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sim","display":"\\sim","kind":"use","occurrences":[21,22,26,30,36,77]},{"id":"symbol-b92ecce9a562b5d7","tex":"\\sim_H","aliases":"∼_H sim","display":"\\sim_H","kind":"use","occurrences":[21,22,26,36]},{"id":"symbol-2c5d47bf0e1e95ba","tex":"\\sim_{H+1}","aliases":"∼_H+1 sim","display":"\\sim_{H+1}","kind":"use","occurrences":[30]},{"id":"symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","occurrences":[140,141]},{"id":"symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","occurrences":[82,84,97]},{"id":"symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","occurrences":[17,20,21,22,31,37,48,117,118,119,142,143,171,173,174,175,178]},{"id":"symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","occurrences":[14,19,60,61,75,123,125,135]},{"id":"symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","occurrences":[128]},{"id":"symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","occurrences":[127,128,132,135]},{"id":"symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","occurrences":[48,51]},{"id":"symbol-529f8d1978a8a0ae","tex":"\\bigcap_{a\\in\\mathcal A}","aliases":"⋂_a∈ A bigcap in mathcal","display":"\\bigcap_{a\\in\\mathcal A}","kind":"use","occurrences":[51]},{"id":"symbol-ee563a8aa1670e17","tex":"\\bigcap_{|w|\\le H}","aliases":"⋂_|w|≤ H bigcap le","display":"\\bigcap_{|w|\\le H}","kind":"use","occurrences":[48]},{"id":"symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","occurrences":[204]},{"id":"symbol-6eb256c5206a08bc","tex":"\\bigcup_{j=0}^{d}","aliases":"⋃_j=0^d bigcup","display":"\\bigcup_{j=0}^{d}","kind":"use","occurrences":[204]},{"id":"symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","occurrences":[8,9,160,161,172]},{"id":"symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","occurrences":[134,198]},{"id":"symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","occurrences":[134,198]},{"id":"symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","occurrences":[28,144,149,198]},{"id":"symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","occurrences":[28]}],"contexts":["A general controlled latent system is\n\\begin{equation}\\label{eq:latent}\n z_{t+1}=F_\\theta(z_t,u_t),\\qquad y_t=O_\\theta(z_t,q_t),\n\\end{equation}\nwhere \\(u_t\\) supplies an operation or observation and \\(q_t\\) specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained operator network, an adapter attached to a frozen language model, or a deliberately constructed finite fixture; these are not interchangeable experimental settings.","\\section{Exact semantics before approximate semantics}\n\\subsection{A task-relative realization}\nLet \\(\\mathcal A\\) be an operation alphabet, \\(Z\\) a latent state space, and \\(Y\\) an observation space. Each \\(a\\in\\mathcal A\\) induces \\(F_a:Z\\to Z\\). For a word \\(w=a_1\\cdots a_k\\), define\n\\[\n F_w=F_{a_k}\\circ\\cdots\\circ F_{a_1}.\n\\]\nA candidate semantic state space \\(S\\), operation maps \\(G_a\\), and abstraction \\(q:Z\\to S\\) should satisfy\n\\begin{equation}\\label{eq:semantics}\n q\\circ F_a=G_a\\circ q,\n \\qquad O=\\bar O\\circ q.\n\\end{equation}\nThese conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the present label establishes neither condition by itself.","semantics before approximate semantics}\n\\subsection{A task-relative realization}\nLet \\(\\mathcal A\\) be an operation alphabet, \\(Z\\) a latent state space, and \\(Y\\) an observation space. Each \\(a\\in\\mathcal A\\) induces \\(F_a:Z\\to Z\\). For a word \\(w=a_1\\cdots a_k\\), define\n\\[\n F_w=F_{a_k}\\circ\\cdots\\circ F_{a_1}.\n\\]\nA candidate semantic state space \\(S\\), operation maps \\(G_a\\), and abstraction \\(q:Z\\to S\\) should satisfy\n\\begin{equation}\\label{eq:semantics}\n q\\circ F_a=G_a\\circ q,\n \\qquad O=\\bar O\\circ q.\n\\end{equation}\nThese conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the present label establishes neither condition by itself.","subsection{A task-relative realization}\nLet \\(\\mathcal A\\) be an operation alphabet, \\(Z\\) a latent state space, and \\(Y\\) an observation space. Each \\(a\\in\\mathcal A\\) induces \\(F_a:Z\\to Z\\). For a word \\(w=a_1\\cdots a_k\\), define\n\\[\n F_w=F_{a_k}\\circ\\cdots\\circ F_{a_1}.\n\\]\nA candidate semantic state space \\(S\\), operation maps \\(G_a\\), and abstraction \\(q:Z\\to S\\) should satisfy\n\\begin{equation}\\label{eq:semantics}\n q\\circ F_a=G_a\\circ q,\n \\qquad O=\\bar O\\circ q.\n\\end{equation}\nThese conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the present label establishes neither condition by itself.","\\begin{proposition}[Semantic continuation]\\label{prop:continuation}\nIf \\eqref{eq:semantics} holds on a forward-invariant domain \\(D\\subseteq Z\\), then\n\\[\n O(F_w(z))=\\bar O(G_w(q(z)))\n \\quad\\text{for every }z\\in D\\text{ and every finite }w.\n\\]\n\\end{proposition}\n\\begin{proof}\nForward invariance keeps every intermediate state inside the domain where the equations hold. Induction gives \\(q(F_w(z))=G_w(q(z))\\). Apply the observation factorization to the endpoint.\n\\end{proof}\nWithout forward invariance, a one-step certificate may leave its own scope. Without observation factorization, the reduced state may preserve transitions but omit an output the consumer needs. Without operation compatibility, matching today's readout says nothing about tomorrow's computation.","An approximate relation such as \\(d(z,z')\\le\\varepsilon\\) is generally not an equivalence relation. For points \\(0,0.75\\varepsilon,1.5\\varepsilon\\), successive pairs meet the bound but the first and last do not. Quotient terminology therefore requires an actual equivalence relation, a specified quantization, or a separate approximate-abstraction formalism. Pairwise tolerance alone is insufficient.","\\section{The two-index completion surface}\n\\subsection{History depth and future horizon}\nLet \\(I\\subseteq Z\\) be the allowed initial states and\n\\[\n R_{\\le L}=\\{F_w(z):z\\in I,\\ |w|\\le L\\}.\n\\]\nDefine exact finite-horizon observational equivalence by\n\\[\n z\\sim_H z'\\iff O(F_w(z))=O(F_w(z'))\\text{ for every }|w|\\le H.\n\\]\nThe finite object is\n\\begin{equation}\\label{eq:surface}\n Q_{L,H}=R_{\\le L}/\\!\\sim_H.\n\\end{equation}\nThe parameter \\(L\\) governs which microstates can be reached; \\(H\\) governs which future observations can distinguish them. A single depth index conceals this difference. The causal-completion and completion-surface audits were designed to expose it. \\cite{ncc,ncs}","surface}\n\\subsection{History depth and future horizon}\nLet \\(I\\subseteq Z\\) be the allowed initial states and\n\\[\n R_{\\le L}=\\{F_w(z):z\\in I,\\ |w|\\le L\\}.\n\\]\nDefine exact finite-horizon observational equivalence by\n\\[\n z\\sim_H z'\\iff O(F_w(z))=O(F_w(z'))\\text{ for every }|w|\\le H.\n\\]\nThe finite object is\n\\begin{equation}\\label{eq:surface}\n Q_{L,H}=R_{\\le L}/\\!\\sim_H.\n\\end{equation}\nThe parameter \\(L\\) governs which microstates can be reached; \\(H\\) governs which future observations can distinguish them. A single depth index conceals this difference. The causal-completion and completion-surface audits were designed to expose it. \\cite{ncc,ncs}","rizon}\nLet \\(I\\subseteq Z\\) be the allowed initial states and\n\\[\n R_{\\le L}=\\{F_w(z):z\\in I,\\ |w|\\le L\\}.\n\\]\nDefine exact finite-horizon observational equivalence by\n\\[\n z\\sim_H z'\\iff O(F_w(z))=O(F_w(z'))\\text{ for every }|w|\\le H.\n\\]\nThe finite object is\n\\begin{equation}\\label{eq:surface}\n Q_{L,H}=R_{\\le L}/\\!\\sim_H.\n\\end{equation}\nThe parameter \\(L\\) governs which microstates can be reached; \\(H\\) governs which future observations can distinguish them. A single depth index conceals this difference. The causal-completion and completion-surface audits were designed to expose it. \\cite{ncc,ncs}","\\begin{proposition}[Refinement and diagonal action]\\label{prop:diagonal}\nIncreasing \\(H\\) refines \\(\\sim_H\\). For every action \\(a\\), there is a well-defined map\n\\[\n Q_{L,H+1}\\longrightarrow Q_{L+1,H},\\qquad [z]\\longmapsto[F_a(z)].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon w","\\begin{proposition}[Refinement and diagonal action]\\label{prop:diagonal}\nIncreasing \\(H\\) refines \\(\\sim_H\\). For every action \\(a\\), there is a well-defined map\n\\[\n Q_{L,H+1}\\longrightarrow Q_{L+1,H},\\qquad [z]\\longmapsto[F_a(z)].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional c","\\begin{proposition}[Refinement and diagonal action]\\label{prop:diagonal}\nIncreasing \\(H\\) refines \\(\\sim_H\\). For every action \\(a\\), there is a well-defined map\n\\[\n Q_{L,H+1}\\longrightarrow Q_{L+1,H},\\qquad [z]\\longmapsto[F_a(z)].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional closure information.","n{proposition}[Refinement and diagonal action]\\label{prop:diagonal}\nIncreasing \\(H\\) refines \\(\\sim_H\\). For every action \\(a\\), there is a well-defined map\n\\[\n Q_{L,H+1}\\longrightarrow Q_{L+1,H},\\qquad [z]\\longmapsto[F_a(z)].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional closure information.","tion]\\label{prop:diagonal}\nIncreasing \\(H\\) refines \\(\\sim_H\\). For every action \\(a\\), there is a well-defined map\n\\[\n Q_{L,H+1}\\longrightarrow Q_{L+1,H},\\qquad [z]\\longmapsto[F_a(z)].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional closure information.","efines \\(\\sim_H\\). For every action \\(a\\), there is a well-defined map\n\\[\n Q_{L,H+1}\\longrightarrow Q_{L+1,H},\\qquad [z]\\longmapsto[F_a(z)].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional closure information.","For every action \\(a\\), there is a well-defined map\n\\[\n Q_{L,H+1}\\longrightarrow Q_{L+1,H},\\qquad [z]\\longmapsto[F_a(z)].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional closure information.","there is a well-defined map\n\\[\n Q_{L,H+1}\\longrightarrow Q_{L+1,H},\\qquad [z]\\longmapsto[F_a(z)].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional closure information.","-defined map\n\\[\n Q_{L,H+1}\\longrightarrow Q_{L+1,H},\\qquad [z]\\longmapsto[F_a(z)].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional closure information.","tarrow Q_{L+1,H},\\qquad [z]\\longmapsto[F_a(z)].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional closure information.","d{proposition}\n\\begin{proof}\nThe set of tested continuations grows with \\(H\\), so equality under all longer-horizon tests implies equality under the shorter set. If \\(z\\sim_{H+1}z'\\), then for every suffix \\(|w|\\le H\\), the continuation consisting of \\(a\\) followed by \\(w\\) has length at most \\(H+1\\). Hence \\(O(F_wF_a(z))=O(F_wF_a(z'))\\), so \\(F_a(z)\\sim_HF_a(z')\\). Both successor states are in \\(R_{\\le L+1}\\).\n\\end{proof}\nThe horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional closure information.","\\subsection{A minimal delayed-observability counterexample}\nLet the concrete state be \\((x,y)\\in\\{0,1\\}^2\\), with present observation \\(O(x,y)=x\\), and let the only operation swap the coordinates. At horizon zero, there are two classes, classified by \\(x\\). At horizon one, there are four: the next observation also reveals \\(y\\).","The states \\((0,0)\\) and \\((0,1)\\) initially look identical but become distinguishable after one operation. Keeping only the current answer destroys future semantics. This exact example is tested by the new finite checkpoint interface. It is the simplest reason a successful final-answer probe should not be called a complete causal state.","\\subsection{Hard and soft observations}\nA hard interface is \\(O_h(z)=\\arg\\max_j \\ell_j(z)\\). A soft interface may be the probability vector or centered logits. Exact agreement under \\(O_h\\) permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","\\section{Linear observable kernels}\n\\subsection{A computable null-space construction}\nFor a switched linear system \\(z_{t+1}=M_{a_t}z_t\\) with observation \\(Cz_t\\), define\n\\[\n N_H=\\bigcap_{|w|\\le H}\\ker(CM_w),\\qquad M_\\varnothing=I.\n\\]\nA direction belongs to \\(N_H\\) exactly when it is invisible under every continuation through horizon \\(H\\).","\\begin{proposition}[Observable-kernel recurrence]\\label{prop:kernel}\nOne has\n\\[\n N_{H+1}=N_H\\cap\\bigcap_{a\\in\\mathcal A}M_a^{-1}N_H.\n\\]\nIf \\(N_{H+1}=N_H\\), then \\(N_H\\) is invariant under every generator and equals \\(N_\\infty\\). In a \\(d\\)-dimensional state space, the decreasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identif","\\begin{proposition}[Observable-kernel recurrence]\\label{prop:kernel}\nOne has\n\\[\n N_{H+1}=N_H\\cap\\bigcap_{a\\in\\mathcal A}M_a^{-1}N_H.\n\\]\nIf \\(N_{H+1}=N_H\\), then \\(N_H\\) is invariant under every generator and equals \\(N_\\infty\\). In a \\(d\\)-dimensional state space, the decreasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest com","\\begin{proposition}[Observable-kernel recurrence]\\label{prop:kernel}\nOne has\n\\[\n N_{H+1}=N_H\\cap\\bigcap_{a\\in\\mathcal A}M_a^{-1}N_H.\n\\]\nIf \\(N_{H+1}=N_H\\), then \\(N_H\\) is invariant under every generator and equals \\(N_\\infty\\). In a \\(d\\)-dimensional state space, the decreasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest common invariant","\\begin{proposition}[Observable-kernel recurrence]\\label{prop:kernel}\nOne has\n\\[\n N_{H+1}=N_H\\cap\\bigcap_{a\\in\\mathcal A}M_a^{-1}N_H.\n\\]\nIf \\(N_{H+1}=N_H\\), then \\(N_H\\) is invariant under every generator and equals \\(N_\\infty\\). In a \\(d\\)-dimensional state space, the decreasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest common invariant subspace contained in \\(\\ker C\\). It is standard finite-dim","\\begin{proposition}[Observable-kernel recurrence]\\label{prop:kernel}\nOne has\n\\[\n N_{H+1}=N_H\\cap\\bigcap_{a\\in\\mathcal A}M_a^{-1}N_H.\n\\]\nIf \\(N_{H+1}=N_H\\), then \\(N_H\\) is invariant under every generator and equals \\(N_\\infty\\). In a \\(d\\)-dimensional state space, the decreasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest common invariant subspace contained in \\(\\ker C\\). It is standard finite-dimensional sys","\\begin{proposition}[Observable-kernel recurrence]\\label{prop:kernel}\nOne has\n\\[\n N_{H+1}=N_H\\cap\\bigcap_{a\\in\\mathcal A}M_a^{-1}N_H.\n\\]\nIf \\(N_{H+1}=N_H\\), then \\(N_H\\) is invariant under every generator and equals \\(N_\\infty\\). In a \\(d\\)-dimensional state space, the decreasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest common invariant subspace contained in \\(\\ker C\\). It is standard finite-dimensional system reasoning presented here because a visually small activation direction is no","Observable-kernel recurrence]\\label{prop:kernel}\nOne has\n\\[\n N_{H+1}=N_H\\cap\\bigcap_{a\\in\\mathcal A}M_a^{-1}N_H.\n\\]\nIf \\(N_{H+1}=N_H\\), then \\(N_H\\) is invariant under every generator and equals \\(N_\\infty\\). In a \\(d\\)-dimensional state space, the decreasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest common invariant subspace contained in \\(\\ker C\\). It is standard finite-dimensional system reasoning presented here because a visually small activation direction is not necessarily a persistent nuisance direction.","_{H+1}=N_H\\cap\\bigcap_{a\\in\\mathcal A}M_a^{-1}N_H.\n\\]\nIf \\(N_{H+1}=N_H\\), then \\(N_H\\) is invariant under every generator and equals \\(N_\\infty\\). In a \\(d\\)-dimensional state space, the decreasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest common invariant subspace contained in \\(\\ker C\\). It is standard finite-dimensional system reasoning presented here because a visually small activation direction is not necessarily a persistent nuisance direction.","\\) is invariant under every generator and equals \\(N_\\infty\\). In a \\(d\\)-dimensional state space, the decreasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest common invariant subspace contained in \\(\\ker C\\). It is standard finite-dimensional system reasoning presented here because a visually small activation direction is not necessarily a persistent nuisance direction.","imensional state space, the decreasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest common invariant subspace contained in \\(\\ker C\\). It is standard finite-dimensional system reasoning presented here because a visually small activation direction is not necessarily a persistent nuisance direction.","reasing sequence stabilizes after at most \\(d\\) strict dimension drops.\n\\end{proposition}\n\\begin{proof}\nInvisibility for a word of length at most \\(H+1\\) includes invisibility for all words of length at most \\(H\\), and invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest common invariant subspace contained in \\(\\ker C\\). It is standard finite-dimensional system reasoning presented here because a visually small activation direction is not necessarily a persistent nuisance direction.","invisibility after one action followed by every suffix of length at most \\(H\\). This is the displayed recurrence. Equality of adjacent spaces gives \\(M_aN_H\\subseteq N_H\\). Since \\(N_H\\subseteq\\ker C\\), induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension.\n\\end{proof}\nThe result identifies the largest common invariant subspace contained in \\(\\ker C\\). It is standard finite-dimensional system reasoning presented here because a visually small activation direction is not necessarily a persistent nuisance direction.","\\subsection{When a linear quotient exists}\n\\begin{proposition}[Quotient-kernel criterion]\\label{prop:quotient}\nLet \\(Q:Z\\to S\\) be a surjective linear map. Linear maps \\(A_a:S\\to S\\) satisfying \\(QM_a=A_aQ\\) exist for all \\(a\\) if and only if \\(\\ker Q\\) is invariant under all \\(M_a\\). They are then unique.\n\\end{proposition}\n\\begin{proof}\nIf the equations hold and \\(z\\in\\ker Q\\), then \\(QM_az=A_aQz=0\\). Conversely, define \\(A_a(Qz)=QM_az\\). Two lifts of the same vector differ by an element of \\(\\ker Q\\); invariance makes the definition independent of the lift. Surjectivity gives existence on all of \\(S\\) and uniqueness.\n\\end{proof}\nTo preserve \\(Cz\\), additionally require \\(\\ker Q\\subseteq\\ker C\\). A candidate quotient justified only by a rank-zero present readout can fail as soon as recurrence returns its supposed","\\subsection{When a linear quotient exists}\n\\begin{proposition}[Quotient-kernel criterion]\\label{prop:quotient}\nLet \\(Q:Z\\to S\\) be a surjective linear map. Linear maps \\(A_a:S\\to S\\) satisfying \\(QM_a=A_aQ\\) exist for all \\(a\\) if and only if \\(\\ker Q\\) is invariant under all \\(M_a\\). They are then unique.\n\\end{proposition}\n\\begin{proof}\nIf the equations hold and \\(z\\in\\ker Q\\), then \\(QM_az=A_aQz=0\\). Conversely, define \\(A_a(Qz)=QM_az\\). Two lifts of the same vector differ by an element of \\(\\ker Q\\); invariance makes the definition independent of the lift. Surjectivity gives existence on all of \\(S\\) and uniqueness.\n\\end{proof}\nTo preserve \\(Cz\\), additionally require \\(\\ker Q\\subseteq\\ker C\\). A candidate quotient justified only by a rank-zero present readout can fail as soon as recurrence returns its supposedly discarded directions to the visible subspace.","uotient exists}\n\\begin{proposition}[Quotient-kernel criterion]\\label{prop:quotient}\nLet \\(Q:Z\\to S\\) be a surjective linear map. Linear maps \\(A_a:S\\to S\\) satisfying \\(QM_a=A_aQ\\) exist for all \\(a\\) if and only if \\(\\ker Q\\) is invariant under all \\(M_a\\). They are then unique.\n\\end{proposition}\n\\begin{proof}\nIf the equations hold and \\(z\\in\\ker Q\\), then \\(QM_az=A_aQz=0\\). Conversely, define \\(A_a(Qz)=QM_az\\). Two lifts of the same vector differ by an element of \\(\\ker Q\\); invariance makes the definition independent of the lift. Surjectivity gives existence on all of \\(S\\) and uniqueness.\n\\end{proof}\nTo preserve \\(Cz\\), additionally require \\(\\ker Q\\subseteq\\ker C\\). A candidate quotient justified only by a rank-zero present readout can fail as soon as recurrence returns its supposedly discarded directions to the visible subspace.","p:quotient}\nLet \\(Q:Z\\to S\\) be a surjective linear map. Linear maps \\(A_a:S\\to S\\) satisfying \\(QM_a=A_aQ\\) exist for all \\(a\\) if and only if \\(\\ker Q\\) is invariant under all \\(M_a\\). They are then unique.\n\\end{proposition}\n\\begin{proof}\nIf the equations hold and \\(z\\in\\ker Q\\), then \\(QM_az=A_aQz=0\\). Conversely, define \\(A_a(Qz)=QM_az\\). Two lifts of the same vector differ by an element of \\(\\ker Q\\); invariance makes the definition independent of the lift. Surjectivity gives existence on all of \\(S\\) and uniqueness.\n\\end{proof}\nTo preserve \\(Cz\\), additionally require \\(\\ker Q\\subseteq\\ker C\\). A candidate quotient justified only by a rank-zero present readout can fail as soon as recurrence returns its supposedly discarded directions to the visible subspace.","exist for all \\(a\\) if and only if \\(\\ker Q\\) is invariant under all \\(M_a\\). They are then unique.\n\\end{proposition}\n\\begin{proof}\nIf the equations hold and \\(z\\in\\ker Q\\), then \\(QM_az=A_aQz=0\\). Conversely, define \\(A_a(Qz)=QM_az\\). Two lifts of the same vector differ by an element of \\(\\ker Q\\); invariance makes the definition independent of the lift. Surjectivity gives existence on all of \\(S\\) and uniqueness.\n\\end{proof}\nTo preserve \\(Cz\\), additionally require \\(\\ker Q\\subseteq\\ker C\\). A candidate quotient justified only by a rank-zero present readout can fail as soon as recurrence returns its supposedly discarded directions to the visible subspace.","is invariant under all \\(M_a\\). They are then unique.\n\\end{proposition}\n\\begin{proof}\nIf the equations hold and \\(z\\in\\ker Q\\), then \\(QM_az=A_aQz=0\\). Conversely, define \\(A_a(Qz)=QM_az\\). Two lifts of the same vector differ by an element of \\(\\ker Q\\); invariance makes the definition independent of the lift. Surjectivity gives existence on all of \\(S\\) and uniqueness.\n\\end{proof}\nTo preserve \\(Cz\\), additionally require \\(\\ker Q\\subseteq\\ker C\\). A candidate quotient justified only by a rank-zero present readout can fail as soon as recurrence returns its supposedly discarded directions to the visible subspace.","\\). They are then unique.\n\\end{proposition}\n\\begin{proof}\nIf the equations hold and \\(z\\in\\ker Q\\), then \\(QM_az=A_aQz=0\\). Conversely, define \\(A_a(Qz)=QM_az\\). Two lifts of the same vector differ by an element of \\(\\ker Q\\); invariance makes the definition independent of the lift. Surjectivity gives existence on all of \\(S\\) and uniqueness.\n\\end{proof}\nTo preserve \\(Cz\\), additionally require \\(\\ker Q\\subseteq\\ker C\\). A candidate quotient justified only by a rank-zero present readout can fail as soon as recurrence returns its supposedly discarded directions to the visible subspace.","\\section{Gauge fixing is not semantic completion}\n\\subsection{Positive rays}\nFor nonzero \\(z\\), positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is \\(z\\sim cz\\) for \\(c>0\\), not identification of \\(z\\) with \\(-z\\). Per-step normalization gives\n\\[\n F_M(z)=\\frac{Mz}{\\norm{Mz}},\n\\]\nprovided \\(Mz\\ne0\\). This fixes radial magnitude, but leaves angular dynamics intact.","\\begin{proposition}[Derivative of normalized dynamics]\\label{prop:derivative}\nFor \\(u=Mz/\\norm{Mz}\\) and \\(Mz\\ne0\\),\n\\[\n DF_M(z)=\\frac{(I-uu^{\\mathsf T})M}{\\norm{Mz}}.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe derivative of \\(x\\mapsto x/\\norm{x}\\) at nonzero \\(x\\) is \\((I-vv^{\\mathsf T})/\\norm{x}\\), with \\(v=x/\\norm{x}\\). Compose it with the linear derivative \\(M\\).\n\\end{proof}\nA bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain, tangent directions, and whether it concerns all tangents or a separately identified semantic fiber.","\\subsection{An exact norm-preserving failure}\n\\begin{counterexample}[Small phase error accumulates]\\label{ex:rotation}\nLet the intended semantic operation be the identity and the learned implementation be a planar rotation \\(R_\\varepsilon\\). Every step preserves Euclidean norm exactly. After \\(k\\) steps the state has rotated by \\(k\\varepsilon\\). With a readout determined by the sign of the first coordinate, a state starting at \\((1,0)\\) eventually changes class for any fixed \\(0<\\varepsilon<\\pi/2\\).\n\\end{counterexample}\nThis counterexample separates numerical stability from correct composition without involving underflow, overflow, or a defective normalization implementation. It also explains why enforcing QR or polar isometry may repair one failure class while leaving the central learning problem untouched.","ing failure}\n\\begin{counterexample}[Small phase error accumulates]\\label{ex:rotation}\nLet the intended semantic operation be the identity and the learned implementation be a planar rotation \\(R_\\varepsilon\\). Every step preserves Euclidean norm exactly. After \\(k\\) steps the state has rotated by \\(k\\varepsilon\\). With a readout determined by the sign of the first coordinate, a state starting at \\((1,0)\\) eventually changes class for any fixed \\(0<\\varepsilon<\\pi/2\\).\n\\end{counterexample}\nThis counterexample separates numerical stability from correct composition without involving underflow, overflow, or a defective normalization implementation. It also explains why enforcing QR or polar isometry may repair one failure class while leaving the central learning problem untouched.","rror accumulates]\\label{ex:rotation}\nLet the intended semantic operation be the identity and the learned implementation be a planar rotation \\(R_\\varepsilon\\). Every step preserves Euclidean norm exactly. After \\(k\\) steps the state has rotated by \\(k\\varepsilon\\). With a readout determined by the sign of the first coordinate, a state starting at \\((1,0)\\) eventually changes class for any fixed \\(0<\\varepsilon<\\pi/2\\).\n\\end{counterexample}\nThis counterexample separates numerical stability from correct composition without involving underflow, overflow, or a defective normalization implementation. It also explains why enforcing QR or polar isometry may repair one failure class while leaving the central learning problem untouched.","\\subsection{Margin protection is finite and local}\n\\begin{proposition}[Readout margin bound]\nSuppose a logit vector has unique winning class \\(i\\) and margin \\(m=\\ell_i-\\max_{j\\ne i}\\ell_j>0\\). Any perturbation \\(\\Delta\\ell\\) with \\(\\norm{\\Delta\\ell}_\\infty<m/2\\) preserves that winner.\n\\end{proposition}\n\\begin{proof}\nFor every competitor \\(j\\), the perturbed difference is at least \\(m-2\\norm{\\Delta\\ell}_\\infty>0\\).\n\\end{proof}\nA finite region with positive margins can hide a continuous path defect for many steps. Correct endpoint labels do not prove the defect is absent. The relevant experiment asks whether repeated paths erode the margin until an ordinary continuation reveals the difference.","\\section{Exact compiled controls and learned operators}\n\\subsection{Anchor transport}\nSuppose \\(B\\) is an invertible matrix whose columns are latent representatives of a finite semantic basis, and \\(P_g\\) is a supplied permutation action. Define\n\\[\n M_g=BP_gB^{-1},\\qquad G=B^{-\\mathsf T}B^{-1}.\n\\]\nThen \\(M_gB=BP_g\\) and \\(M_g^{\\mathsf T}GM_g=G\\). For a permutation representation, every group relation is preserved exactly by conjugacy.","\\begin{proposition}[Exact finite realization]\nThe preceding construction implements the supplied permutation action on every anchor and preserves the \\(G\\)-norm. In the canonical semantic chart \\(B=I\\), it is simply the permutation executor \\(M_g=P_g\\).\n\\end{proposition}\n\\begin{proof}\nThe anchor equation follows by cancellation of \\(B^{-1}B\\). For the metric identity, substitute the definitions and use \\(P_g^{\\mathsf T}P_g=I\\). Products telescope to \\(BP_wB^{-1}\\), so semantic relations transfer.\n\\end{proof}\nThis is a constructive control, not an inference that a learned encoder discovered the same realization. If the true permutation table is supplied to construct \\(M_g\\), exact behavior has been compiled from that table. The learned arm must be evaluated separately.","roposition}[Exact finite realization]\nThe preceding construction implements the supplied permutation action on every anchor and preserves the \\(G\\)-norm. In the canonical semantic chart \\(B=I\\), it is simply the permutation executor \\(M_g=P_g\\).\n\\end{proposition}\n\\begin{proof}\nThe anchor equation follows by cancellation of \\(B^{-1}B\\). For the metric identity, substitute the definitions and use \\(P_g^{\\mathsf T}P_g=I\\). Products telescope to \\(BP_wB^{-1}\\), so semantic relations transfer.\n\\end{proof}\nThis is a constructive control, not an inference that a learned encoder discovered the same realization. If the true permutation table is supplied to construct \\(M_g\\), exact behavior has been compiled from that table. The learned arm must be evaluated separately.","ing construction implements the supplied permutation action on every anchor and preserves the \\(G\\)-norm. In the canonical semantic chart \\(B=I\\), it is simply the permutation executor \\(M_g=P_g\\).\n\\end{proposition}\n\\begin{proof}\nThe anchor equation follows by cancellation of \\(B^{-1}B\\). For the metric identity, substitute the definitions and use \\(P_g^{\\mathsf T}P_g=I\\). Products telescope to \\(BP_wB^{-1}\\), so semantic relations transfer.\n\\end{proof}\nThis is a constructive control, not an inference that a learned encoder discovered the same realization. If the true permutation table is supplied to construct \\(M_g\\), exact behavior has been compiled from that table. The learned arm must be evaluated separately.","in{proof}\nThe anchor equation follows by cancellation of \\(B^{-1}B\\). For the metric identity, substitute the definitions and use \\(P_g^{\\mathsf T}P_g=I\\). Products telescope to \\(BP_wB^{-1}\\), so semantic relations transfer.\n\\end{proof}\nThis is a constructive control, not an inference that a learned encoder discovered the same realization. If the true permutation table is supplied to construct \\(M_g\\), exact behavior has been compiled from that table. The learned arm must be evaluated separately.","\\subsection{Approximate intertwining}\nIf \\(QM_a=A_aQ+E_a\\), repeated composition can accumulate leakage. For latent norm bounded by \\(R\\), semantic operators bounded by \\(a\\), and \\(\\norm{E_a}\\le\\varepsilon\\), the reduced-state discrepancy obeys\n\\[\n e_{t+1}\\le a e_t+\\varepsilon R.\n\\]\nInduction gives\n\\[\n e_t\\le a^t e_0+\\varepsilon R\\sum_{j=0}^{t-1}a^j.\n\\]\nWhen \\(a=1\\), the upper bound grows linearly. When \\(a>1\\), it may grow exponentially. A common invariant metric can control one source of growth, but does not establish zero semantic defect. The companion OSL paper gives the corresponding operator telescoping identity.","n{Approximate intertwining}\nIf \\(QM_a=A_aQ+E_a\\), repeated composition can accumulate leakage. For latent norm bounded by \\(R\\), semantic operators bounded by \\(a\\), and \\(\\norm{E_a}\\le\\varepsilon\\), the reduced-state discrepancy obeys\n\\[\n e_{t+1}\\le a e_t+\\varepsilon R.\n\\]\nInduction gives\n\\[\n e_t\\le a^t e_0+\\varepsilon R\\sum_{j=0}^{t-1}a^j.\n\\]\nWhen \\(a=1\\), the upper bound grows linearly. When \\(a>1\\), it may grow exponentially. A common invariant metric can control one source of growth, but does not establish zero semantic defect. The companion OSL paper gives the corresponding operator telescoping identity.","\\begin{theorem}[Fixed-kernel isometry obstruction]\\label{thm:isometry}\nLet \\(V\\) be a finite-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A","\\begin{theorem}[Fixed-kernel isometry obstruction]\\label{thm:isometry}\nLet \\(V\\) be a finite-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diago","\\begin{theorem}[Fixed-kernel isometry obstruction]\\label{thm:isometry}\nLet \\(V\\) be a finite-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradic","\\begin{theorem}[Fixed-kernel isometry obstruction]\\label{thm:isometry}\nLet \\(V\\) be a finite-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric state","\\begin{theorem}[Fixed-kernel isometry obstruction]\\label{thm:isometry}\nLet \\(V\\) be a finite-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric statement is about inv","\\begin{theorem}[Fixed-kernel isometry obstruction]\\label{thm:isometry}\nLet \\(V\\) be a finite-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric statement is about invariant orthogonal complemen","\\begin{theorem}[Fixed-kernel isometry obstruction]\\label{thm:isometry}\nLet \\(V\\) be a finite-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric statement is about invariant orthogonal complements, not a particular array spelling","\\begin{theorem}[Fixed-kernel isometry obstruction]\\label{thm:isometry}\nLet \\(V\\) be a finite-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric statement is about invariant orthogonal complements, not a particular array spelling.","bel{thm:isometry}\nLet \\(V\\) be a finite-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric statement is about invariant orthogonal complements, not a particular array spelling.","\\(V\\) be a finite-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric statement is about invariant orthogonal complements, not a particular array spelling.","-dimensional inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric statement is about invariant orthogonal complements, not a particular array spelling.","inner-product space, \\(K\\subseteq V\\) a subspace, and \\(U:V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric statement is about invariant orthogonal complements, not a particular array spelling.",":V\\to V\\) an isometry with \\(U(K)\\subseteq K\\). Then \\(U(K)=K\\) and \\(U(K^\\perp)=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric statement is about invariant orthogonal complements, not a particular array spelling.","=K^\\perp\\). Relative to \\(V=K^\\perp\\oplus K\\), \\(U\\) is block diagonal.\n\\end{theorem}\n\\begin{proof}\nAn isometry is injective, so \\(\\dim U(K)=\\dim K\\), and the finite-dimensional inclusion gives equality. Thus \\(U^{-1}(K)=K\\). For \\(x\\in K^\\perp\\), \\(k\\in K\\),\n\\(\\langle Ux,k\\rangle=\\langle x,U^{-1}k\\rangle=0\\). Hence \\(U(K^\\perp)\\subseteq K^\\perp\\); dimensions again give equality.\n\\end{proof}\nIf \\(K\\) is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block \\emph{in this orthogonal decomposition}. A nonorthogonal coordinate chart can display apparent off-diagonal entries without contradicting the theorem. The geometric statement is about invariant orthogonal complements, not a particular array spelling.","\\subsection{Curvature can be invisible to the task}\nConversely, exact semantic behavior does not require the entire ambient space to be path-flat. Consider a block-diagonal action \\(U_g=\\diag(P_g,V_g)\\), observed only through the first block. The \\(P_g\\) may obey all semantic relations while the \\(V_g\\) retain path-dependent history. The task quotient is exact even though a full-state identity loop need not return the hidden block to its initial value.","Norm ratio at 2048 & 1.0000 & 1.0000 & $\\approx0$ & $\\approx0$","\\subsection{The bi-depth surface}\nThe later completion-surface audit examines ordinary history depth \\(L\\le8\\), continuation horizon \\(H\\le8\\), and repeated identity paths through total history depth 70. Its exact registered outcomes must be read before its high aggregate accuracy. \\cite{ncs}","The resulting description is precise:\n\\[\n\\text{entry transient}\\ \\longrightarrow\\ \\text{finite hard decision-cone regime}\\ \\longrightarrow\\ \\text{margin-crossing path drift}.\n\\]\nIt supports neither a globally invariant cone nor a stable quotient of every future probability law. Its value is that it replaces an undifferentiated ``depth failure'' with observable mechanisms and a concrete continuation witness.","\\subsection{Observable rank after recurrence}\nThe learned-checkpoint probe uses all twelve v0.4 checkpoints and 32 primary contexts. It finds rank seven at present observation and rank twelve after one through four continuation steps. Repeat-16 hard-polar log-radial gain remains below \\(2.90\\times10^{-5}\\) in absolute value, while p95 oriented loop angles reach roughly .72--.97 radians. \\cite{nlp}","\\section{From latent computation to an explicit language}\n\\subsection{The intended interface}\nThe working architecture separates three timescales:\n\\[\n\\text{fast latent computation}\\ \\leftrightarrow\\ \\text{typed semantic checkpoint}\\ \\leftrightarrow\\ \\text{persistent evidence ledger}.\n\\]\nA latent state can be provisional and disposable. The checkpoint preserves only distinctions justified for a declared future use. The ledger preserves the origin, interventions, counterexamples, and scope needed to challenge that checkpoint later. This is a proposed integration discipline, not an established end-to-end neural compiler.","\\section{A precise continuation protocol}\nThe remaining scientific target can be stated without claiming a universal language. Fix a task generator, admissible inputs, observation interface, train/development/test separation, architecture family, compute budget, and intervention family before inspecting the primary results. Specify separately the history depth \\(L\\), suffix horizon \\(H\\), and the composition depths used to test a learned operator.","\\section{Objects, meanings, and realizations}\n\\subsection{Three distinct interfaces}\nWe use the following separation throughout:\n\\[\n\\text{latent producer}\\ \\longrightarrow\\ \\text{typed proposal}\\ \\longrightarrow\\ \\text{checked operational consumer}.\n\\]\nThe latent producer can be a neural network, a search procedure, a human, or a fixed fixture. The typed proposal names a carrier, operation, scope, and evidence. The consumer decides what the proposal permits within an explicit language semantics. Success at one arrow does not establish success at the other.","\\begin{definition}[Realization]\nA realization of an operation vocabulary \\(\\mathcal G\\) consists of a state space \\(X\\), maps \\(\\rho_X(g):X\\to X\\) for \\(g\\in\\mathcal G\\), and a declared observation map \\(O_X:X\\to Y\\). For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed.\n\\end{definition}\nThe observation map is part of the claim. Preserving a categorical answer is weaker than preserving all output probabilities, and preserving either can be weaker than preserving future behavior after another operation. The companion Neuralese manuscript develops that distinction in detail. An OSL claim must not silently change its observation interface after evaluation.","A useful abstract operational packet is\n\\[\n p=(\\mathrm{id},\\mathrm{content},\\mathrm{origin},\\mathrm{access},\\mathrm{status},\\mathrm{scope},\\mathrm{evidence}).\n\\]\nThe implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. Endorsement and authorization are separate relations. A packet's numerical confidence does not itself grant either.","\\section{Operator transport and compositional semantics}\n\\subsection{Exact transport}\nLet \\(A_g:X\\to X\\), \\(B_g:Y\\to Y\\), and \\(T:X\\to Y\\) be linear maps. We use column vectors and execute the rightmost map first. The compatibility equation is\n\\begin{equation}\\label{eq:intertwine}\n B_gT=TA_g.\n\\end{equation}\nWhen this equation holds, translating a state and applying the target operator agrees with applying the source operator and then translating. It is a statement about operations, not merely geometric resemblance between embeddings.","\\begin{proposition}[Compositional transport]\\label{prop:transport}\nIf \\eqref{eq:intertwine} holds for every generator, then for every finite word \\(w=g_1\\cdots g_k\\), with \\(A_w=A_{g_k}\\cdots A_{g_1}\\) and analogously for \\(B_w\\), one has \\(B_wT=TA_w\\).\n\\end{proposition}\n\\begin{proof}\nThe empty word gives \\(T=T\\). If the identity holds for \\(w\\), then\n\\(B_gB_wT=B_gTA_w=TA_gA_w\\). Induction on word length proves the result.\n\\end{proof}\nNo invertibility of \\(T\\) is needed. When \\(T\\) is invertible, the target operators are conjugates \\(B_g=TA_gT^{-1}\\). When it is not, the equation may describe an abstraction or information-losing projection. That projection must preserve the observations relevant to the advertised application; the intertwining equation alone does not specify which observations matter.","port}\nIf \\eqref{eq:intertwine} holds for every generator, then for every finite word \\(w=g_1\\cdots g_k\\), with \\(A_w=A_{g_k}\\cdots A_{g_1}\\) and analogously for \\(B_w\\), one has \\(B_wT=TA_w\\).\n\\end{proposition}\n\\begin{proof}\nThe empty word gives \\(T=T\\). If the identity holds for \\(w\\), then\n\\(B_gB_wT=B_gTA_w=TA_gA_w\\). Induction on word length proves the result.\n\\end{proof}\nNo invertibility of \\(T\\) is needed. When \\(T\\) is invertible, the target operators are conjugates \\(B_g=TA_gT^{-1}\\). When it is not, the equation may describe an abstraction or information-losing projection. That projection must preserve the observations relevant to the advertised application; the intertwining equation alone does not specify which observations matter.","ertwine} holds for every generator, then for every finite word \\(w=g_1\\cdots g_k\\), with \\(A_w=A_{g_k}\\cdots A_{g_1}\\) and analogously for \\(B_w\\), one has \\(B_wT=TA_w\\).\n\\end{proposition}\n\\begin{proof}\nThe empty word gives \\(T=T\\). If the identity holds for \\(w\\), then\n\\(B_gB_wT=B_gTA_w=TA_gA_w\\). Induction on word length proves the result.\n\\end{proof}\nNo invertibility of \\(T\\) is needed. When \\(T\\) is invertible, the target operators are conjugates \\(B_g=TA_gT^{-1}\\). When it is not, the equation may describe an abstraction or information-losing projection. That projection must preserve the observations relevant to the advertised application; the intertwining equation alone does not specify which observations matter.","word \\(w=g_1\\cdots g_k\\), with \\(A_w=A_{g_k}\\cdots A_{g_1}\\) and analogously for \\(B_w\\), one has \\(B_wT=TA_w\\).\n\\end{proposition}\n\\begin{proof}\nThe empty word gives \\(T=T\\). If the identity holds for \\(w\\), then\n\\(B_gB_wT=B_gTA_w=TA_gA_w\\). Induction on word length proves the result.\n\\end{proof}\nNo invertibility of \\(T\\) is needed. When \\(T\\) is invertible, the target operators are conjugates \\(B_g=TA_gT^{-1}\\). When it is not, the equation may describe an abstraction or information-losing projection. That projection must preserve the observations relevant to the advertised application; the intertwining equation alone does not specify which observations matter.","\\subsection{Approximate transport}\n\\begin{proposition}[Finite-word error budget]\\label{prop:approx}\nSuppose \\(E_i=B_{g_i}T-TA_{g_i}\\), with \\(\\norm{E_i}\\le\\varepsilon_i\\). Then\n\\[\n B_wT-TA_w=\\sum_{i=1}^{k}B_{g_k}\\cdots B_{g_{i+1}}E_iA_{g_{i-1}}\\cdots A_{g_1}.\n\\]\nConsequently, if every \\(\\norm{A_g}\\le a\\) and \\(\\norm{B_g}\\le b\\),\n\\[\n \\norm{B_wT-TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable compos","\\subsection{Approximate transport}\n\\begin{proposition}[Finite-word error budget]\\label{prop:approx}\nSuppose \\(E_i=B_{g_i}T-TA_{g_i}\\), with \\(\\norm{E_i}\\le\\varepsilon_i\\). Then\n\\[\n B_wT-TA_w=\\sum_{i=1}^{k}B_{g_k}\\cdots B_{g_{i+1}}E_iA_{g_{i-1}}\\cdots A_{g_1}.\n\\]\nConsequently, if every \\(\\norm{A_g}\\le a\\) and \\(\\norm{B_g}\\le b\\),\n\\[\n \\norm{B_wT-TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable composition. If the operators amplify error","\\subsection{Approximate transport}\n\\begin{proposition}[Finite-word error budget]\\label{prop:approx}\nSuppose \\(E_i=B_{g_i}T-TA_{g_i}\\), with \\(\\norm{E_i}\\le\\varepsilon_i\\). Then\n\\[\n B_wT-TA_w=\\sum_{i=1}^{k}B_{g_k}\\cdots B_{g_{i+1}}E_iA_{g_{i-1}}\\cdots A_{g_1}.\n\\]\nConsequently, if every \\(\\norm{A_g}\\le a\\) and \\(\\norm{B_g}\\le b\\),\n\\[\n \\norm{B_wT-TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable composition. If the operators amplify errors, the budget can grow faster. Practical certificates must record their admissible operation","\\subsection{Approximate transport}\n\\begin{proposition}[Finite-word error budget]\\label{prop:approx}\nSuppose \\(E_i=B_{g_i}T-TA_{g_i}\\), with \\(\\norm{E_i}\\le\\varepsilon_i\\). Then\n\\[\n B_wT-TA_w=\\sum_{i=1}^{k}B_{g_k}\\cdots B_{g_{i+1}}E_iA_{g_{i-1}}\\cdots A_{g_1}.\n\\]\nConsequently, if every \\(\\norm{A_g}\\le a\\) and \\(\\norm{B_g}\\le b\\),\n\\[\n \\norm{B_wT-TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable composition. If the operators amplify errors, the budget can grow faster. Practical certificates must record their admissible operation family, norms, and composition horizon rat","\\subsection{Approximate transport}\n\\begin{proposition}[Finite-word error budget]\\label{prop:approx}\nSuppose \\(E_i=B_{g_i}T-TA_{g_i}\\), with \\(\\norm{E_i}\\le\\varepsilon_i\\). Then\n\\[\n B_wT-TA_w=\\sum_{i=1}^{k}B_{g_k}\\cdots B_{g_{i+1}}E_iA_{g_{i-1}}\\cdots A_{g_1}.\n\\]\nConsequently, if every \\(\\norm{A_g}\\le a\\) and \\(\\norm{B_g}\\le b\\),\n\\[\n \\norm{B_wT-TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable composition. If the operators amplify errors, the budget can grow faster. Practical certificates must record their admissible operation family, norms, and composition horizon rather than advertising an","\\subsection{Approximate transport}\n\\begin{proposition}[Finite-word error budget]\\label{prop:approx}\nSuppose \\(E_i=B_{g_i}T-TA_{g_i}\\), with \\(\\norm{E_i}\\le\\varepsilon_i\\). Then\n\\[\n B_wT-TA_w=\\sum_{i=1}^{k}B_{g_k}\\cdots B_{g_{i+1}}E_iA_{g_{i-1}}\\cdots A_{g_1}.\n\\]\nConsequently, if every \\(\\norm{A_g}\\le a\\) and \\(\\norm{B_g}\\le b\\),\n\\[\n \\norm{B_wT-TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable composition. If the operators amplify errors, the budget can grow faster. Practical certificates must record their admissible operation family, norms, and composition horizon rather than advertising an unqualified transport accuracy.","rop:approx}\nSuppose \\(E_i=B_{g_i}T-TA_{g_i}\\), with \\(\\norm{E_i}\\le\\varepsilon_i\\). Then\n\\[\n B_wT-TA_w=\\sum_{i=1}^{k}B_{g_k}\\cdots B_{g_{i+1}}E_iA_{g_{i-1}}\\cdots A_{g_1}.\n\\]\nConsequently, if every \\(\\norm{A_g}\\le a\\) and \\(\\norm{B_g}\\le b\\),\n\\[\n \\norm{B_wT-TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable composition. If the operators amplify errors, the budget can grow faster. Practical certificates must record their admissible operation family, norms, and composition horizon rather than advertising an unqualified transport accuracy.","onsequently, if every \\(\\norm{A_g}\\le a\\) and \\(\\norm{B_g}\\le b\\),\n\\[\n \\norm{B_wT-TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable composition. If the operators amplify errors, the budget can grow faster. Practical certificates must record their admissible operation family, norms, and composition horizon rather than advertising an unqualified transport accuracy.","f every \\(\\norm{A_g}\\le a\\) and \\(\\norm{B_g}\\le b\\),\n\\[\n \\norm{B_wT-TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable composition. If the operators amplify errors, the budget can grow faster. Practical certificates must record their admissible operation family, norms, and composition horizon rather than advertising an unqualified transport accuracy.","e b\\),\n\\[\n \\norm{B_wT-TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable composition. If the operators amplify errors, the budget can grow faster. Practical certificates must record their admissible operation family, norms, and composition horizon rather than advertising an unqualified transport accuracy.","TA_w}\\le\\sum_{i=1}^{k} b^{k-i}\\varepsilon_i a^{i-1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nInsert the intermediate products obtained by moving \\(T\\) past one operator at a time. Consecutive products cancel, leaving the displayed sum. Submultiplicativity and the triangle inequality give the norm bound.\n\\end{proof}\nFor \\(a=b=1\\) and \\(\\varepsilon_i\\le\\varepsilon\\), the bound is \\(k\\varepsilon\\), not \\(\\varepsilon\\). A small one-step residual does not establish unlimited reliable composition. If the operators amplify errors, the budget can grow faster. Practical certificates must record their admissible operation family, norms, and composition horizon rather than advertising an unqualified transport accuracy.","\\subsection{Projectors and changes of chart}\nFor an invertible \\(T\\), define \\(P'=TPT^{-1}\\). If \\(P^2=P\\), then\n\\[\n (P')^2=TP^2T^{-1}=P'.\n\\]\nIdempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric \\(G\\) is transported to \\(G'=T^{-\\mathsf T}GT^{-1}\\), metric-relative identities can be compared consistently. These elementary distinctions matter in a language that allows an operator to be called a projector. A name or an approximate matrix test must not conceal the metric or tolerance under which the predicate is evaluated.","\\section{The operational state machine}\n\\subsection{State and judgment forms}\nA schematic runtime state is\n\\[\n S=(\\Gamma,V,\\mathcal O,\\mathcal P,\\mathcal Q,\\mathcal M,\\mathcal C,\\mathcal A,\\mathcal T),\n\\]\nwhere \\(\\Gamma\\) contains declarations, \\(V\\) vectors, \\(\\mathcal O\\) operators, \\(\\mathcal P\\) packets, \\(\\mathcal Q\\) proof resources, \\(\\mathcal M\\) memories, \\(\\mathcal C\\) capabilities, \\(\\mathcal A\\) planned actions, and \\(\\mathcal T\\) contracts, sessions, and transaction state. The concrete runtime also maintains metadata and audit structures. This tuple is an explanatory grouping, not a replacement wire schema.","hine}\n\\subsection{State and judgment forms}\nA schematic runtime state is\n\\[\n S=(\\Gamma,V,\\mathcal O,\\mathcal P,\\mathcal Q,\\mathcal M,\\mathcal C,\\mathcal A,\\mathcal T),\n\\]\nwhere \\(\\Gamma\\) contains declarations, \\(V\\) vectors, \\(\\mathcal O\\) operators, \\(\\mathcal P\\) packets, \\(\\mathcal Q\\) proof resources, \\(\\mathcal M\\) memories, \\(\\mathcal C\\) capabilities, \\(\\mathcal A\\) planned actions, and \\(\\mathcal T\\) contracts, sessions, and transaction state. The concrete runtime also maintains metadata and audit structures. This tuple is an explanatory grouping, not a replacement wire schema.","A typing judgment \\(\\Gamma\\vdash e:\\tau\\) constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment\n\\[\n \\langle c,S,L\\rangle\\longrightarrow\\langle S',L'\\rangle\n\\]\nrelates a source statement, operational state, and audit ledger to their successors. A statement that rejects still has an outcome and, within the implemented lifecycle, an auditable terminal explanation. Rejection is not equivalent to absence of a trace.","\\section{Bounded discovery and semantic admission}\nThe Discovery Bridge supplies a missing distinction between \\emph{a well-formed proposal} and \\emph{an identified operator}. It accepts a complete finite world \\(W\\), a primitive vocabulary \\(G_W\\), a depth bound \\(d\\), demonstrations \\(D\\), and explicitly claimed laws. All of these belong to the certificate's scope. \\cite{odb}","\\subsection{Transformation classes}\nLet \\(S_W\\) be the full finite legal state set and\n\\[\n \\mathcal P_{W,d}=\\bigcup_{j=0}^{d}G_W^j,\n \\qquad \\sigma_W(p)=\\bigl(p(s)\\bigr)_{s\\in S_W}.\n\\]\nPrograms are identified when their complete state signatures agree. For a demonstration set \\(D=\\{(s_i,t_i)\\}\\), define\n\\[\n \\mathcal C(D)=\\{[p]:p\\in\\mathcal P_{W,d},\\ p(s_i)=t_i\\text{ for all }i\\}.\n\\]\nThe implementation enumerates this bounded universe, quotients equivalent spellings, and tests the advertised laws on every state in \\(S_W\\). It admits only a unique derived transformation that is neither the identity nor an existing primitive and whose listed laws pass.","}^{d}G_W^j,\n \\qquad \\sigma_W(p)=\\bigl(p(s)\\bigr)_{s\\in S_W}.\n\\]\nPrograms are identified when their complete state signatures agree. For a demonstration set \\(D=\\{(s_i,t_i)\\}\\), define\n\\[\n \\mathcal C(D)=\\{[p]:p\\in\\mathcal P_{W,d},\\ p(s_i)=t_i\\text{ for all }i\\}.\n\\]\nThe implementation enumerates this bounded universe, quotients equivalent spellings, and tests the advertised laws on every state in \\(S_W\\). It admits only a unique derived transformation that is neither the identity nor an existing primitive and whose listed laws pass.","\\begin{proposition}[Scope-relative identification]\\label{prop:identification}\nComplete enumeration and exact evaluation identify one transformation precisely when \\(|\\mathcal C(D)|=1\\). If \\(|\\mathcal C(D)|>1\\), two compatible classes have a separating state in \\(S_W\\). Adding the intended output at that state removes at least one of those classes.\n\\end{proposition}\n\\begin{proof}\nDistinct equivalence classes have unequal full signatures. At least one legal input therefore yields unequal outputs. A demonstration choosing one output is inconsistent with a class yielding the other. The uniqueness assertion follows directly from the definition of \\(\\mathcal C(D)\\).\n\\end{proof}\nThe proposition says nothing about programs outside the depth bound or operators outside the vocabulary. ``New'' means new relative to the declared primitive set. It is not a claim of mathematical nov","\\begin{proposition}[Scope-relative identification]\\label{prop:identification}\nComplete enumeration and exact evaluation identify one transformation precisely when \\(|\\mathcal C(D)|=1\\). If \\(|\\mathcal C(D)|>1\\), two compatible classes have a separating state in \\(S_W\\). Adding the intended output at that state removes at least one of those classes.\n\\end{proposition}\n\\begin{proof}\nDistinct equivalence classes have unequal full signatures. At least one legal input therefore yields unequal outputs. A demonstration choosing one output is inconsistent with a class yielding the other. The uniqueness assertion follows directly from the definition of \\(\\mathcal C(D)\\).\n\\end{proof}\nThe proposition says nothing about programs outside the depth bound or operators outside the vocabulary. ``New'' means new relative to the declared primitive set. It is not a claim of mathematical novelty or autonomous concept","\\begin{proposition}[Scope-relative identification]\\label{prop:identification}\nComplete enumeration and exact evaluation identify one transformation precisely when \\(|\\mathcal C(D)|=1\\). If \\(|\\mathcal C(D)|>1\\), two compatible classes have a separating state in \\(S_W\\). Adding the intended output at that state removes at least one of those classes.\n\\end{proposition}\n\\begin{proof}\nDistinct equivalence classes have unequal full signatures. At least one legal input therefore yields unequal outputs. A demonstration choosing one output is inconsistent with a class yielding the other. The uniqueness assertion follows directly from the definition of \\(\\mathcal C(D)\\).\n\\end{proof}\nThe proposition says nothing about programs outside the depth bound or operators outside the vocabulary. ``New'' means new relative to the declared primitive set. It is not a claim of mathematical novelty or autonomous concept invention.","state in \\(S_W\\). Adding the intended output at that state removes at least one of those classes.\n\\end{proposition}\n\\begin{proof}\nDistinct equivalence classes have unequal full signatures. At least one legal input therefore yields unequal outputs. A demonstration choosing one output is inconsistent with a class yielding the other. The uniqueness assertion follows directly from the definition of \\(\\mathcal C(D)\\).\n\\end{proof}\nThe proposition says nothing about programs outside the depth bound or operators outside the vocabulary. ``New'' means new relative to the declared primitive set. It is not a claim of mathematical novelty or autonomous concept invention.","ojector transported between coordinates}\n\\begin{lstlisting}\nmodule coordinate_transport\nspace H dim 2\noperator P : H -> H kind projector matrix [[1,0],[0,0]]\noperator T : H -> H kind linear matrix [[0,-1],[1,0]]\ntransport P via T as P_rot\ncheck_projector P_rot tolerance 1e-12\nvector x in H values [0.0,3.0]\napply y = P_rot(x)\nassert vector y approx [0.0,3.0] tolerance 1e-12\n\\end{lstlisting}\nHere \\(P\\) selects the first coordinate and \\(T\\) rotates the chart. The transported projector selects the second coordinate. The program checks the transported predicate and applies it to a vector in that component. It does not infer a correspondence between two learned models; the transport matrix is supplied.","}\n\\begin{lstlisting}\nmodule coordinate_transport\nspace H dim 2\noperator P : H -> H kind projector matrix [[1,0],[0,0]]\noperator T : H -> H kind linear matrix [[0,-1],[1,0]]\ntransport P via T as P_rot\ncheck_projector P_rot tolerance 1e-12\nvector x in H values [0.0,3.0]\napply y = P_rot(x)\nassert vector y approx [0.0,3.0] tolerance 1e-12\n\\end{lstlisting}\nHere \\(P\\) selects the first coordinate and \\(T\\) rotates the chart. The transported projector selects the second coordinate. The program checks the transported predicate and applies it to a vector in that component. It does not infer a correspondence between two learned models; the transport matrix is supplied.","Let the declared machine be \\(M=(S,\\mathcal A,F,O)\\). The certificate contains a surjection \\(q:S\\to Q\\), quotient transitions \\(\\bar F_a\\), and outputs \\(\\bar O\\), subject to\n\\begin{equation}\\label{eq:finitecertificate}\n q(F_a(s))=\\bar F_a(q(s)),\\qquad O(s)=\\bar O(q(s))\n \\quad(s\\in S,\\ a\\in\\mathcal A).\n\\end{equation}\nThe certificate generator uses partition refinement. The checker instead verifies the displayed equations over every declared state and action. It accepts any sound quotient, including a finer nonminimal one. Minimality is a property of the generator and its tests, not an extra property falsely attributed to the certificate checker.","\\begin{theorem}[Finite semantic checkpoint preservation]\\label{thm:checkpoint}\nIf \\eqref{eq:finitecertificate} holds, then a checkpoint storing \\(q(s)\\) reproduces \\(O(F_w(s))\\) for every finite word \\(w\\) over the declared alphabet, when continued with the quotient transitions. No information inside a fiber of \\(q\\) is needed for those outputs.\n\\end{theorem}\n\\begin{proof}\nInduction on \\(|w|\\) gives \\(q(F_w(s))=\\bar F_w(q(s))\\). Apply the output equation to the terminal state. The argument uses totality of the declared transition maps and the equations at every state, not merely a sampled history panel.\n\\end{proof}\nThis finite theorem permits all word lengths \\emph{for the declared finite model}. It is not an infinite-horizon theorem about a learned implementation that only approximately matches that model.","n{theorem}[Finite semantic checkpoint preservation]\\label{thm:checkpoint}\nIf \\eqref{eq:finitecertificate} holds, then a checkpoint storing \\(q(s)\\) reproduces \\(O(F_w(s))\\) for every finite word \\(w\\) over the declared alphabet, when continued with the quotient transitions. No information inside a fiber of \\(q\\) is needed for those outputs.\n\\end{theorem}\n\\begin{proof}\nInduction on \\(|w|\\) gives \\(q(F_w(s))=\\bar F_w(q(s))\\). Apply the output equation to the terminal state. The argument uses totality of the declared transition maps and the equations at every state, not merely a sampled history panel.\n\\end{proof}\nThis finite theorem permits all word lengths \\emph{for the declared finite model}. It is not an infinite-horizon theorem about a learned implementation that only approximately matches that model.","\\subsection{The D4 fixture}\nThe demonstrator has sixteen concrete states \\((a,b,u)\\), with \\(a\\in\\mathbb Z/4\\mathbb Z\\), \\(b,u\\in\\mathbb Z/2\\mathbb Z\\). Its visible state is \\((a,b)\\); \\(u\\) is a private bit. Operations are\n\\[\n r(a,b,u)=(a+1,b,u),\\quad f(a,b,u)=(-a,1-b,u),\\quad\n v(a,b,u)=(a,b,1-u).\n\\]\nThe first two satisfy \\(r^4=f^2=frfr=1\\). The private operation changes only \\(u\\). Partition refinement recovers eight semantic classes. Discarding the private bit preserves every future declared observation."]}