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Machine Theory: formal development
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326 symbol spellings in 612 mathematical passages across 4 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
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+Introduction passage · 20 passages
Search aliases: +
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
=Introduction passage · 124 passages
Search aliases: =
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
w=a_1\cdots a_k
AIntroduction passage · 39 passages
Search aliases: A
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\mathcal AIntroduction passage · 1 passage
Search aliases: A mathcal
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
\mathcal A
aIntroduction passage · 76 passages
Search aliases: a
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
F_a:X\to X
a_1Introduction passage · 4 passages
Search aliases: a_1
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
w=a_1\cdots a_k
a_kIntroduction passage · 2 passages
Search aliases: a_k
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
w=a_1\cdots a_k
\bar JIntroduction passage · 2 passages
Search aliases: bar J bar
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar J_0Introduction passage · 1 passage
Search aliases: bar J_0 bar
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar J_hIntroduction passage · 2 passages
Search aliases: bar J_h bar
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar J_{h-1}Introduction passage · 1 passage
Search aliases: bar J_h-1 bar
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar zIntroduction passage · 2 passages
Search aliases: bar z bar
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar z_{a,o}Introduction passage · 2 passages
Search aliases: bar z_a,o bar
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar\PsiIntroduction passage · 2 passages
Search aliases: barΨ bar Psi
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\bar\Psi_{a,o}Introduction passage · 2 passages
Search aliases: barΨ_a,o bar Psi
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
cIntroduction passage · 10 passages
Search aliases: c
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
c_aIntroduction passage · 2 passages
Search aliases: c_a
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
DIntroduction passage · 25 passages
Search aliases: D
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
dIntroduction passage · 30 passages
Search aliases: d
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
FIntroduction passage · 12 passages
Search aliases: F
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
F_a:X\to X
F_aIntroduction passage · 6 passages
Search aliases: F_a
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
F_a:X\to X
F_{a_1}Introduction passage · 1 passage
Search aliases: F_a_1
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
F_w=F_{a_k}\cdots F_{a_1}
F_{a_k}Introduction passage · 1 passage
Search aliases: F_a_k
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
F_w=F_{a_k}\cdots F_{a_1}
F_wIntroduction passage · 4 passages
Search aliases: F_w
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
F_w=F_{a_k}\cdots F_{a_1}
hIntroduction passage · 23 passages
Search aliases: h
Introduction passage · core_v0.2.tex, line 655
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,
Exact mathematical passage
J_h(p)
JIntroduction passage · 6 passages
Search aliases: J
Introduction passage · core_v0.2.tex, line 655
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,
Exact mathematical passage
J_h(p)
J_hIntroduction passage · 2 passages
Search aliases: J_h
Introduction passage · core_v0.2.tex, line 655
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,
Exact mathematical passage
J_h(p)
kIntroduction passage · 11 passages
Search aliases: k
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
w=a_1\cdots a_k
\minIntroduction passage · 8 passages
Search aliases: min
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\min_{a\in A(q)}Introduction passage · 1 passage
Search aliases: min_a∈ A(q) min in
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\min_{d\in D(q)}Introduction passage · 1 passage
Search aliases: min_d∈ D(q) min in
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
oIntroduction passage · 42 passages
Search aliases: o
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
pIntroduction passage · 32 passages
Search aliases: p
Introduction passage · core_v0.2.tex, line 655
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,
Exact mathematical passage
J_h(p)
\mathcal QIntroduction passage · 1 passage
Search aliases: Q mathcal
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
qIntroduction passage · 16 passages
Search aliases: q
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
q:X\to Y
\mathcal Q_hIntroduction passage · 1 passage
Search aliases: Q_h mathcal
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
wIntroduction passage · 21 passages
Search aliases: w
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
w=a_1\cdots a_k
XIntroduction passage · 15 passages
Search aliases: X
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
X
YIntroduction passage · 5 passages
Search aliases: Y
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
q:X\to Y
\ellIntroduction passage · 15 passages
Search aliases: ℓ ell
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\ell_dIntroduction passage · 3 passages
Search aliases: ℓ_d ell
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\toIntroduction passage · 18 passages
Search aliases: → to
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
F_a:X\to X
\inIntroduction passage · 18 passages
Search aliases: ∈ in
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\sumIntroduction passage · 16 passages
Search aliases: ∑ sum
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\sum_{o:\bar z_{a,o}(q)>0}Introduction passage · 1 passage
Search aliases: ∑_o:bar z_a,o(q)>0 sum bar
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\cdotsIntroduction passage · 2 passages
Search aliases: ⋯ cdots
Introduction passage · core_v0.2.tex, line 138
One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.
Exact mathematical passage
w=a_1\cdots a_k
\langleIntroduction passage · 2 passages
Search aliases: ⟨ langle
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
\rangleIntroduction passage · 2 passages
Search aliases: ⟩ rangle
Introduction passage · core_v0.2.tex, line 656
Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1(barΨ_a,o(q)), bar J_h(q) · =min{bar J_0(q), min_a∈ A(q) Q_h(q,a)}. An empty experiment catalogue leaves only stopping. Impossible reports do not rece…
Exact mathematical passage
\bar J_0(q)&=\min_{d\in D(q)}\langle q,\ell_d\rangle,\\ \mathcal Q_h(q,a)&=c_a(q)+\sum_{o:\bar z_{a,o}(q)>0} \bar z_{a,o}(q)\bar J_{h-1}\bigl(\bar\Psi_{a,o}(q)\bigr),\\ \bar J_h(q)&=\min\left\{\bar J_0(q),\ \min_{a\in A(q)}\mathcal Q_h(q,a)\right\}.
*Use in source · 22 passages
Search aliases: *
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathbf1Use in source · 2 passages
Search aliases: 1 mathbf
Use in source · core_v0.2.tex, line 439
For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.
Exact mathematical passage
z_{a,o}(p)=\mathbf1^\top K_{a,o}p,\qquad \Psi_{a,o}(p)=K_{a,o}p/z_{a,o}(p)
\mathbf1^\topUse in source · 1 passage
Search aliases: 1^top mathbf top
Use in source · core_v0.2.tex, line 439
For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.
Exact mathematical passage
z_{a,o}(p)=\mathbf1^\top K_{a,o}p,\qquad \Psi_{a,o}(p)=K_{a,o}p/z_{a,o}(p)
\mathbf1_X^\topUse in source · 1 passage
Search aliases: 1_X^top mathbf top
Use in source · core_v0.2.tex, line 463
Because 1_Z^top C=1_X^top, the matrix equality gives equal report masses. Dividing it by a common positive mass gives equal aggregated posteriors. Conversely, equality of masses and positive-mass posteriors gives the unnormalized matrix equality on every prior. If the common mass is zero, both unnormalized vectors are zero by nonnegativity. Point masses finish that direction.
Exact mathematical passage
\mathbf1_Z^\top C=\mathbf1_X^\top
\mathbf1_Z^\topUse in source · 1 passage
Search aliases: 1_Z^top mathbf top
Use in source · core_v0.2.tex, line 463
Because 1_Z^top C=1_X^top, the matrix equality gives equal report masses. Dividing it by a common positive mass gives equal aggregated posteriors. Conversely, equality of masses and positive-mass posteriors gives the unnormalized matrix equality on every prior. If the common mass is zero, both unnormalized vectors are zero by nonnegativity. Point masses finish that direction.
Exact mathematical passage
\mathbf1_Z^\top C=\mathbf1_X^\top
a'Use in source · 1 passage
Search aliases: a'
Use in source · core_v0.2.tex, line 807
After changing a to a', mark a vertex stale when a direct input changes and propagate staleness to all descendants. Call the resulting set Δ.
Exact mathematical passage
a'
a_*Use in source · 2 passages
Search aliases: a_*
Use in source · core_v0.2.tex, line 543
For a true minimizer a_*, R(widehat a)≤widehat R(widehat a)+ξ ≤widehat R(a_*)+ξ≤ R(a_*)+2ξ. The loss range supplies the cap one. Finally, for any other a, subtract the two error bounds to obtain R(a)-R(widehat a)≥widehat R(a)-widehat R(widehat a)-2ξ>0. source[GA], ``A common quantitative grounding certificate''; [MU], ``A joint posterior and recipient decision certificate.'' This is a specialization to a common finite candidate set and bounded loss.
Exact mathematical passage
a_*
A_1Use in source · 1 passage
Search aliases: A_1
Use in source · core_v0.2.tex, line 871
What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.
Exact mathematical passage
Av=(A_1v,\ldots,A_mv)
A_AUse in source · 1 passage
Search aliases: A_A
Use in source · core_v0.2.tex, line 881
Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.
Exact mathematical passage
A_A(b,s)=b+s
A_aUse in source · 5 passages
Search aliases: A_a
Use in source · core_v0.2.tex, line 225
One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be finite dimensional, let T_a^θ be finitely many candidate linear operators, and let A_a be one complete candidate system chosen as reference. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear readout for every candidate.
Exact mathematical passage
A_a
A_BUse in source · 1 passage
Search aliases: A_B
Use in source · core_v0.2.tex, line 881
Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.
Exact mathematical passage
A_B(b,s)=s
a_dUse in source · 1 passage
Search aliases: a_d
Use in source · core_v0.2.tex, line 805
Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined by those components and parent claims. External assumptions are included explicitly as premises.
Exact mathematical passage
a=(a_1,\ldots,a_d)
A_iUse in source · 2 passages
Search aliases: A_i
Use in source · core_v0.2.tex, line 871
What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.
Exact mathematical passage
A_i v
A_jUse in source · 3 passages
Search aliases: A_j
Use in source · core_v0.2.tex, line 849
[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopping time depend on earlier history. Taking expectations gives Pr(A_j)≤δ_j. A union bound gives the result for every finite set of attempts. The finite…
Exact mathematical passage
A_j
A_mUse in source · 1 passage
Search aliases: A_m
Use in source · core_v0.2.tex, line 871
What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.
Exact mathematical passage
Av=(A_1v,\ldots,A_mv)
A_{\rm end}Use in source · 1 passage
Search aliases: A_rm end rm
Use in source · core_v0.2.tex, line 907
A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\{z\in S:\exists a\in A_{\rm end}(z),\ \operatorname{Post}(z,a)\subseteq T_k\}.
A_{\rm run}Use in source · 1 passage
Search aliases: A_rm run rm
Use in source · core_v0.2.tex, line 912
T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
L_0=T_*,\qquad L_{k+1}=\{z\in L_k:\exists a\in A_{\rm run}(z),\ \operatorname{Post}(z,a)\subseteq L_k\},
A_tUse in source · 1 passage
Search aliases: A_t
Use in source · core_v0.2.tex, line 433
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
Exact mathematical passage
K_{a,o}(x',x)=\Pr(X_{t+1}=x',O_{t+1}=o\mid X_t=x,A_t=a)
A_wUse in source · 3 passages
Search aliases: A_w
Use in source · core_v0.2.tex, line 228
Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=0.
Exact mathematical passage
E=\Span\{A_wD_a^\theta v: v\in V,\ a,\theta,\ w\text{ finite}\}.
\addcontentslineUse in source · 1 passage
Search aliases: addcontentsline
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\arraybackslashUse in source · 1 passage
Search aliases: arraybackslash
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
BUse in source · 18 passages
Search aliases: B
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
bUse in source · 31 passages
Search aliases: b
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\bar DUse in source · 1 passage
Search aliases: bar D bar
Use in source · core_v0.2.tex, line 890
[A quotient that preserves legal operations] For a surjection π:X→ Z, quotient labels, enabled sets, and successors satisfying ℓ=barℓπ, D=bar Dπ, π F(x,a)=bar F(π x,a) exist exactly when equal fibres have equal labels, equal enabled sets, and equal projected successors for every enabled action. The quotient data are unique and preserve every legal finite action and label sequence. Factoring each quantity proves necessity. For sufficiency choose a representative of each fibre to define its label,…
Exact mathematical passage
\ell=\bar\ell\pi,\qquad D=\bar D\pi,\qquad \pi F(x,a)=\bar F(\pi x,a)
\bar FUse in source · 3 passages
Search aliases: bar F bar
Use in source · core_v0.2.tex, line 151
[Evidence-initialized behavioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a([x])=[F_ax] and its readout is bar q([x])=q(x). If two initial records are equal, a deterministic predictor has the same initialized state. Applying t…
Exact mathematical passage
\bar F_a([x])=[F_ax]
\bar F_aUse in source · 2 passages
Search aliases: bar F_a bar
Use in source · core_v0.2.tex, line 151
[Evidence-initialized behavioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a([x])=[F_ax] and its readout is bar q([x])=q(x). If two initial records are equal, a deterministic predictor has the same initialized state. Applying t…
Exact mathematical passage
\bar F_a([x])=[F_ax]
\bar KUse in source · 5 passages
Search aliases: bar K bar
Use in source · core_v0.2.tex, line 413
nique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly positive coarse likelihoods for every prior. At a stable partition the probability of reaching each coarse block depends only on the current block. These sums define a stochastic coarse transition and yield C_π K_a^top=bar K_a^top C_π, first on point masses and then by linearity. Likelihoods are constant within final blocks. Grouping the …
Exact mathematical passage
C_\pi K_a^\top=\bar K_a^\top C_\pi
\bar K_{a,o}Use in source · 4 passages
Search aliases: bar K_a,o bar
Use in source · core_v0.2.tex, line 446
[One quotient for reports, posteriors, and feedback] A nonnegative normalized coarse instrument bar K_a,o satisfies CK_a,o=bar K_a,oC for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the report probabilities and the aggregated positive-probability posteriors: z_a,o(p)=bar z_a,o(Cp), CΨ_a,o(p)=barΨ_a,o(Cp). Starting from Cp_0, these identiti…
Exact mathematical passage
\bar K_{a,o}
\bar K_a^\topUse in source · 1 passage
Search aliases: bar K_a^top bar top
Use in source · core_v0.2.tex, line 413
nique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly positive coarse likelihoods for every prior. At a stable partition the probability of reaching each coarse block depends only on the current block. These sums define a stochastic coarse transition and yield C_π K_a^top=bar K_a^top C_π, first on point masses and then by linearity. Likelihoods are constant within final blocks. Grouping the …
Exact mathematical passage
C_\pi K_a^\top=\bar K_a^\top C_\pi
\bar QUse in source · 2 passages
Search aliases: bar Q bar
Use in source · core_v0.2.tex, line 238
[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also predicts the output when the candidate index changes between operations. Assume the first condition. For every a,θ,w, equality as operators on all of V g…
Exact mathematical passage
\bar Q
\bar qUse in source · 5 passages
Search aliases: bar q bar
Use in source · core_v0.2.tex, line 151
havioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a([x])=[F_ax] and its readout is bar q([x])=q(x). If two initial records are equal, a deterministic predictor has the same initialized state. Applying the same subsequent opera…
Exact mathematical passage
\bar q([x])=q(x)
\bar uUse in source · 3 passages
Search aliases: bar u bar
Use in source · core_v0.2.tex, line 465
For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histories. Normalization on positive histories gives the posterior statement. Stopping decisions based on the same history can be included as matching termi…
Exact mathematical passage
u_h,\bar u_h
\bar u_hUse in source · 2 passages
Search aliases: bar u_h bar
Use in source · core_v0.2.tex, line 465
For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histories. Normalization on positive histories gives the posterior statement. Stopping decisions based on the same history can be included as matching termi…
Exact mathematical passage
u_h,\bar u_h
\bar u_\varnothingUse in source · 1 passage
Search aliases: bar u_∅ bar varnothing
Use in source · core_v0.2.tex, line 465
For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histories. Normalization on positive histories gives the posterior statement. Stopping decisions based on the same history can be included as matching termi…
Exact mathematical passage
\bar u_\varnothing=Cu_\varnothing
\bar\ellUse in source · 3 passages
Search aliases: barℓ bar ell
Use in source · core_v0.2.tex, line 415
For minimality, take any admissible refinement η of π_0. It has at least two blocks. Fix a strictly positive fine likelihood ℓ and a strictly positive coarse likelihood barℓ inducing its assumed update. For states x,y in one block and z in a different block, compare priors assigning half their mass to x,z and to y,z. Their coarse priors agree. Equality of the posterior odds gives ℓ(x)/ℓ(z)=barℓ(η(x))/barℓ(η(z))=ℓ(y)/ℓ(z). Thus ℓ(x)=ℓ(y), and η refines the first likelihood split. Coarse likelihoo…
Exact mathematical passage
\bar\ell
\bmodUse in source · 1 passage
Search aliases: bmod
Use in source · core_v0.2.tex, line 122
An agent given only y can predict every future display under repeated flips: after k flips it is y⊕(kbmod2). It cannot determine the current physical bit. Worlds (0,0,0) and (1,1,0) share display zero and have different targets. The obstruction is the fibre of the available record, not a shortage of computational effort.
Exact mathematical passage
y\oplus(k\bmod2)
\bottomruleUse in source · 1 passage
Search aliases: bottomrule
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
CUse in source · 32 passages
Search aliases: C
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathcal CUse in source · 7 passages
Search aliases: C mathcal
Use in source · core_v0.2.tex, line 558
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
\mathcal C
c_0Use in source · 2 passages
Search aliases: c_0
Use in source · core_v0.2.tex, line 578
[Individual acceptability is not a common message] Two cases have the same sender record. In one, the sole acceptable artifact is c_0; in the other it is c_1≠ c_0. Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.
Exact mathematical passage
c_0
C_1Use in source · 1 passage
Search aliases: C_1
Use in source · core_v0.2.tex, line 329
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
C_1,\ldots,C_m
c_1Use in source · 1 passage
Search aliases: c_1
Use in source · core_v0.2.tex, line 578
[Individual acceptability is not a common message] Two cases have the same sender record. In one, the sole acceptable artifact is c_0; in the other it is c_1≠ c_0. Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.
Exact mathematical passage
c_1\ne c_0
c_eUse in source · 2 passages
Search aliases: c_e
Use in source · core_v0.2.tex, line 590
tion, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegative integer resource vector c_e.
Exact mathematical passage
c_e
\mathcal C_hUse in source · 5 passages
Search aliases: C_h mathcal
Use in source · core_v0.2.tex, line 561
Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).
Exact mathematical passage
\mathcal C_h=\bigcap_{x:\alpha(x)=h}\mathcal C(x).
C_iUse in source · 1 passage
Search aliases: C_i
Use in source · core_v0.2.tex, line 330
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
C_mUse in source · 1 passage
Search aliases: C_m
Use in source · core_v0.2.tex, line 329
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
C_1,\ldots,C_m
c_{\max}Use in source · 2 passages
Search aliases: c_max max
Use in source · core_v0.2.tex, line 708
te planning must not share one unqualified guarantee. Fix a finite horizon H and one finite class P of admitted action/report trees, shared by the true and approximate models. The complete loss of a tree is the same function of its coarse-state and action/report path in both models, taking values in [0,B]. For example, a terminal loss in [0,1] and at most H charges bounded by c_max give B=1+Hc_max.
Exact mathematical passage
c_{\max}
C_\piUse in source · 2 passages
Search aliases: C_π pi
Use in source · core_v0.2.tex, line 401
First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet, assume also ∑_oℓ_o(x)=1.
Exact mathematical passage
C_\pi(z,x)=1_{\{\pi(x)=z\}}
\chapterUse in source · 1 passage
Search aliases: chapter
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
d_0Use in source · 3 passages
Search aliases: d_0
Use in source · core_v0.2.tex, line 523
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\TV(p,\widehat p)\le e,\qquad \sup_\omega d(v_\omega,\widehat v_\omega)\le d_0.
D_a^\thetaUse in source · 5 passages
Search aliases: D_a^θ theta
Use in source · core_v0.2.tex, line 227
Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=0.
Exact mathematical passage
D_a^\theta=T_a^\theta-A_a
D_eUse in source · 4 passages
Search aliases: D_e
Use in source · core_v0.2.tex, line 590
One investigation, one resource budget. questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegative integer resource vector c_e.
Exact mathematical passage
D_e\subseteq X
d_HUse in source · 11 passages
Search aliases: d_H
Use in source · core_v0.2.tex, line 485
[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetil…
Exact mathematical passage
\TV(P_H,\widetilde P_H)\le d_H:=\min\left\{1,\delta_0+\sum_{t=0}^{H-1}\epsilon_t\right\}.
D_tUse in source · 2 passages
Search aliases: D_t
Use in source · core_v0.2.tex, line 330
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
D_vUse in source · 1 passage
Search aliases: D_v
Use in source · core_v0.2.tex, line 805
Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined by those components and parent claims. External assumptions are included explicitly as premises.
Exact mathematical passage
D_v
EUse in source · 30 passages
Search aliases: E
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathcal EUse in source · 4 passages
Search aliases: E mathcal
Use in source · core_v0.2.tex, line 175
Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
\mathcal E(T)=ATS
eUse in source · 15 passages
Search aliases: e
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
E_0Use in source · 3 passages
Search aliases: E_0
Use in source · core_v0.2.tex, line 315
[Minimal exact memory for a correction contract] Equality of Σ is a congruence for every update V↦ V∩ E_0. The signature supports the current target set and its exact update after every finite evidence sequence. Any deterministic retained state supporting the same readouts and updates on all candidate subsets must distinguish every two distinct signatures. Thus the image of Σ has the least possible cardinality for this contract. If Σ(V)=Σ(W), then for each E, t((V∩ E_0)∩ E)=t(V∩(E_0∩ E))=t(W∩(E_…
Exact mathematical passage
V\mapsto V\cap E_0
\mathbb FUse in source · 1 passage
Search aliases: F mathbb
Use in source · core_v0.2.tex, line 881
Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.
Exact mathematical passage
\mathbb F_2
fUse in source · 4 passages
Search aliases: f
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathbb F_2Use in source · 1 passage
Search aliases: F_2 mathbb
Use in source · core_v0.2.tex, line 881
Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.
Exact mathematical passage
\mathbb F_2
f_eUse in source · 2 passages
Search aliases: f_e
Use in source · core_v0.2.tex, line 590
, one resource budget questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegative integer resource vector c_e.
Exact mathematical passage
f_e:D_e\to X
GUse in source · 14 passages
Search aliases: G
Use in source · core_v0.2.tex, line 202
A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.
Exact mathematical passage
G_{\tau(a)}
\mathcal GUse in source · 1 passage
Search aliases: G mathcal
Use in source · core_v0.2.tex, line 737
Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that agrees everywhere or returns a context where it disagrees. The teacher's work is part of the development account.
Exact mathematical passage
\mathcal G=(p_0,\ldots,p_{N-1})
gUse in source · 3 passages
Search aliases: g
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
G_aUse in source · 3 passages
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Use in source · core_v0.2.tex, line 238
[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also predicts the output when the candidate index changes between operations. Assume the first condition. For every a,θ,w, equality as operators on all of V g…
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Use in source · core_v0.2.tex, line 598
A terminal artifact a is acceptable only if every fixed recipient succeeds in every state of V. Its worst-case terminal cost is g_a(V), computed componentwise. A controller is a literal finite tree with one record per experiment or terminal node. Stored memory counts all branches, including those not visited in one execution.
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g_a(V)
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Search aliases: G_τ(a) tau
Use in source · core_v0.2.tex, line 202
A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.
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G_{\tau(a)}
G_{\tau(w)}Use in source · 2 passages
Search aliases: G_τ(w) tau
Use in source · core_v0.2.tex, line 210
[Forward transport of a reusable method] If the source domain is invariant, and TF_a=G_τ(a)T, q=bar qT, then for every finite word w, TF_w=G_τ(w)T, qF_w=bar qG_τ(w)T. The empty word gives TI_Z=I_UT. Assume the first equality for a prefix. Invariance allows the next local equality to be applied on its successor, yielding TF_aF_w=G_τ(a)TF_w=G_τ(a)G_τ(w)T. Induction proves the first claim. Composition with the matched readout proves the second. source[MU], Theorem 6.1; the same local-to-continuatio…
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TF_w=G_{\tau(w)}T,\qquad qF_w=\bar qG_{\tau(w)}T.
HUse in source · 19 passages
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Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathcal HUse in source · 2 passages
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Use in source · core_v0.2.tex, line 849
[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopping time depend on earlier history. Taking expectations gives Pr(A_j)≤δ_j. A union bound gives the result for every finite set of attempts. The finite…
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\mathcal H_{j-1}
\mathcal H_{j-1}Use in source · 2 passages
Search aliases: H_j-1 mathcal
Use in source · core_v0.2.tex, line 849
[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopping time depend on earlier history. Taking expectations gives Pr(A_j)≤δ_j. A union bound gives the result for every finite set of attempts. The finite…
Exact mathematical passage
\mathcal H_{j-1}
\hfillUse in source · 1 passage
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Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\hruleUse in source · 1 passage
Search aliases: hrule
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
IUse in source · 12 passages
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Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
iUse in source · 25 passages
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Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
i_1Use in source · 3 passages
Search aliases: i_1
Use in source · core_v0.2.tex, line 641
Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of the frontier, leaving that pair as the whole nondominated frontier. source[SII], Theorem 11.1. The assumptions exclude destructive resets and repairs …
Exact mathematical passage
i_1,i_2
i_2Use in source · 3 passages
Search aliases: i_2
Use in source · core_v0.2.tex, line 641
Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of the frontier, leaving that pair as the whole nondominated frontier. source[SII], Theorem 11.1. The assumptions exclude destructive resets and repairs …
Exact mathematical passage
i_1,i_2
I_UUse in source · 1 passage
Search aliases: I_U
Use in source · core_v0.2.tex, line 215
[Forward transport of a reusable method] If the source domain is invariant, and TF_a=G_τ(a)T, q=bar qT, then for every finite word w, TF_w=G_τ(w)T, qF_w=bar qG_τ(w)T. The empty word gives TI_Z=I_UT. Assume the first equality for a prefix. Invariance allows the next local equality to be applied on its successor, yielding TF_aF_w=G_τ(a)TF_w=G_τ(a)G_τ(w)T. Induction proves the first claim. Composition with the matched readout proves the second. source[MU], Theorem 6.1; the same local-to-continuatio…
Exact mathematical passage
TI_Z=I_UT
I_VUse in source · 4 passages
Search aliases: I_V
Use in source · core_v0.2.tex, line 175
Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
SA=I_V
I_ZUse in source · 4 passages
Search aliases: I_Z
Use in source · core_v0.2.tex, line 179
be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
I_Z
\imUse in source · 1 passage
Search aliases: im operatorname
Use in source · core_v0.2.tex, line 877
:reconstruction There is a linear decoder D with Q=DA if and only if ker A⊆ker Q. Full-state reconstruction is possible exactly when A is injective. The intersection identity ker A=⋂_iker A_i determines which ambiguities sharing removes. If Q=DA, a vector annihilated by A is also annihilated by Q. Conversely, define D(Av)=Qv on im A. If Av=Aw, then v-w∈ker A⊆ker Q, so this definition is independent of the representative. It is linear and extends to the full record space by extending a basis. For…
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\im A
jUse in source · 8 passages
Search aliases: j
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
KUse in source · 18 passages
Search aliases: K
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathcal KUse in source · 8 passages
Search aliases: K mathcal
Use in source · core_v0.2.tex, line 265
Let K be a convex cone in a real vector space, and let λ be linear and strictly positive on every nonzero member of K. Its normalized states satisfy v∈ K and λ v=1. A coordinate change can send K to a cone containing negative coordinates without changing those states' probability meaning. Conversely, an arbitrary signed component of a valid state need not itself be valid.
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\mathcal K
K_aUse in source · 2 passages
Search aliases: K_a
Use in source · core_v0.2.tex, line 401
First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet, assume also ∑_oℓ_o(x)=1.
Exact mathematical passage
K_a(x,x')
K_{a,o}Use in source · 8 passages
Search aliases: K_a,o
Use in source · core_v0.2.tex, line 433
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
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K_{a,o}(x',x)=\Pr(X_{t+1}=x',O_{t+1}=o\mid X_t=x,A_t=a)
K_a^\topUse in source · 2 passages
Search aliases: K_a^top top
Use in source · core_v0.2.tex, line 401
First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet, assume also ∑_oℓ_o(x)=1.
Exact mathematical passage
p\mapsto K_a^\top p
\kerUse in source · 6 passages
Search aliases: ker
Use in source · core_v0.2.tex, line 237
[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also predicts the output when the candidate index changes between operations. Assume the first condition. For every a,θ,w, equality as operators on all of V g…
Exact mathematical passage
E\subseteq\ker Q
LUse in source · 16 passages
Search aliases: L
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
lUse in source · 5 passages
Search aliases: l
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
L_*Use in source · 7 passages
Search aliases: L_*
Use in source · core_v0.2.tex, line 916
T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
L_*
L_0Use in source · 1 passage
Search aliases: L_0
Use in source · core_v0.2.tex, line 912
T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
L_0=T_*,\qquad L_{k+1}=\{z\in L_k:\exists a\in A_{\rm run}(z),\ \operatorname{Post}(z,a)\subseteq L_k\},
L_iUse in source · 1 passage
Search aliases: L_i
Use in source · core_v0.2.tex, line 558
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
L_i
L_kUse in source · 2 passages
Search aliases: L_k
Use in source · core_v0.2.tex, line 912
T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
L_0=T_*,\qquad L_{k+1}=\{z\in L_k:\exists a\in A_{\rm run}(z),\ \operatorname{Post}(z,a)\subseteq L_k\},
L_{k+1}Use in source · 1 passage
Search aliases: L_k+1
Use in source · core_v0.2.tex, line 912
T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
L_0=T_*,\qquad L_{k+1}=\{z\in L_k:\exists a\in A_{\rm run}(z),\ \operatorname{Post}(z,a)\subseteq L_k\},
L_mUse in source · 5 passages
Search aliases: L_m
Use in source · core_v0.2.tex, line 358
Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability events.
Exact mathematical passage
L_m(o)
\LARGEUse in source · 1 passage
Search aliases: LARGE
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\LargeUse in source · 1 passage
Search aliases: Large
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\largeUse in source · 1 passage
Search aliases: large
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\lceilUse in source · 2 passages
Search aliases: lceil
Use in source · core_v0.2.tex, line 636
[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf, so at least n leaves are reachable. A binary tree of query depth d has at most 2^d leaves, giving d≥lceillog_2nrceil. The terminal repair adds one un…
Exact mathematical passage
1+\lceil\log_2n\rceil
\linewidthUse in source · 1 passage
Search aliases: linewidth
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\logUse in source · 2 passages
Search aliases: log
Use in source · core_v0.2.tex, line 636
[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf, so at least n leaves are reachable. A binary tree of query depth d has at most 2^d leaves, giving d≥lceillog_2nrceil. The terminal repair adds one un…
Exact mathematical passage
1+\lceil\log_2n\rceil
\log_2Use in source · 2 passages
Search aliases: log_2 log
Use in source · core_v0.2.tex, line 636
[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf, so at least n leaves are reachable. A binary tree of query depth d has at most 2^d leaves, giving d≥lceillog_2nrceil. The terminal repair adds one un…
Exact mathematical passage
1+\lceil\log_2n\rceil
\LongleftrightarrowUse in source · 1 passage
Search aliases: Longleftrightarrow
Use in source · core_v0.2.tex, line 141
Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.
Exact mathematical passage
x\sim x'\quad\Longleftrightarrow\quad q(F_wx)=q(F_wx')\text{ for every finite word }w.
\LongrightarrowUse in source · 3 passages
Search aliases: Longrightarrow
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
MUse in source · 2 passages
Search aliases: M
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathcal MUse in source · 1 passage
Search aliases: M mathcal
Use in source · core_v0.2.tex, line 358
Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability events.
Exact mathematical passage
M\in\mathcal M
mUse in source · 20 passages
Search aliases: m
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
m_oUse in source · 2 passages
Search aliases: m_o
Use in source · core_v0.2.tex, line 605
athcal P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it is no larger in every resou…
Exact mathematical passage
(v_o,m_o)
\mainmatterUse in source · 1 passage
Search aliases: mainmatter
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\maxUse in source · 7 passages
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Use in source · core_v0.2.tex, line 489
e distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p_E)}. If either event probability exceeds d_H, the other is necessarily…
Exact mathematical passage
\TV\bigl(P_H(Z_H\in\cdot\mid E),\widetilde P_H(Z_H\in\cdot\mid E)\bigr) \le\min\left\{1,\frac{d_H}{\max(p_E,\widetilde p_E)}\right\}.
\max_oUse in source · 1 passage
Search aliases: max_o max
Use in source · core_v0.2.tex, line 606
m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it is no larger in every resource coordinate and strictly smaller in at l…
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\left(c_e+\max_o v_o,\quad1+\sum_o m_o\right).
\midUse in source · 14 passages
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Use in source · core_v0.2.tex, line 369
[When a report preserves every Bayesian correction] The following are equivalent: item For every prior, the posterior after o equals that after z on every positive-probability pair (o,z). item The equality holds for one prior positive on every model. item There is a stochastic kernel ρ(omid z) such that L_m(o)B(o,z)=Q_m(z)ρ(omid z) for all m,o,z. The reconstruction in the last condition refers to the actual joint experiment, not an unrelated simulation with matching marginals. The first conditio…
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\rho(o\mid z)
\midruleUse in source · 1 passage
Search aliases: midrule
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\min_aUse in source · 1 passage
Search aliases: min_a min
Use in source · core_v0.2.tex, line 535
[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add and subtract ∑_ω p(ω)ℓ(a,widehat v_ω). The response term has absolute value at most Kd_0. The expectation of a function in [0,1] changes by at most tot…
Exact mathematical passage
R(\widehat a)-\min_a R(a)\le\min\{1,2\xi\}.
\min_{w\in W}Use in source · 1 passage
Search aliases: min_w∈ W min in
Use in source · core_v0.2.tex, line 836
[Which gains survive every admitted price?] The saving w·Δ is nonnegative for every nonnegative price vector if and only if every coordinate of Δ is nonnegative. If prices lie in the convex hull of a finite nonempty set W, their minimum saving equals min_w∈ Ww·Δ. A componentwise nonnegative vector has nonnegative inner product with every nonnegative vector. If coordinate j is negative, assigning price one only to that coordinate gives a negative result. This proves the first assertion. A point i…
Exact mathematical passage
\min_{w\in W}w\cdot\Delta
NUse in source · 4 passages
Search aliases: N
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
nUse in source · 22 passages
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Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
n'Use in source · 1 passage
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Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
n_iUse in source · 2 passages
Search aliases: n_i
Use in source · core_v0.2.tex, line 829
Semantic improvement and resource improvement are separate. Fix named additive event types. Let K≥0 be an extra preparation vector, let s_i be the old-minus-new per-use vector for workload type i, and let n_i count its uses. The net event difference is Δ=∑_i n_i s_i-K. A nonnegative price vector w converts these events into one declared additive measure. Storage capacity, peak memory, and elapsed time need their own models unless a justified conversion is specified.
Exact mathematical passage
n_i
OUse in source · 3 passages
Search aliases: O
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathcal OUse in source · 1 passage
Search aliases: O mathcal
Use in source · core_v0.2.tex, line 358
Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability events.
Exact mathematical passage
\mathcal O
O_eUse in source · 1 passage
Search aliases: O_e
Use in source · core_v0.2.tex, line 590
belch:inquiry questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegative integer resource vector c_e.
Exact mathematical passage
o_e:D_e\to O_e
o_eUse in source · 2 passages
Search aliases: o_e
Use in source · core_v0.2.tex, line 590
belch:inquiry questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegative integer resource vector c_e.
Exact mathematical passage
o_e:D_e\to O_e
O_{t+1}Use in source · 1 passage
Search aliases: O_t+1
Use in source · core_v0.2.tex, line 433
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
Exact mathematical passage
K_{a,o}(x',x)=\Pr(X_{t+1}=x',O_{t+1}=o\mid X_t=x,A_t=a)
PUse in source · 17 passages
Search aliases: P
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathcal PUse in source · 1 passage
Search aliases: P mathcal
Use in source · core_v0.2.tex, line 601
The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it …
Exact mathematical passage
\mathcal P_h(V)
\mathfrak PUse in source · 2 passages
Search aliases: P mathfrak
Use in source · core_v0.2.tex, line 708
An approximate model supports a different, quantitative claim. Exact policy transport and approximate planning must not share one unqualified guarantee. Fix a finite horizon H and one finite class P of admitted action/report trees, shared by the true and approximate models. The complete loss of a tree is the same function of its coarse-state and action/report path in both models, taking values in [0,B]. For example, a terminal loss in [0,1] and at most H charges bounded by c_max give B=1+Hc_max.
Exact mathematical passage
\mathfrak P
P'Use in source · 2 passages
Search aliases: P'
Use in source · core_v0.2.tex, line 389
The theorem permits removal of an independent nuisance flag. It does not permit removal of every association. Consider two priors on (b,s), with rows indexed by b and columns by s: P=1/2 · 0 0 · 1/2, P'=0 · 1/2 1/2 · 0. Both individual marginals are uniform. A noisy calibration reports s correctly with probability 3/4, independently conditional on the fixed world. The probability of report one is 1/2 under both priors. After that report, however, the target posteriors are 3/4 and 1/4. Separate t…
Exact mathematical passage
P=\begin{pmatrix}1/2&0\\0&1/2\end{pmatrix},\qquad P'=\begin{pmatrix}0&1/2\\1/2&0\end{pmatrix}.
p_0Use in source · 3 passages
Search aliases: p_0
Use in source · core_v0.2.tex, line 458
for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the report probabilities and the aggregated positive-probability posteriors: z_a,o(p)=bar z_a,o(Cp), CΨ_a,o(p)=barΨ_a,o(Cp). Starting from Cp_0, these identities preserve every finite joint action/report law and every corresponding coarse posterior under one common policy that uses only the p…
Exact mathematical passage
Cp_0
P_1Use in source · 1 passage
Search aliases: P_1
Use in source · core_v0.2.tex, line 268
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass is nonneg…
Exact mathematical passage
P_1,\ldots,P_m
p_EUse in source · 7 passages
Search aliases: p_E
Use in source · core_v0.2.tex, line 488
d with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p_E)}. If either e…
Exact mathematical passage
p_E,\widetilde p_E>0
P_HUse in source · 3 passages
Search aliases: P_H
Use in source · core_v0.2.tex, line 484
[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetil…
Exact mathematical passage
P_H,\widetilde P_H
\mathcal P_hUse in source · 1 passage
Search aliases: P_h mathcal
Use in source · core_v0.2.tex, line 601
The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it …
Exact mathematical passage
\mathcal P_h(V)
P_iUse in source · 5 passages
Search aliases: P_i
Use in source · core_v0.2.tex, line 268
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass is nonneg…
Exact mathematical passage
\sum_iP_i=I
P_mUse in source · 1 passage
Search aliases: P_m
Use in source · core_v0.2.tex, line 268
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass is nonneg…
Exact mathematical passage
P_1,\ldots,P_m
p_{N-1}Use in source · 1 passage
Search aliases: p_N-1
Use in source · core_v0.2.tex, line 737
Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that agrees everywhere or returns a context where it disagrees. The teacher's work is part of the development account.
Exact mathematical passage
\mathcal G=(p_0,\ldots,p_{N-1})
P_{\rm av}Use in source · 1 passage
Search aliases: P_rm av rm
Use in source · core_v0.2.tex, line 284
[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero. Normalizing it as a belief is undefined. Clipping its negative entry and normalizing the resulting sum would instead produce (3/5,2/5)^top, a differe…
Exact mathematical passage
P_{\rm av}=\tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad P_{\rm det}=\tfrac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix}.
P_{\rm det}Use in source · 1 passage
Search aliases: P_rm det rm
Use in source · core_v0.2.tex, line 284
[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero. Normalizing it as a belief is undefined. Clipping its negative entry and normalizing the resulting sum would instead produce (3/5,2/5)^top, a differe…
Exact mathematical passage
P_{\rm av}=\tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad P_{\rm det}=\tfrac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix}.
\operatorname{pa}Use in source · 1 passage
Search aliases: pa operatorname
Use in source · core_v0.2.tex, line 805
Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined by those components and parent claims. External assumptions are included explicitly as premises.
Exact mathematical passage
\operatorname{pa}(v)
\parUse in source · 1 passage
Search aliases: par
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\operatorname{Pareto}Use in source · 1 passage
Search aliases: Pareto operatorname
Use in source · core_v0.2.tex, line 602
The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it …
Exact mathematical passage
\operatorname{Pareto}\{(g_a(V),1):a\text{ succeeds throughout }V\}.
\operatorname{Post}Use in source · 2 passages
Search aliases: Post operatorname
Use in source · core_v0.2.tex, line 907
A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\{z\in S:\exists a\in A_{\rm end}(z),\ \operatorname{Post}(z,a)\subseteq T_k\}.
\PowUse in source · 1 passage
Search aliases: Pow P mathcal
Use in source · core_v0.2.tex, line 305
Let E⊆ P(Ω) contain Ω and be closed under intersection. In this chapter the world does not change when an evidence event is received. The update is V↦ V∩ E. Physical actions and their effects will return in Chapter~[source reference].
Exact mathematical passage
\mathcal E\subseteq\Pow(\Omega)
\PrUse in source · 6 passages
Search aliases: Pr
Use in source · core_v0.2.tex, line 359
Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability events.
Exact mathematical passage
\Pr(m,o,z)=p(m)L_m(o)B(o,z),\qquad Q_m(z)=\sum_oL_m(o)B(o,z).
QUse in source · 22 passages
Search aliases: Q
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
Q_mUse in source · 2 passages
Search aliases: Q_m
Use in source · core_v0.2.tex, line 359
Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability events.
Exact mathematical passage
\Pr(m,o,z)=p(m)L_m(o)B(o,z),\qquad Q_m(z)=\sum_oL_m(o)B(o,z).
\questionUse in source · 1 passage
Search aliases: question
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
RUse in source · 14 passages
Search aliases: R
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathbb RUse in source · 2 passages
Search aliases: R mathbb
Use in source · core_v0.2.tex, line 196
[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.
Exact mathematical passage
V=\mathbb R^2
rUse in source · 13 passages
Search aliases: r
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
r'Use in source · 1 passage
Search aliases: r'
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\mathbb R^2Use in source · 1 passage
Search aliases: R^2 mathbb
Use in source · core_v0.2.tex, line 196
[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.
Exact mathematical passage
V=\mathbb R^2
\mathbb R^d_{\ge0}Use in source · 1 passage
Search aliases: R^d_≥0 mathbb ge
Use in source · core_v0.2.tex, line 273
beginproposition[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so it…
Exact mathematical passage
\mathcal K=\mathbb R^d_{\ge0}
\raggedrightUse in source · 1 passage
Search aliases: raggedright
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\rceilUse in source · 2 passages
Search aliases: rceil
Use in source · core_v0.2.tex, line 636
[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf, so at least n leaves are reachable. A binary tree of query depth d has at most 2^d leaves, giving d≥lceillog_2nrceil. The terminal repair adds one un…
Exact mathematical passage
1+\lceil\log_2n\rceil
\rmUse in source · 5 passages
Search aliases: rm
Use in source · core_v0.2.tex, line 284
[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero. Normalizing it as a belief is undefined. Clipping its negative entry and normalizing the resulting sum would instead produce (3/5,2/5)^top, a differe…
Exact mathematical passage
P_{\rm av}=\tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad P_{\rm det}=\tfrac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix}.
SUse in source · 11 passages
Search aliases: S
Use in source · core_v0.2.tex, line 175
Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
S:Z\to V
sUse in source · 34 passages
Search aliases: s
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
s_iUse in source · 2 passages
Search aliases: s_i
Use in source · core_v0.2.tex, line 829
Semantic improvement and resource improvement are separate. Fix named additive event types. Let K≥0 be an extra preparation vector, let s_i be the old-minus-new per-use vector for workload type i, and let n_i count its uses. The net event difference is Δ=∑_i n_i s_i-K. A nonnegative price vector w converts these events into one declared additive measure. Storage capacity, peak memory, and elapsed time need their own models unless a justified conversion is specified.
Exact mathematical passage
s_i
\smallUse in source · 1 passage
Search aliases: small
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\SpanUse in source · 1 passage
Search aliases: Span span operatorname
Use in source · core_v0.2.tex, line 228
Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=0.
Exact mathematical passage
E=\Span\{A_wD_a^\theta v: v\in V,\ a,\theta,\ w\text{ finite}\}.
\supUse in source · 1 passage
Search aliases: sup
Use in source · core_v0.2.tex, line 523
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\TV(p,\widehat p)\le e,\qquad \sup_\omega d(v_\omega,\widehat v_\omega)\le d_0.
\sup_\omegaUse in source · 1 passage
Search aliases: sup_ω sup omega
Use in source · core_v0.2.tex, line 523
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\TV(p,\widehat p)\le e,\qquad \sup_\omega d(v_\omega,\widehat v_\omega)\le d_0.
TUse in source · 40 passages
Search aliases: T
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
tUse in source · 20 passages
Search aliases: t
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
t^{-1}Use in source · 1 passage
Search aliases: t^-1
Use in source · core_v0.2.tex, line 330
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
T^2Use in source · 2 passages
Search aliases: T^2
Use in source · core_v0.2.tex, line 196
[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.
Exact mathematical passage
T^2=I_V
T^\thetaUse in source · 1 passage
Search aliases: T^θ theta
Use in source · core_v0.2.tex, line 254
For example, let Q(x_1,x_2,x_3)=x_1 and let every candidate act as T^θ(x_1,x_2,x_3)=(x_1+x_2,x_2,θ x_3). Then E is contained in the third coordinate and remains invisible. The two-coordinate method (x_1,x_2)↦(x_1+x_2,x_2) is exact, although the hidden dynamics are not identified.
Exact mathematical passage
T^\theta(x_1,x_2,x_3)=(x_1+x_2,x_2,\theta x_3).
T_*Use in source · 9 passages
Search aliases: T_*
Use in source · core_v0.2.tex, line 911
T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
T_*
T_0Use in source · 1 passage
Search aliases: T_0
Use in source · core_v0.2.tex, line 907
A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\{z\in S:\exists a\in A_{\rm end}(z),\ \operatorname{Post}(z,a)\subseteq T_k\}.
T_a^\thetaUse in source · 5 passages
Search aliases: T_a^θ theta
Use in source · core_v0.2.tex, line 225
One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be finite dimensional, let T_a^θ be finitely many candidate linear operators, and let A_a be one complete candidate system chosen as reference. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear readout for every candidate.
Exact mathematical passage
T_a^\theta
t_iUse in source · 1 passage
Search aliases: t_i
Use in source · core_v0.2.tex, line 558
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
t_i(x)
T_kUse in source · 3 passages
Search aliases: T_k
Use in source · core_v0.2.tex, line 907
A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\{z\in S:\exists a\in A_{\rm end}(z),\ \operatorname{Post}(z,a)\subseteq T_k\}.
T_{k+1}Use in source · 1 passage
Search aliases: T_k+1
Use in source · core_v0.2.tex, line 907
A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\{z\in S:\exists a\in A_{\rm end}(z),\ \operatorname{Post}(z,a)\subseteq T_k\}.
T_w^\thetaUse in source · 2 passages
Search aliases: T_w^θ theta
Use in source · core_v0.2.tex, line 236
[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also predicts the output when the candidate index changes between operations. Assume the first condition. For every a,θ,w, equality as operators on all of V g…
Exact mathematical passage
QT_w^\theta=QA_w
\tableofcontentsUse in source · 1 passage
Search aliases: tableofcontents
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\topUse in source · 8 passages
Search aliases: top
Use in source · core_v0.2.tex, line 283
[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1
Exact mathematical passage
p=(3/4,1/4)^\top
\topruleUse in source · 1 passage
Search aliases: toprule
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\TVUse in source · 5 passages
Search aliases: TV d_TV mathrm
Use in source · core_v0.2.tex, line 477
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_0)≤δ_0 with nonnegat…
Exact mathematical passage
\TV\left(\left(\sum_{x':\pi(x')=z'}K_{a,o}(x',x)\right)_{o,z'}, (\widetilde K_{a,o}(z',\pi(x)))_{o,z'}\right)\le\epsilon_t.
UUse in source · 8 passages
Search aliases: U
Use in source · core_v0.2.tex, line 176
Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
\mathcal E(T)\mathcal E(U)=ATSAUS=ATUS=\mathcal E(TU).
uUse in source · 15 passages
Search aliases: u
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
u_hUse in source · 2 passages
Search aliases: u_h
Use in source · core_v0.2.tex, line 465
For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histories. Normalization on positive histories gives the posterior statement. Stopping decisions based on the same history can be included as matching termi…
Exact mathematical passage
u_h,\bar u_h
u_\varnothingUse in source · 1 passage
Search aliases: u_∅ varnothing
Use in source · core_v0.2.tex, line 465
For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histories. Normalization on positive histories gives the posterior statement. Stopping decisions based on the same history can be included as matching termi…
Exact mathematical passage
\bar u_\varnothing=Cu_\varnothing
VUse in source · 37 passages
Search aliases: V
Use in source · core_v0.2.tex, line 175
Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
V,Z
vUse in source · 33 passages
Search aliases: v
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
V_{e,o}Use in source · 2 passages
Search aliases: V_e,o
Use in source · core_v0.2.tex, line 593
A current support V≠∅ records possible states. The experiment is permitted only if V⊆ D_e. After report o, the possible post-experiment states are V_e,o={f_e(x):x∈ V, o_e(x)=o}. Only nonempty report branches are retained. If the target concerns a past value, that value must survive as an immutable tag. Resetting the current state is not recovering its history.
Exact mathematical passage
V_{e,o}=\{f_e(x):x\in V,\ o_e(x)=o\}.
v_iUse in source · 1 passage
Search aliases: v_i
Use in source · core_v0.2.tex, line 269
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass is nonneg…
Exact mathematical passage
v=\sum_{i:w_i>0}w_i v_i,\qquad w_i=\lambda P_iv\ge0,\quad \sum_iw_i=1,\quad v_i=P_iv/w_i.
v_oUse in source · 2 passages
Search aliases: v_o
Use in source · core_v0.2.tex, line 605
athcal P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty branch, form (c_e+max_o v_o,quad1+∑_o m_o). A profile dominates another when it is no larger in every resou…
Exact mathematical passage
(v_o,m_o)
v_\omegaUse in source · 3 passages
Search aliases: v_ω omega
Use in source · core_v0.2.tex, line 522
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
v_\omega
\vfillUse in source · 1 passage
Search aliases: vfill
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
WUse in source · 6 passages
Search aliases: W
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
w_iUse in source · 2 passages
Search aliases: w_i
Use in source · core_v0.2.tex, line 269
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass is nonneg…
Exact mathematical passage
v=\sum_{i:w_i>0}w_i v_i,\qquad w_i=\lambda P_iv\ge0,\quad \sum_iw_i=1,\quad v_i=P_iv/w_i.
\widehat aUse in source · 4 passages
Search aliases: widehat a widehat
Use in source · core_v0.2.tex, line 534
[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add and subtract ∑_ω p(ω)ℓ(a,widehat v_ω). The response term has absolute value at most Kd_0. The expectation of a function in [0,1] changes by at most tot…
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\widehat a
\widehat pUse in source · 3 passages
Search aliases: widehat p widehat
Use in source · core_v0.2.tex, line 522
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\widehat p
\widehat RUse in source · 4 passages
Search aliases: widehat R widehat
Use in source · core_v0.2.tex, line 528
widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
R(a)=\sum_\omega p(\omega)\ell(a,v_\omega),\qquad \widehat R(a)=\sum_\omega\widehat p(\omega)\ell(a,\widehat v_\omega).
\widehat vUse in source · 4 passages
Search aliases: widehat v widehat
Use in source · core_v0.2.tex, line 522
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\widehat v_\omega
\widehat v_\omegaUse in source · 4 passages
Search aliases: widehat v_ω widehat omega
Use in source · core_v0.2.tex, line 522
One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
\widehat v_\omega
\widetilde JUse in source · 3 passages
Search aliases: widetilde J widetilde
Use in source · core_v0.2.tex, line 713
[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certify a zero-risk constraint or preservation of admissibility outside the common class. The expectation of a function in [0,B] changes by at most B times…
Exact mathematical passage
\widetilde J(\pi)
\widetilde KUse in source · 2 passages
Search aliases: widetilde K widetilde
Use in source · core_v0.2.tex, line 476
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_0)≤δ_0 with nonnegat…
Exact mathematical passage
\widetilde K
\widetilde K_{a,o}Use in source · 1 passage
Search aliases: widetilde K_a,o widetilde
Use in source · core_v0.2.tex, line 477
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_0)≤δ_0 with nonnegat…
Exact mathematical passage
\TV\left(\left(\sum_{x':\pi(x')=z'}K_{a,o}(x',x)\right)_{o,z'}, (\widetilde K_{a,o}(z',\pi(x)))_{o,z'}\right)\le\epsilon_t.
\widetilde PUse in source · 3 passages
Search aliases: widetilde P widetilde
Use in source · core_v0.2.tex, line 484
[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetil…
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P_H,\widetilde P_H
\widetilde pUse in source · 7 passages
Search aliases: widetilde p widetilde
Use in source · core_v0.2.tex, line 481
idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative δ_0,ϵ_t. Both models run the same history-based policy. Coarse states recorded in the comparison below are analytical coordina…
Exact mathematical passage
\TV(Cp_0,\widetilde p_0)\le\delta_0
\widetilde p_0Use in source · 1 passage
Search aliases: widetilde p_0 widetilde
Use in source · core_v0.2.tex, line 481
idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative δ_0,ϵ_t. Both models run the same history-based policy. Coarse states recorded in the comparison below are analytical coordina…
Exact mathematical passage
\TV(Cp_0,\widetilde p_0)\le\delta_0
\widetilde p_EUse in source · 6 passages
Search aliases: widetilde p_E widetilde
Use in source · core_v0.2.tex, line 488
d with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p_E)}. If either e…
Exact mathematical passage
p_E,\widetilde p_E>0
\widetilde P_HUse in source · 3 passages
Search aliases: widetilde P_H widetilde
Use in source · core_v0.2.tex, line 484
[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetil…
Exact mathematical passage
P_H,\widetilde P_H
\widetilde\piUse in source · 3 passages
Search aliases: widetildeπ widetilde pi
Use in source · core_v0.2.tex, line 717
[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certify a zero-risk constraint or preservation of admissibility outside the common class. The expectation of a function in [0,B] changes by at most B times…
Exact mathematical passage
\widetilde\pi
xUse in source · 47 passages
Search aliases: x
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
x'Use in source · 8 passages
Search aliases: x'
Use in source · core_v0.2.tex, line 141
Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.
Exact mathematical passage
x\sim x'\quad\Longleftrightarrow\quad q(F_wx)=q(F_wx')\text{ for every finite word }w.
x^2Use in source · 1 passage
Search aliases: x^2
Use in source · core_v0.2.tex, line 278
On the nonnegative coordinate cone, preservation is equivalent to nonnegative matrix entries, as is seen by applying a matrix to every standard basis vector. Because the matrices sum to I, their nonnegative off-diagonal entries must all vanish. An idempotent diagonal entry satisfies x^2=x, hence is zero or one. The diagonal sums are one, so each coordinate is selected by exactly one map. Finally, completeness and orthogonality without positivity do not give the first premise, as the following ex…
Exact mathematical passage
x^2=x
x_1Use in source · 5 passages
Search aliases: x_1
Use in source · core_v0.2.tex, line 196
[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.
Exact mathematical passage
C(x_1,x_2)=x_1
x_2Use in source · 5 passages
Search aliases: x_2
Use in source · core_v0.2.tex, line 196
[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.
Exact mathematical passage
C(x_1,x_2)=x_1
x_3Use in source · 2 passages
Search aliases: x_3
Use in source · core_v0.2.tex, line 253
For example, let Q(x_1,x_2,x_3)=x_1 and let every candidate act as T^θ(x_1,x_2,x_3)=(x_1+x_2,x_2,θ x_3). Then E is contained in the third coordinate and remains invisible. The two-coordinate method (x_1,x_2)↦(x_1+x_2,x_2) is exact, although the hidden dynamics are not identified.
Exact mathematical passage
Q(x_1,x_2,x_3)=x_1
X_tUse in source · 1 passage
Search aliases: X_t
Use in source · core_v0.2.tex, line 433
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
Exact mathematical passage
K_{a,o}(x',x)=\Pr(X_{t+1}=x',O_{t+1}=o\mid X_t=x,A_t=a)
X_{t+1}Use in source · 1 passage
Search aliases: X_t+1
Use in source · core_v0.2.tex, line 433
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
Exact mathematical passage
K_{a,o}(x',x)=\Pr(X_{t+1}=x',O_{t+1}=o\mid X_t=x,A_t=a)
yUse in source · 23 passages
Search aliases: y
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
ZUse in source · 20 passages
Search aliases: Z
Use in source · core_v0.2.tex, line 175
Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary coordinates.
Exact mathematical passage
V,Z
zUse in source · 35 passages
Search aliases: z
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
z'Use in source · 4 passages
Search aliases: z'
Use in source · core_v0.2.tex, line 450
[One quotient for reports, posteriors, and feedback] A nonnegative normalized coarse instrument bar K_a,o satisfies CK_a,o=bar K_a,oC for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the report probabilities and the aggregated positive-probability posteriors: z_a,o(p)=bar z_a,o(Cp), CΨ_a,o(p)=barΨ_a,o(Cp). Starting from Cp_0, these identiti…
Exact mathematical passage
a,o,z'
z_{a,o}Use in source · 2 passages
Search aliases: z_a,o
Use in source · core_v0.2.tex, line 439
For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.
Exact mathematical passage
z_{a,o}(p)=\mathbf1^\top K_{a,o}p,\qquad \Psi_{a,o}(p)=K_{a,o}p/z_{a,o}(p)
Z_HUse in source · 1 passage
Search aliases: Z_H
Use in source · core_v0.2.tex, line 489
e distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetilde p_E)}. If either event probability exceeds d_H, the other is necessarily…
Exact mathematical passage
\TV\bigl(P_H(Z_H\in\cdot\mid E),\widetilde P_H(Z_H\in\cdot\mid E)\bigr) \le\min\left\{1,\frac{d_H}{\max(p_E,\widetilde p_E)}\right\}.
|Use in source · 7 passages
Search aliases: |
Use in source · core_v0.2.tex, line 330
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
\cdotUse in source · 4 passages
Search aliases: · cdot
Use in source · core_v0.2.tex, line 382
probability. For a positive pair, posterior equality yields p(m)L_m(o)B(o,z)=Pr(o,z)p(mmid o) =ρ(omid z)Pr(z)p(mmid z). Substituting Bayes' formula and cancelling positive p(m) gives the factorization. A zero-probability pair under a positive prior has L_m(o)B(o,z)=0 for all m; its conditional weight is zero. If z is impossible for every model, assign any probability row to ρ(·mid z) because both sides vanish.
Exact mathematical passage
\rho(\cdot\mid z)
\alphaUse in source · 3 passages
Search aliases: α alpha
Use in source · core_v0.2.tex, line 558
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
\alpha:X\to H
\betaUse in source · 1 passage
Search aliases: β beta
Use in source · core_v0.2.tex, line 558
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
\beta_i(x)
\beta_iUse in source · 1 passage
Search aliases: β_i beta
Use in source · core_v0.2.tex, line 558
An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Exact mathematical passage
\beta_i(x)
\gammaUse in source · 3 passages
Search aliases: γ gamma
Use in source · core_v0.2.tex, line 465
For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histories. Normalization on positive histories gives the posterior statement. Stopping decisions based on the same history can be included as matching termi…
Exact mathematical passage
\gamma(a\mid h)
\DeltaUse in source · 7 passages
Search aliases: Δ Delta
Use in source · core_v0.2.tex, line 807
After changing a to a', mark a vertex stale when a direct input changes and propagate staleness to all descendants. Call the resulting set Δ.
Exact mathematical passage
\Delta
\deltaUse in source · 10 passages
Search aliases: δ delta
Use in source · core_v0.2.tex, line 481
idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative δ_0,ϵ_t. Both models run the same history-based policy. Coarse states recorded in the comparison below are analytical coordina…
Exact mathematical passage
\TV(Cp_0,\widetilde p_0)\le\delta_0
\delta_0Use in source · 3 passages
Search aliases: δ_0 delta
Use in source · core_v0.2.tex, line 481
idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_0)≤δ_0 with nonnegative δ_0,ϵ_t. Both models run the same history-based policy. Coarse states recorded in the comparison below are analytical coordina…
Exact mathematical passage
\TV(Cp_0,\widetilde p_0)\le\delta_0
\delta_jUse in source · 3 passages
Search aliases: δ_j delta
Use in source · core_v0.2.tex, line 850
[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopping time depend on earlier history. Taking expectations gives Pr(A_j)≤δ_j. A union bound gives the result for every finite set of attempts. The finite…
Exact mathematical passage
\Pr(A_j\mid\mathcal H_{j-1})\le\delta_j\quad\text{almost surely}, \qquad\sum_{j\ge1}\delta_j\le\delta,
\epsilonUse in source · 6 passages
Search aliases: ϵ epsilon
Use in source · core_v0.2.tex, line 477
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_0)≤δ_0 with nonnegat…
Exact mathematical passage
\TV\left(\left(\sum_{x':\pi(x')=z'}K_{a,o}(x',x)\right)_{o,z'}, (\widetilde K_{a,o}(z',\pi(x)))_{o,z'}\right)\le\epsilon_t.
\epsilon_tUse in source · 6 passages
Search aliases: ϵ_t epsilon
Use in source · core_v0.2.tex, line 477
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_0)≤δ_0 with nonnegat…
Exact mathematical passage
\TV\left(\left(\sum_{x':\pi(x')=z'}K_{a,o}(x',x)\right)_{o,z'}, (\widetilde K_{a,o}(z',\pi(x)))_{o,z'}\right)\le\epsilon_t.
\etaUse in source · 8 passages
Search aliases: η eta
Use in source · core_v0.2.tex, line 415
For minimality, take any admissible refinement η of π_0. It has at least two blocks. Fix a strictly positive fine likelihood ℓ and a strictly positive coarse likelihood barℓ inducing its assumed update. For states x,y in one block and z in a different block, compare priors assigning half their mass to x,z and to y,z. Their coarse priors agree. Equality of the posterior odds gives ℓ(x)/ℓ(z)=barℓ(η(x))/barℓ(η(z))=ℓ(y)/ℓ(z). Thus ℓ(x)=ℓ(y), and η refines the first likelihood split. Coarse likelihoo…
Exact mathematical passage
\eta
\thetaUse in source · 15 passages
Search aliases: θ theta
Use in source · core_v0.2.tex, line 225
One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be finite dimensional, let T_a^θ be finitely many candidate linear operators, and let A_a be one complete candidate system chosen as reference. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear readout for every candidate.
Exact mathematical passage
T_a^\theta
\lambdaUse in source · 5 passages
Search aliases: λ lambda
Use in source · core_v0.2.tex, line 265
Let K be a convex cone in a real vector space, and let λ be linear and strictly positive on every nonzero member of K. Its normalized states satisfy v∈ K and λ v=1. A coordinate change can send K to a cone containing negative coordinates without changing those states' probability meaning. Conversely, an arbitrary signed component of a valid state need not itself be valid.
Exact mathematical passage
\lambda
\xiUse in source · 6 passages
Search aliases: ξ xi
Use in source · core_v0.2.tex, line 534
[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add and subtract ∑_ω p(ω)ℓ(a,widehat v_ω). The response term has absolute value at most Kd_0. The expectation of a function in [0,1] changes by at most tot…
Exact mathematical passage
\xi=\min\{1,e+Kd_0\}
\piUse in source · 21 passages
Search aliases: π pi
Use in source · core_v0.2.tex, line 401
First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet, assume also ∑_oℓ_o(x)=1.
Exact mathematical passage
\pi
\pi_*Use in source · 3 passages
Search aliases: π_* pi
Use in source · core_v0.2.tex, line 717
[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certify a zero-risk constraint or preservation of admissibility outside the common class. The expectation of a function in [0,B] changes by at most B times…
Exact mathematical passage
\pi_*
\pi_0Use in source · 4 passages
Search aliases: π_0 pi
Use in source · core_v0.2.tex, line 403
A mandatory partition π_0 with at least two blocks protects the target distinctions. We seek a refinement that supports every prediction and each separate Bayesian update for every prior. The observation is an operation available in its own right. This is stronger than inspecting only a chosen combined transition-and-report schedule.
Exact mathematical passage
\pi_0
\pi_{\rm init}Use in source · 2 passages
Search aliases: π_rm init pi rm
Use in source · core_v0.2.tex, line 406
[Least refinement closed under prediction and correction] First split π_0 by the complete vectors (ℓ_o(x))_o, obtaining π_rm init. Then repeatedly split current blocks by (∑_x'∈ BK_a(x,x'))_a,B, where B runs over the current blocks. The process stops after at most |X|-|π_rm init| strict refinement rounds. Its final partition is the unique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly posit…
Exact mathematical passage
\pi_{\rm init}
\rhoUse in source · 5 passages
Search aliases: ρ rho
Use in source · core_v0.2.tex, line 369
[When a report preserves every Bayesian correction] The following are equivalent: item For every prior, the posterior after o equals that after z on every positive-probability pair (o,z). item The equality holds for one prior positive on every model. item There is a stochastic kernel ρ(omid z) such that L_m(o)B(o,z)=Q_m(z)ρ(omid z) for all m,o,z. The reconstruction in the last condition refers to the actual joint experiment, not an unrelated simulation with matching marginals. The first conditio…
Exact mathematical passage
\rho(o\mid z)
\SigmaUse in source · 4 passages
Search aliases: Σ Sigma
Use in source · core_v0.2.tex, line 309
A state sufficient for every later filter. Define the correction signature Σ(V)=(t(V∩ E))_E∈ E. It records what each admitted future filter would leave of the possible target values.
Exact mathematical passage
\Sigma(V)=\bigl(t(V\cap E)\bigr)_{E\in\mathcal E}.
\tauUse in source · 5 passages
Search aliases: τ tau
Use in source · core_v0.2.tex, line 202
A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.
Exact mathematical passage
G_{\tau(a)}
\PhiUse in source · 5 passages
Search aliases: Φ Phi
Use in source · core_v0.2.tex, line 181
For a compression, reverse the splitting. Let C:V→ Z and R:Z→ V satisfy CR=I_Z, and put P=RC. The compressed operator is Φ(T)=CTR.
Exact mathematical passage
\Phi(T)=CTR
\PsiUse in source · 2 passages
Search aliases: Ψ Psi
Use in source · core_v0.2.tex, line 439
For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.
Exact mathematical passage
z_{a,o}(p)=\mathbf1^\top K_{a,o}p,\qquad \Psi_{a,o}(p)=K_{a,o}p/z_{a,o}(p)
\Psi_{a,o}Use in source · 2 passages
Search aliases: Ψ_a,o Psi
Use in source · core_v0.2.tex, line 439
For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.
Exact mathematical passage
z_{a,o}(p)=\mathbf1^\top K_{a,o}p,\qquad \Psi_{a,o}(p)=K_{a,o}p/z_{a,o}(p)
\OmegaUse in source · 9 passages
Search aliases: Ω Omega
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\omegaUse in source · 10 passages
Search aliases: ω omega
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\omega'Use in source · 1 passage
Search aliases: ω' omega
Use in source · core_v0.2.tex, line 43
[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Histor…
Exact mathematical passage
0.12in]Reasoning\par} \vspace{0.28in} {\LARGE Evidence, Methods,\\and Accountable Revision\par} \vspace{0.45in} {\Large Reviewed Mathematical Core\par} \vspace{0.25in}\hrule\vspace{0.2in} Komorebi P.\hfill Core edition 0.2\par 9 October 2026\hfill Internal proof review\par \vfill {\large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision.\par} \vspace{0.3in} This is a newly written selection, not a concatenation of the research archive. Only the mathematical statements developed and reviewed here belong to its core. Historical experiments, speculative architectures, and unresolved integration claims are not imported as conclusions. \vspace{0.2in} {\small The review record separates prose-proof checking, finite computational checks, historical formal evidence, and document validation. This edition is not independently peer reviewed or proof-assistant verified.} \end{titlepage} \chapter*{The shape of the argument}\addcontentsline{toc}{chapter}{The shape of the argument} A machine can retain an answer without retaining a method. It can possess a method without having evidence to initialize it. It can execute that method correctly while using a record that no longer permits correction. It can improve one part of the computation while making the complete process more expensive. These are not unrelated qualifications. Each is a failure at a particular connection between evidence, representation, and action. This book follows those connections in order. It begins with the information actually available to a reasoner. It asks what state that information can initialize, which operations the state supports, what must be remembered when further evidence arrives, and how one bounded investigation can acquire what is missing. Only then does it discuss improving the investigation and maintaining commitments across several participants. The running example is deliberately small. A sensor reports a physical bit under an unknown convention, with a third nuisance bit that never affects the specified task. The example is enlarged only when a chapter explicitly changes its contract. It is not a model of a whole mind. Its purpose is to make hidden information, permitted actions, and each passage between results inspectable. \begin{center}\begin{tabular}{>{\raggedright\arraybackslash}p{0.08\linewidth}>{\raggedright\arraybackslash}p{0.38\linewidth}>{\raggedright\arraybackslash}p{0.43\linewidth}}\toprule Chapter & Question answered & What the next chapter still needs\\\midrule 1 & What does the record determine? & Operations that can be performed on the retained state.\\ 2 & What survives a change of representation? & A record capable of later correction.\\ 3 & What must memory retain? & Decisions and recipients that use that memory.\\ 4 & What can the retained information justify? & An investigation that can actually acquire it.\\ 5 & What fits one executable budget? & A justified way to improve the discovery procedure.\\ 6 & What search work can be removed? & Evidence that remains applicable after a revision.\\ 7 & Which guarantees and gains survive change? & Explicit treatment of other participants and obligations.\\ 8 & What can be maintained together? & A clear boundary between the proved chain and its applications.\\\bottomrule \end{tabular}\end{center} The second admission block joins the signal-model, Bayesian-refinement, and artifact-planning sources at their common probability interface. Chapters 2, 3 and 5 are expanded in place: no whole source manuscript is appended. There is one notation and one proof sequence. Repeated source arguments appear once. Original source names identify provenance, not new modules assumed to interoperate. The chapter endings explain exactly which output becomes the next chapter's input. \chapter*{What the review label means}\addcontentsline{toc}{chapter}{What the review label means} The earlier 1,919-page collection preserved source documents and navigation. Its own assembly report did not claim a proof review of every imported page. It remains a historical source reader. This book replaces breadth as the admission rule with result-level inspection. A formal statement is admitted here only when its domains and quantifiers are explicit, its proof is given in full, its source relationship is identified, its use in a later chapter is checked, and its applicable finite construction or sharp boundary is exercised by a documented exact check. This is an internal mathematical review. It does not convert a finite check into a proof of an infinite statement. The review is also narrower than accepting a source wholesale. A theorem selected from \emph{Machine Societies} does not import a general claim about spontaneously formed institutions. A linear reconstruction argument does not certify an emergent workspace. A commuting equation does not establish that a neural network learned its terms. Those stronger interpretations are outside the core. The preceding core reran the exact suites of four recent source books. Their receipts remain historical regression evidence, not new runs in this revision. The current build reruns the complete selected-core suite and the additional admission checks. The selection's proofs were reviewed separately. Older formal developments retain their own evidence status, but no historical Lean result is attributed to this rewritten text. No new Lean compilation was performed. The companion review files name every retained result, its assumptions, its source anchor, its computational check, and its remaining limit. They also account for every source document in the former collection. Omission means either that its full evidence was not re-established for this edition or that it is unnecessary to the continuous argument. It is not a claim that the omitted work is false or valueless. \tableofcontents \mainmatter \chapter{From evidence to a reusable state}\label{ch:evidence} \question{The first question is not how much state to store. It is which distinctions are accessible.} Let $\Omega$ be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function $r:\Omega\to H$. A target is a function $t:\Omega\to Y$. All codomains used below are nonempty. We work on attained records $r(\Omega)$, so no arbitrary interpretation of an impossible record is needed. \begin{proposition}[A record determines a target exactly on its fibres]\label{r:factor} There exists $d:r(\Omega)\to Y$ with $d(r(\omega))=t(\omega)$ for every world if and only if \[ r(\omega)=r(\omega')\ \Longrightarrow\ t(\omega)=t(\omega').
\ldotsUse in source · 5 passages
Search aliases: … ldots
Use in source · core_v0.2.tex, line 268
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass is nonneg…
Exact mathematical passage
P_1,\ldots,P_m
\ell^1Use in source · 1 passage
Search aliases: ℓ^1 ell
Use in source · core_v0.2.tex, line 498
For conditioning, let u,v be the restrictions of the two history laws to E, with masses p_E,widetilde p_E. The difference on the complement has ℓ^1 norm at least |p_E-widetilde p_E|. Thus u-v _1+|p_E-widetilde p_E|≤2d_H. Adding and subtracting v/p_E bounds the conditional distance by d_H/p_E. Interchanging the laws gives d_H/widetilde p_E, so the smaller of these bounds has the displayed maximum denominator. Projection to the final state cannot increase the distance. Finally, the probability of …
Exact mathematical passage
\ell^1
\ell_oUse in source · 3 passages
Search aliases: ℓ_o ell
Use in source · core_v0.2.tex, line 401
First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet, assume also ∑_oℓ_o(x)=1.
Exact mathematical passage
\ell_o(x)>0
\mapstoUse in source · 5 passages
Search aliases: ↦ mapsto
Use in source · core_v0.2.tex, line 188
[Exact accounting of the discarded excursion] For any endomorphisms T,U of V, Φ(TU)-Φ(T)Φ(U)=CT(I-P)UR. Thus compression preserves this composition precisely when the discarded excursion on the right vanishes. Full split encoding T↦ ATS preserves composition without that extra condition. Expand the product Φ(T)Φ(U)=CTRCUR=CTPUR and subtract it from CTUR. The identity for full split encoding follows from SA=I_V as computed above. These computations make no approximation and require no orthogonali…
Exact mathematical passage
T\mapsto ATS
\existsUse in source · 2 passages
Search aliases: ∃ exists
Use in source · core_v0.2.tex, line 907
A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\{z\in S:\exists a\in A_{\rm end}(z),\ \operatorname{Post}(z,a)\subseteq T_k\}.
\varnothingUse in source · 4 passages
Search aliases: ∅ varnothing
Use in source · core_v0.2.tex, line 330
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
\sum_iUse in source · 4 passages
Search aliases: ∑_i sum
Use in source · core_v0.2.tex, line 268
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass is nonneg…
Exact mathematical passage
\sum_iP_i=I
\sum_{i:w_i>0}Use in source · 1 passage
Search aliases: ∑_i:w_i>0 sum
Use in source · core_v0.2.tex, line 269
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass is nonneg…
Exact mathematical passage
v=\sum_{i:w_i>0}w_i v_i,\qquad w_i=\lambda P_iv\ge0,\quad \sum_iw_i=1,\quad v_i=P_iv/w_i.
\sum_{j\ge1}Use in source · 1 passage
Search aliases: ∑_j≥1 sum ge
Use in source · core_v0.2.tex, line 850
[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopping time depend on earlier history. Taking expectations gives Pr(A_j)≤δ_j. A union bound gives the result for every finite set of attempts. The finite…
Exact mathematical passage
\Pr(A_j\mid\mathcal H_{j-1})\le\delta_j\quad\text{almost surely}, \qquad\sum_{j\ge1}\delta_j\le\delta,
\sum_oUse in source · 3 passages
Search aliases: ∑_o sum
Use in source · core_v0.2.tex, line 359
Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive-probability events.
Exact mathematical passage
\Pr(m,o,z)=p(m)L_m(o)B(o,z),\qquad Q_m(z)=\sum_oL_m(o)B(o,z).
\sum_{o,x'}Use in source · 1 passage
Search aliases: ∑_o,x' sum
Use in source · core_v0.2.tex, line 436
Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every a,x. This notation is column-oriented: K_a,op is an unnormalized post-experiment distribution. It specifies an experiment's physical change and its o…
Exact mathematical passage
\sum_{o,x'}K_{a,o}(x',x)=1
\sum_{t=0}^{H-1}Use in source · 1 passage
Search aliases: ∑_t=0^H-1 sum
Use in source · core_v0.2.tex, line 485
[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_TV(P_H(Z_H∈·mid E),widetilde P_H(Z_H∈·mid E)) ≤min{1,fracd_Hmax(p_E,widetil…
Exact mathematical passage
\TV(P_H,\widetilde P_H)\le d_H:=\min\left\{1,\delta_0+\sum_{t=0}^{H-1}\epsilon_t\right\}.
\sum_{x':\pi(x')=z'}Use in source · 2 passages
Search aliases: ∑_x':π(x')=z' sum pi
Use in source · core_v0.2.tex, line 451
[One quotient for reports, posteriors, and feedback] A nonnegative normalized coarse instrument bar K_a,o satisfies CK_a,o=bar K_a,oC for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the report probabilities and the aggregated positive-probability posteriors: z_a,o(p)=bar z_a,o(Cp), CΨ_a,o(p)=barΨ_a,o(Cp). Starting from Cp_0, these identiti…
Exact mathematical passage
\sum_{x':\pi(x')=z'}K_{a,o}(x',x)
\sum_{x'\in B}Use in source · 1 passage
Search aliases: ∑_x'∈ B sum in
Use in source · core_v0.2.tex, line 407
[Least refinement closed under prediction and correction] First split π_0 by the complete vectors (ℓ_o(x))_o, obtaining π_rm init. Then repeatedly split current blocks by (∑_x'∈ BK_a(x,x'))_a,B, where B runs over the current blocks. The process stops after at most |X|-|π_rm init| strict refinement rounds. Its final partition is the unique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly posit…
Exact mathematical passage
\left(\sum_{x'\in B}K_a(x,x')\right)_{a,B},
\sum_\omegaUse in source · 2 passages
Search aliases: ∑_ω sum omega
Use in source · core_v0.2.tex, line 528
widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v_ω), widehat R(a)=∑_ωwidehat p(ω)ℓ(a,widehat v_ω).
Exact mathematical passage
R(a)=\sum_\omega p(\omega)\ell(a,v_\omega),\qquad \widehat R(a)=\sum_\omega\widehat p(\omega)\ell(a,\widehat v_\omega).
\capUse in source · 7 passages
Search aliases: ∩ cap
Use in source · core_v0.2.tex, line 305
Let E⊆ P(Ω) contain Ω and be closed under intersection. In this chapter the world does not change when an evidence event is received. The update is V↦ V∩ E. Physical actions and their effects will return in Chapter~[source reference].
Exact mathematical passage
V\mapsto V\cap E
\cupUse in source · 1 passage
Search aliases: ∪ cup
Use in source · core_v0.2.tex, line 907
A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.
Exact mathematical passage
T_0&=G,\\ T_{k+1}&=T_k\cup\{z\in S:\exists a\in A_{\rm end}(z),\ \operatorname{Post}(z,a)\subseteq T_k\}.
\simUse in source · 3 passages
Search aliases: ∼ sim
Use in source · core_v0.2.tex, line 141
Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.
Exact mathematical passage
x\sim x'\quad\Longleftrightarrow\quad q(F_wx)=q(F_wx')\text{ for every finite word }w.
\neUse in source · 4 passages
Search aliases: ≠ ne
Use in source · core_v0.2.tex, line 330
An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.
Exact mathematical passage
D_t=\{(i,y): C_i\cap t^{-1}(y)\ne\varnothing\},\qquad r=|D_t|.
\leUse in source · 16 passages
Search aliases: ≤ le
Use in source · core_v0.2.tex, line 477
Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_0)≤δ_0 with nonnegat…
Exact mathematical passage
\TV\left(\left(\sum_{x':\pi(x')=z'}K_{a,o}(x',x)\right)_{o,z'}, (\widetilde K_{a,o}(z',\pi(x)))_{o,z'}\right)\le\epsilon_t.
\geUse in source · 9 passages
Search aliases: ≥ ge
Use in source · core_v0.2.tex, line 269
[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagonal coordinate selections. Orthogonal completeness alone does not imply a probability mixture. Cone preservation places P_iv in K, so its mass is nonneg…
Exact mathematical passage
v=\sum_{i:w_i>0}w_i v_i,\qquad w_i=\lambda P_iv\ge0,\quad \sum_iw_i=1,\quad v_i=P_iv/w_i.
\subseteqUse in source · 13 passages
Search aliases: ⊆ subseteq
Use in source · core_v0.2.tex, line 237
[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also predicts the output when the candidate index changes between operations. Assume the first condition. For every a,θ,w, equality as operators on all of V g…
Exact mathematical passage
E\subseteq\ker Q
\oplusUse in source · 6 passages
Search aliases: ⊕ oplus
Use in source · core_v0.2.tex, line 117
The sensor used throughout the book. Write a world as ω=(b,s,n)∈{0,1}^3. The physical target bit is b. The sensor convention is s. The nuisance bit n never affects the operations or outputs of the initial contract. The display is y=b⊕ s, where ⊕ is addition modulo two. A permitted control f flips the physical bit and leaves s,n unchanged. Consequently the next display is 1-y.
Exact mathematical passage
y=b\oplus s,
\bigcapUse in source · 2 passages
Search aliases: ⋂ bigcap
Use in source · core_v0.2.tex, line 561
Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).
Exact mathematical passage
\mathcal C_h=\bigcap_{x:\alpha(x)=h}\mathcal C(x).
\bigcap_iUse in source · 1 passage
Search aliases: ⋂_i bigcap
Use in source · core_v0.2.tex, line 874
[Target-relative joint reconstruction] There is a linear decoder D with Q=DA if and only if ker A⊆ker Q. Full-state reconstruction is possible exactly when A is injective. The intersection identity ker A=⋂_iker A_i determines which ambiguities sharing removes. If Q=DA, a vector annihilated by A is also annihilated by Q. Conversely, define D(Av)=Qv on im A. If Av=Aw, then v-w∈ker A⊆ker Q, so this definition is independent of the representative. It is linear and extends to the full record space by…
Exact mathematical passage
\ker A=\bigcap_i\ker A_i
\bigcap_{x:\alpha(x)=h}Use in source · 1 passage
Search aliases: ⋂_x:α(x)=h bigcap alpha
Use in source · core_v0.2.tex, line 561
Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).
Exact mathematical passage
\mathcal C_h=\bigcap_{x:\alpha(x)=h}\mathcal C(x).
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