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Resolution Semantics
All notation · Glossary · Source coverage
243 symbol spellings in 380 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 243 entries are available below.
ANotation table · 26 passages
Search aliases: A
Notation table · Komorebi_Resolution_Semantics.tex, line 722
A · A Boolean algebra of observables, often sentences modulo provable equivalence.
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A
BNotation table · 20 passages
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Notation table · Komorebi_Resolution_Semantics.tex, line 723
B_M · The Boolean algebra of values at one world.
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B_M
B_MNotation table · 6 passages
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Notation table · Komorebi_Resolution_Semantics.tex, line 723
B_M · The Boolean algebra of values at one world.
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B_M
\mathcal CNotation table · 8 passages
Search aliases: C mathcal
Notation table · Komorebi_Resolution_Semantics.tex, line 721
C · A small category of typed world transformations.
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\mathcal C
DNotation table · 23 passages
Search aliases: D
Notation table · Komorebi_Resolution_Semantics.tex, line 725
D_f · Additive defect between transported source evaluation and target evaluation.
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D_f
D_fNotation table · 14 passages
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Notation table · Komorebi_Resolution_Semantics.tex, line 725
D_f · Additive defect between transported source evaluation and target evaluation.
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D_f
ENotation table · 16 passages
Search aliases: E
Notation table · Komorebi_Resolution_Semantics.tex, line 732
E_Π · Weighted real orthogonal projection. Not a Boolean homomorphism.
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E_\Pi
E_\PiNotation table · 5 passages
Search aliases: E_Π Pi
Notation table · Komorebi_Resolution_Semantics.tex, line 732
E_Π · Weighted real orthogonal projection. Not a Boolean homomorphism.
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E_\Pi
fNotation table · 33 passages
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Notation table · Komorebi_Resolution_Semantics.tex, line 725
D_f · Additive defect between transported source evaluation and target evaluation.
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D_f
\InvNotation table · 32 passages
Search aliases: Inv I mathcal
Notation table · Komorebi_Resolution_Semantics.tex, line 726
I · Zero-defect observables. A subalgebra, generally not an ideal.
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\Inv
KNotation table · 6 passages
Search aliases: K
Notation table · Komorebi_Resolution_Semantics.tex, line 727
K · Observables vanishing at every world. An ideal.
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K
kNotation table · 17 passages
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Notation table · Komorebi_Resolution_Semantics.tex, line 733
Q_k · Real multiresolution detail projector for nested partitions.
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Q_k
LNotation table · 19 passages
Search aliases: L
Notation table · Komorebi_Resolution_Semantics.tex, line 730
L_Π,U_Π · Canonical lower and upper invariant approximations.
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L_\Pi,U_\Pi
L_\PiNotation table · 11 passages
Search aliases: L_Π Pi
Notation table · Komorebi_Resolution_Semantics.tex, line 730
L_Π,U_Π · Canonical lower and upper invariant approximations.
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L_\Pi,U_\Pi
MNotation table · 58 passages
Search aliases: M
Notation table · Komorebi_Resolution_Semantics.tex, line 720
midrule Mmodels T · A supplied model, not an assertion that the ambient theory proves such a model exists.
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M\models T
\modelsNotation table · 10 passages
Search aliases: models
Notation table · Komorebi_Resolution_Semantics.tex, line 720
midrule Mmodels T · A supplied model, not an assertion that the ambient theory proves such a model exists.
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M\models T
PNotation table · 22 passages
Search aliases: P
Notation table · Komorebi_Resolution_Semantics.tex, line 731
P_s · Boolean retraction determined by a representative in each finite block.
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P_s
pNotation table · 3 passages
Search aliases: p
Notation table · Komorebi_Resolution_Semantics.tex, line 734
Δ_p,q · Typed discrepancy between parallel transport paths.
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\Delta_{p,q}
P_sNotation table · 5 passages
Search aliases: P_s
Notation table · Komorebi_Resolution_Semantics.tex, line 731
P_s · Boolean retraction determined by a representative in each finite block.
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P_s
QNotation table · 5 passages
Search aliases: Q
Notation table · Komorebi_Resolution_Semantics.tex, line 733
Q_k · Real multiresolution detail projector for nested partitions.
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Q_k
qNotation table · 3 passages
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Notation table · Komorebi_Resolution_Semantics.tex, line 734
Δ_p,q · Typed discrepancy between parallel transport paths.
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\Delta_{p,q}
Q_kNotation table · 5 passages
Search aliases: Q_k
Notation table · Komorebi_Resolution_Semantics.tex, line 733
Q_k · Real multiresolution detail projector for nested partitions.
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Q_k
sNotation table · 13 passages
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Notation table · Komorebi_Resolution_Semantics.tex, line 731
P_s · Boolean retraction determined by a representative in each finite block.
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P_s
TNotation table · 14 passages
Search aliases: T
Notation table · Komorebi_Resolution_Semantics.tex, line 720
midrule Mmodels T · A supplied model, not an assertion that the ambient theory proves such a model exists.
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M\models T
UNotation table · 17 passages
Search aliases: U
Notation table · Komorebi_Resolution_Semantics.tex, line 730
L_Π,U_Π · Canonical lower and upper invariant approximations.
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L_\Pi,U_\Pi
U_\PiNotation table · 11 passages
Search aliases: U_Π Pi
Notation table · Komorebi_Resolution_Semantics.tex, line 730
L_Π,U_Π · Canonical lower and upper invariant approximations.
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L_\Pi,U_\Pi
vNotation table · 16 passages
Search aliases: v
Notation table · Komorebi_Resolution_Semantics.tex, line 724
v_M · Semantic evaluation of observables.
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v_M
v_MNotation table · 10 passages
Search aliases: v_M
Notation table · Komorebi_Resolution_Semantics.tex, line 724
v_M · Semantic evaluation of observables.
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v_M
\DeltaNotation table · 2 passages
Search aliases: Δ Delta
Notation table · Komorebi_Resolution_Semantics.tex, line 734
Δ_p,q · Typed discrepancy between parallel transport paths.
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\Delta_{p,q}
\Delta_{p,q}Notation table · 2 passages
Search aliases: Δ_p,q Delta
Notation table · Komorebi_Resolution_Semantics.tex, line 734
Δ_p,q · Typed discrepancy between parallel transport paths.
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\Delta_{p,q}
\PiNotation table · 30 passages
Search aliases: Π Pi
Notation table · Komorebi_Resolution_Semantics.tex, line 730
L_Π,U_Π · Canonical lower and upper invariant approximations.
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L_\Pi,U_\Pi
+Definition passage · 35 passages
Search aliases: +
Definition passage: Transport defect · Komorebi_Resolution_Semantics.tex, line 176
[Transport defect] For f:M→ N define D_f(a)=B_f(v_M(a))+v_N(a)∈ B_N. The terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.
Exact mathematical passage
\label{eq:defect} D_f(a)=B_f(v_M(a))+v_N(a)\in B_N.
=Definition passage · 100 passages
Search aliases: =
Definition passage: Transport defect · Komorebi_Resolution_Semantics.tex, line 176
[Transport defect] For f:M→ N define D_f(a)=B_f(v_M(a))+v_N(a)∈ B_N. The terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.
Exact mathematical passage
\label{eq:defect} D_f(a)=B_f(v_M(a))+v_N(a)\in B_N.
aDefinition passage · 64 passages
Search aliases: a
Definition passage: Transport defect · Komorebi_Resolution_Semantics.tex, line 176
[Transport defect] For f:M→ N define D_f(a)=B_f(v_M(a))+v_N(a)∈ B_N. The terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.
Exact mathematical passage
\label{eq:defect} D_f(a)=B_f(v_M(a))+v_N(a)\in B_N.
B_fDefinition passage · 8 passages
Search aliases: B_f
Definition passage: Resolution datum · Komorebi_Resolution_Semantics.tex, line 168
[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In Boolean-ring notation, + is symmetric difference, multiplication is conjunction, and neg a=1+a. In particular, a+a=0.
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B_f:B_M\to B_N
B_NDefinition passage · 2 passages
Search aliases: B_N
Definition passage: Resolution datum · Komorebi_Resolution_Semantics.tex, line 168
[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In Boolean-ring notation, + is symmetric difference, multiplication is conjunction, and neg a=1+a. In particular, a+a=0.
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B_f:B_M\to B_N
\BoolDefinition passage · 2 passages
Search aliases: Bool mathbf
Definition passage: Resolution datum · Komorebi_Resolution_Semantics.tex, line 165
[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In Boolean-ring notation, + is symmetric difference, multiplication is conjunction, and neg a=1+a. In particular, a+a=0.
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B:\mathcal C\longrightarrow\Bool,
NDefinition passage · 18 passages
Search aliases: N
Definition passage: Resolution datum · Komorebi_Resolution_Semantics.tex, line 168
[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In Boolean-ring notation, + is symmetric difference, multiplication is conjunction, and neg a=1+a. In particular, a+a=0.
Exact mathematical passage
B_f:B_M\to B_N
v_NDefinition passage · 5 passages
Search aliases: v_N
Definition passage: Transport defect · Komorebi_Resolution_Semantics.tex, line 176
[Transport defect] For f:M→ N define D_f(a)=B_f(v_M(a))+v_N(a)∈ B_N. The terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.
Exact mathematical passage
\label{eq:defect} D_f(a)=B_f(v_M(a))+v_N(a)\in B_N.
\toDefinition passage · 17 passages
Search aliases: → to
Definition passage: Resolution datum · Komorebi_Resolution_Semantics.tex, line 168
[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In Boolean-ring notation, + is symmetric difference, multiplication is conjunction, and neg a=1+a. In particular, a+a=0.
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v_M:A\to B_M
\inDefinition passage · 34 passages
Search aliases: ∈ in
Definition passage: Transport defect · Komorebi_Resolution_Semantics.tex, line 176
[Transport defect] For f:M→ N define D_f(a)=B_f(v_M(a))+v_N(a)∈ B_N. The terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.
Exact mathematical passage
\label{eq:defect} D_f(a)=B_f(v_M(a))+v_N(a)\in B_N.
\longrightarrowDefinition passage · 2 passages
Search aliases: ⟶ longrightarrow
Definition passage: Resolution datum · Komorebi_Resolution_Semantics.tex, line 165
[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In Boolean-ring notation, + is symmetric difference, multiplication is conjunction, and neg a=1+a. In particular, a+a=0.
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B:\mathcal C\longrightarrow\Bool,
E_0Introduction passage · 1 passage
Search aliases: E_0
Introduction passage · Komorebi_Resolution_Semantics.tex, line 504
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
\label{eq:multiresolution} f=E_0f+\sum_{k=1}^mQ_kf+(\id-E_m)f.
E_iIntroduction passage · 2 passages
Search aliases: E_i
Introduction passage · Komorebi_Resolution_Semantics.tex, line 500
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_iE_j=E_{\min(i,j)},\qquad Q_k^2=Q_k,\qquad Q_kQ_l=0\ (k\ne l),
E_jIntroduction passage · 2 passages
Search aliases: E_j
Introduction passage · Komorebi_Resolution_Semantics.tex, line 500
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_iE_j=E_{\min(i,j)},\qquad Q_k^2=Q_k,\qquad Q_kQ_l=0\ (k\ne l),
E_kIntroduction passage · 2 passages
Search aliases: E_k
Introduction passage · Komorebi_Resolution_Semantics.tex, line 499
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_k=E_{\Pi_k}
E_{k-1}Introduction passage · 1 passage
Search aliases: E_k-1
Introduction passage · Komorebi_Resolution_Semantics.tex, line 499
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
Q_k=E_k-E_{k-1}
E_mIntroduction passage · 1 passage
Search aliases: E_m
Introduction passage · Komorebi_Resolution_Semantics.tex, line 504
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
\label{eq:multiresolution} f=E_0f+\sum_{k=1}^mQ_kf+(\id-E_m)f.
E_{\min(i,j)}Introduction passage · 1 passage
Search aliases: E_min(i,j) min
Introduction passage · Komorebi_Resolution_Semantics.tex, line 500
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_iE_j=E_{\min(i,j)},\qquad Q_k^2=Q_k,\qquad Q_kQ_l=0\ (k\ne l),
E_{\Pi_k}Introduction passage · 1 passage
Search aliases: E_Π_k Pi
Introduction passage · Komorebi_Resolution_Semantics.tex, line 499
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_k=E_{\Pi_k}
iIntroduction passage · 20 passages
Search aliases: i
Introduction passage · Komorebi_Resolution_Semantics.tex, line 500
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_iE_j=E_{\min(i,j)},\qquad Q_k^2=Q_k,\qquad Q_kQ_l=0\ (k\ne l),
\idIntroduction passage · 7 passages
Search aliases: id operatorname
Introduction passage · Komorebi_Resolution_Semantics.tex, line 504
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
\label{eq:multiresolution} f=E_0f+\sum_{k=1}^mQ_kf+(\id-E_m)f.
jIntroduction passage · 4 passages
Search aliases: j
Introduction passage · Komorebi_Resolution_Semantics.tex, line 500
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_iE_j=E_{\min(i,j)},\qquad Q_k^2=Q_k,\qquad Q_kQ_l=0\ (k\ne l),
lIntroduction passage · 1 passage
Search aliases: l
Introduction passage · Komorebi_Resolution_Semantics.tex, line 500
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_iE_j=E_{\min(i,j)},\qquad Q_k^2=Q_k,\qquad Q_kQ_l=0\ (k\ne l),
mIntroduction passage · 5 passages
Search aliases: m
Introduction passage · Komorebi_Resolution_Semantics.tex, line 499
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
\Pi_0,\ldots,\Pi_m
\minIntroduction passage · 1 passage
Search aliases: min
Introduction passage · Komorebi_Resolution_Semantics.tex, line 500
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_iE_j=E_{\min(i,j)},\qquad Q_k^2=Q_k,\qquad Q_kQ_l=0\ (k\ne l),
\negIntroduction passage · 4 passages
Search aliases: neg
Introduction passage · Komorebi_Resolution_Semantics.tex, line 170
B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In Boolean-ring notation, + is symmetric difference, multiplication is conjunction, and neg a=1+a. In particular, a+a=0.
Exact mathematical passage
\neg a=1+a
Q_k^2Introduction passage · 1 passage
Search aliases: Q_k^2
Introduction passage · Komorebi_Resolution_Semantics.tex, line 500
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_iE_j=E_{\min(i,j)},\qquad Q_k^2=Q_k,\qquad Q_kQ_l=0\ (k\ne l),
Q_lIntroduction passage · 1 passage
Search aliases: Q_l
Introduction passage · Komorebi_Resolution_Semantics.tex, line 500
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_iE_j=E_{\min(i,j)},\qquad Q_k^2=Q_k,\qquad Q_kQ_l=0\ (k\ne l),
\mathbb RIntroduction passage · 2 passages
Search aliases: R mathbb
Introduction passage · Komorebi_Resolution_Semantics.tex, line 503
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
f\in\mathbb R^W
\mathbb R^WIntroduction passage · 2 passages
Search aliases: R^W mathbb
Introduction passage · Komorebi_Resolution_Semantics.tex, line 503
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
f\in\mathbb R^W
WIntroduction passage · 12 passages
Search aliases: W
Introduction passage · Komorebi_Resolution_Semantics.tex, line 503
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
f\in\mathbb R^W
\Pi_0Introduction passage · 2 passages
Search aliases: Π_0 Pi
Introduction passage · Komorebi_Resolution_Semantics.tex, line 499
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
\Pi_0,\ldots,\Pi_m
\Pi_kIntroduction passage · 1 passage
Search aliases: Π_k Pi
Introduction passage · Komorebi_Resolution_Semantics.tex, line 499
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_k=E_{\Pi_k}
\Pi_mIntroduction passage · 1 passage
Search aliases: Π_m Pi
Introduction passage · Komorebi_Resolution_Semantics.tex, line 499
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
\Pi_0,\ldots,\Pi_m
\ldotsIntroduction passage · 2 passages
Search aliases: … ldots
Introduction passage · Komorebi_Resolution_Semantics.tex, line 499
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
\Pi_0,\ldots,\Pi_m
\sumIntroduction passage · 6 passages
Search aliases: ∑ sum
Introduction passage · Komorebi_Resolution_Semantics.tex, line 504
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
\label{eq:multiresolution} f=E_0f+\sum_{k=1}^mQ_kf+(\id-E_m)f.
\sum_{k=1}^mIntroduction passage · 1 passage
Search aliases: ∑_k=1^m sum
Introduction passage · Komorebi_Resolution_Semantics.tex, line 504
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
\label{eq:multiresolution} f=E_0f+\sum_{k=1}^mQ_kf+(\id-E_m)f.
\neIntroduction passage · 2 passages
Search aliases: ≠ ne
Introduction passage · Komorebi_Resolution_Semantics.tex, line 500
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
E_iE_j=E_{\min(i,j)},\qquad Q_k^2=Q_k,\qquad Q_kQ_l=0\ (k\ne l),
\leqIntroduction passage · 9 passages
Search aliases: ≤ leq
Introduction passage · Komorebi_Resolution_Semantics.tex, line 499
[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of these terms. The blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives E_iE_j=E_i for i≤ j, and the reverse prod…
Exact mathematical passage
1\leq k\leq m
\mathbf1Use in source · 2 passages
Search aliases: 1 mathbf
Use in source · Komorebi_Resolution_Semantics.tex, line 551
φ with names vecτ, the standard forcing truth lemma gives M[G]modelsφ(vecτ_G) Longleftrightarrow bnormφ(vecτ)_ B^M∈ G. Boolean-valued forcing and the role of generic quotient models are developed in [source reference]. The completeness assumption here is internal to M. The map b↦1_b∈ G is a Boolean homomorphism but need not preserve every externally given infinite join.
Exact mathematical passage
b\mapsto\mathbf1_{b\in G}
\mathbf1_{b\in G}Use in source · 1 passage
Search aliases: 1_b∈ G mathbf in
Use in source · Komorebi_Resolution_Semantics.tex, line 551
φ with names vecτ, the standard forcing truth lemma gives M[G]modelsφ(vecτ_G) Longleftrightarrow bnormφ(vecτ)_ B^M∈ G. Boolean-valued forcing and the role of generic quotient models are developed in [source reference]. The completeness assumption here is internal to M. The map b↦1_b∈ G is a Boolean homomorphism but need not preserve every externally given infinite join.
Exact mathematical passage
b\mapsto\mathbf1_{b\in G}
\mathbf1_{M[G]\models\varphi}Use in source · 1 passage
Search aliases: 1_M[G]modelsφ mathbf models varphi
Use in source · Komorebi_Resolution_Semantics.tex, line 557
A changing observable and an invariant fragment. Taking two-valued fibers and an arrow M→ M[G] gives D_f(φ)=1_Mmodelsφ +1_M[G]modelsφ. For suitable forcing, CH can change, so it is not invariant under unrestricted forcing. Both CH and its negation are forceable in the relevant forcing setting [source reference]. This is not a contradiction: the quantifiers of CH are interpreted in worlds with different available sets.
Exact mathematical passage
D_f(\varphi)=\mathbf1_{M\models\varphi} +\mathbf1_{M[G]\models\varphi}.
\mathbf1_{M\models\varphi}Use in source · 1 passage
Search aliases: 1_Mmodelsφ mathbf models varphi
Use in source · Komorebi_Resolution_Semantics.tex, line 557
A changing observable and an invariant fragment. Taking two-valued fibers and an arrow M→ M[G] gives D_f(φ)=1_Mmodelsφ +1_M[G]modelsφ. For suitable forcing, CH can change, so it is not invariant under unrestricted forcing. Both CH and its negation are forceable in the relevant forcing setting [source reference]. This is not a contradiction: the quantifiers of CH are interpreted in worlds with different available sets.
Exact mathematical passage
D_f(\varphi)=\mathbf1_{M\models\varphi} +\mathbf1_{M[G]\models\varphi}.
\mathcal AUse in source · 3 passages
Search aliases: A mathcal
Use in source · Komorebi_Resolution_Semantics.tex, line 454
[Residual support determines ambiguity] Take Π to be the connected components of a finite graph G. Define A_Π(a)=U_Π(a)+L_Π(a). Then A_Π(a) is the indicator of the union of components containing a nonzero edge of da: supp A_Π(a)=Sat_Π(supp da). Here edge support is saturated by taking all vertices in components containing a supported edge. A component is ambiguous exactly when it contains both truth values. If it contains an edge with unequal endpoint values, it is mixed. Conversely a path from …
Exact mathematical passage
\mathcal A_\Pi(a)=U_\Pi(a)+L_\Pi(a).
a_0Use in source · 2 passages
Search aliases: a_0
Use in source · Komorebi_Resolution_Semantics.tex, line 340
With root value a_0=0 the labeling above is recovered. With root value a_0=1 the reconstruction is (1,0,1,0). The derivative cannot distinguish these two complementary labelings. That ambiguity is exactly the one-dimensional invariant sector of this connected graph.
Exact mathematical passage
a_0=0
a_1Use in source · 1 passage
Search aliases: a_1
Use in source · Komorebi_Resolution_Semantics.tex, line 216
[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty product case equal to D_f(1)=0.
Exact mathematical passage
a_1,\ldots,a_n\in A
a_iUse in source · 1 passage
Search aliases: a_i
Use in source · Komorebi_Resolution_Semantics.tex, line 217
[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty product case equal to D_f(1)=0.
Exact mathematical passage
D_f\left(\prod_{i=1}^n a_i\right) =\sum_{k=1}^n\left(\prod_{i<k}w(a_i)\right) D_f(a_k)\left(\prod_{i>k}u(a_i)\right).
a_kUse in source · 2 passages
Search aliases: a_k
Use in source · Komorebi_Resolution_Semantics.tex, line 217
[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty product case equal to D_f(1)=0.
Exact mathematical passage
D_f\left(\prod_{i=1}^n a_i\right) =\sum_{k=1}^n\left(\prod_{i<k}w(a_i)\right) D_f(a_k)\left(\prod_{i>k}u(a_i)\right).
A_MUse in source · 3 passages
Search aliases: A_M
Use in source · Komorebi_Resolution_Semantics.tex, line 228
Transported syntax and parameters. For parameters or changing languages, replace A by a functor A_M with translation maps α_f:A_M→ A_N. The correct defect is D_f(a)=B_fv_M(a)+v_Nα_f(a), a∈ A_M. Assuming both transport systems are functorial, the composition law becomes D_gf(a)=B_gD_f(a)+D_g(α_f(a)). The same cancellation proves it. The fixed-A theory is the special case α_f=id. A single aggregate quotient of A will only be used in that special case. This prevents a silent comparison of formulas …
Exact mathematical passage
A_M
A_NUse in source · 1 passage
Search aliases: A_N
Use in source · Komorebi_Resolution_Semantics.tex, line 228
Transported syntax and parameters. For parameters or changing languages, replace A by a functor A_M with translation maps α_f:A_M→ A_N. The correct defect is D_f(a)=B_fv_M(a)+v_Nα_f(a), a∈ A_M. Assuming both transport systems are functorial, the composition law becomes D_gf(a)=B_gD_f(a)+D_g(α_f(a)). The same cancellation proves it. The fixed-A theory is the special case α_f=id. A single aggregate quotient of A will only be used in that special case. This prevents a silent comparison of formulas …
Exact mathematical passage
\alpha_f:A_M\to A_N
a_NUse in source · 1 passage
Search aliases: a_N
Use in source · Komorebi_Resolution_Semantics.tex, line 524
Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.
Exact mathematical passage
\exists N\ \forall n\geq N\quad a_n=a_N,
a_nUse in source · 4 passages
Search aliases: a_n
Use in source · Komorebi_Resolution_Semantics.tex, line 216
[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty product case equal to D_f(1)=0.
Exact mathematical passage
a_1,\ldots,a_n\in A
a_{n+1}Use in source · 1 passage
Search aliases: a_n+1
Use in source · Komorebi_Resolution_Semantics.tex, line 527
Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.
Exact mathematical passage
a_n+a_{n+1}=0
a_{s(C)}Use in source · 1 passage
Search aliases: a_s(C)
Use in source · Komorebi_Resolution_Semantics.tex, line 410
[Finite resolution envelope] Every Boolean retraction P: F_2^W→ I_Π is obtained by choosing one representative s(C)∈ C per block and setting P_s(a)_w=a_s(C) (w∈ C). For each fixed observable a, its possible projected values are exactly {P_s(a):s chooses one point in each block} ={b∈ I_Π:L_Π(a)≤ b≤ U_Π(a)}. Consequently, L_Π(a)=bigwedge_sP_s(a), U_Π(a)=bigvee_sP_s(a). If a is mixed on exactly k blocks, there are exactly 2^k possible invariant outputs for that a. A unital Boolean homomorphism h: F…
Exact mathematical passage
\label{eq:selector} P_s(a)_w=a_{s(C)}\quad(w\in C).
a_uUse in source · 2 passages
Search aliases: a_u
Use in source · Komorebi_Resolution_Semantics.tex, line 305
Finite integrability and reconstruction. Let G=(W,E) be a finite simple undirected graph. Edges record which worlds are directly compared. Choose an orientation only to index them. For a∈ F_2^W set (da)_{u,v}=a_u+a_v. Unlike the semantic algebra A, the space F_2^W contains every binary labeling. An actual family of models and a selected language may realize only a proper subalgebra of these labelings.
Exact mathematical passage
(da)_{\{u,v\}}=a_u+a_v.
a_vUse in source · 2 passages
Search aliases: a_v
Use in source · Komorebi_Resolution_Semantics.tex, line 305
Finite integrability and reconstruction. Let G=(W,E) be a finite simple undirected graph. Edges record which worlds are directly compared. Choose an orientation only to index them. For a∈ F_2^W set (da)_{u,v}=a_u+a_v. Unlike the semantic algebra A, the space F_2^W contains every binary labeling. An actual family of models and a selected language may realize only a proper subalgebra of these labelings.
Exact mathematical passage
(da)_{\{u,v\}}=a_u+a_v.
\mathcal A_\PiUse in source · 3 passages
Search aliases: A_Π mathcal Pi
Use in source · Komorebi_Resolution_Semantics.tex, line 454
[Residual support determines ambiguity] Take Π to be the connected components of a finite graph G. Define A_Π(a)=U_Π(a)+L_Π(a). Then A_Π(a) is the indicator of the union of components containing a nonzero edge of da: supp A_Π(a)=Sat_Π(supp da). Here edge support is saturated by taking all vertices in components containing a supported edge. A component is ambiguous exactly when it contains both truth values. If it contains an edge with unequal endpoint values, it is mixed. Conversely a path from …
Exact mathematical passage
\mathcal A_\Pi(a)=U_\Pi(a)+L_\Pi(a).
\mathbb BUse in source · 2 passages
Search aliases: B mathbb
Use in source · Komorebi_Resolution_Semantics.tex, line 545
Forcing as a conditional set-theoretic instance. Fix externally a countable transitive MmodelsZFC. Let B∈ M be a Boolean algebra that M regards as complete, and let G be an M-generic ultrafilter. For a sentence φ with names vecτ, the standard forcing truth lemma gives M[G]modelsφ(vecτ_G) Longleftrightarrow bnormφ(vecτ)_ B^M∈ G. Boolean-valued forcing and the role of generic quotient models are developed in [source reference]. The completeness assumption here is internal to M. The map b↦1_b∈ G is…
Exact mathematical passage
\mathbb B\in M
bUse in source · 14 passages
Search aliases: b
Use in source · Komorebi_Resolution_Semantics.tex, line 184
[Defect calculus] Put u=B_f∘ v_M and w=v_N. Then D_f(0) · =D_f(1)=0, D_f(neg a)=D_f(a), D_f(a+b) · =D_f(a)+D_f(b), D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characte…
Exact mathematical passage
D_f(0)&=D_f(1)=0,\qquad D_f(\neg a)=D_f(a),\\ D_f(a+b)&=D_f(a)+D_f(b),\\ D_f(ab)&=D_f(a)u(b)+w(a)D_f(b).\label{eq:leibniz}
B_{f_n\cdots f_1}Use in source · 1 passage
Search aliases: B_f_n⋯ f_1 cdots
Use in source · Komorebi_Resolution_Semantics.tex, line 346
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
\label{eq:pathreconstruction} v_{M_n}(a)=B_{f_n\cdots f_1}v_{M_0}(a) +\sum_{k=1}^n B_{f_n\cdots f_{k+1}}D_{f_k}(a),
B_{f_n\cdots f_{k+1}}Use in source · 1 passage
Search aliases: B_f_n⋯ f_k+1 cdots
Use in source · Komorebi_Resolution_Semantics.tex, line 346
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
\label{eq:pathreconstruction} v_{M_n}(a)=B_{f_n\cdots f_1}v_{M_0}(a) +\sum_{k=1}^n B_{f_n\cdots f_{k+1}}D_{f_k}(a),
B_gUse in source · 3 passages
Search aliases: B_g
Use in source · Komorebi_Resolution_Semantics.tex, line 190
D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(…
Exact mathematical passage
\label{eq:cocycle} D_{gf}(a)=B_g(D_f(a))+D_g(a),\qquad D_{\id_M}(a)=0.
B_{gf}Use in source · 1 passage
Search aliases: B_gf
Use in source · Komorebi_Resolution_Semantics.tex, line 205
Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(a)+v_P(a) · =B_gfv_M(a)+v_P(a). The identity case follows from B_id_M=id.
Exact mathematical passage
B_g(D_f(a))+D_g(a) &=B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(a)+v_P(a)\\ &=B_{gf}v_M(a)+v_P(a).
B_{\id_M}Use in source · 1 passage
Search aliases: B_id_M id operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 210
· =B_gfv_M(a)+v_P(a). The identity case follows from B_id_M=id.
Exact mathematical passage
B_{\id_M}=\id
b_MUse in source · 2 passages
Search aliases: b_M
Use in source · Komorebi_Resolution_Semantics.tex, line 287
Coherent sections and definability. The limit varprojlim_ CB consists of families (b_M)_M satisfying B_f(b_M)=b_N. Thus a∈ I precisely when Ev(a) is a coherent family. But not every coherent family need be represented by a sentence, and distinct sentences can have the same evaluations.
Exact mathematical passage
(b_M)_M
b_NUse in source · 1 passage
Search aliases: b_N
Use in source · Komorebi_Resolution_Semantics.tex, line 287
Coherent sections and definability. The limit varprojlim_ CB consists of families (b_M)_M satisfying B_f(b_M)=b_N. Thus a∈ I precisely when Ev(a) is a coherent family. But not every coherent family need be represented by a sentence, and distinct sentences can have the same evaluations.
Exact mathematical passage
B_f(b_M)=b_N
B_{\operatorname{target}(f)}Use in source · 1 passage
Search aliases: B_target(f) operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 240
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
C^0=\prod_{M\in\Ob(\mathcal C)}B_M,\qquad C^1=\prod_{f\in\Mor(\mathcal C)}B_{\operatorname{target}(f)}.
\bigveeUse in source · 2 passages
Search aliases: bigvee
Use in source · Komorebi_Resolution_Semantics.tex, line 402
The finite resolution envelope theorem. Let Π be a partition of a finite nonempty set W. Let I_Π⊆ F_2^W be the Boolean algebra of functions constant on each block. For w∈ C∈Π define L_Π(a)_w=bigwedge_u∈ Ca_u, U_Π(a)_w=bigvee_u∈ Ca_u. These lower and upper approximations are the classical equivalence-relation rough-set operators [source reference]. Here they give a precise replacement for a missing canonical projector.
Exact mathematical passage
L_\Pi(a)_w=\bigwedge_{u\in C}a_u,\qquad U_\Pi(a)_w=\bigvee_{u\in C}a_u.
\bigvee_sUse in source · 1 passage
Search aliases: bigvee_s bigvee
Use in source · Komorebi_Resolution_Semantics.tex, line 419
is obtained by choosing one representative s(C)∈ C per block and setting P_s(a)_w=a_s(C) (w∈ C). For each fixed observable a, its possible projected values are exactly {P_s(a):s chooses one point in each block} ={b∈ I_Π:L_Π(a)≤ b≤ U_Π(a)}. Consequently, L_Π(a)=bigwedge_sP_s(a), U_Π(a)=bigvee_sP_s(a). If a is mixed on exactly k blocks, there are exactly 2^k possible invariant outputs for that a. A unital Boolean homomorphism h: F_2^W→ F_2 must evaluate at exactly one point. To see this, the singl…
Exact mathematical passage
L_\Pi(a)=\bigwedge_sP_s(a),\qquad U_\Pi(a)=\bigvee_sP_s(a).
\bigvee_{u\in C}Use in source · 1 passage
Search aliases: bigvee_u∈ C bigvee in
Use in source · Komorebi_Resolution_Semantics.tex, line 402
The finite resolution envelope theorem. Let Π be a partition of a finite nonempty set W. Let I_Π⊆ F_2^W be the Boolean algebra of functions constant on each block. For w∈ C∈Π define L_Π(a)_w=bigwedge_u∈ Ca_u, U_Π(a)_w=bigvee_u∈ Ca_u. These lower and upper approximations are the classical equivalence-relation rough-set operators [source reference]. Here they give a precise replacement for a missing canonical projector.
Exact mathematical passage
L_\Pi(a)_w=\bigwedge_{u\in C}a_u,\qquad U_\Pi(a)_w=\bigvee_{u\in C}a_u.
\bigwedgeUse in source · 2 passages
Search aliases: bigwedge
Use in source · Komorebi_Resolution_Semantics.tex, line 402
The finite resolution envelope theorem. Let Π be a partition of a finite nonempty set W. Let I_Π⊆ F_2^W be the Boolean algebra of functions constant on each block. For w∈ C∈Π define L_Π(a)_w=bigwedge_u∈ Ca_u, U_Π(a)_w=bigvee_u∈ Ca_u. These lower and upper approximations are the classical equivalence-relation rough-set operators [source reference]. Here they give a precise replacement for a missing canonical projector.
Exact mathematical passage
L_\Pi(a)_w=\bigwedge_{u\in C}a_u,\qquad U_\Pi(a)_w=\bigvee_{u\in C}a_u.
\bigwedge_sUse in source · 1 passage
Search aliases: bigwedge_s bigwedge
Use in source · Komorebi_Resolution_Semantics.tex, line 419
is obtained by choosing one representative s(C)∈ C per block and setting P_s(a)_w=a_s(C) (w∈ C). For each fixed observable a, its possible projected values are exactly {P_s(a):s chooses one point in each block} ={b∈ I_Π:L_Π(a)≤ b≤ U_Π(a)}. Consequently, L_Π(a)=bigwedge_sP_s(a), U_Π(a)=bigvee_sP_s(a). If a is mixed on exactly k blocks, there are exactly 2^k possible invariant outputs for that a. A unital Boolean homomorphism h: F_2^W→ F_2 must evaluate at exactly one point. To see this, the singl…
Exact mathematical passage
L_\Pi(a)=\bigwedge_sP_s(a),\qquad U_\Pi(a)=\bigvee_sP_s(a).
\bigwedge_{u\in C}Use in source · 1 passage
Search aliases: bigwedge_u∈ C bigwedge in
Use in source · Komorebi_Resolution_Semantics.tex, line 402
The finite resolution envelope theorem. Let Π be a partition of a finite nonempty set W. Let I_Π⊆ F_2^W be the Boolean algebra of functions constant on each block. For w∈ C∈Π define L_Π(a)_w=bigwedge_u∈ Ca_u, U_Π(a)_w=bigvee_u∈ Ca_u. These lower and upper approximations are the classical equivalence-relation rough-set operators [source reference]. Here they give a precise replacement for a missing canonical projector.
Exact mathematical passage
L_\Pi(a)_w=\bigwedge_{u\in C}a_u,\qquad U_\Pi(a)_w=\bigvee_{u\in C}a_u.
\bnormUse in source · 4 passages
Search aliases: bnorm
Use in source · Komorebi_Resolution_Semantics.tex, line 136
res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An indu…
Exact mathematical passage
\bnorm{x=y}=\{i:x_i=y_i\},\qquad \bnorm{R(\vec x)}=\{i:M_i\models R(\vec x_i)\}.
\botUse in source · 3 passages
Search aliases: bot
Use in source · Komorebi_Resolution_Semantics.tex, line 119
There is an equally important distinction between transformation and information refinement. In the information order on {bot,0,1}, where bot≤0 and bot≤1 but 0 and 1 are incomparable, a monotone refinement cannot change a settled 0 into a settled 1. A truth-changing world transformation is therefore not a monotone refinement of a fixed partial truth assignment. It may instead change the quantifier domain, the admissible objects, or the semantic realization.
Exact mathematical passage
\{\bot,0,1\}
\BoxUse in source · 2 passages
Search aliases: Box
Use in source · Komorebi_Resolution_Semantics.tex, line 529
On a reflexive accessibility relation, DiamondBoxφ says that a world can be reached from which every later accessible world satisfies φ. A path that happens to stay at a φ-world does not establish this. A two-state switch system with arrows in both directions and identity loops already separates the notions.
Exact mathematical passage
\Diamond\Box\varphi
CUse in source · 14 passages
Search aliases: C
Use in source · Komorebi_Resolution_Semantics.tex, line 240
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
C^0=\prod_{M\in\Ob(\mathcal C)}B_M,\qquad C^1=\prod_{f\in\Mor(\mathcal C)}B_{\operatorname{target}(f)}.
cUse in source · 7 passages
Search aliases: c
Use in source · Komorebi_Resolution_Semantics.tex, line 245
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
(d^0c)_f=B_f(c_M)+c_N,\qquad \Ev(a)=(v_M(a))_M,\qquad D=d^0\Ev.
C^0Use in source · 1 passage
Search aliases: C^0
Use in source · Komorebi_Resolution_Semantics.tex, line 240
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
C^0=\prod_{M\in\Ob(\mathcal C)}B_M,\qquad C^1=\prod_{f\in\Mor(\mathcal C)}B_{\operatorname{target}(f)}.
C^1Use in source · 1 passage
Search aliases: C^1
Use in source · Komorebi_Resolution_Semantics.tex, line 240
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
C^0=\prod_{M\in\Ob(\mathcal C)}B_M,\qquad C^1=\prod_{f\in\Mor(\mathcal C)}B_{\operatorname{target}(f)}.
c_MUse in source · 1 passage
Search aliases: c_M
Use in source · Komorebi_Resolution_Semantics.tex, line 245
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
(d^0c)_f=B_f(c_M)+c_N,\qquad \Ev(a)=(v_M(a))_M,\qquad D=d^0\Ev.
c_NUse in source · 1 passage
Search aliases: c_N
Use in source · Komorebi_Resolution_Semantics.tex, line 245
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
(d^0c)_f=B_f(c_M)+c_N,\qquad \Ev(a)=(v_M(a))_M,\qquad D=d^0\Ev.
dUse in source · 17 passages
Search aliases: d
Use in source · Komorebi_Resolution_Semantics.tex, line 245
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
(d^0c)_f=B_f(c_M)+c_N,\qquad \Ev(a)=(v_M(a))_M,\qquad D=d^0\Ev.
d^0Use in source · 2 passages
Search aliases: d^0
Use in source · Komorebi_Resolution_Semantics.tex, line 245
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
(d^0c)_f=B_f(c_M)+c_N,\qquad \Ev(a)=(v_M(a))_M,\qquad D=d^0\Ev.
D_{f_k}Use in source · 1 passage
Search aliases: D_f_k
Use in source · Komorebi_Resolution_Semantics.tex, line 346
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
\label{eq:pathreconstruction} v_{M_n}(a)=B_{f_n\cdots f_1}v_{M_0}(a) +\sum_{k=1}^n B_{f_n\cdots f_{k+1}}D_{f_k}(a),
D_gUse in source · 3 passages
Search aliases: D_g
Use in source · Komorebi_Resolution_Semantics.tex, line 190
D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(…
Exact mathematical passage
\label{eq:cocycle} D_{gf}(a)=B_g(D_f(a))+D_g(a),\qquad D_{\id_M}(a)=0.
D_{gf}Use in source · 2 passages
Search aliases: D_gf
Use in source · Komorebi_Resolution_Semantics.tex, line 190
D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(…
Exact mathematical passage
\label{eq:cocycle} D_{gf}(a)=B_g(D_f(a))+D_g(a),\qquad D_{\id_M}(a)=0.
D_{\id_M}Use in source · 1 passage
Search aliases: D_id_M id operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 190
D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(…
Exact mathematical passage
\label{eq:cocycle} D_{gf}(a)=B_g(D_f(a))+D_g(a),\qquad D_{\id_M}(a)=0.
\DiamondUse in source · 2 passages
Search aliases: Diamond
Use in source · Komorebi_Resolution_Semantics.tex, line 529
On a reflexive accessibility relation, DiamondBoxφ says that a world can be reached from which every later accessible world satisfies φ. A path that happens to stay at a φ-world does not establish this. A two-state switch system with arrows in both directions and identity loops already separates the notions.
Exact mathematical passage
\Diamond\Box\varphi
\dimUse in source · 2 passages
Search aliases: dim
Use in source · Komorebi_Resolution_Semantics.tex, line 316
m edges, and c connected components. item ker d consists exactly of the componentwise constant labelings and has dimension c. item An edge field r∈ F_2^E equals da for some a if and only if its sum around every cycle is zero. item Every integrable r has exactly 2^c reconstructions. Fixing one vertex value per component makes the reconstruction unique. item dimim d=n-c and dim( F_2^E/im d)=m-n+c. If da=0, endpoints of every edge have equal values, so values agree along each path. Conversely compo…
Exact mathematical passage
\dim\im d=n-c
eUse in source · 7 passages
Search aliases: e
Use in source · Komorebi_Resolution_Semantics.tex, line 391
A two-world obstruction. [No symmetric Boolean projector] Let A= F_2^2 and I={(0,0),(1,1)}. There is no Boolean retraction P:A→ I that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto I. Write e_1=(1,0) and e_2=(0,1). Swapping coordinates fixes every element of I, so equivariance forces P(e_1)=P(e_2). Additivity then gives P(1,1)=P(e_1+e_2)=P(e_1)+P(e_2)=0, contrary to the retraction condition P(1,1)=(1,1). Every Boolean retraction is addi…
Exact mathematical passage
e_1=(1,0)
e_1Use in source · 6 passages
Search aliases: e_1
Use in source · Komorebi_Resolution_Semantics.tex, line 391
A two-world obstruction. [No symmetric Boolean projector] Let A= F_2^2 and I={(0,0),(1,1)}. There is no Boolean retraction P:A→ I that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto I. Write e_1=(1,0) and e_2=(0,1). Swapping coordinates fixes every element of I, so equivariance forces P(e_1)=P(e_2). Additivity then gives P(1,1)=P(e_1+e_2)=P(e_1)+P(e_2)=0, contrary to the retraction condition P(1,1)=(1,1). Every Boolean retraction is addi…
Exact mathematical passage
e_1=(1,0)
e_2Use in source · 6 passages
Search aliases: e_2
Use in source · Komorebi_Resolution_Semantics.tex, line 391
A two-world obstruction. [No symmetric Boolean projector] Let A= F_2^2 and I={(0,0),(1,1)}. There is no Boolean retraction P:A→ I that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto I. Write e_1=(1,0) and e_2=(0,1). Swapping coordinates fixes every element of I, so equivariance forces P(e_1)=P(e_2). Additivity then gives P(1,1)=P(e_1+e_2)=P(e_1)+P(e_2)=0, contrary to the retraction condition P(1,1)=(1,1). Every Boolean retraction is addi…
Exact mathematical passage
e_2=(0,1)
\EvUse in source · 5 passages
Search aliases: Ev operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 245
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
(d^0c)_f=B_f(c_M)+c_N,\qquad \Ev(a)=(v_M(a))_M,\qquad D=d^0\Ev.
\FUse in source · 21 passages
Search aliases: F F_2 mathbb
Use in source · Komorebi_Resolution_Semantics.tex, line 172
For sentences, A can be the Lindenbaum Boolean algebra of L-sentences modulo provable equivalence over T. An ordinary model gives B_M= F_2 and v_M([φ])=1 precisely when Mmodelsφ. The datum can also be studied without claiming that its valuations arise from models of ZFC.
Exact mathematical passage
B_M=\F
\F^{\mathbb N}Use in source · 1 passage
Search aliases: F^ N F_2^ N F mathbb mathbb mathbb
Use in source · Komorebi_Resolution_Semantics.tex, line 301
For example, a countable sentence algebra cannot cover all coherent families over a countably infinite discrete diagram with B_M= F_2, since that limit is F_2^ N and is uncountable. The correct phrase is therefore definable coherent observables, not ``all global sections.'' No sheaf has yet been defined. A sheaf additionally requires a site and a gluing condition, neither of which follows from a limit alone.
Exact mathematical passage
\F^{\mathbb N}
\F^2Use in source · 3 passages
Search aliases: F^2 F_2^2 F mathbb
Use in source · Komorebi_Resolution_Semantics.tex, line 388
A two-world obstruction. [No symmetric Boolean projector] Let A= F_2^2 and I={(0,0),(1,1)}. There is no Boolean retraction P:A→ I that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto I. Write e_1=(1,0) and e_2=(0,1). Swapping coordinates fixes every element of I, so equivariance forces P(e_1)=P(e_2). Additivity then gives P(1,1)=P(e_1+e_2)=P(e_1)+P(e_2)=0, contrary to the retraction condition P(1,1)=(1,1). Every Boolean retraction is addi…
Exact mathematical passage
A=\F^2
\F^EUse in source · 3 passages
Search aliases: F^E F_2^E F mathbb
Use in source · Komorebi_Resolution_Semantics.tex, line 314
[Finite residual integrability] Suppose G has n vertices, m edges, and c connected components. item ker d consists exactly of the componentwise constant labelings and has dimension c. item An edge field r∈ F_2^E equals da for some a if and only if its sum around every cycle is zero. item Every integrable r has exactly 2^c reconstructions. Fixing one vertex value per component makes the reconstruction unique. item dimim d=n-c and dim( F_2^E/im d)=m-n+c. If da=0, endpoints of every edge have equal…
Exact mathematical passage
r\in\F^E
\F^WUse in source · 5 passages
Search aliases: F^W F_2^W F mathbb
Use in source · Komorebi_Resolution_Semantics.tex, line 304
Finite integrability and reconstruction. Let G=(W,E) be a finite simple undirected graph. Edges record which worlds are directly compared. Choose an orientation only to index them. For a∈ F_2^W set (da)_{u,v}=a_u+a_v. Unlike the semantic algebra A, the space F_2^W contains every binary labeling. An actual family of models and a selected language may realize only a proper subalgebra of these labelings.
Exact mathematical passage
a\in\F^W
f_1Use in source · 2 passages
Search aliases: f_1
Use in source · Komorebi_Resolution_Semantics.tex, line 345
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
M_0\xrightarrow{f_1}\cdots\xrightarrow{f_n}M_n
f_kUse in source · 1 passage
Search aliases: f_k
Use in source · Komorebi_Resolution_Semantics.tex, line 346
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
\label{eq:pathreconstruction} v_{M_n}(a)=B_{f_n\cdots f_1}v_{M_0}(a) +\sum_{k=1}^n B_{f_n\cdots f_{k+1}}D_{f_k}(a),
f_{k+1}Use in source · 1 passage
Search aliases: f_k+1
Use in source · Komorebi_Resolution_Semantics.tex, line 346
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
\label{eq:pathreconstruction} v_{M_n}(a)=B_{f_n\cdots f_1}v_{M_0}(a) +\sum_{k=1}^n B_{f_n\cdots f_{k+1}}D_{f_k}(a),
f_nUse in source · 2 passages
Search aliases: f_n
Use in source · Komorebi_Resolution_Semantics.tex, line 345
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
M_0\xrightarrow{f_1}\cdots\xrightarrow{f_n}M_n
f_uUse in source · 1 passage
Search aliases: f_u
Use in source · Komorebi_Resolution_Semantics.tex, line 475
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
(E_\Pi f)_w= \frac{\sum_{u\in C}\mu_u f_u}{\sum_{u\in C}\mu_u} \quad(w\in C\in\Pi).
f_wUse in source · 2 passages
Search aliases: f_w
Use in source · Komorebi_Resolution_Semantics.tex, line 471
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
\langle f,g\rangle_\mu=\sum_{w\in W}\mu_w f_wg_w.
GUse in source · 10 passages
Search aliases: G
Use in source · Komorebi_Resolution_Semantics.tex, line 304
Finite integrability and reconstruction. Let G=(W,E) be a finite simple undirected graph. Edges record which worlds are directly compared. Choose an orientation only to index them. For a∈ F_2^W set (da)_{u,v}=a_u+a_v. Unlike the semantic algebra A, the space F_2^W contains every binary labeling. An actual family of models and a selected language may realize only a proper subalgebra of these labelings.
Exact mathematical passage
G=(W,E)
gUse in source · 7 passages
Search aliases: g
Use in source · Komorebi_Resolution_Semantics.tex, line 189
D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(…
Exact mathematical passage
M\xrightarrow{f}N\xrightarrow{g}P
g_CUse in source · 1 passage
Search aliases: g_C
Use in source · Komorebi_Resolution_Semantics.tex, line 489
d E_Π is the unique orthogonal projector onto the real vector space of blockwise constant functions. It is positive, fixes 1, and is equivariant under weight-preserving permutations that preserve the partition. For binary a, L_Π(a)≤ E_Π(a)≤ U_Π(a). The formula is blockwise constant and fixes such functions. For every blockwise constant g, ⟨ f-E_Π f,g⟩_μ =∑_C∈Πg_C∑_w∈ Cμ_w(f_w-(E_Π f)_w)=0. This characterizes the orthogonal projector uniquely. Positivity, equivariance, and the interval bounds fol…
Exact mathematical passage
\langle f-E_\Pi f,g\rangle_\mu =\sum_{C\in\Pi}g_C\sum_{w\in C}\mu_w(f_w-(E_\Pi f)_w)=0.
g_wUse in source · 1 passage
Search aliases: g_w
Use in source · Komorebi_Resolution_Semantics.tex, line 471
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
\langle f,g\rangle_\mu=\sum_{w\in W}\mu_w f_wg_w.
hUse in source · 5 passages
Search aliases: h
Use in source · Komorebi_Resolution_Semantics.tex, line 108
[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a…
Exact mathematical passage
h:M\cong N
h^{-1}Use in source · 1 passage
Search aliases: h^-1
Use in source · Komorebi_Resolution_Semantics.tex, line 116
Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a witness through h in one direction and through h^-1 in the other.
Exact mathematical passage
h^{-1}
IUse in source · 9 passages
Search aliases: I
Use in source · Komorebi_Resolution_Semantics.tex, line 128
A common presentation exists, but need not explain anything. [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function …
Exact mathematical passage
I
\id_MUse in source · 2 passages
Search aliases: id_M id operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 190
D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(…
Exact mathematical passage
\label{eq:cocycle} D_{gf}(a)=B_g(D_f(a))+D_g(a),\qquad D_{\id_M}(a)=0.
\imUse in source · 7 passages
Search aliases: im operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 258
[Invariant algebra and residual quotient] The set I=ker D=⋂_f{a:B_fv_M(a)=v_N(a)} is a unital Boolean subalgebra of A. There is a natural exact sequence of F_2-vector spaces 0⟶ I⟶ AxrightarrowDim D⟶0, and therefore a natural additive isomorphism A/ I ≅ im D. Two observables have the same residual profile exactly when their symmetric difference belongs to I. Each defining condition for I is an equalizer of two Boolean homomorphisms. Equalizers are closed under 0, 1, addition, multiplication, and …
Exact mathematical passage
\label{eq:exactsequence} 0\longrightarrow\Inv\longrightarrow A\xrightarrow{D}\im D\longrightarrow0,
\Inv_\PiUse in source · 5 passages
Search aliases: Inv_Π I_Π Inv Pi mathcal Pi
Use in source · Komorebi_Resolution_Semantics.tex, line 401
The finite resolution envelope theorem. Let Π be a partition of a finite nonempty set W. Let I_Π⊆ F_2^W be the Boolean algebra of functions constant on each block. For w∈ C∈Π define L_Π(a)_w=bigwedge_u∈ Ca_u, U_Π(a)_w=bigvee_u∈ Ca_u. These lower and upper approximations are the classical equivalence-relation rough-set operators [source reference]. Here they give a precise replacement for a missing canonical projector.
Exact mathematical passage
\Inv_\Pi\subseteq\F^W
JUse in source · 2 passages
Search aliases: J
Use in source · Komorebi_Resolution_Semantics.tex, line 367
If r is a Boolean retraction, its kernel is an ideal J and the additive decomposition is A= I⊕ J. In these coordinates multiplication is (i,x)(j,y)=(ij, iy+jx+xy). It is not generally a direct-product multiplication. Writing a direct sum symbol must not silently erase these cross terms.
Exact mathematical passage
J
\kerUse in source · 5 passages
Search aliases: ker
Use in source · Komorebi_Resolution_Semantics.tex, line 254
[Invariant algebra and residual quotient] The set I=ker D=⋂_f{a:B_fv_M(a)=v_N(a)} is a unital Boolean subalgebra of A. There is a natural exact sequence of F_2-vector spaces 0⟶ I⟶ AxrightarrowDim D⟶0, and therefore a natural additive isomorphism A/ I ≅ im D. Two observables have the same residual profile exactly when their symmetric difference belongs to I. Each defining condition for I is an equalizer of two Boolean homomorphisms. Equalizers are closed under 0, 1, addition, multiplication, and …
Exact mathematical passage
\Inv=\ker D=\bigcap_f\{a:B_fv_M(a)=v_N(a)\}
L_{\Pi_0}Use in source · 1 passage
Search aliases: L_Π_0 Pi
Use in source · Komorebi_Resolution_Semantics.tex, line 516
For the original Boolean bounds, finer partitions instead give monotone information: L_Π_0(a)≤ L_Π_1(a)≤⋯≤ a ≤⋯≤ U_Π_1(a)≤ U_Π_0(a). The ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.
Exact mathematical passage
L_{\Pi_0}(a)\leq L_{\Pi_1}(a)\leq\cdots\leq a \leq\cdots\leq U_{\Pi_1}(a)\leq U_{\Pi_0}(a).
L_{\Pi_1}Use in source · 1 passage
Search aliases: L_Π_1 Pi
Use in source · Komorebi_Resolution_Semantics.tex, line 516
For the original Boolean bounds, finer partitions instead give monotone information: L_Π_0(a)≤ L_Π_1(a)≤⋯≤ a ≤⋯≤ U_Π_1(a)≤ U_Π_0(a). The ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.
Exact mathematical passage
L_{\Pi_0}(a)\leq L_{\Pi_1}(a)\leq\cdots\leq a \leq\cdots\leq U_{\Pi_1}(a)\leq U_{\Pi_0}(a).
\LongleftrightarrowUse in source · 3 passages
Search aliases: Longleftrightarrow
Use in source · Komorebi_Resolution_Semantics.tex, line 109
[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a…
Exact mathematical passage
M\models\varphi(\vec a)\quad\Longleftrightarrow\quad N\models\varphi(h\vec a).
\mathcal MUse in source · 2 passages
Search aliases: M mathcal
Use in source · Komorebi_Resolution_Semantics.tex, line 128
A common presentation exists, but need not explain anything. [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function …
Exact mathematical passage
\mathcal M
M_0Use in source · 3 passages
Search aliases: M_0
Use in source · Komorebi_Resolution_Semantics.tex, line 345
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
M_0\xrightarrow{f_1}\cdots\xrightarrow{f_n}M_n
M_1Use in source · 1 passage
Search aliases: M_1
Use in source · Komorebi_Resolution_Semantics.tex, line 523
Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.
Exact mathematical passage
M_0\to M_1\to\cdots
M_iUse in source · 8 passages
Search aliases: M_i
Use in source · Komorebi_Resolution_Semantics.tex, line 128
A common presentation exists, but need not explain anything. [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function …
Exact mathematical passage
(M_i)_{i\in I}
M_nUse in source · 3 passages
Search aliases: M_n
Use in source · Komorebi_Resolution_Semantics.tex, line 345
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
M_0\xrightarrow{f_1}\cdots\xrightarrow{f_n}M_n
\MorUse in source · 1 passage
Search aliases: Mor operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 240
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
C^0=\prod_{M\in\Ob(\mathcal C)}B_M,\qquad C^1=\prod_{f\in\Mor(\mathcal C)}B_{\operatorname{target}(f)}.
\mathbb NUse in source · 5 passages
Search aliases: N mathbb
Use in source · Komorebi_Resolution_Semantics.tex, line 301
For example, a countable sentence algebra cannot cover all coherent families over a countably infinite discrete diagram with B_M= F_2, since that limit is F_2^ N and is uncountable. The correct phrase is therefore definable coherent observables, not ``all global sections.'' No sheaf has yet been defined. A sheaf additionally requires a site and a gluing condition, neither of which follows from a limit alone.
Exact mathematical passage
\F^{\mathbb N}
nUse in source · 13 passages
Search aliases: n
Use in source · Komorebi_Resolution_Semantics.tex, line 216
[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty product case equal to D_f(1)=0.
Exact mathematical passage
a_1,\ldots,a_n\in A
\ObUse in source · 1 passage
Search aliases: Ob operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 240
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
C^0=\prod_{M\in\Ob(\mathcal C)}B_M,\qquad C^1=\prod_{f\in\Mor(\mathcal C)}B_{\operatorname{target}(f)}.
P^2Use in source · 2 passages
Search aliases: P^2
Use in source · Komorebi_Resolution_Semantics.tex, line 357
[Additive splitting] Over a metatheory with choice, the inclusion I⊆ A always admits an F_2-linear retraction r:A→ I. For P=ι r and R=id+P, P^2=P, R^2=R, PR=RP=0, a=P(a)+R(a). This is a vector-space decomposition. The retraction need not preserve multiplication or be canonical. Choose a vector-space basis of I and extend it to a basis of A. Let r fix the first basis and kill the added vectors. Then P^2=P. In characteristic two, (id+P)^2=id+P, and P(id+P)=(id+P)P=0.
Exact mathematical passage
P^2=P,\quad R^2=R,\quad PR=RP=0,\quad a=P(a)+R(a).
\PowUse in source · 3 passages
Search aliases: Pow P mathcal
Use in source · Komorebi_Resolution_Semantics.tex, line 128
A common presentation exists, but need not explain anything. [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function …
Exact mathematical passage
\Pow(I)
RUse in source · 4 passages
Search aliases: R
Use in source · Komorebi_Resolution_Semantics.tex, line 136
res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An indu…
Exact mathematical passage
\bnorm{x=y}=\{i:x_i=y_i\},\qquad \bnorm{R(\vec x)}=\{i:M_i\models R(\vec x_i)\}.
rUse in source · 12 passages
Search aliases: r
Use in source · Komorebi_Resolution_Semantics.tex, line 314
[Finite residual integrability] Suppose G has n vertices, m edges, and c connected components. item ker d consists exactly of the componentwise constant labelings and has dimension c. item An edge field r∈ F_2^E equals da for some a if and only if its sum around every cycle is zero. item Every integrable r has exactly 2^c reconstructions. Fixing one vertex value per component makes the reconstruction unique. item dimim d=n-c and dim( F_2^E/im d)=m-n+c. If da=0, endpoints of every edge have equal…
Exact mathematical passage
r\in\F^E
R^2Use in source · 1 passage
Search aliases: R^2
Use in source · Komorebi_Resolution_Semantics.tex, line 357
[Additive splitting] Over a metatheory with choice, the inclusion I⊆ A always admits an F_2-linear retraction r:A→ I. For P=ι r and R=id+P, P^2=P, R^2=R, PR=RP=0, a=P(a)+R(a). This is a vector-space decomposition. The retraction need not preserve multiplication or be canonical. Choose a vector-space basis of I and extend it to a basis of A. Let r fix the first basis and kill the added vectors. Then P^2=P. In characteristic two, (id+P)^2=id+P, and P(id+P)=(id+P)P=0.
Exact mathematical passage
P^2=P,\quad R^2=R,\quad PR=RP=0,\quad a=P(a)+R(a).
SUse in source · 9 passages
Search aliases: S
Use in source · Komorebi_Resolution_Semantics.tex, line 129
ommon presentation exists, but need not explain anything [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symb…
Exact mathematical passage
U_i=\{S\subseteq I:i\in S\}
s^2Use in source · 2 passages
Search aliases: s^2
Use in source · Komorebi_Resolution_Semantics.tex, line 590
[Nontrivial holonomy with an exact defect] Let the one-object category be the group with generator s and relation s^2=1. Put X= F_2^2 and let T_s swap coordinates. This is a valid nontrivial action. Its invariant vectors are (0,0) and (1,1). For x=(1,0), T_sx+x=(1,1)≠0, but T_s(T_sx+x)+(T_sx+x)=0. The second equation is the transported composition law for s^2=1. Thus nontrivial fiber transport is compatible with an exact defect. No contradiction or unexplained curvature is involved.
Exact mathematical passage
s^2=1
\SatUse in source · 1 passage
Search aliases: Sat operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 458
[Residual support determines ambiguity] Take Π to be the connected components of a finite graph G. Define A_Π(a)=U_Π(a)+L_Π(a). Then A_Π(a) is the indicator of the union of components containing a nonzero edge of da: supp A_Π(a)=Sat_Π(supp da). Here edge support is saturated by taking all vertices in components containing a supported edge. A component is ambiguous exactly when it contains both truth values. If it contains an edge with unequal endpoint values, it is mixed. Conversely a path from …
Exact mathematical passage
\supp\mathcal A_\Pi(a)=\Sat_\Pi(\supp da).
\Sat_\PiUse in source · 1 passage
Search aliases: Sat_Π Sat Pi operatorname Pi
Use in source · Komorebi_Resolution_Semantics.tex, line 458
[Residual support determines ambiguity] Take Π to be the connected components of a finite graph G. Define A_Π(a)=U_Π(a)+L_Π(a). Then A_Π(a) is the indicator of the union of components containing a nonzero edge of da: supp A_Π(a)=Sat_Π(supp da). Here edge support is saturated by taking all vertices in components containing a supported edge. A component is ambiguous exactly when it contains both truth values. If it contains an edge with unequal endpoint values, it is mixed. Conversely a path from …
Exact mathematical passage
\supp\mathcal A_\Pi(a)=\Sat_\Pi(\supp da).
\suppUse in source · 1 passage
Search aliases: supp operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 458
[Residual support determines ambiguity] Take Π to be the connected components of a finite graph G. Define A_Π(a)=U_Π(a)+L_Π(a). Then A_Π(a) is the indicator of the union of components containing a nonzero edge of da: supp A_Π(a)=Sat_Π(supp da). Here edge support is saturated by taking all vertices in components containing a supported edge. A component is ambiguous exactly when it contains both truth values. If it contains an edge with unequal endpoint values, it is mixed. Conversely a path from …
Exact mathematical passage
\supp\mathcal A_\Pi(a)=\Sat_\Pi(\supp da).
T_fUse in source · 1 passage
Search aliases: T_f
Use in source · Komorebi_Resolution_Semantics.tex, line 583
To retain information about structures rather than just truth values, give each world a fiber X_M and each arrow an actual transport map T_f:X_M→ X_N. Parallel paths p,q:M→ N can then be compared only because their source and target types agree. For linear fibers define the path discrepancy Δ_p,q=T_p-T_q. If the paths are declared to be the same morphism in the source category and T is a functor, this discrepancy must vanish. A nonzero value means that the paths are distinct morphisms, or that a…
Exact mathematical passage
T_f:X_M\to X_N
T_pUse in source · 1 passage
Search aliases: T_p
Use in source · Komorebi_Resolution_Semantics.tex, line 584
To retain information about structures rather than just truth values, give each world a fiber X_M and each arrow an actual transport map T_f:X_M→ X_N. Parallel paths p,q:M→ N can then be compared only because their source and target types agree. For linear fibers define the path discrepancy Δ_p,q=T_p-T_q. If the paths are declared to be the same morphism in the source category and T is a functor, this discrepancy must vanish. A nonzero value means that the paths are distinct morphisms, or that a…
Exact mathematical passage
\Delta_{p,q}=T_p-T_q.
T_qUse in source · 1 passage
Search aliases: T_q
Use in source · Komorebi_Resolution_Semantics.tex, line 584
To retain information about structures rather than just truth values, give each world a fiber X_M and each arrow an actual transport map T_f:X_M→ X_N. Parallel paths p,q:M→ N can then be compared only because their source and target types agree. For linear fibers define the path discrepancy Δ_p,q=T_p-T_q. If the paths are declared to be the same morphism in the source category and T is a functor, this discrepancy must vanish. A nonzero value means that the paths are distinct morphisms, or that a…
Exact mathematical passage
\Delta_{p,q}=T_p-T_q.
T_sUse in source · 3 passages
Search aliases: T_s
Use in source · Komorebi_Resolution_Semantics.tex, line 590
[Nontrivial holonomy with an exact defect] Let the one-object category be the group with generator s and relation s^2=1. Put X= F_2^2 and let T_s swap coordinates. This is a valid nontrivial action. Its invariant vectors are (0,0) and (1,1). For x=(1,0), T_sx+x=(1,1)≠0, but T_s(T_sx+x)+(T_sx+x)=0. The second equation is the transported composition law for s^2=1. Thus nontrivial fiber transport is compatible with an exact defect. No contradiction or unexplained curvature is involved.
Exact mathematical passage
T_s
\operatorname{target}Use in source · 1 passage
Search aliases: target operatorname
Use in source · Komorebi_Resolution_Semantics.tex, line 240
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
C^0=\prod_{M\in\Ob(\mathcal C)}B_M,\qquad C^1=\prod_{f\in\Mor(\mathcal C)}B_{\operatorname{target}(f)}.
uUse in source · 11 passages
Search aliases: u
Use in source · Komorebi_Resolution_Semantics.tex, line 183
[Defect calculus] Put u=B_f∘ v_M and w=v_N. Then D_f(0) · =D_f(1)=0, D_f(neg a)=D_f(a),
Exact mathematical passage
u=B_f\circ v_M
U_iUse in source · 3 passages
Search aliases: U_i
Use in source · Komorebi_Resolution_Semantics.tex, line 129
ommon presentation exists, but need not explain anything [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symb…
Exact mathematical passage
U_i=\{S\subseteq I:i\in S\}
U_{\Pi_0}Use in source · 1 passage
Search aliases: U_Π_0 Pi
Use in source · Komorebi_Resolution_Semantics.tex, line 516
For the original Boolean bounds, finer partitions instead give monotone information: L_Π_0(a)≤ L_Π_1(a)≤⋯≤ a ≤⋯≤ U_Π_1(a)≤ U_Π_0(a). The ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.
Exact mathematical passage
L_{\Pi_0}(a)\leq L_{\Pi_1}(a)\leq\cdots\leq a \leq\cdots\leq U_{\Pi_1}(a)\leq U_{\Pi_0}(a).
U_{\Pi_1}Use in source · 1 passage
Search aliases: U_Π_1 Pi
Use in source · Komorebi_Resolution_Semantics.tex, line 516
For the original Boolean bounds, finer partitions instead give monotone information: L_Π_0(a)≤ L_Π_1(a)≤⋯≤ a ≤⋯≤ U_Π_1(a)≤ U_Π_0(a). The ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.
Exact mathematical passage
L_{\Pi_0}(a)\leq L_{\Pi_1}(a)\leq\cdots\leq a \leq\cdots\leq U_{\Pi_1}(a)\leq U_{\Pi_0}(a).
v_{M_0}Use in source · 1 passage
Search aliases: v_M_0
Use in source · Komorebi_Resolution_Semantics.tex, line 346
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
\label{eq:pathreconstruction} v_{M_n}(a)=B_{f_n\cdots f_1}v_{M_0}(a) +\sum_{k=1}^n B_{f_n\cdots f_{k+1}}D_{f_k}(a),
v_{M_n}Use in source · 2 passages
Search aliases: v_M_n
Use in source · Komorebi_Resolution_Semantics.tex, line 346
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
\label{eq:pathreconstruction} v_{M_n}(a)=B_{f_n\cdots f_1}v_{M_0}(a) +\sum_{k=1}^n B_{f_n\cdots f_{k+1}}D_{f_k}(a),
v_PUse in source · 1 passage
Search aliases: v_P
Use in source · Komorebi_Resolution_Semantics.tex, line 205
Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(a)+v_P(a) · =B_gfv_M(a)+v_P(a). The identity case follows from B_id_M=id.
Exact mathematical passage
B_g(D_f(a))+D_g(a) &=B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(a)+v_P(a)\\ &=B_{gf}v_M(a)+v_P(a).
v_UUse in source · 1 passage
Search aliases: v_U
Use in source · Komorebi_Resolution_Semantics.tex, line 382
This example can occur as an invariant core of a resolution datum, rather than merely as an arbitrary subalgebra. Take worlds to be the nonprincipal ultrafilters on N, join every ordered pair by an arrow, and use v_U(S)=1 iff S∈ U. All these worlds agree on finite and cofinite sets. Every infinite coinfinite set belongs to some nonprincipal ultrafilter and has its complement in another, by the ultrafilter extension argument. Hence the invariant core is exactly the finite-cofinite algebra. These …
Exact mathematical passage
v_U(S)=1
\varprojlimUse in source · 2 passages
Search aliases: varprojlim
Use in source · Komorebi_Resolution_Semantics.tex, line 287
Coherent sections and definability. The limit varprojlim_ CB consists of families (b_M)_M satisfying B_f(b_M)=b_N. Thus a∈ I precisely when Ev(a) is a coherent family. But not every coherent family need be represented by a sentence, and distinct sentences can have the same evaluations.
Exact mathematical passage
\varprojlim_{\mathcal C}B
\varprojlim_{\mathcal C}Use in source · 2 passages
Search aliases: varprojlim_ C varprojlim mathcal
Use in source · Komorebi_Resolution_Semantics.tex, line 287
Coherent sections and definability. The limit varprojlim_ CB consists of families (b_M)_M satisfying B_f(b_M)=b_N. Thus a∈ I precisely when Ev(a) is a coherent family. But not every coherent family need be represented by a sentence, and distinct sentences can have the same evaluations.
Exact mathematical passage
\varprojlim_{\mathcal C}B
\vec aUse in source · 2 passages
Search aliases: vec a vec
Use in source · Komorebi_Resolution_Semantics.tex, line 108
[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a…
Exact mathematical passage
\vec a
\vec xUse in source · 4 passages
Search aliases: vec x vec
Use in source · Komorebi_Resolution_Semantics.tex, line 108
[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a…
Exact mathematical passage
\varphi(\vec x)
\vec x_iUse in source · 2 passages
Search aliases: vec x_i vec
Use in source · Komorebi_Resolution_Semantics.tex, line 136
res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An indu…
Exact mathematical passage
\bnorm{x=y}=\{i:x_i=y_i\},\qquad \bnorm{R(\vec x)}=\{i:M_i\models R(\vec x_i)\}.
\vec\tauUse in source · 2 passages
Search aliases: vecτ vec tau
Use in source · Komorebi_Resolution_Semantics.tex, line 545
Forcing as a conditional set-theoretic instance. Fix externally a countable transitive MmodelsZFC. Let B∈ M be a Boolean algebra that M regards as complete, and let G be an M-generic ultrafilter. For a sentence φ with names vecτ, the standard forcing truth lemma gives M[G]modelsφ(vecτ_G) Longleftrightarrow bnormφ(vecτ)_ B^M∈ G. Boolean-valued forcing and the role of generic quotient models are developed in [source reference]. The completeness assumption here is internal to M. The map b↦1_b∈ G is…
Exact mathematical passage
\vec\tau
\vec\tau_GUse in source · 1 passage
Search aliases: vecτ_G vec tau
Use in source · Komorebi_Resolution_Semantics.tex, line 546
Forcing as a conditional set-theoretic instance. Fix externally a countable transitive MmodelsZFC. Let B∈ M be a Boolean algebra that M regards as complete, and let G be an M-generic ultrafilter. For a sentence φ with names vecτ, the standard forcing truth lemma gives M[G]modelsφ(vecτ_G) Longleftrightarrow bnormφ(vecτ)_ B^M∈ G. Boolean-valued forcing and the role of generic quotient models are developed in [source reference]. The completeness assumption here is internal to M. The map b↦1_b∈ G is…
Exact mathematical passage
\label{eq:truthlemma} M[G]\models\varphi(\vec\tau_G) \quad\Longleftrightarrow\quad \bnorm{\varphi(\vec\tau)}_{\mathbb B}^{M}\in G.
wUse in source · 15 passages
Search aliases: w
Use in source · Komorebi_Resolution_Semantics.tex, line 183
[Defect calculus] Put u=B_f∘ v_M and w=v_N. Then D_f(0) · =D_f(1)=0, D_f(neg a)=D_f(a),
Exact mathematical passage
w=v_N
XUse in source · 6 passages
Search aliases: X
Use in source · Komorebi_Resolution_Semantics.tex, line 135
and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the sem…
Exact mathematical passage
X=\prod_{i\in I}|M_i|
xUse in source · 9 passages
Search aliases: x
Use in source · Komorebi_Resolution_Semantics.tex, line 136
res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An indu…
Exact mathematical passage
\bnorm{x=y}=\{i:x_i=y_i\},\qquad \bnorm{R(\vec x)}=\{i:M_i\models R(\vec x_i)\}.
x_iUse in source · 3 passages
Search aliases: x_i
Use in source · Komorebi_Resolution_Semantics.tex, line 136
res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An indu…
Exact mathematical passage
\bnorm{x=y}=\{i:x_i=y_i\},\qquad \bnorm{R(\vec x)}=\{i:M_i\models R(\vec x_i)\}.
x_i^2Use in source · 1 passage
Search aliases: x_i^2
Use in source · Komorebi_Resolution_Semantics.tex, line 651
Why the symbolic checks support universal identities. A Boolean polynomial is represented as a finite sum of squarefree monomials over F_2. Addition cancels duplicate monomials. Multiplication unions their variable supports, using x_i^2=x_i. These rules are valid in every Boolean ring. A polynomial identity whose difference reduces to the empty normal form therefore has an algebraic certificate applicable to every Boolean-ring substitution.
Exact mathematical passage
x_i^2=x_i
X_MUse in source · 2 passages
Search aliases: X_M
Use in source · Komorebi_Resolution_Semantics.tex, line 583
To retain information about structures rather than just truth values, give each world a fiber X_M and each arrow an actual transport map T_f:X_M→ X_N. Parallel paths p,q:M→ N can then be compared only because their source and target types agree. For linear fibers define the path discrepancy Δ_p,q=T_p-T_q. If the paths are declared to be the same morphism in the source category and T is a functor, this discrepancy must vanish. A nonzero value means that the paths are distinct morphisms, or that a…
Exact mathematical passage
X_M
X_NUse in source · 1 passage
Search aliases: X_N
Use in source · Komorebi_Resolution_Semantics.tex, line 583
To retain information about structures rather than just truth values, give each world a fiber X_M and each arrow an actual transport map T_f:X_M→ X_N. Parallel paths p,q:M→ N can then be compared only because their source and target types agree. For linear fibers define the path discrepancy Δ_p,q=T_p-T_q. If the paths are declared to be the same morphism in the source category and T is a functor, this discrepancy must vanish. A nonzero value means that the paths are distinct morphisms, or that a…
Exact mathematical passage
T_f:X_M\to X_N
\xrightarrowUse in source · 3 passages
Search aliases: xrightarrow
Use in source · Komorebi_Resolution_Semantics.tex, line 189
D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(…
Exact mathematical passage
M\xrightarrow{f}N\xrightarrow{g}P
yUse in source · 8 passages
Search aliases: y
Use in source · Komorebi_Resolution_Semantics.tex, line 136
res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An indu…
Exact mathematical passage
\bnorm{x=y}=\{i:x_i=y_i\},\qquad \bnorm{R(\vec x)}=\{i:M_i\models R(\vec x_i)\}.
y_iUse in source · 1 passage
Search aliases: y_i
Use in source · Komorebi_Resolution_Semantics.tex, line 136
res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An indu…
Exact mathematical passage
\bnorm{x=y}=\{i:x_i=y_i\},\qquad \bnorm{R(\vec x)}=\{i:M_i\models R(\vec x_i)\}.
\ZFCUse in source · 2 passages
Search aliases: ZFC mathsf
Use in source · Komorebi_Resolution_Semantics.tex, line 101
Worlds, presentations, and the ambient theory. Fix a first-order language L and an L-theory T. The intended case is the membership language with T=ZFC, but the algebraic results do not depend on that choice. Work in an external metatheory sufficient to discuss sets, categories, and satisfaction for set-sized structures.
Exact mathematical passage
T=\ZFC
|Use in source · 2 passages
Search aliases: |
Use in source · Komorebi_Resolution_Semantics.tex, line 135
and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the sem…
Exact mathematical passage
X=\prod_{i\in I}|M_i|
|_{\Inv}Use in source · 1 passage
Search aliases: |_Inv |_ I Inv mathcal
Use in source · Komorebi_Resolution_Semantics.tex, line 291
[The precise section statement] Let K=⋂_Mker v_M. Then K is a Boolean ideal contained in I, and I/K ≅ im(Ev|_ I) ⊆ varprojlim_ CB as Boolean algebras. The kernel of a Boolean-ring homomorphism is an ideal. An observable whose evaluations are all zero has zero defect, so K⊆ I. Restrict Ev to I and apply the Boolean-ring first isomorphism theorem.
Exact mathematical passage
\Inv/K\ \cong\ \im\bigl(\Ev|_{\Inv}\bigr) \ \subseteq\ \varprojlim_{\mathcal C}B
\alphaUse in source · 4 passages
Search aliases: α alpha
Use in source · Komorebi_Resolution_Semantics.tex, line 228
Transported syntax and parameters. For parameters or changing languages, replace A by a functor A_M with translation maps α_f:A_M→ A_N. The correct defect is D_f(a)=B_fv_M(a)+v_Nα_f(a), a∈ A_M. Assuming both transport systems are functorial, the composition law becomes D_gf(a)=B_gD_f(a)+D_g(α_f(a)). The same cancellation proves it. The fixed-A theory is the special case α_f=id. A single aggregate quotient of A will only be used in that special case. This prevents a silent comparison of formulas …
Exact mathematical passage
\alpha_f:A_M\to A_N
\alpha_fUse in source · 4 passages
Search aliases: α_f alpha
Use in source · Komorebi_Resolution_Semantics.tex, line 228
Transported syntax and parameters. For parameters or changing languages, replace A by a functor A_M with translation maps α_f:A_M→ A_N. The correct defect is D_f(a)=B_fv_M(a)+v_Nα_f(a), a∈ A_M. Assuming both transport systems are functorial, the composition law becomes D_gf(a)=B_gD_f(a)+D_g(α_f(a)). The same cancellation proves it. The fixed-A theory is the special case α_f=id. A single aggregate quotient of A will only be used in that special case. This prevents a silent comparison of formulas …
Exact mathematical passage
\alpha_f:A_M\to A_N
\GammaUse in source · 2 passages
Search aliases: Γ Gamma
Use in source · Komorebi_Resolution_Semantics.tex, line 276
[Generating transformations suffice] If a family Γ generates the arrows of C under composition, then I=⋂_f∈Γker D_f. Adding admissible transformations can only shrink the invariant core. The composition law propagates vanishing along finite composites. Identity defects vanish. The monotonicity statement is an immediate property of intersections.
Exact mathematical passage
\Gamma
\iotaUse in source · 1 passage
Search aliases: ι iota
Use in source · Komorebi_Resolution_Semantics.tex, line 356
[Additive splitting] Over a metatheory with choice, the inclusion I⊆ A always admits an F_2-linear retraction r:A→ I. For P=ι r and R=id+P, P^2=P, R^2=R, PR=RP=0, a=P(a)+R(a). This is a vector-space decomposition. The retraction need not preserve multiplication or be canonical. Choose a vector-space basis of I and extend it to a basis of A. Let r fix the first basis and kill the added vectors. Then P^2=P. In characteristic two, (id+P)^2=id+P, and P(id+P)=(id+P)P=0.
Exact mathematical passage
P=\iota r
\muUse in source · 4 passages
Search aliases: μ mu
Use in source · Komorebi_Resolution_Semantics.tex, line 470
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
\mu_w
\mu_uUse in source · 1 passage
Search aliases: μ_u mu
Use in source · Komorebi_Resolution_Semantics.tex, line 475
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
(E_\Pi f)_w= \frac{\sum_{u\in C}\mu_u f_u}{\sum_{u\in C}\mu_u} \quad(w\in C\in\Pi).
\mu_wUse in source · 3 passages
Search aliases: μ_w mu
Use in source · Komorebi_Resolution_Semantics.tex, line 470
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
\mu_w
\Pi_1Use in source · 1 passage
Search aliases: Π_1 Pi
Use in source · Komorebi_Resolution_Semantics.tex, line 516
For the original Boolean bounds, finer partitions instead give monotone information: L_Π_0(a)≤ L_Π_1(a)≤⋯≤ a ≤⋯≤ U_Π_1(a)≤ U_Π_0(a). The ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.
Exact mathematical passage
L_{\Pi_0}(a)\leq L_{\Pi_1}(a)\leq\cdots\leq a \leq\cdots\leq U_{\Pi_1}(a)\leq U_{\Pi_0}(a).
\varphiUse in source · 14 passages
Search aliases: φ varphi
Use in source · Komorebi_Resolution_Semantics.tex, line 108
[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a…
Exact mathematical passage
\varphi(\vec x)
\mapstoUse in source · 2 passages
Search aliases: ↦ mapsto
Use in source · Komorebi_Resolution_Semantics.tex, line 146
Modulo U_i, equality is exactly equality of the ith coordinate. The map [x]_U_i↦ x_i is well-defined, injective, and surjective. It preserves functions and relations, so it is an isomorphism to M_i. Equation~[source reference] also gives Boolean value I to each axiom satisfied by every coordinate structure.
Exact mathematical passage
[x]_{U_i}\mapsto x_i
\forallUse in source · 2 passages
Search aliases: ∀ forall
Use in source · Komorebi_Resolution_Semantics.tex, line 524
Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.
Exact mathematical passage
\exists N\ \forall n\geq N\quad a_n=a_N,
\existsUse in source · 2 passages
Search aliases: ∃ exists
Use in source · Komorebi_Resolution_Semantics.tex, line 524
Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.
Exact mathematical passage
\exists N\ \forall n\geq N\quad a_n=a_N,
\prodUse in source · 3 passages
Search aliases: ∏ prod
Use in source · Komorebi_Resolution_Semantics.tex, line 135
and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the sem…
Exact mathematical passage
X=\prod_{i\in I}|M_i|
\prod_{f\in\Mor(\mathcal C)}Use in source · 1 passage
Search aliases: ∏_f∈Mor( C) prod in Mor mathcal prod in operatorname mathcal
Use in source · Komorebi_Resolution_Semantics.tex, line 240
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
C^0=\prod_{M\in\Ob(\mathcal C)}B_M,\qquad C^1=\prod_{f\in\Mor(\mathcal C)}B_{\operatorname{target}(f)}.
\prod_{i<k}Use in source · 1 passage
Search aliases: ∏_i<k prod
Use in source · Komorebi_Resolution_Semantics.tex, line 217
[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty product case equal to D_f(1)=0.
Exact mathematical passage
D_f\left(\prod_{i=1}^n a_i\right) =\sum_{k=1}^n\left(\prod_{i<k}w(a_i)\right) D_f(a_k)\left(\prod_{i>k}u(a_i)\right).
\prod_{i=1}^nUse in source · 1 passage
Search aliases: ∏_i=1^n prod
Use in source · Komorebi_Resolution_Semantics.tex, line 217
[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty product case equal to D_f(1)=0.
Exact mathematical passage
D_f\left(\prod_{i=1}^n a_i\right) =\sum_{k=1}^n\left(\prod_{i<k}w(a_i)\right) D_f(a_k)\left(\prod_{i>k}u(a_i)\right).
\prod_{i>k}Use in source · 1 passage
Search aliases: ∏_i>k prod
Use in source · Komorebi_Resolution_Semantics.tex, line 217
[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty product case equal to D_f(1)=0.
Exact mathematical passage
D_f\left(\prod_{i=1}^n a_i\right) =\sum_{k=1}^n\left(\prod_{i<k}w(a_i)\right) D_f(a_k)\left(\prod_{i>k}u(a_i)\right).
\prod_{i\in I}Use in source · 1 passage
Search aliases: ∏_i∈ I prod in
Use in source · Komorebi_Resolution_Semantics.tex, line 135
and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the sem…
Exact mathematical passage
X=\prod_{i\in I}|M_i|
\prod_{M\in\Ob(\mathcal C)}Use in source · 1 passage
Search aliases: ∏_M∈Ob( C) prod in Ob mathcal prod in operatorname mathcal
Use in source · Komorebi_Resolution_Semantics.tex, line 240
What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.
Exact mathematical passage
C^0=\prod_{M\in\Ob(\mathcal C)}B_M,\qquad C^1=\prod_{f\in\Mor(\mathcal C)}B_{\operatorname{target}(f)}.
\sum_{C\in\Pi}Use in source · 1 passage
Search aliases: ∑_C∈Π sum in Pi
Use in source · Komorebi_Resolution_Semantics.tex, line 489
d E_Π is the unique orthogonal projector onto the real vector space of blockwise constant functions. It is positive, fixes 1, and is equivariant under weight-preserving permutations that preserve the partition. For binary a, L_Π(a)≤ E_Π(a)≤ U_Π(a). The formula is blockwise constant and fixes such functions. For every blockwise constant g, ⟨ f-E_Π f,g⟩_μ =∑_C∈Πg_C∑_w∈ Cμ_w(f_w-(E_Π f)_w)=0. This characterizes the orthogonal projector uniquely. Positivity, equivariance, and the interval bounds fol…
Exact mathematical passage
\langle f-E_\Pi f,g\rangle_\mu =\sum_{C\in\Pi}g_C\sum_{w\in C}\mu_w(f_w-(E_\Pi f)_w)=0.
\sum_{k=1}^nUse in source · 2 passages
Search aliases: ∑_k=1^n sum
Use in source · Komorebi_Resolution_Semantics.tex, line 217
[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty product case equal to D_f(1)=0.
Exact mathematical passage
D_f\left(\prod_{i=1}^n a_i\right) =\sum_{k=1}^n\left(\prod_{i<k}w(a_i)\right) D_f(a_k)\left(\prod_{i>k}u(a_i)\right).
\sum_{u\in C}Use in source · 1 passage
Search aliases: ∑_u∈ C sum in
Use in source · Komorebi_Resolution_Semantics.tex, line 475
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
(E_\Pi f)_w= \frac{\sum_{u\in C}\mu_u f_u}{\sum_{u\in C}\mu_u} \quad(w\in C\in\Pi).
\sum_{w\in C}Use in source · 1 passage
Search aliases: ∑_w∈ C sum in
Use in source · Komorebi_Resolution_Semantics.tex, line 489
d E_Π is the unique orthogonal projector onto the real vector space of blockwise constant functions. It is positive, fixes 1, and is equivariant under weight-preserving permutations that preserve the partition. For binary a, L_Π(a)≤ E_Π(a)≤ U_Π(a). The formula is blockwise constant and fixes such functions. For every blockwise constant g, ⟨ f-E_Π f,g⟩_μ =∑_C∈Πg_C∑_w∈ Cμ_w(f_w-(E_Π f)_w)=0. This characterizes the orthogonal projector uniquely. Positivity, equivariance, and the interval bounds fol…
Exact mathematical passage
\langle f-E_\Pi f,g\rangle_\mu =\sum_{C\in\Pi}g_C\sum_{w\in C}\mu_w(f_w-(E_\Pi f)_w)=0.
\sum_{w\in W}Use in source · 1 passage
Search aliases: ∑_w∈ W sum in
Use in source · Komorebi_Resolution_Semantics.tex, line 471
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
\langle f,g\rangle_\mu=\sum_{w\in W}\mu_w f_wg_w.
\circUse in source · 1 passage
Search aliases: ∘ circ
Use in source · Komorebi_Resolution_Semantics.tex, line 183
[Defect calculus] Put u=B_f∘ v_M and w=v_N. Then D_f(0) · =D_f(1)=0, D_f(neg a)=D_f(a),
Exact mathematical passage
u=B_f\circ v_M
\capUse in source · 1 passage
Search aliases: ∩ cap
Use in source · Komorebi_Resolution_Semantics.tex, line 376
[Completeness of the ambient algebra is not enough] Let A= P( N) and let I be the finite-cofinite subalgebra. There is no Boolean retraction r:A→ I fixing I. Indeed, for every S⊆ N and n∈ N, r(S)∩{n}=r(S∩{n})=S∩{n}. Thus r(S) has exactly the same members as S, so r(S)=S. An infinite coinfinite S gives a contradiction.
Exact mathematical passage
r(S)\cap\{n\}=r(S\cap\{n\})=S\cap\{n\}.
\congUse in source · 3 passages
Search aliases: ≅ cong
Use in source · Komorebi_Resolution_Semantics.tex, line 108
[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a…
Exact mathematical passage
h:M\cong N
\geqUse in source · 1 passage
Search aliases: ≥ geq
Use in source · Komorebi_Resolution_Semantics.tex, line 524
Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.
Exact mathematical passage
\exists N\ \forall n\geq N\quad a_n=a_N,
\subseteqUse in source · 7 passages
Search aliases: ⊆ subseteq
Use in source · Komorebi_Resolution_Semantics.tex, line 129
ommon presentation exists, but need not explain anything [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symb…
Exact mathematical passage
U_i=\{S\subseteq I:i\in S\}
\oplusUse in source · 1 passage
Search aliases: ⊕ oplus
Use in source · Komorebi_Resolution_Semantics.tex, line 367
If r is a Boolean retraction, its kernel is an ideal J and the additive decomposition is A= I⊕ J. In these coordinates multiplication is (i,x)(j,y)=(ij, iy+jx+xy). It is not generally a direct-product multiplication. Writing a direct sum symbol must not silently erase these cross terms.
Exact mathematical passage
A=\Inv\oplus J
\bigcapUse in source · 3 passages
Search aliases: ⋂ bigcap
Use in source · Komorebi_Resolution_Semantics.tex, line 254
[Invariant algebra and residual quotient] The set I=ker D=⋂_f{a:B_fv_M(a)=v_N(a)} is a unital Boolean subalgebra of A. There is a natural exact sequence of F_2-vector spaces 0⟶ I⟶ AxrightarrowDim D⟶0, and therefore a natural additive isomorphism A/ I ≅ im D. Two observables have the same residual profile exactly when their symmetric difference belongs to I. Each defining condition for I is an equalizer of two Boolean homomorphisms. Equalizers are closed under 0, 1, addition, multiplication, and …
Exact mathematical passage
\Inv=\ker D=\bigcap_f\{a:B_fv_M(a)=v_N(a)\}
\bigcap_fUse in source · 1 passage
Search aliases: ⋂_f bigcap
Use in source · Komorebi_Resolution_Semantics.tex, line 254
[Invariant algebra and residual quotient] The set I=ker D=⋂_f{a:B_fv_M(a)=v_N(a)} is a unital Boolean subalgebra of A. There is a natural exact sequence of F_2-vector spaces 0⟶ I⟶ AxrightarrowDim D⟶0, and therefore a natural additive isomorphism A/ I ≅ im D. Two observables have the same residual profile exactly when their symmetric difference belongs to I. Each defining condition for I is an equalizer of two Boolean homomorphisms. Equalizers are closed under 0, 1, addition, multiplication, and …
Exact mathematical passage
\Inv=\ker D=\bigcap_f\{a:B_fv_M(a)=v_N(a)\}
\bigcap_{f\in\Gamma}Use in source · 1 passage
Search aliases: ⋂_f∈Γ bigcap in Gamma
Use in source · Komorebi_Resolution_Semantics.tex, line 277
[Generating transformations suffice] If a family Γ generates the arrows of C under composition, then I=⋂_f∈Γker D_f. Adding admissible transformations can only shrink the invariant core. The composition law propagates vanishing along finite composites. Identity defects vanish. The monotonicity statement is an immediate property of intersections.
Exact mathematical passage
\Inv=\bigcap_{f\in\Gamma}\ker D_f.
\bigcap_MUse in source · 1 passage
Search aliases: ⋂_M bigcap
Use in source · Komorebi_Resolution_Semantics.tex, line 290
[The precise section statement] Let K=⋂_Mker v_M. Then K is a Boolean ideal contained in I, and I/K ≅ im(Ev|_ I) ⊆ varprojlim_ CB as Boolean algebras. The kernel of a Boolean-ring homomorphism is an ideal. An observable whose evaluations are all zero has zero defect, so K⊆ I. Restrict Ev to I and apply the Boolean-ring first isomorphism theorem.
Exact mathematical passage
K=\bigcap_M\ker v_M
\bigcupUse in source · 1 passage
Search aliases: ⋃ bigcup
Use in source · Komorebi_Resolution_Semantics.tex, line 144
R(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An induction on formulas gives bnormφ(vec x)={i:M_imodelsφ(vec x_i)}. Only the quantifier step requires comment. A coordinate i belongs to ⋃_y∈ Xbnormφ(y,vec x) precisely when a witness exists in M_i. To see this from right to left, fill the other coordinates using a chosen family of default elements. To obtain fullness, choose witnes…
Exact mathematical passage
\bigcup_{y\in X}\bnorm{\varphi(y,\vec x)}
\bigcup_{y\in X}Use in source · 1 passage
Search aliases: ⋃_y∈ X bigcup in
Use in source · Komorebi_Resolution_Semantics.tex, line 144
R(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An induction on formulas gives bnormφ(vec x)={i:M_imodelsφ(vec x_i)}. Only the quantifier step requires comment. A coordinate i belongs to ⋃_y∈ Xbnormφ(y,vec x) precisely when a witness exists in M_i. To see this from right to left, fill the other coordinates using a chosen family of default elements. To obtain fullness, choose witnes…
Exact mathematical passage
\bigcup_{y\in X}\bnorm{\varphi(y,\vec x)}
\cdotsUse in source · 4 passages
Search aliases: ⋯ cdots
Use in source · Komorebi_Resolution_Semantics.tex, line 345
Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.
Exact mathematical passage
M_0\xrightarrow{f_1}\cdots\xrightarrow{f_n}M_n
\langleUse in source · 2 passages
Search aliases: ⟨ langle
Use in source · Komorebi_Resolution_Semantics.tex, line 471
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
\langle f,g\rangle_\mu=\sum_{w\in W}\mu_w f_wg_w.
\rangleUse in source · 2 passages
Search aliases: ⟩ rangle
Use in source · Komorebi_Resolution_Semantics.tex, line 471
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
\langle f,g\rangle_\mu=\sum_{w\in W}\mu_w f_wg_w.
\rangle_\muUse in source · 2 passages
Search aliases: ⟩_μ rangle mu
Use in source · Komorebi_Resolution_Semantics.tex, line 471
A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).
Exact mathematical passage
\langle f,g\rangle_\mu=\sum_{w\in W}\mu_w f_wg_w.
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