Notation by work

Machine Theory Compendium

All notation · Glossary · Source coverage

110 symbol spellings in 132 mathematical passages across 3 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 110 entries are available below.

*Use in source · 5 passages

Search aliases: *

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Use in source · connected_synthesis.tex, line 247
ng widehat R has regret at most 2η relative to a minimizer of R when the action set is finite and nonempty. Insert the intermediate expectation ∑widehatμ_ωℓ(a,y_ω). The first difference is at most e because the integrand is in [0,1]. The second is at most Kd_0. Risks themselves lie in [0,1], giving the cap. If widehat a and a^* minimize the approximate and exact risks, respectively, then R(widehat a)≤widehat R(widehat a)+η≤widehat R(a^*)+η≤ R(a^*)+2η.
Exact mathematical passage
```
a^*
```

+Use in source · 4 passages

Search aliases: +

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Use in source · connected_synthesis.tex, line 169
This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution studies hidden-return defects and exact memory alternatives. Large Signal Model Foundations places those mechanisms inside its refinement and predict…
Exact mathematical passage
```
\begin{pmatrix}z_{t+1}\\u_{t+1}\end{pmatrix}
=
\begin{pmatrix}A&B\\C&D\end{pmatrix}
\begin{pmatrix}z_t\\u_t\end{pmatrix}.
```

=Use in source · 33 passages

Search aliases: =

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Use in source · connected_synthesis.tex, line 116
A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.
Exact mathematical passage
```
\mathfrak C=(\Omega,\Pi,\mathcal Q,\mathcal A,\mathcal E,\mathcal B,\mathcal V).
```

AUse in source · 6 passages

Search aliases: A

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Use in source · connected_synthesis.tex, line 166
For a linear map C:X→ Z and a known operation A:X→ X, an induced operation B satisfying CA=BC exists precisely when ker C is invariant under A, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by C remain collapsed after the operation. If it fails, the discarded distinction can return through the retained state.
Exact mathematical passage
```
A:X\to X
```

\mathcal AUse in source · 2 passages

Search aliases: A mathcal

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Use in source · connected_synthesis.tex, line 116
A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.
Exact mathematical passage
```
\mathfrak C=(\Omega,\Pi,\mathcal Q,\mathcal A,\mathcal E,\mathcal B,\mathcal V).
```

aUse in source · 14 passages

Search aliases: a

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Use in source · connected_synthesis.tex, line 139
Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machin…
Exact mathematical passage
```
F_a
```

a^*Use in source · 2 passages

Search aliases: a^*

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Use in source · connected_synthesis.tex, line 247
ng widehat R has regret at most 2η relative to a minimizer of R when the action set is finite and nonempty. Insert the intermediate expectation ∑widehatμ_ωℓ(a,y_ω). The first difference is at most e because the integrand is in [0,1]. The second is at most Kd_0. Risks themselves lie in [0,1], giving the cap. If widehat a and a^* minimize the approximate and exact risks, respectively, then R(widehat a)≤widehat R(widehat a)+η≤widehat R(a^*)+η≤ R(a^*)+2η.
Exact mathematical passage
```
a^*
```

A_0Use in source · 1 passage

Search aliases: A_0

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Use in source · connected_synthesis.tex, line 179
One common quotient across uncertain dynamics. Consider two possible linear realizations on R^3, A_0=1 · 1 · 0 0 · 1 · 0 0 · 0 · 2, A_1=1 · 1 · 0 0 · 1 · 0 1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1
Exact mathematical passage
```
A_0=\begin{pmatrix}1&1&0\\0&1&0\\0&0&2\end{pmatrix},\qquad
A_1=\begin{pmatrix}1&1&0\\0&1&0\\1&0&3\end{pmatrix}.
```

A_1Use in source · 1 passage

Search aliases: A_1

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Use in source · connected_synthesis.tex, line 179
One common quotient across uncertain dynamics. Consider two possible linear realizations on R^3, A_0=1 · 1 · 0 0 · 1 · 0 0 · 0 · 2, A_1=1 · 1 · 0 0 · 1 · 0 1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1
Exact mathematical passage
```
A_0=\begin{pmatrix}1&1&0\\0&1&0\\0&0&2\end{pmatrix},\qquad
A_1=\begin{pmatrix}1&1&0\\0&1&0\\1&0&3\end{pmatrix}.
```

A_iUse in source · 1 passage

Search aliases: A_i

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Use in source · connected_synthesis.tex, line 184
1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1 0 · 1. Consequently the output after k operations is x_1+kx_2 for either fixed model. In this particular example it also holds for arbitrary switching, since the same local intertwining equation holds for both operations. The uncertain third-coordinate dynamics remain irrelevant to the protected output.
Exact mathematical passage
```
pA_i=Gp,\qquad G=\begin{pmatrix}1&1\\0&1\end{pmatrix}.
```

BUse in source · 4 passages

Search aliases: B

Link to this entry ·

Use in source · connected_synthesis.tex, line 166
For a linear map C:X→ Z and a known operation A:X→ X, an induced operation B satisfying CA=BC exists precisely when ker C is invariant under A, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by C remain collapsed after the operation. If it fails, the discarded distinction can return through the retained state.
Exact mathematical passage
```
B
```

\mathcal BUse in source · 2 passages

Search aliases: B mathcal

Link to this entry ·

Use in source · connected_synthesis.tex, line 116
A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.
Exact mathematical passage
```
\mathfrak C=(\Omega,\Pi,\mathcal Q,\mathcal A,\mathcal E,\mathcal B,\mathcal V).
```

bUse in source · 16 passages

Search aliases: b

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Use in source · connected_synthesis.tex, line 322
Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i succeeds precisely when b=i.
Exact mathematical passage
```
\omega=(b,\theta,u)\in\F_2^3.
```

\bar qUse in source · 3 passages

Search aliases: bar q bar

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Use in source · connected_synthesis.tex, line 201
[Two local bridges compose over finite continuations] Assume the domains used by both equations are preserved under all admitted operations and that a protected output has the form q=bar q Tr. Then, for every admitted finite word w, TrF_w=H_wTr, qF_w=bar q H_wTr. For one operation, TrF_a=TG_ar=H_aTr. The empty word gives the identity. Applying the one-operation relation successively proves the first equation by induction, and composing with bar q proves the second. Domain invariance ensures that…
Exact mathematical passage
```
q=\bar q\,Tr
```

CUse in source · 5 passages

Search aliases: C

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Use in source · connected_synthesis.tex, line 166
For a linear map C:X→ Z and a known operation A:X→ X, an induced operation B satisfying CA=BC exists precisely when ker C is invariant under A, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by C remain collapsed after the operation. If it fails, the discarded distinction can return through the retained state.
Exact mathematical passage
```
C:X\to Z
```

\mathfrak CUse in source · 1 passage

Search aliases: C mathfrak

Link to this entry ·

Use in source · connected_synthesis.tex, line 116
A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.
Exact mathematical passage
```
\mathfrak C=(\Omega,\Pi,\mathcal Q,\mathcal A,\mathcal E,\mathcal B,\mathcal V).
```

DUse in source · 1 passage

Search aliases: D

Link to this entry ·

Use in source · connected_synthesis.tex, line 169
This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution studies hidden-return defects and exact memory alternatives. Large Signal Model Foundations places those mechanisms inside its refinement and predict…
Exact mathematical passage
```
\begin{pmatrix}z_{t+1}\\u_{t+1}\end{pmatrix}
=
\begin{pmatrix}A&B\\C&D\end{pmatrix}
\begin{pmatrix}z_t\\u_t\end{pmatrix}.
```

dUse in source · 5 passages

Search aliases: d

Link to this entry ·

Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
d(y_\omega,\widehat y_\omega)\leq d_0
```

d_0Use in source · 3 passages

Search aliases: d_0

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Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
d(y_\omega,\widehat y_\omega)\leq d_0
```

EUse in source · 2 passages

Search aliases: E

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Use in source · connected_synthesis.tex, line 219
For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have been contradicted is part of a correct update, not an exceptional case to omit from the theorem. srcGA
Exact mathematical passage
```
V\mapsto V\cap E
```

\mathcal EUse in source · 3 passages

Search aliases: E mathcal

Link to this entry ·

Use in source · connected_synthesis.tex, line 116
A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.
Exact mathematical passage
```
\mathfrak C=(\Omega,\Pi,\mathcal Q,\mathcal A,\mathcal E,\mathcal B,\mathcal V).
```

eUse in source · 3 passages

Search aliases: e

Link to this entry ·

Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
\TV(\mu,\widehat\mu)\leq e
```

FUse in source · 7 passages

Search aliases: F

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Use in source · connected_synthesis.tex, line 139
Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machin…
Exact mathematical passage
```
F_a
```

\FUse in source · 1 passage

Search aliases: F mathbb

Link to this entry ·

Use in source · connected_synthesis.tex, line 322
Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i succeeds precisely when b=i.
Exact mathematical passage
```
\omega=(b,\theta,u)\in\F_2^3.
```

\F_2^3Use in source · 1 passage

Search aliases: F_2^3 F mathbb

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Use in source · connected_synthesis.tex, line 322
Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i succeeds precisely when b=i.
Exact mathematical passage
```
\omega=(b,\theta,u)\in\F_2^3.
```

F_aUse in source · 4 passages

Search aliases: F_a

Link to this entry ·

Use in source · connected_synthesis.tex, line 139
Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machin…
Exact mathematical passage
```
F_a
```

F_wUse in source · 3 passages

Search aliases: F_w

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Use in source · connected_synthesis.tex, line 140
Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machin…
Exact mathematical passage
```
x\sim y\quad\Longleftrightarrow\quad
q(F_wx)=q(F_wy)\quad\text{for every admitted finite word }w.
```

GUse in source · 5 passages

Search aliases: G

Link to this entry ·

Use in source · connected_synthesis.tex, line 184
1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1 0 · 1. Consequently the output after k operations is x_1+kx_2 for either fixed model. In this particular example it also holds for arbitrary switching, since the same local intertwining equation holds for both operations. The uncertain third-coordinate dynamics remain irrelevant to the protected output.
Exact mathematical passage
```
pA_i=Gp,\qquad G=\begin{pmatrix}1&1\\0&1\end{pmatrix}.
```

G_aUse in source · 3 passages

Search aliases: G_a

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Use in source · connected_synthesis.tex, line 194
For a common invariant domain, let r:X→ Z encode the relevant state, let G_a act on Z, and let T:Z→ W translate that state into another representation with operations H_a. The desired equations are rF_a=G_ar, TG_a=H_aT. The first connects the source process to its representation. The second connects that representation to the recipient interface.
Exact mathematical passage
```
G_a
```

HUse in source · 4 passages

Search aliases: H

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Use in source · connected_synthesis.tex, line 194
For a common invariant domain, let r:X→ Z encode the relevant state, let G_a act on Z, and let T:Z→ W translate that state into another representation with operations H_a. The desired equations are rF_a=G_ar, TG_a=H_aT. The first connects the source process to its representation. The second connects that representation to the recipient interface.
Exact mathematical passage
```
H_a
```

hUse in source · 1 passage

Search aliases: h

Link to this entry ·

Use in source · connected_synthesis.tex, line 121
A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer is not automatically the complete state of the others.
Exact mathematical passage
```
h_t
```

H_aUse in source · 3 passages

Search aliases: H_a

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Use in source · connected_synthesis.tex, line 194
For a common invariant domain, let r:X→ Z encode the relevant state, let G_a act on Z, and let T:Z→ W translate that state into another representation with operations H_a. The desired equations are rF_a=G_ar, TG_a=H_aT. The first connects the source process to its representation. The second connects that representation to the recipient interface.
Exact mathematical passage
```
H_a
```

h_tUse in source · 1 passage

Search aliases: h_t

Link to this entry ·

Use in source · connected_synthesis.tex, line 121
A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer is not automatically the complete state of the others.
Exact mathematical passage
```
h_t
```

H_wUse in source · 1 passage

Search aliases: H_w

Link to this entry ·

Use in source · connected_synthesis.tex, line 202
[Two local bridges compose over finite continuations] Assume the domains used by both equations are preserved under all admitted operations and that a protected output has the form q=bar q Tr. Then, for every admitted finite word w, TrF_w=H_wTr, qF_w=bar q H_wTr. For one operation, TrF_a=TG_ar=H_aTr. The empty word gives the identity. Applying the one-operation relation successively proves the first equation by induction, and composing with bar q proves the second. Domain invariance ensures that…
Exact mathematical passage
```
TrF_w=H_wTr,\qquad qF_w=\bar q\,H_wTr.
```

iUse in source · 6 passages

Search aliases: i

Link to this entry ·

Use in source · connected_synthesis.tex, line 149
[Evidence must initialize the predictive class] Let p:X→ X/∼ be the behavioral quotient above. There exists an initializer i:r(X)→ X/∼ with i(r(x))=p(x) for every x if and only if r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because a function has one value on an equal input. For sufficiency, assign i(r(x))=p(x); the hypothesis makes the assignment independent of the representativ…
Exact mathematical passage
```
i:r(X)\to X/{\sim}
```

KUse in source · 3 passages

Search aliases: K

Link to this entry ·

Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
K
```

kUse in source · 2 passages

Search aliases: k

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Use in source · connected_synthesis.tex, line 187
0 · 1. Consequently the output after k operations is x_1+kx_2 for either fixed model. In this particular example it also holds for arbitrary switching, since the same local intertwining equation holds for both operations. The uncertain third-coordinate dynamics remain irrelevant to the protected output.
Exact mathematical passage
```
k
```

\keropUse in source · 1 passage

Search aliases: kerop ker operatorname

Link to this entry ·

Use in source · connected_synthesis.tex, line 166
For a linear map C:X→ Z and a known operation A:X→ X, an induced operation B satisfying CA=BC exists precisely when ker C is invariant under A, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by C remain collapsed after the operation. If it fails, the discarded distinction can return through the retained state.
Exact mathematical passage
```
\kerop C
```

\LongleftrightarrowUse in source · 1 passage

Search aliases: Longleftrightarrow

Link to this entry ·

Use in source · connected_synthesis.tex, line 140
Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machin…
Exact mathematical passage
```
x\sim y\quad\Longleftrightarrow\quad
q(F_wx)=q(F_wy)\quad\text{for every admitted finite word }w.
```

\LongrightarrowUse in source · 1 passage

Search aliases: Longrightarrow

Link to this entry ·

Use in source · connected_synthesis.tex, line 150
[Evidence must initialize the predictive class] Let p:X→ X/∼ be the behavioral quotient above. There exists an initializer i:r(X)→ X/∼ with i(r(x))=p(x) for every x if and only if r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because a function has one value on an equal input. For sufficiency, assign i(r(x))=p(x); the hypothesis makes the assignment independent of the representativ…
Exact mathematical passage
```
r(x)=r(y)\quad\Longrightarrow\quad x\sim y.
```

\minUse in source · 1 passage

Search aliases: min

Link to this entry ·

Use in source · connected_synthesis.tex, line 241
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
|R(a)-\widehat R(a)|\leq\eta:=\min\{1,e+Kd_0\},
```

\overline FUse in source · 2 passages

Search aliases: overline F overline

Link to this entry ·

Use in source · connected_synthesis.tex, line 156
r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because a function has one value on an equal input. For sufficiency, assign i(r(x))=p(x); the hypothesis makes the assignment independent of the representative x. The quotient update overline F_a(p(x))=p(F_ax) is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives pF_w=overline…
Exact mathematical passage
```
\overline F_a(p(x))=p(F_ax)
```

\overline F_aUse in source · 1 passage

Search aliases: overline F_a overline

Link to this entry ·

Use in source · connected_synthesis.tex, line 156
r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because a function has one value on an equal input. For sufficiency, assign i(r(x))=p(x); the hypothesis makes the assignment independent of the representative x. The quotient update overline F_a(p(x))=p(F_ax) is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives pF_w=overline…
Exact mathematical passage
```
\overline F_a(p(x))=p(F_ax)
```

\overline F_wUse in source · 1 passage

Search aliases: overline F_w overline

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Use in source · connected_synthesis.tex, line 156
induced updates. Necessity follows because a function has one value on an equal input. For sufficiency, assign i(r(x))=p(x); the hypothesis makes the assignment independent of the representative x. The quotient update overline F_a(p(x))=p(F_ax) is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives pF_w=overline F_wp, and the output factors through p. Thus the initialized quotient returns the protected output after every word.
Exact mathematical passage
```
pF_w=\overline F_wp
```

pUse in source · 8 passages

Search aliases: p

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Use in source · connected_synthesis.tex, line 149
[Evidence must initialize the predictive class] Let p:X→ X/∼ be the behavioral quotient above. There exists an initializer i:r(X)→ X/∼ with i(r(x))=p(x) for every x if and only if r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because a function has one value on an equal input. For sufficiency, assign i(r(x))=p(x); the hypothesis makes the assignment independent of the representativ…
Exact mathematical passage
```
p:X\to X/{\sim}
```

\mathcal QUse in source · 2 passages

Search aliases: Q mathcal

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Use in source · connected_synthesis.tex, line 116
A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.
Exact mathematical passage
```
\mathfrak C=(\Omega,\Pi,\mathcal Q,\mathcal A,\mathcal E,\mathcal B,\mathcal V).
```

qUse in source · 5 passages

Search aliases: q

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Use in source · connected_synthesis.tex, line 139
Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machin…
Exact mathematical passage
```
q
```

RUse in source · 9 passages

Search aliases: R

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Use in source · connected_synthesis.tex, line 241
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
|R(a)-\widehat R(a)|\leq\eta:=\min\{1,e+Kd_0\},
```

\mathbb RUse in source · 3 passages

Search aliases: R mathbb

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Use in source · connected_synthesis.tex, line 178
One common quotient across uncertain dynamics. Consider two possible linear realizations on R^3, A_0=1 · 1 · 0
Exact mathematical passage
```
\mathbb R^3
```

rUse in source · 12 passages

Search aliases: r

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Use in source · connected_synthesis.tex, line 146
The missing link is initialization. Let r(x) be the observer's actual initial record. A quotient may exist mathematically while a decoder from r(x) does not.
Exact mathematical passage
```
r(x)
```

\mathbb R^3Use in source · 1 passage

Search aliases: R^3 mathbb

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Use in source · connected_synthesis.tex, line 178
One common quotient across uncertain dynamics. Consider two possible linear realizations on R^3, A_0=1 · 1 · 0
Exact mathematical passage
```
\mathbb R^3
```

\mathbb R^dUse in source · 1 passage

Search aliases: R^d mathbb

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Use in source · connected_synthesis.tex, line 304
[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at one of its listed points. Nonnegative coordinates give a nonnegative scalar product. A negative coordinate is exposed by assigning positive price onl…
Exact mathematical passage
```
\Delta\in\mathbb R^d
```

\mathbb R^d_{\geq0}Use in source · 1 passage

Search aliases: R^d_≥0 mathbb geq

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Use in source · connected_synthesis.tex, line 304
[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at one of its listed points. Nonnegative coordinates give a nonnegative scalar product. A negative coordinate is exposed by assigning positive price onl…
Exact mathematical passage
```
w\in\mathbb R^d_{\geq0}
```

R_*Use in source · 3 passages

Search aliases: R_*

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Use in source · connected_synthesis.tex, line 382
Revision can change the proof obligation. Suppose a newly admitted repair R_* succeeds for either value of b. Then the task no longer requires identifying b before acting. The old statement that both information channels are necessary applied to the previous repair language. It does not apply after R_* is introduced.
Exact mathematical passage
```
R_*
```

R_bUse in source · 1 passage

Search aliases: R_b

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Use in source · connected_synthesis.tex, line 335
Second: the repair question changes the required distinction. When the task becomes selecting R_b, the old display is insufficient. Worlds (0,0,u) and (1,1,u) have the same display and different successful repairs. Repeating the normal reading changes no information. Calibration supplies the missing convention, after which b=s⊕θ. The refinement is a genuine information acquisition. Merely adding a field named θ to the retained record would not initialize it.
Exact mathematical passage
```
R_b
```

R_iUse in source · 1 passage

Search aliases: R_i

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Use in source · connected_synthesis.tex, line 325
the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i succeeds precisely when b=i.
Exact mathematical passage
```
R_i
```

sUse in source · 6 passages

Search aliases: s

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Use in source · connected_synthesis.tex, line 325
Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i succeeds precisely when b=i.
Exact mathematical passage
```
s=b\oplus\theta
```

TUse in source · 8 passages

Search aliases: T

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Use in source · connected_synthesis.tex, line 194
For a common invariant domain, let r:X→ Z encode the relevant state, let G_a act on Z, and let T:Z→ W translate that state into another representation with operations H_a. The desired equations are rF_a=G_ar, TG_a=H_aT. The first connects the source process to its representation. The second connects that representation to the recipient interface.
Exact mathematical passage
```
T:Z\to W
```

tUse in source · 8 passages

Search aliases: t

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Use in source · connected_synthesis.tex, line 121
A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer is not automatically the complete state of the others.
Exact mathematical passage
```
x_t
```

\topUse in source · 1 passage

Search aliases: top

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Use in source · connected_synthesis.tex, line 304
[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at one of its listed points. Nonnegative coordinates give a nonnegative scalar product. A negative coordinate is exposed by assigning positive price onl…
Exact mathematical passage
```
w^\top\Delta\geq0
```

\TVUse in source · 1 passage

Search aliases: TV d_TV mathrm

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Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
\TV(\mu,\widehat\mu)\leq e
```

uUse in source · 9 passages

Search aliases: u

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Use in source · connected_synthesis.tex, line 169
This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution studies hidden-return defects and exact memory alternatives. Large Signal Model Foundations places those mechanisms inside its refinement and predict…
Exact mathematical passage
```
\begin{pmatrix}z_{t+1}\\u_{t+1}\end{pmatrix}
=
\begin{pmatrix}A&B\\C&D\end{pmatrix}
\begin{pmatrix}z_t\\u_t\end{pmatrix}.
```

u_tUse in source · 3 passages

Search aliases: u_t

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Use in source · connected_synthesis.tex, line 169
This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution studies hidden-return defects and exact memory alternatives. Large Signal Model Foundations places those mechanisms inside its refinement and predict…
Exact mathematical passage
```
\begin{pmatrix}z_{t+1}\\u_{t+1}\end{pmatrix}
=
\begin{pmatrix}A&B\\C&D\end{pmatrix}
\begin{pmatrix}z_t\\u_t\end{pmatrix}.
```

u_{t+1}Use in source · 1 passage

Search aliases: u_t+1

Link to this entry ·

Use in source · connected_synthesis.tex, line 169
This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution studies hidden-return defects and exact memory alternatives. Large Signal Model Foundations places those mechanisms inside its refinement and predict…
Exact mathematical passage
```
\begin{pmatrix}z_{t+1}\\u_{t+1}\end{pmatrix}
=
\begin{pmatrix}A&B\\C&D\end{pmatrix}
\begin{pmatrix}z_t\\u_t\end{pmatrix}.
```

VUse in source · 4 passages

Search aliases: V

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Use in source · connected_synthesis.tex, line 219
For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have been contradicted is part of a correct update, not an exceptional case to omit from the theorem. srcGA
Exact mathematical passage
```
V\subseteq\Omega
```

\mathcal VUse in source · 2 passages

Search aliases: V mathcal

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Use in source · connected_synthesis.tex, line 116
A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.
Exact mathematical passage
```
\mathfrak C=(\Omega,\Pi,\mathcal Q,\mathcal A,\mathcal E,\mathcal B,\mathcal V).
```

WUse in source · 1 passage

Search aliases: W

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Use in source · connected_synthesis.tex, line 194
For a common invariant domain, let r:X→ Z encode the relevant state, let G_a act on Z, and let T:Z→ W translate that state into another representation with operations H_a. The desired equations are rF_a=G_ar, TG_a=H_aT. The first connects the source process to its representation. The second connects that representation to the recipient interface.
Exact mathematical passage
```
T:Z\to W
```

wUse in source · 6 passages

Search aliases: w

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Use in source · connected_synthesis.tex, line 140
Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machin…
Exact mathematical passage
```
x\sim y\quad\Longleftrightarrow\quad
q(F_wx)=q(F_wy)\quad\text{for every admitted finite word }w.
```

w^\topUse in source · 1 passage

Search aliases: w^top top

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Use in source · connected_synthesis.tex, line 304
[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at one of its listed points. Nonnegative coordinates give a nonnegative scalar product. A negative coordinate is exposed by assigning positive price onl…
Exact mathematical passage
```
w^\top\Delta\geq0
```

\widehat aUse in source · 2 passages

Search aliases: widehat a widehat

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Use in source · connected_synthesis.tex, line 247
n action minimizing widehat R has regret at most 2η relative to a minimizer of R when the action set is finite and nonempty. Insert the intermediate expectation ∑widehatμ_ωℓ(a,y_ω). The first difference is at most e because the integrand is in [0,1]. The second is at most Kd_0. Risks themselves lie in [0,1], giving the cap. If widehat a and a^* minimize the approximate and exact risks, respectively, then R(widehat a)≤widehat R(widehat a)+η≤widehat R(a^*)+η≤ R(a^*)+2η.
Exact mathematical passage
```
\widehat a
```

\widehat RUse in source · 4 passages

Search aliases: widehat R widehat

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Use in source · connected_synthesis.tex, line 241
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
|R(a)-\widehat R(a)|\leq\eta:=\min\{1,e+Kd_0\},
```

\widehat yUse in source · 2 passages

Search aliases: widehat y widehat

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Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
d(y_\omega,\widehat y_\omega)\leq d_0
```

\widehat y_\omegaUse in source · 2 passages

Search aliases: widehat y_ω widehat omega

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Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
d(y_\omega,\widehat y_\omega)\leq d_0
```

\widehat\muUse in source · 4 passages

Search aliases: widehatμ widehat mu

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Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
\widehat\mu
```

\widehat\mu_\omegaUse in source · 2 passages

Search aliases: widehatμ_ω widehat mu omega

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Use in source · connected_synthesis.tex, line 244
res on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a minimizer of R when the action set is finite and nonempty. Insert t…
Exact mathematical passage
```
\widehat R(a)=\sum_\omega\widehat\mu_\omega\ell(a,\widehat y_\omega)
```

XUse in source · 6 passages

Search aliases: X

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Use in source · connected_synthesis.tex, line 139
Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machin…
Exact mathematical passage
```
X
```

xUse in source · 13 passages

Search aliases: x

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Use in source · connected_synthesis.tex, line 121
A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer is not automatically the complete state of the others.
Exact mathematical passage
```
x_t
```

x_1Use in source · 3 passages

Search aliases: x_1

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Use in source · connected_synthesis.tex, line 183
1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1
Exact mathematical passage
```
p(x_1,x_2,x_3)=(x_1,x_2)
```

x_2Use in source · 2 passages

Search aliases: x_2

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Use in source · connected_synthesis.tex, line 183
1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1
Exact mathematical passage
```
p(x_1,x_2,x_3)=(x_1,x_2)
```

x_3Use in source · 1 passage

Search aliases: x_3

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Use in source · connected_synthesis.tex, line 183
1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1
Exact mathematical passage
```
p(x_1,x_2,x_3)=(x_1,x_2)
```

x_tUse in source · 1 passage

Search aliases: x_t

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Use in source · connected_synthesis.tex, line 121
A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer is not automatically the complete state of the others.
Exact mathematical passage
```
x_t
```

yUse in source · 6 passages

Search aliases: y

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Use in source · connected_synthesis.tex, line 140
Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machin…
Exact mathematical passage
```
x\sim y\quad\Longleftrightarrow\quad
q(F_wx)=q(F_wy)\quad\text{for every admitted finite word }w.
```

y_\omegaUse in source · 3 passages

Search aliases: y_ω omega

Link to this entry ·

Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
d(y_\omega,\widehat y_\omega)\leq d_0
```

ZUse in source · 4 passages

Search aliases: Z

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Use in source · connected_synthesis.tex, line 166
For a linear map C:X→ Z and a known operation A:X→ X, an induced operation B satisfying CA=BC exists precisely when ker C is invariant under A, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by C remain collapsed after the operation. If it fails, the discarded distinction can return through the retained state.
Exact mathematical passage
```
C:X\to Z
```

zUse in source · 4 passages

Search aliases: z

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Use in source · connected_synthesis.tex, line 121
A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer is not automatically the complete state of the others.
Exact mathematical passage
```
z_t
```

z_tUse in source · 2 passages

Search aliases: z_t

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Use in source · connected_synthesis.tex, line 121
A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer is not automatically the complete state of the others.
Exact mathematical passage
```
z_t
```

z_{t+1}Use in source · 1 passage

Search aliases: z_t+1

Link to this entry ·

Use in source · connected_synthesis.tex, line 169
This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution studies hidden-return defects and exact memory alternatives. Large Signal Model Foundations places those mechanisms inside its refinement and predict…
Exact mathematical passage
```
\begin{pmatrix}z_{t+1}\\u_{t+1}\end{pmatrix}
=
\begin{pmatrix}A&B\\C&D\end{pmatrix}
\begin{pmatrix}z_t\\u_t\end{pmatrix}.
```

|Use in source · 1 passage

Search aliases: |

Link to this entry ·

Use in source · connected_synthesis.tex, line 241
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
|R(a)-\widehat R(a)|\leq\eta:=\min\{1,e+Kd_0\},
```

\DeltaUse in source · 3 passages

Search aliases: Δ Delta

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Use in source · connected_synthesis.tex, line 304
[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at one of its listed points. Nonnegative coordinates give a nonnegative scalar product. A negative coordinate is exposed by assigning positive price onl…
Exact mathematical passage
```
\Delta\in\mathbb R^d
```

\etaUse in source · 3 passages

Search aliases: η eta

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Use in source · connected_synthesis.tex, line 241
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
|R(a)-\widehat R(a)|\leq\eta:=\min\{1,e+Kd_0\},
```

\thetaUse in source · 13 passages

Search aliases: θ theta

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Use in source · connected_synthesis.tex, line 322
Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i succeeds precisely when b=i.
Exact mathematical passage
```
\omega=(b,\theta,u)\in\F_2^3.
```

\muUse in source · 3 passages

Search aliases: μ mu

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Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
\mu
```

\mu_\omegaUse in source · 1 passage

Search aliases: μ_ω mu omega

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Use in source · connected_synthesis.tex, line 244
Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a minimizer of R when the acti…
Exact mathematical passage
```
R(a)=\sum_\omega\mu_\omega\ell(a,y_\omega)
```

\PiUse in source · 2 passages

Search aliases: Π Pi

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Use in source · connected_synthesis.tex, line 116
A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.
Exact mathematical passage
```
\mathfrak C=(\Omega,\Pi,\mathcal Q,\mathcal A,\mathcal E,\mathcal B,\mathcal V).
```

\SigmaUse in source · 1 passage

Search aliases: Σ Sigma

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Use in source · connected_synthesis.tex, line 220
For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have been contradicted is part of a correct update, not an exceptional case to omit from the theorem. srcGA
Exact mathematical passage
```
\Sigma(V)=\bigl(t(V\cap E)\bigr)_{E\in\mathcal E}.
```

\OmegaUse in source · 3 passages

Search aliases: Ω Omega

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Use in source · connected_synthesis.tex, line 116
A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.
Exact mathematical passage
```
\mathfrak C=(\Omega,\Pi,\mathcal Q,\mathcal A,\mathcal E,\mathcal B,\mathcal V).
```

\omegaUse in source · 5 passages

Search aliases: ω omega

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Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
d(y_\omega,\widehat y_\omega)\leq d_0
```

\ellUse in source · 4 passages

Search aliases: ℓ ell

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Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
\ell(a,y)\in[0,1]
```

\toUse in source · 6 passages

Search aliases: → to

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Use in source · connected_synthesis.tex, line 149
[Evidence must initialize the predictive class] Let p:X→ X/∼ be the behavioral quotient above. There exists an initializer i:r(X)→ X/∼ with i(r(x))=p(x) for every x if and only if r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because a function has one value on an equal input. For sufficiency, assign i(r(x))=p(x); the hypothesis makes the assignment independent of the representativ…
Exact mathematical passage
```
p:X\to X/{\sim}
```

\mapstoUse in source · 1 passage

Search aliases: ↦ mapsto

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Use in source · connected_synthesis.tex, line 219
For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have been contradicted is part of a correct update, not an exceptional case to omit from the theorem. srcGA
Exact mathematical passage
```
V\mapsto V\cap E
```

\inUse in source · 5 passages

Search aliases: ∈ in

Link to this entry ·

Use in source · connected_synthesis.tex, line 220
For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have been contradicted is part of a correct update, not an exceptional case to omit from the theorem. srcGA
Exact mathematical passage
```
\Sigma(V)=\bigl(t(V\cap E)\bigr)_{E\in\mathcal E}.
```

\sumUse in source · 3 passages

Search aliases: ∑ sum

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Use in source · connected_synthesis.tex, line 244
Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a minimizer of R when the acti…
Exact mathematical passage
```
R(a)=\sum_\omega\mu_\omega\ell(a,y_\omega)
```

\sum_\omegaUse in source · 2 passages

Search aliases: ∑_ω sum omega

Link to this entry ·

Use in source · connected_synthesis.tex, line 244
Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a minimizer of R when the acti…
Exact mathematical passage
```
R(a)=\sum_\omega\mu_\omega\ell(a,y_\omega)
```

\capUse in source · 2 passages

Search aliases: ∩ cap

Link to this entry ·

Use in source · connected_synthesis.tex, line 219
For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have been contradicted is part of a correct update, not an exceptional case to omit from the theorem. srcGA
Exact mathematical passage
```
V\mapsto V\cap E
```

\simUse in source · 4 passages

Search aliases: ∼ sim

Link to this entry ·

Use in source · connected_synthesis.tex, line 140
Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machin…
Exact mathematical passage
```
x\sim y\quad\Longleftrightarrow\quad
q(F_wx)=q(F_wy)\quad\text{for every admitted finite word }w.
```

\leqUse in source · 4 passages

Search aliases: ≤ leq

Link to this entry ·

Use in source · connected_synthesis.tex, line 240
[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(a,widehat y_ω). An action minimizing widehat R has regret at most 2η relative to a mi…
Exact mathematical passage
```
\TV(\mu,\widehat\mu)\leq e
```

\geqUse in source · 2 passages

Search aliases: ≥ geq

Link to this entry ·

Use in source · connected_synthesis.tex, line 304
[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at one of its listed points. Nonnegative coordinates give a nonnegative scalar product. A negative coordinate is exposed by assigning positive price onl…
Exact mathematical passage
```
w^\top\Delta\geq0
```

\subseteqUse in source · 1 passage

Search aliases: ⊆ subseteq

Link to this entry ·

Use in source · connected_synthesis.tex, line 219
For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have been contradicted is part of a correct update, not an exceptional case to omit from the theorem. srcGA
Exact mathematical passage
```
V\subseteq\Omega
```

\oplusUse in source · 3 passages

Search aliases: ⊕ oplus

Link to this entry ·

Use in source · connected_synthesis.tex, line 325
Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i succeeds precisely when b=i.
Exact mathematical passage
```
s=b\oplus\theta
```

\longrightarrowUse in source · 1 passage

Search aliases: ⟶ longrightarrow

Link to this entry ·

Use in source · connected_synthesis.tex, line 298
The combined lesson is neither that every representation change speeds computation nor that compression is futile. It is that the complete path must be accounted for: acquire ⟶ represent ⟶ select ⟶ verify ⟶ execute ⟶ maintain. The arrows can have different units. Model observations, interpreter visits, physical time, peak memory, and stored records should not be summed without an explicit conversion model.
Exact mathematical passage
```
\text{acquire}\ \longrightarrow\ \text{represent}\ \longrightarrow\ \text{select}\ \longrightarrow\ \text{verify}\ \longrightarrow\ \text{execute}\ \longrightarrow\ \text{maintain}.
```

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