Notation by work
Grounded Abstraction
All notation · Glossary · Source coverage
371 symbol spellings in 820 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 371 entries are available below.
aNotation table · 94 passages
Search aliases: a
Notation table · Grounded_Abstraction_v1.0.0.tex, line 81
F_a, q · A state transition and the task observation it must preserve.
Exact mathematical passage
F_a
\CNotation table · 7 passages
Search aliases: C mathcal
Notation table · Grounded_Abstraction_v1.0.0.tex, line 82
x∼_ Cy · Equality of every task observation after every permitted finite word.
Exact mathematical passage
x\sim_{\C}y
ENotation table · 19 passages
Search aliases: E
Notation table · Grounded_Abstraction_v1.0.0.tex, line 84
E∈ E · An admitted evidence event used to filter V.
Exact mathematical passage
E\in\E
\ENotation table · 8 passages
Search aliases: E mathcal
Notation table · Grounded_Abstraction_v1.0.0.tex, line 84
E∈ E · An admitted evidence event used to filter V.
Exact mathematical passage
E\in\E
E_\inftyNotation table · 8 passages
Search aliases: E_∞ infty
Notation table · Grounded_Abstraction_v1.0.0.tex, line 89
E_∞ · Propagated span of all model-discrepancy directions.
Exact mathematical passage
E_\infty
FNotation table · 8 passages
Search aliases: F
Notation table · Grounded_Abstraction_v1.0.0.tex, line 81
F_a, q · A state transition and the task observation it must preserve.
Exact mathematical passage
F_a
F_aNotation table · 4 passages
Search aliases: F_a
Notation table · Grounded_Abstraction_v1.0.0.tex, line 81
F_a, q · A state transition and the task observation it must preserve.
Exact mathematical passage
F_a
hNotation table · 26 passages
Search aliases: h
Notation table · Grounded_Abstraction_v1.0.0.tex, line 79
K^π(hmidω) · Probability of observable record h under one permitted protocol.
Exact mathematical passage
K^\pi(h\mid\omega)
INotation table · 12 passages
Search aliases: I
Notation table · Grounded_Abstraction_v1.0.0.tex, line 88
I_Q · Initial vectors with common output continuations across fixed models.
Exact mathematical passage
I_Q
iNotation table · 46 passages
Search aliases: i
Notation table · Grounded_Abstraction_v1.0.0.tex, line 90
p_i,y · Joint probability of a likelihood profile and a target value.
Exact mathematical passage
p_{i,y}
I_QNotation table · 8 passages
Search aliases: I_Q
Notation table · Grounded_Abstraction_v1.0.0.tex, line 88
I_Q · Initial vectors with common output continuations across fixed models.
Exact mathematical passage
I_Q
KNotation table · 10 passages
Search aliases: K
Notation table · Grounded_Abstraction_v1.0.0.tex, line 79
K^π(hmidω) · Probability of observable record h under one permitted protocol.
Exact mathematical passage
K^\pi(h\mid\omega)
K^\piNotation table · 3 passages
Search aliases: K^π pi
Notation table · Grounded_Abstraction_v1.0.0.tex, line 79
K^π(hmidω) · Probability of observable record h under one permitted protocol.
Exact mathematical passage
K^\pi(h\mid\omega)
\midNotation table · 6 passages
Search aliases: mid
Notation table · Grounded_Abstraction_v1.0.0.tex, line 79
K^π(hmidω) · Probability of observable record h under one permitted protocol.
Exact mathematical passage
K^\pi(h\mid\omega)
pNotation table · 29 passages
Search aliases: p
Notation table · Grounded_Abstraction_v1.0.0.tex, line 90
p_i,y · Joint probability of a likelihood profile and a target value.
Exact mathematical passage
p_{i,y}
p_{i,y}Notation table · 6 passages
Search aliases: p_i,y
Notation table · Grounded_Abstraction_v1.0.0.tex, line 90
p_i,y · Joint probability of a likelihood profile and a target value.
Exact mathematical passage
p_{i,y}
QNotation table · 55 passages
Search aliases: Q
Notation table · Grounded_Abstraction_v1.0.0.tex, line 88
I_Q · Initial vectors with common output continuations across fixed models.
Exact mathematical passage
I_Q
qNotation table · 17 passages
Search aliases: q
Notation table · Grounded_Abstraction_v1.0.0.tex, line 81
F_a, q · A state transition and the task observation it must preserve.
Exact mathematical passage
q
rNotation table · 21 passages
Search aliases: r
Notation table · Grounded_Abstraction_v1.0.0.tex, line 80
r:X→ Z · Retained evidence or an available initial code. Its access is explicit.
Exact mathematical passage
r:X\to Z
SNotation table · 19 passages
Search aliases: S
Notation table · Grounded_Abstraction_v1.0.0.tex, line 86
U, S_a · A calibrated input subspace and a known operator restriction.
Exact mathematical passage
S_a
S_aNotation table · 14 passages
Search aliases: S_a
Notation table · Grounded_Abstraction_v1.0.0.tex, line 86
U, S_a · A calibrated input subspace and a known operator restriction.
Exact mathematical passage
S_a
UNotation table · 40 passages
Search aliases: U
Notation table · Grounded_Abstraction_v1.0.0.tex, line 86
U, S_a · A calibrated input subspace and a known operator restriction.
Exact mathematical passage
U
U_\inftyNotation table · 6 passages
Search aliases: U_∞ infty
Notation table · Grounded_Abstraction_v1.0.0.tex, line 87
U_∞ · The maximal domain on which calibrated continuation is determined.
Exact mathematical passage
U_\infty
VNotation table · 53 passages
Search aliases: V
Notation table · Grounded_Abstraction_v1.0.0.tex, line 83
V⊆Ω · A set of candidate worlds, including the possible empty set.
Exact mathematical passage
V\subseteq\Omega
XNotation table · 10 passages
Search aliases: X
Notation table · Grounded_Abstraction_v1.0.0.tex, line 80
r:X→ Z · Retained evidence or an available initial code. Its access is explicit.
Exact mathematical passage
r:X\to Z
xNotation table · 27 passages
Search aliases: x
Notation table · Grounded_Abstraction_v1.0.0.tex, line 82
x∼_ Cy · Equality of every task observation after every permitted finite word.
Exact mathematical passage
x\sim_{\C}y
yNotation table · 27 passages
Search aliases: y
Notation table · Grounded_Abstraction_v1.0.0.tex, line 82
x∼_ Cy · Equality of every task observation after every permitted finite word.
Exact mathematical passage
x\sim_{\C}y
ZNotation table · 16 passages
Search aliases: Z
Notation table · Grounded_Abstraction_v1.0.0.tex, line 80
r:X→ Z · Retained evidence or an available initial code. Its access is explicit.
Exact mathematical passage
r:X\to Z
\piNotation table · 7 passages
Search aliases: π pi
Notation table · Grounded_Abstraction_v1.0.0.tex, line 79
K^π(hmidω) · Probability of observable record h under one permitted protocol.
Exact mathematical passage
K^\pi(h\mid\omega)
\SigmaNotation table · 5 passages
Search aliases: Σ Sigma
Notation table · Grounded_Abstraction_v1.0.0.tex, line 85
Σ(V) · The complete signature of future task-valued corrections.
Exact mathematical passage
\Sigma(V)
\OmegaNotation table · 24 passages
Search aliases: Ω Omega
Notation table · Grounded_Abstraction_v1.0.0.tex, line 78
midrule ω∈Ω · A candidate world, including any fixed model or historical tag.
Exact mathematical passage
\omega\in\Omega
\omegaNotation table · 34 passages
Search aliases: ω omega
Notation table · Grounded_Abstraction_v1.0.0.tex, line 78
midrule ω∈Ω · A candidate world, including any fixed model or historical tag.
Exact mathematical passage
\omega\in\Omega
\toNotation table · 26 passages
Search aliases: → to
Notation table · Grounded_Abstraction_v1.0.0.tex, line 80
r:X→ Z · Retained evidence or an available initial code. Its access is explicit.
Exact mathematical passage
r:X\to Z
\inNotation table · 47 passages
Search aliases: ∈ in
Notation table · Grounded_Abstraction_v1.0.0.tex, line 78
midrule ω∈Ω · A candidate world, including any fixed model or historical tag.
Exact mathematical passage
\omega\in\Omega
\inftyNotation table · 21 passages
Search aliases: ∞ infty
Notation table · Grounded_Abstraction_v1.0.0.tex, line 87
U_∞ · The maximal domain on which calibrated continuation is determined.
Exact mathematical passage
U_\infty
\simNotation table · 9 passages
Search aliases: ∼ sim
Notation table · Grounded_Abstraction_v1.0.0.tex, line 82
x∼_ Cy · Equality of every task observation after every permitted finite word.
Exact mathematical passage
x\sim_{\C}y
\sim_{\C}Notation table · 1 passage
Search aliases: ∼_C ∼_ C sim C sim mathcal
Notation table · Grounded_Abstraction_v1.0.0.tex, line 82
x∼_ Cy · Equality of every task observation after every permitted finite word.
Exact mathematical passage
x\sim_{\C}y
\subseteqNotation table · 24 passages
Search aliases: ⊆ subseteq
Notation table · Grounded_Abstraction_v1.0.0.tex, line 83
V⊆Ω · A set of candidate worlds, including the possible empty set.
Exact mathematical passage
V\subseteq\Omega
=Definition passage · 191 passages
Search aliases: =
Definition passage: Exact initialization from available evidence · Grounded_Abstraction_v1.0.0.tex, line 206
[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation names, but no additional world observations or hidden state.
Exact mathematical passage
\bar q\,G_w\,d\,r(x)=q(F_wx)
\bar qDefinition passage · 3 passages
Search aliases: bar q bar
Definition passage: Exact initialization from available evidence · Grounded_Abstraction_v1.0.0.tex, line 205
[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation names, but no additional world observations or hidden state.
Exact mathematical passage
\bar q:Z\to Y
dDefinition passage · 34 passages
Search aliases: d
Definition passage: Exact initialization from available evidence · Grounded_Abstraction_v1.0.0.tex, line 205
[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation names, but no additional world observations or hidden state.
Exact mathematical passage
d:H\to Z
F_wDefinition passage · 3 passages
Search aliases: F_w
Definition passage: Exact initialization from available evidence · Grounded_Abstraction_v1.0.0.tex, line 206
[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation names, but no additional world observations or hidden state.
Exact mathematical passage
\bar q\,G_w\,d\,r(x)=q(F_wx)
GDefinition passage · 18 passages
Search aliases: G
Definition passage: Exact initialization from available evidence · Grounded_Abstraction_v1.0.0.tex, line 205
[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation names, but no additional world observations or hidden state.
Exact mathematical passage
G_a:Z\to Z
G_aDefinition passage · 8 passages
Search aliases: G_a
Definition passage: Exact initialization from available evidence · Grounded_Abstraction_v1.0.0.tex, line 205
[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation names, but no additional world observations or hidden state.
Exact mathematical passage
G_a:Z\to Z
G_wDefinition passage · 3 passages
Search aliases: G_w
Definition passage: Exact initialization from available evidence · Grounded_Abstraction_v1.0.0.tex, line 206
[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation names, but no additional world observations or hidden state.
Exact mathematical passage
\bar q\,G_w\,d\,r(x)=q(F_wx)
HDefinition passage · 29 passages
Search aliases: H
Definition passage: Finite evidence contract · Grounded_Abstraction_v1.0.0.tex, line 176
Laws, records, and reusable predictive state. [Finite evidence contract] A finite evidence contract has a nonempty candidate set Ω, a family Π of permitted protocols, a finite record space H_π for each protocol, and kernels K^π(hmidω). A target t:Ω→ Y states what is to be identified. Any hidden model index that is fixed during a protocol is included in ω. The protocol's action choices are functions of its permitted records and independent randomization, not of the unobserved world.
Exact mathematical passage
H_\pi
H_\piDefinition passage · 2 passages
Search aliases: H_π pi
Definition passage: Finite evidence contract · Grounded_Abstraction_v1.0.0.tex, line 176
Laws, records, and reusable predictive state. [Finite evidence contract] A finite evidence contract has a nonempty candidate set Ω, a family Π of permitted protocols, a finite record space H_π for each protocol, and kernels K^π(hmidω). A target t:Ω→ Y states what is to be identified. Any hidden model index that is fixed during a protocol is included in ω. The protocol's action choices are functions of its permitted records and independent randomization, not of the unobserved world.
Exact mathematical passage
H_\pi
tDefinition passage · 47 passages
Search aliases: t
Definition passage: Finite evidence contract · Grounded_Abstraction_v1.0.0.tex, line 176
Laws, records, and reusable predictive state. [Finite evidence contract] A finite evidence contract has a nonempty candidate set Ω, a family Π of permitted protocols, a finite record space H_π for each protocol, and kernels K^π(hmidω). A target t:Ω→ Y states what is to be identified. Any hidden model index that is fixed during a protocol is included in ω. The protocol's action choices are functions of its permitted records and independent randomization, not of the unobserved world.
Exact mathematical passage
t:\Omega\to Y
wDefinition passage · 28 passages
Search aliases: w
Definition passage: Exact initialization from available evidence · Grounded_Abstraction_v1.0.0.tex, line 206
[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation names, but no additional world observations or hidden state.
Exact mathematical passage
\bar q\,G_w\,d\,r(x)=q(F_wx)
YDefinition passage · 10 passages
Search aliases: Y
Definition passage: Finite evidence contract · Grounded_Abstraction_v1.0.0.tex, line 176
Laws, records, and reusable predictive state. [Finite evidence contract] A finite evidence contract has a nonempty candidate set Ω, a family Π of permitted protocols, a finite record space H_π for each protocol, and kernels K^π(hmidω). A target t:Ω→ Y states what is to be identified. Any hidden model index that is fixed during a protocol is included in ω. The protocol's action choices are functions of its permitted records and independent randomization, not of the unobserved world.
Exact mathematical passage
t:\Omega\to Y
\PiDefinition passage · 4 passages
Search aliases: Π Pi
Definition passage: Finite evidence contract · Grounded_Abstraction_v1.0.0.tex, line 176
Laws, records, and reusable predictive state. [Finite evidence contract] A finite evidence contract has a nonempty candidate set Ω, a family Π of permitted protocols, a finite record space H_π for each protocol, and kernels K^π(hmidω). A target t:Ω→ Y states what is to be identified. Any hidden model index that is fixed during a protocol is included in ω. The protocol's action choices are functions of its permitted records and independent randomization, not of the unobserved world.
Exact mathematical passage
\Pi
\Sigma_{\E,t}Definition passage · 2 passages
Search aliases: Σ_E,t Σ_ E,t Sigma E Sigma mathcal
Definition passage: Correction signature · Grounded_Abstraction_v1.0.0.tex, line 312
[Correction signature] The correction signature of V is Σ_ E,t(V)=(Q(V∩ E))_E∈ E.tag3 Two candidate sets have the same signature when every admitted evidence filter leaves the same possible target values, including whether no candidate remains.
Exact mathematical passage
\Sigma_{\E,t}(V)=\bigl(Q(V\cap E)\bigr)_{E\in\E}.\tag{3}
\capDefinition passage · 11 passages
Search aliases: ∩ cap
Definition passage: Correction signature · Grounded_Abstraction_v1.0.0.tex, line 312
[Correction signature] The correction signature of V is Σ_ E,t(V)=(Q(V∩ E))_E∈ E.tag3 Two candidate sets have the same signature when every admitted evidence filter leaves the same possible target values, including whether no candidate remains.
Exact mathematical passage
\Sigma_{\E,t}(V)=\bigl(Q(V\cap E)\bigr)_{E\in\E}.\tag{3}
+Introduction passage · 43 passages
Search aliases: +
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 615
Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.
Exact mathematical passage
U_0=U,\qquad U_{j+1}=\{v\in U:S_av\in U_j\text{ for every }a\in\A\}.\tag{7}
AIntroduction passage · 50 passages
Search aliases: A
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
A_1,\ldots,A_m
\AIntroduction passage · 3 passages
Search aliases: A mathcal
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 615
Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.
Exact mathematical passage
U_0=U,\qquad U_{j+1}=\{v\in U:S_av\in U_j\text{ for every }a\in\A\}.\tag{7}
A^{n-1-j}Introduction passage · 1 passage
Search aliases: A^n-1-j
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 669
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
\|T_wv-S_wv\| \le\eta\|v\|\sum_{j=0}^{n-1}A^{n-1-j}B^j.\tag{9}
A_1Introduction passage · 2 passages
Search aliases: A_1
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
A_1,\ldots,A_m
a_1Introduction passage · 4 passages
Search aliases: a_1
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 676
plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives…
Exact mathematical passage
z_j=S_{a_j}\cdots S_{a_1}v\in L
A_iIntroduction passage · 7 passages
Search aliases: A_i
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 512
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
p_{i,y}(\mu)=\sum_{\omega\in A_i,\ t(\omega)=y}\mu(\omega), \qquad (i,y)\in D_t=\{(i,y):A_i\cap t^{-1}(y)\ne\varnothing\}.
a_jIntroduction passage · 2 passages
Search aliases: a_j
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 676
plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives…
Exact mathematical passage
z_j=S_{a_j}\cdots S_{a_1}v\in L
A_mIntroduction passage · 2 passages
Search aliases: A_m
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
A_1,\ldots,A_m
BIntroduction passage · 44 passages
Search aliases: B
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 669
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
\|T_wv-S_wv\| \le\eta\|v\|\sum_{j=0}^{n-1}A^{n-1-j}B^j.\tag{9}
B^jIntroduction passage · 2 passages
Search aliases: B^j
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 669
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
\|T_wv-S_wv\| \le\eta\|v\|\sum_{j=0}^{n-1}A^{n-1-j}B^j.\tag{9}
DIntroduction passage · 30 passages
Search aliases: D
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 512
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
p_{i,y}(\mu)=\sum_{\omega\in A_i,\ t(\omega)=y}\mu(\omega), \qquad (i,y)\in D_t=\{(i,y):A_i\cap t^{-1}(y)\ne\varnothing\}.
D_tIntroduction passage · 4 passages
Search aliases: D_t
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 512
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
p_{i,y}(\mu)=\sum_{\omega\in A_i,\ t(\omega)=y}\mu(\omega), \qquad (i,y)\in D_t=\{(i,y):A_i\cap t^{-1}(y)\ne\varnothing\}.
\dimIntroduction passage · 4 passages
Search aliases: dim
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 619
Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.
Exact mathematical passage
\dim U
eIntroduction passage · 55 passages
Search aliases: e
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
(\ell_{e,o}(\omega))_{e,o}
jIntroduction passage · 38 passages
Search aliases: j
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 615
Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.
Exact mathematical passage
U_0=U,\qquad U_{j+1}=\{v\in U:S_av\in U_j\text{ for every }a\in\A\}.\tag{7}
kIntroduction passage · 28 passages
Search aliases: k
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 882
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
k\ge1
LIntroduction passage · 27 passages
Search aliases: L
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 668
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
v\in L
\lceilIntroduction passage · 2 passages
Search aliases: lceil
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 886
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
k\lceil\log_2(N+1)\rceil
\logIntroduction passage · 1 passage
Search aliases: log
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 886
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
k\lceil\log_2(N+1)\rceil
\log_2Introduction passage · 1 passage
Search aliases: log_2 log
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 886
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
k\lceil\log_2(N+1)\rceil
mIntroduction passage · 14 passages
Search aliases: m
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
A_1,\ldots,A_m
NIntroduction passage · 53 passages
Search aliases: N
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 882
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
N\ge1
nIntroduction passage · 37 passages
Search aliases: n
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 668
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
n\ge1
n_1Introduction passage · 1 passage
Search aliases: n_1
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 883
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
\widehat p=(n_1/N,\ldots,n_k/N),\qquad n_i\in\{0,\ldots,N\},\quad \sum_i n_i=N.
n_iIntroduction passage · 3 passages
Search aliases: n_i
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 883
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
\widehat p=(n_1/N,\ldots,n_k/N),\qquad n_i\in\{0,\ldots,N\},\quad \sum_i n_i=N.
n_kIntroduction passage · 1 passage
Search aliases: n_k
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 883
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
\widehat p=(n_1/N,\ldots,n_k/N),\qquad n_i\in\{0,\ldots,N\},\quad \sum_i n_i=N.
oIntroduction passage · 15 passages
Search aliases: o
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
(\ell_{e,o}(\omega))_{e,o}
\rceilIntroduction passage · 2 passages
Search aliases: rceil
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 886
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
k\lceil\log_2(N+1)\rceil
S_{a_1}Introduction passage · 1 passage
Search aliases: S_a_1
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 676
plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives…
Exact mathematical passage
z_j=S_{a_j}\cdots S_{a_1}v\in L
S_{a_j}Introduction passage · 1 passage
Search aliases: S_a_j
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 676
plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives…
Exact mathematical passage
z_j=S_{a_j}\cdots S_{a_1}v\in L
S_wIntroduction passage · 1 passage
Search aliases: S_w
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 669
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
\|T_wv-S_wv\| \le\eta\|v\|\sum_{j=0}^{n-1}A^{n-1-j}B^j.\tag{9}
TIntroduction passage · 34 passages
Search aliases: T
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 669
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
\|T_wv-S_wv\| \le\eta\|v\|\sum_{j=0}^{n-1}A^{n-1-j}B^j.\tag{9}
t^{-1}Introduction passage · 3 passages
Search aliases: t^-1
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 512
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
p_{i,y}(\mu)=\sum_{\omega\in A_i,\ t(\omega)=y}\mu(\omega), \qquad (i,y)\in D_t=\{(i,y):A_i\cap t^{-1}(y)\ne\varnothing\}.
T_{a_1}Introduction passage · 1 passage
Search aliases: T_a_1
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 676
rox For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives (9). Ambient bounde…
Exact mathematical passage
x_j=T_{a_j}\cdots T_{a_1}v
T_{a_j}Introduction passage · 1 passage
Search aliases: T_a_j
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 676
rox For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives (9). Ambient bounde…
Exact mathematical passage
x_j=T_{a_j}\cdots T_{a_1}v
T_wIntroduction passage · 3 passages
Search aliases: T_w
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 669
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
\|T_wv-S_wv\| \le\eta\|v\|\sum_{j=0}^{n-1}A^{n-1-j}B^j.\tag{9}
U_0Introduction passage · 2 passages
Search aliases: U_0
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 615
Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.
Exact mathematical passage
U_0=U,\qquad U_{j+1}=\{v\in U:S_av\in U_j\text{ for every }a\in\A\}.\tag{7}
U_jIntroduction passage · 2 passages
Search aliases: U_j
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 615
Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.
Exact mathematical passage
U_0=U,\qquad U_{j+1}=\{v\in U:S_av\in U_j\text{ for every }a\in\A\}.\tag{7}
U_{j+1}Introduction passage · 1 passage
Search aliases: U_j+1
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 615
Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.
Exact mathematical passage
U_0=U,\qquad U_{j+1}=\{v\in U:S_av\in U_j\text{ for every }a\in\A\}.\tag{7}
vIntroduction passage · 64 passages
Search aliases: v
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 615
Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.
Exact mathematical passage
U_0=U,\qquad U_{j+1}=\{v\in U:S_av\in U_j\text{ for every }a\in\A\}.\tag{7}
\widehat pIntroduction passage · 9 passages
Search aliases: widehat p widehat
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 883
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
\widehat p=(n_1/N,\ldots,n_k/N),\qquad n_i\in\{0,\ldots,N\},\quad \sum_i n_i=N.
x_jIntroduction passage · 3 passages
Search aliases: x_j
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 676
rox For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives (9). Ambient bounde…
Exact mathematical passage
x_j=T_{a_j}\cdots T_{a_1}v
x_{j+1}Introduction passage · 1 passage
Search aliases: x_j+1
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 677
≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives (9). Ambient boundedness applies to x_j-z_j even when that error h…
Exact mathematical passage
\|x_{j+1}-z_{j+1}\| \le A\|x_j-z_j\|+\eta\|z_j\| \le A\|x_j-z_j\|+\eta B^j\|v\|.
zIntroduction passage · 21 passages
Search aliases: z
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 676
plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives…
Exact mathematical passage
z_j=S_{a_j}\cdots S_{a_1}v\in L
z_jIntroduction passage · 4 passages
Search aliases: z_j
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 676
plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives…
Exact mathematical passage
z_j=S_{a_j}\cdots S_{a_1}v\in L
z_{j+1}Introduction passage · 1 passage
Search aliases: z_j+1
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 677
≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives (9). Ambient boundedness applies to x_j-z_j even when that error h…
Exact mathematical passage
\|x_{j+1}-z_{j+1}\| \le A\|x_j-z_j\|+\eta\|z_j\| \le A\|x_j-z_j\|+\eta B^j\|v\|.
|Introduction passage · 9 passages
Search aliases: |
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 516
ete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexity claim would also have to account for their representation and…
Exact mathematical passage
r=|D_t|
\DeltaIntroduction passage · 6 passages
Search aliases: Δ Delta
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 882
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
\Delta_k
\Delta_kIntroduction passage · 3 passages
Search aliases: Δ_k Delta
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 882
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
\Delta_k
\etaIntroduction passage · 18 passages
Search aliases: η eta
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 669
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
\|T_wv-S_wv\| \le\eta\|v\|\sum_{j=0}^{n-1}A^{n-1-j}B^j.\tag{9}
\lambdaIntroduction passage · 4 passages
Search aliases: λ lambda
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
\lambda_{e,o,i}
\lambda_{e,o,i}Introduction passage · 2 passages
Search aliases: λ_e,o,i lambda
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
\lambda_{e,o,i}
\muIntroduction passage · 12 passages
Search aliases: μ mu
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
\mu
\ldotsIntroduction passage · 12 passages
Search aliases: … ldots
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
A_1,\ldots,A_m
\ellIntroduction passage · 28 passages
Search aliases: ℓ ell
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
(\ell_{e,o}(\omega))_{e,o}
\ell_{e,o}Introduction passage · 5 passages
Search aliases: ℓ_e,o ell
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 511
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
(\ell_{e,o}(\omega))_{e,o}
\varnothingIntroduction passage · 1 passage
Search aliases: ∅ varnothing
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 512
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
p_{i,y}(\mu)=\sum_{\omega\in A_i,\ t(\omega)=y}\mu(\omega), \qquad (i,y)\in D_t=\{(i,y):A_i\cap t^{-1}(y)\ne\varnothing\}.
\sumIntroduction passage · 31 passages
Search aliases: ∑ sum
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 512
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
p_{i,y}(\mu)=\sum_{\omega\in A_i,\ t(\omega)=y}\mu(\omega), \qquad (i,y)\in D_t=\{(i,y):A_i\cap t^{-1}(y)\ne\varnothing\}.
\sum_iIntroduction passage · 11 passages
Search aliases: ∑_i sum
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 883
A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil bits. This is an upper bound for this specified encoding, not a minimum description length. It excludes the likelihood table, program, temporary arith…
Exact mathematical passage
\widehat p=(n_1/N,\ldots,n_k/N),\qquad n_i\in\{0,\ldots,N\},\quad \sum_i n_i=N.
\sum_{j=0}^{n-1}Introduction passage · 2 passages
Search aliases: ∑_j=0^n-1 sum
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 669
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
\|T_wv-S_wv\| \le\eta\|v\|\sum_{j=0}^{n-1}A^{n-1-j}B^j.\tag{9}
\sum_{\omega\in A_i,\ t(\omega)=y}Introduction passage · 1 passage
Search aliases: ∑_ω∈ A_i, t(ω)=y sum omega in omega
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 512
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
p_{i,y}(\mu)=\sum_{\omega\in A_i,\ t(\omega)=y}\mu(\omega), \qquad (i,y)\in D_t=\{(i,y):A_i\cap t^{-1}(y)\ne\varnothing\}.
\neIntroduction passage · 11 passages
Search aliases: ≠ ne
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 512
Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entries summing to one. Exact real probabilities are not a fixed r-bit code. A bit-complexi…
Exact mathematical passage
p_{i,y}(\mu)=\sum_{\omega\in A_i,\ t(\omega)=y}\mu(\omega), \qquad (i,y)\in D_t=\{(i,y):A_i\cap t^{-1}(y)\ne\varnothing\}.
\leIntroduction passage · 32 passages
Search aliases: ≤ le
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 669
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
\|T_wv-S_wv\| \le\eta\|v\|\sum_{j=0}^{n-1}A^{n-1-j}B^j.\tag{9}
\geIntroduction passage · 16 passages
Search aliases: ≥ ge
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 668
[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Itera…
Exact mathematical passage
n\ge1
\cdotsIntroduction passage · 3 passages
Search aliases: ⋯ cdots
Introduction passage · Grounded_Abstraction_v1.0.0.tex, line 676
plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j +η B^j v . The initial discrepancy is zero. Iterating this scalar recurrence gives…
Exact mathematical passage
z_j=S_{a_j}\cdots S_{a_1}v\in L
*Use in source · 5 passages
Search aliases: *
Use in source · Grounded_Abstraction_v1.0.0.tex, line 454
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\Pr\{\omega_*\in B(H)\}\ge1-\delta_{\mathrm{id}}.
\mathbf1Use in source · 1 passage
Search aliases: 1 mathbf
Use in source · Grounded_Abstraction_v1.0.0.tex, line 544
For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1. Each monomial is the likelihood product of an admitted fixed experiment-report word. Consequently p_i,y(μ)=∑_t(ω)=yμ(ω)P_i(ω) is a linear combination…
Exact mathematical passage
\mathbf1_{A_i}(\omega)
\mathbf1_{A_i}Use in source · 1 passage
Search aliases: 1_A_i mathbf
Use in source · Grounded_Abstraction_v1.0.0.tex, line 544
For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1. Each monomial is the likelihood product of an admitted fixed experiment-report word. Consequently p_i,y(μ)=∑_t(ω)=yμ(ω)P_i(ω) is a linear combination…
Exact mathematical passage
\mathbf1_{A_i}(\omega)
A^2Use in source · 1 passage
Search aliases: A^2
Use in source · Grounded_Abstraction_v1.0.0.tex, line 756
1 · 0, Q=1 · 1, v=binom11. Both fixed systems have output sequence 2,1,0,0,… from v, since A^2=B^2=0. But applying A first and B second gives QBAv=1, not zero. That hybrid trajectory belongs to neither candidate model. Treating fixed uncertainty as a freely changing matrix would incorrectly reject the valid output prediction.
Exact mathematical passage
A^2=B^2=0
a_*Use in source · 2 passages
Search aliases: a_*
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1104
Let a_* minimize R. Then R(widehat a)≤widehat R(widehat a)+ξ ≤widehat R(a_*)+ξ≤ R(a_*)+2ξ. For any other decision, the approximate gap loses at most 2ξ when the two risks are replaced by their true values, proving the strict-gap assertion. The threshold assertion is the upper risk bound itself. Each step uses the same posterior and surrogate responses, which is why separately valid certificates about different histories or targets cannot be substituted.
Exact mathematical passage
a_*
A_aUse in source · 11 passages
Search aliases: A_a
Use in source · Grounded_Abstraction_v1.0.0.tex, line 714
A finite test for agreement under a fixed unknown model. Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y. Define I_Q={v∈ V:QT_w^θ v=QA_wv for every θ∈Θ and every word w}. This is an initial-condition subspace. It is not assumed invariant under the physica…
Exact mathematical passage
A_a=T_a^0
A_HUse in source · 2 passages
Search aliases: A_H
Use in source · Grounded_Abstraction_v1.0.0.tex, line 947
let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lceil frackA_H2(τ-ε B_H)rceil} tag12 is sufficient for target and cell error at most τ at every time th…
Exact mathematical passage
A_H=\sum_{j=0}^{H}\gamma^j,\qquad B_H=\gamma\sum_{j=0}^{H-1}\gamma^j.
a_mUse in source · 2 passages
Search aliases: a_m
Use in source · Grounded_Abstraction_v1.0.0.tex, line 197
For a word w=a_1⋯ a_m, F_w applies a_1 first and a_m last. The empty word is the identity. Define x∼_ C x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite w. This is a comparison in a common observation language. It does not require that different models share arbitrary latent coordinates. The behavioral class is a canonical response profile even when its implementation is not identifiable.
Exact mathematical passage
w=a_1\cdots a_m
A_tUse in source · 1 passage
Search aliases: A_t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 966
Proposition~[source reference] and the rounding bound give d_TV(p_t+1,widehat p_t+1) ≤γd_TV(p_t,widehat p_t)+γε+η_N. Iterating the uncapped scalar bound proves (11); total variation is also at most one. Aggregating cell coordinates into target coordinates cannot increase total variation, by the triangle inequality. Substituting e_0≤η_N gives η_N A_t+ε B_t. These quantities are nondecreasing in t because γ≥1 and all terms are nonnegative. Inequality (12) makes the horizon-H value at most τ. It th…
Exact mathematical passage
\eta_N A_t+\varepsilon B_t
A_uUse in source · 1 passage
Search aliases: A_u
Use in source · Grounded_Abstraction_v1.0.0.tex, line 864
, let any competing linear predictor be given. Its reachable span is generated by vectors G_wdR(v). Associate such a vector with [A_wv]∈ W/N and extend by linearity. To see this association is well defined, suppose a finite linear combination of the predictor vectors is zero. Applying any later word and the predictor readout shows that the matching combination of reference vectors has zero QA_u output for every u. That combination therefore belongs to N. The induced map from the competitor's rea…
Exact mathematical passage
QA_u
A_wUse in source · 11 passages
Search aliases: A_w
Use in source · Grounded_Abstraction_v1.0.0.tex, line 715
ent under a fixed unknown model Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y. Define I_Q={v∈ V:QT_w^θ v=QA_wv for every θ∈Θ and every word w}. This is an initial-condition subspace. It is not assumed invariant under the physical maps. The common coordi…
Exact mathematical passage
I_Q=\{v\in V:QT_w^\theta v=QA_wv\text{ for every }\theta\in\Theta\text{ and every word }w\}.
\mathrm{after}Use in source · 1 passage
Search aliases: after mathrm
Use in source · Grounded_Abstraction_v1.0.0.tex, line 162
A state-changing measurement. A different operation Z sets s to zero and then reports the new display. The report is θ. If the old display y_old was retained, then s_old=y_old⊕ y_after Z. The current physical state is zero, not s_old. A correct account distinguishes the recovered historical value from the new state it will act on.
Exact mathematical passage
s_{\mathrm{old}}=y_{\mathrm{old}}\oplus y_{\mathrm{after}\ Z}.
\mathcal BUse in source · 7 passages
Search aliases: B mathcal
Use in source · Grounded_Abstraction_v1.0.0.tex, line 505
Let Ω be finite and nonempty, let t:Ω→ Y have finite image, and let B be a finite collection of experiments. Experiment e has a finite outcome set and specified likelihoods ℓ_e,o(ω)≥0, ∑_oℓ_e,o(ω)=1. The world is fixed. Every experiment remains available, and each new report has this conditional law given the world, the chosen experiment, and all earlier reports. This is a fresh conditional-observation premise, not a claim that copied readings become new samples. No experiment changes the physic…
Exact mathematical passage
\mathcal B
bUse in source · 28 passages
Search aliases: b
Use in source · Grounded_Abstraction_v1.0.0.tex, line 289
Why experimentwise evidence need not compose. [Pairwise separation without a successful investigation] There are three possible initial cases 0,1,2. Test a reports whether the case is 0, and test b reports whether it is 1. Either first test then destroys the accessible case state. Every later test returns a fixed zero report. The target remains the initial case, stored only as an inaccessible historical tag.
Exact mathematical passage
b
B'Use in source · 2 passages
Search aliases: B'
Use in source · Grounded_Abstraction_v1.0.0.tex, line 451
[Ambiguity radius and an explicit prediction certificate] For every nonempty finite B, inf_z∈ R^d max_ω∈ B z-v(ω) _∞=ρ(B), and m(B) attains the infimum. If B'⊆ B is nonempty then ρ(B')≤ρ(B).
Exact mathematical passage
B'\subseteq B
B^2Use in source · 1 passage
Search aliases: B^2
Use in source · Grounded_Abstraction_v1.0.0.tex, line 756
1 · 0, Q=1 · 1, v=binom11. Both fixed systems have output sequence 2,1,0,0,… from v, since A^2=B^2=0. But applying A first and B second gives QBAv=1, not zero. That hybrid trajectory belongs to neither candidate model. Treating fixed uncertainty as a freely changing matrix would incorrectly reject the valid output prediction.
Exact mathematical passage
A^2=B^2=0
B_a^\thetaUse in source · 3 passages
Search aliases: B_a^θ theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 721
For each θ, form the paired maps and difference observation B_a^θ=T_a^θ · 0 0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.
Exact mathematical passage
B_a^\theta=\begin{pmatrix}T_a^\theta&0\\0&A_a\end{pmatrix},\qquad C=\begin{pmatrix}Q&-Q\end{pmatrix},\qquad \Delta v=(v,v).
\mathcal B_{\bar\ell_{e,o}}Use in source · 1 passage
Search aliases: B_barℓ_e,o mathcal bar ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 934
After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment selection may depend on earlier retained records and independent randomness. It may not introduce a hidden world-dependent selector whose information is …
Exact mathematical passage
\widehat p_{t+1}=\mathcal Q_N\!\left(\mathcal B_{\bar\ell_{e,o}}(\widehat p_t)\right).
B_cUse in source · 2 passages
Search aliases: B_c
Use in source · Grounded_Abstraction_v1.0.0.tex, line 489
What compression does to the candidate set. For a fixed raw-record channel, let B(h) be its exact support set. If only c(h) is retained, the correct support is B_c(z)=⋃_h:c(h)=zB(h). Indeed, a world can produce the coarse record exactly when it can produce at least one raw record in that fibre. Thus B(h)⊆ B_c(c(h)), and the ambiguity radius cannot decrease for the same target vector. This is an information statement, not a command to retain every byte. The correction signature in Section~[source…
Exact mathematical passage
B_c(z)=\bigcup_{h:c(h)=z}B(h).
B_HUse in source · 4 passages
Search aliases: B_H
Use in source · Grounded_Abstraction_v1.0.0.tex, line 947
let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lceil frackA_H2(τ-ε B_H)rceil} tag12 is sufficient for target and cell error at most τ at every time th…
Exact mathematical passage
A_H=\sum_{j=0}^{H}\gamma^j,\qquad B_H=\gamma\sum_{j=0}^{H-1}\gamma^j.
B_tUse in source · 4 passages
Search aliases: B_t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 966
Proposition~[source reference] and the rounding bound give d_TV(p_t+1,widehat p_t+1) ≤γd_TV(p_t,widehat p_t)+γε+η_N. Iterating the uncapped scalar bound proves (11); total variation is also at most one. Aggregating cell coordinates into target coordinates cannot increase total variation, by the triangle inequality. Substituting e_0≤η_N gives η_N A_t+ε B_t. These quantities are nondecreasing in t because γ≥1 and all terms are nonnegative. Inequality (12) makes the horizon-H value at most τ. It th…
Exact mathematical passage
\eta_N A_t+\varepsilon B_t
B_w^\thetaUse in source · 2 passages
Search aliases: B_w^θ theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 734
[Finite output-agreement test with persistent model tags] The descending paired-kernel sequence stabilizes in at most 2n strict dimension drops. Its terminal subspace is K_∞^θ=⋂_wker(CB_w^θ), I_Q=⋂_θ∈ΘΔ^-1K_∞^θ. Thus exact all-word output agreement for the finite candidate family is decidable by finite linear algebra. If an initial vector is excluded, a disagreeing candidate and a distinguishing word can be extracted. Words of length at most 2n suffice for this test. The conclusion compares fixe…
Exact mathematical passage
K_\infty^\theta=\bigcap_w\ker(CB_w^\theta),\qquad I_Q=\bigcap_{\theta\in\Theta}\Delta^{-1}K_\infty^\theta.
\mathcal B_{\widetilde\ell}Use in source · 2 passages
Search aliases: B_widetildeℓ mathcal widetilde ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 902
[Rounding and positive-likelihood update bounds] For every p∈Δ_k, d_TV(p, Q_N(p))≤frack2N. Let p,q∈Δ_k, let 0<β≤1, and let every entry of ℓ and widetildeℓ lie in [β,1]. If max_i|ℓ_i-widetildeℓ_i|≤ε, then d_TV( B_ℓ(p), B_widetildeℓ(q)) ≤ fracd_TV(p,q)+εβ. Each right-hand side may also be replaced by its minimum with one. The probability-field storage bound above applies to the retained rounded record, not to all working memory. The remainder rule produces nonnegative integers summing to N. Each r…
Exact mathematical passage
\TV\bigl(\mathcal B_\ell(p),\mathcal B_{\widetilde\ell}(q)\bigr) \le \frac{\TV(p,q)+\varepsilon}{\beta}.
B_\piUse in source · 1 passage
Search aliases: B_π pi
Use in source · Grounded_Abstraction_v1.0.0.tex, line 183
[Law identification and finite-record identification] The target is a function of the complete family of protocol laws if and only if it is constant on the ∼_Π classes. For a fixed protocol π, an error-free rule d:H_π→ Y exists if and only if t is constant on every nonempty support set B_π(h)={ω:K^π(hmidω)>0}. Allowing a randomized output rule does not weaken this condition for probability-one correctness in every world. For the first claim, factor t through the map sending a world to its tuple …
Exact mathematical passage
B_\pi(h)=\{\omega:K^\pi(h\mid\omega)>0\}.
\mathcal B_\ellUse in source · 4 passages
Search aliases: B_ℓ mathcal ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 891
For a likelihood vector ℓ=(ℓ_i) define the Bayes map B_ℓ(p)_i=fracp_iℓ_i∑_jp_jℓ_j whenever its denominator is positive.
Exact mathematical passage
\mathcal B_\ell(p)_i=\frac{p_i\ell_i}{\sum_jp_j\ell_j}
\bar QUse in source · 5 passages
Search aliases: bar Q bar
Use in source · Grounded_Abstraction_v1.0.0.tex, line 789
following conditions are equivalent. item QT_w^θ=QA_w on all of V, for every fixed model θ and every word w. item E_∞⊆ker Q. item There exists a subspace N⊆ker Q, invariant under every A_a, containing every im D_a^θ. item There is a surjective linear representation r:V→ Z, shared linear updates G_a, and a readout bar Q such that rT_a^θ=G_ar for all a,θ, Q=bar Qr. When the conditions hold, V/E_∞ gives one such quotient. The smallest-dimensional common linear quotient is V/N_0, where N_0=⋂_wker(QA…
Exact mathematical passage
\bar Q
\bar vUse in source · 7 passages
Search aliases: bar v bar
Use in source · Grounded_Abstraction_v1.0.0.tex, line 992
Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g,…
Exact mathematical passage
\begin{pmatrix}v_{t+1}\\h_{t+1}\end{pmatrix} =\begin{pmatrix}G_t&B_t\\C_t&H_t\end{pmatrix} \begin{pmatrix}v_t\\h_t\end{pmatrix}, \qquad \bar v_{t+1}=G_t\bar v_t. \tag{13}
\bar v_0Use in source · 2 passages
Search aliases: bar v_0 bar
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1004
tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g, B_t ≤ b, C_t ≤ c, H_t ≤ d, g,b,c,d≥0. Let v_0 ≤ V_0, h_0 ≤ H_0, and v_0-bar v_0 ≤ e_0. These initial bounds are supplied evidence or declared model assumptions. Put M=g · b
Exact mathematical passage
\|v_0-\bar v_0\|\le e_0
\bar v_tUse in source · 5 passages
Search aliases: bar v_t bar
Use in source · Grounded_Abstraction_v1.0.0.tex, line 992
Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g,…
Exact mathematical passage
\begin{pmatrix}v_{t+1}\\h_{t+1}\end{pmatrix} =\begin{pmatrix}G_t&B_t\\C_t&H_t\end{pmatrix} \begin{pmatrix}v_t\\h_t\end{pmatrix}, \qquad \bar v_{t+1}=G_t\bar v_t. \tag{13}
\bar v_{t+1}Use in source · 2 passages
Search aliases: bar v_t+1 bar
Use in source · Grounded_Abstraction_v1.0.0.tex, line 992
Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g,…
Exact mathematical passage
\begin{pmatrix}v_{t+1}\\h_{t+1}\end{pmatrix} =\begin{pmatrix}G_t&B_t\\C_t&H_t\end{pmatrix} \begin{pmatrix}v_t\\h_t\end{pmatrix}, \qquad \bar v_{t+1}=G_t\bar v_t. \tag{13}
\bar yUse in source · 6 passages
Search aliases: bar y bar
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1074
Fix an actual evidence history. Let μ be the exact posterior on the stipulated finite world set and let p_i=μ(C_i). Suppose the retained distribution is widehat p with d_TV(p,widehat p)≤ e. For a declared later action word or response horizon, let y(ω) be the actual model response and let bar y_i be the response used for cell i. Assume sup_ω∈ C_i y(ω)-bar y_i ≤ d for every cell i. tag16 The cell need not identify the physical state or candidate operator. It must support this common response allo…
Exact mathematical passage
\bar y_i
\bar y_iUse in source · 6 passages
Search aliases: bar y_i bar
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1074
Fix an actual evidence history. Let μ be the exact posterior on the stipulated finite world set and let p_i=μ(C_i). Suppose the retained distribution is widehat p with d_TV(p,widehat p)≤ e. For a declared later action word or response horizon, let y(ω) be the actual model response and let bar y_i be the response used for cell i. Assume sup_ω∈ C_i y(ω)-bar y_i ≤ d for every cell i. tag16 The cell need not identify the physical state or candidate operator. It must support this common response allo…
Exact mathematical passage
\bar y_i
\bar\ellUse in source · 4 passages
Search aliases: barℓ bar ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 926
Choose one valid representative report law barℓ_e,o,i per cell. Assume, for every admitted experiment, report, cell, and world in that cell, β≤ ℓ_e,o(ω),barℓ_e,o,i≤1, |ℓ_e,o(ω)-barℓ_e,o,i|≤ε. tag10 Both the actual and representative report laws sum to one over reports. Exact profile cells satisfy this condition with ε=0 when their likelihoods have the positive floor. A coarser partition can be used with a nonzero ε, but target-constant cells are still required for the target readout.
Exact mathematical passage
\bar\ell_{e,o,i}
\bar\ell_{e,o}Use in source · 1 passage
Search aliases: barℓ_e,o bar ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 934
After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment selection may depend on earlier retained records and independent randomness. It may not introduce a hidden world-dependent selector whose information is …
Exact mathematical passage
\widehat p_{t+1}=\mathcal Q_N\!\left(\mathcal B_{\bar\ell_{e,o}}(\widehat p_t)\right).
\bar\ell_{e,o,i}Use in source · 3 passages
Search aliases: barℓ_e,o,i bar ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 926
Choose one valid representative report law barℓ_e,o,i per cell. Assume, for every admitted experiment, report, cell, and world in that cell, β≤ ℓ_e,o(ω),barℓ_e,o,i≤1, |ℓ_e,o(ω)-barℓ_e,o,i|≤ε. tag10 Both the actual and representative report laws sum to one over reports. Exact profile cells satisfy this condition with ε=0 when their likelihoods have the positive floor. A coarser partition can be used with a nonzero ε, but target-constant cells are still required for the target readout.
Exact mathematical passage
\bar\ell_{e,o,i}
CUse in source · 13 passages
Search aliases: C
Use in source · Grounded_Abstraction_v1.0.0.tex, line 721
For each θ, form the paired maps and difference observation B_a^θ=T_a^θ · 0 0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.
Exact mathematical passage
B_a^\theta=\begin{pmatrix}T_a^\theta&0\\0&A_a\end{pmatrix},\qquad C=\begin{pmatrix}Q&-Q\end{pmatrix},\qquad \Delta v=(v,v).
cUse in source · 13 passages
Search aliases: c
Use in source · Grounded_Abstraction_v1.0.0.tex, line 273
every pair in D is separated by at least one admitted test. Executing all tests recovers t, and a selected nonadaptive set J suffices exactly when it covers D. item Some pair in D has identical reports under every admitted test. No finite adaptive protocol using this unchanged test interface and r guarantees t in every world. If test j has nonnegative fixed cost c_j, the minimum cost of one fixed nonadaptive test set, used for all initial codes in case 2, is precisely min_J: J covers D ∑_j∈ Jc_j…
Exact mathematical passage
c_j
C_1Use in source · 1 passage
Search aliases: C_1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 924
Approximate likelihood profiles. Let Ω be a finite static world set with prior μ_0. A fixed partition C_1,…,C_k refines the target: t(ω)=t_i throughout C_i. Each admitted experiment e is nondestructive and has a finite report alphabet. Its actual conditional probabilities are ℓ_e,o(ω). Reports are freshly generated from those likelihoods conditional on the fixed world and the chosen experiments.
Exact mathematical passage
C_1,\ldots,C_k
C_iUse in source · 3 passages
Search aliases: C_i
Use in source · Grounded_Abstraction_v1.0.0.tex, line 924
Approximate likelihood profiles. Let Ω be a finite static world set with prior μ_0. A fixed partition C_1,…,C_k refines the target: t(ω)=t_i throughout C_i. Each admitted experiment e is nondestructive and has a finite report alphabet. Its actual conditional probabilities are ℓ_e,o(ω). Reports are freshly generated from those likelihoods conditional on the fixed world and the chosen experiments.
Exact mathematical passage
C_i
c_jUse in source · 2 passages
Search aliases: c_j
Use in source · Grounded_Abstraction_v1.0.0.tex, line 273
every pair in D is separated by at least one admitted test. Executing all tests recovers t, and a selected nonadaptive set J suffices exactly when it covers D. item Some pair in D has identical reports under every admitted test. No finite adaptive protocol using this unchanged test interface and r guarantees t in every world. If test j has nonnegative fixed cost c_j, the minimum cost of one fixed nonadaptive test set, used for all initial codes in case 2, is precisely min_J: J covers D ∑_j∈ Jc_j…
Exact mathematical passage
c_j
C_kUse in source · 1 passage
Search aliases: C_k
Use in source · Grounded_Abstraction_v1.0.0.tex, line 924
Approximate likelihood profiles. Let Ω be a finite static world set with prior μ_0. A fixed partition C_1,…,C_k refines the target: t(ω)=t_i throughout C_i. Each admitted experiment e is nondestructive and has a finite report alphabet. Its actual conditional probabilities are ℓ_e,o(ω). Reports are freshly generated from those likelihoods conditional on the fixed world and the chosen experiments.
Exact mathematical passage
C_1,\ldots,C_k
C_tUse in source · 2 passages
Search aliases: C_t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 992
Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g,…
Exact mathematical passage
\begin{pmatrix}v_{t+1}\\h_{t+1}\end{pmatrix} =\begin{pmatrix}G_t&B_t\\C_t&H_t\end{pmatrix} \begin{pmatrix}v_t\\h_t\end{pmatrix}, \qquad \bar v_{t+1}=G_t\bar v_t. \tag{13}
\mathcal DUse in source · 2 passages
Search aliases: D mathcal
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1081
Let the permitted decision set D be finite and nonempty. Its loss L(y,a) takes values in [0,1] and is K-Lipschitz in y, uniformly over a. Define R(a)=∑_ωμ(ω)L(y(ω),a), widehat R(a)=∑_iwidehat p_iL(bar y_i,a). These are risks under the stated model posterior. They are not frequentist coverage probabilities or guarantees on the loss of a single realized world.
Exact mathematical passage
\mathcal D
D^{-1}Use in source · 1 passage
Search aliases: D^-1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1044
Under the three strict inequalities, (I-M)^-1=D^-11-d · b c · 1-g has nonnegative entries. Let a=(I-M)^-1(1,1)^T. Both coordinates are positive and Ma=a-(1,1)^T<a. Hence M is a strict contraction in the weighted maximum norm with weights a, and its geometric series converges to (I-M)^-1. Summing the nonnegative right-hand side of (14) gives frace_01-g+fracb1-g∑_j≥0(M^ju_0)_2. The second coordinate of (I-M)^-1u_0 is (cV_0+(1-g)H_0)/D, proving (15). The common comparison applies at each step and t…
Exact mathematical passage
(I-M)^{-1}=D^{-1}\begin{pmatrix}1-d&b\\c&1-g\end{pmatrix}
d^2Use in source · 3 passages
Search aliases: d^2
Use in source · Grounded_Abstraction_v1.0.0.tex, line 972
[A tiny rounded-away mass can control a later posterior] Take two worlds with prior (1-d,d), where 0<d<1/(2N). The remainder rule gives the retained record (1,0). A binary report has likelihoods (d^2,1) for outcome one. After that outcome, the actual posterior mass of the second world is fracdd+(1-d)d^2=frac11+d-d^2, which tends to one as d tends to zero. The rounded posterior remains zero on that world. The initial total-variation error is only d. There is no common positive likelihood floor ac…
Exact mathematical passage
(d^2,1)
d_3Use in source · 1 passage
Search aliases: d_3
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1134
, (v_0,h_0)=(s,1). tag17 The predictor for cell (s,θ) uses bar v_t=2^-ts and therefore reports bar y_i=s/8. The nuisance bit n affects the physical response but remains unidentified by the reference observations. With g=d=1/2, b=1/64, c=1/8, and V_0=H_0=1, the comparison determinant is 127/512>0. Directly evaluating (14) at three prediction steps yields d_3=frac48132 768. Choose between numerical predictions a=0 and a=1/8, with loss L(y,a)=min{1,|y-a|}. It is bounded and 1-Lipschitz. The common …
Exact mathematical passage
d_3=\frac{481}{32\,768}.
D_a^\thetaUse in source · 7 passages
Search aliases: D_a^θ theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 773
When uncertain dynamics admit one common observable quotient. Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies. Let E_0=∑_a,θim D_a^θ, E_j+1=E_j+∑_a A_aE_j, E_∞=⋃_jE_j. This increasing sequence stabilizes after at most n strict d…
Exact mathematical passage
D_a^\theta=T_a^\theta-A_a.
d_tUse in source · 1 passage
Search aliases: d_t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1013
[Visible-error bounds from a common block comparison] Every admitted execution satisfying (13) obeys v_t-bar v_t ≤ d_t :=g^te_0+b∑_j=0^t-1g^t-1-j(u_j)_2. tag14 In particular, when b=0 and e_0=0, the visible prediction is exact for every finite time, without a stability requirement on d or c.
Exact mathematical passage
\|v_t-\bar v_t\|\le d_t :=g^te_0+b\sum_{j=0}^{t-1}g^{t-1-j}(u_j)_2. \tag{14}
E_0Use in source · 5 passages
Search aliases: E_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 319
[The coarsest exact memory for admitted corrections] Equality of correction signatures is a congruence for all updates V↦ V∩ E_0. The signature therefore supports well-defined updates and a readout of Q(V). It preserves that readout after every finite sequence of admitted filters.
Exact mathematical passage
V\mapsto V\cap E_0
e_0Use in source · 10 passages
Search aliases: e_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 940
[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lcei…
Exact mathematical passage
e_0
e_1Use in source · 13 passages
Search aliases: e_1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 254
We first separate these cases in a setting where observations do not interfere with one another. Let Ω be finite, let r:Ω→ Z be retained evidence, and let t:Ω→ Y be the revised target. Let E={e_1,…,e_m} be a finite family of total, deterministic, nondestructive tests. Each test reads the same fixed world and leaves it unchanged. All tests remain available throughout the investigation. Equivalently, a reset may be allowed if it genuinely restores the same unknown world, including the relevant his…
Exact mathematical passage
\E=\{e_1,\ldots,e_m\}
e_2Use in source · 7 passages
Search aliases: e_2
Use in source · Grounded_Abstraction_v1.0.0.tex, line 640
A two-dimensional calibration with a one-dimensional reusable core. [Calibration does not close under continuation] Let V have basis e_1,e_2,e_3, let U=span(e_1,e_2), and let the single calibrated operation satisfy Se_1=e_1, Se_2=e_3. The first continuation is known on all of U. The next calibrated domain is U_1=span(e_1), which is already invariant. One completion has T_0e_3=0 and another has T_1e_3=e_1. Both agree on every supplied calibration. Starting from e_2, their first outputs agree at e…
Exact mathematical passage
e_1,e_2,e_3
e_3Use in source · 10 passages
Search aliases: e_3
Use in source · Grounded_Abstraction_v1.0.0.tex, line 417
For odd n, majority voting is the likelihood-ratio rule. Its error in either world is e_n=∑_j=0^(n-1)/2binomnj(3/4)^j(1/4)^n-j. The first values are e_1=1/4, e_3=5/32, and e_5=53/512. These are exact analytical probabilities, not observed error rates from an agent experiment.
Exact mathematical passage
e_3=5/32
e_5Use in source · 1 passage
Search aliases: e_5
Use in source · Grounded_Abstraction_v1.0.0.tex, line 417
For odd n, majority voting is the likelihood-ratio rule. Its error in either world is e_n=∑_j=0^(n-1)/2binomnj(3/4)^j(1/4)^n-j. The first values are e_1=1/4, e_3=5/32, and e_5=53/512. These are exact analytical probabilities, not observed error rates from an agent experiment.
Exact mathematical passage
e_5=53/512
E_jUse in source · 1 passage
Search aliases: E_j
Use in source · Grounded_Abstraction_v1.0.0.tex, line 777
ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies. Let E_0=∑_a,θim D_a^θ, E_j+1=E_j+∑_a A_aE_j, E_∞=⋃_jE_j. This increasing sequence stabilizes after at most n strict dimension increases. Its limit is…
Exact mathematical passage
E_0=\sum_{a,\theta}\im D_a^\theta,\qquad E_{j+1}=E_j+\sum_a A_aE_j,\qquad E_\infty=\bigcup_jE_j.
e_jUse in source · 2 passages
Search aliases: e_j
Use in source · Grounded_Abstraction_v1.0.0.tex, line 257
For J⊆{1,…,m}, write R_J(ω)=(r(ω),(e_j(ω))_j∈ J). Let the unresolved target pairs be D={{ω,ω'}:r(ω)=r(ω'), t(ω)≠ t(ω')}. A test covers such a pair when it gives different reports in the two worlds.
Exact mathematical passage
R_J(\omega)=\bigl(r(\omega),(e_j(\omega))_{j\in J}\bigr).
E_{j+1}Use in source · 1 passage
Search aliases: E_j+1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 777
ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies. Let E_0=∑_a,θim D_a^θ, E_j+1=E_j+∑_a A_aE_j, E_∞=⋃_jE_j. This increasing sequence stabilizes after at most n strict dimension increases. Its limit is…
Exact mathematical passage
E_0=\sum_{a,\theta}\im D_a^\theta,\qquad E_{j+1}=E_j+\sum_a A_aE_j,\qquad E_\infty=\bigcup_jE_j.
e_kUse in source · 1 passage
Search aliases: e_k
Use in source · Grounded_Abstraction_v1.0.0.tex, line 531
For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. The empty word gives the current target distribution.
Exact mathematical passage
w=((e_1,o_1),\ldots,(e_k,o_k))
e_mUse in source · 1 passage
Search aliases: e_m
Use in source · Grounded_Abstraction_v1.0.0.tex, line 254
We first separate these cases in a setting where observations do not interfere with one another. Let Ω be finite, let r:Ω→ Z be retained evidence, and let t:Ω→ Y be the revised target. Let E={e_1,…,e_m} be a finite family of total, deterministic, nondestructive tests. Each test reads the same fixed world and leaves it unchanged. All tests remain available throughout the investigation. Equivalently, a reset may be allowed if it genuinely restores the same unknown world, including the relevant his…
Exact mathematical passage
\E=\{e_1,\ldots,e_m\}
e_nUse in source · 1 passage
Search aliases: e_n
Use in source · Grounded_Abstraction_v1.0.0.tex, line 414
For odd n, majority voting is the likelihood-ratio rule. Its error in either world is e_n=∑_j=0^(n-1)/2binomnj(3/4)^j(1/4)^n-j. The first values are e_1=1/4, e_3=5/32, and e_5=53/512. These are exact analytical probabilities, not observed error rates from an agent experiment.
Exact mathematical passage
e_n=\sum_{j=0}^{(n-1)/2}\binom{n}{j}(3/4)^j(1/4)^{n-j}.
e_tUse in source · 1 passage
Search aliases: e_t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 941
[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lcei…
Exact mathematical passage
\TV(p_t,\widehat p_t)\le e_t :=\min\!\left\{1,\ \gamma^t e_0+(\gamma\varepsilon+\eta_N) \sum_{j=0}^{t-1}\gamma^j\right\}. \tag{11}
\mathbb FUse in source · 1 passage
Search aliases: F mathbb
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1259
Small exact illustrations and reproduction. Three small exact suites check the declared constructions with integers, sets, rational arithmetic, and F_2. The retained observable suite compares paired row-space closure with direct word enumeration, tests common-quotient and initialization criteria, and compares likelihood-profile updates with raw-world Bayes updates. It includes explicit switching, hidden-return, zero-probability, and memory-migration controls. The two retained suites are unchange…
Exact mathematical passage
\mathbb F_2
fUse in source · 10 passages
Search aliases: f
Use in source · Grounded_Abstraction_v1.0.0.tex, line 401
(P_0,P_1)=frac12∑_h|P_0(h)-P_1(h)|.tag4 A deterministic function or a common stochastic processing channel applied to the transcript cannot increase this upper bound. Convert the estimator into a binary test by reporting world zero when it outputs the first target and world one otherwise. A correct target output always gives a correct binary decision. If f(h)∈[0,1] is the test's chance of declaring world zero, its success is frac12(1+∑_h f(h)(P_0(h)-P_1(h))). The sum is at most the sum of positi…
Exact mathematical passage
f(h)\in[0,1]
\mathbb F_2Use in source · 1 passage
Search aliases: F_2 mathbb
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1259
Small exact illustrations and reproduction. Three small exact suites check the declared constructions with integers, sets, rational arithmetic, and F_2. The retained observable suite compares paired row-space closure with direct word enumeration, tests common-quotient and initialization criteria, and compares likelihood-profile updates with raw-world Bayes updates. It includes explicit switching, hidden-return, zero-probability, and memory-migration controls. The two retained suites are unchange…
Exact mathematical passage
\mathbb F_2
f_iUse in source · 6 passages
Search aliases: f_i
Use in source · Grounded_Abstraction_v1.0.0.tex, line 911
For the update, first hold ℓ fixed. Given 0≤ f_i≤1, put a=∑_i B_ℓ(q)_if_i and Z_p=∑_ip_iℓ_i. Then ∑_i( B_ℓ(p)_i- B_ℓ(q)_i)f_i =frac∑_i(p_i-q_i)ℓ_i(f_i-a)Z_p. Every value ℓ_i(f_i-a) lies in [-a,1-a], an interval of length one. Since p-q has total mass zero, its pairing with a function of oscillation at most one is bounded in absolute value by d_TV(p,q). Also Z_p≥β. Taking the supremum over f bounds the fixed-likelihood difference by d_TV(p,q)/β.
Exact mathematical passage
0\le f_i\le1
gUse in source · 12 passages
Search aliases: g
Use in source · Grounded_Abstraction_v1.0.0.tex, line 405
it outputs the first target and world one otherwise. A correct target output always gives a correct binary decision. If f(h)∈[0,1] is the test's chance of declaring world zero, its success is frac12(1+∑_h f(h)(P_0(h)-P_1(h))). The sum is at most the sum of positive differences, which equals total variation. This proves (4). If B(h,z) is a common processing channel and g(z)∈[0,1] a processed-record test, then f(h)=∑_zB(h,z)g(z) is a raw-record test with the same two expectations. Thus every proce…
Exact mathematical passage
g(z)\in[0,1]
g^tUse in source · 1 passage
Search aliases: g^t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1013
[Visible-error bounds from a common block comparison] Every admitted execution satisfying (13) obeys v_t-bar v_t ≤ d_t :=g^te_0+b∑_j=0^t-1g^t-1-j(u_j)_2. tag14 In particular, when b=0 and e_0=0, the visible prediction is exact for every finite time, without a stability requirement on d or c.
Exact mathematical passage
\|v_t-\bar v_t\|\le d_t :=g^te_0+b\sum_{j=0}^{t-1}g^{t-1-j}(u_j)_2. \tag{14}
g^{t-1-j}Use in source · 1 passage
Search aliases: g^t-1-j
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1013
[Visible-error bounds from a common block comparison] Every admitted execution satisfying (13) obeys v_t-bar v_t ≤ d_t :=g^te_0+b∑_j=0^t-1g^t-1-j(u_j)_2. tag14 In particular, when b=0 and e_0=0, the visible prediction is exact for every finite time, without a stability requirement on d or c.
Exact mathematical passage
\|v_t-\bar v_t\|\le d_t :=g^te_0+b\sum_{j=0}^{t-1}g^{t-1-j}(u_j)_2. \tag{14}
G_tUse in source · 4 passages
Search aliases: G_t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 992
Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g,…
Exact mathematical passage
\begin{pmatrix}v_{t+1}\\h_{t+1}\end{pmatrix} =\begin{pmatrix}G_t&B_t\\C_t&H_t\end{pmatrix} \begin{pmatrix}v_t\\h_t\end{pmatrix}, \qquad \bar v_{t+1}=G_t\bar v_t. \tag{13}
H_0Use in source · 5 passages
Search aliases: H_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1004
v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g, B_t ≤ b, C_t ≤ c, H_t ≤ d, g,b,c,d≥0. Let v_0 ≤ V_0, h_0 ≤ H_0, and v_0-bar v_0 ≤ e_0. These initial bounds are supplied evidence or declared model assumptions. Put M=g · b
Exact mathematical passage
\|h_0\|\le H_0
h_0Use in source · 3 passages
Search aliases: h_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1004
v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g, B_t ≤ b, C_t ≤ c, H_t ≤ d, g,b,c,d≥0. Let v_0 ≤ V_0, h_0 ≤ H_0, and v_0-bar v_0 ≤ e_0. These initial bounds are supplied evidence or declared model assumptions. Put M=g · b
Exact mathematical passage
\|h_0\|\le H_0
H_tUse in source · 2 passages
Search aliases: H_t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 992
Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g,…
Exact mathematical passage
\begin{pmatrix}v_{t+1}\\h_{t+1}\end{pmatrix} =\begin{pmatrix}G_t&B_t\\C_t&H_t\end{pmatrix} \begin{pmatrix}v_t\\h_t\end{pmatrix}, \qquad \bar v_{t+1}=G_t\bar v_t. \tag{13}
h_tUse in source · 5 passages
Search aliases: h_t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 991
Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1
Exact mathematical passage
(v_t,h_t)
h_{t+1}Use in source · 2 passages
Search aliases: h_t+1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 992
Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g,…
Exact mathematical passage
\begin{pmatrix}v_{t+1}\\h_{t+1}\end{pmatrix} =\begin{pmatrix}G_t&B_t\\C_t&H_t\end{pmatrix} \begin{pmatrix}v_t\\h_t\end{pmatrix}, \qquad \bar v_{t+1}=G_t\bar v_t. \tag{13}
\mathrm{id}Use in source · 3 passages
Search aliases: id mathrm
Use in source · Grounded_Abstraction_v1.0.0.tex, line 453
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\delta_{\mathrm{id}},\delta_{\mathrm{model}}\in[0,1]
\imUse in source · 2 passages
Search aliases: im operatorname
Use in source · Grounded_Abstraction_v1.0.0.tex, line 777
ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies. Let E_0=∑_a,θim D_a^θ, E_j+1=E_j+∑_a A_aE_j, E_∞=⋃_jE_j. This increasing sequence stabilizes after at most n strict dimension increases. Its limit is…
Exact mathematical passage
E_0=\sum_{a,\theta}\im D_a^\theta,\qquad E_{j+1}=E_j+\sum_a A_aE_j,\qquad E_\infty=\bigcup_jE_j.
\infUse in source · 1 passage
Search aliases: inf
Use in source · Grounded_Abstraction_v1.0.0.tex, line 448
[Ambiguity radius and an explicit prediction certificate] For every nonempty finite B, inf_z∈ R^d max_ω∈ B z-v(ω) _∞=ρ(B), and m(B) attains the infimum. If B'⊆ B is nonempty then ρ(B')≤ρ(B).
Exact mathematical passage
\inf_{z\in\mathbb R^d}\ \max_{\omega\in B}\|z-v(\omega)\|_\infty=\rho(B),
\inf_{z\in\mathbb R^d}Use in source · 1 passage
Search aliases: inf_z∈ R^d inf in mathbb
Use in source · Grounded_Abstraction_v1.0.0.tex, line 448
[Ambiguity radius and an explicit prediction certificate] For every nonempty finite B, inf_z∈ R^d max_ω∈ B z-v(ω) _∞=ρ(B), and m(B) attains the infimum. If B'⊆ B is nonempty then ρ(B')≤ρ(B).
Exact mathematical passage
\inf_{z\in\mathbb R^d}\ \max_{\omega\in B}\|z-v(\omega)\|_\infty=\rho(B),
JUse in source · 5 passages
Search aliases: J
Use in source · Grounded_Abstraction_v1.0.0.tex, line 256
For J⊆{1,…,m}, write R_J(ω)=(r(ω),(e_j(ω))_j∈ J). Let the unresolved target pairs be D={{ω,ω'}:r(ω)=r(ω'), t(ω)≠ t(ω')}. A test covers such a pair when it gives different reports in the two worlds.
Exact mathematical passage
J\subseteq\{1,\ldots,m\}
K_0^\thetaUse in source · 1 passage
Search aliases: K_0^θ theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 727
0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.
Exact mathematical passage
K_0^\theta=\ker C,\qquad K_{j+1}^\theta=K_j^\theta\cap\bigcap_a(B_a^\theta)^{-1}K_j^\theta.
K_{j+1}^\thetaUse in source · 1 passage
Search aliases: K_j+1^θ theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 727
0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.
Exact mathematical passage
K_0^\theta=\ker C,\qquad K_{j+1}^\theta=K_j^\theta\cap\bigcap_a(B_a^\theta)^{-1}K_j^\theta.
K_j^\thetaUse in source · 2 passages
Search aliases: K_j^θ theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 727
0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.
Exact mathematical passage
K_0^\theta=\ker C,\qquad K_{j+1}^\theta=K_j^\theta\cap\bigcap_a(B_a^\theta)^{-1}K_j^\theta.
K_\infty^\thetaUse in source · 1 passage
Search aliases: K_∞^θ infty theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 734
[Finite output-agreement test with persistent model tags] The descending paired-kernel sequence stabilizes in at most 2n strict dimension drops. Its terminal subspace is K_∞^θ=⋂_wker(CB_w^θ), I_Q=⋂_θ∈ΘΔ^-1K_∞^θ. Thus exact all-word output agreement for the finite candidate family is decidable by finite linear algebra. If an initial vector is excluded, a disagreeing candidate and a distinguishing word can be extracted. Words of length at most 2n suffice for this test. The conclusion compares fixe…
Exact mathematical passage
K_\infty^\theta=\bigcap_w\ker(CB_w^\theta),\qquad I_Q=\bigcap_{\theta\in\Theta}\Delta^{-1}K_\infty^\theta.
\kerUse in source · 13 passages
Search aliases: ker
Use in source · Grounded_Abstraction_v1.0.0.tex, line 727
0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.
Exact mathematical passage
K_0^\theta=\ker C,\qquad K_{j+1}^\theta=K_j^\theta\cap\bigcap_a(B_a^\theta)^{-1}K_j^\theta.
\lfloorUse in source · 1 passage
Search aliases: lfloor
Use in source · Grounded_Abstraction_v1.0.0.tex, line 888
Define Q_N(p) by first setting n_i=lfloor Np_irfloor. Distribute the remaining N-∑_i n_i units to the largest fractional remainders, with ties broken by a fixed cell order. This makes the normalization exact after every rounding. It is not independent coordinate rounding followed by an unspecified repair.
Exact mathematical passage
n_i=\lfloor Np_i\rfloor
\LongleftrightarrowUse in source · 1 passage
Search aliases: Longleftrightarrow
Use in source · Grounded_Abstraction_v1.0.0.tex, line 198
For a word w=a_1⋯ a_m, F_w applies a_1 first and a_m last. The empty word is the identity. Define x∼_ C x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite w. This is a comparison in a common observation language. It does not require that different models share arbitrary latent coordinates. The behavioral class is a canonical response profile even when its implementation is not identifiable.
Exact mathematical passage
x\sim_\C x'\quad\Longleftrightarrow\quad q(F_wx)=q(F_wx')\quad\text{for every finite }w.
\LongrightarrowUse in source · 1 passage
Search aliases: Longrightarrow
Use in source · Grounded_Abstraction_v1.0.0.tex, line 214
[Evidence sufficient for one exact predictive realization] An exact reusable predictor exists if and only if r(x)=r(x') Longrightarrow x∼_ C x'.tag1 When it exists, the behavioral quotient X/∼_ C supplies a predictor with the fewest initialized states. Every predictor initialized for all x∈ X requires at least |X/∼_ C| distinct values in d(r(X)). Equal initial codes yield equal initialized states. The same operation sequence must then produce equal predicted outputs. Exactness forces equality of…
Exact mathematical passage
r(x)=r(x')\quad\Longrightarrow\quad x\sim_\C x'.\tag{1}
MUse in source · 15 passages
Search aliases: M
Use in source · Grounded_Abstraction_v1.0.0.tex, line 532
For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. The empty word gives the current target distribution.
Exact mathematical passage
M_{\mu,y}(w)=\sum_{t(\omega)=y}\mu(\omega) \prod_{j=1}^k\ell_{e_j,o_j}(\omega).
M^jUse in source · 3 passages
Search aliases: M^j
Use in source · Grounded_Abstraction_v1.0.0.tex, line 692
η · M, η>0, M≥0. The one-step error on the unit calibrated vector is η, independent of M. After two steps the error is η(1+M). It can be arbitrarily large while the local calibration error is fixed. For n steps the second coordinate is η∑_j=0^n-1M^j.
Exact mathematical passage
\eta\sum_{j=0}^{n-1}M^j
M^{t+1}Use in source · 1 passage
Search aliases: M^t+1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1033
+fracb(cV_0+(1-g)H_0)(1-g)D. tag15 These are sufficient common bounds. Failure of the inequality for D is not a proof that a particular signed system is unstable. Conversely, the two diagonal conditions alone are insufficient for a uniform conclusion over the stated block class. The norm triangle inequality and the common block bounds give v_t+1 h_t+1 ≤ M v_t h_t ≤ M^t+1u_0 componentwise. The error satisfies v_t+1-bar v_t+1=G_t(v_t-bar v_t)+B_th_t. Induction gives (14), including its empty sum a…
Exact mathematical passage
\begin{pmatrix}\|v_{t+1}\|\\\|h_{t+1}\|\end{pmatrix} \le M\begin{pmatrix}\|v_t\|\\\|h_t\|\end{pmatrix} \le M^{t+1}u_0
m_jUse in source · 1 passage
Search aliases: m_j
Use in source · Grounded_Abstraction_v1.0.0.tex, line 440
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
m_j(B)=\frac{\ell_j(B)+u_j(B)}2,\qquad \rho(B)=\frac12\max_j\bigl(u_j(B)-\ell_j(B)\bigr).\tag{5}
M_{\mu,y}Use in source · 2 passages
Search aliases: M_μ,y mu
Use in source · Grounded_Abstraction_v1.0.0.tex, line 532
For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. The empty word gives the current target distribution.
Exact mathematical passage
M_{\mu,y}(w)=\sum_{t(\omega)=y}\mu(\omega) \prod_{j=1}^k\ell_{e_j,o_j}(\omega).
\maxUse in source · 5 passages
Search aliases: max
Use in source · Grounded_Abstraction_v1.0.0.tex, line 436
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
\ell_j(B)=\min_{\omega\in B}v_j(\omega),\qquad u_j(B)=\max_{\omega\in B}v_j(\omega),
\max_iUse in source · 1 passage
Search aliases: max_i max
Use in source · Grounded_Abstraction_v1.0.0.tex, line 901
[Rounding and positive-likelihood update bounds] For every p∈Δ_k, d_TV(p, Q_N(p))≤frack2N. Let p,q∈Δ_k, let 0<β≤1, and let every entry of ℓ and widetildeℓ lie in [β,1]. If max_i|ℓ_i-widetildeℓ_i|≤ε, then d_TV( B_ℓ(p), B_widetildeℓ(q)) ≤ fracd_TV(p,q)+εβ. Each right-hand side may also be replaced by its minimum with one. The probability-field storage bound above applies to the retained rounded record, not to all working memory. The remainder rule produces nonnegative integers summing to N. Each r…
Exact mathematical passage
\max_i|\ell_i-\widetilde\ell_i|\le\varepsilon
\max_jUse in source · 1 passage
Search aliases: max_j max
Use in source · Grounded_Abstraction_v1.0.0.tex, line 440
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
m_j(B)=\frac{\ell_j(B)+u_j(B)}2,\qquad \rho(B)=\frac12\max_j\bigl(u_j(B)-\ell_j(B)\bigr).\tag{5}
\max_{\omega\in B}Use in source · 2 passages
Search aliases: max_ω∈ B max omega in
Use in source · Grounded_Abstraction_v1.0.0.tex, line 436
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
\ell_j(B)=\min_{\omega\in B}v_j(\omega),\qquad u_j(B)=\max_{\omega\in B}v_j(\omega),
\minUse in source · 5 passages
Search aliases: min
Use in source · Grounded_Abstraction_v1.0.0.tex, line 274
nonadaptive set J suffices exactly when it covers D. item Some pair in D has identical reports under every admitted test. No finite adaptive protocol using this unchanged test interface and r guarantees t in every world. If test j has nonnegative fixed cost c_j, the minimum cost of one fixed nonadaptive test set, used for all initial codes in case 2, is precisely min_J: J covers D ∑_j∈ Jc_j.tag2 This is not a claim that an optimal adaptive protocol has the same cost. If D is empty, the target is…
Exact mathematical passage
\min_{J:\,J\text{ covers }D}\ \sum_{j\in J}c_j.\tag{2}
\min_{J:\,J\text{ covers }D}Use in source · 1 passage
Search aliases: min_J: J covers D min text
Use in source · Grounded_Abstraction_v1.0.0.tex, line 274
nonadaptive set J suffices exactly when it covers D. item Some pair in D has identical reports under every admitted test. No finite adaptive protocol using this unchanged test interface and r guarantees t in every world. If test j has nonnegative fixed cost c_j, the minimum cost of one fixed nonadaptive test set, used for all initial codes in case 2, is precisely min_J: J covers D ∑_j∈ Jc_j.tag2 This is not a claim that an optimal adaptive protocol has the same cost. If D is empty, the target is…
Exact mathematical passage
\min_{J:\,J\text{ covers }D}\ \sum_{j\in J}c_j.\tag{2}
\min_{\omega\in B}Use in source · 1 passage
Search aliases: min_ω∈ B min omega in
Use in source · Grounded_Abstraction_v1.0.0.tex, line 436
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
\ell_j(B)=\min_{\omega\in B}v_j(\omega),\qquad u_j(B)=\max_{\omega\in B}v_j(\omega),
\mathrm{model}Use in source · 3 passages
Search aliases: model mathrm
Use in source · Grounded_Abstraction_v1.0.0.tex, line 453
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\delta_{\mathrm{id}},\delta_{\mathrm{model}}\in[0,1]
N_0Use in source · 9 passages
Search aliases: N_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 794
eqker Q, invariant under every A_a, containing every im D_a^θ. item There is a surjective linear representation r:V→ Z, shared linear updates G_a, and a readout bar Q such that rT_a^θ=G_ar for all a,θ, Q=bar Qr. When the conditions hold, V/E_∞ gives one such quotient. The smallest-dimensional common linear quotient is V/N_0, where N_0=⋂_wker(QA_w) is the reference system's unobservable subspace. Its dimension is n-dim N_0. The shared quotient also preserves outputs if the candidate model is chan…
Exact mathematical passage
V/N_0
OUse in source · 2 passages
Search aliases: O
Use in source · Grounded_Abstraction_v1.0.0.tex, line 342
An exact count when evidence identifies a partition cell. Suppose A_1,…,A_m are a partition of Ω into nonempty cells, and E consists exactly of all unions of those cells. Define the attainable evidence--target pairs D_t={(i,y): y∈ t(A_i)}, r=|D_t|=∑_i|t(A_i)|. A candidate set induces an occupancy record O(V)={(i,t(ω)):ω∈ V∩ A_i}.
Exact mathematical passage
O(V)=\{(i,t(\omega)):\omega\in V\cap A_i\}.
o_1Use in source · 1 passage
Search aliases: o_1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 531
For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. The empty word gives the current target distribution.
Exact mathematical passage
w=((e_1,o_1),\ldots,(e_k,o_k))
o_jUse in source · 1 passage
Search aliases: o_j
Use in source · Grounded_Abstraction_v1.0.0.tex, line 532
For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. The empty word gives the current target distribution.
Exact mathematical passage
M_{\mu,y}(w)=\sum_{t(\omega)=y}\mu(\omega) \prod_{j=1}^k\ell_{e_j,o_j}(\omega).
o_kUse in source · 1 passage
Search aliases: o_k
Use in source · Grounded_Abstraction_v1.0.0.tex, line 531
For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. The empty word gives the current target distribution.
Exact mathematical passage
w=((e_1,o_1),\ldots,(e_k,o_k))
\mathrm{old}Use in source · 3 passages
Search aliases: old mathrm
Use in source · Grounded_Abstraction_v1.0.0.tex, line 161
A state-changing measurement. A different operation Z sets s to zero and then reports the new display. The report is θ. If the old display y_old was retained, then s_old=y_old⊕ y_after Z. The current physical state is zero, not s_old. A correct account distinguishes the recovered historical value from the new state it will act on.
Exact mathematical passage
y_{\mathrm{old}}
PUse in source · 7 passages
Search aliases: P
Use in source · Grounded_Abstraction_v1.0.0.tex, line 237
immutable worlds, one with a permanently zero output and one with a permanently one output. The available initial record is the same symbol in both. Within each world separately, a one-state predictor exists: it simply prints that world's output. There is nevertheless no predictor that can be initialized from the common record and be correct in both. The obstruction is the order of quantifiers, ∀ω ∃ P_ω versus ∃ P ∀ω. The first does not provide a selector for the second. The behavioral quotient …
Exact mathematical passage
\forall\omega\ \exists P_\omega \qquad\text{versus}\qquad \exists P\ \forall\omega.
\mathcal PUse in source · 6 passages
Search aliases: P mathcal
Use in source · Grounded_Abstraction_v1.0.0.tex, line 570
What a revised evidence language requires. Let P_0 be the partition of worlds into the nonempty old profile--target cells. Let P_1 be the corresponding partition after likelihoods or the target have changed, still on the same finite Ω. The old record stores the probability of each P_0 cell. The new record would store the probability of each P_1 cell.
Exact mathematical passage
\mathcal P_0
p'_{i,y}Use in source · 1 passage
Search aliases: p'_i,y
Use in source · Grounded_Abstraction_v1.0.0.tex, line 520
[Exact target correction from likelihood-profile memory] Under the fixed-world likelihood model, the record p=(p_i,y) supports every finite sequence of report predictions and target-posterior corrections. Its next-report probability and update are z_e,o=∑_i,yλ_e,o,ip_i,y, p'_i,y=fracλ_e,o,ip_i,yz_e,o (z_e,o>0), and its target readout is Pr(t=y)=∑_i p_i,y. A zero-probability report is marked impossible rather than assigned a normalized posterior.
Exact mathematical passage
z_{e,o}=\sum_{i,y}\lambda_{e,o,i}p_{i,y},\qquad p'_{i,y}=\frac{\lambda_{e,o,i}p_{i,y}}{z_{e,o}}\quad(z_{e,o}>0),
P_0Use in source · 3 passages
Search aliases: P_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 390
For a fixed protocol, let two worlds with different targets generate finite transcript laws P_0 and P_1. An estimator sees only that transcript and independent randomness. Its average success under equal prior probabilities is bounded by the discrimination in the entire record.
Exact mathematical passage
P_0
\mathcal P_0Use in source · 3 passages
Search aliases: P_0 mathcal
Use in source · Grounded_Abstraction_v1.0.0.tex, line 570
What a revised evidence language requires. Let P_0 be the partition of worlds into the nonempty old profile--target cells. Let P_1 be the corresponding partition after likelihoods or the target have changed, still on the same finite Ω. The old record stores the probability of each P_0 cell. The new record would store the probability of each P_1 cell.
Exact mathematical passage
\mathcal P_0
p_0Use in source · 3 passages
Search aliases: p_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 933
After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment selection may depend on earlier retained records and independent randomness. It may not introduce a hidden world-dependent selector whose information is …
Exact mathematical passage
\widehat p_0=\mathcal Q_N(p_0)
P_1Use in source · 3 passages
Search aliases: P_1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 390
For a fixed protocol, let two worlds with different targets generate finite transcript laws P_0 and P_1. An estimator sees only that transcript and independent randomness. Its average success under equal prior probabilities is bounded by the discrimination in the entire record.
Exact mathematical passage
P_1
\mathcal P_1Use in source · 3 passages
Search aliases: P_1 mathcal
Use in source · Grounded_Abstraction_v1.0.0.tex, line 570
What a revised evidence language requires. Let P_0 be the partition of worlds into the nonempty old profile--target cells. Let P_1 be the corresponding partition after likelihoods or the target have changed, still on the same finite Ω. The old record stores the probability of each P_0 cell. The new record would store the probability of each P_1 cell.
Exact mathematical passage
\mathcal P_1
P_iUse in source · 2 passages
Search aliases: P_i
Use in source · Grounded_Abstraction_v1.0.0.tex, line 539
For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1. Each monomial is the likelihood product of an admitted fixed experiment-report word. Consequently p_i,y(μ)=∑_t(ω)=yμ(ω)P_i(ω) is a linear combination…
Exact mathematical passage
P_i(\omega)=\prod_{j\ne i} \frac{\ell_{b(i,j)}(\omega)-\lambda_{b(i,j),j}} {\lambda_{b(i,j),i}-\lambda_{b(i,j),j}}
p_iUse in source · 6 passages
Search aliases: p_i
Use in source · Grounded_Abstraction_v1.0.0.tex, line 888
Define Q_N(p) by first setting n_i=lfloor Np_irfloor. Distribute the remaining N-∑_i n_i units to the largest fractional remainders, with ties broken by a fixed cell order. This makes the normalization exact after every rounding. It is not independent coordinate rounding followed by an unspecified repair.
Exact mathematical passage
n_i=\lfloor Np_i\rfloor
p_jUse in source · 1 passage
Search aliases: p_j
Use in source · Grounded_Abstraction_v1.0.0.tex, line 891
For a likelihood vector ℓ=(ℓ_i) define the Bayes map B_ℓ(p)_i=fracp_iℓ_i∑_jp_jℓ_j whenever its denominator is positive.
Exact mathematical passage
\mathcal B_\ell(p)_i=\frac{p_i\ell_i}{\sum_jp_j\ell_j}
p_tUse in source · 3 passages
Search aliases: p_t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 933
After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment selection may depend on earlier retained records and independent randomness. It may not introduce a hidden world-dependent selector whose information is …
Exact mathematical passage
p_t
p_{t+1}Use in source · 1 passage
Search aliases: p_t+1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 962
Proposition~[source reference] and the rounding bound give d_TV(p_t+1,widehat p_t+1) ≤γd_TV(p_t,widehat p_t)+γε+η_N. Iterating the uncapped scalar bound proves (11); total variation is also at most one. Aggregating cell coordinates into target coordinates cannot increase total variation, by the triangle inequality. Substituting e_0≤η_N gives η_N A_t+ε B_t. These quantities are nondecreasing in t because γ≥1 and all terms are nonnegative. Inequality (12) makes the horizon-H value at most τ. It th…
Exact mathematical passage
\TV(p_{t+1},\widehat p_{t+1}) \le\gamma\TV(p_t,\widehat p_t)+\gamma\varepsilon+\eta_N.
P_\omegaUse in source · 1 passage
Search aliases: P_ω omega
Use in source · Grounded_Abstraction_v1.0.0.tex, line 237
immutable worlds, one with a permanently zero output and one with a permanently one output. The available initial record is the same symbol in both. Within each world separately, a one-state predictor exists: it simply prints that world's output. There is nevertheless no predictor that can be initialized from the common record and be correct in both. The obstruction is the order of quantifiers, ∀ω ∃ P_ω versus ∃ P ∀ω. The first does not provide a selector for the second. The behavioral quotient …
Exact mathematical passage
\forall\omega\ \exists P_\omega \qquad\text{versus}\qquad \exists P\ \forall\omega.
\PowUse in source · 1 passage
Search aliases: Pow P mathcal
Use in source · Grounded_Abstraction_v1.0.0.tex, line 308
Let Ω be a finite model set and t:Ω→ Y a fixed target. For a candidate set V⊆Ω, let Q(V)=t(V)={t(ω):ω∈ V}. The empty set is permitted. It represents a contradiction between the admitted model family and the evidence, not a successful identification. Let E⊆ P(Ω) be a finite family of evidence events containing Ω and closed under intersections. Filtering by E sends V to V∩ E. The events are known, truthfully interpreted, and do not change the underlying model in this section.
Exact mathematical passage
\E\subseteq\Pow(\Omega)
\PrUse in source · 5 passages
Search aliases: Pr
Use in source · Grounded_Abstraction_v1.0.0.tex, line 454
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\Pr\{\omega_*\in B(H)\}\ge1-\delta_{\mathrm{id}}.
\Pr_{\mu_+}Use in source · 1 passage
Search aliases: Pr_μ_+ Pr mu
Use in source · Grounded_Abstraction_v1.0.0.tex, line 558
Consider a prior μ_+ assigning mass 1/2 to (s,θ)=(0,0),(1,1) and a prior μ_- assigning mass 1/2 to (1,0),(0,1), with any fixed nuisance convention. Both have uniform s and uniform θ. Both predict the next report 1 with probability 1/2. After that report, however, Pr_μ_+(s=1mid o=1)=tfrac34, Pr_μ_-(s=1mid o=1)=tfrac14. After k fresh reports all equal to one, these become 3^k/(1+3^k) and 1/(1+3^k). The difference is not a numerical error. It is the association that the separate marginals failed to…
Exact mathematical passage
\Pr_{\mu_+}(s=1\mid o=1)=\tfrac34,\qquad \Pr_{\mu_-}(s=1\mid o=1)=\tfrac14.
\Pr_{\mu_-}Use in source · 1 passage
Search aliases: Pr_μ_- Pr mu
Use in source · Grounded_Abstraction_v1.0.0.tex, line 558
Consider a prior μ_+ assigning mass 1/2 to (s,θ)=(0,0),(1,1) and a prior μ_- assigning mass 1/2 to (1,0),(0,1), with any fixed nuisance convention. Both have uniform s and uniform θ. Both predict the next report 1 with probability 1/2. After that report, however, Pr_μ_+(s=1mid o=1)=tfrac34, Pr_μ_-(s=1mid o=1)=tfrac14. After k fresh reports all equal to one, these become 3^k/(1+3^k) and 1/(1+3^k). The difference is not a numerical error. It is the association that the separate marginals failed to…
Exact mathematical passage
\Pr_{\mu_+}(s=1\mid o=1)=\tfrac34,\qquad \Pr_{\mu_-}(s=1\mid o=1)=\tfrac14.
\mathbb QUse in source · 3 passages
Search aliases: Q mathbb
Use in source · Grounded_Abstraction_v1.0.0.tex, line 750
Why the unknown model must not change between steps. [A switching adversary creates a nonexistent failure] Over Q, let there be one action and two possible fixed matrices A=0 · 1
Exact mathematical passage
\mathbb Q
\mathcal QUse in source · 4 passages
Search aliases: Q mathcal
Use in source · Grounded_Abstraction_v1.0.0.tex, line 888
Define Q_N(p) by first setting n_i=lfloor Np_irfloor. Distribute the remaining N-∑_i n_i units to the largest fractional remainders, with ties broken by a fixed cell order. This makes the normalization exact after every rounding. It is not independent coordinate rounding followed by an unspecified repair.
Exact mathematical passage
\mathcal Q_N(p)
\mathbb Q^2Use in source · 1 passage
Search aliases: Q^2 mathbb
Use in source · Grounded_Abstraction_v1.0.0.tex, line 870
Initial dimension, dynamic memory, and hidden state. In Counterexample~[source reference], set L=span{(1,1)} and R(x)=x_1. The reference reaches (1,1) and (1,0), so W= Q^2. The rows Q=(1,1) and QA=(0,1) have trivial common kernel, giving N=0. A two-dimensional linear predictor is therefore necessary even for this one-dimensional initial family: it must reproduce 2t,t,0,0,… for every multiplier t.
Exact mathematical passage
W=\mathbb Q^2
q_iUse in source · 2 passages
Search aliases: q_i
Use in source · Grounded_Abstraction_v1.0.0.tex, line 912
For the update, first hold ℓ fixed. Given 0≤ f_i≤1, put a=∑_i B_ℓ(q)_if_i and Z_p=∑_ip_iℓ_i. Then ∑_i( B_ℓ(p)_i- B_ℓ(q)_i)f_i =frac∑_i(p_i-q_i)ℓ_i(f_i-a)Z_p. Every value ℓ_i(f_i-a) lies in [-a,1-a], an interval of length one. Since p-q has total mass zero, its pairing with a function of oscillation at most one is bounded in absolute value by d_TV(p,q). Also Z_p≥β. Taking the supremum over f bounds the fixed-likelihood difference by d_TV(p,q)/β.
Exact mathematical passage
\sum_i\bigl(\mathcal B_\ell(p)_i-\mathcal B_\ell(q)_i\bigr)f_i =\frac{\sum_i(p_i-q_i)\ell_i(f_i-a)}{Z_p}.
\mathcal Q_NUse in source · 4 passages
Search aliases: Q_N mathcal
Use in source · Grounded_Abstraction_v1.0.0.tex, line 888
Define Q_N(p) by first setting n_i=lfloor Np_irfloor. Distribute the remaining N-∑_i n_i units to the largest fractional remainders, with ties broken by a fixed cell order. This makes the normalization exact after every rounding. It is not independent coordinate rounding followed by an unspecified repair.
Exact mathematical passage
\mathcal Q_N(p)
RUse in source · 16 passages
Search aliases: R
Use in source · Grounded_Abstraction_v1.0.0.tex, line 257
For J⊆{1,…,m}, write R_J(ω)=(r(ω),(e_j(ω))_j∈ J). Let the unresolved target pairs be D={{ω,ω'}:r(ω)=r(ω'), t(ω)≠ t(ω')}. A test covers such a pair when it gives different reports in the two worlds.
Exact mathematical passage
R_J(\omega)=\bigl(r(\omega),(e_j(\omega))_{j\in J}\bigr).
\mathbb RUse in source · 3 passages
Search aliases: R mathbb
Use in source · Grounded_Abstraction_v1.0.0.tex, line 435
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
v:\Omega\to\mathbb R^d
\mathbb R^dUse in source · 3 passages
Search aliases: R^d mathbb
Use in source · Grounded_Abstraction_v1.0.0.tex, line 435
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
v:\Omega\to\mathbb R^d
R_JUse in source · 1 passage
Search aliases: R_J
Use in source · Grounded_Abstraction_v1.0.0.tex, line 257
For J⊆{1,…,m}, write R_J(ω)=(r(ω),(e_j(ω))_j∈ J). Let the unresolved target pairs be D={{ω,ω'}:r(ω)=r(ω'), t(ω)≠ t(ω')}. A test covers such a pair when it gives different reports in the two worlds.
Exact mathematical passage
R_J(\omega)=\bigl(r(\omega),(e_j(\omega))_{j\in J}\bigr).
\rfloorUse in source · 1 passage
Search aliases: rfloor
Use in source · Grounded_Abstraction_v1.0.0.tex, line 888
Define Q_N(p) by first setting n_i=lfloor Np_irfloor. Distribute the remaining N-∑_i n_i units to the largest fractional remainders, with ties broken by a fixed cell order. This makes the normalization exact after every rounding. It is not independent coordinate rounding followed by an unspecified repair.
Exact mathematical passage
n_i=\lfloor Np_i\rfloor
sUse in source · 29 passages
Search aliases: s
Use in source · Grounded_Abstraction_v1.0.0.tex, line 144
A calibrated bit with an irrelevant hidden register. Consider eight possible worlds ω=(s,θ,k)∈{0,1}^3. The physical bit is s. The sensor convention is θ, and k is a nuisance bit that none of the admitted operations or questions uses. The display is y=s⊕θ. The ordinary action F flips s and leaves the other bits fixed. Its effect on the display is exactly y↦1-y, whatever the unknown convention. Reading the current display therefore initializes a two-state representation that predicts every finite …
Exact mathematical passage
\omega=(s,\theta,k)\in\{0,1\}^3.
S_bUse in source · 1 passage
Search aliases: S_b
Use in source · Grounded_Abstraction_v1.0.0.tex, line 631
ppose v∉ U_n-1. If v∉ U, the one-step proposition supplies two extensions that disagree within one action. Otherwise the recursive definition gives a word of length j≤ n-1 with a first exit from U. Its intermediate vectors before exit lie in U, so its exit vector z∉ U is the same under every compatible extension family. Choose any next action b. Two extensions of S_b can be made to disagree on z by adding a linear map that vanishes on U. Keep the other action maps the same. Even if action b occu…
Exact mathematical passage
S_b
s_{\mathrm{old}}Use in source · 2 passages
Search aliases: s_old mathrm
Use in source · Grounded_Abstraction_v1.0.0.tex, line 162
A state-changing measurement. A different operation Z sets s to zero and then reports the new display. The report is θ. If the old display y_old was retained, then s_old=y_old⊕ y_after Z. The current physical state is zero, not s_old. A correct account distinguishes the recovered historical value from the new state it will act on.
Exact mathematical passage
s_{\mathrm{old}}=y_{\mathrm{old}}\oplus y_{\mathrm{after}\ Z}.
\SpanUse in source · 9 passages
Search aliases: Span span operatorname
Use in source · Grounded_Abstraction_v1.0.0.tex, line 595
Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}. The calibration is consistent precisely when every linear relation among the inputs is respected by the outputs, for each action. Under that conditi…
Exact mathematical passage
U=\Span\{x_1,\ldots,x_m\}.
\supUse in source · 2 passages
Search aliases: sup
Use in source · Grounded_Abstraction_v1.0.0.tex, line 458
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\sup_{\omega\in B(H)}\|v(\omega)-\widehat v_H(\omega)\|_\infty\le\eta.
\sup_{\omega\in B(H)}Use in source · 1 passage
Search aliases: sup_ω∈ B(H) sup omega in
Use in source · Grounded_Abstraction_v1.0.0.tex, line 458
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\sup_{\omega\in B(H)}\|v(\omega)-\widehat v_H(\omega)\|_\infty\le\eta.
\sup_{\omega\in C_i}Use in source · 1 passage
Search aliases: sup_ω∈ C_i sup omega in
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1075
Fix an actual evidence history. Let μ be the exact posterior on the stipulated finite world set and let p_i=μ(C_i). Suppose the retained distribution is widehat p with d_TV(p,widehat p)≤ e. For a declared later action word or response horizon, let y(ω) be the actual model response and let bar y_i be the response used for cell i. Assume sup_ω∈ C_i y(ω)-bar y_i ≤ d for every cell i. tag16 The cell need not identify the physical state or candidate operator. It must support this common response allo…
Exact mathematical passage
\sup_{\omega\in C_i}\|y(\omega)-\bar y_i\|\le d \quad\text{for every cell }i. \tag{16}
T_0Use in source · 2 passages
Search aliases: T_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 644
one-dimensional reusable core [Calibration does not close under continuation] Let V have basis e_1,e_2,e_3, let U=span(e_1,e_2), and let the single calibrated operation satisfy Se_1=e_1, Se_2=e_3. The first continuation is known on all of U. The next calibrated domain is U_1=span(e_1), which is already invariant. One completion has T_0e_3=0 and another has T_1e_3=e_1. Both agree on every supplied calibration. Starting from e_2, their first outputs agree at e_3, but their second outputs are respe…
Exact mathematical passage
T_0e_3=0
T_1Use in source · 3 passages
Search aliases: T_1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 644
ore [Calibration does not close under continuation] Let V have basis e_1,e_2,e_3, let U=span(e_1,e_2), and let the single calibrated operation satisfy Se_1=e_1, Se_2=e_3. The first continuation is known on all of U. The next calibrated domain is U_1=span(e_1), which is already invariant. One completion has T_0e_3=0 and another has T_1e_3=e_1. Both agree on every supplied calibration. Starting from e_2, their first outputs agree at e_3, but their second outputs are respectively zero and e_1.
Exact mathematical passage
T_1e_3=e_1
T_aUse in source · 9 passages
Search aliases: T_a
Use in source · Grounded_Abstraction_v1.0.0.tex, line 594
Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}. The calibration is consistent precisely when every linear relation among the inputs is respected by the outputs, for each action. Under that conditi…
Exact mathematical passage
T_a:V\to V
T_a'Use in source · 2 passages
Search aliases: T_a'
Use in source · Grounded_Abstraction_v1.0.0.tex, line 606
For a fixed action a, the value T_av is the same for every linear extension of S_a if and only if v∈ U. The common value is then S_av. If v∈ U, all extensions agree by definition. If v∉ U, choose a linear functional ℓ that vanishes on U and satisfies ℓ(v)=1, and choose a nonzero vector b∈ V. Given any extension T_a, the map T_a'(x)=T_a(x)+ℓ(x)b is another extension, but T_a'v=T_av+b≠ T_av. Such a functional exists by extending a basis of U with v.
Exact mathematical passage
T_a'(x)=T_a(x)+\ell(x)b
T_a^0Use in source · 1 passage
Search aliases: T_a^0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 714
A finite test for agreement under a fixed unknown model. Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y. Define I_Q={v∈ V:QT_w^θ v=QA_wv for every θ∈Θ and every word w}. This is an initial-condition subspace. It is not assumed invariant under the physica…
Exact mathematical passage
A_a=T_a^0
T_a^\thetaUse in source · 8 passages
Search aliases: T_a^θ theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 714
A finite test for agreement under a fixed unknown model. Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y. Define I_Q={v∈ V:QT_w^θ v=QA_wv for every θ∈Θ and every word w}. This is an initial-condition subspace. It is not assumed invariant under the physica…
Exact mathematical passage
T_a^\theta:V\to V
t_iUse in source · 1 passage
Search aliases: t_i
Use in source · Grounded_Abstraction_v1.0.0.tex, line 924
Approximate likelihood profiles. Let Ω be a finite static world set with prior μ_0. A fixed partition C_1,…,C_k refines the target: t(ω)=t_i throughout C_i. Each admitted experiment e is nondestructive and has a finite report alphabet. Its actual conditional probabilities are ℓ_e,o(ω). Reports are freshly generated from those likelihoods conditional on the fixed world and the chosen experiments.
Exact mathematical passage
t(\omega)=t_i
T_nUse in source · 1 passage
Search aliases: T_n
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1128
The later response is produced by three applications of one fixed candidate matrix, T_n=1/2 · (-1)^n/64 1/8 · 1/2, (v_0,h_0)=(s,1). tag17 The predictor for cell (s,θ) uses bar v_t=2^-ts and therefore reports bar y_i=s/8. The nuisance bit n affects the physical response but remains unidentified by the reference observations. With g=d=1/2, b=1/64, c=1/8, and V_0=H_0=1, the comparison determinant is 127/512>0. Directly evaluating (14) at three prediction steps yields d_3=frac48132 768. Choose betwe…
Exact mathematical passage
T_n=\begin{pmatrix}1/2&(-1)^n/64\\1/8&1/2\end{pmatrix}, \qquad (v_0,h_0)=(s,1). \tag{17}
T_w^\thetaUse in source · 5 passages
Search aliases: T_w^θ theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 715
ent under a fixed unknown model Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y. Define I_Q={v∈ V:QT_w^θ v=QA_wv for every θ∈Θ and every word w}. This is an initial-condition subspace. It is not assumed invariant under the physical maps. The common coordi…
Exact mathematical passage
I_Q=\{v\in V:QT_w^\theta v=QA_wv\text{ for every }\theta\in\Theta\text{ and every word }w\}.
\TVUse in source · 9 passages
Search aliases: TV d_TV mathrm
Use in source · Grounded_Abstraction_v1.0.0.tex, line 394
[A finite-record discrimination bound] For two distinct target values and equal world probabilities, every transcript-based estimator has average success at most frac1+d_TV(P_0,P_1)2, d_TV(P_0,P_1)=frac12∑_h|P_0(h)-P_1(h)|.tag4 A deterministic function or a common stochastic processing channel applied to the transcript cannot increase this upper bound. Convert the estimator into a binary test by reporting world zero when it outputs the first target and world one otherwise. A correct target outpu…
Exact mathematical passage
\frac{1+\TV(P_0,P_1)}{2},\qquad \TV(P_0,P_1)=\frac12\sum_h|P_0(h)-P_1(h)|.\tag{4}
uUse in source · 10 passages
Search aliases: u
Use in source · Grounded_Abstraction_v1.0.0.tex, line 436
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
\ell_j(B)=\min_{\omega\in B}v_j(\omega),\qquad u_j(B)=\max_{\omega\in B}v_j(\omega),
u_0Use in source · 4 passages
Search aliases: u_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1005
observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g, B_t ≤ b, C_t ≤ c, H_t ≤ d, g,b,c,d≥0. Let v_0 ≤ V_0, h_0 ≤ H_0, and v_0-bar v_0 ≤ e_0. These initial bounds are supplied evidence or declared model assumptions. Put M=g · b c · d, u_0=V_0 H_0, u_j=M^ju_0. Powers of this nonnegative comparison matrix are bounds on norms. They are not a replacement physical dynamics inferred from measurements.
Exact mathematical passage
M=\begin{pmatrix}g&b\\c&d\end{pmatrix},\qquad u_0=\begin{pmatrix}V_0\\H_0\end{pmatrix},\qquad u_j=M^ju_0.
U_1Use in source · 1 passage
Search aliases: U_1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 644
A two-dimensional calibration with a one-dimensional reusable core. [Calibration does not close under continuation] Let V have basis e_1,e_2,e_3, let U=span(e_1,e_2), and let the single calibrated operation satisfy Se_1=e_1, Se_2=e_3. The first continuation is known on all of U. The next calibrated domain is U_1=span(e_1), which is already invariant. One completion has T_0e_3=0 and another has T_1e_3=e_1. Both agree on every supplied calibration. Starting from e_2, their first outputs agree at e…
Exact mathematical passage
U_1=\Span(e_1)
U_{h-1}Use in source · 2 passages
Search aliases: U_h-1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 703
[No enlargement under unrestricted linear completion] If Q≠0, then for every h≥1, the vectors QT_wv are independent of the compatible extension family for every word of length at most h if and only if v∈ U_h-1. In particular, the all-word output-identifiable domain is exactly U_∞. If Q=0, every initial vector is trivially output-identifiable. On U_h-1 the full vectors are determined by Theorem~[source reference], so their Q outputs are determined. Outside that subspace, follow the same first-exi…
Exact mathematical passage
v\in U_{h-1}
u_jUse in source · 5 passages
Search aliases: u_j
Use in source · Grounded_Abstraction_v1.0.0.tex, line 436
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
\ell_j(B)=\min_{\omega\in B}v_j(\omega),\qquad u_j(B)=\max_{\omega\in B}v_j(\omega),
U_{n-1}Use in source · 3 passages
Search aliases: U_n-1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 623
[The maximal calibrated continuation domain] For every integer n≥1 and every v∈ V, the values T_wv for all operation words of length at most n are independent of the chosen compatible extension family if and only if v∈ U_n-1. Consequently, every finite full-state continuation is identifiable exactly on U_∞. This is the largest subspace L⊆ U such that S_aL⊆ L for every a. On this subspace the known restrictions themselves realize the complete operation family. Membership in U_n-1 means that every…
Exact mathematical passage
v\in U_{n-1}.
V_0Use in source · 7 passages
Search aliases: V_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 362
Association can be the missing state. [Equal marginals with different corrected answers] In the sensor model, let V_0={(s,θ,k):s⊕θ=0}, V_1={(s,θ,k):s⊕θ=1}. Take the target to be s and the evidence cells to be the two values of θ. Both candidate sets currently allow physical bits {0,1}, and both allow conventions {0,1}. Yet after the reference reports θ=0, V_0 requires s=0 and V_1 requires s=1.
Exact mathematical passage
V_0=\{(s,\theta,k):s\oplus\theta=0\},\qquad V_1=\{(s,\theta,k):s\oplus\theta=1\}.
v_0Use in source · 4 passages
Search aliases: v_0
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1004
ix, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g, B_t ≤ b, C_t ≤ c, H_t ≤ d, g,b,c,d≥0. Let v_0 ≤ V_0, h_0 ≤ H_0, and v_0-bar v_0 ≤ e_0. These initial bounds are supplied evidence or declared model assumptions. Put M=g · b
Exact mathematical passage
\|v_0\|\le V_0
V_1Use in source · 2 passages
Search aliases: V_1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 362
Association can be the missing state. [Equal marginals with different corrected answers] In the sensor model, let V_0={(s,θ,k):s⊕θ=0}, V_1={(s,θ,k):s⊕θ=1}. Take the target to be s and the evidence cells to be the two values of θ. Both candidate sets currently allow physical bits {0,1}, and both allow conventions {0,1}. Yet after the reference reports θ=0, V_0 requires s=0 and V_1 requires s=1.
Exact mathematical passage
V_0=\{(s,\theta,k):s\oplus\theta=0\},\qquad V_1=\{(s,\theta,k):s\oplus\theta=1\}.
v_jUse in source · 1 passage
Search aliases: v_j
Use in source · Grounded_Abstraction_v1.0.0.tex, line 436
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
\ell_j(B)=\min_{\omega\in B}v_j(\omega),\qquad u_j(B)=\max_{\omega\in B}v_j(\omega),
v_tUse in source · 8 passages
Search aliases: v_t
Use in source · Grounded_Abstraction_v1.0.0.tex, line 991
Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1
Exact mathematical passage
(v_t,h_t)
v_{t+1}Use in source · 3 passages
Search aliases: v_t+1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 992
Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g,…
Exact mathematical passage
\begin{pmatrix}v_{t+1}\\h_{t+1}\end{pmatrix} =\begin{pmatrix}G_t&B_t\\C_t&H_t\end{pmatrix} \begin{pmatrix}v_t\\h_t\end{pmatrix}, \qquad \bar v_{t+1}=G_t\bar v_t. \tag{13}
WUse in source · 13 passages
Search aliases: W
Use in source · Grounded_Abstraction_v1.0.0.tex, line 324
Conversely, any retained representation with deterministic filter updates and a readout reproducing Q after every such sequence must distinguish every two candidate sets with different signatures. Thus the image of Σ_ E,t is the smallest exact state space, in cardinality, for this correction contract on all candidate subsets. If Σ(V)=Σ(W), then for every E∈ E, Q((V∩ E_0)∩ E)=Q(V∩(E_0∩ E)) =Q(W∩(E_0∩ E)). Intersection closure puts E_0∩ E in E, so the equality is available from the original signat…
Exact mathematical passage
\Sigma(V)=\Sigma(W)
\widehat aUse in source · 3 passages
Search aliases: widehat a widehat
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1093
[A common quantitative grounding certificate] Under (16), put ξ=min{1,e+Kd}. Then |R(a)-widehat R(a)|≤ξ for every a∈ D. If widehat a minimizes widehat R, its true posterior regret is at most 2ξ. If it is the unique approximate minimizer and widehat R(a)-widehat R(widehat a)>2ξ for every a≠widehat a, then it is the unique true minimizer. A decision issued only when widehat R(a)+ξ≤τ has true posterior risk at most the declared threshold τ.
Exact mathematical passage
\widehat a
\widehat mUse in source · 3 passages
Search aliases: widehat m widehat
Use in source · Grounded_Abstraction_v1.0.0.tex, line 461
model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of probability at least 1-δ_id-δ_model. The rule …
Exact mathematical passage
\widehat m
\widehat p_0Use in source · 2 passages
Search aliases: widehat p_0 widehat
Use in source · Grounded_Abstraction_v1.0.0.tex, line 933
After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment selection may depend on earlier retained records and independent randomness. It may not introduce a hidden world-dependent selector whose information is …
Exact mathematical passage
\widehat p_0=\mathcal Q_N(p_0)
\widehat p_iUse in source · 1 passage
Search aliases: widehat p_i widehat
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1082
Let the permitted decision set D be finite and nonempty. Its loss L(y,a) takes values in [0,1] and is K-Lipschitz in y, uniformly over a. Define R(a)=∑_ωμ(ω)L(y(ω),a), widehat R(a)=∑_iwidehat p_iL(bar y_i,a). These are risks under the stated model posterior. They are not frequentist coverage probabilities or guarantees on the loss of a single realized world.
Exact mathematical passage
R(a)=\sum_{\omega}\mu(\omega)L(y(\omega),a),\qquad \widehat R(a)=\sum_i\widehat p_iL(\bar y_i,a).
\widehat p_tUse in source · 3 passages
Search aliases: widehat p_t widehat
Use in source · Grounded_Abstraction_v1.0.0.tex, line 934
After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment selection may depend on earlier retained records and independent randomness. It may not introduce a hidden world-dependent selector whose information is …
Exact mathematical passage
\widehat p_{t+1}=\mathcal Q_N\!\left(\mathcal B_{\bar\ell_{e,o}}(\widehat p_t)\right).
\widehat p_{t+1}Use in source · 2 passages
Search aliases: widehat p_t+1 widehat
Use in source · Grounded_Abstraction_v1.0.0.tex, line 934
After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment selection may depend on earlier retained records and independent randomness. It may not introduce a hidden world-dependent selector whose information is …
Exact mathematical passage
\widehat p_{t+1}=\mathcal Q_N\!\left(\mathcal B_{\bar\ell_{e,o}}(\widehat p_t)\right).
\widehat RUse in source · 8 passages
Search aliases: widehat R widehat
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1082
Let the permitted decision set D be finite and nonempty. Its loss L(y,a) takes values in [0,1] and is K-Lipschitz in y, uniformly over a. Define R(a)=∑_ωμ(ω)L(y(ω),a), widehat R(a)=∑_iwidehat p_iL(bar y_i,a). These are risks under the stated model posterior. They are not frequentist coverage probabilities or guarantees on the loss of a single realized world.
Exact mathematical passage
R(a)=\sum_{\omega}\mu(\omega)L(y(\omega),a),\qquad \widehat R(a)=\sum_i\widehat p_iL(\bar y_i,a).
\widehat vUse in source · 3 passages
Search aliases: widehat v widehat
Use in source · Grounded_Abstraction_v1.0.0.tex, line 458
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\sup_{\omega\in B(H)}\|v(\omega)-\widehat v_H(\omega)\|_\infty\le\eta.
\widehat v_HUse in source · 3 passages
Search aliases: widehat v_H widehat
Use in source · Grounded_Abstraction_v1.0.0.tex, line 458
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\sup_{\omega\in B(H)}\|v(\omega)-\widehat v_H(\omega)\|_\infty\le\eta.
\widehat\rhoUse in source · 3 passages
Search aliases: widehatρ widehat rho
Use in source · Grounded_Abstraction_v1.0.0.tex, line 461
se a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of probability at least 1-δ_id-δ_model. The rule abstains on an emp…
Exact mathematical passage
\widehat\rho
\widetilde\ellUse in source · 5 passages
Search aliases: widetildeℓ widetilde ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 901
[Rounding and positive-likelihood update bounds] For every p∈Δ_k, d_TV(p, Q_N(p))≤frack2N. Let p,q∈Δ_k, let 0<β≤1, and let every entry of ℓ and widetildeℓ lie in [β,1]. If max_i|ℓ_i-widetildeℓ_i|≤ε, then d_TV( B_ℓ(p), B_widetildeℓ(q)) ≤ fracd_TV(p,q)+εβ. Each right-hand side may also be replaced by its minimum with one. The probability-field storage bound above applies to the retained rounded record, not to all working memory. The remainder rule produces nonnegative integers summing to N. Each r…
Exact mathematical passage
\widetilde\ell
\widetilde\ell_iUse in source · 2 passages
Search aliases: widetildeℓ_i widetilde ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 901
[Rounding and positive-likelihood update bounds] For every p∈Δ_k, d_TV(p, Q_N(p))≤frack2N. Let p,q∈Δ_k, let 0<β≤1, and let every entry of ℓ and widetildeℓ lie in [β,1]. If max_i|ℓ_i-widetildeℓ_i|≤ε, then d_TV( B_ℓ(p), B_widetildeℓ(q)) ≤ fracd_TV(p,q)+εβ. Each right-hand side may also be replaced by its minimum with one. The probability-field storage bound above applies to the retained rounded record, not to all working memory. The remainder rule produces nonnegative integers summing to N. Each r…
Exact mathematical passage
\max_i|\ell_i-\widetilde\ell_i|\le\varepsilon
x'Use in source · 4 passages
Search aliases: x'
Use in source · Grounded_Abstraction_v1.0.0.tex, line 198
For a word w=a_1⋯ a_m, F_w applies a_1 first and a_m last. The empty word is the identity. Define x∼_ C x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite w. This is a comparison in a common observation language. It does not require that different models share arbitrary latent coordinates. The behavioral class is a canonical response profile even when its implementation is not identifiable.
Exact mathematical passage
x\sim_\C x'\quad\Longleftrightarrow\quad q(F_wx)=q(F_wx')\quad\text{for every finite }w.
x_1Use in source · 9 passages
Search aliases: x_1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 594
Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}. The calibration is consistent precisely when every linear relation among the inputs is respected by the outputs, for each action. Under that conditi…
Exact mathematical passage
x_1,\ldots,x_m
x_2Use in source · 5 passages
Search aliases: x_2
Use in source · Grounded_Abstraction_v1.0.0.tex, line 816
2 · -1 · 3, Q=1 · 0 · 0. The hidden third-coordinate dynamics are different, and T_1 can make that coordinate grow. Nevertheless the discrepancy range is span{e_3}, which the reference leaves inside itself and Q annihilates. The common quotient retains (x_1,x_2), with update G(x_1,x_2)=(x_1+x_2,x_2). The output at step k is x_1+kx_2 under both models, and even under switching between them. The output alone is not an initialized reusable state: equal x_1 with different x_2 gives different next ou…
Exact mathematical passage
(x_1,x_2)
x_3Use in source · 1 passage
Search aliases: x_3
Use in source · Grounded_Abstraction_v1.0.0.tex, line 820
ancy range is span{e_3}, which the reference leaves inside itself and Q annihilates. The common quotient retains (x_1,x_2), with update G(x_1,x_2)=(x_1+x_2,x_2). The output at step k is x_1+kx_2 under both models, and even under switching between them. The output alone is not an initialized reusable state: equal x_1 with different x_2 gives different next outputs. Ignoring x_3 is justified by the no-return structure, not by its small size or its being temporarily unobserved.
Exact mathematical passage
x_3
x_iUse in source · 1 passage
Search aliases: x_i
Use in source · Grounded_Abstraction_v1.0.0.tex, line 598
supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}. The calibration is consistent precisely when every linear relation among the inputs is respected by the outputs, for each action. Under that condition it defines maps S_a:U→ V with S_ax_i=y_a,i. Every S_a has linear extensions to V by extending a basis. Unless stated otherwise, all such extensions are admissible. There …
Exact mathematical passage
S_ax_i=y_{a,i}
x_mUse in source · 2 passages
Search aliases: x_m
Use in source · Grounded_Abstraction_v1.0.0.tex, line 594
Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}. The calibration is consistent precisely when every linear relation among the inputs is respected by the outputs, for each action. Under that conditi…
Exact mathematical passage
x_1,\ldots,x_m
y_{a,1}Use in source · 1 passage
Search aliases: y_a,1
Use in source · Grounded_Abstraction_v1.0.0.tex, line 594
Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}. The calibration is consistent precisely when every linear relation among the inputs is respected by the outputs, for each action. Under that conditi…
Exact mathematical passage
y_{a,1},\ldots,y_{a,m}
y_{a,i}Use in source · 1 passage
Search aliases: y_a,i
Use in source · Grounded_Abstraction_v1.0.0.tex, line 598
supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}. The calibration is consistent precisely when every linear relation among the inputs is respected by the outputs, for each action. Under that condition it defines maps S_a:U→ V with S_ax_i=y_a,i. Every S_a has linear extensions to V by extending a basis. Unless stated otherwise, all such extensions are admissible. There …
Exact mathematical passage
S_ax_i=y_{a,i}
y_{a,m}Use in source · 1 passage
Search aliases: y_a,m
Use in source · Grounded_Abstraction_v1.0.0.tex, line 594
Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}. The calibration is consistent precisely when every linear relation among the inputs is respected by the outputs, for each action. Under that conditi…
Exact mathematical passage
y_{a,1},\ldots,y_{a,m}
y_{\mathrm{after}\ Z}Use in source · 1 passage
Search aliases: y_after Z mathrm
Use in source · Grounded_Abstraction_v1.0.0.tex, line 162
A state-changing measurement. A different operation Z sets s to zero and then reports the new display. The report is θ. If the old display y_old was retained, then s_old=y_old⊕ y_after Z. The current physical state is zero, not s_old. A correct account distinguishes the recovered historical value from the new state it will act on.
Exact mathematical passage
s_{\mathrm{old}}=y_{\mathrm{old}}\oplus y_{\mathrm{after}\ Z}.
y_{\mathrm{old}}Use in source · 2 passages
Search aliases: y_old mathrm
Use in source · Grounded_Abstraction_v1.0.0.tex, line 161
A state-changing measurement. A different operation Z sets s to zero and then reports the new display. The report is θ. If the old display y_old was retained, then s_old=y_old⊕ y_after Z. The current physical state is zero, not s_old. A correct account distinguishes the recovered historical value from the new state it will act on.
Exact mathematical passage
y_{\mathrm{old}}
z_{e,o}Use in source · 1 passage
Search aliases: z_e,o
Use in source · Grounded_Abstraction_v1.0.0.tex, line 520
[Exact target correction from likelihood-profile memory] Under the fixed-world likelihood model, the record p=(p_i,y) supports every finite sequence of report predictions and target-posterior corrections. Its next-report probability and update are z_e,o=∑_i,yλ_e,o,ip_i,y, p'_i,y=fracλ_e,o,ip_i,yz_e,o (z_e,o>0), and its target readout is Pr(t=y)=∑_i p_i,y. A zero-probability report is marked impossible rather than assigned a normalized posterior.
Exact mathematical passage
z_{e,o}=\sum_{i,y}\lambda_{e,o,i}p_{i,y},\qquad p'_{i,y}=\frac{\lambda_{e,o,i}p_{i,y}}{z_{e,o}}\quad(z_{e,o}>0),
Z_pUse in source · 3 passages
Search aliases: Z_p
Use in source · Grounded_Abstraction_v1.0.0.tex, line 911
For the update, first hold ℓ fixed. Given 0≤ f_i≤1, put a=∑_i B_ℓ(q)_if_i and Z_p=∑_ip_iℓ_i. Then ∑_i( B_ℓ(p)_i- B_ℓ(q)_i)f_i =frac∑_i(p_i-q_i)ℓ_i(f_i-a)Z_p. Every value ℓ_i(f_i-a) lies in [-a,1-a], an interval of length one. Since p-q has total mass zero, its pairing with a function of oscillation at most one is bounded in absolute value by d_TV(p,q). Also Z_p≥β. Taking the supremum over f bounds the fixed-likelihood difference by d_TV(p,q)/β.
Exact mathematical passage
Z_p=\sum_ip_i\ell_i
\cdotUse in source · 1 passage
Search aliases: · cdot
Use in source · Grounded_Abstraction_v1.0.0.tex, line 706
able. On U_h-1 the full vectors are determined by Theorem~[source reference], so their Q outputs are determined. Outside that subspace, follow the same first-exit construction as in that proof. At an uncalibrated input z, choose a linear functional ℓ vanishing on U with ℓ(z)=1. Because Q≠0, choose b with Qb≠0. Replacing the next action by T_a+ℓ(·)b preserves all calibration data and every earlier step of the first-exit path, but changes the next observed output by Qb. This produces disagreement …
Exact mathematical passage
T_a+\ell(\cdot)b
\betaUse in source · 10 passages
Search aliases: β beta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 901
[Rounding and positive-likelihood update bounds] For every p∈Δ_k, d_TV(p, Q_N(p))≤frack2N. Let p,q∈Δ_k, let 0<β≤1, and let every entry of ℓ and widetildeℓ lie in [β,1]. If max_i|ℓ_i-widetildeℓ_i|≤ε, then d_TV( B_ℓ(p), B_widetildeℓ(q)) ≤ fracd_TV(p,q)+εβ. Each right-hand side may also be replaced by its minimum with one. The probability-field storage bound above applies to the retained rounded record, not to all working memory. The remainder rule produces nonnegative integers summing to N. Each r…
Exact mathematical passage
0<\beta\le1
\beta^{-1}Use in source · 1 passage
Search aliases: β^-1 beta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 940
[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lcei…
Exact mathematical passage
\gamma=\beta^{-1}
\gammaUse in source · 5 passages
Search aliases: γ gamma
Use in source · Grounded_Abstraction_v1.0.0.tex, line 940
[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lcei…
Exact mathematical passage
\gamma=\beta^{-1}
\gamma^jUse in source · 2 passages
Search aliases: γ^j gamma
Use in source · Grounded_Abstraction_v1.0.0.tex, line 941
[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lcei…
Exact mathematical passage
\TV(p_t,\widehat p_t)\le e_t :=\min\!\left\{1,\ \gamma^t e_0+(\gamma\varepsilon+\eta_N) \sum_{j=0}^{t-1}\gamma^j\right\}. \tag{11}
\gamma^tUse in source · 1 passage
Search aliases: γ^t gamma
Use in source · Grounded_Abstraction_v1.0.0.tex, line 941
[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lcei…
Exact mathematical passage
\TV(p_t,\widehat p_t)\le e_t :=\min\!\left\{1,\ \gamma^t e_0+(\gamma\varepsilon+\eta_N) \sum_{j=0}^{t-1}\gamma^j\right\}. \tag{11}
\deltaUse in source · 9 passages
Search aliases: δ delta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 453
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\delta_{\mathrm{id}},\delta_{\mathrm{model}}\in[0,1]
\Delta^{-1}Use in source · 1 passage
Search aliases: Δ^-1 Delta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 734
[Finite output-agreement test with persistent model tags] The descending paired-kernel sequence stabilizes in at most 2n strict dimension drops. Its terminal subspace is K_∞^θ=⋂_wker(CB_w^θ), I_Q=⋂_θ∈ΘΔ^-1K_∞^θ. Thus exact all-word output agreement for the finite candidate family is decidable by finite linear algebra. If an initial vector is excluded, a disagreeing candidate and a distinguishing word can be extracted. Words of length at most 2n suffice for this test. The conclusion compares fixe…
Exact mathematical passage
K_\infty^\theta=\bigcap_w\ker(CB_w^\theta),\qquad I_Q=\bigcap_{\theta\in\Theta}\Delta^{-1}K_\infty^\theta.
\delta_{\mathrm{id}}Use in source · 3 passages
Search aliases: δ_id delta mathrm
Use in source · Grounded_Abstraction_v1.0.0.tex, line 453
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\delta_{\mathrm{id}},\delta_{\mathrm{model}}\in[0,1]
\delta_{\mathrm{model}}Use in source · 3 passages
Search aliases: δ_model delta mathrm
Use in source · Grounded_Abstraction_v1.0.0.tex, line 453
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\delta_{\mathrm{id}},\delta_{\mathrm{model}}\in[0,1]
\varepsilonUse in source · 18 passages
Search aliases: ε varepsilon
Use in source · Grounded_Abstraction_v1.0.0.tex, line 453
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\eta,\varepsilon
\eta_NUse in source · 6 passages
Search aliases: η_N eta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 940
[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lcei…
Exact mathematical passage
\eta_N=k/(2N)
\ThetaUse in source · 5 passages
Search aliases: Θ Theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 714
A finite test for agreement under a fixed unknown model. Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y. Define I_Q={v∈ V:QT_w^θ v=QA_wv for every θ∈Θ and every word w}. This is an initial-condition subspace. It is not assumed invariant under the physica…
Exact mathematical passage
\Theta
\thetaUse in source · 49 passages
Search aliases: θ theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 144
A calibrated bit with an irrelevant hidden register. Consider eight possible worlds ω=(s,θ,k)∈{0,1}^3. The physical bit is s. The sensor convention is θ, and k is a nuisance bit that none of the admitted operations or questions uses. The display is y=s⊕θ. The ordinary action F flips s and leaves the other bits fixed. Its effect on the display is exactly y↦1-y, whatever the unknown convention. Reading the current display therefore initializes a two-state representation that predicts every finite …
Exact mathematical passage
\omega=(s,\theta,k)\in\{0,1\}^3.
\lambda_{b(i,j),i}Use in source · 2 passages
Search aliases: λ_b(i,j),i lambda
Use in source · Grounded_Abstraction_v1.0.0.tex, line 538
For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1. Each monomial is the likelihood product of an admitted fixed experiment-report word. Consequently p_i,y(μ)=∑_t(ω)=yμ(ω)P_i(ω) is a linear combination…
Exact mathematical passage
\lambda_{b(i,j),i}\ne\lambda_{b(i,j),j}
\lambda_{b(i,j),j}Use in source · 2 passages
Search aliases: λ_b(i,j),j lambda
Use in source · Grounded_Abstraction_v1.0.0.tex, line 538
For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1. Each monomial is the likelihood product of an admitted fixed experiment-report word. Consequently p_i,y(μ)=∑_t(ω)=yμ(ω)P_i(ω) is a linear combination…
Exact mathematical passage
\lambda_{b(i,j),i}\ne\lambda_{b(i,j),j}
\mu_+Use in source · 2 passages
Search aliases: μ_+ mu
Use in source · Grounded_Abstraction_v1.0.0.tex, line 557
Consider a prior μ_+ assigning mass 1/2 to (s,θ)=(0,0),(1,1) and a prior μ_- assigning mass 1/2 to (1,0),(0,1), with any fixed nuisance convention. Both have uniform s and uniform θ. Both predict the next report 1 with probability 1/2. After that report, however, Pr_μ_+(s=1mid o=1)=tfrac34, Pr_μ_-(s=1mid o=1)=tfrac14. After k fresh reports all equal to one, these become 3^k/(1+3^k) and 1/(1+3^k). The difference is not a numerical error. It is the association that the separate marginals failed to…
Exact mathematical passage
\mu_+
\mu_-Use in source · 2 passages
Search aliases: μ_- mu
Use in source · Grounded_Abstraction_v1.0.0.tex, line 557
Consider a prior μ_+ assigning mass 1/2 to (s,θ)=(0,0),(1,1) and a prior μ_- assigning mass 1/2 to (1,0),(0,1), with any fixed nuisance convention. Both have uniform s and uniform θ. Both predict the next report 1 with probability 1/2. After that report, however, Pr_μ_+(s=1mid o=1)=tfrac34, Pr_μ_-(s=1mid o=1)=tfrac14. After k fresh reports all equal to one, these become 3^k/(1+3^k) and 1/(1+3^k). The difference is not a numerical error. It is the association that the separate marginals failed to…
Exact mathematical passage
\mu_-
\mu_0Use in source · 1 passage
Search aliases: μ_0 mu
Use in source · Grounded_Abstraction_v1.0.0.tex, line 924
Approximate likelihood profiles. Let Ω be a finite static world set with prior μ_0. A fixed partition C_1,…,C_k refines the target: t(ω)=t_i throughout C_i. Each admitted experiment e is nondestructive and has a finite report alphabet. Its actual conditional probabilities are ℓ_e,o(ω). Reports are freshly generated from those likelihoods conditional on the fixed world and the chosen experiments.
Exact mathematical passage
\mu_0
\xiUse in source · 10 passages
Search aliases: ξ xi
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1089
[A common quantitative grounding certificate] Under (16), put ξ=min{1,e+Kd}. Then |R(a)-widehat R(a)|≤ξ for every a∈ D. If widehat a minimizes widehat R, its true posterior regret is at most 2ξ. If it is the unique approximate minimizer and widehat R(a)-widehat R(widehat a)>2ξ for every a≠widehat a, then it is the unique true minimizer. A decision issued only when widehat R(a)+ξ≤τ has true posterior risk at most the declared threshold τ.
Exact mathematical passage
\xi=\min\{1,e+Kd\}
\rhoUse in source · 3 passages
Search aliases: ρ rho
Use in source · Grounded_Abstraction_v1.0.0.tex, line 440
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
m_j(B)=\frac{\ell_j(B)+u_j(B)}2,\qquad \rho(B)=\frac12\max_j\bigl(u_j(B)-\ell_j(B)\bigr).\tag{5}
\tauUse in source · 6 passages
Search aliases: τ tau
Use in source · Grounded_Abstraction_v1.0.0.tex, line 951
[ d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lceil frackA_H2(τ-ε B_H)rceil} tag12 is sufficient for target and cell error at most τ at every time through H. Empty sums are zero. This is a finite-horizon retained-record guarantee, not an unbounded-time o…
Exact mathematical passage
\tau>\varepsilon B_H
\omega'Use in source · 2 passages
Search aliases: ω' omega
Use in source · Grounded_Abstraction_v1.0.0.tex, line 179
Define ω∼_Πω' when their record laws agree for every π∈Π. This relation concerns the complete records, including action choices when those are observable. Equality of individual observation marginals is insufficient.
Exact mathematical passage
\omega\sim_\Pi\omega'
\omega_*Use in source · 3 passages
Search aliases: ω_* omega
Use in source · Grounded_Abstraction_v1.0.0.tex, line 454
Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical vector z satisfies z-widehat m _∞≤ε, then z-v(ω_*) _∞≤widehatρ+η+εtag6 on an event of prob…
Exact mathematical passage
\Pr\{\omega_*\in B(H)\}\ge1-\delta_{\mathrm{id}}.
\ell^1Use in source · 1 passage
Search aliases: ℓ^1 ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 688
[An uncalibrated direction amplifies a small residual] Let L=span(e_1), let Se_1=e_1, and in the ℓ^1 norm define T=1 · 0
Exact mathematical passage
\ell^1
\ell_{b(i,j)}Use in source · 1 passage
Search aliases: ℓ_b(i,j) ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 539
For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1. Each monomial is the likelihood product of an admitted fixed experiment-report word. Consequently p_i,y(μ)=∑_t(ω)=yμ(ω)P_i(ω) is a linear combination…
Exact mathematical passage
P_i(\omega)=\prod_{j\ne i} \frac{\ell_{b(i,j)}(\omega)-\lambda_{b(i,j),j}} {\lambda_{b(i,j),i}-\lambda_{b(i,j),j}}
\ell_{e_j,o_j}Use in source · 1 passage
Search aliases: ℓ_e_j,o_j ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 532
For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. The empty word gives the current target distribution.
Exact mathematical passage
M_{\mu,y}(w)=\sum_{t(\omega)=y}\mu(\omega) \prod_{j=1}^k\ell_{e_j,o_j}(\omega).
\ell_iUse in source · 7 passages
Search aliases: ℓ_i ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 890
For a likelihood vector ℓ=(ℓ_i) define the Bayes map B_ℓ(p)_i=fracp_iℓ_i∑_jp_jℓ_j whenever its denominator is positive.
Exact mathematical passage
\ell=(\ell_i)
\ell_jUse in source · 4 passages
Search aliases: ℓ_j ell
Use in source · Grounded_Abstraction_v1.0.0.tex, line 436
Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is not an uncertainty measure invariant under arbitrary nonlinear reparameterization.
Exact mathematical passage
\ell_j(B)=\min_{\omega\in B}v_j(\omega),\qquad u_j(B)=\max_{\omega\in B}v_j(\omega),
\mapstoUse in source · 3 passages
Search aliases: ↦ mapsto
Use in source · Grounded_Abstraction_v1.0.0.tex, line 151
rrelevant hidden register Consider eight possible worlds ω=(s,θ,k)∈{0,1}^3. The physical bit is s. The sensor convention is θ, and k is a nuisance bit that none of the admitted operations or questions uses. The display is y=s⊕θ. The ordinary action F flips s and leaves the other bits fixed. Its effect on the display is exactly y↦1-y, whatever the unknown convention. Reading the current display therefore initializes a two-state representation that predicts every finite sequence of flips.
Exact mathematical passage
y\mapsto1-y
\forallUse in source · 1 passage
Search aliases: ∀ forall
Use in source · Grounded_Abstraction_v1.0.0.tex, line 237
immutable worlds, one with a permanently zero output and one with a permanently one output. The available initial record is the same symbol in both. Within each world separately, a one-state predictor exists: it simply prints that world's output. There is nevertheless no predictor that can be initialized from the common record and be correct in both. The obstruction is the order of quantifiers, ∀ω ∃ P_ω versus ∃ P ∀ω. The first does not provide a selector for the second. The behavioral quotient …
Exact mathematical passage
\forall\omega\ \exists P_\omega \qquad\text{versus}\qquad \exists P\ \forall\omega.
\existsUse in source · 1 passage
Search aliases: ∃ exists
Use in source · Grounded_Abstraction_v1.0.0.tex, line 237
immutable worlds, one with a permanently zero output and one with a permanently one output. The available initial record is the same symbol in both. Within each world separately, a one-state predictor exists: it simply prints that world's output. There is nevertheless no predictor that can be initialized from the common record and be correct in both. The obstruction is the order of quantifiers, ∀ω ∃ P_ω versus ∃ P ∀ω. The first does not provide a selector for the second. The behavioral quotient …
Exact mathematical passage
\forall\omega\ \exists P_\omega \qquad\text{versus}\qquad \exists P\ \forall\omega.
\notinUse in source · 4 passages
Search aliases: ∉ notin
Use in source · Grounded_Abstraction_v1.0.0.tex, line 606
[One-step identification from exact calibration] For a fixed action a, the value T_av is the same for every linear extension of S_a if and only if v∈ U. The common value is then S_av. If v∈ U, all extensions agree by definition. If v∉ U, choose a linear functional ℓ that vanishes on U and satisfies ℓ(v)=1, and choose a nonzero vector b∈ V. Given any extension T_a, the map T_a'(x)=T_a(x)+ℓ(x)b is another extension, but T_a'v=T_av+b≠ T_av. Such a functional exists by extending a basis of U with v.
Exact mathematical passage
v\notin U
\prodUse in source · 2 passages
Search aliases: ∏ prod
Use in source · Grounded_Abstraction_v1.0.0.tex, line 532
For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. The empty word gives the current target distribution.
Exact mathematical passage
M_{\mu,y}(w)=\sum_{t(\omega)=y}\mu(\omega) \prod_{j=1}^k\ell_{e_j,o_j}(\omega).
\prod_{j=1}^kUse in source · 1 passage
Search aliases: ∏_j=1^k prod
Use in source · Grounded_Abstraction_v1.0.0.tex, line 532
For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. The empty word gives the current target distribution.
Exact mathematical passage
M_{\mu,y}(w)=\sum_{t(\omega)=y}\mu(\omega) \prod_{j=1}^k\ell_{e_j,o_j}(\omega).
\prod_{j\ne i}Use in source · 1 passage
Search aliases: ∏_j≠ i prod ne
Use in source · Grounded_Abstraction_v1.0.0.tex, line 539
For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1. Each monomial is the likelihood product of an admitted fixed experiment-report word. Consequently p_i,y(μ)=∑_t(ω)=yμ(ω)P_i(ω) is a linear combination…
Exact mathematical passage
P_i(\omega)=\prod_{j\ne i} \frac{\ell_{b(i,j)}(\omega)-\lambda_{b(i,j),j}} {\lambda_{b(i,j),i}-\lambda_{b(i,j),j}}
\sum_aUse in source · 1 passage
Search aliases: ∑_a sum
Use in source · Grounded_Abstraction_v1.0.0.tex, line 777
ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies. Let E_0=∑_a,θim D_a^θ, E_j+1=E_j+∑_a A_aE_j, E_∞=⋃_jE_j. This increasing sequence stabilizes after at most n strict dimension increases. Its limit is…
Exact mathematical passage
E_0=\sum_{a,\theta}\im D_a^\theta,\qquad E_{j+1}=E_j+\sum_a A_aE_j,\qquad E_\infty=\bigcup_jE_j.
\sum_{a,\theta}Use in source · 1 passage
Search aliases: ∑_a,θ sum theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 777
ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies. Let E_0=∑_a,θim D_a^θ, E_j+1=E_j+∑_a A_aE_j, E_∞=⋃_jE_j. This increasing sequence stabilizes after at most n strict dimension increases. Its limit is…
Exact mathematical passage
E_0=\sum_{a,\theta}\im D_a^\theta,\qquad E_{j+1}=E_j+\sum_a A_aE_j,\qquad E_\infty=\bigcup_jE_j.
\sum_hUse in source · 2 passages
Search aliases: ∑_h sum
Use in source · Grounded_Abstraction_v1.0.0.tex, line 394
[A finite-record discrimination bound] For two distinct target values and equal world probabilities, every transcript-based estimator has average success at most frac1+d_TV(P_0,P_1)2, d_TV(P_0,P_1)=frac12∑_h|P_0(h)-P_1(h)|.tag4 A deterministic function or a common stochastic processing channel applied to the transcript cannot increase this upper bound. Convert the estimator into a binary test by reporting world zero when it outputs the first target and world one otherwise. A correct target outpu…
Exact mathematical passage
\frac{1+\TV(P_0,P_1)}{2},\qquad \TV(P_0,P_1)=\frac12\sum_h|P_0(h)-P_1(h)|.\tag{4}
\sum_{i,y}Use in source · 1 passage
Search aliases: ∑_i,y sum
Use in source · Grounded_Abstraction_v1.0.0.tex, line 520
[Exact target correction from likelihood-profile memory] Under the fixed-world likelihood model, the record p=(p_i,y) supports every finite sequence of report predictions and target-posterior corrections. Its next-report probability and update are z_e,o=∑_i,yλ_e,o,ip_i,y, p'_i,y=fracλ_e,o,ip_i,yz_e,o (z_e,o>0), and its target readout is Pr(t=y)=∑_i p_i,y. A zero-probability report is marked impossible rather than assigned a normalized posterior.
Exact mathematical passage
z_{e,o}=\sum_{i,y}\lambda_{e,o,i}p_{i,y},\qquad p'_{i,y}=\frac{\lambda_{e,o,i}p_{i,y}}{z_{e,o}}\quad(z_{e,o}>0),
\sum_jUse in source · 1 passage
Search aliases: ∑_j sum
Use in source · Grounded_Abstraction_v1.0.0.tex, line 891
For a likelihood vector ℓ=(ℓ_i) define the Bayes map B_ℓ(p)_i=fracp_iℓ_i∑_jp_jℓ_j whenever its denominator is positive.
Exact mathematical passage
\mathcal B_\ell(p)_i=\frac{p_i\ell_i}{\sum_jp_j\ell_j}
\sum_{j=0}^{(n-1)/2}Use in source · 1 passage
Search aliases: ∑_j=0^(n-1)/2 sum
Use in source · Grounded_Abstraction_v1.0.0.tex, line 414
For odd n, majority voting is the likelihood-ratio rule. Its error in either world is e_n=∑_j=0^(n-1)/2binomnj(3/4)^j(1/4)^n-j. The first values are e_1=1/4, e_3=5/32, and e_5=53/512. These are exact analytical probabilities, not observed error rates from an agent experiment.
Exact mathematical passage
e_n=\sum_{j=0}^{(n-1)/2}\binom{n}{j}(3/4)^j(1/4)^{n-j}.
\sum_{j=0}^{H}Use in source · 1 passage
Search aliases: ∑_j=0^H sum
Use in source · Grounded_Abstraction_v1.0.0.tex, line 947
let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lceil frackA_H2(τ-ε B_H)rceil} tag12 is sufficient for target and cell error at most τ at every time th…
Exact mathematical passage
A_H=\sum_{j=0}^{H}\gamma^j,\qquad B_H=\gamma\sum_{j=0}^{H-1}\gamma^j.
\sum_{j=0}^{H-1}Use in source · 1 passage
Search aliases: ∑_j=0^H-1 sum
Use in source · Grounded_Abstraction_v1.0.0.tex, line 947
let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lceil frackA_H2(τ-ε B_H)rceil} tag12 is sufficient for target and cell error at most τ at every time th…
Exact mathematical passage
A_H=\sum_{j=0}^{H}\gamma^j,\qquad B_H=\gamma\sum_{j=0}^{H-1}\gamma^j.
\sum_{j=0}^{t-1}Use in source · 3 passages
Search aliases: ∑_j=0^t-1 sum
Use in source · Grounded_Abstraction_v1.0.0.tex, line 941
[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lcei…
Exact mathematical passage
\TV(p_t,\widehat p_t)\le e_t :=\min\!\left\{1,\ \gamma^t e_0+(\gamma\varepsilon+\eta_N) \sum_{j=0}^{t-1}\gamma^j\right\}. \tag{11}
\sum_{j\in J}Use in source · 1 passage
Search aliases: ∑_j∈ J sum in
Use in source · Grounded_Abstraction_v1.0.0.tex, line 274
nonadaptive set J suffices exactly when it covers D. item Some pair in D has identical reports under every admitted test. No finite adaptive protocol using this unchanged test interface and r guarantees t in every world. If test j has nonnegative fixed cost c_j, the minimum cost of one fixed nonadaptive test set, used for all initial codes in case 2, is precisely min_J: J covers D ∑_j∈ Jc_j.tag2 This is not a claim that an optimal adaptive protocol has the same cost. If D is empty, the target is…
Exact mathematical passage
\min_{J:\,J\text{ covers }D}\ \sum_{j\in J}c_j.\tag{2}
\sum_{j\ge0}Use in source · 1 passage
Search aliases: ∑_j≥0 sum ge
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1045
c · 1-g has nonnegative entries. Let a=(I-M)^-1(1,1)^T. Both coordinates are positive and Ma=a-(1,1)^T<a. Hence M is a strict contraction in the weighted maximum norm with weights a, and its geometric series converges to (I-M)^-1. Summing the nonnegative right-hand side of (14) gives frace_01-g+fracb1-g∑_j≥0(M^ju_0)_2. The second coordinate of (I-M)^-1u_0 is (cV_0+(1-g)H_0)/D, proving (15). The common comparison applies at each step and therefore also covers switching when that switching is expl…
Exact mathematical passage
\frac{e_0}{1-g}+\frac{b}{1-g}\sum_{j\ge0}(M^ju_0)_2.
\sum_oUse in source · 1 passage
Search aliases: ∑_o sum
Use in source · Grounded_Abstraction_v1.0.0.tex, line 506
Let Ω be finite and nonempty, let t:Ω→ Y have finite image, and let B be a finite collection of experiments. Experiment e has a finite outcome set and specified likelihoods ℓ_e,o(ω)≥0, ∑_oℓ_e,o(ω)=1. The world is fixed. Every experiment remains available, and each new report has this conditional law given the world, the chosen experiment, and all earlier reports. This is a fresh conditional-observation premise, not a claim that copied readings become new samples. No experiment changes the physic…
Exact mathematical passage
\ell_{e,o}(\omega)\ge0,\qquad \sum_o\ell_{e,o}(\omega)=1.
\sum_{t(\omega)=y}Use in source · 2 passages
Search aliases: ∑_t(ω)=y sum omega
Use in source · Grounded_Abstraction_v1.0.0.tex, line 532
For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. The empty word gives the current target distribution.
Exact mathematical passage
M_{\mu,y}(w)=\sum_{t(\omega)=y}\mu(\omega) \prod_{j=1}^k\ell_{e_j,o_j}(\omega).
\sum_{t=0}^{\infty}Use in source · 1 passage
Search aliases: ∑_t=0^∞ sum infty
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1024
If g<1, d<1, and D=(1-g)(1-d)-bc>0, then the total visible discrepancy is bounded by ∑_t=0^∞ v_t-bar v_t ≤ frace_01-g +fracb(cV_0+(1-g)H_0)(1-g)D. tag15 These are sufficient common bounds. Failure of the inequality for D is not a proof that a particular signed system is unstable. Conversely, the two diagonal conditions alone are insufficient for a uniform conclusion over the stated block class. The norm triangle inequality and the common block bounds give v_t+1
Exact mathematical passage
\sum_{t=0}^{\infty}\|v_t-\bar v_t\| \le \frac{e_0}{1-g} +\frac{b\bigl(cV_0+(1-g)H_0\bigr)}{(1-g)D}. \tag{15}
\sum_zUse in source · 1 passage
Search aliases: ∑_z sum
Use in source · Grounded_Abstraction_v1.0.0.tex, line 405
herwise. A correct target output always gives a correct binary decision. If f(h)∈[0,1] is the test's chance of declaring world zero, its success is frac12(1+∑_h f(h)(P_0(h)-P_1(h))). The sum is at most the sum of positive differences, which equals total variation. This proves (4). If B(h,z) is a common processing channel and g(z)∈[0,1] a processed-record test, then f(h)=∑_zB(h,z)g(z) is a raw-record test with the same two expectations. Thus every processed decision was already available from the…
Exact mathematical passage
f(h)=\sum_zB(h,z)g(z)
\sum_{\omega}Use in source · 1 passage
Search aliases: ∑_ω sum omega
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1082
Let the permitted decision set D be finite and nonempty. Its loss L(y,a) takes values in [0,1] and is K-Lipschitz in y, uniformly over a. Define R(a)=∑_ωμ(ω)L(y(ω),a), widehat R(a)=∑_iwidehat p_iL(bar y_i,a). These are risks under the stated model posterior. They are not frequentist coverage probabilities or guarantees on the loss of a single realized world.
Exact mathematical passage
R(a)=\sum_{\omega}\mu(\omega)L(y(\omega),a),\qquad \widehat R(a)=\sum_i\widehat p_iL(\bar y_i,a).
\sim_\CUse in source · 6 passages
Search aliases: ∼_C ∼_ C sim C sim mathcal
Use in source · Grounded_Abstraction_v1.0.0.tex, line 198
For a word w=a_1⋯ a_m, F_w applies a_1 first and a_m last. The empty word is the identity. Define x∼_ C x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite w. This is a comparison in a common observation language. It does not require that different models share arbitrary latent coordinates. The behavioral class is a canonical response profile even when its implementation is not identifiable.
Exact mathematical passage
x\sim_\C x'\quad\Longleftrightarrow\quad q(F_wx)=q(F_wx')\quad\text{for every finite }w.
\sim_\PiUse in source · 2 passages
Search aliases: ∼_Π sim Pi
Use in source · Grounded_Abstraction_v1.0.0.tex, line 179
Define ω∼_Πω' when their record laws agree for every π∈Π. This relation concerns the complete records, including action choices when those are observable. Equality of individual observation marginals is insufficient.
Exact mathematical passage
\omega\sim_\Pi\omega'
\approxUse in source · 5 passages
Search aliases: ≈ approx
Use in source · Grounded_Abstraction_v1.0.0.tex, line 1146
midrule a=0 · 23991/262144 ≈0.09152 · 173737/1835008 ≈0.09468
Exact mathematical passage
23991/262144\ \approx0.09152
\oplusUse in source · 5 passages
Search aliases: ⊕ oplus
Use in source · Grounded_Abstraction_v1.0.0.tex, line 148
A calibrated bit with an irrelevant hidden register. Consider eight possible worlds ω=(s,θ,k)∈{0,1}^3. The physical bit is s. The sensor convention is θ, and k is a nuisance bit that none of the admitted operations or questions uses. The display is y=s⊕θ. The ordinary action F flips s and leaves the other bits fixed. Its effect on the display is exactly y↦1-y, whatever the unknown convention. Reading the current display therefore initializes a two-state representation that predicts every finite …
Exact mathematical passage
y=s\oplus\theta.
\bigcapUse in source · 3 passages
Search aliases: ⋂ bigcap
Use in source · Grounded_Abstraction_v1.0.0.tex, line 727
0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.
Exact mathematical passage
K_0^\theta=\ker C,\qquad K_{j+1}^\theta=K_j^\theta\cap\bigcap_a(B_a^\theta)^{-1}K_j^\theta.
\bigcap_aUse in source · 1 passage
Search aliases: ⋂_a bigcap
Use in source · Grounded_Abstraction_v1.0.0.tex, line 727
0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.
Exact mathematical passage
K_0^\theta=\ker C,\qquad K_{j+1}^\theta=K_j^\theta\cap\bigcap_a(B_a^\theta)^{-1}K_j^\theta.
\bigcap_wUse in source · 2 passages
Search aliases: ⋂_w bigcap
Use in source · Grounded_Abstraction_v1.0.0.tex, line 734
[Finite output-agreement test with persistent model tags] The descending paired-kernel sequence stabilizes in at most 2n strict dimension drops. Its terminal subspace is K_∞^θ=⋂_wker(CB_w^θ), I_Q=⋂_θ∈ΘΔ^-1K_∞^θ. Thus exact all-word output agreement for the finite candidate family is decidable by finite linear algebra. If an initial vector is excluded, a disagreeing candidate and a distinguishing word can be extracted. Words of length at most 2n suffice for this test. The conclusion compares fixe…
Exact mathematical passage
K_\infty^\theta=\bigcap_w\ker(CB_w^\theta),\qquad I_Q=\bigcap_{\theta\in\Theta}\Delta^{-1}K_\infty^\theta.
\bigcap_{\theta\in\Theta}Use in source · 1 passage
Search aliases: ⋂_θ∈Θ bigcap theta in Theta
Use in source · Grounded_Abstraction_v1.0.0.tex, line 734
[Finite output-agreement test with persistent model tags] The descending paired-kernel sequence stabilizes in at most 2n strict dimension drops. Its terminal subspace is K_∞^θ=⋂_wker(CB_w^θ), I_Q=⋂_θ∈ΘΔ^-1K_∞^θ. Thus exact all-word output agreement for the finite candidate family is decidable by finite linear algebra. If an initial vector is excluded, a disagreeing candidate and a distinguishing word can be extracted. Words of length at most 2n suffice for this test. The conclusion compares fixe…
Exact mathematical passage
K_\infty^\theta=\bigcap_w\ker(CB_w^\theta),\qquad I_Q=\bigcap_{\theta\in\Theta}\Delta^{-1}K_\infty^\theta.
\bigcupUse in source · 2 passages
Search aliases: ⋃ bigcup
Use in source · Grounded_Abstraction_v1.0.0.tex, line 489
What compression does to the candidate set. For a fixed raw-record channel, let B(h) be its exact support set. If only c(h) is retained, the correct support is B_c(z)=⋃_h:c(h)=zB(h). Indeed, a world can produce the coarse record exactly when it can produce at least one raw record in that fibre. Thus B(h)⊆ B_c(c(h)), and the ambiguity radius cannot decrease for the same target vector. This is an information statement, not a command to retain every byte. The correction signature in Section~[source…
Exact mathematical passage
B_c(z)=\bigcup_{h:c(h)=z}B(h).
\bigcup_{h:c(h)=z}Use in source · 1 passage
Search aliases: ⋃_h:c(h)=z bigcup
Use in source · Grounded_Abstraction_v1.0.0.tex, line 489
What compression does to the candidate set. For a fixed raw-record channel, let B(h) be its exact support set. If only c(h) is retained, the correct support is B_c(z)=⋃_h:c(h)=zB(h). Indeed, a world can produce the coarse record exactly when it can produce at least one raw record in that fibre. Thus B(h)⊆ B_c(c(h)), and the ambiguity radius cannot decrease for the same target vector. This is an information statement, not a command to retain every byte. The correction signature in Section~[source…
Exact mathematical passage
B_c(z)=\bigcup_{h:c(h)=z}B(h).
\bigcup_jUse in source · 1 passage
Search aliases: ⋃_j bigcup
Use in source · Grounded_Abstraction_v1.0.0.tex, line 777
ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies. Let E_0=∑_a,θim D_a^θ, E_j+1=E_j+∑_a A_aE_j, E_∞=⋃_jE_j. This increasing sequence stabilizes after at most n strict dimension increases. Its limit is…
Exact mathematical passage
E_0=\sum_{a,\theta}\im D_a^\theta,\qquad E_{j+1}=E_j+\sum_a A_aE_j,\qquad E_\infty=\bigcup_jE_j.
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