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{₇}{{$_7$}}1 {₈}{{$_8$}}1 {₉}{{$_9$}}1}\n\\lstset{style=LeanCode}\n\\newcommand{\\bayescover}[2]{\\hypersetup{pageanchor=false}\\begin{titlepage}\\color{navy}\n \\vspace*{16mm}{\\large KOMOREBI\\quad /\\quad REALIZATION CALCULUS\\par}\n \\vspace{17mm}{\\Huge\\bfseries Certified Bayesian","$}}1 {Γ}{{$\\Gamma$}}1 {Σ}{{$\\Sigma$}}1 {∧}{{$\\land$}}1 {∨}{{$\\lor$}}1 {¬}{{$\\neg$}}1 {×}{{$\\times$}}1 {⊆}{{$\\subseteq$}}1 {⊂}{{$\\subset$}}1 {⟨}{{$\\langle$}}1 {⟩}{{$\\rangle$}}1 {∑}{{$\\sum$}}1 {∏}{{$\\prod$}}1 {∣}{{$\\mid$}}1 {∘}{{$\\circ$}}1 {↑}{{$\\uparrow$}}1 {↓}{{$\\downarrow$}}1 {▸}{{$\\triangleright$}}1 {₀}{{$_0$}}1 {₁}{{$_1$}}1 {₂}{{$_2$}}1 {₃}{{$_3$}}1 {₄}{{$_4$}}1 {₅}{{$_5$}}1 {₆}{{$_6$}}1 {₇}{{$_7$}}1 {₈}{{$_8$}}1 {₉}{{$_9$}}1}\n\\lstset{style=LeanCode}\n\\newcommand{\\bayescover}[2]{\\hypersetup{pageanchor=false}\\begin{titlepage}\\color{navy}\n \\vspace*{16mm}{\\large KOMOREBI\\quad /\\quad REALIZATION CALCULUS\\par}\n \\vspace{17mm}{\\Huge\\bfseries Certified Bayesian","amma$}}1 {Σ}{{$\\Sigma$}}1 {∧}{{$\\land$}}1 {∨}{{$\\lor$}}1 {¬}{{$\\neg$}}1 {×}{{$\\times$}}1 {⊆}{{$\\subseteq$}}1 {⊂}{{$\\subset$}}1 {⟨}{{$\\langle$}}1 {⟩}{{$\\rangle$}}1 {∑}{{$\\sum$}}1 {∏}{{$\\prod$}}1 {∣}{{$\\mid$}}1 {∘}{{$\\circ$}}1 {↑}{{$\\uparrow$}}1 {↓}{{$\\downarrow$}}1 {▸}{{$\\triangleright$}}1 {₀}{{$_0$}}1 {₁}{{$_1$}}1 {₂}{{$_2$}}1 {₃}{{$_3$}}1 {₄}{{$_4$}}1 {₅}{{$_5$}}1 {₆}{{$_6$}}1 {₇}{{$_7$}}1 {₈}{{$_8$}}1 {₉}{{$_9$}}1}\n\\lstset{style=LeanCode}\n\\newcommand{\\bayescover}[2]{\\hypersetup{pageanchor=false}\\begin{titlepage}\\color{navy}\n \\vspace*{16mm}{\\large KOMOREBI\\quad /\\quad REALIZATION CALCULUS\\par}\n \\vspace{17mm}{\\Huge\\bfseries Certified Bayesian","{$\\Sigma$}}1 {∧}{{$\\land$}}1 {∨}{{$\\lor$}}1 {¬}{{$\\neg$}}1 {×}{{$\\times$}}1 {⊆}{{$\\subseteq$}}1 {⊂}{{$\\subset$}}1 {⟨}{{$\\langle$}}1 {⟩}{{$\\rangle$}}1 {∑}{{$\\sum$}}1 {∏}{{$\\prod$}}1 {∣}{{$\\mid$}}1 {∘}{{$\\circ$}}1 {↑}{{$\\uparrow$}}1 {↓}{{$\\downarrow$}}1 {▸}{{$\\triangleright$}}1 {₀}{{$_0$}}1 {₁}{{$_1$}}1 {₂}{{$_2$}}1 {₃}{{$_3$}}1 {₄}{{$_4$}}1 {₅}{{$_5$}}1 {₆}{{$_6$}}1 {₇}{{$_7$}}1 {₈}{{$_8$}}1 {₉}{{$_9$}}1}\n\\lstset{style=LeanCode}\n\\newcommand{\\bayescover}[2]{\\hypersetup{pageanchor=false}\\begin{titlepage}\\color{navy}\n \\vspace*{16mm}{\\large KOMOREBI\\quad /\\quad REALIZATION CALCULUS\\par}\n \\vspace{17mm}{\\Huge\\bfseries Certified Bayesian","The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be $\\R^n$ and the observation space be $\\R^p$. A state $X$ has prior law $\\mathcal N(m,P)$, and its observation is\n\\begin{equation}\n Y=HX+V,\\qquad V\\sim\\mathcal N(0,R),\\qquad V\\text{ independent of }X.\n \\label{eq:foundation-model}\n\\end{equation}\nHere $H\\in\\R^{p\\times n}$ may be rectangular, and $P\\in\\R^{n\\times n}$ may be singular. The assumptions are $P\\succeq0$ and $R\\succ0$. A singular prior is an ordinary Gaussian probability law concentrated on an affine subspace. The argument does not replace that law by a nonsingular approximation or assign an artificial inverse to its covariance.","e inference problem concerns one observation of a finite-dimensional latent state. Let the state space be $\\R^n$ and the observation space be $\\R^p$. A state $X$ has prior law $\\mathcal N(m,P)$, and its observation is\n\\begin{equation}\n Y=HX+V,\\qquad V\\sim\\mathcal N(0,R),\\qquad V\\text{ independent of }X.\n \\label{eq:foundation-model}\n\\end{equation}\nHere $H\\in\\R^{p\\times n}$ may be rectangular, and $P\\in\\R^{n\\times n}$ may be singular. The assumptions are $P\\succeq0$ and $R\\succ0$. A singular prior is an ordinary Gaussian probability law concentrated on an affine subspace. The argument does not replace that law by a nonsingular approximation or assign an artificial inverse to its covariance.","imensional latent state. Let the state space be $\\R^n$ and the observation space be $\\R^p$. A state $X$ has prior law $\\mathcal N(m,P)$, and its observation is\n\\begin{equation}\n Y=HX+V,\\qquad V\\sim\\mathcal N(0,R),\\qquad V\\text{ independent of }X.\n \\label{eq:foundation-model}\n\\end{equation}\nHere $H\\in\\R^{p\\times n}$ may be rectangular, and $P\\in\\R^{n\\times n}$ may be singular. The assumptions are $P\\succeq0$ and $R\\succ0$. A singular prior is an ordinary Gaussian probability law concentrated on an affine subspace. The argument does not replace that law by a nonsingular approximation or assign an artificial inverse to its covariance.","t state. Let the state space be $\\R^n$ and the observation space be $\\R^p$. A state $X$ has prior law $\\mathcal N(m,P)$, and its observation is\n\\begin{equation}\n Y=HX+V,\\qquad V\\sim\\mathcal N(0,R),\\qquad V\\text{ independent of }X.\n \\label{eq:foundation-model}\n\\end{equation}\nHere $H\\in\\R^{p\\times n}$ may be rectangular, and $P\\in\\R^{n\\times n}$ may be singular. The assumptions are $P\\succeq0$ and $R\\succ0$. A singular prior is an ordinary Gaussian probability law concentrated on an affine subspace. The argument does not replace that law by a nonsingular approximation or assign an artificial inverse to its covariance.","We use Euclidean vector norms and the squared Frobenius energy\n\\begin{equation}\n E(A)=\\Frob{A}^2=\\sum_{i,j}A_{ij}^2.\n \\label{eq:foundation-energy}\n\\end{equation}\nThe distinction between these norms matters. A matrix energy is a finite sum of scalar squares that can be evaluated exactly on rational inputs. An operator norm is used in the proof that this energy controls distribution error. The certificate never asks a producer to supply an unverified operator norm or an approximate eigenvalue calculation.","For an observed value $y$, put\n\\begin{equation}\n d=y-Hm,\\qquad S=HPH^\\top+R,\\qquad K=PH^\\top S^{-1}.\n \\label{eq:foundation-innovation}\n\\end{equation}\nThe matrix $S$ is the innovation covariance. It is positive definite because, for every nonzero $z\\in\\R^p$,\n\\[\n z^\\top Sz=(H^\\top z)^\\top P(H^\\top z)+z^\\top Rz>0.\n\\]\nConsequently the inverse in the definition of $K$ exists even when $P$ has a nontrivial nullspace or $H$ has deficient rank. Every inverse used by the posterior argument is an inverse on the observation space. This is the structural reason singular priors are supported.","lue $y$, put\n\\begin{equation}\n d=y-Hm,\\qquad S=HPH^\\top+R,\\qquad K=PH^\\top S^{-1}.\n \\label{eq:foundation-innovation}\n\\end{equation}\nThe matrix $S$ is the innovation covariance. It is positive definite because, for every nonzero $z\\in\\R^p$,\n\\[\n z^\\top Sz=(H^\\top z)^\\top P(H^\\top z)+z^\\top Rz>0.\n\\]\nConsequently the inverse in the definition of $K$ exists even when $P$ has a nontrivial nullspace or $H$ has deficient rank. Every inverse used by the posterior argument is an inverse on the observation space. This is the structural reason singular priors are supported.","The completed executable contract represents its model by rational matrices and vectors. Its input consists of $H$, $m$, $y$, a prior factor $L\\in\\Q^{n\\times r}$, a noise factor $N\\in\\Q^{p\\times s}$, and a rational number $\\sigma$. The covariance matrices are derived from these inputs:\n\\begin{equation}\n P=LL^\\top,\\qquad R=NN^\\top+\\sigma I_p,\\qquad \\sigma>0.\n \\label{eq:foundation-factor-model}\n\\end{equation}\nThe factor dimensions $r$ and $s$ are finite and need not equal the state or observation dimension. The factors may have dependent columns or zero entries. The prior factor is allowed to have rank below $n$. The positive floor is checked rather than inferred from a decimal approximation.","represents its model by rational matrices and vectors. Its input consists of $H$, $m$, $y$, a prior factor $L\\in\\Q^{n\\times r}$, a noise factor $N\\in\\Q^{p\\times s}$, and a rational number $\\sigma$. The covariance matrices are derived from these inputs:\n\\begin{equation}\n P=LL^\\top,\\qquad R=NN^\\top+\\sigma I_p,\\qquad \\sigma>0.\n \\label{eq:foundation-factor-model}\n\\end{equation}\nThe factor dimensions $r$ and $s$ are finite and need not equal the state or observation dimension. The factors may have dependent columns or zero entries. The prior factor is allowed to have rank below $n$. The positive floor is checked rather than inferred from a decimal approximation.","ts its model by rational matrices and vectors. Its input consists of $H$, $m$, $y$, a prior factor $L\\in\\Q^{n\\times r}$, a noise factor $N\\in\\Q^{p\\times s}$, and a rational number $\\sigma$. The covariance matrices are derived from these inputs:\n\\begin{equation}\n P=LL^\\top,\\qquad R=NN^\\top+\\sigma I_p,\\qquad \\sigma>0.\n \\label{eq:foundation-factor-model}\n\\end{equation}\nThe factor dimensions $r$ and $s$ are finite and need not equal the state or observation dimension. The factors may have dependent columns or zero entries. The prior factor is allowed to have rank below $n$. The positive floor is checked rather than inferred from a decimal approximation.","number $\\sigma$. The covariance matrices are derived from these inputs:\n\\begin{equation}\n P=LL^\\top,\\qquad R=NN^\\top+\\sigma I_p,\\qquad \\sigma>0.\n \\label{eq:foundation-factor-model}\n\\end{equation}\nThe factor dimensions $r$ and $s$ are finite and need not equal the state or observation dimension. The factors may have dependent columns or zero entries. The prior factor is allowed to have rank below $n$. The positive floor is checked rather than inferred from a decimal approximation.","This representation makes the covariance hypotheses concrete. For every $x$,\n\\[\n x^\\top LL^\\top x=\\|L^\\top x\\|^2\\ge0,\n \\qquad\n x^\\top Rx=\\|N^\\top x\\|^2+\\sigma\\|x\\|^2.\n\\]\nThus $P$ is positive semidefinite and $R\\succeq\\sigma I_p\\succ0$. These conclusions are proved from the factors; they are not additional statistical assumptions hidden inside a covariance record. The real semantics interprets each rational coordinate through the exact inclusion $\\Q\\hookrightarrow\\R$. There is no intermediate conversion through hardware floating point.","s concrete. For every $x$,\n\\[\n x^\\top LL^\\top x=\\|L^\\top x\\|^2\\ge0,\n \\qquad\n x^\\top Rx=\\|N^\\top x\\|^2+\\sigma\\|x\\|^2.\n\\]\nThus $P$ is positive semidefinite and $R\\succeq\\sigma I_p\\succ0$. These conclusions are proved from the factors; they are not additional statistical assumptions hidden inside a covariance record. The real semantics interprets each rational coordinate through the exact inclusion $\\Q\\hookrightarrow\\R$. There is no intermediate conversion through hardware floating point.","The same $\\sigma$ that certifies positive noise also bounds the inverse innovation operator. Adding the positive semidefinite term $HPH^\\top$ gives\n\\begin{equation}\n z^\\top Sz\\ge\\sigma\\|z\\|^2\\qquad\\text{for all }z\\in\\R^p.\n \\label{eq:foundation-coercivity}\n\\end{equation}\nLet $x=S^{-1}v$. Then $Sx=v$, and the Cauchy--Schwarz inequality yields\n\\[\n \\sigma\\|x\\|^2\\le x^\\top Sx=x^\\top v\\le\\|x\\|\\|v\\|.\n\\]\nIf $x=0$ the desired bound is immediate. Otherwise division by $\\sigma\\|x\\|$ gives $\\|S^{-1}v\\|\\le\\sigma^{-1}\\|v\\|$. Hence\n\\begin{equation}\n \\|S^{-1}\\|_{\\mathrm{op}}\\le\\sigma^{-1}.\n \\label{eq:foundation-inverse-bound}\n\\end{equation}\nThis elementary estimate supplies an explicit certificate constant without computin","The same $\\sigma$ that certifies positive noise also bounds the inverse innovation operator. Adding the positive semidefinite term $HPH^\\top$ gives\n\\begin{equation}\n z^\\top Sz\\ge\\sigma\\|z\\|^2\\qquad\\text{for all }z\\in\\R^p.\n \\label{eq:foundation-coercivity}\n\\end{equation}\nLet $x=S^{-1}v$. Then $Sx=v$, and the Cauchy--Schwarz inequality yields\n\\[\n \\sigma\\|x\\|^2\\le x^\\top Sx=x^\\top v\\le\\|x\\|\\|v\\|.\n\\]\nIf $x=0$ the desired bound is immediate. Otherwise division by $\\sigma\\|x\\|$ gives $\\|S^{-1}v\\|\\le\\sigma^{-1}\\|v\\|$. Hence\n\\begin{equation}\n \\|S^{-1}\\|_{\\mathrm{op}}\\le\\sigma^{-1}.\n \\label{eq:foundation-inverse-bound}\n\\end{equation}\nThis elementary estimate supplies an explicit certificate constant without computing the inverse. The inverse exists in the semantic proof, while the executable stopping test uses $\\sigma$, an innovation vec","The same $\\sigma$ that certifies positive noise also bounds the inverse innovation operator. Adding the positive semidefinite term $HPH^\\top$ gives\n\\begin{equation}\n z^\\top Sz\\ge\\sigma\\|z\\|^2\\qquad\\text{for all }z\\in\\R^p.\n \\label{eq:foundation-coercivity}\n\\end{equation}\nLet $x=S^{-1}v$. Then $Sx=v$, and the Cauchy--Schwarz inequality yields\n\\[\n \\sigma\\|x\\|^2\\le x^\\top Sx=x^\\top v\\le\\|x\\|\\|v\\|.\n\\]\nIf $x=0$ the desired bound is immediate. Otherwise division by $\\sigma\\|x\\|$ gives $\\|S^{-1}v\\|\\le\\sigma^{-1}\\|v\\|$. Hence\n\\begin{equation}\n \\|S^{-1}\\|_{\\mathrm{op}}\\le\\sigma^{-1}.\n \\label{eq:foundation-inverse-bound}\n\\end{equation}\nThis elementary estimate supplies an explicit certificate constant without computing the inverse. The inverse exists in the semantic proof, while the executable stopping test uses $\\sigma$, an innovation vector, and a residual. This separation allows an approximate gain to be certified independently of the method that pro","The same $\\sigma$ that certifies positive noise also bounds the inverse innovation operator. Adding the positive semidefinite term $HPH^\\top$ gives\n\\begin{equation}\n z^\\top Sz\\ge\\sigma\\|z\\|^2\\qquad\\text{for all }z\\in\\R^p.\n \\label{eq:foundation-coercivity}\n\\end{equation}\nLet $x=S^{-1}v$. Then $Sx=v$, and the Cauchy--Schwarz inequality yields\n\\[\n \\sigma\\|x\\|^2\\le x^\\top Sx=x^\\top v\\le\\|x\\|\\|v\\|.\n\\]\nIf $x=0$ the desired bound is immediate. Otherwise division by $\\sigma\\|x\\|$ gives $\\|S^{-1}v\\|\\le\\sigma^{-1}\\|v\\|$. Hence\n\\begin{equation}\n \\|S^{-1}\\|_{\\mathrm{op}}\\le\\sigma^{-1}.\n \\label{eq:foundation-inverse-bound}\n\\end{equation}\nThis elementary estimate supplies an explicit certificate constant without computing the inverse. The inverse exists in the semantic proof, while the executable stopping test uses $\\sigma$, an innovation vector, and a residual. This separation allows an approximate gain to be certified independently of the method that produced it.","same $\\sigma$ that certifies positive noise also bounds the inverse innovation operator. Adding the positive semidefinite term $HPH^\\top$ gives\n\\begin{equation}\n z^\\top Sz\\ge\\sigma\\|z\\|^2\\qquad\\text{for all }z\\in\\R^p.\n \\label{eq:foundation-coercivity}\n\\end{equation}\nLet $x=S^{-1}v$. Then $Sx=v$, and the Cauchy--Schwarz inequality yields\n\\[\n \\sigma\\|x\\|^2\\le x^\\top Sx=x^\\top v\\le\\|x\\|\\|v\\|.\n\\]\nIf $x=0$ the desired bound is immediate. Otherwise division by $\\sigma\\|x\\|$ gives $\\|S^{-1}v\\|\\le\\sigma^{-1}\\|v\\|$. Hence\n\\begin{equation}\n \\|S^{-1}\\|_{\\mathrm{op}}\\le\\sigma^{-1}.\n \\label{eq:foundation-inverse-bound}\n\\end{equation}\nThis elementary estimate supplies an explicit certificate constant without computing the inverse. The inverse exists in the semantic proof, while the executable stopping test uses $\\sigma$, an innovation vector, and a residual. This separation allows an approximate gain to be certified independently of the method that produced it.","inverse innovation operator. Adding the positive semidefinite term $HPH^\\top$ gives\n\\begin{equation}\n z^\\top Sz\\ge\\sigma\\|z\\|^2\\qquad\\text{for all }z\\in\\R^p.\n \\label{eq:foundation-coercivity}\n\\end{equation}\nLet $x=S^{-1}v$. Then $Sx=v$, and the Cauchy--Schwarz inequality yields\n\\[\n \\sigma\\|x\\|^2\\le x^\\top Sx=x^\\top v\\le\\|x\\|\\|v\\|.\n\\]\nIf $x=0$ the desired bound is immediate. Otherwise division by $\\sigma\\|x\\|$ gives $\\|S^{-1}v\\|\\le\\sigma^{-1}\\|v\\|$. Hence\n\\begin{equation}\n \\|S^{-1}\\|_{\\mathrm{op}}\\le\\sigma^{-1}.\n \\label{eq:foundation-inverse-bound}\n\\end{equation}\nThis elementary estimate supplies an explicit certificate constant without computing the inverse. The inverse exists in the semantic proof, while the executable stopping test uses $\\sigma$, an innovation vector, and a residual. This separation allows an approximate gain to be certified independently of the method that produced it.","perator. Adding the positive semidefinite term $HPH^\\top$ gives\n\\begin{equation}\n z^\\top Sz\\ge\\sigma\\|z\\|^2\\qquad\\text{for all }z\\in\\R^p.\n \\label{eq:foundation-coercivity}\n\\end{equation}\nLet $x=S^{-1}v$. Then $Sx=v$, and the Cauchy--Schwarz inequality yields\n\\[\n \\sigma\\|x\\|^2\\le x^\\top Sx=x^\\top v\\le\\|x\\|\\|v\\|.\n\\]\nIf $x=0$ the desired bound is immediate. Otherwise division by $\\sigma\\|x\\|$ gives $\\|S^{-1}v\\|\\le\\sigma^{-1}\\|v\\|$. Hence\n\\begin{equation}\n \\|S^{-1}\\|_{\\mathrm{op}}\\le\\sigma^{-1}.\n \\label{eq:foundation-inverse-bound}\n\\end{equation}\nThis elementary estimate supplies an explicit certificate constant without computing the inverse. The inverse exists in the semantic proof, while the executable stopping test uses $\\sigma$, an innovation vector, and a residual. This separation allows an approximate gain to be certified independently of the method that produced it.","p$ gives\n\\begin{equation}\n z^\\top Sz\\ge\\sigma\\|z\\|^2\\qquad\\text{for all }z\\in\\R^p.\n \\label{eq:foundation-coercivity}\n\\end{equation}\nLet $x=S^{-1}v$. Then $Sx=v$, and the Cauchy--Schwarz inequality yields\n\\[\n \\sigma\\|x\\|^2\\le x^\\top Sx=x^\\top v\\le\\|x\\|\\|v\\|.\n\\]\nIf $x=0$ the desired bound is immediate. Otherwise division by $\\sigma\\|x\\|$ gives $\\|S^{-1}v\\|\\le\\sigma^{-1}\\|v\\|$. Hence\n\\begin{equation}\n \\|S^{-1}\\|_{\\mathrm{op}}\\le\\sigma^{-1}.\n \\label{eq:foundation-inverse-bound}\n\\end{equation}\nThis elementary estimate supplies an explicit certificate constant without computing the inverse. The inverse exists in the semantic proof, while the executable stopping test uses $\\sigma$, an innovation vector, and a residual. This separation allows an approximate gain to be certified independently of the method that produced it.","esired bound is immediate. Otherwise division by $\\sigma\\|x\\|$ gives $\\|S^{-1}v\\|\\le\\sigma^{-1}\\|v\\|$. Hence\n\\begin{equation}\n \\|S^{-1}\\|_{\\mathrm{op}}\\le\\sigma^{-1}.\n \\label{eq:foundation-inverse-bound}\n\\end{equation}\nThis elementary estimate supplies an explicit certificate constant without computing the inverse. The inverse exists in the semantic proof, while the executable stopping test uses $\\sigma$, an innovation vector, and a residual. This separation allows an approximate gain to be certified independently of the method that produced it.","The exact posterior formula is\n\\begin{equation}\n \\nu_y=\\mathcal N(m+Kd,C_*),\\qquad C_*=P-KHP.\n \\label{eq:foundation-exact-posterior}\n\\end{equation}\nWriting this formula does not by itself establish conditional probability semantics. A matrix identity could describe a plausible mean and covariance without proving that its Gaussian is a disintegration of the prior and likelihood. The source constructs that disintegration at the level of probability measures. This section records the argument and the precise meaning of the word posterior.","The covariance $C_*$ admits the Joseph representation\n\\begin{equation}\n C_*=(I_n-KH)P(I_n-KH)^\\top+KRK^\\top.\n \\label{eq:foundation-exact-joseph}\n\\end{equation}\nEach term on the right is positive semidefinite. Therefore $C_*\\succeq0$, including at the singular boundary. This avoids any proof based on $P^{-1}$ or $C_*^{-1}$.","\\begin{lemma}[Exact Joseph identity]\n\\label{lem:foundation-exact-joseph}\nUnder the assumptions of Section~\\ref{sec:foundation-model}, the covariance in \\eqref{eq:foundation-exact-posterior} equals \\eqref{eq:foundation-exact-joseph}.\n\\end{lemma}\n\\begin{proof}\nThe innovation equation gives $KS=PH^\\top$. Since $P$ and $S$ are symmetric, transposition gives $SK^\\top=HP$. Thus\n\\[\n KSK^\\top=KHP=PH^\\top K^\\top.\n\\]\nExpanding the right side of \\eqref{eq:foundation-exact-joseph} and collecting its two quadratic terms gives\n\\begin{align*}\n &(I_n-KH)P(I_n-KH)^\\top+KRK^\\top","the covariance in \\eqref{eq:foundation-exact-posterior} equals \\eqref{eq:foundation-exact-joseph}.\n\\end{lemma}\n\\begin{proof}\nThe innovation equation gives $KS=PH^\\top$. Since $P$ and $S$ are symmetric, transposition gives $SK^\\top=HP$. Thus\n\\[\n KSK^\\top=KHP=PH^\\top K^\\top.\n\\]\nExpanding the right side of \\eqref{eq:foundation-exact-joseph} and collecting its two quadratic terms gives\n\\begin{align*}\n &(I_n-KH)P(I_n-KH)^\\top+KRK^\\top\\\\\n &\\quad=P-KHP-PH^\\top K^\\top+KSK^\\top\\\\\n &\\quad=P-KHP.\n\\end{align*}\nThe last line is $C_*$. Positivity follows because each unexpanded term is a covariance congruence of a positive semidefinite matrix.\n\\end{proof}","&\\quad=P-KHP.\n\\end{align*}\nThe last line is $C_*$. Positivity follows because each unexpanded term is a covariance congruence of a positive semidefinite matrix.\n\\end{proof}","The forward construction samples $X\\sim\\mathcal N(m,P)$ and then samples $Y$ with conditional law $\\mathcal N(HX,R)$. Its joint mean, in the order $(Y,X)$, is $(Hm,m)$. Its covariance is\n\\begin{equation}\n \\begin{pmatrix}\n S&HP","The forward construction samples $X\\sim\\mathcal N(m,P)$ and then samples $Y$ with conditional law $\\mathcal N(HX,R)$. Its joint mean, in the order $(Y,X)$, is $(Hm,m)$. Its covariance is\n\\begin{equation}\n \\begin{pmatrix}\n S&HP\\\\\n PH^\\top&P\n \\end{pmatrix}.\n \\label{eq:foundation-joint-cov}\n\\end{equation}\nThis joint law is Gaussian because it is an affine image of independent Gaussian inputs. The argument remains valid when the input covariance has zero eigenvalues.","For the reverse construction, sample $Y\\sim\\mathcal N(Hm,S)$ and an independent $E_*\\sim\\mathcal N(0,C_*)$. Define\n\\[\n Z=m+K(Y-Hm)+E_*.\n\\]\nConditional on $Y=y$, the law of $Z$ is precisely $\\nu_y$. The joint mean of $(Y,Z)$ is again $(Hm,m)$. Its cross covariance is $SK^\\top=HP$, and its state covariance is\n\\[\n KSK^\\top+C_*=KHP+(P-KHP)=P.\n\\]\nIt therefore has exactly the covariance in \\eqref{eq:foundation-joint-cov}. Both constructions yield Gaussian measures with the same mean and covariance, so they yield the same joint probability measure. This is a full law identity, rather than a statement that the two constructions merely agree in their first two moments without Gaussianity.","\\begin{theorem}[Gaussian disintegration]\n\\label{thm:foundation-disintegration}\nLet $\\ell(x,\\cdot)=\\mathcal N(Hx,R)$ be the observation kernel and let $\\eta=\\mathcal N(Hm,S)$ be the evidence law. Then the explicit kernel $y\\mapsto\\nu_y$ satisfies\n\\begin{equation}\n \\eta(dy)\\,\\nu_y(dx)\n   =\\mathcal N(m,P)(dx)\\,\\ell(x,dy)\n \\label{eq:foundation-disintegration}\n\\end{equation}\nafter exchanging the order of the two coordinates on the right. In particular, $\\eta$ is the actual observation marginal and $\\nu$ is a posterior version for this prior and likelihood.\n\\end{theorem}\n\\begin{proof}\nThe forward and reverse constructions just described have the same joint Gaussian mean and covariance. The reverse construction is the product of the evidence measure with its affine Gaussian conditional kernel. The forward cons","\\begin{theorem}[Gaussian disintegration]\n\\label{thm:foundation-disintegration}\nLet $\\ell(x,\\cdot)=\\mathcal N(Hx,R)$ be the observation kernel and let $\\eta=\\mathcal N(Hm,S)$ be the evidence law. Then the explicit kernel $y\\mapsto\\nu_y$ satisfies\n\\begin{equation}\n \\eta(dy)\\,\\nu_y(dx)\n   =\\mathcal N(m,P)(dx)\\,\\ell(x,dy)\n \\label{eq:foundation-disintegration}\n\\end{equation}\nafter exchanging the order of the two coordinates on the right. In particular, $\\eta$ is the actual observation marginal and $\\nu$ is a posterior version for this prior and likelihood.\n\\end{theorem}\n\\begin{proof}\nThe forward and reverse constructions just described have the same joint Gaussian mean and covariance. The reverse construction is the product of the evidence measure with its affine Gaussian conditional kernel. The forward construction is the product of the prior measure with the like","\\begin{theorem}[Gaussian disintegration]\n\\label{thm:foundation-disintegration}\nLet $\\ell(x,\\cdot)=\\mathcal N(Hx,R)$ be the observation kernel and let $\\eta=\\mathcal N(Hm,S)$ be the evidence law. Then the explicit kernel $y\\mapsto\\nu_y$ satisfies\n\\begin{equation}\n \\eta(dy)\\,\\nu_y(dx)\n   =\\mathcal N(m,P)(dx)\\,\\ell(x,dy)\n \\label{eq:foundation-disintegration}\n\\end{equation}\nafter exchanging the order of the two coordinates on the right. In particular, $\\eta$ is the actual observation marginal and $\\nu$ is a posterior version for this prior and likelihood.\n\\end{theorem}\n\\begin{proof}\nThe forward and reverse constructions just described have the same joint Gaussian mean and covariance. The reverse construction is the product of the evidence measure with its affine Gaussian conditional kernel. The forward construction is the product of the prior measure with the likelihood kernel. Their joint equality is \\eqref{eq:foundation-di","\\begin{theorem}[Gaussian disintegration]\n\\label{thm:foundation-disintegration}\nLet $\\ell(x,\\cdot)=\\mathcal N(Hx,R)$ be the observation kernel and let $\\eta=\\mathcal N(Hm,S)$ be the evidence law. Then the explicit kernel $y\\mapsto\\nu_y$ satisfies\n\\begin{equation}\n \\eta(dy)\\,\\nu_y(dx)\n   =\\mathcal N(m,P)(dx)\\,\\ell(x,dy)\n \\label{eq:foundation-disintegration}\n\\end{equation}\nafter exchanging the order of the two coordinates on the right. In particular, $\\eta$ is the actual observation marginal and $\\nu$ is a posterior version for this prior and likelihood.\n\\end{theorem}\n\\begin{proof}\nThe forward and reverse constructions just described have the same joint Gaussian mean and covariance. The reverse construction is the product of the evidence measure with its affine Gaussian conditional kernel. The forward construction is the product of the prior measure with the likelihood kernel. Their joint equality is \\eqref{eq:foundation-disintegration}. Taking the observation marginal yields the stated evidence law, and equality of the joint measures is the def","oundation-disintegration}\nLet $\\ell(x,\\cdot)=\\mathcal N(Hx,R)$ be the observation kernel and let $\\eta=\\mathcal N(Hm,S)$ be the evidence law. Then the explicit kernel $y\\mapsto\\nu_y$ satisfies\n\\begin{equation}\n \\eta(dy)\\,\\nu_y(dx)\n   =\\mathcal N(m,P)(dx)\\,\\ell(x,dy)\n \\label{eq:foundation-disintegration}\n\\end{equation}\nafter exchanging the order of the two coordinates on the right. In particular, $\\eta$ is the actual observation marginal and $\\nu$ is a posterior version for this prior and likelihood.\n\\end{theorem}\n\\begin{proof}\nThe forward and reverse constructions just described have the same joint Gaussian mean and covariance. The reverse construction is the product of the evidence measure with its affine Gaussian conditional kernel. The forward construction is the product of the prior measure with the likelihood kernel. Their joint equality is \\eqref{eq:foundation-disintegration}. Taking the observation marginal yields the stated evidence law, and equality of the joint measures is the defining disintegration property.\n\\end{proof}","mathcal N(Hx,R)$ be the observation kernel and let $\\eta=\\mathcal N(Hm,S)$ be the evidence law. Then the explicit kernel $y\\mapsto\\nu_y$ satisfies\n\\begin{equation}\n \\eta(dy)\\,\\nu_y(dx)\n   =\\mathcal N(m,P)(dx)\\,\\ell(x,dy)\n \\label{eq:foundation-disintegration}\n\\end{equation}\nafter exchanging the order of the two coordinates on the right. In particular, $\\eta$ is the actual observation marginal and $\\nu$ is a posterior version for this prior and likelihood.\n\\end{theorem}\n\\begin{proof}\nThe forward and reverse constructions just described have the same joint Gaussian mean and covariance. The reverse construction is the product of the evidence measure with its affine Gaussian conditional kernel. The forward construction is the product of the prior measure with the likelihood kernel. Their joint equality is \\eqref{eq:foundation-disintegration}. Taking the observation marginal yields the stated evidence law, and equality of the joint measures is the defining disintegration property.\n\\end{proof}","For a supplied rational observation $y$, the executable contract uses the explicit version $\\nu_y$. That version is defined at every such $y$ and is already proved to disintegrate the model. The solver's distance theorem is pointwise against this version. It does not prove pointwise equality to every possible modification of a canonical posterior kernel on evidence-null sets. The distinction is standard for conditional laws and does not weaken the law-level certificate that the program actually returns.","In particular, observing a single real vector in a continuous model does not require the event $Y=y$ to have positive probability. Conditioning is encoded by a kernel whose integration reconstructs the joint law. The proof cannot be replaced by dividing the probability of a zero-probability event. This is one reason the measure-level construction is essential when the inference contract claims a posterior distribution guarantee.","Let $B\\in\\R^{n\\times p}$ be an arbitrary gain candidate. It need not solve the innovation equation and need not have been produced by the packet algorithm. Define\n\\begin{align}\n m_B&=m+Bd,\\label{eq:foundation-approx-mean}","Let $B\\in\\R^{n\\times p}$ be an arbitrary gain candidate. It need not solve the innovation equation and need not have been produced by the packet algorithm. Define\n\\begin{align}\n m_B&=m+Bd,\\label{eq:foundation-approx-mean}\\\\\n C_B&=(I_n-BH)P(I_n-BH)^\\top+BRB^\\top,\\label{eq:foundation-approx-cov}\\\\\n \\mu_B&=\\mathcal N(m_B,C_B).\\label{eq:foundation-approx-law}\n\\end{align}\nEvery $C_B$ is positive semidefinite, since it is a sum of covariance congruences. Thus every candidate gain determines a genuine Gaussian law. This is useful for certification: an imperfect candidate is a valid distribution with a measurable error, rather than a covariance formula that can become negative merely because $B$ is inexact.","\\mu_B&=\\mathcal N(m_B,C_B).\\label{eq:foundation-approx-law}\n\\end{align}\nEvery $C_B$ is positive semidefinite, since it is a sum of covariance congruences. Thus every candidate gain determines a genuine Gaussian law. This is useful for certification: an imperfect candidate is a valid distribution with a measurable error, rather than a covariance formula that can become negative merely because $B$ is inexact.","One should not replace \\eqref{eq:foundation-approx-cov} by $P-BHP$. The latter is the exact posterior covariance only when the required innovation identities hold. For an arbitrary candidate it need not even be symmetric. The Joseph form makes the mathematical object independent of those identities, and the final verifier certifies this same object.","Define the gain-equation residual\n\\begin{equation}\n F(B)=PH^\\top-BS.\n \\label{eq:foundation-residual}\n\\end{equation}\nIt can be computed from the model and $B$ without using $K$. The exact target satisfies $KS=PH^\\top$, so\n\\begin{equation}\n F(B)=(K-B)S,\\qquad K-B=F(B)S^{-1}.\n \\label{eq:foundation-gap-residual}\n\\end{equation}\nThese are exact identities, not perturbative approximations. They explain how a computable defect in a linear equation controls an uncomputed target gain.","\\begin{proposition}[Excess covariance]\n\\label{prop:foundation-excess-cov}\nFor every gain $B$,\n\\begin{equation}\n C_B-C_*=(K-B)S(K-B)^\\top\n          =F(B)S^{-1}F(B)^\\top\\succeq0.\n \\label{eq:foundation-excess-cov}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nExpansion of the Joseph form gives\n\\[\n C_B=P-BHP-PH^\\top B^\\top+BSB^\\top.\n\\]\nSubtract $C_*=P-KHP$. Expanding $(K-B)S(K-B)^\\top$ and using $KS=PH^\\top$, $SK^\\top=HP$, and $KSK^\\top=KHP$ gives the same expression. The second equality follows from \\eqref{eq:foundation-gap-residual} and the symmetry of $S$. Since $S\\succ0$, its covariance congruence by $K-B$ is positive semidefinite.\n\\end{proof}","osition}[Excess covariance]\n\\label{prop:foundation-excess-cov}\nFor every gain $B$,\n\\begin{equation}\n C_B-C_*=(K-B)S(K-B)^\\top\n          =F(B)S^{-1}F(B)^\\top\\succeq0.\n \\label{eq:foundation-excess-cov}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nExpansion of the Joseph form gives\n\\[\n C_B=P-BHP-PH^\\top B^\\top+BSB^\\top.\n\\]\nSubtract $C_*=P-KHP$. Expanding $(K-B)S(K-B)^\\top$ and using $KS=PH^\\top$, $SK^\\top=HP$, and $KSK^\\top=KHP$ gives the same expression. The second equality follows from \\eqref{eq:foundation-gap-residual} and the symmetry of $S$. Since $S\\succ0$, its covariance congruence by $K-B$ is positive semidefinite.\n\\end{proof}","variance]\n\\label{prop:foundation-excess-cov}\nFor every gain $B$,\n\\begin{equation}\n C_B-C_*=(K-B)S(K-B)^\\top\n          =F(B)S^{-1}F(B)^\\top\\succeq0.\n \\label{eq:foundation-excess-cov}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nExpansion of the Joseph form gives\n\\[\n C_B=P-BHP-PH^\\top B^\\top+BSB^\\top.\n\\]\nSubtract $C_*=P-KHP$. Expanding $(K-B)S(K-B)^\\top$ and using $KS=PH^\\top$, $SK^\\top=HP$, and $KSK^\\top=KHP$ gives the same expression. The second equality follows from \\eqref{eq:foundation-gap-residual} and the symmetry of $S$. Since $S\\succ0$, its covariance congruence by $K-B$ is positive semidefinite.\n\\end{proof}","(B)^\\top\\succeq0.\n \\label{eq:foundation-excess-cov}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nExpansion of the Joseph form gives\n\\[\n C_B=P-BHP-PH^\\top B^\\top+BSB^\\top.\n\\]\nSubtract $C_*=P-KHP$. Expanding $(K-B)S(K-B)^\\top$ and using $KS=PH^\\top$, $SK^\\top=HP$, and $KSK^\\top=KHP$ gives the same expression. The second equality follows from \\eqref{eq:foundation-gap-residual} and the symmetry of $S$. Since $S\\succ0$, its covariance congruence by $K-B$ is positive semidefinite.\n\\end{proof}","cceq0.\n \\label{eq:foundation-excess-cov}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nExpansion of the Joseph form gives\n\\[\n C_B=P-BHP-PH^\\top B^\\top+BSB^\\top.\n\\]\nSubtract $C_*=P-KHP$. Expanding $(K-B)S(K-B)^\\top$ and using $KS=PH^\\top$, $SK^\\top=HP$, and $KSK^\\top=KHP$ gives the same expression. The second equality follows from \\eqref{eq:foundation-gap-residual} and the symmetry of $S$. Since $S\\succ0$, its covariance congruence by $K-B$ is positive semidefinite.\n\\end{proof}","\\end{equation}\n\\end{proposition}\n\\begin{proof}\nExpansion of the Joseph form gives\n\\[\n C_B=P-BHP-PH^\\top B^\\top+BSB^\\top.\n\\]\nSubtract $C_*=P-KHP$. Expanding $(K-B)S(K-B)^\\top$ and using $KS=PH^\\top$, $SK^\\top=HP$, and $KSK^\\top=KHP$ gives the same expression. The second equality follows from \\eqref{eq:foundation-gap-residual} and the symmetry of $S$. Since $S\\succ0$, its covariance congruence by $K-B$ is positive semidefinite.\n\\end{proof}","Take independent centered Gaussian variables $U\\sim\\mathcal N(0,P)$ and $V\\sim\\mathcal N(0,R)$. For each gain $B$, define the mathematical affine random variable\n\\begin{equation}\n T_B=m+Bd+(I_n-BH)U-BV.\n \\label{eq:foundation-shared-sample}\n\\end{equation}\nIts mean is $m_B$ and its covariance is $C_B$, so its law is $\\mu_B$. At $B=K$, the exact Joseph identity identifies its law with $\\nu_y$. The pair $(T_K,T_B)$, using the same inputs $(U,V)$, is therefore an actual transport plan between the exact posterior version and the approximate law.","independent centered Gaussian variables $U\\sim\\mathcal N(0,P)$ and $V\\sim\\mathcal N(0,R)$. For each gain $B$, define the mathematical affine random variable\n\\begin{equation}\n T_B=m+Bd+(I_n-BH)U-BV.\n \\label{eq:foundation-shared-sample}\n\\end{equation}\nIts mean is $m_B$ and its covariance is $C_B$, so its law is $\\mu_B$. At $B=K$, the exact Joseph identity identifies its law with $\\nu_y$. The pair $(T_K,T_B)$, using the same inputs $(U,V)$, is therefore an actual transport plan between the exact posterior version and the approximate law.","ables $U\\sim\\mathcal N(0,P)$ and $V\\sim\\mathcal N(0,R)$. For each gain $B$, define the mathematical affine random variable\n\\begin{equation}\n T_B=m+Bd+(I_n-BH)U-BV.\n \\label{eq:foundation-shared-sample}\n\\end{equation}\nIts mean is $m_B$ and its covariance is $C_B$, so its law is $\\mu_B$. At $B=K$, the exact Joseph identity identifies its law with $\\nu_y$. The pair $(T_K,T_B)$, using the same inputs $(U,V)$, is therefore an actual transport plan between the exact posterior version and the approximate law.","Let $\\Delta=K-B$. Subtraction in \\eqref{eq:foundation-shared-sample} gives\n\\[\n T_K-T_B=\\Delta d-\\Delta(HU+V).\n\\]\nThe random vector $HU+V$ is centered and has covariance $S$. The difference therefore has mean $\\Delta d$ and covariance $\\Delta S\\Delta^\\top$. Its expected squared Euclidean norm is\n\\begin{equation}\n \\E\\|T_K-T_B\\|^2\n   =\\|\\Delta d\\|^2+\\tr(\\Delta S\\Delta^\\top).\n \\label{eq:foundation-coupling-cost}\n\\end{equation}\nThe source proves both marginals and this cost. It does not assume that a producer has supplied a coupling with the desired properties. The load-bearing declarations are \\path{gainSample_law}, \\path{gainSample_exact_posterior_law}, and \\path{inexactGainCoupling_isTransportPlan} in \\path{","Let $\\Delta=K-B$. Subtraction in \\eqref{eq:foundation-shared-sample} gives\n\\[\n T_K-T_B=\\Delta d-\\Delta(HU+V).\n\\]\nThe random vector $HU+V$ is centered and has covariance $S$. The difference therefore has mean $\\Delta d$ and covariance $\\Delta S\\Delta^\\top$. Its expected squared Euclidean norm is\n\\begin{equation}\n \\E\\|T_K-T_B\\|^2\n   =\\|\\Delta d\\|^2+\\tr(\\Delta S\\Delta^\\top).\n \\label{eq:foundation-coupling-cost}\n\\end{equation}\nThe source proves both marginals and this cost. It does not assume that a producer has supplied a coupling with the desired properties. The load-bearing declarations are \\path{gainSample_law}, \\path{gainSample_exact_posterior_law}, and \\path{inexactGainCoupling_isTransportPlan} in \\path{FDGainSampler.lean}, followed by \\path{inexactGainCoupling_cost_gap} in \\path{FDInexactGainCoup","Let $\\Delta=K-B$. Subtraction in \\eqref{eq:foundation-shared-sample} gives\n\\[\n T_K-T_B=\\Delta d-\\Delta(HU+V).\n\\]\nThe random vector $HU+V$ is centered and has covariance $S$. The difference therefore has mean $\\Delta d$ and covariance $\\Delta S\\Delta^\\top$. Its expected squared Euclidean norm is\n\\begin{equation}\n \\E\\|T_K-T_B\\|^2\n   =\\|\\Delta d\\|^2+\\tr(\\Delta S\\Delta^\\top).\n \\label{eq:foundation-coupling-cost}\n\\end{equation}\nThe source proves both marginals and this cost. It does not assume that a producer has supplied a coupling with the desired properties. The load-bearing declarations are \\path{gainSample_law}, \\path{gainSample_exact_posterior_law}, and \\path{inexactGainCoupling_isTransportPlan} in \\path{FDGainSampler.lean}, followed by \\path{inexactGainCoupling_cost_gap} in \\path{FDInexactGainCoupling.lean}.","The extreme case $P=0$ makes this especially clear. The prior is the point mass at $m$, and the model's observation carries no uncertainty about the state. The exact gain is $K=0$, and the posterior version is still the point mass at $m$. For an arbitrary candidate $B$, however, the Joseph approximation has mean $m+Bd$ and covariance $BRB^\\top$. These are valid Gaussian parameters, but they need not describe a point mass. Its residual is $F(B)=-BR$, so the residual certificate measures the error introduced by that candidate rather than declaring every valid Gaussian approximation exact.","y clear. The prior is the point mass at $m$, and the model's observation carries no uncertainty about the state. The exact gain is $K=0$, and the posterior version is still the point mass at $m$. For an arbitrary candidate $B$, however, the Joseph approximation has mean $m+Bd$ and covariance $BRB^\\top$. These are valid Gaussian parameters, but they need not describe a point mass. Its residual is $F(B)=-BR$, so the residual certificate measures the error introduced by that candidate rather than declaring every valid Gaussian approximation exact.","For a candidate $B=0$, both its mean and covariance are exactly those of the point mass, and its residual energy is zero. This boundary calculation shows why the proof separates covariance validity from posterior accuracy. The Joseph formula supplies validity for every gain. The residual supplies accuracy for the selected gain. Neither property is inferred from the other.","For probability laws with finite second moment, write\n\\[\n \\Wtwo(\\nu,\\mu)^2\n =\\inf_{\\pi\\in\\Pi(\\nu,\\mu)}\\int\\|x-z\\|^2\\,d\\pi(x,z),\n\\]\nwhere $\\Pi(\\nu,\\mu)$ is the set of transport plans with the stated marginals. The formal source defines the squared quantity as a nonnegative extended-real infimum and obtains the distance by a square-root operation. The explicit finite Gaussian coupling proves that the quantities considered here are finite. The notation $\\Wtwo$ in the paper is the corresponding finite distance.","[\n \\Wtwo(\\nu,\\mu)^2\n =\\inf_{\\pi\\in\\Pi(\\nu,\\mu)}\\int\\|x-z\\|^2\\,d\\pi(x,z),\n\\]\nwhere $\\Pi(\\nu,\\mu)$ is the set of transport plans with the stated marginals. The formal source defines the squared quantity as a nonnegative extended-real infimum and obtains the distance by a square-root operation. The explicit finite Gaussian coupling proves that the quantities considered here are finite. The notation $\\Wtwo$ in the paper is the corresponding finite distance.","Using \\eqref{eq:foundation-gap-residual} in \\eqref{eq:foundation-coupling-cost} yields the more informative, but not directly inverse-free, bound\n\\begin{equation}\n \\Wtwo(\\nu_y,\\mu_B)^2\n \\le\\|F(B)S^{-1}d\\|^2\n       +\\tr\\bigl(F(B)S^{-1}F(B)^\\top\\bigr).\n \\label{eq:foundation-residual-exact-cost}\n\\end{equation}\nThe source declaration \\path{posterior_wasserstein2Sq_le_gain_residual} proves this inequality from the exhibited transport plan. The displayed right side is its exact cost. It is not asserted to equal the optimal squared Wasserstein distance.","\\begin{lemma}[Frobenius energy controls the residual cost]\n\\label{lem:foundation-energy-majorant}\nIf $F\\in\\R^{n\\times p}$ and $S\\succeq\\sigma I_p\\succ0$, then\n\\begin{equation}\n \\|FS^{-1}d\\|^2+\\tr(FS^{-1}F^\\top)\n \\le E(F)\\left(\\sigma^{-1}+\\sigma^{-2}\\|d\\|^2\\right).\n \\label{eq:foundation-cost-majorant}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor a vector $v$, rowwise Cauchy--Schwarz gives\n\\[\n \\|Fv\\|^2=\\sum_i\\left(\\sum_jF_{ij}v_j\\right)^2\n \\le\\sum_i\\left(\\sum_jF_{ij}^2\\right)\\|v\\|^2\n =E(F)\\|v\\|^2.\n\\]\nApply this with $v=S^{-1}d$ and use \\eqref{eq:foundation-inverse-bound} to obtain\n\\[\n \\|FS^{-1}d\\|^2\\le E(F)\\sigma^{-2}\\|d\\|^2.\n\\]\nIf $f_i$ is the transpose of the $i$th row of $F$, then\n\\[\n \\tr(FS^{-1}F^\\top)=\\sum_i f_i^\\top S^{-1}f_i\n \\le\\sum_i\\|f_i\\|\\|S^{-1}f_i\\|\n \\le\\sigma^{-1}\\sum_i\\|f_i\\|^2.\n\\]\nThe last sum i","\\begin{lemma}[Frobenius energy controls the residual cost]\n\\label{lem:foundation-energy-majorant}\nIf $F\\in\\R^{n\\times p}$ and $S\\succeq\\sigma I_p\\succ0$, then\n\\begin{equation}\n \\|FS^{-1}d\\|^2+\\tr(FS^{-1}F^\\top)\n \\le E(F)\\left(\\sigma^{-1}+\\sigma^{-2}\\|d\\|^2\\right).\n \\label{eq:foundation-cost-majorant}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor a vector $v$, rowwise Cauchy--Schwarz gives\n\\[\n \\|Fv\\|^2=\\sum_i\\left(\\sum_jF_{ij}v_j\\right)^2\n \\le\\sum_i\\left(\\sum_jF_{ij}^2\\right)\\|v\\|^2\n =E(F)\\|v\\|^2.\n\\]\nApply this with $v=S^{-1}d$ and use \\eqref{eq:foundation-inverse-bound} to obtain\n\\[\n \\|FS^{-1}d\\|^2\\le E(F)\\sigma^{-2}\\|d\\|^2.\n\\]\nIf $f_i$ is the transpose of the $i$th row of $F$, then\n\\[\n \\tr(FS^{-1}F^\\top)=\\sum_i f_i^\\top S^{-1}f_i\n \\le\\sum_i\\|f_i\\|\\|S^{-1}f_i\\|\n \\le\\sigma^{-1}\\sum_i\\|f_i\\|^2.\n\\]\nThe last sum is $E(F)$. Adding the two estima","\\begin{lemma}[Frobenius energy controls the residual cost]\n\\label{lem:foundation-energy-majorant}\nIf $F\\in\\R^{n\\times p}$ and $S\\succeq\\sigma I_p\\succ0$, then\n\\begin{equation}\n \\|FS^{-1}d\\|^2+\\tr(FS^{-1}F^\\top)\n \\le E(F)\\left(\\sigma^{-1}+\\sigma^{-2}\\|d\\|^2\\right).\n \\label{eq:foundation-cost-majorant}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor a vector $v$, rowwise Cauchy--Schwarz gives\n\\[\n \\|Fv\\|^2=\\sum_i\\left(\\sum_jF_{ij}v_j\\right)^2\n \\le\\sum_i\\left(\\sum_jF_{ij}^2\\right)\\|v\\|^2\n =E(F)\\|v\\|^2.\n\\]\nApply this with $v=S^{-1}d$ and use \\eqref{eq:foundation-inverse-bound} to obtain\n\\[\n \\|FS^{-1}d\\|^2\\le E(F)\\sigma^{-2}\\|d\\|^2.\n\\]\nIf $f_i$ is the transpose of the $i$th row of $F$, then\n\\[\n \\tr(FS^{-1}F^\\top)=\\sum_i f_i^\\top S^{-1}f_i\n \\le\\sum_i\\|f_i\\|\\|S^{-1}f_i\\|\n \\le\\sigma^{-1}\\sum_i\\|f_i\\|^2.\n\\]\nThe last sum is $E(F)$. Adding the two estimates proves the result.\n\\end{proof}","and $S\\succeq\\sigma I_p\\succ0$, then\n\\begin{equation}\n \\|FS^{-1}d\\|^2+\\tr(FS^{-1}F^\\top)\n \\le E(F)\\left(\\sigma^{-1}+\\sigma^{-2}\\|d\\|^2\\right).\n \\label{eq:foundation-cost-majorant}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor a vector $v$, rowwise Cauchy--Schwarz gives\n\\[\n \\|Fv\\|^2=\\sum_i\\left(\\sum_jF_{ij}v_j\\right)^2\n \\le\\sum_i\\left(\\sum_jF_{ij}^2\\right)\\|v\\|^2\n =E(F)\\|v\\|^2.\n\\]\nApply this with $v=S^{-1}d$ and use \\eqref{eq:foundation-inverse-bound} to obtain\n\\[\n \\|FS^{-1}d\\|^2\\le E(F)\\sigma^{-2}\\|d\\|^2.\n\\]\nIf $f_i$ is the transpose of the $i$th row of $F$, then\n\\[\n \\tr(FS^{-1}F^\\top)=\\sum_i f_i^\\top S^{-1}f_i\n \\le\\sum_i\\|f_i\\|\\|S^{-1}f_i\\|\n \\le\\sigma^{-1}\\sum_i\\|f_i\\|^2.\n\\]\nThe last sum is $E(F)$. Adding the two estimates proves the result.\n\\end{proof}","|^2+\\tr(FS^{-1}F^\\top)\n \\le E(F)\\left(\\sigma^{-1}+\\sigma^{-2}\\|d\\|^2\\right).\n \\label{eq:foundation-cost-majorant}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor a vector $v$, rowwise Cauchy--Schwarz gives\n\\[\n \\|Fv\\|^2=\\sum_i\\left(\\sum_jF_{ij}v_j\\right)^2\n \\le\\sum_i\\left(\\sum_jF_{ij}^2\\right)\\|v\\|^2\n =E(F)\\|v\\|^2.\n\\]\nApply this with $v=S^{-1}d$ and use \\eqref{eq:foundation-inverse-bound} to obtain\n\\[\n \\|FS^{-1}d\\|^2\\le E(F)\\sigma^{-2}\\|d\\|^2.\n\\]\nIf $f_i$ is the transpose of the $i$th row of $F$, then\n\\[\n \\tr(FS^{-1}F^\\top)=\\sum_i f_i^\\top S^{-1}f_i\n \\le\\sum_i\\|f_i\\|\\|S^{-1}f_i\\|\n \\le\\sigma^{-1}\\sum_i\\|f_i\\|^2.\n\\]\nThe last sum is $E(F)$. Adding the two estimates proves the result.\n\\end{proof}","sigma^{-2}\\|d\\|^2\\right).\n \\label{eq:foundation-cost-majorant}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor a vector $v$, rowwise Cauchy--Schwarz gives\n\\[\n \\|Fv\\|^2=\\sum_i\\left(\\sum_jF_{ij}v_j\\right)^2\n \\le\\sum_i\\left(\\sum_jF_{ij}^2\\right)\\|v\\|^2\n =E(F)\\|v\\|^2.\n\\]\nApply this with $v=S^{-1}d$ and use \\eqref{eq:foundation-inverse-bound} to obtain\n\\[\n \\|FS^{-1}d\\|^2\\le E(F)\\sigma^{-2}\\|d\\|^2.\n\\]\nIf $f_i$ is the transpose of the $i$th row of $F$, then\n\\[\n \\tr(FS^{-1}F^\\top)=\\sum_i f_i^\\top S^{-1}f_i\n \\le\\sum_i\\|f_i\\|\\|S^{-1}f_i\\|\n \\le\\sigma^{-1}\\sum_i\\|f_i\\|^2.\n\\]\nThe last sum is $E(F)$. Adding the two estimates proves the result.\n\\end{proof}","bel{eq:foundation-cost-majorant}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor a vector $v$, rowwise Cauchy--Schwarz gives\n\\[\n \\|Fv\\|^2=\\sum_i\\left(\\sum_jF_{ij}v_j\\right)^2\n \\le\\sum_i\\left(\\sum_jF_{ij}^2\\right)\\|v\\|^2\n =E(F)\\|v\\|^2.\n\\]\nApply this with $v=S^{-1}d$ and use \\eqref{eq:foundation-inverse-bound} to obtain\n\\[\n \\|FS^{-1}d\\|^2\\le E(F)\\sigma^{-2}\\|d\\|^2.\n\\]\nIf $f_i$ is the transpose of the $i$th row of $F$, then\n\\[\n \\tr(FS^{-1}F^\\top)=\\sum_i f_i^\\top S^{-1}f_i\n \\le\\sum_i\\|f_i\\|\\|S^{-1}f_i\\|\n \\le\\sigma^{-1}\\sum_i\\|f_i\\|^2.\n\\]\nThe last sum is $E(F)$. Adding the two estimates proves the result.\n\\end{proof}","tion-cost-majorant}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor a vector $v$, rowwise Cauchy--Schwarz gives\n\\[\n \\|Fv\\|^2=\\sum_i\\left(\\sum_jF_{ij}v_j\\right)^2\n \\le\\sum_i\\left(\\sum_jF_{ij}^2\\right)\\|v\\|^2\n =E(F)\\|v\\|^2.\n\\]\nApply this with $v=S^{-1}d$ and use \\eqref{eq:foundation-inverse-bound} to obtain\n\\[\n \\|FS^{-1}d\\|^2\\le E(F)\\sigma^{-2}\\|d\\|^2.\n\\]\nIf $f_i$ is the transpose of the $i$th row of $F$, then\n\\[\n \\tr(FS^{-1}F^\\top)=\\sum_i f_i^\\top S^{-1}f_i\n \\le\\sum_i\\|f_i\\|\\|S^{-1}f_i\\|\n \\le\\sigma^{-1}\\sum_i\\|f_i\\|^2.\n\\]\nThe last sum is $E(F)$. Adding the two estimates proves the result.\n\\end{proof}","ajorant}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor a vector $v$, rowwise Cauchy--Schwarz gives\n\\[\n \\|Fv\\|^2=\\sum_i\\left(\\sum_jF_{ij}v_j\\right)^2\n \\le\\sum_i\\left(\\sum_jF_{ij}^2\\right)\\|v\\|^2\n =E(F)\\|v\\|^2.\n\\]\nApply this with $v=S^{-1}d$ and use \\eqref{eq:foundation-inverse-bound} to obtain\n\\[\n \\|FS^{-1}d\\|^2\\le E(F)\\sigma^{-2}\\|d\\|^2.\n\\]\nIf $f_i$ is the transpose of the $i$th row of $F$, then\n\\[\n \\tr(FS^{-1}F^\\top)=\\sum_i f_i^\\top S^{-1}f_i\n \\le\\sum_i\\|f_i\\|\\|S^{-1}f_i\\|\n \\le\\sigma^{-1}\\sum_i\\|f_i\\|^2.\n\\]\nThe last sum is $E(F)$. Adding the two estimates proves the result.\n\\end{proof}","_{ij}v_j\\right)^2\n \\le\\sum_i\\left(\\sum_jF_{ij}^2\\right)\\|v\\|^2\n =E(F)\\|v\\|^2.\n\\]\nApply this with $v=S^{-1}d$ and use \\eqref{eq:foundation-inverse-bound} to obtain\n\\[\n \\|FS^{-1}d\\|^2\\le E(F)\\sigma^{-2}\\|d\\|^2.\n\\]\nIf $f_i$ is the transpose of the $i$th row of $F$, then\n\\[\n \\tr(FS^{-1}F^\\top)=\\sum_i f_i^\\top S^{-1}f_i\n \\le\\sum_i\\|f_i\\|\\|S^{-1}f_i\\|\n \\le\\sigma^{-1}\\sum_i\\|f_i\\|^2.\n\\]\nThe last sum is $E(F)$. Adding the two estimates proves the result.\n\\end{proof}","The formal estimate \\path{quadratic_residual_cost_le_energy} in \\path{FDResidualEnergyCertificate.lean} is slightly more general: it accepts an auxiliary operator $Q$ with a proved bound $\\|Q\\|_{\\mathrm{op}}\\le c$. Its application to $Q=S^{-1}$ uses the established noise-floor bound and gives the formula above.","\\begin{theorem}[Residual-energy posterior certificate]\n\\label{thm:foundation-residual-certificate}\nFor the model in \\eqref{eq:foundation-model}, with $R\\succeq\\sigma I_p$ and $\\sigma>0$, every gain $B$ satisfies\n\\begin{equation}\n \\Wtwo(\\nu_y,\\mu_B)^2\\le a(B),\\qquad\n a(B)=E(F(B))\\frac{\\sigma+\\|d\\|^2}{\\sigma^2}.\n \\label{eq:foundation-score}\n\\end{equation}\nIn particular, for $\\varepsilon\\ge0$, the stopping condition\n\\begin{equation}\n E(F(B))(\\sigma+\\|d\\|^2)\\le\\varepsilon^2\\sigma^2\n \\label{eq:foundation-stop}\n\\end{equation}\nimplies $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\nThe common-noise transport plan gives \\eqref{eq:foundation-residual-exact-cost}. Lemma~\\ref{lem:foundation-energy-majorant} bounds its cost by $a(B)$, establishing the first statement. Since $\\sigma^2>0$, \\eqref{eq:foundation-stop} is equivalent to $a(B)\\le\\varepsilon^2$","\\begin{theorem}[Residual-energy posterior certificate]\n\\label{thm:foundation-residual-certificate}\nFor the model in \\eqref{eq:foundation-model}, with $R\\succeq\\sigma I_p$ and $\\sigma>0$, every gain $B$ satisfies\n\\begin{equation}\n \\Wtwo(\\nu_y,\\mu_B)^2\\le a(B),\\qquad\n a(B)=E(F(B))\\frac{\\sigma+\\|d\\|^2}{\\sigma^2}.\n \\label{eq:foundation-score}\n\\end{equation}\nIn particular, for $\\varepsilon\\ge0$, the stopping condition\n\\begin{equation}\n E(F(B))(\\sigma+\\|d\\|^2)\\le\\varepsilon^2\\sigma^2\n \\label{eq:foundation-stop}\n\\end{equation}\nimplies $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\nThe common-noise transport plan gives \\eqref{eq:foundation-residual-exact-cost}. Lemma~\\ref{lem:foundation-energy-majorant} bounds its cost by $a(B)$, establishing the first statement. Since $\\sigma^2>0$, \\eqref{eq:foundation-stop} is equivalent to $a(B)\\le\\varepsilon^2$. Taking the no","\\begin{theorem}[Residual-energy posterior certificate]\n\\label{thm:foundation-residual-certificate}\nFor the model in \\eqref{eq:foundation-model}, with $R\\succeq\\sigma I_p$ and $\\sigma>0$, every gain $B$ satisfies\n\\begin{equation}\n \\Wtwo(\\nu_y,\\mu_B)^2\\le a(B),\\qquad\n a(B)=E(F(B))\\frac{\\sigma+\\|d\\|^2}{\\sigma^2}.\n \\label{eq:foundation-score}\n\\end{equation}\nIn particular, for $\\varepsilon\\ge0$, the stopping condition\n\\begin{equation}\n E(F(B))(\\sigma+\\|d\\|^2)\\le\\varepsilon^2\\sigma^2\n \\label{eq:foundation-stop}\n\\end{equation}\nimplies $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\nThe common-noise transport plan gives \\eqref{eq:foundation-residual-exact-cost}. Lemma~\\ref{lem:foundation-energy-majorant} bounds its cost by $a(B)$, establishing the first statement. Since $\\sigma^2>0$, \\eqref{eq:foundation-stop} is equivalent to $a(B)\\le\\varepsilon^2$. Taking the nonnegative square","\\begin{theorem}[Residual-energy posterior certificate]\n\\label{thm:foundation-residual-certificate}\nFor the model in \\eqref{eq:foundation-model}, with $R\\succeq\\sigma I_p$ and $\\sigma>0$, every gain $B$ satisfies\n\\begin{equation}\n \\Wtwo(\\nu_y,\\mu_B)^2\\le a(B),\\qquad\n a(B)=E(F(B))\\frac{\\sigma+\\|d\\|^2}{\\sigma^2}.\n \\label{eq:foundation-score}\n\\end{equation}\nIn particular, for $\\varepsilon\\ge0$, the stopping condition\n\\begin{equation}\n E(F(B))(\\sigma+\\|d\\|^2)\\le\\varepsilon^2\\sigma^2\n \\label{eq:foundation-stop}\n\\end{equation}\nimplies $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\nThe common-noise transport plan gives \\eqref{eq:foundation-residual-exact-cost}. Lemma~\\ref{lem:foundation-energy-majorant} bounds its cost by $a(B)$, establishing the first statement. Since $\\sigma^2>0$, \\eqref{eq:foundation-stop} is equivalent to $a(B)\\le\\varepsilon^2$. Taking the nonnegative square root gives the asserted distance bound.\n\\end{proof}","osterior certificate]\n\\label{thm:foundation-residual-certificate}\nFor the model in \\eqref{eq:foundation-model}, with $R\\succeq\\sigma I_p$ and $\\sigma>0$, every gain $B$ satisfies\n\\begin{equation}\n \\Wtwo(\\nu_y,\\mu_B)^2\\le a(B),\\qquad\n a(B)=E(F(B))\\frac{\\sigma+\\|d\\|^2}{\\sigma^2}.\n \\label{eq:foundation-score}\n\\end{equation}\nIn particular, for $\\varepsilon\\ge0$, the stopping condition\n\\begin{equation}\n E(F(B))(\\sigma+\\|d\\|^2)\\le\\varepsilon^2\\sigma^2\n \\label{eq:foundation-stop}\n\\end{equation}\nimplies $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\nThe common-noise transport plan gives \\eqref{eq:foundation-residual-exact-cost}. Lemma~\\ref{lem:foundation-energy-majorant} bounds its cost by $a(B)$, establishing the first statement. Since $\\sigma^2>0$, \\eqref{eq:foundation-stop} is equivalent to $a(B)\\le\\varepsilon^2$. Taking the nonnegative square root gives the asserted distance bound.\n\\end{proof}","n-model}, with $R\\succeq\\sigma I_p$ and $\\sigma>0$, every gain $B$ satisfies\n\\begin{equation}\n \\Wtwo(\\nu_y,\\mu_B)^2\\le a(B),\\qquad\n a(B)=E(F(B))\\frac{\\sigma+\\|d\\|^2}{\\sigma^2}.\n \\label{eq:foundation-score}\n\\end{equation}\nIn particular, for $\\varepsilon\\ge0$, the stopping condition\n\\begin{equation}\n E(F(B))(\\sigma+\\|d\\|^2)\\le\\varepsilon^2\\sigma^2\n \\label{eq:foundation-stop}\n\\end{equation}\nimplies $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\nThe common-noise transport plan gives \\eqref{eq:foundation-residual-exact-cost}. Lemma~\\ref{lem:foundation-energy-majorant} bounds its cost by $a(B)$, establishing the first statement. Since $\\sigma^2>0$, \\eqref{eq:foundation-stop} is equivalent to $a(B)\\le\\varepsilon^2$. Taking the nonnegative square root gives the asserted distance bound.\n\\end{proof}","end{equation}\nIn particular, for $\\varepsilon\\ge0$, the stopping condition\n\\begin{equation}\n E(F(B))(\\sigma+\\|d\\|^2)\\le\\varepsilon^2\\sigma^2\n \\label{eq:foundation-stop}\n\\end{equation}\nimplies $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\nThe common-noise transport plan gives \\eqref{eq:foundation-residual-exact-cost}. Lemma~\\ref{lem:foundation-energy-majorant} bounds its cost by $a(B)$, establishing the first statement. Since $\\sigma^2>0$, \\eqref{eq:foundation-stop} is equivalent to $a(B)\\le\\varepsilon^2$. Taking the nonnegative square root gives the asserted distance bound.\n\\end{proof}","0$, the stopping condition\n\\begin{equation}\n E(F(B))(\\sigma+\\|d\\|^2)\\le\\varepsilon^2\\sigma^2\n \\label{eq:foundation-stop}\n\\end{equation}\nimplies $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\nThe common-noise transport plan gives \\eqref{eq:foundation-residual-exact-cost}. Lemma~\\ref{lem:foundation-energy-majorant} bounds its cost by $a(B)$, establishing the first statement. Since $\\sigma^2>0$, \\eqref{eq:foundation-stop} is equivalent to $a(B)\\le\\varepsilon^2$. Taking the nonnegative square root gives the asserted distance bound.\n\\end{proof}","a+\\|d\\|^2)\\le\\varepsilon^2\\sigma^2\n \\label{eq:foundation-stop}\n\\end{equation}\nimplies $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{theorem}\n\\begin{proof}\nThe common-noise transport plan gives \\eqref{eq:foundation-residual-exact-cost}. Lemma~\\ref{lem:foundation-energy-majorant} bounds its cost by $a(B)$, establishing the first statement. Since $\\sigma^2>0$, \\eqref{eq:foundation-stop} is equivalent to $a(B)\\le\\varepsilon^2$. Taking the nonnegative square root gives the asserted distance bound.\n\\end{proof}","ns are \\path{posterior_wasserstein2Sq_le_residual_energy} and \\path{posterior_wasserstein2_le_of_residual_tolerance}. Their hypotheses include the covariance assumptions, positivity of the floor, and the floor's quadratic interpretation. In the factor-coded program, those hypotheses are discharged by the represented factors and the checked positive scalar. The user does not supply the conclusion $a(B)\\ge\\Wtwo^2$ as an input promise.","The score depends on the proposed gain through $F(B)$, on the observation through $d$, and on conditioning through $\\sigma$. The same residual energy can lead to different certified errors for different observations. A large innovation can amplify a gain discrepancy in the posterior mean. A small noise floor makes a residual less informative about the gain error. These are visible features of the bound, rather than implementation effects hidden in a tolerance parameter.","The stopping test is sufficient and can be conservative. A score above $\\varepsilon^2$ does not prove that the true distance exceeds $\\varepsilon$. The coupling may have a nonoptimal cost, the inverse norm bound may be loose, and later interval evaluation may further enlarge an upper bound. Conversely, a passing verified score does prove the tolerance. The solver's convergence theorem concerns this explicit sufficient test, so its correctness does not depend on predicting how tight it will be on a particular application.","If $F(B)=0$, invertibility of $S$ gives $B=K$, and the approximation is exactly the posterior version. The source proves this as \\path{gainPosteriorLaw_eq_of_residual_zero}. No inference from approximately zero floating point entries is used in that theorem. For a nonzero tolerance the theorem uses the whole squared energy, allowing a candidate to pass before the gain equation is solved exactly.","The completed solver uses a posterior-law tolerance $\\varepsilon>0$ and a parameter-representation radius tolerance $\\delta>0$. The first bounds $\\Wtwo(\\nu_y,\\mu_B)$. The second bounds the radii of outward-rounded intervals containing entries of $m_B$ and $C_B$. The latter arrays are descriptions of the approximation already certified by the former theorem.","An interval enclosing an entry of $C_B$ is not, by itself, an interval enclosing the corresponding entry of $C_*$. Nor does a radius at most $\\delta$ certify a Gaussian formed by treating all interval centers as exact parameters. The represented gain and its Joseph law remain the semantic object. Keeping that object fixed is what permits the solver, the final certificate, and the output enclosure proof to compose without changing the meaning of the returned distribution midway through the computation.","A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let $\\{Q_a:a\\in\\mathcal A\\}$ act on the state space and $\\{D_b:b\\in\\mathcal B\\}$ act on the observation space. For each family, the structural conditions are\n\\begin{align}\n Q_a^\\top&=Q_a,& Q_a^2&=Q_a,& Q_aQ_{a'}&=0\\quad(a\\ne a'),\\label{eq:foundation-output-packets}","A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let $\\{Q_a:a\\in\\mathcal A\\}$ act on the state space and $\\{D_b:b\\in\\mathcal B\\}$ act on the observation space. For each family, the structural conditions are\n\\begin{align}\n Q_a^\\top&=Q_a,& Q_a^2&=Q_a,& Q_aQ_{a'}&=0\\quad(a\\ne a'),\\label{eq:foundation-output-packets}\\\\\n D_b^\\top&=D_b,& D_b^2&=D_b,& D_bD_{b'}&=0\\quad(b\\ne b').\\label{eq:foundation-input-packets}\n\\end{align}\nCompleteness is a separate condition:\n\\begin{equation}\n \\sum_{a\\in\\mathcal A}Q_a=I_n,\\qquad\n \\sum_{b\\in\\mathcal B}D_b=I_p.\n \\label{eq:foundation-packet-complete}\n\\end{equation}\nThe self-adjoint condition makes these Euclidean orthogonal projectors. Idempotence alone would not give the required energy geometry. Completeness prevents a certificate from forgetting a residual direction outside the selected packet spans.","D_b^\\top&=D_b,& D_b^2&=D_b,& D_bD_{b'}&=0\\quad(b\\ne b').\\label{eq:foundation-input-packets}\n\\end{align}\nCompleteness is a separate condition:\n\\begin{equation}\n \\sum_{a\\in\\mathcal A}Q_a=I_n,\\qquad\n \\sum_{b\\in\\mathcal B}D_b=I_p.\n \\label{eq:foundation-packet-complete}\n\\end{equation}\nThe self-adjoint condition makes these Euclidean orthogonal projectors. Idempotence alone would not give the required energy geometry. Completeness prevents a certificate from forgetting a residual direction outside the selected packet spans.","The two-sided residual packet is\n\\begin{equation}\n F_{ab}(B)=Q_aF(B)D_b.\n \\label{eq:foundation-packet-residual}\n\\end{equation}\nEach packet remains an $n\\times p$ matrix, although it is supported on the output and input subspaces selected by its projectors. The notation does not assume a coordinate partition or require projectors to be diagonal. Rational rotated projectors are allowed when their exact entries satisfy the declared identities.","Use the Frobenius inner product $\\langle A,C\\rangle_{\\mathrm F}=\\tr(A^\\top C)$. For $a\\ne a'$, self-adjointness and disjointness give\n\\[\n \\langle Q_aA,Q_{a'}A\\rangle_{\\mathrm F}\n =\\tr(A^\\top Q_aQ_{a'}A)=0.\n\\]\nConsequently the energy of a finite sum of output packets is the sum of their energies. The input analogue follows by transposing the matrices. Applying the two decompositions successively gives complete two-sided accounting.","\\begin{theorem}[Complete packet decomposition]\n\\label{thm:foundation-packet-energy}\nIf both families satisfy \\eqref{eq:foundation-output-packets}--\\eqref{eq:foundation-packet-complete}, then every $A\\in\\R^{n\\times p}$ satisfies\n\\begin{equation}\n E(A)=\\sum_{a\\in\\mathcal A}\\sum_{b\\in\\mathcal B}E(Q_aAD_b).\n \\label{eq:foundation-packet-energy}\n\\end{equation}\nMore generally, for subsets $U\\subseteq\\mathcal A$ and $V\\subseteq\\mathcal B$,\n\\begin{equation}\n E\\!\\left(\\left(\\sum_{a\\in U}Q_a\\right)A\n             \\left(\\sum_{b\\in V}D_b\\right)\\right)\n =\\sum_{a\\in U}\\sum_{b\\in V}E(Q_aAD_b).\n \\label{eq:foundation-parent-energy}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of any subfamily of the $Q_a$ is again self-adjoint and idempotent: expansion of its square removes the off-diagonal products, and the remaining diagonal products equal the original projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((","\\begin{theorem}[Complete packet decomposition]\n\\label{thm:foundation-packet-energy}\nIf both families satisfy \\eqref{eq:foundation-output-packets}--\\eqref{eq:foundation-packet-complete}, then every $A\\in\\R^{n\\times p}$ satisfies\n\\begin{equation}\n E(A)=\\sum_{a\\in\\mathcal A}\\sum_{b\\in\\mathcal B}E(Q_aAD_b).\n \\label{eq:foundation-packet-energy}\n\\end{equation}\nMore generally, for subsets $U\\subseteq\\mathcal A$ and $V\\subseteq\\mathcal B$,\n\\begin{equation}\n E\\!\\left(\\left(\\sum_{a\\in U}Q_a\\right)A\n             \\left(\\sum_{b\\in V}D_b\\right)\\right)\n =\\sum_{a\\in U}\\sum_{b\\in V}E(Q_aAD_b).\n \\label{eq:foundation-parent-energy}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of any subfamily of the $Q_a$ is again self-adjoint and idempotent: expansion of its square removes the off-diagonal products, and the remaining diagonal products equal the original projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Trans","\\begin{theorem}[Complete packet decomposition]\n\\label{thm:foundation-packet-energy}\nIf both families satisfy \\eqref{eq:foundation-output-packets}--\\eqref{eq:foundation-packet-complete}, then every $A\\in\\R^{n\\times p}$ satisfies\n\\begin{equation}\n E(A)=\\sum_{a\\in\\mathcal A}\\sum_{b\\in\\mathcal B}E(Q_aAD_b).\n \\label{eq:foundation-packet-energy}\n\\end{equation}\nMore generally, for subsets $U\\subseteq\\mathcal A$ and $V\\subseteq\\mathcal B$,\n\\begin{equation}\n E\\!\\left(\\left(\\sum_{a\\in U}Q_a\\right)A\n             \\left(\\sum_{b\\in V}D_b\\right)\\right)\n =\\sum_{a\\in U}\\sum_{b\\in V}E(Q_aAD_b).\n \\label{eq:foundation-parent-energy}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of any subfamily of the $Q_a$ is again self-adjoint and idempotent: expansion of its square removes the off-diagonal products, and the remaining diagonal products equal the original projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication b…","m}[Complete packet decomposition]\n\\label{thm:foundation-packet-energy}\nIf both families satisfy \\eqref{eq:foundation-output-packets}--\\eqref{eq:foundation-packet-complete}, then every $A\\in\\R^{n\\times p}$ satisfies\n\\begin{equation}\n E(A)=\\sum_{a\\in\\mathcal A}\\sum_{b\\in\\mathcal B}E(Q_aAD_b).\n \\label{eq:foundation-packet-energy}\n\\end{equation}\nMore generally, for subsets $U\\subseteq\\mathcal A$ and $V\\subseteq\\mathcal B$,\n\\begin{equation}\n E\\!\\left(\\left(\\sum_{a\\in U}Q_a\\right)A\n             \\left(\\sum_{b\\in V}D_b\\right)\\right)\n =\\sum_{a\\in U}\\sum_{b\\in V}E(Q_aAD_b).\n \\label{eq:foundation-parent-energy}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of any subfamily of the $Q_a$ is again self-adjoint and idempotent: expansion of its square removes the off-diagonal products, and the remaining diagonal products equal the original projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication by the sum ove…","l{thm:foundation-packet-energy}\nIf both families satisfy \\eqref{eq:foundation-output-packets}--\\eqref{eq:foundation-packet-complete}, then every $A\\in\\R^{n\\times p}$ satisfies\n\\begin{equation}\n E(A)=\\sum_{a\\in\\mathcal A}\\sum_{b\\in\\mathcal B}E(Q_aAD_b).\n \\label{eq:foundation-packet-energy}\n\\end{equation}\nMore generally, for subsets $U\\subseteq\\mathcal A$ and $V\\subseteq\\mathcal B$,\n\\begin{equation}\n E\\!\\left(\\left(\\sum_{a\\in U}Q_a\\right)A\n             \\left(\\sum_{b\\in V}D_b\\right)\\right)\n =\\sum_{a\\in U}\\sum_{b\\in V}E(Q_aAD_b).\n \\label{eq:foundation-parent-energy}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of any subfamily of the $Q_a$ is again self-adjoint and idempotent: expansion of its square removes the off-diagonal products, and the remaining diagonal products equal the original projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication by the sum over $V$ into left multiplication, proving…","aAD_b).\n \\label{eq:foundation-packet-energy}\n\\end{equation}\nMore generally, for subsets $U\\subseteq\\mathcal A$ and $V\\subseteq\\mathcal B$,\n\\begin{equation}\n E\\!\\left(\\left(\\sum_{a\\in U}Q_a\\right)A\n             \\left(\\sum_{b\\in V}D_b\\right)\\right)\n =\\sum_{a\\in U}\\sum_{b\\in V}E(Q_aAD_b).\n \\label{eq:foundation-parent-energy}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of any subfamily of the $Q_a$ is again self-adjoint and idempotent: expansion of its square removes the off-diagonal products, and the remaining diagonal products equal the original projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication by the sum over $V$ into left multiplication, proving the input decomposition. Combining these proves \\eqref{eq:foundation-parent-energy}. Setting $U=\\mathcal A$ and $V=\\mathcal B$, completeness reduces the matrix inside its energy to $A$, proving \\eqref{eq:foundation-packet-energy}.\n\\end{proof}","\\left(\\sum_{b\\in V}D_b\\right)\\right)\n =\\sum_{a\\in U}\\sum_{b\\in V}E(Q_aAD_b).\n \\label{eq:foundation-parent-energy}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of any subfamily of the $Q_a$ is again self-adjoint and idempotent: expansion of its square removes the off-diagonal products, and the remaining diagonal products equal the original projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication by the sum over $V$ into left multiplication, proving the input decomposition. Combining these proves \\eqref{eq:foundation-parent-energy}. Setting $U=\\mathcal A$ and $V=\\mathcal B$, completeness reduces the matrix inside its energy to $A$, proving \\eqref{eq:foundation-packet-energy}.\n\\end{proof}","\\label{eq:foundation-parent-energy}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of any subfamily of the $Q_a$ is again self-adjoint and idempotent: expansion of its square removes the off-diagonal products, and the remaining diagonal products equal the original projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication by the sum over $V$ into left multiplication, proving the input decomposition. Combining these proves \\eqref{eq:foundation-parent-energy}. Setting $U=\\mathcal A$ and $V=\\mathcal B$, completeness reduces the matrix inside its energy to $A$, proving \\eqref{eq:foundation-packet-energy}.\n\\end{proof}","tion-parent-energy}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of any subfamily of the $Q_a$ is again self-adjoint and idempotent: expansion of its square removes the off-diagonal products, and the remaining diagonal products equal the original projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication by the sum over $V$ into left multiplication, proving the input decomposition. Combining these proves \\eqref{eq:foundation-parent-energy}. Setting $U=\\mathcal A$ and $V=\\mathcal B$, completeness reduces the matrix inside its energy to $A$, proving \\eqref{eq:foundation-packet-energy}.\n\\end{proof}","adjoint and idempotent: expansion of its square removes the off-diagonal products, and the remaining diagonal products equal the original projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication by the sum over $V$ into left multiplication, proving the input decomposition. Combining these proves \\eqref{eq:foundation-parent-energy}. Setting $U=\\mathcal A$ and $V=\\mathcal B$, completeness reduces the matrix inside its energy to $A$, proving \\eqref{eq:foundation-packet-energy}.\n\\end{proof}","iginal projectors. For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication by the sum over $V$ into left multiplication, proving the input decomposition. Combining these proves \\eqref{eq:foundation-parent-energy}. Setting $U=\\mathcal A$ and $V=\\mathcal B$, completeness reduces the matrix inside its energy to $A$, proving \\eqref{eq:foundation-packet-energy}.\n\\end{proof}","For a self-adjoint idempotent $Q$,\n\\[\n E(QA)=\\tr((QA)^\\top QA)=\\tr(A^\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication by the sum over $V$ into left multiplication, proving the input decomposition. Combining these proves \\eqref{eq:foundation-parent-energy}. Setting $U=\\mathcal A$ and $V=\\mathcal B$, completeness reduces the matrix inside its energy to $A$, proving \\eqref{eq:foundation-packet-energy}.\n\\end{proof}","\\top QA).\n\\]\nThe trace is linear, so replacing $Q$ by a sum over $U$ proves the output decomposition. Transposition preserves energy and turns right multiplication by the sum over $V$ into left multiplication, proving the input decomposition. Combining these proves \\eqref{eq:foundation-parent-energy}. Setting $U=\\mathcal A$ and $V=\\mathcal B$, completeness reduces the matrix inside its energy to $A$, proving \\eqref{eq:foundation-packet-energy}.\n\\end{proof}","Suppose a producer supplies values $u_{ab}$ and $v$ with proved inequalities\n\\begin{equation}\n E(F_{ab}(B))\\le u_{ab},\\qquad \\|d\\|^2\\le v.\n \\label{eq:foundation-packet-majorants}\n\\end{equation}\nCompleteness and nonnegative energy imply\n\\[\n E(F(B))\\le U,\\qquad U=\\sum_{a,b}u_{ab}.\n\\]\nEach $u_{ab}$ is nonnegative because it bounds a nonnegative quantity. Likewise $v\\ge0$. The score\n\\begin{equation}\n \\widehat a(B)=U\\frac{\\sigma+v}{\\sigma^2}\n \\label{eq:foundation-majorant-score}\n\\end{equation}\ntherefore bounds $a(B)$ and hence the squared posterior distance. If\n\\begin{equation}\n U(\\sigma+v)\\le\\varepsilon^2\\sigma^2,\\qquad\\varepsilon\\ge0,\n \\label{eq:foundation-majorant-stop}\n\\end{equation}\nthe same law tolerance follows.","_{ab},\\qquad \\|d\\|^2\\le v.\n \\label{eq:foundation-packet-majorants}\n\\end{equation}\nCompleteness and nonnegative energy imply\n\\[\n E(F(B))\\le U,\\qquad U=\\sum_{a,b}u_{ab}.\n\\]\nEach $u_{ab}$ is nonnegative because it bounds a nonnegative quantity. Likewise $v\\ge0$. The score\n\\begin{equation}\n \\widehat a(B)=U\\frac{\\sigma+v}{\\sigma^2}\n \\label{eq:foundation-majorant-score}\n\\end{equation}\ntherefore bounds $a(B)$ and hence the squared posterior distance. If\n\\begin{equation}\n U(\\sigma+v)\\le\\varepsilon^2\\sigma^2,\\qquad\\varepsilon\\ge0,\n \\label{eq:foundation-majorant-stop}\n\\end{equation}\nthe same law tolerance follows.","\\end{equation}\nCompleteness and nonnegative energy imply\n\\[\n E(F(B))\\le U,\\qquad U=\\sum_{a,b}u_{ab}.\n\\]\nEach $u_{ab}$ is nonnegative because it bounds a nonnegative quantity. Likewise $v\\ge0$. The score\n\\begin{equation}\n \\widehat a(B)=U\\frac{\\sigma+v}{\\sigma^2}\n \\label{eq:foundation-majorant-score}\n\\end{equation}\ntherefore bounds $a(B)$ and hence the squared posterior distance. If\n\\begin{equation}\n U(\\sigma+v)\\le\\varepsilon^2\\sigma^2,\\qquad\\varepsilon\\ge0,\n \\label{eq:foundation-majorant-stop}\n\\end{equation}\nthe same law tolerance follows.","\\begin{corollary}[Packet-majorant tolerance]\n\\label{cor:foundation-packet-tolerance}\nUnder the model and complete packet hypotheses, \\eqref{eq:foundation-packet-majorants} and \\eqref{eq:foundation-majorant-stop} imply $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{corollary}\n\\begin{proof}\nSum the packet inequalities and apply Theorem~\\ref{thm:foundation-packet-energy} to obtain $E(F(B))\\le U$. Since $\\sigma+\\|d\\|^2\\ge0$, and $U\\ge0$,\n\\[\n E(F(B))(\\sigma+\\|d\\|^2)\n \\le U(\\sigma+\\|d\\|^2)\n \\le U(\\sigma+v)\n \\le\\varepsilon^2\\sigma^2.\n\\]\nTheorem~\\ref{thm:foundation-residual-certificate} now applies.\n\\end{proof}","-majorant tolerance]\n\\label{cor:foundation-packet-tolerance}\nUnder the model and complete packet hypotheses, \\eqref{eq:foundation-packet-majorants} and \\eqref{eq:foundation-majorant-stop} imply $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{corollary}\n\\begin{proof}\nSum the packet inequalities and apply Theorem~\\ref{thm:foundation-packet-energy} to obtain $E(F(B))\\le U$. Since $\\sigma+\\|d\\|^2\\ge0$, and $U\\ge0$,\n\\[\n E(F(B))(\\sigma+\\|d\\|^2)\n \\le U(\\sigma+\\|d\\|^2)\n \\le U(\\sigma+v)\n \\le\\varepsilon^2\\sigma^2.\n\\]\nTheorem~\\ref{thm:foundation-residual-certificate} now applies.\n\\end{proof}","tolerance]\n\\label{cor:foundation-packet-tolerance}\nUnder the model and complete packet hypotheses, \\eqref{eq:foundation-packet-majorants} and \\eqref{eq:foundation-majorant-stop} imply $\\Wtwo(\\nu_y,\\mu_B)\\le\\varepsilon$.\n\\end{corollary}\n\\begin{proof}\nSum the packet inequalities and apply Theorem~\\ref{thm:foundation-packet-energy} to obtain $E(F(B))\\le U$. Since $\\sigma+\\|d\\|^2\\ge0$, and $U\\ge0$,\n\\[\n E(F(B))(\\sigma+\\|d\\|^2)\n \\le U(\\sigma+\\|d\\|^2)\n \\le U(\\sigma+v)\n \\le\\varepsilon^2\\sigma^2.\n\\]\nTheorem~\\ref{thm:foundation-residual-certificate} now applies.\n\\end{proof}","Majorants allow different arithmetic representations to feed the same semantic theorem. An exact rational evaluator can choose $u_{ab}=E(F_{ab}(B))$. An outward-rounded evaluator can instead supply a certified upper endpoint. The law proof needs the inequality, not an assumption that every local value was calculated with unlimited precision. The arithmetic chapters explain how the final dyadic evaluator proves those inequalities for its own operations.","For example, take the coordinate projectors onto the two axes of $\\R^2$. A residual matrix with a nonzero second row has zero energy in the first output packet. Certifying that first packet alone says nothing about the omitted row. Completeness and a bound for every packet are what change local information into a global posterior guarantee. The exact checker tests those structural and local obligations; it does not accept a producer's assertion that a chosen subset happened to be representative.","The canonical selective loop uses the output-packet energies $E(Q_aF(B))$ to choose which gain component to update. Its final residual certificate may use both output and input packets. These roles are compatible but different: output packets guide the iterative correction, while complete two-sided packets organize the final verification. Neither role makes the final distribution error a collection of independent per-packet posterior distances.","The rational checker turns the previous inequalities into an inspectable decision boundary. A candidate contains a rational gain $B$, rational packet bounds $u_{ab}$, and a rational innovation bound $v$. In the source these are the fields \\path{gain}, \\path{packetUpper}, and \\path{innovationUpper} of \\path{RationalCandidate}. Its validity proposition requires five groups of conditions: a positive noise floor, valid complete output packets, valid complete input packets, every local residual-energy inequality, and the innovation-energy inequality.","The packet check is exact. It checks symmetry, idempotence, pairwise disjointness, and the sum-to-identity condition by finite rational equalities. These identities are not inferred from near-equality of floating point arrays. Exact inclusion into $\\R$ preserves matrix products, transposes, finite sums, and order. A rational family that passes therefore constructs a real self-adjoint complete packet family with the geometry required in Section~\\ref{sec:foundation-packets}.","The first Boolean function is\n\\[\n \\texttt{checkCandidate}(M,Q,D,c)\n   =\\texttt{decide}(\\texttt{ValidCandidate}(M,Q,D,c)).\n\\]\nIt recomputes the finite obligations for the explicit model, packets, and candidate. A claimed successful producer run is not one of its inputs. The equivalence \\path{checkCandidate_eq_true} states that its true result is exactly candidate validity.","The second Boolean function checks $\\varepsilon\\ge0$ and\n\\[\n \\left(\\sum_{a,b}u_{ab}\\right)(\\sigma+v)\n    \\le\\varepsilon^2\\sigma^2.\n\\]\nThis function is \\path{checkTolerance}. The displayed cross-multiplied form avoids evaluating square roots and does not require division in the tolerance decision. Positivity of $\\sigma$, obtained through candidate validity, justifies interpreting this test as a score bound. A tolerance check alone is not a substitute for the structural and local validity checks.","\\begin{theorem}[Sound exact candidate checking]\n\\label{thm:foundation-checked-tolerance}\nLet $M$ be a factor-coded rational model, $Q,D$ rational packet families, and $c$ a rational candidate. If both \\path{checkCandidate M Q D c} and \\path{checkTolerance M c epsilon} return true, then the exact real posterior version represented by $M$ and the exact real Joseph Gaussian represented by the gain of $c$ satisfy\n\\begin{equation}\n \\Wtwo(M.\\mathrm{posterior},M.\\mathrm{approximation}(c.\\mathrm{gain}))\n \\le\\varepsilon.\n \\label{eq:foundation-rational-soundness}\n\\end{equation}\nHere $\\varepsilon$ is the real interpretation of the rational tolerance.\n\\end{theorem}\n\\begin{proof}\nThe first check gives positivity of the floor and the four finite packet identities for each family. The factors prove","\\begin{theorem}[Sound exact candidate checking]\n\\label{thm:foundation-checked-tolerance}\nLet $M$ be a factor-coded rational model, $Q,D$ rational packet families, and $c$ a rational candidate. If both \\path{checkCandidate M Q D c} and \\path{checkTolerance M c epsilon} return true, then the exact real posterior version represented by $M$ and the exact real Joseph Gaussian represented by the gain of $c$ satisfy\n\\begin{equation}\n \\Wtwo(M.\\mathrm{posterior},M.\\mathrm{approximation}(c.\\mathrm{gain}))\n \\le\\varepsilon.\n \\label{eq:foundation-rational-soundness}\n\\end{equation}\nHere $\\varepsilon$ is the real interpretation of the rational tolerance.\n\\end{theorem}\n\\begin{proof}\nThe first check gives positivity of the floor and the four finite packet identities for each family. The factors prove the covariance and noise-floor hypothese","\\begin{theorem}[Sound exact candidate checking]\n\\label{thm:foundation-checked-tolerance}\nLet $M$ be a factor-coded rational model, $Q,D$ rational packet families, and $c$ a rational candidate. If both \\path{checkCandidate M Q D c} and \\path{checkTolerance M c epsilon} return true, then the exact real posterior version represented by $M$ and the exact real Joseph Gaussian represented by the gain of $c$ satisfy\n\\begin{equation}\n \\Wtwo(M.\\mathrm{posterior},M.\\mathrm{approximation}(c.\\mathrm{gain}))\n \\le\\varepsilon.\n \\label{eq:foundation-rational-soundness}\n\\end{equation}\nHere $\\varepsilon$ is the real interpretation of the rational tolerance.\n\\end{theorem}\n\\begin{proof}\nThe first check gives positivity of the floor and the four finite packet identities for each family. The factors prove the covariance and noise-floor hypotheses. The rational-to-real interpreta","\\begin{theorem}[Sound exact candidate checking]\n\\label{thm:foundation-checked-tolerance}\nLet $M$ be a factor-coded rational model, $Q,D$ rational packet families, and $c$ a rational candidate. If both \\path{checkCandidate M Q D c} and \\path{checkTolerance M c epsilon} return true, then the exact real posterior version represented by $M$ and the exact real Joseph Gaussian represented by the gain of $c$ satisfy\n\\begin{equation}\n \\Wtwo(M.\\mathrm{posterior},M.\\mathrm{approximation}(c.\\mathrm{gain}))\n \\le\\varepsilon.\n \\label{eq:foundation-rational-soundness}\n\\end{equation}\nHere $\\varepsilon$ is the real interpretation of the rational tolerance.\n\\end{theorem}\n\\begin{proof}\nThe first check gives positivity of the floor and the four finite packet identities for each family. The factors prove the covariance and noise-floor hypotheses. The rational-to-real interpretation lemmas turn every checked local inequality into a bound on the corresponding real packet energy, and turn the innovation inequality into a bound on $\\|d\\|^2$. Thus","egin{theorem}[Sound exact candidate checking]\n\\label{thm:foundation-checked-tolerance}\nLet $M$ be a factor-coded rational model, $Q,D$ rational packet families, and $c$ a rational candidate. If both \\path{checkCandidate M Q D c} and \\path{checkTolerance M c epsilon} return true, then the exact real posterior version represented by $M$ and the exact real Joseph Gaussian represented by the gain of $c$ satisfy\n\\begin{equation}\n \\Wtwo(M.\\mathrm{posterior},M.\\mathrm{approximation}(c.\\mathrm{gain}))\n \\le\\varepsilon.\n \\label{eq:foundation-rational-soundness}\n\\end{equation}\nHere $\\varepsilon$ is the real interpretation of the rational tolerance.\n\\end{theorem}\n\\begin{proof}\nThe first check gives positivity of the floor and the four finite packet identities for each family. The factors prove the covariance and noise-floor hypotheses. The rational-to-real interpretation lemmas turn every checked local inequality into a bound on the corresponding real packet energy, and turn the innovation inequality into a bound on $\\|d\\|^2$. Thus the real majorant score in \\eqref{eq:foundation-majorant-score} …","andidate checking]\n\\label{thm:foundation-checked-tolerance}\nLet $M$ be a factor-coded rational model, $Q,D$ rational packet families, and $c$ a rational candidate. If both \\path{checkCandidate M Q D c} and \\path{checkTolerance M c epsilon} return true, then the exact real posterior version represented by $M$ and the exact real Joseph Gaussian represented by the gain of $c$ satisfy\n\\begin{equation}\n \\Wtwo(M.\\mathrm{posterior},M.\\mathrm{approximation}(c.\\mathrm{gain}))\n \\le\\varepsilon.\n \\label{eq:foundation-rational-soundness}\n\\end{equation}\nHere $\\varepsilon$ is the real interpretation of the rational tolerance.\n\\end{theorem}\n\\begin{proof}\nThe first check gives positivity of the floor and the four finite packet identities for each family. The factors prove the covariance and noise-floor hypotheses. The rational-to-real interpretation lemmas turn every checked local inequality into a bound on the corresponding real packet energy, and turn the innovation inequality into a bound on $\\|d\\|^2$. Thus the real majorant score in \\eqref{eq:foundation-majorant-score} equals the interpretation o…","candidate. If both \\path{checkCandidate M Q D c} and \\path{checkTolerance M c epsilon} return true, then the exact real posterior version represented by $M$ and the exact real Joseph Gaussian represented by the gain of $c$ satisfy\n\\begin{equation}\n \\Wtwo(M.\\mathrm{posterior},M.\\mathrm{approximation}(c.\\mathrm{gain}))\n \\le\\varepsilon.\n \\label{eq:foundation-rational-soundness}\n\\end{equation}\nHere $\\varepsilon$ is the real interpretation of the rational tolerance.\n\\end{theorem}\n\\begin{proof}\nThe first check gives positivity of the floor and the four finite packet identities for each family. The factors prove the covariance and noise-floor hypotheses. The rational-to-real interpretation lemmas turn every checked local inequality into a bound on the corresponding real packet energy, and turn the innovation inequality into a bound on $\\|d\\|^2$. Thus the real majorant score in \\eqref{eq:foundation-majorant-score} equals the interpretation of the rational candidate score. Corollary~\\ref{cor:foundation-packet-tolerance}, applied to the second check, gives the law tolerance. The model semantic…","the rational tolerance.\n\\end{theorem}\n\\begin{proof}\nThe first check gives positivity of the floor and the four finite packet identities for each family. The factors prove the covariance and noise-floor hypotheses. The rational-to-real interpretation lemmas turn every checked local inequality into a bound on the corresponding real packet energy, and turn the innovation inequality into a bound on $\\|d\\|^2$. Thus the real majorant score in \\eqref{eq:foundation-majorant-score} equals the interpretation of the rational candidate score. Corollary~\\ref{cor:foundation-packet-tolerance}, applied to the second check, gives the law tolerance. The model semantics identifies the two laws in its conclusion with those in \\eqref{eq:foundation-rational-soundness}.\n\\end{proof}","Consider the rational model\n\\[\n P=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix},\\qquad\n H=\\begin{pmatrix}1&1\\end{pmatrix},\\qquad\n R=1,\\qquad m=\\begin{pmatrix}0\\\\0\\end{pmatrix},\\qquad y=1.\n\\]\nIt is factor-coded by $L=(1,0)^\\top$, a zero noise factor, and $\\sigma=1$. The second state coordinate is deterministic under the prior. Nevertheless, the observation operator includes that coordinate, and all model formulas remain well typed. The innovation covariance is $S=2$, the innovation is $d=1$, and\n\\[\n K=\\begin{pmatrix}1/2","0\\end{pmatrix},\\qquad y=1.\n\\]\nIt is factor-coded by $L=(1,0)^\\top$, a zero noise factor, and $\\sigma=1$. The second state coordinate is deterministic under the prior. Nevertheless, the observation operator includes that coordinate, and all model formulas remain well typed. The innovation covariance is $S=2$, the innovation is $d=1$, and\n\\[\n K=\\begin{pmatrix}1/2","0\\end{pmatrix},\\qquad y=1.\n\\]\nIt is factor-coded by $L=(1,0)^\\top$, a zero noise factor, and $\\sigma=1$. The second state coordinate is deterministic under the prior. Nevertheless, the observation operator includes that coordinate, and all model formulas remain well typed. The innovation covariance is $S=2$, the innovation is $d=1$, and\n\\[\n K=\\begin{pmatrix}1/2\\\\0\\end{pmatrix},\\qquad\n C_*=\\begin{pmatrix}1/2&0\\\\0&0\\end{pmatrix}.\n\\]\nThus the explicit posterior is concentrated on the first coordinate axis with mean $(1/2,0)^\\top$.","0&0\\end{pmatrix}.\n\\]\nThus the explicit posterior is concentrated on the first coordinate axis with mean $(1/2,0)^\\top$.","Choose $B=(1/3,0)^\\top$. The approximate mean and Joseph covariance are\n\\[\n m_B=\\begin{pmatrix}1/3","Choose $B=(1/3,0)^\\top$. The approximate mean and Joseph covariance are\n\\[\n m_B=\\begin{pmatrix}1/3\\\\0\\end{pmatrix},\\qquad\n C_B=\\begin{pmatrix}5/9&0\\\\0&0\\end{pmatrix}.\n\\]\nThe residual is $F(B)=(1/3,0)^\\top$, with energy $1/9$. Its exact common-noise coupling cost is\n\\[\n \\|F(B)S^{-1}d\\|^2+\\tr(F(B)S^{-1}F(B)^\\top)\n =\\frac1{36}+\\frac1{18}=\\frac1{12}.\n\\]\nThe inverse-free energy score is larger:\n\\[\n a(B)=\\frac19\\frac{1+1}{1^2}=\\frac29.\n\\]\nBoth are valid upper bounds on the squared law distance. The difference illustrates the cost of replacing the exact inverse expression by an explicit noise-floor majorant. No inverse prior or inverse posterior covariance was used at any stage.","0&0\\end{pmatrix}.\n\\]\nThe residual is $F(B)=(1/3,0)^\\top$, with energy $1/9$. Its exact common-noise coupling cost is\n\\[\n \\|F(B)S^{-1}d\\|^2+\\tr(F(B)S^{-1}F(B)^\\top)\n =\\frac1{36}+\\frac1{18}=\\frac1{12}.\n\\]\nThe inverse-free energy score is larger:\n\\[\n a(B)=\\frac19\\frac{1+1}{1^2}=\\frac29.\n\\]\nBoth are valid upper bounds on the squared law distance. The difference illustrates the cost of replacing the exact inverse expression by an explicit noise-floor majorant. No inverse prior or inverse posterior covariance was used at any stage.","For coordinate output packets and the single identity input packet, the local residual energies are $1/9$ and $0$. Their complete sum is exactly the full energy. If the rational tolerance is $\\varepsilon=1/2$, the squared tolerance is $1/4$, and $2/9\\le1/4$. The exact certificate therefore proves the posterior-law tolerance for this candidate. This is a worked arithmetic specialization of the general theorems, not a claim that a fresh Lean instance of these particular constants has been compiled in this publication run.","The example also separates three objects that can otherwise be confused: the target conditional Gaussian $\\nu_y$, the gain-defined Joseph approximation $\\mu_B$, and any finite interval arrays used to describe the latter. The posterior theorem relates the first two probability laws. The output theorem encloses parameters of the second. Their composition provides a usable represented approximation whose law error and representation error are stated separately.","The problem is one Gaussian observation update. Write the prior as\n\\[\nX\\sim\\mathcal N(m,P),\\qquad Y=HX+\\eta,\n\\]\nwhere the centered Gaussian noise $\\eta$ is independent of $X$. The representation uses rational factors\n\\[\nP=LL^{\\mathsf T},\\qquad R=NN^{\\mathsf T}+\\sigma I,\n\\qquad \\sigma>0.\n\\]\nAll data supplied to the executable construction, including $m,H,L,N,\\sigma$ and the observation $y$, are rational. Finite state and observation index types determine the matrix shapes. There is no requirement that $H$ be square or that $P$ be invertible. The innovation matrix is\n\\[\nS=HPH^{\\mathsf T}+R.\n\\]\nIts positive noise floor supports the inverse used in the exact posterior semantics. The executable iterative solver avoids computing that inverse to construct its returned gain.",",\n\\]\nwhere the centered Gaussian noise $\\eta$ is independent of $X$. The representation uses rational factors\n\\[\nP=LL^{\\mathsf T},\\qquad R=NN^{\\mathsf T}+\\sigma I,\n\\qquad \\sigma>0.\n\\]\nAll data supplied to the executable construction, including $m,H,L,N,\\sigma$ and the observation $y$, are rational. Finite state and observation index types determine the matrix shapes. There is no requirement that $H$ be square or that $P$ be invertible. The innovation matrix is\n\\[\nS=HPH^{\\mathsf T}+R.\n\\]\nIts positive noise floor supports the inverse used in the exact posterior semantics. The executable iterative solver avoids computing that inverse to construct its returned gain.","d Gaussian noise $\\eta$ is independent of $X$. The representation uses rational factors\n\\[\nP=LL^{\\mathsf T},\\qquad R=NN^{\\mathsf T}+\\sigma I,\n\\qquad \\sigma>0.\n\\]\nAll data supplied to the executable construction, including $m,H,L,N,\\sigma$ and the observation $y$, are rational. Finite state and observation index types determine the matrix shapes. There is no requirement that $H$ be square or that $P$ be invertible. The innovation matrix is\n\\[\nS=HPH^{\\mathsf T}+R.\n\\]\nIts positive noise floor supports the inverse used in the exact posterior semantics. The executable iterative solver avoids computing that inverse to construct its returned gain.",". The representation uses rational factors\n\\[\nP=LL^{\\mathsf T},\\qquad R=NN^{\\mathsf T}+\\sigma I,\n\\qquad \\sigma>0.\n\\]\nAll data supplied to the executable construction, including $m,H,L,N,\\sigma$ and the observation $y$, are rational. Finite state and observation index types determine the matrix shapes. There is no requirement that $H$ be square or that $P$ be invertible. The innovation matrix is\n\\[\nS=HPH^{\\mathsf T}+R.\n\\]\nIts positive noise floor supports the inverse used in the exact posterior semantics. The executable iterative solver avoids computing that inverse to construct its returned gain.","For a rational gain $B$, the Joseph approximation has mean and covariance\n\\begin{align*}\nm_B&=m+B(y-Hm),","For a rational gain $B$, the Joseph approximation has mean and covariance\n\\begin{align*}\nm_B&=m+B(y-Hm),\\\\\nP_B&=(I-BH)P(I-BH)^{\\mathsf T}+BRB^{\\mathsf T}.\n\\end{align*}\nIts covariance is positive semidefinite by construction. The approximation is a measure, rather than a collection of unqualified decimal entries. The exact gain $K=PH^{\\mathsf T}S^{-1}$ identifies the familiar Gaussian posterior parameters. Later sections prove that this explicit Gaussian kernel disintegrates the actual forward joint law. This gives a posterior version defined at every supplied observation. Agreement with Mathlib's canonical conditional kernel is an evidence-almost-everywhere statement, as is appropriate for conditional distributions.","P_B&=(I-BH)P(I-BH)^{\\mathsf T}+BRB^{\\mathsf T}.\n\\end{align*}\nIts covariance is positive semidefinite by construction. The approximation is a measure, rather than a collection of unqualified decimal entries. The exact gain $K=PH^{\\mathsf T}S^{-1}$ identifies the familiar Gaussian posterior parameters. Later sections prove that this explicit Gaussian kernel disintegrates the actual forward joint law. This gives a posterior version defined at every supplied observation. Agreement with Mathlib's canonical conditional kernel is an evidence-almost-everywhere statement, as is appropriate for conditional distributions.","\\begin{theorem}[Publication contract of the canonical solver]\\label{thm:publication-contract}\nFix a finite-dimensional rational factor-coded model with $\\sigma>0$, and complete, mutually orthogonal self-adjoint rational output and input packet families. Supply an output-packet seed and a finite scan containing every output label. Let $\\epsilon,\\delta>0$ be rational. An arbitrary rational initial gain and an arbitrary finite list of requested precision probes are permitted.","The canonical program \\path{Komorebi.CachedPosterior.runCached} returns a result $r$. If $B$ is its candidate gain and $\\Pi_y$ is the explicitly proved Gaussian posterior version, then\n\\[\n\\Wtwo(\\Pi_y,\\mathcal N(m_B,P_B))\\leq\\epsilon.\n\\]\nThe candidate and tolerance predicates hold. The reported mean and covariance intervals enclose $m_B$ and $P_B$, respectively, and every reported radius is at most $\\delta$. Its parameter precision is the program's computed output precision, and its arrays equal the declared evaluator at that precision.","e canonical program \\path{Komorebi.CachedPosterior.runCached} returns a result $r$. If $B$ is its candidate gain and $\\Pi_y$ is the explicitly proved Gaussian posterior version, then\n\\[\n\\Wtwo(\\Pi_y,\\mathcal N(m_B,P_B))\\leq\\epsilon.\n\\]\nThe candidate and tolerance predicates hold. The reported mean and covariance intervals enclose $m_B$ and $P_B$, respectively, and every reported radius is at most $\\delta$. Its parameter precision is the program's computed output precision, and its arrays equal the declared evaluator at that precision.","ath{cached_solver_reaches_posterior_tolerance}. That theorem combines initialization, cache preservation, greedy-trajectory equivalence, finite fuel, and a sufficient fallback verification precision. The Gaussian semantic bridge converts its residual certificate into the Wasserstein conclusion. The output theorem \\path{represent_spec} provides parameter enclosure and the radius bound at positive $\\delta$. The definition of the result record supplies both reported-evaluator equalities. The separate source declaration \\path{runCached_sound} derives these properties from an actual successful return. The mathematical steps are developed in the following chapters and traced to their exact declarations in the companion.\n\\end{proof}","The two tolerances have different purposes. The parameter tolerance $\\delta$ is an enclosure radius for the parameters of the returned approximation. It is not an entrywise error against the exact posterior covariance. The law tolerance $\\epsilon$ compares the whole approximation with the posterior in $W_2$. Neither conclusion certifies a Gaussian formed by discarding every interval radius and using its centers as exact parameters. Such a replacement would need a separate perturbation argument, including a positive-semidefinite covariance obligation.","\\subsection{Why the packet representation matters}\nThe residual is $F(B)=PH^{\\mathsf T}-BS$. Orthogonal output projectors $Q_a$ split it into components $G_a=Q_aF(B)$. Completeness makes the sum of their squared Frobenius energies equal the energy of the full residual. Selecting a largest-energy component therefore chooses a block containing a definite fraction of the remaining error.","The exact cache retains these components, their energies, and their total. For a selected label $a$, the update changes $B$ by $\\tau G_a$, changes $G_a$ by right multiplication with $I-\\tau S$, and leaves every other cached component unchanged. The proof of that last statement uses orthogonality of the output projectors. It is an algebraic locality result. Keeping unrelated cache entries untouched is justified by their semantic invariant.","The naive rounded-cache counterexample explains the distinction. In a scalar model, exact gain and residual both equal $1/3$ after one step. Rounding both centers independently to zero fractional bits makes both stored values zero. The stored residual score then says zero, although the true residual at the stored gain has score two. The independent final interval verifier rejects that candidate. Sound final verification survives, but the invalid cache cannot supply the exact-cache progress proof.","\\subsection{Relation to established methods}\nMaximum-energy packet selection has a direct optimization interpretation. For the quadratic objective\n\\[\n\\phi(B)=\\tfrac12\\tr(BSB^{\\mathsf T})-\\langle B,PH^{\\mathsf T}\\rangle_{\\mathrm F},\n\\]\nthe gradient is $BS-PH^{\\mathsf T}=-F(B)$. The selected packet has largest projected-gradient norm. This is a block Gauss--Southwell mechanism, and the residual cache is maintained by exact linear identities. Greedy coordinate and block selection are established methods \\cite{nutini2015,nutini2022}. Pathwise Gaussian conditioning and Wasserstein approximation analysis also have substantial prior literature \\cite{wilson2021}. Formally checked interval reasoning for rounded arithmetic is an established methodology \\cite{daumas2007}.","Throughout this section, the state dimension and observation dimension are finite.\nThe supplied rational model has\n\\[\n P=LL^{\\mathsf T},\\qquad R=NN^{\\mathsf T}+\\sigma I,\n \\qquad \\sigma>0,\n\\]\nand observation operator $H$. Put\n\\[\n S=HPH^{\\mathsf T}+R,\\quad v=y-Hm,\\quad\n F(B)=PH^{\\mathsf T}-BS,\\quad E(X)=\\Frob{X}^{2}.\n\\]\nThe output packet family $(Q_a)_{a\\in A}$ consists of rational self-adjoint,\nmutually orthogonal projectors with sum $I$. The input family $(D_b)_{b\\in D}$\nhas the same properties on the observation space. The output family controls\nselection and cache maintenance. The input family participates in the independent\nfinal certificate. No commutation relation between $S$ and the input projectors\nis assumed.","ension and observation dimension are finite.\nThe supplied rational model has\n\\[\n P=LL^{\\mathsf T},\\qquad R=NN^{\\mathsf T}+\\sigma I,\n \\qquad \\sigma>0,\n\\]\nand observation operator $H$. Put\n\\[\n S=HPH^{\\mathsf T}+R,\\quad v=y-Hm,\\quad\n F(B)=PH^{\\mathsf T}-BS,\\quad E(X)=\\Frob{X}^{2}.\n\\]\nThe output packet family $(Q_a)_{a\\in A}$ consists of rational self-adjoint,\nmutually orthogonal projectors with sum $I$. The input family $(D_b)_{b\\in D}$\nhas the same properties on the observation space. The output family controls\nselection and cache maintenance. The input family participates in the independent\nfinal certificate. No commutation relation between $S$ and the input projectors\nis assumed.","dimension are finite.\nThe supplied rational model has\n\\[\n P=LL^{\\mathsf T},\\qquad R=NN^{\\mathsf T}+\\sigma I,\n \\qquad \\sigma>0,\n\\]\nand observation operator $H$. Put\n\\[\n S=HPH^{\\mathsf T}+R,\\quad v=y-Hm,\\quad\n F(B)=PH^{\\mathsf T}-BS,\\quad E(X)=\\Frob{X}^{2}.\n\\]\nThe output packet family $(Q_a)_{a\\in A}$ consists of rational self-adjoint,\nmutually orthogonal projectors with sum $I$. The input family $(D_b)_{b\\in D}$\nhas the same properties on the observation space. The output family controls\nselection and cache maintenance. The input family participates in the independent\nfinal certificate. No commutation relation between $S$ and the input projectors\nis assumed.","}-BS,\\quad E(X)=\\Frob{X}^{2}.\n\\]\nThe output packet family $(Q_a)_{a\\in A}$ consists of rational self-adjoint,\nmutually orthogonal projectors with sum $I$. The input family $(D_b)_{b\\in D}$\nhas the same properties on the observation space. The output family controls\nselection and cache maintenance. The input family participates in the independent\nfinal certificate. No commutation relation between $S$ and the input projectors\nis assumed.","Preprocessing forms three model-dependent constants:\n\\begin{equation}\n \\tau=\\frac{1}{\\sigma+\\tr S},\\qquad T=I-\\tau S,\n \\qquad w=\\frac{\\sigma+\\lVert v\\rVert^{2}}{\\sigma^{2}}.\n \\label{eq:solver-kernel}\n\\end{equation}\nAll are rational. They are the fields of the record\n\\path{Komorebi.CachedPosterior.Kernel}, produced by\n\\path{Komorebi.CachedPosterior.prepare}. The correctness theorems use this actual\npreparation, rather than an arbitrary record whose fields happen to have the same\nnames. A wrong transition matrix or weight would invalidate the correspondence\nbetween the stored score and the posterior certificate.","\\begin{definition}[Cache validity]\nA cache state is a tuple $c=(B,(G_a),(e_a),t)$ satisfying\n\\begin{equation}\n G_a=Q_aF(B),\\qquad e_a=E(G_a),\\qquad t=\\sum_{a\\in A}e_a.\n \\label{eq:solver-invariant}\n\\end{equation}\nThe cached score is $s(c)=tw$.\n\\end{definition}","Validity is an exact equality invariant. It does not mean that a stored residual\nis approximately near a true residual, nor that its energy is merely an upper\nbound. Initialization evaluates $F(B_0)$ once, projects it onto every output\npacket, computes every energy, and sums them. The theorem\n\\path{initialState_valid} follows directly from the definitions. It requires no\nprediction about future iterates.","Orthogonal complete packets account for the whole energy:\n\\[\n \\sum_{a\\in A}E(Q_aF)=E(F).\n\\]\nConsequently a valid cache has $t=E(F(B))$ and\n\\begin{equation}\n s(c)=E(F(B))\\frac{\\sigma+\\lVert v\\rVert^2}{\\sigma^2}=a(B).\n \\label{eq:solver-score}\n\\end{equation}\nThe right-hand side is the exact residual score used by the posterior error\ncertificate. The declarations \\path{total_eq_residual_energy} and\n\\path{score_eq} establish these two identifications. The total is therefore a\nmeaningful global stopping signal even though only one of its summands changes\nin an iteration.","For a selected label $a$, the executed update is\n\\begin{align}\n B'&=B+\\tau G_a, &G'_a&=G_aT, &e'_a&=E(G'_a),\\nonumber","For a selected label $a$, the executed update is\n\\begin{align}\n B'&=B+\\tau G_a, &G'_a&=G_aT, &e'_a&=E(G'_a),\\nonumber\\\\\n G'_b&=G_b\\quad(b\\ne a), &e'_b&=e_b\\quad(b\\ne a),\n &t'&=t-e_a+e'_a.\n \\label{eq:solver-cache-step}\n\\end{align}\nThis is the definition \\path{Komorebi.CachedPosterior.step}. It computes the new\nselected residual from the old selected residual and the precomputed transition.\nIt retains unselected entries and updates the scalar total by a difference. The\nimplementation does not define its new state by recomputing $F(B')$ and projecting\nthat matrix onto every packet.","ad(b\\ne a), &e'_b&=e_b\\quad(b\\ne a),\n &t'&=t-e_a+e'_a.\n \\label{eq:solver-cache-step}\n\\end{align}\nThis is the definition \\path{Komorebi.CachedPosterior.step}. It computes the new\nselected residual from the old selected residual and the precomputed transition.\nIt retains unselected entries and updates the scalar total by a difference. The\nimplementation does not define its new state by recomputing $F(B')$ and projecting\nthat matrix onto every packet.","\\begin{proposition}[Preservation by the executed update]\nIf the output packets satisfy the stated projector assumptions, $c$ is valid,\nand the kernel is \\path{prepare M}, then the state in\n\\eqref{eq:solver-cache-step} is valid.\n\\end{proposition}\n\\begin{proof}\nBy cache validity, $G_a=Q_aF(B)$, so the gain update is the exact packet step\n$B'=B+\\tau Q_aF(B)$. Expand its residual:\n\\[\n F(B')=F(B)-\\tau Q_aF(B)S.\n\\]\nMultiplication on the left by the selected projector gives\n\\[\n Q_aF(B')=Q_aF(B)-\\tau Q_a^2F(B)S\n          =G_a(I-\\tau S)=G_aT.\n\\]\nFor $b\\ne a$, mutual orthogonality gives $Q_bQ_a=0$ and hence\n$Q_bF(B')=Q_bF(B)=G_b$. This proves every residual equality in the new cache.\nIts energy equalities hold because the selected energy is evaluated from its new\nresidual and the remaining residual-energy pairs are retain","\\begin{proposition}[Preservation by the executed update]\nIf the output packets satisfy the stated projector assumptions, $c$ is valid,\nand the kernel is \\path{prepare M}, then the state in\n\\eqref{eq:solver-cache-step} is valid.\n\\end{proposition}\n\\begin{proof}\nBy cache validity, $G_a=Q_aF(B)$, so the gain update is the exact packet step\n$B'=B+\\tau Q_aF(B)$. Expand its residual:\n\\[\n F(B')=F(B)-\\tau Q_aF(B)S.\n\\]\nMultiplication on the left by the selected projector gives\n\\[\n Q_aF(B')=Q_aF(B)-\\tau Q_a^2F(B)S\n          =G_a(I-\\tau S)=G_aT.\n\\]\nFor $b\\ne a$, mutual orthogonality gives $Q_bQ_a=0$ and hence\n$Q_bF(B')=Q_bF(B)=G_b$. This proves every residual equality in the new cache.\nIts energy equalities hold because the selected energy is evaluated from its new\nresidual and the remaining residual-energy pairs are retained. Finally, replacing\none term in a finite sum changes that sum by exactly $-e_a+e'_a$. Thus\n$t'=\\sum_b e'_b$. All three clauses of validity are preserved.\n\\end{proof}","kets satisfy the stated projector assumptions, $c$ is valid,\nand the kernel is \\path{prepare M}, then the state in\n\\eqref{eq:solver-cache-step} is valid.\n\\end{proposition}\n\\begin{proof}\nBy cache validity, $G_a=Q_aF(B)$, so the gain update is the exact packet step\n$B'=B+\\tau Q_aF(B)$. Expand its residual:\n\\[\n F(B')=F(B)-\\tau Q_aF(B)S.\n\\]\nMultiplication on the left by the selected projector gives\n\\[\n Q_aF(B')=Q_aF(B)-\\tau Q_a^2F(B)S\n          =G_a(I-\\tau S)=G_aT.\n\\]\nFor $b\\ne a$, mutual orthogonality gives $Q_bQ_a=0$ and hence\n$Q_bF(B')=Q_bF(B)=G_b$. This proves every residual equality in the new cache.\nIts energy equalities hold because the selected energy is evaluated from its new\nresidual and the remaining residual-energy pairs are retained. Finally, replacing\none term in a finite sum changes that sum by exactly $-e_a+e'_a$. Thus\n$t'=\\sum_b e'_b$. All three clauses of validity are preserved.\n\\end{proof}","l is \\path{prepare M}, then the state in\n\\eqref{eq:solver-cache-step} is valid.\n\\end{proposition}\n\\begin{proof}\nBy cache validity, $G_a=Q_aF(B)$, so the gain update is the exact packet step\n$B'=B+\\tau Q_aF(B)$. Expand its residual:\n\\[\n F(B')=F(B)-\\tau Q_aF(B)S.\n\\]\nMultiplication on the left by the selected projector gives\n\\[\n Q_aF(B')=Q_aF(B)-\\tau Q_a^2F(B)S\n          =G_a(I-\\tau S)=G_aT.\n\\]\nFor $b\\ne a$, mutual orthogonality gives $Q_bQ_a=0$ and hence\n$Q_bF(B')=Q_bF(B)=G_b$. This proves every residual equality in the new cache.\nIts energy equalities hold because the selected energy is evaluated from its new\nresidual and the remaining residual-energy pairs are retained. Finally, replacing\none term in a finite sum changes that sum by exactly $-e_a+e'_a$. Thus\n$t'=\\sum_b e'_b$. All three clauses of validity are preserved.\n\\end{proof}","in\n\\eqref{eq:solver-cache-step} is valid.\n\\end{proposition}\n\\begin{proof}\nBy cache validity, $G_a=Q_aF(B)$, so the gain update is the exact packet step\n$B'=B+\\tau Q_aF(B)$. Expand its residual:\n\\[\n F(B')=F(B)-\\tau Q_aF(B)S.\n\\]\nMultiplication on the left by the selected projector gives\n\\[\n Q_aF(B')=Q_aF(B)-\\tau Q_a^2F(B)S\n          =G_a(I-\\tau S)=G_aT.\n\\]\nFor $b\\ne a$, mutual orthogonality gives $Q_bQ_a=0$ and hence\n$Q_bF(B')=Q_bF(B)=G_b$. This proves every residual equality in the new cache.\nIts energy equalities hold because the selected energy is evaluated from its new\nresidual and the remaining residual-energy pairs are retained. Finally, replacing\none term in a finite sum changes that sum by exactly $-e_a+e'_a$. Thus\n$t'=\\sum_b e'_b$. All three clauses of validity are preserved.\n\\end{proof}","cache-step} is valid.\n\\end{proposition}\n\\begin{proof}\nBy cache validity, $G_a=Q_aF(B)$, so the gain update is the exact packet step\n$B'=B+\\tau Q_aF(B)$. Expand its residual:\n\\[\n F(B')=F(B)-\\tau Q_aF(B)S.\n\\]\nMultiplication on the left by the selected projector gives\n\\[\n Q_aF(B')=Q_aF(B)-\\tau Q_a^2F(B)S\n          =G_a(I-\\tau S)=G_aT.\n\\]\nFor $b\\ne a$, mutual orthogonality gives $Q_bQ_a=0$ and hence\n$Q_bF(B')=Q_bF(B)=G_b$. This proves every residual equality in the new cache.\nIts energy equalities hold because the selected energy is evaluated from its new\nresidual and the remaining residual-energy pairs are retained. Finally, replacing\none term in a finite sum changes that sum by exactly $-e_a+e'_a$. Thus\n$t'=\\sum_b e'_b$. All three clauses of validity are preserved.\n\\end{proof}","a^2F(B)S\n          =G_a(I-\\tau S)=G_aT.\n\\]\nFor $b\\ne a$, mutual orthogonality gives $Q_bQ_a=0$ and hence\n$Q_bF(B')=Q_bF(B)=G_b$. This proves every residual equality in the new cache.\nIts energy equalities hold because the selected energy is evaluated from its new\nresidual and the remaining residual-energy pairs are retained. Finally, replacing\none term in a finite sum changes that sum by exactly $-e_a+e'_a$. Thus\n$t'=\\sum_b e'_b$. All three clauses of validity are preserved.\n\\end{proof}","=G_a(I-\\tau S)=G_aT.\n\\]\nFor $b\\ne a$, mutual orthogonality gives $Q_bQ_a=0$ and hence\n$Q_bF(B')=Q_bF(B)=G_b$. This proves every residual equality in the new cache.\nIts energy equalities hold because the selected energy is evaluated from its new\nresidual and the remaining residual-energy pairs are retained. Finally, replacing\none term in a finite sum changes that sum by exactly $-e_a+e'_a$. Thus\n$t'=\\sum_b e'_b$. All three clauses of validity are preserved.\n\\end{proof}","The locality is on the output side. For an input projector $D_b$,\n\\[\n F(B')D_b=F(B)D_b-\\tau Q_aF(B)SD_b.\n\\]\nThe factor $SD_b$ can mix input directions. The source declaration\n\\path{input_packet_effect} makes this effect explicit. Complete input packets\ncan still account for certificate energy, but an input packet decomposition does\nnot by itself make the iteration independently local in both indices.","\\begin{proposition}[Exact selection and gain equality]\nFor a valid cache, cached selection returns the same label as a scan of the\nrecomputed energies $E(Q_aF(B))$. Starting from an initialized cache and using the\nprepared kernel, its gain after $n$ updates equals the recomputed greedy gain after\n$n$ updates, for every $n\\in\\N$.\n\\end{proposition}\n\\begin{proof}\nEach stored energy equals the corresponding recomputed energy by\n\\eqref{eq:solver-invariant}. Both scans therefore receive the same energy\nfunction, the same seed, the same list, and the same strict comparison. Their\noutputs are equal, including ties. This is \\path{chosen_eq}. The cache gain update\nthen equals the recomputed greedy step by \\path{advance_gain}.","The initial gains agree at $n=0$. Suppose they agree at $n$. The cache at that\nindex is valid by repeated application of \\path{step_valid}. Equality of\nselection and equality of the gain step give equality at $n+1$. Induction proves\nthe entire trajectory identity. The formal statements are\n\\path{trajectory_valid} and \\path{trajectory_gain}.\n\\end{proof}","Write $A_{\\mathrm{count}}=|A|$ for the positive number of output labels, and set\n\\begin{equation}\n \\rho=1-\\frac{\\sigma}{\\sigma+\\tr S},\\qquad\n q=\\rho^2,\\qquad r=1-\\frac{1-q}{A_{\\mathrm{count}}}.\n \\label{eq:solver-rates}\n\\end{equation}\nThe model factorization and positive noise floor imply that $S$ is positive\nsemidefinite with quadratic lower bound $\\sigma$. Its trace is nonnegative, and\n$\\sigma+\\tr S$ is a valid positive spectral cap. Thus $0\\le\\rho<1$, $0\\le q<1$,\nand $0\\le r<1$. No eigenvalue oracle is needed to construct these rational\nconstants.","ve number of output labels, and set\n\\begin{equation}\n \\rho=1-\\frac{\\sigma}{\\sigma+\\tr S},\\qquad\n q=\\rho^2,\\qquad r=1-\\frac{1-q}{A_{\\mathrm{count}}}.\n \\label{eq:solver-rates}\n\\end{equation}\nThe model factorization and positive noise floor imply that $S$ is positive\nsemidefinite with quadratic lower bound $\\sigma$. Its trace is nonnegative, and\n$\\sigma+\\tr S$ is a valid positive spectral cap. Thus $0\\le\\rho<1$, $0\\le q<1$,\nand $0\\le r<1$. No eigenvalue oracle is needed to construct these rational\nconstants.","utput labels, and set\n\\begin{equation}\n \\rho=1-\\frac{\\sigma}{\\sigma+\\tr S},\\qquad\n q=\\rho^2,\\qquad r=1-\\frac{1-q}{A_{\\mathrm{count}}}.\n \\label{eq:solver-rates}\n\\end{equation}\nThe model factorization and positive noise floor imply that $S$ is positive\nsemidefinite with quadratic lower bound $\\sigma$. Its trace is nonnegative, and\n$\\sigma+\\tr S$ is a valid positive spectral cap. Thus $0\\le\\rho<1$, $0\\le q<1$,\nand $0\\le r<1$. No eigenvalue oracle is needed to construct these rational\nconstants.","d set\n\\begin{equation}\n \\rho=1-\\frac{\\sigma}{\\sigma+\\tr S},\\qquad\n q=\\rho^2,\\qquad r=1-\\frac{1-q}{A_{\\mathrm{count}}}.\n \\label{eq:solver-rates}\n\\end{equation}\nThe model factorization and positive noise floor imply that $S$ is positive\nsemidefinite with quadratic lower bound $\\sigma$. Its trace is nonnegative, and\n$\\sigma+\\tr S$ is a valid positive spectral cap. Thus $0\\le\\rho<1$, $0\\le q<1$,\nand $0\\le r<1$. No eigenvalue oracle is needed to construct these rational\nconstants.","The transition contracts every residual matrix:\n\\[\n E(XT)\\le qE(X).\n\\]\nIf label $a$ is updated, its residual energy contracts by this inequality and\nall other output residuals remain unchanged. Orthogonal energy accounting gives\n\\[\n E(F(B'))\\le E(F(B))-(1-q)e_a.\n\\]\nBecause $a$ is a maximum, $e_a\\ge E(F(B))/A_{\\mathrm{count}}$. It follows that\n\\begin{equation}\n E(F(B'))\\le rE(F(B)),\\qquad\n a(B_n)\\le r^na(B_0).\n \\label{eq:solver-geometric}\n\\end{equation}\nThe energy share is \\path{selected_energy_share}; one-step contraction is\n\\path{greedy_energy_contract}; and the score estimate is\n\\path{greedy_score_geometric}. The estimate allows singular $P$, arbitrary initial\nrational $B_0$, rectangular $H$, and rational rotated packets under the same\nprojector assumption","The transition contracts every residual matrix:\n\\[\n E(XT)\\le qE(X).\n\\]\nIf label $a$ is updated, its residual energy contracts by this inequality and\nall other output residuals remain unchanged. Orthogonal energy accounting gives\n\\[\n E(F(B'))\\le E(F(B))-(1-q)e_a.\n\\]\nBecause $a$ is a maximum, $e_a\\ge E(F(B))/A_{\\mathrm{count}}$. It follows that\n\\begin{equation}\n E(F(B'))\\le rE(F(B)),\\qquad\n a(B_n)\\le r^na(B_0).\n \\label{eq:solver-geometric}\n\\end{equation}\nThe energy share is \\path{selected_energy_share}; one-step contraction is\n\\path{greedy_energy_contract}; and the score estimate is\n\\path{greedy_score_geometric}. The estimate allows singular $P$, arbitrary initial\nrational $B_0$, rectangular $H$, and rational rotated packets under the same\nprojector assumptions.","B))-(1-q)e_a.\n\\]\nBecause $a$ is a maximum, $e_a\\ge E(F(B))/A_{\\mathrm{count}}$. It follows that\n\\begin{equation}\n E(F(B'))\\le rE(F(B)),\\qquad\n a(B_n)\\le r^na(B_0).\n \\label{eq:solver-geometric}\n\\end{equation}\nThe energy share is \\path{selected_energy_share}; one-step contraction is\n\\path{greedy_energy_contract}; and the score estimate is\n\\path{greedy_score_geometric}. The estimate allows singular $P$, arbitrary initial\nrational $B_0$, rectangular $H$, and rational rotated packets under the same\nprojector assumptions.","a maximum, $e_a\\ge E(F(B))/A_{\\mathrm{count}}$. It follows that\n\\begin{equation}\n E(F(B'))\\le rE(F(B)),\\qquad\n a(B_n)\\le r^na(B_0).\n \\label{eq:solver-geometric}\n\\end{equation}\nThe energy share is \\path{selected_energy_share}; one-step contraction is\n\\path{greedy_energy_contract}; and the score estimate is\n\\path{greedy_score_geometric}. The estimate allows singular $P$, arbitrary initial\nrational $B_0$, rectangular $H$, and rational rotated packets under the same\nprojector assumptions.","E(F(B))/A_{\\mathrm{count}}$. It follows that\n\\begin{equation}\n E(F(B'))\\le rE(F(B)),\\qquad\n a(B_n)\\le r^na(B_0).\n \\label{eq:solver-geometric}\n\\end{equation}\nThe energy share is \\path{selected_energy_share}; one-step contraction is\n\\path{greedy_energy_contract}; and the score estimate is\n\\path{greedy_score_geometric}. The estimate allows singular $P$, arbitrary initial\nrational $B_0$, rectangular $H$, and rational rotated packets under the same\nprojector assumptions.","A convergent sequence alone would not define a terminating program with an\nexplicit natural-number budget. The source supplies a logarithm-free fuel:\n\\begin{equation}\n N=\\left\\lceil\n \\frac{a(B_0)}{(\\varepsilon/2)^2(1-r)}\n \\right\\rceil_{\\N},\\qquad \\varepsilon>0.\n \\label{eq:solver-fuel}\n\\end{equation}\nHere $\\lceil\\cdot\\rceil_{\\N}$ is the natural-number ceiling used by\n\\path{toleranceFuel}. The canonical \\path{adaptiveFuel} evaluates this expression\nfrom the actual model, initial gain, tolerance, and number of labels.","\\begin{lemma}[Rational fuel sufficiency]\nFor $0\\le r<1$ and $n\\ge N$, the geometric envelope obeys\n$r^na(B_0)\\le\\varepsilon^2/4$.\n\\end{lemma}\n\\begin{proof}\nThe source first proves\n\\[\n r^n\\bigl(1+n(1-r)\\bigr)\\le1.\n\\]\nAt the induction step, the inequality\n$r(1+(n+1)(1-r))\\le1+n(1-r)$ reduces to nonnegativity of\n$(n+1)(1-r)^2$. The natural ceiling in \\eqref{eq:solver-fuel} gives\n$a(B_0)\\le n(\\varepsilon/2)^2(1-r)$. Multiplying by the nonnegative\n$r^n$ and applying the preceding bound yields the required squared tolerance.\nThe proof remains valid when the contraction factor is zero. The source\nstatements are \\path{geometric_bernoulli}, \\path{geometric_fuel_sufficient}, and\n\\path{greedy_score_at_fuel}.\n\\end{proof}","le r<1$ and $n\\ge N$, the geometric envelope obeys\n$r^na(B_0)\\le\\varepsilon^2/4$.\n\\end{lemma}\n\\begin{proof}\nThe source first proves\n\\[\n r^n\\bigl(1+n(1-r)\\bigr)\\le1.\n\\]\nAt the induction step, the inequality\n$r(1+(n+1)(1-r))\\le1+n(1-r)$ reduces to nonnegativity of\n$(n+1)(1-r)^2$. The natural ceiling in \\eqref{eq:solver-fuel} gives\n$a(B_0)\\le n(\\varepsilon/2)^2(1-r)$. Multiplying by the nonnegative\n$r^n$ and applying the preceding bound yields the required squared tolerance.\nThe proof remains valid when the contraction factor is zero. The source\nstatements are \\path{geometric_bernoulli}, \\path{geometric_fuel_sufficient}, and\n\\path{greedy_score_at_fuel}.\n\\end{proof}","\\path{search} tests the exact cached score against $\\varepsilon^2/4$ at the current\nstate. If the test fails and fuel remains, it advances once and decrements fuel.\nAt fuel zero it still tests the last permitted state. This convention permits\nsuccess after exactly $N$ updates. \\path{search_score} proves that every returned\nstate passes the score condition. \\path{search_origin} proves that a returned\nstate is an actual trajectory state at some $k\\le N$. Finally,\n\\path{search_complete} proves that a passing score at the last permitted state\nforces the search to return some state, possibly earlier.","st $\\varepsilon^2/4$ at the current\nstate. If the test fails and fuel remains, it advances once and decrements fuel.\nAt fuel zero it still tests the last permitted state. This convention permits\nsuccess after exactly $N$ updates. \\path{search_score} proves that every returned\nstate passes the score condition. \\path{search_origin} proves that a returned\nstate is an actual trajectory state at some $k\\le N$. Finally,\n\\path{search_complete} proves that a passing score at the last permitted state\nforces the search to return some state, possibly earlier.","Let $h_p=2^{-p}$. The dyadic evaluator propagates rational centers and outward\nradii through its arithmetic graph. Its budget calculation supplies a nonnegative\ncoefficient $C_{\\mathrm{ver}}(M,Q,D,B)$ independent of $p$, with\n\\begin{equation}\n \\widehat a_p(B)\\le a(B)+C_{\\mathrm{ver}}(M,Q,D,B)h_p.\n \\label{eq:solver-inflation}\n\\end{equation}\nHere $\\widehat a_p$ is the actual rounded candidate score. The full expression for\nthe coefficient is \\path{scoreRoundCoeff}, built from packet residual energy and\ninnovation-vector magnitude and radius budgets. It bounds the entire score graph,\nrather than rounding only its final exact answer. The theorem\n\\path{rounded_score_inflation} proves \\eqref{eq:solver-inflation}","Let $h_p=2^{-p}$. The dyadic evaluator propagates rational centers and outward\nradii through its arithmetic graph. Its budget calculation supplies a nonnegative\ncoefficient $C_{\\mathrm{ver}}(M,Q,D,B)$ independent of $p$, with\n\\begin{equation}\n \\widehat a_p(B)\\le a(B)+C_{\\mathrm{ver}}(M,Q,D,B)h_p.\n \\label{eq:solver-inflation}\n\\end{equation}\nHere $\\widehat a_p$ is the actual rounded candidate score. The full expression for\nthe coefficient is \\path{scoreRoundCoeff}, built from packet residual energy and\ninnovation-vector magnitude and radius budgets. It bounds the entire score graph,\nrather than rounding only its final exact answer. The theorem\n\\path{rounded_score_inflation} proves \\eqref{eq:solver-inflation}.","The common precision selector is the computable integer expression\n\\begin{equation}\n \\operatorname{precisionFor}(C,d)\n =\\log^{\\N}_2\\left(\\left\\lceil C/d\\right\\rceil_{\\N}\\right)+1,\n \\qquad d>0,\n \\label{eq:solver-precision}\n\\end{equation}\nwhere $\\log^{\\N}_2$ denotes the natural-number logarithm \\path{Nat.log}, including its\nspecified value at zero. It is not a floating-point logarithm. For every precision\nat least this selected count, \\path{coefficient_mesh_le} gives $C2^{-p}\\le d$.\nThe proof uses integer power and ceiling inequalities. The additional bit is a\nsufficient choice, not a claim that the bit count is minimal.","in{equation}\n \\operatorname{precisionFor}(C,d)\n =\\log^{\\N}_2\\left(\\left\\lceil C/d\\right\\rceil_{\\N}\\right)+1,\n \\qquad d>0,\n \\label{eq:solver-precision}\n\\end{equation}\nwhere $\\log^{\\N}_2$ denotes the natural-number logarithm \\path{Nat.log}, including its\nspecified value at zero. It is not a floating-point logarithm. For every precision\nat least this selected count, \\path{coefficient_mesh_le} gives $C2^{-p}\\le d$.\nThe proof uses integer power and ceiling inequalities. The additional bit is a\nsufficient choice, not a claim that the bit count is minimal.","For final verification, the solver appends\n\\[\n p_* = \\operatorname{sufficientPrecision}(M,Q,D,B,\\varepsilon^2/4).\n\\]\nAt its returned search state, $a(B)\\le\\varepsilon^2/4$. The fallback guarantees\ninflation at most another $\\varepsilon^2/4$, so\n\\[\n \\widehat a_{p_*}(B)\\le\\varepsilon^2/2\\le\\varepsilon^2.\n\\]\nPositive $\\varepsilon$ gives a positive inflation allowance. Valid model and\npacket setup give the candidate check, while this inequality gives the tolerance\ncheck. The declarations \\path{certifyRounded_of_slack} and\n\\path{certifyGain_complete} establish that the fallback must succeed.","The stricter search threshold supplies room for rounded verification. Searching\nonly until an exact score happened to equal $\\varepsilon^2$ would leave no positive\nallowance for an enclosing evaluator. The construction deliberately separates\nprogress from verification slack. Its inherited geometric fuel is computed for\n$\\varepsilon/2$, which is exactly the square-root tolerance corresponding to\n$\\varepsilon^2/4$.","\\begin{theorem}[Successful and complete cached gain construction]\nFor rational factor-coded model data with $\\sigma>0$, valid complete rational\noutput and input packets, a seed and complete output scan, any positive rational\n$\\varepsilon$, any finite precision probe list, and any rational initial gain,\n\\path{solveCached} returns a candidate and verification precision. The candidate\nand tolerance checks pass, and its gain is an actual greedy iterate at an index\nwithin the computed fuel. Its Joseph Gaussian has Wasserstein distance at most\n$\\varepsilon$ from the explicit posterior version established by the Bayesian\nsemantics.\n\\end{theorem}","utput and input packets, a seed and complete output scan, any positive rational\n$\\varepsilon$, any finite precision probe list, and any rational initial gain,\n\\path{solveCached} returns a candidate and verification precision. The candidate\nand tolerance checks pass, and its gain is an actual greedy iterate at an index\nwithin the computed fuel. Its Joseph Gaussian has Wasserstein distance at most\n$\\varepsilon$ from the explicit posterior version established by the Bayesian\nsemantics.\n\\end{theorem}","A useful posterior output contains more than a passing gain. For the returned\nrational gain $B$, the parameter stage represents\n\\begin{align}\n \\mu_B&=m+B(y-Hm),\\nonumber","A useful posterior output contains more than a passing gain. For the returned\nrational gain $B$, the parameter stage represents\n\\begin{align}\n \\mu_B&=m+B(y-Hm),\\nonumber\\\\\n V_B&=(I-BH)P(I-BH)^{\\mathsf T}+BRB^{\\mathsf T}.\n \\label{eq:solver-joseph-output}\n\\end{align}\nThese are the mean and Joseph covariance of the same Gaussian law certified by\nthe Wasserstein statement. The covariance is positive semidefinite under the model\nhypotheses even when the prior is singular. The declaration\n\\path{approximation_eq_output_gaussian} identifies this parameterized Gaussian\nwith the approximation law used in the posterior theorem.","The evaluator returns a ball for each mean coordinate and each covariance entry.\nFor exact rational entry $x$, a ball $(c,r)$ means $|x-c|\\le r$. Its radius is an\narithmetic enclosure error for that parameter of the returned approximation.\nThe output tolerance $\\delta>0$ bounds every such radius. It is independent of\nthe posterior Wasserstein tolerance $\\varepsilon$.","Each output arithmetic expression has a precision-independent budget coefficient.\nThe source sums all mean coefficients and all covariance coefficients to obtain\n$C_{\\mathrm{out}}$, defined by \\path{outputCoefficient}. Every individual coefficient is\nnonnegative and no greater than this sum. It chooses\n\\[\n p_{\\mathrm{out}}=\\operatorname{precisionFor}(C_{\\mathrm{out}},\\delta)\n\\]\nand evaluates the output graph at that precision. Since every entry radius is\nbounded by its coefficient times $2^{-p_{\\mathrm{out}}}$, the same integer\nprecision bounds all radii by $\\delta$. The declarations \\path{mean_coeff_le},\n\\path{covariance_coeff_le}, and \\path{evaluate_radii_le} establish this argument.\n\\path{represent_spec} joins entrywise enclosure and the requested radius bound.","sums all mean coefficients and all covariance coefficients to obtain\n$C_{\\mathrm{out}}$, defined by \\path{outputCoefficient}. Every individual coefficient is\nnonnegative and no greater than this sum. It chooses\n\\[\n p_{\\mathrm{out}}=\\operatorname{precisionFor}(C_{\\mathrm{out}},\\delta)\n\\]\nand evaluates the output graph at that precision. Since every entry radius is\nbounded by its coefficient times $2^{-p_{\\mathrm{out}}}$, the same integer\nprecision bounds all radii by $\\delta$. The declarations \\path{mean_coeff_le},\n\\path{covariance_coeff_le}, and \\path{evaluate_radii_le} establish this argument.\n\\path{represent_spec} joins entrywise enclosure and the requested radius bound.","{\\mathrm{out}}$, defined by \\path{outputCoefficient}. Every individual coefficient is\nnonnegative and no greater than this sum. It chooses\n\\[\n p_{\\mathrm{out}}=\\operatorname{precisionFor}(C_{\\mathrm{out}},\\delta)\n\\]\nand evaluates the output graph at that precision. Since every entry radius is\nbounded by its coefficient times $2^{-p_{\\mathrm{out}}}$, the same integer\nprecision bounds all radii by $\\delta$. The declarations \\path{mean_coeff_le},\n\\path{covariance_coeff_le}, and \\path{evaluate_radii_le} establish this argument.\n\\path{represent_spec} joins entrywise enclosure and the requested radius bound.","The result record stores verification precision, the checked candidate, parameter\nprecision, and the evaluated arrays. Its contract \\path{ResultValid} contains\nseven clauses: the candidate check passes, the tolerance check passes, the\nWasserstein bound holds, the arrays enclose the returned approximation's parameters,\nall radii are at most $\\delta$, the reported parameter precision equals the actual\nprecision selector, and the arrays equal the actual evaluator at that precision.\nThe final two clauses make the reported representation part of the contract,\nrather than allowing arbitrary arrays that happen to satisfy a looser description.","\\begin{theorem}[Complete first-release output]\nUnder the valid-input hypotheses above and $\\delta>0$,\n\\path{Komorebi.CachedPosterior.runCached} returns a result satisfying\n\\path{ResultValid}. Every successful return is sound even without setup-validity\nproofs supplied by the caller. A nonpositive output-radius tolerance is rejected.\n\\end{theorem}","Two interpretations must remain separate. The intervals enclose the parameters\nof the certified Joseph Gaussian. They do not certify another Gaussian obtained\nby treating interval centers as exact mean and covariance. In particular, rounded\ncovariance centers are not independently proved positive semidefinite. Likewise,\n$\\delta$ is not a coordinate error bound against the exact posterior. The\nWasserstein guarantee compares laws; the interval guarantee encloses the parameters\nof one of those laws. A theorem combining either guarantee with a new operation\nmust account for that operation explicitly.","The program is mathematically finite for every positive rational pair\n$(\\varepsilon,\\delta)$ under its declared assumptions. A reference implementation\ncan still encounter a time, memory, or user-supplied resource cap. The Python tools\nreport \\path{RESOURCE_LIMIT} when their caps are reached. Such a cap is a limit of\nthe execution environment, not a counterexample to the finite-success theorem.\nThe result contract also contains no sampler and no guarantee for an external\nfloating-point implementation.","The coordinate fixture has a two-dimensional state and one-dimensional\nobservation:\n\\[\n P=\\begin{pmatrix}4&0\\\\0&1\\end{pmatrix},\\quad H=\\begin{pmatrix}1&1\\end{pmatrix},\n \\quad R=1,\\quad m=\\begin{pmatrix}0\\\\0\\end{pmatrix},\\quad y=1.\n\\]\nIt is represented by $L=\\operatorname{diag}(2,1)$, $N=0$, and $\\sigma=1$.\nThe two output projectors select the coordinate axes, and the single input\nprojector is $1$. Start with $B_0=0$, seed $0$, scan $[0,1]$, probes $[4,12]$,\n$\\varepsilon=1/20$, and $\\delta=1/1000$.","0\\end{pmatrix},\\quad y=1.\n\\]\nIt is represented by $L=\\operatorname{diag}(2,1)$, $N=0$, and $\\sigma=1$.\nThe two output projectors select the coordinate axes, and the single input\nprojector is $1$. Start with $B_0=0$, seed $0$, scan $[0,1]$, probes $[4,12]$,\n$\\varepsilon=1/20$, and $\\delta=1/1000$.","Here $S=6$, $\\tau=1/7$, $T=1/7$, and $w=2$. The initial residual is $(4,1)^{\\mathsf T}$,\nwith packet energies $16$ and $1$. Updating one coordinate multiplies its residual\nby $1/7$ and its energy by $1/49$. The other coordinate remains exactly unchanged.\nThe maximum-energy schedule therefore alternates between labels $0$ and $1$.\nThe complete pre-update trace is\n\\begin{center}\n\\begin{tabular}{rrrrr}\n\\toprule\nUpdates already made & Selected label & $e_0$ & $e_1$ & $t$","al residual is $(4,1)^{\\mathsf T}$,\nwith packet energies $16$ and $1$. Updating one coordinate multiplies its residual\nby $1/7$ and its energy by $1/49$. The other coordinate remains exactly unchanged.\nThe maximum-energy schedule therefore alternates between labels $0$ and $1$.\nThe complete pre-update trace is\n\\begin{center}\n\\begin{tabular}{rrrrr}\n\\toprule\nUpdates already made & Selected label & $e_0$ & $e_1$ & $t$","ual is $(4,1)^{\\mathsf T}$,\nwith packet energies $16$ and $1$. Updating one coordinate multiplies its residual\nby $1/7$ and its energy by $1/49$. The other coordinate remains exactly unchanged.\nThe maximum-energy schedule therefore alternates between labels $0$ and $1$.\nThe complete pre-update trace is\n\\begin{center}\n\\begin{tabular}{rrrrr}\n\\toprule\nUpdates already made & Selected label & $e_0$ & $e_1$ & $t$","(4,1)^{\\mathsf T}$,\nwith packet energies $16$ and $1$. Updating one coordinate multiplies its residual\nby $1/7$ and its energy by $1/49$. The other coordinate remains exactly unchanged.\nThe maximum-energy schedule therefore alternates between labels $0$ and $1$.\nThe complete pre-update trace is\n\\begin{center}\n\\begin{tabular}{rrrrr}\n\\toprule\nUpdates already made & Selected label & $e_0$ & $e_1$ & $t$","\\midrule\n0 & 0 & $16$ & $1$ & $17$","1 & 1 & $16/49$ & $1$ & $65/49$","2 & 0 & $16/49$ & $1/49$ & $17/49$","3 & 1 & $16/2401$ & $1/49$ & $65/2401$","4 & 0 & $16/2401$ & $1/2401$ & $17/2401$","5 & 1 & $16/117649$ & $1/2401$ & $65/117649$","After six updates, each coordinate has been selected three times. The geometric\nsum in its gain gives\n\\[\n B_6=\\begin{pmatrix}228/343\\\\57/343\\end{pmatrix},\\qquad\n F(B_6)=\\begin{pmatrix}4/343\\\\1/343\\end{pmatrix},\\qquad\n a(B_6)=\\frac{34}{117649}.\n\\]\nThe strict search threshold is $\\varepsilon^2/4=1/1600$. At the previous state,\n$a(B_5)=130/117649>1/1600$, while $a(B_6)<1/1600$. Thus six is the actual update\ncount for this stopping rule. The formal fuel is $410093$; the program uses only\nsix of these allowed updates. This large difference illustrates the conservative\nfuel construction, not an ambiguity in the observed trace.","1/343\\end{pmatrix},\\qquad\n a(B_6)=\\frac{34}{117649}.\n\\]\nThe strict search threshold is $\\varepsilon^2/4=1/1600$. At the previous state,\n$a(B_5)=130/117649>1/1600$, while $a(B_6)<1/1600$. Thus six is the actual update\ncount for this stopping rule. The formal fuel is $410093$; the program uses only\nsix of these allowed updates. This large difference illustrates the conservative\nfuel construction, not an ambiguity in the observed trace.","At verification precision $4$, the rounded score is $25/16$ and the attempt fails.\nAt precision $12$, it is $481/1048576$, which is below the required final threshold\n$\\varepsilon^2=1/400$. The solver therefore returns verification precision $12$.\nA failed coarse enclosure is a failed certificate attempt, not evidence that the\ngain itself is inaccurate. Sufficiently high precision bounds the enclosure inflation.","The exact posterior mean is $(2/3,1/6)^{\\mathsf T}$, while the returned Joseph mean\nis the gain $B_6$ because $m=0$ and $y=1$. Parameter output uses precision $32$.\nIts mean centers are\n\\[\n \\begin{pmatrix}2854963683/4294967296","The exact posterior mean is $(2/3,1/6)^{\\mathsf T}$, while the returned Joseph mean\nis the gain $B_6$ because $m=0$ and $y=1$. Parameter output uses precision $32$.\nIts mean centers are\n\\[\n \\begin{pmatrix}2854963683/4294967296\\\\89217615/536870912\\end{pmatrix}\n\\]\nwith radius $1/4294967296$ in each coordinate. The covariance centers are\n\\[\n \\begin{pmatrix}\n 5726720411/4294967296 & -1431643597/2147483648","89217615/536870912\\end{pmatrix}\n\\]\nwith radius $1/4294967296$ in each coordinate. The covariance centers are\n\\[\n \\begin{pmatrix}\n 5726720411/4294967296 & -1431643597/2147483648","89217615/536870912\\end{pmatrix}\n\\]\nwith radius $1/4294967296$ in each coordinate. The covariance centers are\n\\[\n \\begin{pmatrix}\n 5726720411/4294967296 & -1431643597/2147483648\\\\\n -1431643597/2147483648 & 3579145495/4294967296\n \\end{pmatrix},\n\\]\nand the corresponding radii are\n\\[\n \\begin{pmatrix}\n 1/536870912&7/4294967296","-1431643597/2147483648 & 3579145495/4294967296\n \\end{pmatrix},\n\\]\nand the corresponding radii are\n\\[\n \\begin{pmatrix}\n 1/536870912&7/4294967296\\\\\n 7/4294967296&1/536870912\n \\end{pmatrix}.\n\\]\nEvery radius is smaller than $1/1000$. These fractions describe output arithmetic,\nnot uncertainty intervals for model estimation. They enclose exactly the Joseph\nmean and covariance at $B_6$.","7/4294967296&1/536870912\n \\end{pmatrix}.\n\\]\nEvery radius is smaller than $1/1000$. These fractions describe output arithmetic,\nnot uncertainty intervals for model estimation. They enclose exactly the Joseph\nmean and covariance at $B_6$.","The exact rational matrix enclosed by those covariance balls is\n\\[\n V_{B_6}=\\frac{1}{117649}\n \\begin{pmatrix}156868&-78432\\\\-78432&98041\\end{pmatrix}.\n\\]\nIt can be checked directly by substituting $B_6$ into\n\\eqref{eq:solver-joseph-output}. Another useful check starts from the exact\nposterior covariance\n$V_* = \\left(\\begin{smallmatrix}4/3&-2/3","-78432&98041\\end{pmatrix}.\n\\]\nIt can be checked directly by substituting $B_6$ into\n\\eqref{eq:solver-joseph-output}. Another useful check starts from the exact\nposterior covariance\n$V_* = \\left(\\begin{smallmatrix}4/3&-2/3","-78432&98041\\end{pmatrix}.\n\\]\nIt can be checked directly by substituting $B_6$ into\n\\eqref{eq:solver-joseph-output}. Another useful check starts from the exact\nposterior covariance\n$V_* = \\left(\\begin{smallmatrix}4/3&-2/3\\\\-2/3&5/6\\end{smallmatrix}\\right)$.\nIf $B_*=(2/3,1/6)^{\\mathsf T}$ is the exact gain, expansion of the Joseph\nexpression gives\n\\[\n V_{B_6}=V_*+6(B_6-B_*)(B_6-B_*)^{\\mathsf T}.\n\\]\nThe gain discrepancy is\n$B_*-B_6=(2/1029,1/2058)^{\\mathsf T}$, which produces the displayed matrix.\nThis second calculation checks the sign of the off-diagonal entries and explains\nwhy the approximate covariance has an additional positive semidefinite term.\nThe posterior accuracy certificate concerns the complete mean-covariance law,\nwhile the dyadic output stage faithfully represents these rational parameters.\nThe larger output budget can still choose many more bits than the smallest\npossible entrywise enclosure would require. Its purpose is a gua","-2/3&5/6\\end{smallmatrix}\\right)$.\nIf $B_*=(2/3,1/6)^{\\mathsf T}$ is the exact gain, expansion of the Joseph\nexpression gives\n\\[\n V_{B_6}=V_*+6(B_6-B_*)(B_6-B_*)^{\\mathsf T}.\n\\]\nThe gain discrepancy is\n$B_*-B_6=(2/1029,1/2058)^{\\mathsf T}$, which produces the displayed matrix.\nThis second calculation checks the sign of the off-diagonal entries and explains\nwhy the approximate covariance has an additional positive semidefinite term.\nThe posterior accuracy certificate concerns the complete mean-covariance law,\nwhile the dyadic output stage faithfully represents these rational parameters.\nThe larger output budget can still choose many more bits than the smallest\npossible entrywise enclosure would require. Its purpose is a guaranteed common\nprecision, not an","-2/3&5/6\\end{smallmatrix}\\right)$.\nIf $B_*=(2/3,1/6)^{\\mathsf T}$ is the exact gain, expansion of the Joseph\nexpression gives\n\\[\n V_{B_6}=V_*+6(B_6-B_*)(B_6-B_*)^{\\mathsf T}.\n\\]\nThe gain discrepancy is\n$B_*-B_6=(2/1029,1/2058)^{\\mathsf T}$, which produces the displayed matrix.\nThis second calculation checks the sign of the off-diagonal entries and explains\nwhy the approximate covariance has an additional positive semidefinite term.\nThe posterior accuracy certificate concerns the complete mean-covariance law,\nwhile the dyadic output stage faithfully represents these rational parameters.\nThe larger output budget can still choose many more bits than the smallest\npossible entrywise enclosure would require. Its purpose is a guaranteed common\nprecision, not an optimal encoding of this particular covariance.","The rotated fixture makes the prior singular and the packets non-coordinate:\n\\[\n P=J=\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix},\\quad H=I,\n \\quad R=I,\\quad m=0,\\quad y=\\begin{pmatrix}1\\\\-1\\end{pmatrix}.\n\\]\nIt uses the rectangular prior factor $L=(1,1)^{\\mathsf T}$, zero noise factor,\nand $\\sigma=1$. Both output and input packet families are\n\\[\n Q_+=\\frac12\\begin{pmatrix}1&1","-1\\end{pmatrix}.\n\\]\nIt uses the rectangular prior factor $L=(1,1)^{\\mathsf T}$, zero noise factor,\nand $\\sigma=1$. Both output and input packet families are\n\\[\n Q_+=\\frac12\\begin{pmatrix}1&1","-1\\end{pmatrix}.\n\\]\nIt uses the rectangular prior factor $L=(1,1)^{\\mathsf T}$, zero noise factor,\nand $\\sigma=1$. Both output and input packet families are\n\\[\n Q_+=\\frac12\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix},\\qquad\n Q_-=\\frac12\\begin{pmatrix}1&-1\\\\-1&1\\end{pmatrix}.\n\\]\nThese project onto the common and antisymmetric directions. They are rational,\nself-adjoint, orthogonal, and complete. The prior has rank one, but the innovation\nmatrix $S=I+J$ is positive definite because of the positive noise floor.","-1&1\\end{pmatrix}.\n\\]\nThese project onto the common and antisymmetric directions. They are rational,\nself-adjoint, orthogonal, and complete. The prior has rank one, but the innovation\nmatrix $S=I+J$ is positive definite because of the positive noise floor.","Take $B_0=0$, seed $0$, scan $[0,1]$, probes $[4,12]$, $\\varepsilon=1/4$, and\n$\\delta=1/1000$. Here $\\tr S=4$, $\\tau=1/5$, and\n$T=I-(I+J)/5$. The initial residual is $J$, entirely in the $Q_+$ output packet.\nSince $JS=3J$, each selected update multiplies this residual by $2/5$. The\n$Q_-$ residual remains zero. Thus every selected label is $0$ and\n\\[\n (e_+,e_-)=(4(4/25)^n,0),\\qquad B_n=\\frac{1-(2/5)^n}{3}J.\n\\]\nThe first four pre-update energy pairs are $(4,0)$, $(16/25,0)$,\n$(64/625,0)$, and $(256/15625,0)$.","varepsilon=1/4$, and\n$\\delta=1/1000$. Here $\\tr S=4$, $\\tau=1/5$, and\n$T=I-(I+J)/5$. The initial residual is $J$, entirely in the $Q_+$ output packet.\nSince $JS=3J$, each selected update multiplies this residual by $2/5$. The\n$Q_-$ residual remains zero. Thus every selected label is $0$ and\n\\[\n (e_+,e_-)=(4(4/25)^n,0),\\qquad B_n=\\frac{1-(2/5)^n}{3}J.\n\\]\nThe first four pre-update energy pairs are $(4,0)$, $(16/25,0)$,\n$(64/625,0)$, and $(256/15625,0)$.","n=1/4$, and\n$\\delta=1/1000$. Here $\\tr S=4$, $\\tau=1/5$, and\n$T=I-(I+J)/5$. The initial residual is $J$, entirely in the $Q_+$ output packet.\nSince $JS=3J$, each selected update multiplies this residual by $2/5$. The\n$Q_-$ residual remains zero. Thus every selected label is $0$ and\n\\[\n (e_+,e_-)=(4(4/25)^n,0),\\qquad B_n=\\frac{1-(2/5)^n}{3}J.\n\\]\nThe first four pre-update energy pairs are $(4,0)$, $(16/25,0)$,\n$(64/625,0)$, and $(256/15625,0)$.","\\delta=1/1000$. Here $\\tr S=4$, $\\tau=1/5$, and\n$T=I-(I+J)/5$. The initial residual is $J$, entirely in the $Q_+$ output packet.\nSince $JS=3J$, each selected update multiplies this residual by $2/5$. The\n$Q_-$ residual remains zero. Thus every selected label is $0$ and\n\\[\n (e_+,e_-)=(4(4/25)^n,0),\\qquad B_n=\\frac{1-(2/5)^n}{3}J.\n\\]\nThe first four pre-update energy pairs are $(4,0)$, $(16/25,0)$,\n$(64/625,0)$, and $(256/15625,0)$.","re $\\tr S=4$, $\\tau=1/5$, and\n$T=I-(I+J)/5$. The initial residual is $J$, entirely in the $Q_+$ output packet.\nSince $JS=3J$, each selected update multiplies this residual by $2/5$. The\n$Q_-$ residual remains zero. Thus every selected label is $0$ and\n\\[\n (e_+,e_-)=(4(4/25)^n,0),\\qquad B_n=\\frac{1-(2/5)^n}{3}J.\n\\]\nThe first four pre-update energy pairs are $(4,0)$, $(16/25,0)$,\n$(64/625,0)$, and $(256/15625,0)$.","After four updates,\n\\[\n B_4=\\frac{203}{625}J,\\qquad F(B_4)=\\frac{16}{625}J,\n \\qquad a(B_4)=\\frac{3072}{390625}.\n\\]\nThe score weight is $w=3$, because $\\lVert y\\rVert^2=2$. The strict margin is\n$\\varepsilon^2/4=1/64$. At the previous state the score is $768/15625>1/64$;\nafter the fourth it is below $1/64$. The computed fuel is $4267$. Precision $4$\nproduces score $435/64$ and fails; precision $12$ produces $9111/1048576$ and\npasses the final threshold $1/16$.","updates,\n\\[\n B_4=\\frac{203}{625}J,\\qquad F(B_4)=\\frac{16}{625}J,\n \\qquad a(B_4)=\\frac{3072}{390625}.\n\\]\nThe score weight is $w=3$, because $\\lVert y\\rVert^2=2$. The strict margin is\n$\\varepsilon^2/4=1/64$. At the previous state the score is $768/15625>1/64$;\nafter the fourth it is below $1/64$. The computed fuel is $4267$. Precision $4$\nproduces score $435/64$ and fails; precision $12$ produces $9111/1048576$ and\npasses the final threshold $1/16$.","4)=\\frac{16}{625}J,\n \\qquad a(B_4)=\\frac{3072}{390625}.\n\\]\nThe score weight is $w=3$, because $\\lVert y\\rVert^2=2$. The strict margin is\n$\\varepsilon^2/4=1/64$. At the previous state the score is $768/15625>1/64$;\nafter the fourth it is below $1/64$. The computed fuel is $4267$. Precision $4$\nproduces score $435/64$ and fails; precision $12$ produces $9111/1048576$ and\npasses the final threshold $1/16$.","The exact posterior mean is zero because $Jy=0$, and the returned approximation's\nmean is also zero. Its Joseph covariance is\n\\[\n V_{B_4}=\\frac{130379}{390625}J.\n\\]\nThe exact posterior covariance is $J/3$. Both laws remain supported in the prior's\none-dimensional common-direction subspace. This example therefore combines a\nsingular prior, rotated packet selection, a nontrivial covariance approximation,\nand a zero mean without using an inverse of the singular prior covariance.","At output precision $32$, each mean center is zero with radius $1/2147483648$.\nEvery covariance entry has center $179191533/536870912$ and radius\n$7/4294967296$. All output radii satisfy $\\delta=1/1000$. The equality of these\ncenters reflects the exact common-direction structure of the returned Joseph\ncovariance. It does not turn the rounded center matrix into the object of the\nposterior theorem.","Exact cache validity cannot be reused unchanged after independently rounding its\ncomponents. The release contains a scalar counterexample with\n$P=H=R=1$, $\\sigma=1$, $m=0$, $y=1$, and $B_0=0$. Then $S=2$, $\\tau=1/3$,\n$w=2$, and the first exact update is\n\\[\n B_1=1/3,\\qquad F(B_1)=1/3.\n\\]\nThe naive route separately rounds the gain center and residual center downward\nto zero fractional bits. Both become zero. It then computes the stored energy\nfrom the rounded residual, obtaining a stored score of zero.","The actual residual at its stored gain is instead\n\\[\n F(0)=1,\n \\qquad a(0)=2.\n\\]\nThus the state claims exact internal convergence while its true residual is\nnonzero. The equality $G=QF(B)$ has been destroyed by rounding independently\nrelated quantities. The source module \\path{CachedRoundingBoundary.lean} proves\n\\path{naive_score_zero}, \\path{true_score_two}, and \\path{naive_cache_invalid}.\nFor $\\varepsilon=1/10$ and probe $[12]$, it also proves\n\\path{naive_final_verification_rejected}: \\path{certifyGain} returns no candidate.","tored gain is instead\n\\[\n F(0)=1,\n \\qquad a(0)=2.\n\\]\nThus the state claims exact internal convergence while its true residual is\nnonzero. The equality $G=QF(B)$ has been destroyed by rounding independently\nrelated quantities. The source module \\path{CachedRoundingBoundary.lean} proves\n\\path{naive_score_zero}, \\path{true_score_two}, and \\path{naive_cache_invalid}.\nFor $\\varepsilon=1/10$ and probe $[12]$, it also proves\n\\path{naive_final_verification_rejected}: \\path{certifyGain} returns no candidate.","Core 1.0.0 already retains a different, completed rounded solver. Its update is a\ngenuinely rounded full-gain Richardson step. It neither claims exact selective\ncache invariants nor selects a largest cached packet. Its declarations expose\nthe whole-update defect instead. If $\\widetilde B_{n+1}$ is the rounded update\nand $B_{n+1}^{\\mathrm{ideal}}$ is the exact full step at the current gain, then\n\\[\n F(\\widetilde B_{n+1})=F(\\widetilde B_n)T\n -(\\widetilde B_{n+1}-B_{n+1}^{\\mathrm{ideal}})S.\n\\]\nThis is \\path{rounded_residual_step}. The defect is explicitly included in the\ntracking and termination argument.","Writing $d$ for the number of gain entries and $h_t=2^{-t}$, the source establishes\nan all-iteration tracking bound\n\\[\n E(\\widetilde B_n-B_n^{\\mathrm{exact}})\n \\le\\frac{6d h_t^2}{(1-q)^2}.\n\\]\nIts residual-score comparison has the form\n\\[\n a(\\widetilde B_n)\\le2a(B_n^{\\mathrm{exact}})+C_{\\mathrm{round}}h_t^2,\n \\qquad\n C_{\\mathrm{round}}=\n \\frac{12dE(S)w}{(1-q)^2}.\n\\]\nThe source names are \\path{roundedGain_tracking}, \\path{roundoffScoreCoeff}, and\n\\path{rounded_score_compare}. The first term is ideal progress. The second is a\nprecision-dependent floor. Both are needed.","The chosen update precision \\path{updateAccuracy} places the floor below\n$\\varepsilon^2/8$. The rounded fuel \\path{roundedFuel} drives the exact full-step\nscore below $(\\varepsilon/4)^2=\\varepsilon^2/16$. The factor of two in the\ncomparison therefore gives another $\\varepsilon^2/8$, so the actual rounded\niterate reaches score at most $\\varepsilon^2/4$. The same independent final\nprecision fallback can certify it. \\path{solveRounded_terminates} proves finite\nsuccessful return, and \\path{rounded_solver_gain_origin} identifies the result\nas an actual rounded iterate within its declared fuel.","\\midrule\ncoordinate & 2.38 & 2.46 & 2.33 & $1.94$","rotated & 4.38 & 4.45 & 4.28 & $2.16$","sparse4 & 13.84 & 14.52 & 13.79 & $2.51$","dense4 & 16.34 & 17.96 & 16.34 & $3.11$","sparse8 & 171.98 & 174.30 & 169.87 & $4.62$","dense8 & 200.69 & 235.15 & 195.44 & $5.61$","conditioned4 & 19.16 & 21.36 & 17.46 & $3.08$","The cache improves selective iteration in these cases, yet these measurements\nestablish no overall advantage over the efficient full baseline. Several cases\nrequire substantially more selective updates than full updates. The dense8 case,\nfor example, takes $67$ cached selective updates and $9$ full updates. Final\ncertification and output account for a large part of total production cost. The\nbounded result supports a narrower conclusion: preserving a packet cache avoids\nunnecessary selective recomputation, while end-to-end performance depends on more\nthan the selective update loop.","Every method-case pair also exported an explicit final candidate certificate,\nfor $21$ checked benchmark outputs. Their archived Lean checks certify those\nexplicit rational gains and posterior tolerances. They do not certify the Python\ntiming trace, counters, or numeric parameter arrays from the benchmark runs. The\ntwo generated coordinate and rotated modules have the stronger finite-program and\noutput-array checks described above. A universal theorem about the mathematical\noutput evaluator is also distinct from a universal refinement theorem for a\nPython implementation of that evaluator."]}