k∈{1,2,3,4}. A grant consumes one available slot token. Copying a message that describes a grant does not create an additional token. The resource invariant is therefore ∑_i∈{A,B,C,D}q_i(e)≤4, with no slot name assigned to two offices. In the worked trace all four offices demand th…
Exact mathematical passage
\sum_{i\in\{A,B,C,D\}}q_i(e)\leq4,
q_tUse in source · 2 passages
Search aliases: q_t
Use in source · Machine_Societies_v1.0.0.tex, line 686
hcal U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assigns nonnegative voting weights, q_t is an approval threshold, and e_t is a configuration ep…
Exact mathematical passage
B_t=(M_t,V_t,c_t,w_t,q_t,e_t),
\mathsf{quantity}Use in source · 1 passage
Search aliases: quantity mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 192
ubsectionTwo rules and an inherited obligation Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The exchange creates a registered obligation o=(id,debtor=j,creditor=k, quantity=1,due=e+1). At the next epoch the creditor receives two slots and the debtor n…
Exact mathematical passage
o=(\mathsf{id},\mathsf{debtor}=j,\mathsf{creditor}=k, \mathsf{quantity}=1,\mathsf{due}=e+1).
\mathbb RUse in source · 1 passage
Search aliases: R mathbb
Use in source · Machine_Societies_v1.0.0.tex, line 2008
A transparent comparison model can make one class of failure mathematically tractable. Let x_t∈ R_≥0^n record nonnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞.…
Exact mathematical passage
x_t\in\mathbb R_{\ge0}^n
\mathcal RUse in source · 5 passages
Search aliases: R mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 273
A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distribution, observation maps, and coalition family. It is not equivalent to any one general-purpose partial-information-decomposition statistic~[source re…
Exact mathematical passage
\mathcal R_\ell(O)=\inf_f\E\bigl[\ell(f(O),Y)\bigr].
r'Use in source · 1 passage
Search aliases: r'
Use in source · Machine_Societies_v1.0.0.tex, line 2732
[.5em] What state must persist? · The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens. [.5em] Can a due debt be erased? · The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are required to preserve pending commitments. [.5em] Who can block change? · Two registered offices block a three-of-four rule under independent approval …
Exact mathematical passage
.5em] What state must persist? & The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens.\\[.5em] Can a due debt be erased? & The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are required to preserve pending commitments.\\[.5em] Who can block change? & Two registered offices block a three-of-four rule under independent approval rights. A hidden administrator changes this calculation.\\[.5em] Can every request be safely served? & Resource conservation can hold even when private information or conflicting deadlines prevent timely service. Viability with full information is not decentralized implementability.\\[.5em] What crosses a replacement? & An authorized office handover preserves debt and credential semantics. Copying a model does not duplicate an office's budget or approval right.\\[.5em] Can two jurisdictions merge? & Their translator must preserve the agreed action semantics, permissions, obligations, resources, and evidence coordinates. Nonzero residuals remain visible until an authorized repair is found.\\[.5em] How long does the organization last? & Lifetime is first exit from a declared operational contract under a specified transition law. It is neither occupant longevity nor mere archive retention.\\ \bottomrule \end{longtable} The common event mechanism and constitutional envelope are supplied. Within that envelope, offices can choose rules, incur obligations, revise decisions, and replace occupants. This demonstrates how the theory's objects fit together. It does not prove that the envelope itself arose endogenously or that these rules are uniquely desirable. \subsection{The additional boundary, language, and knowledge contract} An admission changes the membership configuration, not merely the occupant list. A message is interpreted using its type, issuer, and epoch. A resource approval creates the specified ledger event only after its admissibility check. A debtor replacement transports the obligation monitor but does not manufacture new approval rights. A notification history records which office has received which adoption record; an available ledger entry is not automatically common knowledge. For concurrent amendments, the contract states whether ordering is significant, whether a joinable normal form is required, and which external effects remain visible. For federation, the role-label transformations must be checked around ordered loops. A compatible allocation on one invariant subset does not certify synchronization of every role. For inquiry, the commons records unresolved scheduling questions independently of which questions it chooses to fund. For founding authority, it preserves the resource owner's stipulated grant separately from its own subsequent votes. Together these clauses specify what it would mean to preserve this analytical institution across translations and replacements. They do not claim that every useful society needs this exact architecture or that the four-office example exhibits autonomous institutional formation. \subsection{Distributed realization, enforcement, and adaptive staffing} In the atomic reference commons, a unique slot token prevents two successful grants of the same slot. In its replicated variant, two local approvals can each look valid while their merged grant set violates exclusivity. The institution must therefore distinguish a provisional reservation from a committed grant and identify the primitive that prevents double execution. A convergent archive is not that primitive by itself. Accurate urgency reporting is a different problem. A committed monitor can discourage a specified profitable misreport only within the discrimination and penalty budget of Section~\ref{sec:enforcement}. Its false-positive limit, evidence access, and remedy authority remain separate. When an averaged model supplies the observation laws, the margin must include its approximation error. A fixed review schedule and a data-selected review schedule cannot share the same mixing certificate without further work. The commons also distinguishes an urgency vote from the signal supporting it. Decision-only publication can reproduce a mistaken cascade; preservation of independent evidence changes the information available to later reviewers, not the legal effect of an already committed vote. Finally, creating a specialist verifier changes the capability graph and the distribution of authority. A staffing certificate covers all protected roles jointly, while the handover procedure must still deliver credentials, pending debts, and evidence before service deadlines. These clauses supplement the reference semantics rather than claim that the society has generated its own infrastructure without assumptions. \subsection{The joint continuation contract} The bounded audit-and-service variant in Section~\ref{sec:joint-synthesis} combines three protections that cannot be checked in isolation: the common monitor must deter every listed deviation, its false-positive rate must meet the cap, and its resource use must leave enough capacity for all due work. The current public possibility set must also admit the same protocol in every compatible complete state. A continuation certificate records this common choice and all possible successors. A progress rank distinguishes eventual certified service from indefinite safe waiting. An amendment is acceptable under this particular continuation contract only when its current authority is valid and its resulting states retain certified continuations. A transport certificate additionally checks public observations, enabled profiles, safe and goal labels, and successor sets. Replacing an office, changing an audit channel, or adding a slot changes specified parts of that contract. Such a change may repair an obstruction, but does not retroactively fulfill an earlier unmet promise. \FloatBarrier \subsection{The accountability extension} Section~\ref{sec:accountable-commons} deliberately changes the audit catalogue to satisfy its pairwise privacy constraint. It separately includes an outside claim, an admitted appeal, a genuine-execution fee, and a funded final record with revocation. The resulting eight-unit settlement requirement must not be confused with the five-unit service-and-audit variant. The supplied reference trace checks one declared five-profile settlement, while the stopping theorem characterizes an entire finite ending-preserving running kernel. Both are model-relative results, not certificates for an unspecified deployment. \section{A finite algebraic example of reusable state}\label{sec:worked} The following analytical example makes the preceding distinctions concrete. Four components initially hold independent fair bits $(a,b,c,d)$. Two local workspaces expose \[ p=a\oplus b,\qquad q=c\oplus d,
r_{-i}Use in source · 1 passage
Search aliases: r_-i
Use in source · Machine_Societies_v1.0.0.tex, line 869
A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',r_-i)-Φ(r_i,r_-i). This is the familiar potential-game condition~[source reference]; it is not generically satisfied by social institutions.
Exact mathematical passage
u_i(r_i',r_{-i})-u_i(r_i,r_{-i}) =\Phi(r_i',r_{-i})-\Phi(r_i,r_{-i}).
R_0Use in source · 1 passage
Search aliases: R_0
Use in source · Machine_Societies_v1.0.0.tex, line 2200
Let G⊆ S_rm safe be a stipulated completion set. The target is certified completion, meaning that the current possibility set lies wholly in G. Let G={B∈ W_*:B⊆ G}, A_*(B)={u∈ A(B):Succ(B,u)⊆ W_*}. Define R_0= G and R_k+1=R_k∪ {B∈ W_*:∃ u∈ A_*(B), Succ(B,u)⊆ R_k}. The increasing sequence stabilizes at R_*. For B∈ R_*, let r(B) be its first index of entry.
Exact mathematical passage
R_0=\mathcal G
r_1Use in source · 1 passage
Search aliases: r_1
Use in source · Machine_Societies_v1.0.0.tex, line 868
A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',r_-i)-Φ(r_i,r_-i). This is the familiar potential-game condition~[source reference]; it is not generically satisfied by social institutions.
Exact mathematical passage
r=(r_1,\ldots,r_n)
r_FUse in source · 3 passages
Search aliases: r_F
Use in source · Machine_Societies_v1.0.0.tex, line 1462
Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not describe all cultural evolution. It establishes that selection for reproduction alone does not logically impose selection for accuracy. An institution nee…
Exact mathematical passage
r_T,r_F>0
r_F^tUse in source · 1 passage
Search aliases: r_F^t
Use in source · Machine_Societies_v1.0.0.tex, line 1463
Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not describe all cultural evolution. It establishes that selection for reproduction alone does not logically impose selection for accuracy. An institution nee…
Exact mathematical passage
\frac{N_T r_T^t}{N_T r_T^t+N_F r_F^t} =\frac{1}{1+(N_F/N_T)(r_F/r_T)^t}\longrightarrow0.
\mathcal R_iUse in source · 1 passage
Search aliases: R_i mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 868
A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',r_-i)-Φ(r_i,r_-i). This is the familiar potential-game condition~[source reference]; it is not generically satisfied by social institutions.
Exact mathematical passage
r_i\in\mathcal R_i
r_iUse in source · 2 passages
Search aliases: r_i
Use in source · Machine_Societies_v1.0.0.tex, line 868
A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',r_-i)-Φ(r_i,r_-i). This is the familiar potential-game condition~[source reference]; it is not generically satisfied by social institutions.
Exact mathematical passage
r_i\in\mathcal R_i
r_i'Use in source · 1 passage
Search aliases: r_i'
Use in source · Machine_Societies_v1.0.0.tex, line 869
A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',r_-i)-Φ(r_i,r_-i). This is the familiar potential-game condition~[source reference]; it is not generically satisfied by social institutions.
Exact mathematical passage
u_i(r_i',r_{-i})-u_i(r_i,r_{-i}) =\Phi(r_i',r_{-i})-\Phi(r_i,r_{-i}).
r_{ij}^iUse in source · 2 passages
Search aliases: r_ij^i
Use in source · Machine_Societies_v1.0.0.tex, line 548
Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentation of its global sections. These compatibility conditions follow the elementary logic of gluing local descriptions into a shared global state.
Exact mathematical passage
r_{ij}^i
r_{ij}^jUse in source · 2 passages
Search aliases: r_ij^j
Use in source · Machine_Societies_v1.0.0.tex, line 548
Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentation of its global sections. These compatibility conditions follow the elementary logic of gluing local descriptions into a shared global state.
Exact mathematical passage
r_{ij}^j
r_jUse in source · 4 passages
Search aliases: r_j
Use in source · Machine_Societies_v1.0.0.tex, line 2469
A bounded allocation model makes the capacity obligation precise. Let J be a finite set of admitted appeals and let the integer epochs be 1,…,T. At epoch t, there are c_t≥0 reserved review slots after accounting for all other protected work. An appeal j has an evidence-ready release epoch r_j and deadline d_j, with 1≤ r_j≤ d_j≤ T. It requires one slot containing the complete stipulated review and disposition operation. The evidence and qualified authorized reviewer are assumed available througho…
Exact mathematical passage
r_j
R_kUse in source · 4 passages
Search aliases: R_k
Use in source · Machine_Societies_v1.0.0.tex, line 2201
Let G⊆ S_rm safe be a stipulated completion set. The target is certified completion, meaning that the current possibility set lies wholly in G. Let G={B∈ W_*:B⊆ G}, A_*(B)={u∈ A(B):Succ(B,u)⊆ W_*}. Define R_0= G and R_k+1=R_k∪ {B∈ W_*:∃ u∈ A_*(B), Succ(B,u)⊆ R_k}. The increasing sequence stabilizes at R_*. For B∈ R_*, let r(B) be its first index of entry.
Exact mathematical passage
\label{eq:progress-attractor} R_{k+1}=R_k\cup \{B\in W_*:\exists u\in A_*(B),\ \operatorname{Succ}(B,u)\subseteq R_k\}.
R_{k+1}Use in source · 1 passage
Search aliases: R_k+1
Use in source · Machine_Societies_v1.0.0.tex, line 2201
Let G⊆ S_rm safe be a stipulated completion set. The target is certified completion, meaning that the current possibility set lies wholly in G. Let G={B∈ W_*:B⊆ G}, A_*(B)={u∈ A(B):Succ(B,u)⊆ W_*}. Define R_0= G and R_k+1=R_k∪ {B∈ W_*:∃ u∈ A_*(B), Succ(B,u)⊆ R_k}. The increasing sequence stabilizes at R_*. For B∈ R_*, let r(B) be its first index of entry.
Exact mathematical passage
\label{eq:progress-attractor} R_{k+1}=R_k\cup \{B\in W_*:\exists u\in A_*(B),\ \operatorname{Succ}(B,u)\subseteq R_k\}.
r_nUse in source · 1 passage
Search aliases: r_n
Use in source · Machine_Societies_v1.0.0.tex, line 868
A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',r_-i)-Φ(r_i,r_-i). This is the familiar potential-game condition~[source reference]; it is not generically satisfied by social institutions.
Exact mathematical passage
r=(r_1,\ldots,r_n)
r_{\rm end}Use in source · 2 passages
Search aliases: r_rm end rm
Use in source · Machine_Societies_v1.0.0.tex, line 2511
T_k+1 · =T_k∪{B∈ B:B⊆ S_rm safe, ∃ u∈ A_rm end(B), Succ(B,u)⊆ T_k}. Let T_* be the stable set and r_rm end(B) the first entry index. Then define L_0=T_*, L_k+1={B∈ T_*:∃ u∈ A_rm run(B), Succ(B,u)⊆ L_k}, with stable set L_*. These are finite constructions with the same potentially exponential possibility space as before.
Exact mathematical passage
r_{\rm end}(B)
R_{\rm repair}Use in source · 1 passage
Search aliases: R_rm repair rm
Use in source · Machine_Societies_v1.0.0.tex, line 257
A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may be unidentified. They should not be replaced by zero or hidden inside a favorable weighted average.
Exact mathematical passage
\mathcal C_{\mathcal T}=(\varepsilon_{\rm pred},\varepsilon_{\rm dyn},\varepsilon_{\rm int}, \Delta_{\rm coal},R_{\rm repair},D_{\rm lineage},G_{\rm formation}).
r_TUse in source · 3 passages
Search aliases: r_T
Use in source · Machine_Societies_v1.0.0.tex, line 1462
Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not describe all cultural evolution. It establishes that selection for reproduction alone does not logically impose selection for accuracy. An institution nee…
Exact mathematical passage
r_T,r_F>0
r_T^tUse in source · 1 passage
Search aliases: r_T^t
Use in source · Machine_Societies_v1.0.0.tex, line 1463
Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not describe all cultural evolution. It establishes that selection for reproduction alone does not logically impose selection for accuracy. An institution nee…
Exact mathematical passage
\frac{N_T r_T^t}{N_T r_T^t+N_F r_F^t} =\frac{1}{1+(N_F/N_T)(r_F/r_T)^t}\longrightarrow0.
\mathcal R_\ellUse in source · 2 passages
Search aliases: R_ℓ mathcal ell
Use in source · Machine_Societies_v1.0.0.tex, line 273
A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distribution, observation maps, and coalition family. It is not equivalent to any one general-purpose partial-information-decomposition statistic~[source re…
Exact mathematical passage
\mathcal R_\ell(O)=\inf_f\E\bigl[\ell(f(O),Y)\bigr].
\mathbb R_{\ge0}^nUse in source · 1 passage
Search aliases: R_≥0^n mathbb ge
Use in source · Machine_Societies_v1.0.0.tex, line 2008
A transparent comparison model can make one class of failure mathematically tractable. Let x_t∈ R_≥0^n record nonnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞.…
Exact mathematical passage
x_t\in\mathbb R_{\ge0}^n
\rceilUse in source · 2 passages
Search aliases: rceil
Use in source · Machine_Societies_v1.0.0.tex, line 457
[Exact finite-memory lower bound] Every deterministic summary that supports exact continuation on these histories requires at least N distinct states, hence at least lceillog_2Nrceil fixed-length bits. If the permitted continuation family is closed under prefixing and the history equivalence is a transition congruence, the quotient classes themselves provide an exact state representation. If a summary assigns the same state to two inequivalent histories, some permitted continuation requires diff…
Exact mathematical passage
\lceil\log_2N\rceil
\mathsf{Reach}Use in source · 2 passages
Search aliases: Reach mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 1896
[Common knowledge as a reachability condition] Let Reach_J(w) be the worlds reachable from w by a finite chain of relations ∼_i with i∈ J, including the zero-length chain. Then wmodels C_Jφ Longleftrightarrow Reach_J(w)⊆{v:vmodelsφ}. Induction on n shows that E_J^nφ requires φ at the endpoint of every chain of n indistinguishability steps. Requiring this for every finite n is exactly the stated reachability inclusion.
Exact mathematical passage
\mathsf{Reach}_J(w)
\mathsf{Reach}_JUse in source · 2 passages
Search aliases: Reach_J mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 1896
[Common knowledge as a reachability condition] Let Reach_J(w) be the worlds reachable from w by a finite chain of relations ∼_i with i∈ J, including the zero-length chain. Then wmodels C_Jφ Longleftrightarrow Reach_J(w)⊆{v:vmodelsφ}. Induction on n shows that E_J^nφ requires φ at the endpoint of every chain of n indistinguishability steps. Requiring this for every finite n is exactly the stated reachability inclusion.
Exact mathematical passage
\mathsf{Reach}_J(w)
\mathsf{Remedy}Use in source · 2 passages
Search aliases: Remedy mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2343
A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. Standing specifies who may initiate a challenge and on which grounds. Remedy specifies the actions an authorized review can actually require. The cont…
Exact mathematical passage
\mathcal B=(Z_{\rm ext},O_{\rm ext},\mathsf{Auth},\mathsf{Effect}, \mathsf{Standing},\mathsf{Remedy}).
\mathsf{Revision}Use in source · 1 passage
Search aliases: Revision mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2539
An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a component, or unavailable. Those statuses should not all become the Boolean value verified. A failed guard means that continued reliance is not justified by t…
Exact mathematical passage
\mathcal K=(\mathsf{Claim},\mathsf{Assumptions},\mathsf{Evidence}, \mathsf{Domain},\mathsf{Guard},\mathsf{Fallback},\mathsf{Revision}).
\rfloorUse in source · 1 passage
Search aliases: rfloor
Use in source · Machine_Societies_v1.0.0.tex, line 903
[Minimum profitable coordinated seed] Starting from universal nonadoption, a simultaneously adopting coalition of size m makes each member strictly better off exactly when b(m-1)>c. When such a coalition exists, the smallest size is m_* = lfloorfrac cbrfloor+2≤ n. Once at least m_* adopters are present, every remaining nonadopter has a profitable unilateral entry. Under a schedule that eventually updates every remaining agent, strict best responses reach universal adoption without an adopter exi…
Exact mathematical passage
m_* = \left\lfloor\frac cb\right\rfloor+2\leq n.
s'Use in source · 3 passages
Search aliases: s'
Use in source · Machine_Societies_v1.0.0.tex, line 1115
Let s=(x,I) be the full operating state and let R(s,s') be the admitted transition relation, including ordinary actions and amendments. A safety property is a predicate Ψ(s).
Exact mathematical passage
R(s,s')
s^\starUse in source · 2 passages
Search aliases: s^⋆ star
Use in source · Machine_Societies_v1.0.0.tex, line 1579
[A robust margin for reduced-model enforcement] Let s^⋆(o) sanction precisely where overline P_D(o)>overline P_C(o) and put overline d=d_TV(overline P_D,overline P_C). In the model of Section~[source reference], this fixed policy makes compliance weakly preferable whenever g≤ F(overline d-ε_C-ε_D). Conversely, if g>Fmin{1,overline d+ε_C+ε_D}, no observation-based sanction with penalty F deters the deviation. Also, E_P_Cs^⋆≤ E_overline P_Cs^⋆+ε_C. The expectation of a [0,1]-valued function change…
Exact mathematical passage
s^\star(o)
s_0Use in source · 1 passage
Search aliases: s_0
Use in source · Machine_Societies_v1.0.0.tex, line 1118
[Constitutional inductive safety] If Ψ(s_0) and every admitted transition satisfies Ψ(s)wedge R(s,s')LongrightarrowΨ(s'), then Ψ(s_t) holds for every state reachable by finitely many admitted transitions. Induction on the number of admitted transitions.
Exact mathematical passage
\Psi(s_0)
S_{1,t}Use in source · 1 passage
Search aliases: S_1,t
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
S_3Use in source · 2 passages
Search aliases: S_3
Use in source · Machine_Societies_v1.0.0.tex, line 1660
A three-jurisdiction example. Take G=S_3, acting on three role labels. Let the translations on the oriented triangle A→ B→ C→ A be g_AB=(12), g_BC=(23), g_CA=e_G. Every edge is a valid bijection. With the specified composition convention, the round trip is (23)(12)≠ e_G. Thus no single assignment of complete role frames realizes all three interfaces. Replacing the final edge by ((23)(12))^-1 repairs this loop, but whether that edit is authorized is a separate institutional question.
Exact mathematical passage
G=S_3
S_AUse in source · 3 passages
Search aliases: S_A
Use in source · Machine_Societies_v1.0.0.tex, line 787
Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated office names, configuration epochs, and reference records. The translation of a message sequence is constructed step by step from the evolving source st…
Exact mathematical passage
S_A,S_B
S_BUse in source · 2 passages
Search aliases: S_B
Use in source · Machine_Societies_v1.0.0.tex, line 787
Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated office names, configuration epochs, and reference records. The translation of a message sequence is constructed step by step from the evolving source st…
Exact mathematical passage
S_A,S_B
S_iUse in source · 1 passage
Search aliases: S_i
Use in source · Machine_Societies_v1.0.0.tex, line 237
One possible factorization of a microstate is X_t=(S_1,t,…,S_n,t,M_t,C_t,A_t,Θ_t), where S_i is local process state, M durable storage, C messages and channel state, A artifacts, and Θ the current configuration, including executable rules. These categories describe a storage and access organization. They need not be disjoint physical objects.
Exact mathematical passage
S_i
S_{n,t}Use in source · 1 passage
Search aliases: S_n,t
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
s_tUse in source · 5 passages
Search aliases: s_t
Use in source · Machine_Societies_v1.0.0.tex, line 1122
[Constitutional inductive safety] If Ψ(s_0) and every admitted transition satisfies Ψ(s)wedge R(s,s')LongrightarrowΨ(s'), then Ψ(s_t) holds for every state reachable by finitely many admitted transitions. Induction on the number of admitted transitions.
Exact mathematical passage
\Psi(s_t)
\mathsf{safe}Use in source · 3 passages
Search aliases: safe mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 1131
[An unchecked change of the checker] Let the institutional state contain a validator mode v∈{strict,open} and a resource policy p∈{safe,unsafe}. Under v=strict, the current policy must remain safe, but the validator mode itself is unprotected. Thus (strict,safe)→(open,safe) is admitted. Under v=open, any policy transition is admitted, including (open,safe)→(open,unsafe). The original current-policy check does not imply that all reachable states remain safe. To prove that property, the inductive …
Exact mathematical passage
p\in\{\mathsf{safe},\mathsf{unsafe}\}
\sectionUse in source · 2 passages
Search aliases: section
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\mathsf{settled}Use in source · 2 passages
Search aliases: settled mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 1328
For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy follows from admitting discharge in this particular contract.
Exact mathematical passage
\mathbf G\bigl(\mathsf{created}(o)\Rightarrow \mathbf F_{\leq D}\mathsf{settled}(o)\bigr).
\mathsf{Standing}Use in source · 2 passages
Search aliases: Standing mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2343
A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. Standing specifies who may initiate a challenge and on which grounds. Remedy specifies the actions an authorized review can actually require. The cont…
Exact mathematical passage
\mathcal B=(Z_{\rm ext},O_{\rm ext},\mathsf{Auth},\mathsf{Effect}, \mathsf{Standing},\mathsf{Remedy}).
\mathsf{status}Use in source · 1 passage
Search aliases: status mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 1322
tibility, and knowledge Temporal obligations, contracts, and office succession. Permissions describe available actions; obligations constrain continuations. The debt created by an urgency exchange is not satisfied merely because the creditor remains authorized to ask for service. It requires a particular future event. A finite obligation record can be written as o=(id,i,j,φ,d,authority,status), where i and j are the debtor and beneficiary offices, φ specifies a fulfillment event, and d is a dead…
Exact mathematical passage
o=(\mathsf{id},i,j,\varphi,d,\mathsf{authority},\mathsf{status}),
\mathsf{strict}Use in source · 3 passages
Search aliases: strict mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 1131
[An unchecked change of the checker] Let the institutional state contain a validator mode v∈{strict,open} and a resource policy p∈{safe,unsafe}. Under v=strict, the current policy must remain safe, but the validator mode itself is unprotected. Thus (strict,safe)→(open,safe) is admitted. Under v=open, any policy transition is admitted, including (open,safe)→(open,unsafe). The original current-policy check does not imply that all reachable states remain safe. To prove that property, the inductive …
Exact mathematical passage
v\in\{\mathsf{strict},\mathsf{open}\}
\subsectionUse in source · 2 passages
Search aliases: subsection
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\succUse in source · 1 passage
Search aliases: succ
Use in source · Machine_Societies_v1.0.0.tex, line 1089
Social choice and principled disagreement. Plurality cannot be retained while silently replacing all preferences with one externally supplied reward. Even agents that agree on every fact can rank institutional alternatives differently. Let the alternatives be three policies R,U,V and let three voters have strict preferences Rsucc_1 Usucc_1 V, Usucc_2 Vsucc_2 R, Vsucc_3 Rsucc_3 U.
Exact mathematical passage
R\succ_1 U\succ_1 V,\qquad U\succ_2 V\succ_2 R,\qquad V\succ_3 R\succ_3 U.
\succ_1Use in source · 1 passage
Search aliases: succ_1 succ
Use in source · Machine_Societies_v1.0.0.tex, line 1089
Social choice and principled disagreement. Plurality cannot be retained while silently replacing all preferences with one externally supplied reward. Even agents that agree on every fact can rank institutional alternatives differently. Let the alternatives be three policies R,U,V and let three voters have strict preferences Rsucc_1 Usucc_1 V, Usucc_2 Vsucc_2 R, Vsucc_3 Rsucc_3 U.
Exact mathematical passage
R\succ_1 U\succ_1 V,\qquad U\succ_2 V\succ_2 R,\qquad V\succ_3 R\succ_3 U.
\succ_2Use in source · 1 passage
Search aliases: succ_2 succ
Use in source · Machine_Societies_v1.0.0.tex, line 1089
Social choice and principled disagreement. Plurality cannot be retained while silently replacing all preferences with one externally supplied reward. Even agents that agree on every fact can rank institutional alternatives differently. Let the alternatives be three policies R,U,V and let three voters have strict preferences Rsucc_1 Usucc_1 V, Usucc_2 Vsucc_2 R, Vsucc_3 Rsucc_3 U.
Exact mathematical passage
R\succ_1 U\succ_1 V,\qquad U\succ_2 V\succ_2 R,\qquad V\succ_3 R\succ_3 U.
\succ_3Use in source · 1 passage
Search aliases: succ_3 succ
Use in source · Machine_Societies_v1.0.0.tex, line 1089
Social choice and principled disagreement. Plurality cannot be retained while silently replacing all preferences with one externally supplied reward. Even agents that agree on every fact can rank institutional alternatives differently. Let the alternatives be three policies R,U,V and let three voters have strict preferences Rsucc_1 Usucc_1 V, Usucc_2 Vsucc_2 R, Vsucc_3 Rsucc_3 U.
Exact mathematical passage
R\succ_1 U\succ_1 V,\qquad U\succ_2 V\succ_2 R,\qquad V\succ_3 R\succ_3 U.
\sup_{e\in\mathcal E}Use in source · 1 passage
Search aliases: sup_e∈ E sup in mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 653
For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared family, not universal inertness. An enforceable institution can also alter which actions are institutionally permitted while producing the same observed …
Exact mathematical passage
\label{eq:inst-effect} \Delta_{\mathrm{inst}}^{\mathcal E,T}(I,I') =\sup_{e\in\mathcal E}\TV\bigl(\Law_e(Y_{0:T}\mid\lambda(I)),\Law_e(Y_{0:T}\mid\lambda(I'))\bigr).
\sup_{i,x}Use in source · 1 passage
Search aliases: sup_i,x sup
Use in source · Machine_Societies_v1.0.0.tex, line 1525
Assume K_i has a stationary law μ_i and sup_i,xd_TV(K_i^m(x,·),μ_i)≤ε. The actual one-review law from (i,x) and its stationary average are P(i,x;j) · =∑_y K_i^m(x,y)R_i(y,j),
Exact mathematical passage
\label{eq:uniform-fast-mixing} \sup_{i,x}\TV\bigl(K_i^m(x,\cdot),\mu_i\bigr)\leq\varepsilon.
\sup_{x,u}Use in source · 1 passage
Search aliases: sup_x,u sup
Use in source · Machine_Societies_v1.0.0.tex, line 338
Finite-horizon error accounting. For finite distributions, d_TV(μ,ν)=tfrac12∑_z|μ(z)-ν(z)|. Suppose candidate kernels satisfy sup_x,ud_TV(push K_u(x,·),overline K_u(π(x),·)) ≤ε. This uniform condition is stronger than a small average error on the trajectories seen during training.
Exact mathematical passage
\label{eq:eps} \sup_{x,u}\TV\bigl(\push K_u(x,\cdot),\overline K_u(\pi(x),\cdot)\bigr) \leq\varepsilon.
\sup_zUse in source · 1 passage
Search aliases: sup_z sup
Use in source · Machine_Societies_v1.0.0.tex, line 1721
For a translation T:Z_i→ Z_j, define ε_rm dyn(T) · =sup_z,ad_j(TF_i^a(z),F_j^aT(z)), ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The symmetric permission residual measures both changes and is delib…
Exact mathematical passage
\begin{split} \varepsilon_{\rm dyn}(T) &=\sup_{z,a}d_j\bigl(TF_i^a(z),F_j^aT(z)\bigr),\\ \varepsilon_c(T) &=\sup_z\delta_c\bigl(g_j^cT(z),g_i^c(z)\bigr) \quad(c=P,O,R,E). \end{split}
\sup_{z,a}Use in source · 1 passage
Search aliases: sup_z,a sup
Use in source · Machine_Societies_v1.0.0.tex, line 1721
For a translation T:Z_i→ Z_j, define ε_rm dyn(T) · =sup_z,ad_j(TF_i^a(z),F_j^aT(z)), ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The symmetric permission residual measures both changes and is delib…
Exact mathematical passage
\begin{split} \varepsilon_{\rm dyn}(T) &=\sup_{z,a}d_j\bigl(TF_i^a(z),F_j^aT(z)\bigr),\\ \varepsilon_c(T) &=\sup_z\delta_c\bigl(g_j^cT(z),g_i^c(z)\bigr) \quad(c=P,O,R,E). \end{split}
\sup_{z\in Z_j}Use in source · 1 passage
Search aliases: sup_z∈ Z_j sup in
Use in source · Machine_Societies_v1.0.0.tex, line 1374
Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε_j=sup_z∈ Z_jd_j+1(T_jF_j(z),F_j+1T_j(z)). An exact zero defect means the representation change intertwines the declared operation. It does not mean a…
Exact mathematical passage
\label{eq:transport-defect} \varepsilon_j=\sup_{z\in Z_j}d_{j+1}\bigl(T_jF_j(z),F_{j+1}T_j(z)\bigr).
\sup_{\tau\in\mathcal T}Use in source · 1 passage
Search aliases: sup_τ∈ T sup tau in mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 415
For approximate comparison, use the pseudometric d_ T(x,x')=sup_τ∈ T d_TV( L_x(Y^+midτ), L_x'(Y^+midτ)). A finite test suite may make distinct implementations distance zero. That is indistinguishability under those tests, not universal sameness.
Exact mathematical passage
d_{\mathcal T}(x,x')=\sup_{\tau\in\mathcal T} \TV\bigl(\Law_x(Y^+\mid\tau),\Law_{x'}(Y^+\mid\tau)\bigr).
T_0Use in source · 2 passages
Search aliases: T_0
Use in source · Machine_Societies_v1.0.0.tex, line 1380
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))
Exact mathematical passage
T_{0:m}=T_{m-1}\cdots T_0
T_{0:m}Use in source · 2 passages
Search aliases: T_0:m
Use in source · Machine_Societies_v1.0.0.tex, line 1380
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))
Exact mathematical passage
T_{0:m}=T_{m-1}\cdots T_0
T_{0:m-1}Use in source · 1 passage
Search aliases: T_0:m-1
Use in source · Machine_Societies_v1.0.0.tex, line 1390
· ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second by ε_m-1. The recurrence D_m≤ L_m-1D_m-1+ε_m-1 unfolds to~[source reference].
Exact mathematical passage
T_{m-1}F_{m-1}T_{0:m-1}(z)
T_{AB}Use in source · 1 passage
Search aliases: T_AB
Use in source · Machine_Societies_v1.0.0.tex, line 806
tion of finite institutional conversations] Under Definition~[source reference], every finite source conversation has an admitted prefix exactly when its stepwise translation has that admitted prefix from the translated initial state. Corresponding reached states are related by T, and corresponding event records agree. Two faithful transports compose faithfully using τ_AC(s,m)=τ_BC(T_AB(s),τ_AB(s,m)). The initial states are related by definition. At each step, equivalence of definedness gives th…
Exact mathematical passage
\tau_{AC}(s,m)=\tau_{BC}(T_{AB}(s),\tau_{AB}(s,m)).
t_iUse in source · 4 passages
Search aliases: t_i
Use in source · Machine_Societies_v1.0.0.tex, line 864
In an idealized transferable-utility setting there is a corresponding conditional observation. If the discounted net advantages b_i satisfy ∑_i b_i≥0, then there exist budget-balanced transfers t_i with ∑_i t_i=0 and b_i+t_i≥0 for all i: choose nonnegative w_i summing to ∑_i b_i and set t_i=w_i-b_i. This algebra does not supply liquidity, a transfer mechanism, consent, enforceability, or a right to redistribute. The difference between mathematical feasible compensation and institutionally permit…
Exact mathematical passage
t_i
T_{ij}Use in source · 2 passages
Search aliases: T_ij
Use in source · Machine_Societies_v1.0.0.tex, line 1630
Even when a global static state exists, a federation may fail dynamically. Local rule applications could commute in one context and conflict in another. A translator T_ij:Z_i→ Z_j should preserve the operations relevant to the interaction, approximately satisfying T_ijF_i≃ F_jT_ij. A federation requires both representational compatibility and admissible operational coordination. Static agreement alone does not define an executable constitution. Nor does faithful gluing prove that the institution…
Exact mathematical passage
T_{ij}:Z_i\to Z_j
T_jUse in source · 3 passages
Search aliases: T_j
Use in source · Machine_Societies_v1.0.0.tex, line 1373
Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε_j=sup_z∈ Z_jd_j+1(T_jF_j(z),F_j+1T_j(z)). An exact zero defect means the representation change intertwines the declared operation. It does not mean a…
Exact mathematical passage
T_j:Z_j\to Z_{j+1}
T_kUse in source · 2 passages
Search aliases: T_k
Use in source · Machine_Societies_v1.0.0.tex, line 2505
Define T_0 · ={B∈ B:B⊆ G}, T_k+1 · =T_k∪{B∈ B:B⊆ S_rm safe, ∃ u∈ A_rm end(B), Succ(B,u)⊆ T_k}. Let T_* be the stable set and r_rm end(B) the first entry index. Then define L_0=T_*, L_k+1={B∈ T_*:∃ u∈ A_rm run(B), Succ(B,u)⊆ L_k}, with stable set L_*. These are finite constructions with the same potentially exponential possibility space as before.
Exact mathematical passage
T_0&=\{B\in\mathcal B:B\subseteq G\},\label{eq:stop-target}\\ T_{k+1}&=T_k\cup\{B\in\mathcal B:B\subseteq S_{\rm safe},\ \exists u\in A_{\rm end}(B),\ \operatorname{Succ}(B,u)\subseteq T_k\}. \label{eq:stop-attractor}
T_{k+1}Use in source · 1 passage
Search aliases: T_k+1
Use in source · Machine_Societies_v1.0.0.tex, line 2505
Define T_0 · ={B∈ B:B⊆ G}, T_k+1 · =T_k∪{B∈ B:B⊆ S_rm safe, ∃ u∈ A_rm end(B), Succ(B,u)⊆ T_k}. Let T_* be the stable set and r_rm end(B) the first entry index. Then define L_0=T_*, L_k+1={B∈ T_*:∃ u∈ A_rm run(B), Succ(B,u)⊆ L_k}, with stable set L_*. These are finite constructions with the same potentially exponential possibility space as before.
Exact mathematical passage
T_0&=\{B\in\mathcal B:B\subseteq G\},\label{eq:stop-target}\\ T_{k+1}&=T_k\cup\{B\in\mathcal B:B\subseteq S_{\rm safe},\ \exists u\in A_{\rm end}(B),\ \operatorname{Succ}(B,u)\subseteq T_k\}. \label{eq:stop-attractor}
T_{m-1}Use in source · 3 passages
Search aliases: T_m-1
Use in source · Machine_Societies_v1.0.0.tex, line 1380
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))
Exact mathematical passage
T_{0:m}=T_{m-1}\cdots T_0
UUse in source · 32 passages
Search aliases: U
Use in source · Machine_Societies_v1.0.0.tex, line 191
Two rules and an inherited obligation. Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The exchange creates a registered obligation o=(id,debtor=j,creditor=k, quantity=1,due=e+1). At the next epoch the creditor receives two slots and the debtor none, wit…
Exact mathematical passage
U
\UUse in source · 2 passages
Search aliases: U mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
u_iUse in source · 4 passages
Search aliases: u_i
Use in source · Machine_Societies_v1.0.0.tex, line 868
A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',r_-i)-Φ(r_i,r_-i). This is the familiar potential-game condition~[source reference]; it is not generically satisfied by social institutions.
Exact mathematical passage
u_i(r)
U_tUse in source · 1 passage
Search aliases: U_t
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\mathsf{unsafe}Use in source · 2 passages
Search aliases: unsafe mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 1131
[An unchecked change of the checker] Let the institutional state contain a validator mode v∈{strict,open} and a resource policy p∈{safe,unsafe}. Under v=strict, the current policy must remain safe, but the validator mode itself is unprotected. Thus (strict,safe)→(open,safe) is admitted. Under v=open, any policy transition is admitted, including (open,safe)→(open,unsafe). The original current-policy check does not imply that all reachable states remain safe. To prove that property, the inductive …
Exact mathematical passage
p\in\{\mathsf{safe},\mathsf{unsafe}\}
V_0Use in source · 2 passages
Search aliases: V_0
Use in source · Machine_Societies_v1.0.0.tex, line 1140
Define V_0=S and recursively, for h≥0, V_h+1={s∈ S:∃ a∈ A(s) such that suppK(·mid s,a)⊆ V_h}. [Finite-horizon almost-sure institutional viability] A state s belongs to V_h if and only if there exists an admissible policy under which every state visited over the next h transitions, including the present state, lies in S with probability one. Policies may use the full observed state and the remaining horizon. For s∈ V_h, a time-dependent Markov policy suffices. The claim at h=0 is V_0=S. Suppose i…
Exact mathematical passage
V_0=S
v_0Use in source · 6 passages
Search aliases: v_0
Use in source · Machine_Societies_v1.0.0.tex, line 1641
Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
\gamma=(v_0,\ldots,v_k)
V_1Use in source · 1 passage
Search aliases: V_1
Use in source · Machine_Societies_v1.0.0.tex, line 1031
-monitor model, Γ_α=min_λ∈Δ_m (FV_α(P_λ,P_C)-g_λ). Both extrema are attained. Consequently, one monitor weakly deters all listed deviations if and only if FV_α(P_λ,P_C)≥ g_λ for every λ∈Δ_m. Without an additional false-positive restriction, α=1 and V_1(P,P_C)=d_TV(P,P_C). For any fixed s, the minimum of the numbers D_j(s) is the minimum of ∑_jλ_jD_j(s) over Δ_m. This expression is bilinear in s and λ, and both domains are nonempty compact polytopes. Finite minimax, equivalently linear programmin…
Exact mathematical passage
V_1(P,P_C)=\TV(P,P_C)
v_1Use in source · 2 passages
Search aliases: v_1
Use in source · Machine_Societies_v1.0.0.tex, line 702
[Weight-conserving refinement of controlled votes] Fix a weighted threshold rule with threshold q. Replace a voting position v of weight w(v) by positions v_1,…,v_k controlled by the same decision center, with nonnegative weights satisfying ∑_j w(v_j)=w(v). Leave all other weights and the threshold unchanged. Then every coalition of decision centers has the same total voting weight and the same approval status before and after the replacement. A coalition not containing the controller of v gains…
Exact mathematical passage
v_1,\ldots,v_k
V_hUse in source · 4 passages
Search aliases: V_h
Use in source · Machine_Societies_v1.0.0.tex, line 1141
Define V_0=S and recursively, for h≥0, V_h+1={s∈ S:∃ a∈ A(s) such that suppK(·mid s,a)⊆ V_h}. [Finite-horizon almost-sure institutional viability] A state s belongs to V_h if and only if there exists an admissible policy under which every state visited over the next h transitions, including the present state, lies in S with probability one. Policies may use the full observed state and the remaining horizon. For s∈ V_h, a time-dependent Markov policy suffices. The claim at h=0 is V_0=S. Suppose i…
Exact mathematical passage
\label{eq:viability} V_{h+1}=\Bigl\{s\in S:\exists a\in A(s)\;\text{such that}\; \operatorname{supp}K(\cdot\mid s,a)\subseteq V_h\Bigr\}.
V_{h+1}Use in source · 2 passages
Search aliases: V_h+1
Use in source · Machine_Societies_v1.0.0.tex, line 1141
Define V_0=S and recursively, for h≥0, V_h+1={s∈ S:∃ a∈ A(s) such that suppK(·mid s,a)⊆ V_h}. [Finite-horizon almost-sure institutional viability] A state s belongs to V_h if and only if there exists an admissible policy under which every state visited over the next h transitions, including the present state, lies in S with probability one. Policies may use the full observed state and the remaining horizon. For s∈ V_h, a time-dependent Markov policy suffices. The claim at h=0 is V_0=S. Suppose i…
Exact mathematical passage
\label{eq:viability} V_{h+1}=\Bigl\{s\in S:\exists a\in A(s)\;\text{such that}\; \operatorname{supp}K(\cdot\mid s,a)\subseteq V_h\Bigr\}.
v_iUse in source · 1 passage
Search aliases: v_i
Use in source · Machine_Societies_v1.0.0.tex, line 2013
nnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-step monotonic estimate followed by induction. Under uniformly bounded disturbance normw_t…
Exact mathematical passage
\norm{x}_{v,\infty}=\max_i x_i/v_i
v_jUse in source · 2 passages
Search aliases: v_j
Use in source · Machine_Societies_v1.0.0.tex, line 702
[Weight-conserving refinement of controlled votes] Fix a weighted threshold rule with threshold q. Replace a voting position v of weight w(v) by positions v_1,…,v_k controlled by the same decision center, with nonnegative weights satisfying ∑_j w(v_j)=w(v). Leave all other weights and the threshold unchanged. Then every coalition of decision centers has the same total voting weight and the same approval status before and after the replacement. A coalition not containing the controller of v gains…
Exact mathematical passage
\sum_j w(v_j)=w(v)
v_kUse in source · 5 passages
Search aliases: v_k
Use in source · Machine_Societies_v1.0.0.tex, line 702
[Weight-conserving refinement of controlled votes] Fix a weighted threshold rule with threshold q. Replace a voting position v of weight w(v) by positions v_1,…,v_k controlled by the same decision center, with nonnegative weights satisfying ∑_j w(v_j)=w(v). Leave all other weights and the threshold unchanged. Then every coalition of decision centers has the same total voting weight and the same approval status before and after the replacement. A coalition not containing the controller of v gains…
Exact mathematical passage
v_1,\ldots,v_k
v_{k-1}Use in source · 1 passage
Search aliases: v_k-1
Use in source · Machine_Societies_v1.0.0.tex, line 1642
Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on every edge.
Exact mathematical passage
H_\gamma=g_{v_{k-1}v_k}\cdots g_{v_0v_1}.
V_TUse in source · 1 passage
Search aliases: V_T
Use in source · Machine_Societies_v1.0.0.tex, line 843
[Institutional amortization] Under stationary costs and infinite-horizon discounting, the institution's expected discounted cost advantage over the baseline is V=-B+fracg1-δ. For a finite horizon T, it is V_T=-B+g(1-δ^T)/(1-δ). The institution is cost-improving under these assumptions exactly when the corresponding value is positive. Subtract institutional operating and maintenance costs from baseline costs in every period, discount, sum the finite or infinite geometric series, and subtract B.
Exact mathematical passage
V_T=-B+g(1-\delta^T)/(1-\delta)
V_\alphaUse in source · 4 passages
Search aliases: V_α alpha
Use in source · Machine_Societies_v1.0.0.tex, line 1020
For a probability vector λ∈Δ_m, set P_λ=∑_jλ_jP_j and g_λ=∑_jλ_jg_j. Write V_α(P,P_C)=max_s∈ M_α⟨ P-P_C,s⟩. This is the same constrained observation advantage used in the single-deviation analysis, now evaluated on mixtures of deviation laws.
Exact mathematical passage
V_\alpha(P,P_C)=\max_{s\in\mathcal M_\alpha}\langle P-P_C,s\rangle.
\operatorname{Var}Use in source · 1 passage
Search aliases: Var operatorname
Use in source · Machine_Societies_v1.0.0.tex, line 1793
Suppose errors E_1,…,E_n in scalar estimates are centered, have common variance σ^2>0, and have common pairwise correlation ρ∈[0,1]. Then Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equal-variance estimators, the variance-equivalent effective sample size is n_eff=fracn1+(n-1)ρ. If the errors are perfectly correlated, adding copies doe…
Exact mathematical passage
\label{eq:correlated} \operatorname{Var}\left(\frac1n\sum_{i=1}^n E_i\right) =\frac{\sigma^2}{n}\bigl(1+(n-1)\rho\bigr).
\veeUse in source · 5 passages
Search aliases: vee
Use in source · Machine_Societies_v1.0.0.tex, line 1205
Convergent records and valid decisions. Let (L,vee) be a join-semilattice, with order a≤ b when avee b=b. A replica holds ℓ_i∈ L. In the following restricted model a local update joins a declared contribution δ into the state, and a merge replaces ℓ_i by ℓ_iveeℓ_j. Contributions are fixed elements of L once created. This is a state-merge construction of the kind used in conflict-free replicated data types~[source reference]. The algebra concerns accepted data, not the authenticity of its source.
Exact mathematical passage
(L,\vee)
\mathcal WUse in source · 5 passages
Search aliases: W mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 523
Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.
Exact mathematical passage
\psi:\Z\to\mathcal W
W_AUse in source · 2 passages
Search aliases: W_A
Use in source · Machine_Societies_v1.0.0.tex, line 1929
Consider models on W_A,W_B with the same named agents and atomic predicates. Let T:W_A→ W_B be surjective and preserve the truth of each atom in both directions. For every agent i, require w∼_i^A v · Longrightarrow T(w)∼_i^B T(v) (forth),
Exact mathematical passage
W_A,W_B
W_BUse in source · 2 passages
Search aliases: W_B
Use in source · Machine_Societies_v1.0.0.tex, line 1929
Consider models on W_A,W_B with the same named agents and atomic predicates. Let T:W_A→ W_B be surjective and preserve the truth of each atom in both directions. For every agent i, require w∼_i^A v · Longrightarrow T(w)∼_i^B T(v) (forth),
Exact mathematical passage
W_A,W_B
w_eUse in source · 2 passages
Search aliases: w_e
Use in source · Machine_Societies_v1.0.0.tex, line 1772
[Cycle residual and minimum authorized repair] The XOR of edge labels around a cycle is unchanged by every local convention change above. On a single simple cycle with odd XOR, suppose an editable subset of edges can be flipped at nonnegative costs w_e, with other edges protected. An authorized repair exists exactly when the editable subset is nonempty. The minimum cost is then min_e w_e over editable edges. If all edges are protected, no change of local binary conventions can produce a compatib…
Exact mathematical passage
w_e
W_iUse in source · 1 passage
Search aliases: W_i
Use in source · Machine_Societies_v1.0.0.tex, line 548
Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentation of its global sections. These compatibility conditions follow the elementary logic of gluing local descriptions into a shared global state.
Exact mathematical passage
W_i
w_iUse in source · 3 passages
Search aliases: w_i
Use in source · Machine_Societies_v1.0.0.tex, line 549
Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentation of its global sections. These compatibility conditions follow the elementary logic of gluing local descriptions into a shared global state.
Exact mathematical passage
W_{\rm glue}=\{(w_i):r_{ij}^i(w_i)=r_{ij}^j(w_j)\text{ on every declared overlap}\}.
w_jUse in source · 2 passages
Search aliases: w_j
Use in source · Machine_Societies_v1.0.0.tex, line 549
Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentation of its global sections. These compatibility conditions follow the elementary logic of gluing local descriptions into a shared global state.
Exact mathematical passage
W_{\rm glue}=\{(w_i):r_{ij}^i(w_i)=r_{ij}^j(w_j)\text{ on every declared overlap}\}.
W_pUse in source · 1 passage
Search aliases: W_p
Use in source · Machine_Societies_v1.0.0.tex, line 1917
An ideal public announcement is a strong update assumption. For a propositional fact p whose valuation is unchanged by the announcement, an ideal truthful public update restricts W to W_p={w:wmodels p} and restricts every ∼_i to that set. This update models that the event, its interpretation, and its receipt are public to the group. In the restricted model, every reachable world satisfies p, so Proposition~[source reference] gives C_Jp. The conclusion follows from the update semantics; it is not…
Exact mathematical passage
W_p=\{w:w\models p\}
W_{\rm glue}Use in source · 1 passage
Search aliases: W_rm glue rm
Use in source · Machine_Societies_v1.0.0.tex, line 549
Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentation of its global sections. These compatibility conditions follow the elementary logic of gluing local descriptions into a shared global state.
Exact mathematical passage
W_{\rm glue}=\{(w_i):r_{ij}^i(w_i)=r_{ij}^j(w_j)\text{ on every declared overlap}\}.
w_tUse in source · 5 passages
Search aliases: w_t
Use in source · Machine_Societies_v1.0.0.tex, line 686
hcal U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assigns nonnegative voting weights, q_t is an approval threshold, and e_t is a configuration ep…
Exact mathematical passage
B_t=(M_t,V_t,c_t,w_t,q_t,e_t),
\mathsf{wait}Use in source · 1 passage
Search aliases: wait mathsf
Use in source · Machine_Societies_v1.0.0.tex, line 2238
[A safe amendment can destroy every safe continuation] There are four publicly distinguishable states r,ℓ,g,b. The safe set is {r,ℓ,g}. At r, profile deliver moves to g, while profile freeze moves to ℓ. At g, wait loops safely. At ℓ, the only admissible profile advances an outstanding deadline and moves to breach state b. The amendment freeze preserves the immediate safe-state predicate but not W_*. No certificate proving unbounded continuation from r can choose it. The relevant defect is loss o…
Exact mathematical passage
\mathsf{wait}
\wedgeUse in source · 1 passage
Search aliases: wedge
Use in source · Machine_Societies_v1.0.0.tex, line 1119
[Constitutional inductive safety] If Ψ(s_0) and every admitted transition satisfies Ψ(s)wedge R(s,s')LongrightarrowΨ(s'), then Ψ(s_t) holds for every state reachable by finitely many admitted transitions. Induction on the number of admitted transitions.
Exact mathematical passage
\Psi(s)\wedge R(s,s')\Longrightarrow\Psi(s'),
\widehat KUse in source · 2 passages
Search aliases: widehat K widehat
Use in source · Machine_Societies_v1.0.0.tex, line 527
Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.
Exact mathematical passage
\TV(\psi_{\#}\overline K_u(z,\cdot),\widehat K_u(\psi z,\cdot))\leq\varepsilon_2.
\widehat K_uUse in source · 2 passages
Search aliases: widehat K_u widehat
Use in source · Machine_Societies_v1.0.0.tex, line 527
Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.
Exact mathematical passage
\TV(\psi_{\#}\overline K_u(z,\cdot),\widehat K_u(\psi z,\cdot))\leq\varepsilon_2.
\widehat PUse in source · 4 passages
Search aliases: widehat P widehat
Use in source · Machine_Societies_v1.0.0.tex, line 1052
Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different for different deviations. They must concern the complete record available to the sanction mechanism, including any effects of review scheduling.
Exact mathematical passage
\widehat P_C,\widehat P_j
\widehat P_CUse in source · 4 passages
Search aliases: widehat P_C widehat
Use in source · Machine_Societies_v1.0.0.tex, line 1052
Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different for different deviations. They must concern the complete record available to the sanction mechanism, including any effects of review scheduling.
Exact mathematical passage
\widehat P_C,\widehat P_j
\widehat P_jUse in source · 4 passages
Search aliases: widehat P_j widehat
Use in source · Machine_Societies_v1.0.0.tex, line 1052
Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different for different deviations. They must concern the complete record available to the sanction mechanism, including any effects of review scheduling.
Exact mathematical passage
\widehat P_C,\widehat P_j
\widehat\thetaUse in source · 2 passages
Search aliases: widehatθ widehat theta
Use in source · Machine_Societies_v1.0.0.tex, line 1853
An evidence-preserving comparison. Now change only the information contract. Suppose authentic individual signals are available to a decision rule that selects their majority. It need not infer signals from prior actions. Let widehatθ_n be any majority choice with an arbitrary fixed rule for ties.
Exact mathematical passage
\widehat\theta_n
\widehat\theta_nUse in source · 2 passages
Search aliases: widehatθ_n widehat theta
Use in source · Machine_Societies_v1.0.0.tex, line 1853
An evidence-preserving comparison. Now change only the information contract. Suppose authentic individual signals are available to a decision rule that selects their majority. It need not infer signals from prior actions. Let widehatθ_n be any majority choice with an arbitrary fixed rule for ties.
Exact mathematical passage
\widehat\theta_n
\widetilde gUse in source · 2 passages
Search aliases: widetilde g widetilde
Use in source · Machine_Societies_v1.0.0.tex, line 1701
_i^-1). [Loop defect bounds unavoidable edge error] For a closed walk γ with k≥1 traversals, d(H_γ,e_G)≤∑_(i,j) traversed by γδ_ij. In particular, some traversed edge has error at least d(H_γ,e_G)/k for every candidate frame family. Gauge-transform each edge to widetilde g_ij=h_j^-1g_ijh_i. Bi-invariance gives d(widetilde g_ij,e_G)=δ_ij. The transformed loop product is h_r^-1H_γ h_r, at the same distance from identity as H_γ. Repeatedly applying the triangle inequality and invariance to a produc…
Exact mathematical passage
\widetilde g_{ij}=h_j^{-1}g_{ij}h_i
\widetilde g_{ij}Use in source · 2 passages
Search aliases: widetilde g_ij widetilde
Use in source · Machine_Societies_v1.0.0.tex, line 1701
_i^-1). [Loop defect bounds unavoidable edge error] For a closed walk γ with k≥1 traversals, d(H_γ,e_G)≤∑_(i,j) traversed by γδ_ij. In particular, some traversed edge has error at least d(H_γ,e_G)/k for every candidate frame family. Gauge-transform each edge to widetilde g_ij=h_j^-1g_ijh_i. Bi-invariance gives d(widetilde g_ij,e_G)=δ_ij. The transformed loop product is h_r^-1H_γ h_r, at the same distance from identity as H_γ. Repeatedly applying the triangle inequality and invariance to a produc…
Exact mathematical passage
\widetilde g_{ij}=h_j^{-1}g_{ij}h_i
x'Use in source · 9 passages
Search aliases: x'
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
x_0Use in source · 1 passage
Search aliases: x_0
Use in source · Machine_Societies_v1.0.0.tex, line 2014
n explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-step monotonic estimate followed by induction. Under uniformly bounded disturbance normw_t_v,∞≤bar w, the last term is at most bar w(1-κ^t)/(1-κ). Thi…
Exact mathematical passage
\label{eq:propagation-bound} \norm{x_t}_{v,\infty}\le \kappa^t\norm{x_0}_{v,\infty} +\sum_{j=0}^{t-1}\kappa^{t-1-j}\norm{w_j}_{v,\infty}.
x_AUse in source · 1 passage
Search aliases: x_A
Use in source · Machine_Societies_v1.0.0.tex, line 1683
[Nontrivial frame defect with a compatible state] On a triangle, let one oriented edge carry (12)∈ S_3 and the other two carry identity. Holonomy is nontrivial, so global frames do not exist. Nevertheless, the constant state x_A=x_B=x_C=3 is compatible because (12) fixes 3. Failure to align all role meanings is stronger than failure to agree on one unused or invariant role.
Exact mathematical passage
x_A=x_B=x_C=3
x_BUse in source · 1 passage
Search aliases: x_B
Use in source · Machine_Societies_v1.0.0.tex, line 1683
[Nontrivial frame defect with a compatible state] On a triangle, let one oriented edge carry (12)∈ S_3 and the other two carry identity. Holonomy is nontrivial, so global frames do not exist. Nevertheless, the constant state x_A=x_B=x_C=3 is compatible because (12) fixes 3. Failure to align all role meanings is stronger than failure to agree on one unused or invariant role.
Exact mathematical passage
x_A=x_B=x_C=3
x_CUse in source · 1 passage
Search aliases: x_C
Use in source · Machine_Societies_v1.0.0.tex, line 1683
[Nontrivial frame defect with a compatible state] On a triangle, let one oriented edge carry (12)∈ S_3 and the other two carry identity. Holonomy is nontrivial, so global frames do not exist. Nevertheless, the constant state x_A=x_B=x_C=3 is compatible because (12) fixes 3. Failure to align all role meanings is stronger than failure to agree on one unused or invariant role.
Exact mathematical passage
x_A=x_B=x_C=3
x_iUse in source · 9 passages
Search aliases: x_i
Use in source · Machine_Societies_v1.0.0.tex, line 1605
Consider an undirected graph G=(V,E) whose vertices are institutional positions. Each position carries a binary coordinate x_i∈ F_2 and each edge {i,j} imposes the relative constraint x_i⊕ x_j=b_ij, b_ij∈ F_2. The coordinates might represent a toy authorization bit or a binary convention, but the mathematical model itself makes no claim about a realistic legal code.
Exact mathematical passage
x_i\in\mathbb F_2
x_i'Use in source · 1 passage
Search aliases: x_i'
Use in source · Machine_Societies_v1.0.0.tex, line 1766
The binary jurisdiction model admits a sharper statement. A local convention change x_i'=x_i⊕ c_i transforms its edge labels to b_ij'=b_ij⊕ c_i⊕ c_j.
Exact mathematical passage
x_i'=x_i\oplus c_i
x_jUse in source · 3 passages
Search aliases: x_j
Use in source · Machine_Societies_v1.0.0.tex, line 1606
Consider an undirected graph G=(V,E) whose vertices are institutional positions. Each position carries a binary coordinate x_i∈ F_2 and each edge {i,j} imposes the relative constraint x_i⊕ x_j=b_ij, b_ij∈ F_2. The coordinates might represent a toy authorization bit or a binary convention, but the mathematical model itself makes no claim about a realistic legal code.
Exact mathematical passage
\label{eq:edge-rule} x_i\oplus x_j=b_{ij},\qquad b_{ij}\in\mathbb F_2.
x_rUse in source · 4 passages
Search aliases: x_r
Use in source · Machine_Societies_v1.0.0.tex, line 1676
[Compatible states and holonomy fixed points] A compatible state family exists if and only if there is x_r∈ D fixed by every holonomy based at the root r. Each such x_r uniquely determines a compatible family by transport along a spanning tree. For a free group action, existence of a compatible family implies identity holonomy; without freeness that implication can fail. A compatible family transported around a closed walk returns to its starting value, so every based holonomy fixes x_r. Convers…
Exact mathematical passage
x_r\in D
x_tUse in source · 3 passages
Search aliases: x_t
Use in source · Machine_Societies_v1.0.0.tex, line 2008
A transparent comparison model can make one class of failure mathematically tractable. Let x_t∈ R_≥0^n record nonnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞.…
Exact mathematical passage
x_t\in\mathbb R_{\ge0}^n
X_{t+1}Use in source · 2 passages
Search aliases: X_t+1
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
x_{t+1}Use in source · 1 passage
Search aliases: x_t+1
Use in source · Machine_Societies_v1.0.0.tex, line 2009
A transparent comparison model can make one class of failure mathematically tractable. Let x_t∈ R_≥0^n record nonnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞.…
Exact mathematical passage
\label{eq:propagation} x_{t+1}\le Bx_t+w_t \quad\text{componentwise},\qquad B\ge0,
\YUse in source · 1 passage
Search aliases: Y mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
y'Use in source · 1 passage
Search aliases: y'
Use in source · Machine_Societies_v1.0.0.tex, line 2732
[.5em] What state must persist? · The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens. [.5em] Can a due debt be erased? · The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are required to preserve pending commitments. [.5em] Who can block change? · Two registered offices block a three-of-four rule under independent approval …
Exact mathematical passage
.5em] What state must persist? & The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens.\\[.5em] Can a due debt be erased? & The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are required to preserve pending commitments.\\[.5em] Who can block change? & Two registered offices block a three-of-four rule under independent approval rights. A hidden administrator changes this calculation.\\[.5em] Can every request be safely served? & Resource conservation can hold even when private information or conflicting deadlines prevent timely service. Viability with full information is not decentralized implementability.\\[.5em] What crosses a replacement? & An authorized office handover preserves debt and credential semantics. Copying a model does not duplicate an office's budget or approval right.\\[.5em] Can two jurisdictions merge? & Their translator must preserve the agreed action semantics, permissions, obligations, resources, and evidence coordinates. Nonzero residuals remain visible until an authorized repair is found.\\[.5em] How long does the organization last? & Lifetime is first exit from a declared operational contract under a specified transition law. It is neither occupant longevity nor mere archive retention.\\ \bottomrule \end{longtable} The common event mechanism and constitutional envelope are supplied. Within that envelope, offices can choose rules, incur obligations, revise decisions, and replace occupants. This demonstrates how the theory's objects fit together. It does not prove that the envelope itself arose endogenously or that these rules are uniquely desirable. \subsection{The additional boundary, language, and knowledge contract} An admission changes the membership configuration, not merely the occupant list. A message is interpreted using its type, issuer, and epoch. A resource approval creates the specified ledger event only after its admissibility check. A debtor replacement transports the obligation monitor but does not manufacture new approval rights. A notification history records which office has received which adoption record; an available ledger entry is not automatically common knowledge. For concurrent amendments, the contract states whether ordering is significant, whether a joinable normal form is required, and which external effects remain visible. For federation, the role-label transformations must be checked around ordered loops. A compatible allocation on one invariant subset does not certify synchronization of every role. For inquiry, the commons records unresolved scheduling questions independently of which questions it chooses to fund. For founding authority, it preserves the resource owner's stipulated grant separately from its own subsequent votes. Together these clauses specify what it would mean to preserve this analytical institution across translations and replacements. They do not claim that every useful society needs this exact architecture or that the four-office example exhibits autonomous institutional formation. \subsection{Distributed realization, enforcement, and adaptive staffing} In the atomic reference commons, a unique slot token prevents two successful grants of the same slot. In its replicated variant, two local approvals can each look valid while their merged grant set violates exclusivity. The institution must therefore distinguish a provisional reservation from a committed grant and identify the primitive that prevents double execution. A convergent archive is not that primitive by itself. Accurate urgency reporting is a different problem. A committed monitor can discourage a specified profitable misreport only within the discrimination and penalty budget of Section~\ref{sec:enforcement}. Its false-positive limit, evidence access, and remedy authority remain separate. When an averaged model supplies the observation laws, the margin must include its approximation error. A fixed review schedule and a data-selected review schedule cannot share the same mixing certificate without further work. The commons also distinguishes an urgency vote from the signal supporting it. Decision-only publication can reproduce a mistaken cascade; preservation of independent evidence changes the information available to later reviewers, not the legal effect of an already committed vote. Finally, creating a specialist verifier changes the capability graph and the distribution of authority. A staffing certificate covers all protected roles jointly, while the handover procedure must still deliver credentials, pending debts, and evidence before service deadlines. These clauses supplement the reference semantics rather than claim that the society has generated its own infrastructure without assumptions. \subsection{The joint continuation contract} The bounded audit-and-service variant in Section~\ref{sec:joint-synthesis} combines three protections that cannot be checked in isolation: the common monitor must deter every listed deviation, its false-positive rate must meet the cap, and its resource use must leave enough capacity for all due work. The current public possibility set must also admit the same protocol in every compatible complete state. A continuation certificate records this common choice and all possible successors. A progress rank distinguishes eventual certified service from indefinite safe waiting. An amendment is acceptable under this particular continuation contract only when its current authority is valid and its resulting states retain certified continuations. A transport certificate additionally checks public observations, enabled profiles, safe and goal labels, and successor sets. Replacing an office, changing an audit channel, or adding a slot changes specified parts of that contract. Such a change may repair an obstruction, but does not retroactively fulfill an earlier unmet promise. \FloatBarrier \subsection{The accountability extension} Section~\ref{sec:accountable-commons} deliberately changes the audit catalogue to satisfy its pairwise privacy constraint. It separately includes an outside claim, an admitted appeal, a genuine-execution fee, and a funded final record with revocation. The resulting eight-unit settlement requirement must not be confused with the five-unit service-and-audit variant. The supplied reference trace checks one declared five-profile settlement, while the stopping theorem characterizes an entire finite ending-preserving running kernel. Both are model-relative results, not certificates for an unspecified deployment. \section{A finite algebraic example of reusable state}\label{sec:worked} The following analytical example makes the preceding distinctions concrete. Four components initially hold independent fair bits $(a,b,c,d)$. Two local workspaces expose \[ p=a\oplus b,\qquad q=c\oplus d,
Y_{0:T}Use in source · 1 passage
Search aliases: Y_0:T
Use in source · Machine_Societies_v1.0.0.tex, line 653
For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared family, not universal inertness. An enforceable institution can also alter which actions are institutionally permitted while producing the same observed …
Exact mathematical passage
\label{eq:inst-effect} \Delta_{\mathrm{inst}}^{\mathcal E,T}(I,I') =\sup_{e\in\mathcal E}\TV\bigl(\Law_e(Y_{0:T}\mid\lambda(I)),\Law_e(Y_{0:T}\mid\lambda(I'))\bigr).
ZUse in source · 30 passages
Search aliases: Z
Use in source · Machine_Societies_v1.0.0.tex, line 176
Every organizational state in this paper remains a function of a complete computational realization. If Z_t=π(X_t), then Z_t is not information outside X_t. The purpose of a higher-level description is to preserve relevant operations with less descriptive complexity. An institutional description must additionally preserve who may act, which commitments remain outstanding, and what counts as an authorized amendment. These are requirements on the abstraction, not a new substance added to the machi…
Exact mathematical passage
Z_t=\pi(X_t)
\mathcal ZUse in source · 1 passage
Search aliases: Z mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 980
Compression can erase enforceable distinctions. Let an observation channel B replace o∈ O by a record z∈ Z. It may be deterministic compression or a stochastic reporting channel, but must be the same channel under compliance and deviation.
Exact mathematical passage
z\in\mathcal Z
Z_0Use in source · 2 passages
Search aliases: Z_0
Use in source · Machine_Societies_v1.0.0.tex, line 1380
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))
Exact mathematical passage
z\in Z_0
z_0Use in source · 2 passages
Search aliases: z_0
Use in source · Machine_Societies_v1.0.0.tex, line 2550
[Guarded certificate use and authorized handover] Assume that the selector, at every reachable decision epoch, either uses a local policy with a sound positive guard or performs a handover satisfying the preceding conditions. Starting from z_0∈ C_v_0, every decision state belongs to its current certified domain and every intermediate handover state remains in C.
Exact mathematical passage
z_0\in C_{v_0}
Z_{0:t}Use in source · 1 passage
Search aliases: Z_0:t
Use in source · Machine_Societies_v1.0.0.tex, line 318
eedspace8baselineskip [Controlled lumpability] A common kernel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-model with initial law pushμ_0. For outputs to agree as well, require Q_u(x,·) to be constant on each fibre. The common fibre value in~[source referen…
Exact mathematical passage
\alpha_t(u\mid Z_{0:t})
Z_1Use in source · 2 passages
Search aliases: Z_1
Use in source · Machine_Societies_v1.0.0.tex, line 1739
[Composition of typed institutional residuals] Let T:Z_0→ Z_1 and U:Z_1→ Z_2 have finite residuals under the same semantic coordinates. Suppose U is L-Lipschitz. Then ε_rm dyn(UT) ≤ Lε_rm dyn(T)+ε_rm dyn(U), and, separately for every c∈{P,O,R,E}, ε_c(UT)≤ε_c(T)+ε_c(U). Consequently translations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continui…
Exact mathematical passage
T:Z_0\to Z_1
Z_2Use in source · 1 passage
Search aliases: Z_2
Use in source · Machine_Societies_v1.0.0.tex, line 1739
[Composition of typed institutional residuals] Let T:Z_0→ Z_1 and U:Z_1→ Z_2 have finite residuals under the same semantic coordinates. Suppose U is L-Lipschitz. Then ε_rm dyn(UT) ≤ Lε_rm dyn(T)+ε_rm dyn(U), and, separately for every c∈{P,O,R,E}, ε_c(UT)≤ε_c(T)+ε_c(U). Consequently translations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continui…
Exact mathematical passage
U:Z_1\to Z_2
Z_iUse in source · 2 passages
Search aliases: Z_i
Use in source · Machine_Societies_v1.0.0.tex, line 1630
Even when a global static state exists, a federation may fail dynamically. Local rule applications could commute in one context and conflict in another. A translator T_ij:Z_i→ Z_j should preserve the operations relevant to the interaction, approximately satisfying T_ijF_i≃ F_jT_ij. A federation requires both representational compatibility and admissible operational coordination. Static agreement alone does not define an executable constitution. Nor does faithful gluing prove that the institution…
Exact mathematical passage
T_{ij}:Z_i\to Z_j
Z_jUse in source · 8 passages
Search aliases: Z_j
Use in source · Machine_Societies_v1.0.0.tex, line 1373
Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε_j=sup_z∈ Z_jd_j+1(T_jF_j(z),F_j+1T_j(z)). An exact zero defect means the representation change intertwines the declared operation. It does not mean a…
Exact mathematical passage
(Z_j,d_j)
Z_{j+1}Use in source · 1 passage
Search aliases: Z_j+1
Use in source · Machine_Societies_v1.0.0.tex, line 1373
Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε_j=sup_z∈ Z_jd_j+1(T_jF_j(z),F_j+1T_j(z)). An exact zero defect means the representation change intertwines the declared operation. It does not mean a…
Exact mathematical passage
T_j:Z_j\to Z_{j+1}
Z_{\rm ext}Use in source · 2 passages
Search aliases: Z_rm ext rm
Use in source · Machine_Societies_v1.0.0.tex, line 2343
A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. Standing specifies who may initiate a challenge and on which grounds. Remedy specifies the actions an authorized review can actually require. The cont…
Exact mathematical passage
\mathcal B=(Z_{\rm ext},O_{\rm ext},\mathsf{Auth},\mathsf{Effect}, \mathsf{Standing},\mathsf{Remedy}).
Z_tUse in source · 7 passages
Search aliases: Z_t
Use in source · Machine_Societies_v1.0.0.tex, line 176
Every organizational state in this paper remains a function of a complete computational realization. If Z_t=π(X_t), then Z_t is not information outside X_t. The purpose of a higher-level description is to preserve relevant operations with less descriptive complexity. An institutional description must additionally preserve who may act, which commitments remain outstanding, and what counts as an authorized amendment. These are requirements on the abstraction, not a new substance added to the machi…
Exact mathematical passage
Z_t=\pi(X_t)
Z_{t+1}Use in source · 2 passages
Search aliases: Z_t+1
Use in source · Machine_Societies_v1.0.0.tex, line 327
[Deterministic microstate, non-Markov macrostate] Let X_t+1=X_t+1pmod 3 with uniform initial distribution. Set π(0)=π(1)=a and π(2)=b. Stationarity gives P(Z_t+1=bmid Z_t=a)=1/2. Yet P(Z_t+1=bmid Z_t=a,Z_t-1=b)=0, whereas the same probability conditional on Z_t=a,Z_t-1=a is 1. Thus the apparent randomness comes with memory: the one-step marginal does not define a valid autonomous first-order Markov model.
Exact mathematical passage
\Prob(Z_{t+1}=b\mid Z_t=a)=1/2
Z_{t-1}Use in source · 2 passages
Search aliases: Z_t-1
Use in source · Machine_Societies_v1.0.0.tex, line 328
[Deterministic microstate, non-Markov macrostate] Let X_t+1=X_t+1pmod 3 with uniform initial distribution. Set π(0)=π(1)=a and π(2)=b. Stationarity gives P(Z_t+1=bmid Z_t=a)=1/2. Yet P(Z_t+1=bmid Z_t=a,Z_t-1=b)=0, whereas the same probability conditional on Z_t=a,Z_t-1=a is 1. Thus the apparent randomness comes with memory: the one-step marginal does not define a valid autonomous first-order Markov model.
Exact mathematical passage
\Prob(Z_{t+1}=b\mid Z_t=a,Z_{t-1}=b)=0,
\timesUse in source · 1 passage
Search aliases: × times
Use in source · Machine_Societies_v1.0.0.tex, line 788
Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated office names, configuration epochs, and reference records. The translation of a message sequence is constructed step by step from the evolving source st…
Exact mathematical passage
\tau:S_A\times\mathcal M_A\to\mathcal M_B.
\alpha_tUse in source · 1 passage
Search aliases: α_t alpha
Use in source · Machine_Societies_v1.0.0.tex, line 318
eedspace8baselineskip [Controlled lumpability] A common kernel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-model with initial law pushμ_0. For outputs to agree as well, require Q_u(x,·) to be constant on each fibre. The common fibre value in~[source referen…
Exact mathematical passage
\alpha_t(u\mid Z_{0:t})
\betaUse in source · 4 passages
Search aliases: β beta
Use in source · Machine_Societies_v1.0.0.tex, line 1550
A transparent kernel family is K_i=(1-β_i)Id+β_i 1μ_i, 0<β_i≤1, where every row of 1μ_i is μ_i. Its mth power is (1-β_i)^mId+[1-(1-β_i)^m] 1μ_i. Thus the mixing error is bounded by (1-β_i)^m. For a fair two-state μ_i and β_i=1/2, the exact worst-row error is 2^-m-1. With m=8, the linear bound over twenty reviews is 20/512=5/128. These are analytical values for the stated kernel, not estimated mixing times for neural agents.
Exact mathematical passage
K_i=(1-\beta_i)\mathrm{Id}+\beta_i\mathbf 1\mu_i, \qquad 0<\beta_i\leq1,
\beta_iUse in source · 4 passages
Search aliases: β_i beta
Use in source · Machine_Societies_v1.0.0.tex, line 1550
A transparent kernel family is K_i=(1-β_i)Id+β_i 1μ_i, 0<β_i≤1, where every row of 1μ_i is μ_i. Its mth power is (1-β_i)^mId+[1-(1-β_i)^m] 1μ_i. Thus the mixing error is bounded by (1-β_i)^m. For a fair two-state μ_i and β_i=1/2, the exact worst-row error is 2^-m-1. With m=8, the linear bound over twenty reviews is 20/512=5/128. These are analytical values for the stated kernel, not estimated mixing times for neural agents.
Exact mathematical passage
K_i=(1-\beta_i)\mathrm{Id}+\beta_i\mathbf 1\mu_i, \qquad 0<\beta_i\leq1,
\gammaUse in source · 14 passages
Search aliases: γ gamma
Use in source · Machine_Societies_v1.0.0.tex, line 1511
Office continuity is distinct from occupant survival. Suppose, separately, that each original occupant is replaced at each epoch with independent probability γ. Its probability of still occupying its role after t epochs is (1-γ)^t. Only if handovers preserve the institutional state and do not change the assumed Q can the institutional survival law above remain unchanged. Independence between turnover and degradation is a simplifying assumption, not an implication of copying state.
Exact mathematical passage
\gamma
\Gamma_\alphaUse in source · 5 passages
Search aliases: Γ_α Gamma alpha
Use in source · Machine_Societies_v1.0.0.tex, line 1012
th probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous margin by D_j(s)=F⟨ P_j-P_C,s⟩-g_j, Γ_α=max_s∈ M_αmin_1≤ j≤ mD_j(s). Weak deterrence of all listed deviations means Γ_α≥0. Strict deterrence means Γ_α>0. At equality a participant can remain indifferent. A rand…
Exact mathematical passage
\label{eq:joint-margin} D_j(s)=F\langle P_j-P_C,s\rangle-g_j, \qquad \Gamma_\alpha=\max_{s\in\mathcal M_\alpha}\min_{1\leq j\leq m}D_j(s).
\deltaUse in source · 18 passages
Search aliases: δ delta
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\delta^TUse in source · 1 passage
Search aliases: δ^T delta
Use in source · Machine_Societies_v1.0.0.tex, line 843
[Institutional amortization] Under stationary costs and infinite-horizon discounting, the institution's expected discounted cost advantage over the baseline is V=-B+fracg1-δ. For a finite horizon T, it is V_T=-B+g(1-δ^T)/(1-δ). The institution is cost-improving under these assumptions exactly when the corresponding value is positive. Subtract institutional operating and maintenance costs from baseline costs in every period, discount, sum the finite or infinite geometric series, and subtract B.
Exact mathematical passage
V_T=-B+g(1-\delta^T)/(1-\delta)
\Delta_{\mathcal A}Use in source · 1 passage
Search aliases: Δ_ A Delta mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 277
A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distribution, observation maps, and coalition family. It is not equivalent to any one general-purpose partial-information-decomposition statistic~[source re…
Exact mathematical passage
\label{eq:coalition} \Delta_{\mathcal A}=\min_{S\in\mathcal A}\mathcal R_\ell(O_S) -\mathcal R_\ell(O_V)
\delta_1Use in source · 2 passages
Search aliases: δ_1 delta
Use in source · Machine_Societies_v1.0.0.tex, line 1208
[Quiescent convergence and invariant preservation] Assume finitely many replicas begin at ℓ_0, only finitely many contributions δ_1,…,δ_N are created, and every contribution eventually becomes included in every replica. Then each replica eventually equals ℓ^⋆=ℓ_0veeδ_1vee⋯veeδ_N, independently of merge order and duplicate deliveries. Moreover, let V⊆ L contain the initial state. If every permitted local update maps V into V and V is closed under vee, every local and merged state remains in V. At…
Exact mathematical passage
\delta_1,\ldots,\delta_N
\delta_cUse in source · 2 passages
Search aliases: δ_c delta
Use in source · Machine_Societies_v1.0.0.tex, line 1713
Common semantic coordinates. For each component type c∈{P,O,R,E}, let g_j^c:Z_j⟶(E_c,δ_c) be a contract observable in a common metric codomain. P records permissions over the common action alphabet; O records obligation-monitor state; R records resources in common physical or accounting units; and E records the required evidentiary and provenance structure. A change of units, action names, or obligation identifiers requires an explicitly specified alignment before these comparisons can be made.
Exact mathematical passage
g_j^c:Z_j\longrightarrow(E_c,\delta_c)
\delta_{F_u(x)}Use in source · 1 passage
Search aliases: δ_F_u(x) delta
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\delta_{ij}Use in source · 4 passages
Search aliases: δ_ij delta
Use in source · Machine_Societies_v1.0.0.tex, line 1690
A quantitative lower bound on interface error. Suppose G has a bi-invariant metric d, so both left and right multiplication preserve distance. For candidate frames h_i, write δ_ij=d(g_ij,h_jh_i^-1). [Loop defect bounds unavoidable edge error] For a closed walk γ with k≥1 traversals, d(H_γ,e_G)≤∑_(i,j) traversed by γδ_ij. In particular, some traversed edge has error at least d(H_γ,e_G)/k for every candidate frame family. Gauge-transform each edge to widetilde g_ij=h_j^-1g_ijh_i. Bi-invariance giv…
Exact mathematical passage
\delta_{ij}=d(g_{ij},h_jh_i^{-1}).
\Delta_{\mathrm{inst}}^{\mathcal E,T}Use in source · 1 passage
Search aliases: Δ_inst^ E,T Delta mathrm mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 653
For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared family, not universal inertness. An enforceable institution can also alter which actions are institutionally permitted while producing the same observed …
Exact mathematical passage
\label{eq:inst-effect} \Delta_{\mathrm{inst}}^{\mathcal E,T}(I,I') =\sup_{e\in\mathcal E}\TV\bigl(\Law_e(Y_{0:T}\mid\lambda(I)),\Law_e(Y_{0:T}\mid\lambda(I'))\bigr).
\Delta_mUse in source · 4 passages
Search aliases: Δ_m Delta
Use in source · Machine_Societies_v1.0.0.tex, line 1019
For a probability vector λ∈Δ_m, set P_λ=∑_jλ_jP_j and g_λ=∑_jλ_jg_j. Write V_α(P,P_C)=max_s∈ M_α⟨ P-P_C,s⟩. This is the same constrained observation advantage used in the single-deviation analysis, now evaluated on mixtures of deviation laws.
Exact mathematical passage
\lambda\in\Delta_m
\delta_NUse in source · 2 passages
Search aliases: δ_N delta
Use in source · Machine_Societies_v1.0.0.tex, line 1208
[Quiescent convergence and invariant preservation] Assume finitely many replicas begin at ℓ_0, only finitely many contributions δ_1,…,δ_N are created, and every contribution eventually becomes included in every replica. Then each replica eventually equals ℓ^⋆=ℓ_0veeδ_1vee⋯veeδ_N, independently of merge order and duplicate deliveries. Moreover, let V⊆ L contain the initial state. If every permitted local update maps V into V and V is closed under vee, every local and merged state remains in V. At…
Exact mathematical passage
\delta_1,\ldots,\delta_N
\Delta_{\rm coal}Use in source · 1 passage
Search aliases: Δ_rm coal Delta rm
Use in source · Machine_Societies_v1.0.0.tex, line 257
A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may be unidentified. They should not be replaced by zero or hidden inside a favorable weighted average.
Exact mathematical passage
\mathcal C_{\mathcal T}=(\varepsilon_{\rm pred},\varepsilon_{\rm dyn},\varepsilon_{\rm int}, \Delta_{\rm coal},R_{\rm repair},D_{\rm lineage},G_{\rm formation}).
\epsilonUse in source · 5 passages
Search aliases: ϵ epsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1053
Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different for different deviations. They must concern the complete record available to the sanction mechanism, including any effects of review scheduling.
Exact mathematical passage
\TV(P_C,\widehat P_C)\leq\epsilon_C, \qquad \TV(P_j,\widehat P_j)\leq\epsilon_j.
\varepsilon_0Use in source · 1 passage
Search aliases: ε_0 varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1390
· ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second by ε_m-1. The recurrence D_m≤ L_m-1D_m-1+ε_m-1 unfolds to~[source reference].
Exact mathematical passage
\varepsilon_0
\varepsilon_1Use in source · 2 passages
Search aliases: ε_1 varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 524
Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.
Exact mathematical passage
\TV(\push K_u(x,\cdot),\overline K_u(\pi x,\cdot))\leq\varepsilon_1,
\varepsilon_2Use in source · 2 passages
Search aliases: ε_2 varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 527
Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.
Exact mathematical passage
\TV(\psi_{\#}\overline K_u(z,\cdot),\widehat K_u(\psi z,\cdot))\leq\varepsilon_2.
\varepsilon_aUse in source · 1 passage
Search aliases: ε_a varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1573
Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bound and the same declared endpoint-to-evidence channel is used in the actual and averaged model.
Exact mathematical passage
\TV(P_a,\overline P_a)\leq\varepsilon_a,\qquad a\in\{C,D\}.
\epsilon_CUse in source · 5 passages
Search aliases: ϵ_C epsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1053
Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different for different deviations. They must concern the complete record available to the sanction mechanism, including any effects of review scheduling.
Exact mathematical passage
\TV(P_C,\widehat P_C)\leq\epsilon_C, \qquad \TV(P_j,\widehat P_j)\leq\epsilon_j.
\varepsilon_CUse in source · 5 passages
Search aliases: ε_C varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1580
[A robust margin for reduced-model enforcement] Let s^⋆(o) sanction precisely where overline P_D(o)>overline P_C(o) and put overline d=d_TV(overline P_D,overline P_C). In the model of Section~[source reference], this fixed policy makes compliance weakly preferable whenever g≤ F(overline d-ε_C-ε_D). Conversely, if g>Fmin{1,overline d+ε_C+ε_D}, no observation-based sanction with penalty F deters the deviation. Also, E_P_Cs^⋆≤ E_overline P_Cs^⋆+ε_C. The expectation of a [0,1]-valued function change…
Exact mathematical passage
g\leq F(\overline d-\varepsilon_C-\varepsilon_D).
\varepsilon_cUse in source · 2 passages
Search aliases: ε_c varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1721
For a translation T:Z_i→ Z_j, define ε_rm dyn(T) · =sup_z,ad_j(TF_i^a(z),F_j^aT(z)), ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The symmetric permission residual measures both changes and is delib…
Exact mathematical passage
\begin{split} \varepsilon_{\rm dyn}(T) &=\sup_{z,a}d_j\bigl(TF_i^a(z),F_j^aT(z)\bigr),\\ \varepsilon_c(T) &=\sup_z\delta_c\bigl(g_j^cT(z),g_i^c(z)\bigr) \quad(c=P,O,R,E). \end{split}
\varepsilon_DUse in source · 4 passages
Search aliases: ε_D varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1580
[A robust margin for reduced-model enforcement] Let s^⋆(o) sanction precisely where overline P_D(o)>overline P_C(o) and put overline d=d_TV(overline P_D,overline P_C). In the model of Section~[source reference], this fixed policy makes compliance weakly preferable whenever g≤ F(overline d-ε_C-ε_D). Conversely, if g>Fmin{1,overline d+ε_C+ε_D}, no observation-based sanction with penalty F deters the deviation. Also, E_P_Cs^⋆≤ E_overline P_Cs^⋆+ε_C. The expectation of a [0,1]-valued function change…
Exact mathematical passage
g\leq F(\overline d-\varepsilon_C-\varepsilon_D).
\varepsilon_EUse in source · 1 passage
Search aliases: ε_E varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1731
ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The symmetric permission residual measures both changes and is deliberately stronger.
Exact mathematical passage
\mathfrak D(T)= (\varepsilon_{\rm dyn},\varepsilon_P,\varepsilon_O, \varepsilon_R,\varepsilon_E).
\epsilon_jUse in source · 3 passages
Search aliases: ϵ_j epsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1053
Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different for different deviations. They must concern the complete record available to the sanction mechanism, including any effects of review scheduling.
Exact mathematical passage
\TV(P_C,\widehat P_C)\leq\epsilon_C, \qquad \TV(P_j,\widehat P_j)\leq\epsilon_j.
\varepsilon_jUse in source · 5 passages
Search aliases: ε_j varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 539
[Compositional one-step closure] The composite abstraction ψπ satisfies d_TV((ψπ)_#K_u(x,·), widehat K_u(ψπ x,·)) ≤ε_1+ε_2. For L levels, the one-step defect is at most ∑_j=1^Lε_j. In particular, exact abstractions compose exactly. Insert ψ_#overline K_u(π x,·), use the triangle inequality, and use nonexpansiveness of total variation under deterministic pushforward. Induction gives the multilevel result.
Exact mathematical passage
\sum_{j=1}^L\varepsilon_j
\varepsilon_{m-1}Use in source · 2 passages
Search aliases: ε_m-1 varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1390
ith empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second by ε_m-1. The recurrence D_m≤ L_m-1D_m-1+ε_m-1 unfolds to~[source reference].
Exact mathematical passage
\varepsilon_{m-1}
\varepsilon_OUse in source · 1 passage
Search aliases: ε_O varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1731
ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The symmetric permission residual measures both changes and is deliberately stronger.
Exact mathematical passage
\mathfrak D(T)= (\varepsilon_{\rm dyn},\varepsilon_P,\varepsilon_O, \varepsilon_R,\varepsilon_E).
\varepsilon_PUse in source · 1 passage
Search aliases: ε_P varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1731
ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The symmetric permission residual measures both changes and is deliberately stronger.
Exact mathematical passage
\mathfrak D(T)= (\varepsilon_{\rm dyn},\varepsilon_P,\varepsilon_O, \varepsilon_R,\varepsilon_E).
\varepsilon_RUse in source · 1 passage
Search aliases: ε_R varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 1731
ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The symmetric permission residual measures both changes and is deliberately stronger.
Exact mathematical passage
\mathfrak D(T)= (\varepsilon_{\rm dyn},\varepsilon_P,\varepsilon_O, \varepsilon_R,\varepsilon_E).
\varepsilon_{\rm dyn}Use in source · 7 passages
Search aliases: ε_rm dyn varepsilon rm
Use in source · Machine_Societies_v1.0.0.tex, line 257
A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may be unidentified. They should not be replaced by zero or hidden inside a favorable weighted average.
Exact mathematical passage
\mathcal C_{\mathcal T}=(\varepsilon_{\rm pred},\varepsilon_{\rm dyn},\varepsilon_{\rm int}, \Delta_{\rm coal},R_{\rm repair},D_{\rm lineage},G_{\rm formation}).
\varepsilon_{\rm int}Use in source · 2 passages
Search aliases: ε_rm int varepsilon rm
Use in source · Machine_Societies_v1.0.0.tex, line 257
A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may be unidentified. They should not be replaced by zero or hidden inside a favorable weighted average.
Exact mathematical passage
\mathcal C_{\mathcal T}=(\varepsilon_{\rm pred},\varepsilon_{\rm dyn},\varepsilon_{\rm int}, \Delta_{\rm coal},R_{\rm repair},D_{\rm lineage},G_{\rm formation}).
\varepsilon_{\rm pred}Use in source · 1 passage
Search aliases: ε_rm pred varepsilon rm
Use in source · Machine_Societies_v1.0.0.tex, line 257
A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may be unidentified. They should not be replaced by zero or hidden inside a favorable weighted average.
Exact mathematical passage
\mathcal C_{\mathcal T}=(\varepsilon_{\rm pred},\varepsilon_{\rm dyn},\varepsilon_{\rm int}, \Delta_{\rm coal},R_{\rm repair},D_{\rm lineage},G_{\rm formation}).
\varepsilon_tUse in source · 2 passages
Search aliases: ε_t varepsilon
Use in source · Machine_Societies_v1.0.0.tex, line 2461
Writing λ_t=e^ε_t gives the additive parameter ∑_tε_t. The proposition requires a fixed finite number of releases and uniform conditional bounds. A data-selected stopping rule must itself be represented in the transcript or controlled by an appropriate conditional argument. Privileged side information not covered by the declared record is not protected by a certificate for that record alone. The preceding counterexample fails the conditional premise at the second release, even though both uncond…
Exact mathematical passage
\lambda_t=e^{\varepsilon_t}
\ThetaUse in source · 2 passages
Search aliases: Θ Theta
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\Theta_tUse in source · 1 passage
Search aliases: Θ_t Theta
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\kappa^tUse in source · 3 passages
Search aliases: κ^t kappa
Use in source · Machine_Societies_v1.0.0.tex, line 351
ition[Approximate closure over time] Under~[source reference], let the actual and reduced processes use the same predeclared open-loop control sequence. If their initial macro-laws are within e_0 in total variation, then at time t they are within e_t≤min{1,e_0+tε}. If each overline K_u contracts total variation by at most κ<1, then e_t≤κ^t e_0+εfrac1-κ^t1-κ. With matched macro-history policies, the weaker path-law bound min{1,e_0+tε} also holds, but the contraction claim needs a separate closed-…
Exact mathematical passage
\label{eq:contract} e_t\leq\kappa^t e_0+\varepsilon\frac{1-\kappa^t}{1-\kappa}.
\kappa^{t-1-j}Use in source · 1 passage
Search aliases: κ^t-1-j kappa
Use in source · Machine_Societies_v1.0.0.tex, line 2014
n explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-step monotonic estimate followed by induction. Under uniformly bounded disturbance normw_t_v,∞≤bar w, the last term is at most bar w(1-κ^t)/(1-κ). Thi…
Exact mathematical passage
\label{eq:propagation-bound} \norm{x_t}_{v,\infty}\le \kappa^t\norm{x_0}_{v,\infty} +\sum_{j=0}^{t-1}\kappa^{t-1-j}\norm{w_j}_{v,\infty}.
\lambda^zUse in source · 1 passage
Search aliases: λ^z lambda
Use in source · Machine_Societies_v1.0.0.tex, line 1832
Set λ=p/(1-p)>1. Before a cascade starts, define an integer public score z so that the likelihood ratio of the observed action history is λ^z. A new private signal changes its owner's posterior exponent by +1 or -1. At z∈{-1,0,1} the stated tie rule makes the action reveal that signal. Once z reaches 2, even a contrary signal leaves a strictly positive exponent, so the action is 1 regardless of private evidence. Every later such action then has probability one under either state and adds no furt…
Exact mathematical passage
\lambda^z
\lambda_jUse in source · 3 passages
Search aliases: λ_j lambda
Use in source · Machine_Societies_v1.0.0.tex, line 1019
For a probability vector λ∈Δ_m, set P_λ=∑_jλ_jP_j and g_λ=∑_jλ_jg_j. Write V_α(P,P_C)=max_s∈ M_α⟨ P-P_C,s⟩. This is the same constrained observation advantage used in the single-deviation analysis, now evaluated on mixtures of deviation laws.
Exact mathematical passage
P_\lambda=\sum_j\lambda_jP_j
\lambda_tUse in source · 4 passages
Search aliases: λ_t lambda
Use in source · Machine_Societies_v1.0.0.tex, line 2451
[Conditional disclosure accounting] Consider m releases on finite alphabets. At each common prefix h, suppose their conditional next-release laws P_C^t(·mid h) and P_D^t(·mid h) satisfy both pointwise inequalities with factor λ_t≥1, uniformly over prefixes that can occur. Then the joint transcript laws satisfy~[source reference] with λ=∏_t=1^mλ_t. The bound also holds after applying the same stochastic postprocessing to the transcript under both cases. In particular, any sanction computed only f…
Exact mathematical passage
\lambda_t\ge1
\lambda_t^{-1}Use in source · 1 passage
Search aliases: λ_t^-1 lambda
Use in source · Machine_Societies_v1.0.0.tex, line 2458
omputed only from that transcript obeys Theorem~[source reference] with this product factor. The conditional inequalities imply common support at each stage. Induction therefore gives common support for prefixes under the two cases. For a positive-probability full transcript, the ratio of its probabilities is the product of conditional ratios, each between λ_t^-1 and λ_t. Multiplying gives the joint pointwise inequalities. Zero-probability transcripts occur under both cases and satisfy them as w…
Exact mathematical passage
\lambda_t^{-1}
\mu_0Use in source · 1 passage
Search aliases: μ_0 mu
Use in source · Machine_Societies_v1.0.0.tex, line 318
rnel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-model with initial law pushμ_0. For outputs to agree as well, require Q_u(x,·) to be constant on each fibre. The common fibre value in~[source reference] defines overline K_u(z,z'). Conditional on any macro-h…
Exact mathematical passage
\push\mu_0
\mu_iUse in source · 8 passages
Search aliases: μ_i mu
Use in source · Machine_Societies_v1.0.0.tex, line 1524
Assume K_i has a stationary law μ_i and sup_i,xd_TV(K_i^m(x,·),μ_i)≤ε. The actual one-review law from (i,x) and its stationary average are P(i,x;j) · =∑_y K_i^m(x,y)R_i(y,j),
Exact mathematical passage
\mu_i
\nuUse in source · 4 passages
Search aliases: ν nu
Use in source · Machine_Societies_v1.0.0.tex, line 337
Finite-horizon error accounting. For finite distributions, d_TV(μ,ν)=tfrac12∑_z|μ(z)-ν(z)|. Suppose candidate kernels satisfy sup_x,ud_TV(push K_u(x,·),overline K_u(π(x),·)) ≤ε. This uniform condition is stronger than a small average error on the trajectories seen during training.
Exact mathematical passage
\TV(\mu,\nu)=\tfrac12\sum_z|\mu(z)-\nu(z)|
\rhoUse in source · 5 passages
Search aliases: ρ rho
Use in source · Machine_Societies_v1.0.0.tex, line 1481
[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random variable gives Eτ=∑_t≥0 P(τ>t). Under ρ(Q)<1, the matrix geometric series converges to (Id-Q)^-1, giving the result.
Exact mathematical passage
\rho(Q)<1
\sigmaUse in source · 4 passages
Search aliases: σ sigma
Use in source · Machine_Societies_v1.0.0.tex, line 1792
Suppose errors E_1,…,E_n in scalar estimates are centered, have common variance σ^2>0, and have common pairwise correlation ρ∈[0,1]. Then Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equal-variance estimators, the variance-equivalent effective sample size is n_eff=fracn1+(n-1)ρ. If the errors are perfectly correlated, adding copies doe…
Exact mathematical passage
\sigma^2>0
\sigma^2Use in source · 2 passages
Search aliases: σ^2 sigma
Use in source · Machine_Societies_v1.0.0.tex, line 1792
Suppose errors E_1,…,E_n in scalar estimates are centered, have common variance σ^2>0, and have common pairwise correlation ρ∈[0,1]. Then Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equal-variance estimators, the variance-equivalent effective sample size is n_eff=fracn1+(n-1)ρ. If the errors are perfectly correlated, adding copies doe…
Exact mathematical passage
\sigma^2>0
\tau_{AB}Use in source · 1 passage
Search aliases: τ_AB tau
Use in source · Machine_Societies_v1.0.0.tex, line 806
tion of finite institutional conversations] Under Definition~[source reference], every finite source conversation has an admitted prefix exactly when its stepwise translation has that admitted prefix from the translated initial state. Corresponding reached states are related by T, and corresponding event records agree. Two faithful transports compose faithfully using τ_AC(s,m)=τ_BC(T_AB(s),τ_AB(s,m)). The initial states are related by definition. At each step, equivalence of definedness gives th…
Exact mathematical passage
\tau_{AC}(s,m)=\tau_{BC}(T_{AB}(s),\tau_{AB}(s,m)).
\tau_{AC}Use in source · 1 passage
Search aliases: τ_AC tau
Use in source · Machine_Societies_v1.0.0.tex, line 806
tion of finite institutional conversations] Under Definition~[source reference], every finite source conversation has an admitted prefix exactly when its stepwise translation has that admitted prefix from the translated initial state. Corresponding reached states are related by T, and corresponding event records agree. Two faithful transports compose faithfully using τ_AC(s,m)=τ_BC(T_AB(s),τ_AB(s,m)). The initial states are related by definition. At each step, equivalence of definedness gives th…
Exact mathematical passage
\tau_{AC}(s,m)=\tau_{BC}(T_{AB}(s),\tau_{AB}(s,m)).
\tau_{BC}Use in source · 1 passage
Search aliases: τ_BC tau
Use in source · Machine_Societies_v1.0.0.tex, line 806
tion of finite institutional conversations] Under Definition~[source reference], every finite source conversation has an admitted prefix exactly when its stepwise translation has that admitted prefix from the translated initial state. Corresponding reached states are related by T, and corresponding event records agree. Two faithful transports compose faithfully using τ_AC(s,m)=τ_BC(T_AB(s),τ_AB(s,m)). The initial states are related by definition. At each step, equivalence of definedness gives th…
Exact mathematical passage
\tau_{AC}(s,m)=\tau_{BC}(T_{AB}(s),\tau_{AB}(s,m)).
\PhiUse in source · 7 passages
Search aliases: Φ Phi
Use in source · Machine_Societies_v1.0.0.tex, line 868
A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',r_-i)-Φ(r_i,r_-i). This is the familiar potential-game condition~[source reference]; it is not generically satisfied by social institutions.
Exact mathematical passage
\Phi(r)
\varphiUse in source · 16 passages
Search aliases: φ varphi
Use in source · Machine_Societies_v1.0.0.tex, line 1322
tibility, and knowledge Temporal obligations, contracts, and office succession. Permissions describe available actions; obligations constrain continuations. The debt created by an urgency exchange is not satisfied merely because the creditor remains authorized to ask for service. It requires a particular future event. A finite obligation record can be written as o=(id,i,j,φ,d,authority,status), where i and j are the debtor and beneficiary offices, φ specifies a fulfillment event, and d is a dead…
Exact mathematical passage
o=(\mathsf{id},i,j,\varphi,d,\mathsf{authority},\mathsf{status}),
\PsiUse in source · 4 passages
Search aliases: Ψ Psi
Use in source · Machine_Societies_v1.0.0.tex, line 1115
Let s=(x,I) be the full operating state and let R(s,s') be the admitted transition relation, including ordinary actions and amendments. A safety property is a predicate Ψ(s).
Exact mathematical passage
\Psi(s)
\psiUse in source · 5 passages
Search aliases: ψ psi
Use in source · Machine_Societies_v1.0.0.tex, line 523
Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.
Exact mathematical passage
\psi:\Z\to\mathcal W
\psi_{\#}Use in source · 2 passages
Search aliases: ψ_# psi
Use in source · Machine_Societies_v1.0.0.tex, line 527
Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.
Exact mathematical passage
\TV(\psi_{\#}\overline K_u(z,\cdot),\widehat K_u(\psi z,\cdot))\leq\varepsilon_2.
\ldotsUse in source · 16 passages
Search aliases: … ldots
Use in source · Machine_Societies_v1.0.0.tex, line 205
[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is rejected; no new entitlement is created. These transitions are specified rather than discovered. The example is a miniature institutional realization…
Exact mathematical passage
.35em] 1 & Three offices adopt $U$; $A$ requests urgency and $B$ consents & $(2,0,1,1)$ & $A$ owes $B$ one slot at epoch 2.\\[.35em] 2 & $A$ is replaced by an authorized successor; repayment takes precedence & $(0,2,1,1)$ & The same obligation is fulfilled, not erased.\\[.35em] 3 & A proposed external grant would duplicate a slot token & $(1,1,1,1)$ & The proposal is rejected; no new entitlement is created.\\ \bottomrule \end{tabularx} \end{center} These transitions are specified rather than discovered. The example is a miniature institutional realization, not a demonstration of autonomous formation. Its purpose is to make several different mathematical obligations visible in one system. \subsection{What an institutional summary must retain} The name of the current rule is insufficient state. Two records may both say $R$ while only one contains a debt due this epoch. Their correct next allocations differ. A summary that keeps the next allocation but omits the amendment threshold can also be inadequate: the same three votes may authorize different future rules under different constitutions. Likewise, identical text with different active credentials can confer different decision rights. The public ledger is not an additional, society-wide preference function. Offices may disagree about whether $R$ or $U$ is desirable, about how urgency should be certified, or about who should pay the cost of maintaining the ledger. An atomic allocation mechanism can enforce the resource invariant without settling those disagreements. Private demand, operative authority, and the shared event record are distinct parts of the system. We return to this commons when considering institution formation, governance-preserving summaries, deadline monitors, blocking coalitions, replacement, and federation. Appendix~\ref{sec:worked} retains a smaller algebraic example that isolates reusable state from social interpretation. The extensions later in the paper make the same commons sensitive to membership epochs, message types, notification histories, and the order of amendments. Its offices may preserve a shared allocation method while disagreeing about which new scheduling question deserves resources. These are analytical variations of one declared example, not reports of a population experiment. \section{A controlled realization model}\label{sec:model} For the controlled realization results, assume finite nonempty sets of microstates $\X$, external controls $\U$, and reported outcomes $\Y$. The finite assumption keeps measurability and regular-conditional-probability issues out of the exposition. Continuous-state generalizations require additional hypotheses. A realization has transition kernel \begin{equation}\label{eq:micro} K_u(x,x')=\Prob(X_{t+1}=x'\mid X_t=x,U_t=u), \end{equation} and output kernel $Q_u(x,y)$. Both are row-stochastic. In this presentation, the current report and next state are sampled conditionally independently given $(x,u)$. A dependent report/transition mechanism can instead be specified by a joint kernel on $(x',y)$; the corresponding aggregation test must preserve that joint law, not just its two marginals. A deterministic implementation is the special case $K_u(x,\cdot)=\delta_{F_u(x)}$. The controls can encode tasks, supplied evidence, and permitted interventions. Scheduling and environment dynamics must be included in $X_t$ or in the stated control model rather than silently omitted. One possible factorization of a microstate is \[ X_t=(S_{1,t},\ldots,S_{n,t},M_t,C_t,A_t,\Theta_t),
\ell^1Use in source · 1 passage
Search aliases: ℓ^1 ell
Use in source · Machine_Societies_v1.0.0.tex, line 1718
Possible metrics include the discrete metric on permission vectors, a declared metric on obligation-monitor states, an ℓ^1 distance on common-unit resource vectors, and symmetric-difference cardinality on required evidence records. For evidence, record equality is structural preservation under a trust model, not a truth certificate. If the payload or inferential dependencies matter, they must be included in the observable rather than only its identifier.
Exact mathematical passage
\ell^1
\ell^\starUse in source · 2 passages
Search aliases: ℓ^⋆ ell star
Use in source · Machine_Societies_v1.0.0.tex, line 1209
[Quiescent convergence and invariant preservation] Assume finitely many replicas begin at ℓ_0, only finitely many contributions δ_1,…,δ_N are created, and every contribution eventually becomes included in every replica. Then each replica eventually equals ℓ^⋆=ℓ_0veeδ_1vee⋯veeδ_N, independently of merge order and duplicate deliveries. Moreover, let V⊆ L contain the initial state. If every permitted local update maps V into V and V is closed under vee, every local and merged state remains in V. At…
Exact mathematical passage
\ell^\star=\ell_0\vee\delta_1\vee\cdots\vee\delta_N,
\ell_0Use in source · 3 passages
Search aliases: ℓ_0 ell
Use in source · Machine_Societies_v1.0.0.tex, line 1208
[Quiescent convergence and invariant preservation] Assume finitely many replicas begin at ℓ_0, only finitely many contributions δ_1,…,δ_N are created, and every contribution eventually becomes included in every replica. Then each replica eventually equals ℓ^⋆=ℓ_0veeδ_1vee⋯veeδ_N, independently of merge order and duplicate deliveries. Moreover, let V⊆ L contain the initial state. If every permitted local update maps V into V and V is closed under vee, every local and merged state remains in V. At…
Exact mathematical passage
\ell_0
\ell_iUse in source · 3 passages
Search aliases: ℓ_i ell
Use in source · Machine_Societies_v1.0.0.tex, line 1205
Convergent records and valid decisions. Let (L,vee) be a join-semilattice, with order a≤ b when avee b=b. A replica holds ℓ_i∈ L. In the following restricted model a local update joins a declared contribution δ into the state, and a merge replaces ℓ_i by ℓ_iveeℓ_j. Contributions are fixed elements of L once created. This is a state-merge construction of the kind used in conflict-free replicated data types~[source reference]. The algebra concerns accepted data, not the authenticity of its source.
Exact mathematical passage
\ell_i\in L
\ell_jUse in source · 1 passage
Search aliases: ℓ_j ell
Use in source · Machine_Societies_v1.0.0.tex, line 1205
Convergent records and valid decisions. Let (L,vee) be a join-semilattice, with order a≤ b when avee b=b. A replica holds ℓ_i∈ L. In the following restricted model a local update joins a declared contribution δ into the state, and a merge replaces ℓ_i by ℓ_iveeℓ_j. Contributions are fixed elements of L once created. This is a state-merge construction of the kind used in conflict-free replicated data types~[source reference]. The algebra concerns accepted data, not the authenticity of its source.
Exact mathematical passage
\ell_i\vee\ell_j
\to^*Use in source · 6 passages
Search aliases: →^* to
Use in source · Machine_Societies_v1.0.0.tex, line 1166
Fix a bounded amendment episode. Let s→ s' mean one permitted amendment step, and let →^* be its reflexive transitive closure. The episode state contains any pending proposals, version counters, permissions, and consequence records required by the contract. It is terminating if it has no infinite chain of actual steps, and locally confluent if s→ a, s→ b Longrightarrow ∃ c: a→^*c and b→^*c. Confluence imposes the same condition on arbitrary finite descendants of s. A normal form has no outgoing …
Exact mathematical passage
\to^*
\mapstoUse in source · 2 passages
Search aliases: ↦ mapsto
Use in source · Machine_Societies_v1.0.0.tex, line 1651
[Exact frame synchronization] Global frames exist if and only if H_γ=e_G for every closed walk. When they exist, they are unique up to common right multiplication h_i↦ h_i k. It suffices to check the rooted fundamental loops associated with one spanning tree. If g_ij=h_jh_i^-1, adjacent factors cancel in every path product, giving H_γ=h_v_kh_v_0^-1 and hence identity on a closed walk. Conversely, choose a root r, set h_r=e_G, and define h_i as the product on the unique tree path from r to i. Tre…
Exact mathematical passage
h_i\mapsto h_i k
\RightarrowUse in source · 2 passages
Search aliases: ⇒ Rightarrow
Use in source · Machine_Societies_v1.0.0.tex, line 1328
For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy follows from admitting discharge in this particular contract.
Exact mathematical passage
\mathbf G\bigl(\mathsf{created}(o)\Rightarrow \mathbf F_{\leq D}\mathsf{settled}(o)\bigr).
\notinUse in source · 1 passage
Search aliases: ∉ notin
Use in source · Machine_Societies_v1.0.0.tex, line 1474
Let Q be the substochastic transition matrix on V, so the missing row mass represents exit. Let μ be an initial row probability vector supported on V and define τ=inf{t≥1:s_t∉ V}.
Exact mathematical passage
\tau=\inf\{t\geq1:s_t\notin V\}
\prodUse in source · 3 passages
Search aliases: ∏ prod
Use in source · Machine_Societies_v1.0.0.tex, line 497
[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.
Exact mathematical passage
e_T\leq e_0\prod_{t=0}^{T-1}a_t+ \sum_{j=0}^{T-1}b_j\prod_{t=j+1}^{T-1}a_t.
\prod_{k=j+1}^{m-1}Use in source · 1 passage
Search aliases: ∏_k=j+1^m-1 prod
Use in source · Machine_Societies_v1.0.0.tex, line 1381
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z)) · ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source r…
Exact mathematical passage
\label{eq:cumulative-transport} \begin{split} &d_m\bigl(T_{0:m}F_0(z),F_mT_{0:m}(z)\bigr)\\ &\qquad\le\sum_{j=0}^{m-1}\varepsilon_j\prod_{k=j+1}^{m-1}L_k, \end{split}
\prod_{t=0}^{T-1}Use in source · 1 passage
Search aliases: ∏_t=0^T-1 prod
Use in source · Machine_Societies_v1.0.0.tex, line 497
[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.
Exact mathematical passage
e_T\leq e_0\prod_{t=0}^{T-1}a_t+ \sum_{j=0}^{T-1}b_j\prod_{t=j+1}^{T-1}a_t.
\prod_{t=1}^{m}Use in source · 1 passage
Search aliases: ∏_t=1^m prod
Use in source · Machine_Societies_v1.0.0.tex, line 2452
tional disclosure accounting] Consider m releases on finite alphabets. At each common prefix h, suppose their conditional next-release laws P_C^t(·mid h) and P_D^t(·mid h) satisfy both pointwise inequalities with factor λ_t≥1, uniformly over prefixes that can occur. Then the joint transcript laws satisfy~[source reference] with λ=∏_t=1^mλ_t. The bound also holds after applying the same stochastic postprocessing to the transcript under both cases. In particular, any sanction computed only from th…
Exact mathematical passage
\lambda=\prod_{t=1}^{m}\lambda_t.
\prod_{t=j+1}^{T-1}Use in source · 1 passage
Search aliases: ∏_t=j+1^T-1 prod
Use in source · Machine_Societies_v1.0.0.tex, line 497
[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.
Exact mathematical passage
e_T\leq e_0\prod_{t=0}^{T-1}a_t+ \sum_{j=0}^{T-1}b_j\prod_{t=j+1}^{T-1}a_t.
\sum_{(i,j)\text{ traversed by }\gamma}Use in source · 1 passage
Search aliases: ∑_(i,j) traversed by γ sum text gamma
Use in source · Machine_Societies_v1.0.0.tex, line 1695
A quantitative lower bound on interface error. Suppose G has a bi-invariant metric d, so both left and right multiplication preserve distance. For candidate frames h_i, write δ_ij=d(g_ij,h_jh_i^-1). [Loop defect bounds unavoidable edge error] For a closed walk γ with k≥1 traversals, d(H_γ,e_G)≤∑_(i,j) traversed by γδ_ij. In particular, some traversed edge has error at least d(H_γ,e_G)/k for every candidate frame family. Gauge-transform each edge to widetilde g_ij=h_j^-1g_ijh_i. Bi-invariance giv…
Exact mathematical passage
d(H_\gamma,e_G)\le\sum_{(i,j)\text{ traversed by }\gamma}\delta_{ij}.
\sum_iUse in source · 5 passages
Search aliases: ∑_i sum
Use in source · Machine_Societies_v1.0.0.tex, line 864
In an idealized transferable-utility setting there is a corresponding conditional observation. If the discounted net advantages b_i satisfy ∑_i b_i≥0, then there exist budget-balanced transfers t_i with ∑_i t_i=0 and b_i+t_i≥0 for all i: choose nonnegative w_i summing to ∑_i b_i and set t_i=w_i-b_i. This algebra does not supply liquidity, a transfer mechanism, consent, enforceability, or a right to redistribute. The difference between mathematical feasible compensation and institutionally permit…
Exact mathematical passage
\sum_i b_i\ge0
\sum_{i=1}^nUse in source · 1 passage
Search aliases: ∑_i=1^n sum
Use in source · Machine_Societies_v1.0.0.tex, line 1793
Suppose errors E_1,…,E_n in scalar estimates are centered, have common variance σ^2>0, and have common pairwise correlation ρ∈[0,1]. Then Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equal-variance estimators, the variance-equivalent effective sample size is n_eff=fracn1+(n-1)ρ. If the errors are perfectly correlated, adding copies doe…
Exact mathematical passage
\label{eq:correlated} \operatorname{Var}\left(\frac1n\sum_{i=1}^n E_i\right) =\frac{\sigma^2}{n}\bigl(1+(n-1)\rho\bigr).
\sum_{i\in\{A,B,C,D\}}Use in source · 1 passage
Search aliases: ∑_i∈{A,B,C,D} sum in
Use in source · Machine_Societies_v1.0.0.tex, line 185
public record contains the current allocation rule, valid credentials, accepted urgency certificates, outstanding service obligations, and the amendment history. Slots have unique names (e,k), where e is the epoch and k∈{1,2,3,4}. A grant consumes one available slot token. Copying a message that describes a grant does not create an additional token. The resource invariant is therefore ∑_i∈{A,B,C,D}q_i(e)≤4, with no slot name assigned to two offices. In the worked trace all four offices demand th…
Exact mathematical passage
\sum_{i\in\{A,B,C,D\}}q_i(e)\leq4,
\sum_jUse in source · 6 passages
Search aliases: ∑_j sum
Use in source · Machine_Societies_v1.0.0.tex, line 545
The same argument applies separately to output kernels. Combining this proposition with Proposition~[source reference] yields a finite-horizon bound of e_0+t∑_jε_j, capped at one, under the corresponding controls. This elementary estimate belongs to the broader subject of compositional abstraction error~[source reference]. It does not establish global causal closure unless the intervention implementations also compose compatibly.
Exact mathematical passage
e_0+t\sum_j\varepsilon_j
\sum_{j=0}^{m-1}Use in source · 1 passage
Search aliases: ∑_j=0^m-1 sum
Use in source · Machine_Societies_v1.0.0.tex, line 1381
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z)) · ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source r…
Exact mathematical passage
\label{eq:cumulative-transport} \begin{split} &d_m\bigl(T_{0:m}F_0(z),F_mT_{0:m}(z)\bigr)\\ &\qquad\le\sum_{j=0}^{m-1}\varepsilon_j\prod_{k=j+1}^{m-1}L_k, \end{split}
\sum_{j=0}^{T-1}Use in source · 1 passage
Search aliases: ∑_j=0^T-1 sum
Use in source · Machine_Societies_v1.0.0.tex, line 497
[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.
Exact mathematical passage
e_T\leq e_0\prod_{t=0}^{T-1}a_t+ \sum_{j=0}^{T-1}b_j\prod_{t=j+1}^{T-1}a_t.
\sum_{j=0}^{t-1}Use in source · 1 passage
Search aliases: ∑_j=0^t-1 sum
Use in source · Machine_Societies_v1.0.0.tex, line 2014
n explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-step monotonic estimate followed by induction. Under uniformly bounded disturbance normw_t_v,∞≤bar w, the last term is at most bar w(1-κ^t)/(1-κ). Thi…
Exact mathematical passage
\label{eq:propagation-bound} \norm{x_t}_{v,\infty}\le \kappa^t\norm{x_0}_{v,\infty} +\sum_{j=0}^{t-1}\kappa^{t-1-j}\norm{w_j}_{v,\infty}.
\sum_{j=1}^LUse in source · 1 passage
Search aliases: ∑_j=1^L sum
Use in source · Machine_Societies_v1.0.0.tex, line 539
[Compositional one-step closure] The composite abstraction ψπ satisfies d_TV((ψπ)_#K_u(x,·), widehat K_u(ψπ x,·)) ≤ε_1+ε_2. For L levels, the one-step defect is at most ∑_j=1^Lε_j. In particular, exact abstractions compose exactly. Insert ψ_#overline K_u(π x,·), use the triangle inequality, and use nonexpansiveness of total variation under deterministic pushforward. Induction gives the multilevel result.
Exact mathematical passage
\sum_{j=1}^L\varepsilon_j
\sum_{j\ne i}Use in source · 1 passage
Search aliases: ∑_j≠ i sum ne
Use in source · Machine_Societies_v1.0.0.tex, line 887
A formation barrier despite mutual benefit. The stationary cost comparison does not describe a route to adoption. Consider n≥2 agents choosing a_i∈{0,1}. An adopter pays cost c>0 and receives benefit b>0 from each other adopter: u_i(a)=a_i(b∑_j≠ ia_j-c). Nonadopters receive zero. This is a finite network-coordination model, not a universal utility function for societies.
Exact mathematical passage
u_i(a)=a_i\left(b\sum_{j\ne i}a_j-c\right).
\sum_{o\in O}Use in source · 1 passage
Search aliases: ∑_o∈ O sum in
Use in source · Machine_Societies_v1.0.0.tex, line 1011
empty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous margin by D_j(s)=F⟨ P_j-P_C,s⟩-g_j, Γ_α=max_s∈ M…
Exact mathematical passage
\langle P,s\rangle=\sum_{o\in O}P(o)s(o)
\sum_{r=0}^{\infty}Use in source · 1 passage
Search aliases: ∑_r=0^∞ sum infty
Use in source · Machine_Societies_v1.0.0.tex, line 1843
a pair locks correctly with probability p^2, locks incorrectly with probability (1-p)^2, and returns to zero with probability 2p(1-p). Conditional independence gives independent repetitions of this two-step cycle until absorption. Since 2p(1-p)<1, nonabsorption through r cycles has the stated probability and tends to zero. Summing the probability of eventual incorrect absorption gives ∑_r=0^∞[2p(1-p)]^r(1-p)^2 =frac(1-p)^21-2p(1-p), which is~[source reference]. In an absorbing regime every subse…
Exact mathematical passage
\sum_{r=0}^{\infty}[2p(1-p)]^r(1-p)^2 =\frac{(1-p)^2}{1-2p(1-p)},
\sum_tUse in source · 1 passage
Search aliases: ∑_t sum
Use in source · Machine_Societies_v1.0.0.tex, line 2461
Writing λ_t=e^ε_t gives the additive parameter ∑_tε_t. The proposition requires a fixed finite number of releases and uniform conditional bounds. A data-selected stopping rule must itself be represented in the transcript or controlled by an appropriate conditional argument. Privileged side information not covered by the declared record is not protected by a certificate for that record alone. The preceding counterexample fails the conditional premise at the second release, even though both uncond…
Exact mathematical passage
\sum_t\varepsilon_t
\sum_{t=a}^{b}Use in source · 1 passage
Search aliases: ∑_t=a^b sum
Use in source · Machine_Societies_v1.0.0.tex, line 2477
All admitted appeals can be assigned to distinct review slots in their windows if and only if, for every subset U⊆ J, |U|≤∑_t∈⋃_j∈ U{r_j,…,d_j}c_t. Equivalently, because the windows are intervals, it suffices to check every integer interval [a,b]⊆[1,T]: |{j:r_j≥ a, d_j≤ b}| ≤∑_t=a^bc_t. Replace epoch t by c_t distinct slot vertices and join appeal j to the vertices in its window. An assignment is a matching covering the appeals. The matching criterion used in Theorem~[source reference], equivale…
Exact mathematical passage
\label{eq:appeal-interval} \bigl|\{j:r_j\ge a,\ d_j\le b\}\bigr| \le\sum_{t=a}^{b}c_t.
\sum_{t\in\bigcup_{j\in U}\{r_j,\ldots,d_j\}}Use in source · 1 passage
Search aliases: ∑_t∈⋃_j∈ U{r_j,…,d_j} sum in bigcup in ldots
Use in source · Machine_Societies_v1.0.0.tex, line 2473
[Capacity for a finite appeal contract] All admitted appeals can be assigned to distinct review slots in their windows if and only if, for every subset U⊆ J, |U|≤∑_t∈⋃_j∈ U{r_j,…,d_j}c_t. Equivalently, because the windows are intervals, it suffices to check every integer interval [a,b]⊆[1,T]: |{j:r_j≥ a, d_j≤ b}| ≤∑_t=a^bc_t. Replace epoch t by c_t distinct slot vertices and join appeal j to the vertices in its window. An assignment is a matching covering the appeals. The matching criterion used…
Exact mathematical passage
\label{eq:appeal-hall} |U|\le\sum_{t\in\bigcup_{j\in U}\{r_j,\ldots,d_j\}}c_t.
\sum_{t\geq0}Use in source · 1 passage
Search aliases: ∑_t≥0 sum geq
Use in source · Machine_Societies_v1.0.0.tex, line 1487
geq0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random variable gives Eτ=∑_t≥0 P(τ>t). Under ρ(Q)<1, the matrix geometric series converges to (Id-Q)^-1, giving the result.
Exact mathematical passage
\E\tau=\sum_{t\geq0}\Prob(\tau>t)
\sum_{w:\pi(w)=z'}Use in source · 1 passage
Search aliases: ∑_w:π(w)=z' sum pi
Use in source · Machine_Societies_v1.0.0.tex, line 314
[Controlled lumpability] A common kernel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-model with initial law pushμ_0. For outputs to agree as well, require Q_u(x,·) to be constant on each fibre. The common fibre value in~[source reference] defines overline K…
Exact mathematical passage
\label{eq:lump} \sum_{w:\pi(w)=z'}K_u(x,w) =\sum_{w:\pi(w)=z'}K_u(x',w).
\sum_yUse in source · 1 passage
Search aliases: ∑_y sum
Use in source · Machine_Societies_v1.0.0.tex, line 1529
Assume K_i has a stationary law μ_i and sup_i,xd_TV(K_i^m(x,·),μ_i)≤ε. The actual one-review law from (i,x) and its stationary average are P(i,x;j) · =∑_y K_i^m(x,y)R_i(y,j), overline R(i,j) · =∑_yμ_i(y)R_i(y,j). The endpoint is the only fast input to this particular revision rule. A rule that uses the full trajectory, accumulated costs, or a data-dependent stopping time requires a different state or a different averaging argument.
Exact mathematical passage
P(i,x;j)&=\sum_y K_i^m(x,y)R_i(y,j),\label{eq:actual-review}\\ \overline R(i,j)&=\sum_y\mu_i(y)R_i(y,j).\label{eq:averaged-review}
\sum_zUse in source · 2 passages
Search aliases: ∑_z sum
Use in source · Machine_Societies_v1.0.0.tex, line 337
Finite-horizon error accounting. For finite distributions, d_TV(μ,ν)=tfrac12∑_z|μ(z)-ν(z)|. Suppose candidate kernels satisfy sup_x,ud_TV(push K_u(x,·),overline K_u(π(x),·)) ≤ε. This uniform condition is stronger than a small average error on the trajectories seen during training.
Exact mathematical passage
\TV(\mu,\nu)=\tfrac12\sum_z|\mu(z)-\nu(z)|
\setminusUse in source · 2 passages
Search aliases: ∖ setminus
Use in source · Machine_Societies_v1.0.0.tex, line 690
nter controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assigns nonnegative voting weights, q_t is an approval threshold, and e_t is a configuration epoch. Admission and exit proposals determine M_t+1=(M_t∖ E_t)∪ A_t, E_t⊆ M_t, A_t⊆ U∖ M_t, but this equation alone specifies neither their authorization nor th…
Exact mathematical passage
M_{t+1}=(M_t\setminus E_t)\cup A_t, \qquad E_t\subseteq M_t,\quad A_t\subseteq\mathcal U\setminus M_t,
\circUse in source · 2 passages
Search aliases: ∘ circ
Use in source · Machine_Societies_v1.0.0.tex, line 675
The order of such operations can matter. Let G set a permission bit to 1 (grant) and R set it to 0 (revoke). Then G∘ R yields 1 and R∘ G yields 0 on every input. This simple noncommutativity has practical conceptual force: a log of individual events without their authorized order may fail to reconstruct the actual institutional state. Revision histories must preserve operations and order, not only a multiset of decisions.
Exact mathematical passage
G\circ R
\inftyUse in source · 5 passages
Search aliases: ∞ infty
Use in source · Machine_Societies_v1.0.0.tex, line 1380
[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))
Exact mathematical passage
\varepsilon_j<\infty
\cupUse in source · 3 passages
Search aliases: ∪ cup
Use in source · Machine_Societies_v1.0.0.tex, line 690
nter controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assigns nonnegative voting weights, q_t is an approval threshold, and e_t is a configuration epoch. Admission and exit proposals determine M_t+1=(M_t∖ E_t)∪ A_t, E_t⊆ M_t, A_t⊆ U∖ M_t, but this equation alone specifies neither their authorization nor th…
Exact mathematical passage
M_{t+1}=(M_t\setminus E_t)\cup A_t, \qquad E_t\subseteq M_t,\quad A_t\subseteq\mathcal U\setminus M_t,
\sim_AUse in source · 1 passage
Search aliases: ∼_A sim
Use in source · Machine_Societies_v1.0.0.tex, line 1907
[Everyone knows, but it is not common knowledge] Let W={0,1,2} and let φ hold exactly at 0,1. Agent A has partition {0,1}mid{2}, and agent B has partition {0}mid{1,2}. At world 0, both agents know φ. However, the chain 0∼_A1∼_B2 reaches a world where it is false. Thus E_{A,B}φ holds at 0 but C_{A,B}φ does not.
Exact mathematical passage
0\sim_A1\sim_B2
\sim_BUse in source · 1 passage
Search aliases: ∼_B sim
Use in source · Machine_Societies_v1.0.0.tex, line 1907
[Everyone knows, but it is not common knowledge] Let W={0,1,2} and let φ hold exactly at 0,1. Agent A has partition {0,1}mid{2}, and agent B has partition {0}mid{1,2}. At world 0, both agents know φ. However, the chain 0∼_A1∼_B2 reaches a world where it is false. Thus E_{A,B}φ holds at 0 but C_{A,B}φ does not.
Exact mathematical passage
0\sim_A1\sim_B2
\sim_iUse in source · 4 passages
Search aliases: ∼_i sim
Use in source · Machine_Societies_v1.0.0.tex, line 1886
Fix one configuration epoch and a nonempty group J of named decision centers. Let W be a finite set of possible worlds. Each equivalence relation ∼_i records the worlds agent i cannot distinguish using its permitted information. A world may include the epoch, relevant history, and current records. This is an idealized semantics of information, not a claim that an implemented agent can compute every logical consequence.
Exact mathematical passage
\sim_i
\sim_i^AUse in source · 1 passage
Search aliases: ∼_i^A sim
Use in source · Machine_Societies_v1.0.0.tex, line 1930
Consider models on W_A,W_B with the same named agents and atomic predicates. Let T:W_A→ W_B be surjective and preserve the truth of each atom in both directions. For every agent i, require w∼_i^A v · Longrightarrow T(w)∼_i^B T(v) (forth), T(w)∼_i^B z · Longrightarrow ∃ v: w∼_i^A v and T(v)=z (back). These are the usual bounded-morphism conditions, stated here as a transport contract.
Exact mathematical passage
w\sim_i^A v&\ \Longrightarrow\ T(w)\sim_i^B T(v) \quad\text{(forth)},\\ T(w)\sim_i^B z&\ \Longrightarrow\ \exists v:\ w\sim_i^A v\text{ and }T(v)=z \quad\text{(back)}.
\sim_i^BUse in source · 1 passage
Search aliases: ∼_i^B sim
Use in source · Machine_Societies_v1.0.0.tex, line 1930
Consider models on W_A,W_B with the same named agents and atomic predicates. Let T:W_A→ W_B be surjective and preserve the truth of each atom in both directions. For every agent i, require w∼_i^A v · Longrightarrow T(w)∼_i^B T(v) (forth), T(w)∼_i^B z · Longrightarrow ∃ v: w∼_i^A v and T(v)=z (back). These are the usual bounded-morphism conditions, stated here as a transport contract.
Exact mathematical passage
w\sim_i^A v&\ \Longrightarrow\ T(w)\sim_i^B T(v) \quad\text{(forth)},\\ T(w)\sim_i^B z&\ \Longrightarrow\ \exists v:\ w\sim_i^A v\text{ and }T(v)=z \quad\text{(back)}.
\simeqUse in source · 1 passage
Search aliases: ≃ simeq
Use in source · Machine_Societies_v1.0.0.tex, line 1630
Even when a global static state exists, a federation may fail dynamically. Local rule applications could commute in one context and conflict in another. A translator T_ij:Z_i→ Z_j should preserve the operations relevant to the interaction, approximately satisfying T_ijF_i≃ F_jT_ij. A federation requires both representational compatibility and admissible operational coordination. Static agreement alone does not define an executable constitution. Nor does faithful gluing prove that the institution…
Exact mathematical passage
T_{ij}F_i\simeq F_jT_{ij}
\equivUse in source · 2 passages
Search aliases: ≡ equiv
Use in source · Machine_Societies_v1.0.0.tex, line 407
For a family T of finite continuation experiments, define exact behavioral equivalence by x≡_ Tx' Longleftrightarrow L_x(Y^+midτ)= L_x'(Y^+midτ) for every τ∈ T. Equality of these indexed laws is reflexive, symmetric, and transitive. It therefore defines a quotient. This equivalence is relative to the chosen experiments. It does not by itself equal bisimulation, and it says nothing about untested interventions or provenance.
Exact mathematical passage
x\equiv_{\mathcal T}x'\quad\Longleftrightarrow\quad \Law_x(Y^+\mid\tau)=\Law_{x'}(Y^+\mid\tau) \quad\text{for every }\tau\in\mathcal T.
\equiv_{\mathcal T}Use in source · 1 passage
Search aliases: ≡_ T equiv mathcal
Use in source · Machine_Societies_v1.0.0.tex, line 407
For a family T of finite continuation experiments, define exact behavioral equivalence by x≡_ Tx' Longleftrightarrow L_x(Y^+midτ)= L_x'(Y^+midτ) for every τ∈ T. Equality of these indexed laws is reflexive, symmetric, and transitive. It therefore defines a quotient. This equivalence is relative to the chosen experiments. It does not by itself equal bisimulation, and it says nothing about untested interventions or provenance.
Exact mathematical passage
x\equiv_{\mathcal T}x'\quad\Longleftrightarrow\quad \Law_x(Y^+\mid\tau)=\Law_{x'}(Y^+\mid\tau) \quad\text{for every }\tau\in\mathcal T.
\equiv_kUse in source · 1 passage
Search aliases: ≡_k equiv
Use in source · Machine_Societies_v1.0.0.tex, line 762
he smallest exact summary under a fixed contract The conditions yield a finite construction rather than only an existence criterion. Begin with the partition B_0 that identifies exactly the states sharing (ℓ(s),D(s)). At each refinement, identify s,t only when they were previously in the same block and every action in their common enabled set leads to the same previous block: s≡_k+1t Longleftrightarrow s≡_kt and F(s,a)≡_kF(t,a) for all a∈ D(s).
Exact mathematical passage
s\equiv_{k+1}t\quad\Longleftrightarrow\quad s\equiv_kt\ \text{ and }\ F(s,a)\equiv_kF(t,a) \ \text{for all }a\in D(s).
\equiv_{k+1}Use in source · 1 passage
Search aliases: ≡_k+1 equiv
Use in source · Machine_Societies_v1.0.0.tex, line 762
he smallest exact summary under a fixed contract The conditions yield a finite construction rather than only an existence criterion. Begin with the partition B_0 that identifies exactly the states sharing (ℓ(s),D(s)). At each refinement, identify s,t only when they were previously in the same block and every action in their common enabled set leads to the same previous block: s≡_k+1t Longleftrightarrow s≡_kt and F(s,a)≡_kF(t,a) for all a∈ D(s).
Exact mathematical passage
s\equiv_{k+1}t\quad\Longleftrightarrow\quad s\equiv_kt\ \text{ and }\ F(s,a)\equiv_kF(t,a) \ \text{for all }a\in D(s).
\leUse in source · 21 passages
Search aliases: ≤ le
Use in source · Machine_Societies_v1.0.0.tex, line 836
Suppose a coordination protocol without institution has expected per-period cost c_0. A proposed institutional protocol has operating cost c_1, per-period maintenance m, and a one-time setup cost B. With discount factor 0≤δ<1, write g=c_0-c_1-m.
Exact mathematical passage
0\le\delta<1
\geUse in source · 13 passages
Search aliases: ≥ ge
Use in source · Machine_Societies_v1.0.0.tex, line 864
In an idealized transferable-utility setting there is a corresponding conditional observation. If the discounted net advantages b_i satisfy ∑_i b_i≥0, then there exist budget-balanced transfers t_i with ∑_i t_i=0 and b_i+t_i≥0 for all i: choose nonnegative w_i summing to ∑_i b_i and set t_i=w_i-b_i. This algebra does not supply liquidity, a transfer mechanism, consent, enforceability, or a right to redistribute. The difference between mathematical feasible compensation and institutionally permit…
Exact mathematical passage
\sum_i b_i\ge0
\oplusUse in source · 15 passages
Search aliases: ⊕ oplus
Use in source · Machine_Societies_v1.0.0.tex, line 266
Relational information: access, synergy, and usefulness. Let A,B be independent fair bits and Y=A⊕ B. Then I(Y;A)=I(Y;B)=0, I(Y;A,B)=1. This elementary example shows that joint observation may reveal information absent from each marginal observation. It does not show that the information is absent from the complete system, that a mysterious carrier has appeared, or that a protocol with ordinary messages cannot recover it.
Exact mathematical passage
Y=A\oplus B
\bigcup_{j\in U}Use in source · 1 passage
Search aliases: ⋃_j∈ U bigcup in
Use in source · Machine_Societies_v1.0.0.tex, line 2473
[Capacity for a finite appeal contract] All admitted appeals can be assigned to distinct review slots in their windows if and only if, for every subset U⊆ J, |U|≤∑_t∈⋃_j∈ U{r_j,…,d_j}c_t. Equivalently, because the windows are intervals, it suffices to check every integer interval [a,b]⊆[1,T]: |{j:r_j≥ a, d_j≤ b}| ≤∑_t=a^bc_t. Replace epoch t by c_t distinct slot vertices and join appeal j to the vertices in its window. An assignment is a matching covering the appeals. The matching criterion used…
Exact mathematical passage
\label{eq:appeal-hall} |U|\le\sum_{t\in\bigcup_{j\in U}\{r_j,\ldots,d_j\}}c_t.
\starUse in source · 5 passages
Search aliases: ⋆ star
Use in source · Machine_Societies_v1.0.0.tex, line 383
[Implementation dependence] If every P_λ is within ε of a proposed macro-intervention law P^⋆_z', then η(x,z')≤2ε. Conversely, if two admissible implementations are distance η apart, no single law can approximate both within less than η/2. Both claims follow from the triangle inequality for total variation.
Exact mathematical passage
P^\star_{z'}
\cdotsUse in source · 3 passages
Search aliases: ⋯ cdots
Use in source · Machine_Societies_v1.0.0.tex, line 1209
[Quiescent convergence and invariant preservation] Assume finitely many replicas begin at ℓ_0, only finitely many contributions δ_1,…,δ_N are created, and every contribution eventually becomes included in every replica. Then each replica eventually equals ℓ^⋆=ℓ_0veeδ_1vee⋯veeδ_N, independently of merge order and duplicate deliveries. Moreover, let V⊆ L contain the initial state. If every permitted local update maps V into V and V is closed under vee, every local and merged state remains in V. At…
Exact mathematical passage
\ell^\star=\ell_0\vee\delta_1\vee\cdots\vee\delta_N,
\langleUse in source · 6 passages
Search aliases: ⟨ langle
Use in source · Machine_Societies_v1.0.0.tex, line 1008
The common-monitor problem. Let O be a finite nonempty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous…
Exact mathematical passage
\mathcal M_\alpha=\{s\in[0,1]^O:\langle P_C,s\rangle\leq\alpha\}.
\rangleUse in source · 6 passages
Search aliases: ⟩ rangle
Use in source · Machine_Societies_v1.0.0.tex, line 1008
The common-monitor problem. Let O be a finite nonempty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous…
Exact mathematical passage
\mathcal M_\alpha=\{s\in[0,1]^O:\langle P_C,s\rangle\leq\alpha\}.
\longrightarrowUse in source · 2 passages
Search aliases: ⟶ longrightarrow
Use in source · Machine_Societies_v1.0.0.tex, line 1463
Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not describe all cultural evolution. It establishes that selection for reproduction alone does not logically impose selection for accuracy. An institution nee…
Exact mathematical passage
\frac{N_T r_T^t}{N_T r_T^t+N_F r_F^t} =\frac{1}{1+(N_F/N_T)(r_F/r_T)^t}\longrightarrow0.
Source coverage
Automatic extraction covers TeX and Markdown math delimiters and common equation environments in these selected files. It does not interpret diagrams, images, PDF text, program identifiers, or historical archives. Occurrences are navigation aids, not a proof review.
Download all entries, passages and source checksums (JSON) · Collection coverage
- monographs/machine-societies/README.md · 0 mathematical passages
- monographs/machine-societies/publication/Machine_Societies_v1.0.0.tex · 1619 mathematical passages
- monographs/machine-societies/publication/release_review.tex · 0 mathematical passages
- monographs/machine-societies/publication/statement_index.tex · 0 mathematical passages
PDF reading editions (1)
The text index does not extract PDF glyphs. Check the reading edition for notation in figures, tables, and PDF-only material.