OSL and Neuralese

Search aliases: G mathcal

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• Definition passage: Realization · OSL_Final_Research_Edition.tex, line 43

  [Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The 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 op…

  Exact mathematical passage

  \mathcal G

Search aliases: g

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• Definition passage: Realization · OSL_Final_Research_Edition.tex, line 43

  [Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The 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 op…

  Exact mathematical passage

  \rho_X(g):X\to X

Search aliases: O

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• Definition passage: Realization · OSL_Final_Research_Edition.tex, line 43

  [Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The 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 op…

  Exact mathematical passage

  O_X:X\to Y

Search aliases: O_X

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• Definition passage: Realization · OSL_Final_Research_Edition.tex, line 43

  [Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The 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 op…

  Exact mathematical passage

  O_X:X\to Y

Search aliases: X

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• Definition passage: Realization · OSL_Final_Research_Edition.tex, line 43

  [Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The 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 op…

  Exact mathematical passage

  X

Search aliases: Y

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• Definition passage: Realization · OSL_Final_Research_Edition.tex, line 43

  [Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The 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 op…

  Exact mathematical passage

  O_X:X\to Y

Search aliases: ρ rho

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• Definition passage: Realization · OSL_Final_Research_Edition.tex, line 43

  [Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The 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 op…

  Exact mathematical passage

  \rho_X(g):X\to X

Search aliases: ρ_X rho

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• Definition passage: Realization · OSL_Final_Research_Edition.tex, line 43

  [Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The 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 op…

  Exact mathematical passage

  \rho_X(g):X\to X

Search aliases: → to

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• Definition passage: Realization · OSL_Final_Research_Edition.tex, line 43

  [Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The 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 op…

  Exact mathematical passage

  \rho_X(g):X\to X

Search aliases: ∈ in

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• Definition passage: Realization · OSL_Final_Research_Edition.tex, line 43

  [Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The 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 op…

  Exact mathematical passage

  g\in\mathcal G

Search aliases: +

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: =

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: A

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• Use in source · Neuralese_Final_Research_Edition.tex, line 132

  When a linear quotient exists. [Quotient-kernel criterion] Let Q:Z→ S be a surjective linear map. Linear maps A_a:S→ 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. If the equations hold and z∈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. To preserv…

  Exact mathematical passage

  A_a:S\to S

Search aliases: A mathcal

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• Use in source · Neuralese_Final_Research_Edition.tex, line 45

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  \mathcal A

Search aliases: a

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• Use in source · Neuralese_Final_Research_Edition.tex, line 45

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  a\in\mathcal A

Search aliases: a^i-1

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• Use in source · OSL_Final_Research_Edition.tex, line 83

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  
   \norm{B_wT-TA_w}\le\sum_{i=1}^{k} b^{k-i}\varepsilon_i a^{i-1}.
  

Search aliases: a^j

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• Use in source · Neuralese_Final_Research_Edition.tex, line 197

  Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When 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 o…

  Exact mathematical passage

  
   e_t\le a^t e_0+\varepsilon R\sum_{j=0}^{t-1}a^j.
  

Search aliases: a^t

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• Use in source · Neuralese_Final_Research_Edition.tex, line 197

  Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When 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 o…

  Exact mathematical passage

  
   e_t\le a^t e_0+\varepsilon R\sum_{j=0}^{t-1}a^j.
  

Search aliases: a_1

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• Use in source · Neuralese_Final_Research_Edition.tex, line 45

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  w=a_1\cdots a_k

Search aliases: A_a

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• Use in source · Neuralese_Final_Research_Edition.tex, line 132

  When a linear quotient exists. [Quotient-kernel criterion] Let Q:Z→ S be a surjective linear map. Linear maps A_a:S→ 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. If the equations hold and z∈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. To preserv…

  Exact mathematical passage

  A_a:S\to S

Search aliases: A_g

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• Use in source · OSL_Final_Research_Edition.tex, line 61

  Operator transport and compositional semantics. Exact transport. Let A_g:X→ X, B_g:Y→ Y, and T:X→ Y be linear maps. We use column vectors and execute the rightmost map first. The compatibility equation is B_gT=TA_g. When 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.

  Exact mathematical passage

  A_g:X\to X

Search aliases: A_g_1

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• Use in source · OSL_Final_Research_Edition.tex, line 68

  [Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No 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 project…

  Exact mathematical passage

  A_w=A_{g_k}\cdots A_{g_1}

Search aliases: A_g_i

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• Use in source · OSL_Final_Research_Edition.tex, line 78

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  E_i=B_{g_i}T-TA_{g_i}

Search aliases: A_g_i-1

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• Use in source · OSL_Final_Research_Edition.tex, line 79

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  
   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}.
  

Search aliases: A_g_k

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• Use in source · OSL_Final_Research_Edition.tex, line 68

  [Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No 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 project…

  Exact mathematical passage

  A_w=A_{g_k}\cdots A_{g_1}

Search aliases: a_k

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• Use in source · Neuralese_Final_Research_Edition.tex, line 45

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  w=a_1\cdots a_k

Search aliases: a_t

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• Use in source · Neuralese_Final_Research_Edition.tex, line 112

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  z_{t+1}=M_{a_t}z_t

Search aliases: A_w

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• Use in source · OSL_Final_Research_Edition.tex, line 68

  [Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No 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 project…

  Exact mathematical passage

  A_w=A_{g_k}\cdots A_{g_1}

Search aliases: access mathrm

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• Use in source · OSL_Final_Research_Edition.tex, line 51

  A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The 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.

  Exact mathematical passage

  
   p=(\mathrm{id},\mathrm{content},\mathrm{origin},\mathrm{access},\mathrm{status},\mathrm{scope},\mathrm{evidence}).
  

Search aliases: arg

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• Use in source · Neuralese_Final_Research_Edition.tex, line 106

  Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_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.

  Exact mathematical passage

  O_h(z)=\arg\max_j \ell_j(z)

Search aliases: B

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• Use in source · Neuralese_Final_Research_Edition.tex, line 177

  Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exactly by conjugacy.

  Exact mathematical passage

  B

Search aliases: b

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• Use in source · OSL_Final_Research_Edition.tex, line 82

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  \norm{B_g}\le b

Search aliases: B^- T mathsf

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• Use in source · Neuralese_Final_Research_Edition.tex, line 178

  Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exactly by conjugacy.

  Exact mathematical passage

  
   M_g=BP_gB^{-1},\qquad G=B^{-\mathsf T}B^{-1}.
  

Search aliases: B^-1

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• Use in source · Neuralese_Final_Research_Edition.tex, line 178

  Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exactly by conjugacy.

  Exact mathematical passage

  
   M_g=BP_gB^{-1},\qquad G=B^{-\mathsf T}B^{-1}.
  

Search aliases: b^k-i

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• Use in source · OSL_Final_Research_Edition.tex, line 83

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  
   \norm{B_wT-TA_w}\le\sum_{i=1}^{k} b^{k-i}\varepsilon_i a^{i-1}.
  

Search aliases: B_g

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• Use in source · OSL_Final_Research_Edition.tex, line 61

  Operator transport and compositional semantics. Exact transport. Let A_g:X→ X, B_g:Y→ Y, and T:X→ Y be linear maps. We use column vectors and execute the rightmost map first. The compatibility equation is B_gT=TA_g. When 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.

  Exact mathematical passage

  B_g:Y\to Y

Search aliases: B_g_i

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• Use in source · OSL_Final_Research_Edition.tex, line 78

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  E_i=B_{g_i}T-TA_{g_i}

Search aliases: B_g_i+1

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• Use in source · OSL_Final_Research_Edition.tex, line 79

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  
   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}.
  

Search aliases: B_g_k

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• Use in source · OSL_Final_Research_Edition.tex, line 79

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  
   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}.
  

Search aliases: B_w

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• Use in source · OSL_Final_Research_Edition.tex, line 68

  [Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No 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 project…

  Exact mathematical passage

  B_w

Search aliases: bar F bar

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• Use in source · OSL_Final_Research_Edition.tex, line 258

  Let the declared machine be M=(S, A,F,O). The certificate contains a surjection q:S→ Q, quotient transitions bar F_a, and outputs bar O, subject to q(F_a(s))=bar F_a(q(s)), O(s)=bar O(q(s)) (s∈ S, a∈ A). The 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 …

  Exact mathematical passage

  \bar F_a

Search aliases: bar F_a bar

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• Use in source · OSL_Final_Research_Edition.tex, line 258

  Let the declared machine be M=(S, A,F,O). The certificate contains a surjection q:S→ Q, quotient transitions bar F_a, and outputs bar O, subject to q(F_a(s))=bar F_a(q(s)), O(s)=bar O(q(s)) (s∈ S, a∈ A). The 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 …

  Exact mathematical passage

  \bar F_a

Search aliases: bar F_w bar

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• Use in source · OSL_Final_Research_Edition.tex, line 269

  ntheorem[Finite semantic checkpoint preservation] If [source reference] 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. Induction 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 samp…

  Exact mathematical passage

  q(F_w(s))=\bar F_w(q(s))

Search aliases: bar O bar

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• Use in source · Neuralese_Final_Research_Edition.tex, line 50

  subsectionA task-relative realization Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the present label establishes neither condit…

  Exact mathematical passage

  \label{eq:semantics}
   q\circ F_a=G_a\circ q,
   \qquad O=\bar O\circ q.
  

Search aliases: C

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• Use in source · Neuralese_Final_Research_Edition.tex, line 112

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  Cz_t

Search aliases: C mathcal

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• Use in source · OSL_Final_Research_Edition.tex, line 104

  The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and 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.

  Exact mathematical passage

  
   S=(\Gamma,V,\mathcal O,\mathcal P,\mathcal Q,\mathcal M,\mathcal C,\mathcal A,\mathcal T),
  

Search aliases: c

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• Use in source · Neuralese_Final_Research_Edition.tex, line 143

  Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact.

  Exact mathematical passage

  z\sim cz

Search aliases: content mathrm

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• Use in source · OSL_Final_Research_Edition.tex, line 51

  A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The 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.

  Exact mathematical passage

  
   p=(\mathrm{id},\mathrm{content},\mathrm{origin},\mathrm{access},\mathrm{status},\mathrm{scope},\mathrm{evidence}).
  

Search aliases: D

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• Use in source · Neuralese_Final_Research_Edition.tex, line 57

  [Semantic continuation] If [source reference] holds on a forward-invariant domain D⊆ Z, then O(F_w(z))=bar O(G_w(q(z))) for every z∈ D and every finite w. Forward 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. Without forward invariance, a one-step certificate may leave its own scope. Without observation factorization, the reduced state may preserve transitions but omi…

  Exact mathematical passage

  D\subseteq Z

Search aliases: d

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• Use in source · Neuralese_Final_Research_Edition.tex, line 71

  An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, 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.

  Exact mathematical passage

  d(z,z')\le\varepsilon

Search aliases: diag

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• Use in source · Neuralese_Final_Research_Edition.tex, line 217

  Curvature can be invisible to the task. Conversely, 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.

  Exact mathematical passage

  U_g=\diag(P_g,V_g)

Search aliases: dim

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• Use in source · Neuralese_Final_Research_Edition.tex, line 209

  [Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality. If K is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-k…

  Exact mathematical passage

  \dim U(K)=\dim K

Search aliases: E

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• Use in source · Neuralese_Final_Research_Edition.tex, line 192

  Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When 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 o…

  Exact mathematical passage

  QM_a=A_aQ+E_a

Search aliases: e

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• Use in source · Neuralese_Final_Research_Edition.tex, line 193

  Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When 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 o…

  Exact mathematical passage

  
   e_{t+1}\le a e_t+\varepsilon R.
  

Search aliases: e_0

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• Use in source · Neuralese_Final_Research_Edition.tex, line 197

  Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When 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 o…

  Exact mathematical passage

  
   e_t\le a^t e_0+\varepsilon R\sum_{j=0}^{t-1}a^j.
  

Search aliases: E_a

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• Use in source · Neuralese_Final_Research_Edition.tex, line 192

  Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When 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 o…

  Exact mathematical passage

  QM_a=A_aQ+E_a

Search aliases: E_i

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• Use in source · OSL_Final_Research_Edition.tex, line 78

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  E_i=B_{g_i}T-TA_{g_i}

Search aliases: e_t

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• Use in source · Neuralese_Final_Research_Edition.tex, line 193

  Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When 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 o…

  Exact mathematical passage

  
   e_{t+1}\le a e_t+\varepsilon R.
  

Search aliases: e_t+1

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• Use in source · Neuralese_Final_Research_Edition.tex, line 193

  Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When 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 o…

  Exact mathematical passage

  
   e_{t+1}\le a e_t+\varepsilon R.
  

Search aliases: evidence mathrm

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• Use in source · OSL_Final_Research_Edition.tex, line 51

  A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The 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.

  Exact mathematical passage

  
   p=(\mathrm{id},\mathrm{content},\mathrm{origin},\mathrm{access},\mathrm{status},\mathrm{scope},\mathrm{evidence}).
  

Search aliases: F

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: f

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• Use in source · OSL_Final_Research_Edition.tex, line 275

  The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The 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.

  Exact mathematical passage

  
   r(a,b,u)=(a+1,b,u),\quad f(a,b,u)=(-a,1-b,u),\quad
   v(a,b,u)=(a,b,1-u).
  

Search aliases: f^2

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• Use in source · OSL_Final_Research_Edition.tex, line 279

  The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The 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.

  Exact mathematical passage

  r^4=f^2=frfr=1

Search aliases: F_a

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• Use in source · Neuralese_Final_Research_Edition.tex, line 45

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  F_a:Z\to Z

Search aliases: F_a_1

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• Use in source · Neuralese_Final_Research_Edition.tex, line 46

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  
   F_w=F_{a_k}\circ\cdots\circ F_{a_1}.
  

Search aliases: F_a_k

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• Use in source · Neuralese_Final_Research_Edition.tex, line 46

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  
   F_w=F_{a_k}\circ\cdots\circ F_{a_1}.
  

Search aliases: F_M

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• Use in source · Neuralese_Final_Research_Edition.tex, line 144

  Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact.

  Exact mathematical passage

  
   F_M(z)=\frac{Mz}{\norm{Mz}},
  

Search aliases: F_w

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• Use in source · Neuralese_Final_Research_Edition.tex, line 46

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  
   F_w=F_{a_k}\circ\cdots\circ F_{a_1}.
  

Search aliases: F_θ theta

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: G

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• Use in source · Neuralese_Final_Research_Edition.tex, line 49

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  G_a

Search aliases: G'

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• Use in source · OSL_Final_Research_Edition.tex, line 97

  Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-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…

  Exact mathematical passage

  G'=T^{-\mathsf T}GT^{-1}

Search aliases: g_1

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• Use in source · OSL_Final_Research_Edition.tex, line 68

  [Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No 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 project…

  Exact mathematical passage

  w=g_1\cdots g_k

Search aliases: G_a

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• Use in source · Neuralese_Final_Research_Edition.tex, line 49

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  G_a

Search aliases: g_i

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• Use in source · OSL_Final_Research_Edition.tex, line 78

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  E_i=B_{g_i}T-TA_{g_i}

Search aliases: g_i+1

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• Use in source · OSL_Final_Research_Edition.tex, line 79

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  
   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}.
  

Search aliases: g_i-1

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• Use in source · OSL_Final_Research_Edition.tex, line 79

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  
   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}.
  

Search aliases: g_k

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• Use in source · OSL_Final_Research_Edition.tex, line 68

  [Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No 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 project…

  Exact mathematical passage

  w=g_1\cdots g_k

Search aliases: G_W

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• Use in source · OSL_Final_Research_Edition.tex, line 175

  Bounded discovery and semantic admission. The Discovery Bridge supplies a missing distinction between a well-formed proposal and 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. [source reference]

  Exact mathematical passage

  G_W

Search aliases: G_w

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• Use in source · Neuralese_Final_Research_Edition.tex, line 58

  [Semantic continuation] If [source reference] holds on a forward-invariant domain D⊆ Z, then O(F_w(z))=bar O(G_w(q(z))) for every z∈ D and every finite w. Forward 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. Without forward invariance, a one-step certificate may leave its own scope. Without observation factorization, the reduced state may preserve transitions but omi…

  Exact mathematical passage

  
   O(F_w(z))=\bar O(G_w(q(z)))
   \quad\text{for every }z\in D\text{ and every finite }w.
  

Search aliases: G_W^j

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• Use in source · OSL_Final_Research_Edition.tex, line 179

  Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The 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 …

  Exact mathematical passage

  
   \mathcal P_{W,d}=\bigcup_{j=0}^{d}G_W^j,
   \qquad \sigma_W(p)=\bigl(p(s)\bigr)_{s\in S_W}.
  

Search aliases: H

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• Use in source · Neuralese_Final_Research_Edition.tex, line 80

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  
   z\sim_H z'\iff O(F_w(z))=O(F_w(z'))\text{ for every }|w|\le H.
  

Search aliases: h

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• Use in source · Neuralese_Final_Research_Edition.tex, line 106

  Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_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.

  Exact mathematical passage

  O_h(z)=\arg\max_j \ell_j(z)

Search aliases: I

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• Use in source · Neuralese_Final_Research_Edition.tex, line 75

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  I\subseteq Z

Search aliases: i

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• Use in source · Neuralese_Final_Research_Edition.tex, line 168

  Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A 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 unt…

  Exact mathematical passage

  i

Search aliases: id mathrm

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• Use in source · OSL_Final_Research_Edition.tex, line 51

  A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The 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.

  Exact mathematical passage

  
   p=(\mathrm{id},\mathrm{content},\mathrm{origin},\mathrm{access},\mathrm{status},\mathrm{scope},\mathrm{evidence}).
  

Search aliases: iff

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• Use in source · Neuralese_Final_Research_Edition.tex, line 80

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  
   z\sim_H z'\iff O(F_w(z))=O(F_w(z'))\text{ for every }|w|\le H.
  

Search aliases: j

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• Use in source · Neuralese_Final_Research_Edition.tex, line 106

  Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_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.

  Exact mathematical passage

  O_h(z)=\arg\max_j \ell_j(z)

Search aliases: K

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• Use in source · Neuralese_Final_Research_Edition.tex, line 206

  [Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality. If K is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-k…

  Exact mathematical passage

  K\subseteq V

Search aliases: k

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• Use in source · Neuralese_Final_Research_Edition.tex, line 45

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  w=a_1\cdots a_k

Search aliases: K^⊥ perp

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• Use in source · Neuralese_Final_Research_Edition.tex, line 206

  [Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality. If K is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-k…

  Exact mathematical passage

  U(K^\perp)=K^\perp

Search aliases: ker

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• Use in source · Neuralese_Final_Research_Edition.tex, line 113

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  
   N_H=\bigcap_{|w|\le H}\ker(CM_w),\qquad M_\varnothing=I.
  

Search aliases: L

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• Use in source · Neuralese_Final_Research_Edition.tex, line 76

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  
   R_{\le L}=\{F_w(z):z\in I,\ |w|\le L\}.
  

Search aliases: L'

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• Use in source · OSL_Final_Research_Edition.tex, line 110

  A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates 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.

  Exact mathematical passage

  
   \langle c,S,L\rangle\longrightarrow\langle S',L'\rangle
  

Search aliases: M

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• Use in source · Neuralese_Final_Research_Edition.tex, line 112

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  z_{t+1}=M_{a_t}z_t

Search aliases: M mathcal

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• Use in source · OSL_Final_Research_Edition.tex, line 104

  The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and 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.

  Exact mathematical passage

  
   S=(\Gamma,V,\mathcal O,\mathcal P,\mathcal Q,\mathcal M,\mathcal C,\mathcal A,\mathcal T),
  

Search aliases: m

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• Use in source · Neuralese_Final_Research_Edition.tex, line 168

  Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A 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 unt…

  Exact mathematical passage

  m=\ell_i-\max_{j\ne i}\ell_j>0

Search aliases: M_a

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• Use in source · Neuralese_Final_Research_Edition.tex, line 126

  imensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility 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⊆ N_H. Since N_H⊆ker C, induction makes every longer word invisible. Each strict inclusion decreases a finite nonnegative dimension. The result …

  Exact mathematical passage

  M_aN_H\subseteq N_H

Search aliases: M_a^-1

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• Use in source · Neuralese_Final_Research_Edition.tex, line 120

  [Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility 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_…

  Exact mathematical passage

  
   N_{H+1}=N_H\cap\bigcap_{a\in\mathcal A}M_a^{-1}N_H.
  

Search aliases: M_a_t

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• Use in source · Neuralese_Final_Research_Edition.tex, line 112

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  z_{t+1}=M_{a_t}z_t

Search aliases: M_g

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• Use in source · Neuralese_Final_Research_Edition.tex, line 178

  Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exactly by conjugacy.

  Exact mathematical passage

  
   M_g=BP_gB^{-1},\qquad G=B^{-\mathsf T}B^{-1}.
  

Search aliases: M_g^ T mathsf

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• Use in source · Neuralese_Final_Research_Edition.tex, line 181

  Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exactly by conjugacy.

  Exact mathematical passage

  M_g^{\mathsf T}GM_g=G

Search aliases: M_w

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• Use in source · Neuralese_Final_Research_Edition.tex, line 113

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  
   N_H=\bigcap_{|w|\le H}\ker(CM_w),\qquad M_\varnothing=I.
  

Search aliases: M_∅ varnothing

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• Use in source · Neuralese_Final_Research_Edition.tex, line 113

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  
   N_H=\bigcap_{|w|\le H}\ker(CM_w),\qquad M_\varnothing=I.
  

Search aliases: max

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• Use in source · Neuralese_Final_Research_Edition.tex, line 106

  Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_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.

  Exact mathematical passage

  O_h(z)=\arg\max_j \ell_j(z)

Search aliases: max_j max

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• Use in source · Neuralese_Final_Research_Edition.tex, line 106

  Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_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.

  Exact mathematical passage

  O_h(z)=\arg\max_j \ell_j(z)

Search aliases: max_j≠ i max ne

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• Use in source · Neuralese_Final_Research_Edition.tex, line 168

  Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A 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 unt…

  Exact mathematical passage

  m=\ell_i-\max_{j\ne i}\ell_j>0

Search aliases: N

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• Use in source · Neuralese_Final_Research_Edition.tex, line 113

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  
   N_H=\bigcap_{|w|\le H}\ker(CM_w),\qquad M_\varnothing=I.
  

Search aliases: N_H

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• Use in source · Neuralese_Final_Research_Edition.tex, line 113

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  
   N_H=\bigcap_{|w|\le H}\ker(CM_w),\qquad M_\varnothing=I.
  

Search aliases: N_H+1

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• Use in source · Neuralese_Final_Research_Edition.tex, line 120

  [Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility 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_…

  Exact mathematical passage

  
   N_{H+1}=N_H\cap\bigcap_{a\in\mathcal A}M_a^{-1}N_H.
  

Search aliases: N_∞ infty

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• Use in source · Neuralese_Final_Research_Edition.tex, line 123

  [Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility 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_…

  Exact mathematical passage

  N_\infty

Search aliases: norm

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• Use in source · Neuralese_Final_Research_Edition.tex, line 144

  Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact.

  Exact mathematical passage

  
   F_M(z)=\frac{Mz}{\norm{Mz}},
  

Search aliases: O mathcal

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• Use in source · OSL_Final_Research_Edition.tex, line 104

  The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and 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.

  Exact mathematical passage

  
   S=(\Gamma,V,\mathcal O,\mathcal P,\mathcal Q,\mathcal M,\mathcal C,\mathcal A,\mathcal T),
  

Search aliases: O_h

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• Use in source · Neuralese_Final_Research_Edition.tex, line 106

  Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_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.

  Exact mathematical passage

  O_h(z)=\arg\max_j \ell_j(z)

Search aliases: O_θ theta

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: origin mathrm

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• Use in source · OSL_Final_Research_Edition.tex, line 51

  A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The 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.

  Exact mathematical passage

  
   p=(\mathrm{id},\mathrm{content},\mathrm{origin},\mathrm{access},\mathrm{status},\mathrm{scope},\mathrm{evidence}).
  

Search aliases: P

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• Use in source · Neuralese_Final_Research_Edition.tex, line 177

  Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exactly by conjugacy.

  Exact mathematical passage

  P_g

Search aliases: P mathcal

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• Use in source · OSL_Final_Research_Edition.tex, line 104

  The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and 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.

  Exact mathematical passage

  
   S=(\Gamma,V,\mathcal O,\mathcal P,\mathcal Q,\mathcal M,\mathcal C,\mathcal A,\mathcal T),
  

Search aliases: p

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• Use in source · OSL_Final_Research_Edition.tex, line 51

  A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The 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.

  Exact mathematical passage

  
   p=(\mathrm{id},\mathrm{content},\mathrm{origin},\mathrm{access},\mathrm{status},\mathrm{scope},\mathrm{evidence}).
  

Search aliases: P'

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• Use in source · OSL_Final_Research_Edition.tex, line 93

  Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-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…

  Exact mathematical passage

  P'=TPT^{-1}

Search aliases: P^2

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• Use in source · OSL_Final_Research_Edition.tex, line 93

  Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-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…

  Exact mathematical passage

  P^2=P

Search aliases: P_g

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• Use in source · Neuralese_Final_Research_Edition.tex, line 177

  Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exactly by conjugacy.

  Exact mathematical passage

  P_g

Search aliases: P_g^ T mathsf

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• Use in source · Neuralese_Final_Research_Edition.tex, line 187

  roposition[Exact finite realization] The 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. The anchor equation follows by cancellation of B^-1B. For the metric identity, substitute the definitions and use P_g^ TP_g=I. Products telescope to BP_wB^-1, so semantic relations transfer. This is a constructive control, not an inference that a learned encoder disc…

  Exact mathematical passage

  P_g^{\mathsf T}P_g=I

Search aliases: P_w

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• Use in source · Neuralese_Final_Research_Edition.tex, line 187

  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. The anchor equation follows by cancellation of B^-1B. For the metric identity, substitute the definitions and use P_g^ TP_g=I. Products telescope to BP_wB^-1, so semantic relations transfer. This is a constructive control, not an inference that a learned encoder discovered the same realization. If the true permut…

  Exact mathematical passage

  BP_wB^{-1}

Search aliases: P_W,d mathcal

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• Use in source · OSL_Final_Research_Edition.tex, line 179

  Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The 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 …

  Exact mathematical passage

  
   \mathcal P_{W,d}=\bigcup_{j=0}^{d}G_W^j,
   \qquad \sigma_W(p)=\bigl(p(s)\bigr)_{s\in S_W}.
  

Search aliases: Q

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• Use in source · Neuralese_Final_Research_Edition.tex, line 84

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  \label{eq:surface}
   Q_{L,H}=R_{\le L}/\!\sim_H.
  

Search aliases: Q mathcal

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• Use in source · OSL_Final_Research_Edition.tex, line 104

  The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and 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.

  Exact mathematical passage

  
   S=(\Gamma,V,\mathcal O,\mathcal P,\mathcal Q,\mathcal M,\mathcal C,\mathcal A,\mathcal T),
  

Search aliases: q

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: Q_L+1,H

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• Use in source · Neuralese_Final_Research_Edition.tex, line 91

  [Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ 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)∼_HF_a(z'). Both successor states are in R_≤ L+1. The horizon decreases alo…

  Exact mathematical passage

  
   Q_{L,H+1}\longrightarrow Q_{L+1,H},\qquad [z]\longmapsto[F_a(z)].
  

Search aliases: Q_L,H

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• Use in source · Neuralese_Final_Research_Edition.tex, line 84

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  \label{eq:surface}
   Q_{L,H}=R_{\le L}/\!\sim_H.
  

Search aliases: Q_L,H+1

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• Use in source · Neuralese_Final_Research_Edition.tex, line 91

  [Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ 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)∼_HF_a(z'). Both successor states are in R_≤ L+1. The horizon decreases alo…

  Exact mathematical passage

  
   Q_{L,H+1}\longrightarrow Q_{L+1,H},\qquad [z]\longmapsto[F_a(z)].
  

Search aliases: q_t

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: R

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• Use in source · Neuralese_Final_Research_Edition.tex, line 76

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  
   R_{\le L}=\{F_w(z):z\in I,\ |w|\le L\}.
  

Search aliases: r

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• Use in source · OSL_Final_Research_Edition.tex, line 275

  The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The 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.

  Exact mathematical passage

  
   r(a,b,u)=(a+1,b,u),\quad f(a,b,u)=(-a,1-b,u),\quad
   v(a,b,u)=(a,b,1-u).
  

Search aliases: r^4

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• Use in source · OSL_Final_Research_Edition.tex, line 279

  The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The 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.

  Exact mathematical passage

  r^4=f^2=frfr=1

Search aliases: R_ε varepsilon

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• Use in source · Neuralese_Final_Research_Edition.tex, line 162

  An exact norm-preserving failure. [Small phase error accumulates] Let the intended semantic operation be the identity and the learned implementation be a planar rotation R_ε. Every step preserves Euclidean norm exactly. After k steps the state has rotated by kε. With a readout determined by the sign of the first coordinate, a state starting at (1,0) eventually changes class for any fixed 0<ε<π/2. This counterexample separates numerical stability from correct composition without involving underfl…

  Exact mathematical passage

  R_\varepsilon

Search aliases: R_≤ L le

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• Use in source · Neuralese_Final_Research_Edition.tex, line 76

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  
   R_{\le L}=\{F_w(z):z\in I,\ |w|\le L\}.
  

Search aliases: R_≤ L+1 le

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• Use in source · Neuralese_Final_Research_Edition.tex, line 96

  dproposition The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ 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)∼_HF_a(z'). Both successor states are in R_≤ L+1. The horizon decreases along this map. A finite-horizon state cannot automatically be updated forever at the same horizon without additional closure inf…

  Exact mathematical passage

  R_{\le L+1}

Search aliases: S

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• Use in source · Neuralese_Final_Research_Edition.tex, line 49

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  S

Search aliases: s

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• Use in source · OSL_Final_Research_Edition.tex, line 179

  Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The 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 …

  Exact mathematical passage

  
   \mathcal P_{W,d}=\bigcup_{j=0}^{d}G_W^j,
   \qquad \sigma_W(p)=\bigl(p(s)\bigr)_{s\in S_W}.
  

Search aliases: S'

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• Use in source · OSL_Final_Research_Edition.tex, line 110

  A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates 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.

  Exact mathematical passage

  
   \langle c,S,L\rangle\longrightarrow\langle S',L'\rangle
  

Search aliases: s_i

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• Use in source · OSL_Final_Research_Edition.tex, line 183

  Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The 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 …

  Exact mathematical passage

  D=\{(s_i,t_i)\}

Search aliases: S_W

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• Use in source · OSL_Final_Research_Edition.tex, line 178

  Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The 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 …

  Exact mathematical passage

  S_W

Search aliases: scope mathrm

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• Use in source · OSL_Final_Research_Edition.tex, line 51

  A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The 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.

  Exact mathematical passage

  
   p=(\mathrm{id},\mathrm{content},\mathrm{origin},\mathrm{access},\mathrm{status},\mathrm{scope},\mathrm{evidence}).
  

Search aliases: status mathrm

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• Use in source · OSL_Final_Research_Edition.tex, line 51

  A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The 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.

  Exact mathematical passage

  
   p=(\mathrm{id},\mathrm{content},\mathrm{origin},\mathrm{access},\mathrm{status},\mathrm{scope},\mathrm{evidence}).
  

Search aliases: T

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• Use in source · OSL_Final_Research_Edition.tex, line 61

  Operator transport and compositional semantics. Exact transport. Let A_g:X→ X, B_g:Y→ Y, and T:X→ Y be linear maps. We use column vectors and execute the rightmost map first. The compatibility equation is B_gT=TA_g. When 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.

  Exact mathematical passage

  T:X\to Y

Search aliases: T mathcal

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• Use in source · OSL_Final_Research_Edition.tex, line 104

  The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and 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.

  Exact mathematical passage

  
   S=(\Gamma,V,\mathcal O,\mathcal P,\mathcal Q,\mathcal M,\mathcal C,\mathcal A,\mathcal T),
  

Search aliases: T mathsf

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• Use in source · Neuralese_Final_Research_Edition.tex, line 151

  [Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A 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.

  Exact mathematical passage

  
   DF_M(z)=\frac{(I-uu^{\mathsf T})M}{\norm{Mz}}.
  

Search aliases: t

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: T^- T mathsf

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• Use in source · OSL_Final_Research_Edition.tex, line 97

  Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-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…

  Exact mathematical passage

  G'=T^{-\mathsf T}GT^{-1}

Search aliases: T^-1

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• Use in source · OSL_Final_Research_Edition.tex, line 74

  word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No 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 i…

  Exact mathematical passage

  B_g=TA_gT^{-1}

Search aliases: t_i

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• Use in source · OSL_Final_Research_Edition.tex, line 183

  Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The 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 …

  Exact mathematical passage

  D=\{(s_i,t_i)\}

Search aliases: U

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• Use in source · Neuralese_Final_Research_Edition.tex, line 206

  [Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality. If K is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-k…

  Exact mathematical passage

  U:V\to V

Search aliases: u

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: u^ T mathsf

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• Use in source · Neuralese_Final_Research_Edition.tex, line 151

  [Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A 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.

  Exact mathematical passage

  
   DF_M(z)=\frac{(I-uu^{\mathsf T})M}{\norm{Mz}}.
  

Search aliases: U^-1

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• Use in source · Neuralese_Final_Research_Edition.tex, line 209

  belthm:isometry Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality. If K is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block in this …

  Exact mathematical passage

  U^{-1}(K)=K

Search aliases: U_g

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• Use in source · Neuralese_Final_Research_Edition.tex, line 217

  Curvature can be invisible to the task. Conversely, 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.

  Exact mathematical passage

  U_g=\diag(P_g,V_g)

Search aliases: u_t

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: V

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• Use in source · Neuralese_Final_Research_Edition.tex, line 206

  [Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality. If K is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-k…

  Exact mathematical passage

  V

Search aliases: v

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• Use in source · Neuralese_Final_Research_Edition.tex, line 156

  [Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A 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.

  Exact mathematical passage

  (I-vv^{\mathsf T})/\norm{x}

Search aliases: v^ T mathsf

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• Use in source · Neuralese_Final_Research_Edition.tex, line 156

  [Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A 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.

  Exact mathematical passage

  (I-vv^{\mathsf T})/\norm{x}

Search aliases: V_g

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• Use in source · Neuralese_Final_Research_Edition.tex, line 217

  Curvature can be invisible to the task. Conversely, 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.

  Exact mathematical passage

  U_g=\diag(P_g,V_g)

Search aliases: vdash

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• Use in source · OSL_Final_Research_Edition.tex, line 109

  A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates 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.

  Exact mathematical passage

  \Gamma\vdash e:\tau

Search aliases: W

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• Use in source · OSL_Final_Research_Edition.tex, line 175

  Bounded discovery and semantic admission. The Discovery Bridge supplies a missing distinction between a well-formed proposal and 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. [source reference]

  Exact mathematical passage

  W

Search aliases: w

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• Use in source · Neuralese_Final_Research_Edition.tex, line 45

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  w=a_1\cdots a_k

Search aliases: x

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• Use in source · Neuralese_Final_Research_Edition.tex, line 101

  A minimal delayed-observability counterexample. Let the concrete state be (x,y)∈{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.

  Exact mathematical passage

  (x,y)\in\{0,1\}^2

Search aliases: y

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: y_t

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: Z

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• Use in source · Neuralese_Final_Research_Edition.tex, line 45

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  Z

Search aliases: Z mathbb

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• Use in source · OSL_Final_Research_Edition.tex, line 274

  The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The 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.

  Exact mathematical passage

  a\in\mathbb Z/4\mathbb Z

Search aliases: z

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: z'

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• Use in source · Neuralese_Final_Research_Edition.tex, line 71

  An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, 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.

  Exact mathematical passage

  d(z,z')\le\varepsilon

Search aliases: z_t

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: z_t+1

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: |

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• Use in source · Neuralese_Final_Research_Edition.tex, line 76

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  
   R_{\le L}=\{F_w(z):z\in I,\ |w|\le L\}.
  

Search aliases: × times

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• Use in source · Neuralese_Final_Research_Edition.tex, line 319

  Observable rank after recurrence. The 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 2.90×10^-5 in absolute value, while p95 oriented loop angles reach roughly .72--.97 radians. [source reference]

  Exact mathematical passage

  2.90\times10^{-5}

Search aliases: Γ Gamma

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• Use in source · OSL_Final_Research_Edition.tex, line 104

  The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and 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.

  Exact mathematical passage

  
   S=(\Gamma,V,\mathcal O,\mathcal P,\mathcal Q,\mathcal M,\mathcal C,\mathcal A,\mathcal T),
  

Search aliases: Δ Delta

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• Use in source · Neuralese_Final_Research_Edition.tex, line 168

  Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A 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 unt…

  Exact mathematical passage

  \Delta\ell

Search aliases: ε varepsilon

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• Use in source · Neuralese_Final_Research_Edition.tex, line 71

  An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, 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.

  Exact mathematical passage

  d(z,z')\le\varepsilon

Search aliases: ε_i varepsilon

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• Use in source · OSL_Final_Research_Edition.tex, line 78

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  \norm{E_i}\le\varepsilon_i

Search aliases: θ theta

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• Use in source · Neuralese_Final_Research_Edition.tex, line 26

  A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where 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 experimenta…

  Exact mathematical passage

  \label{eq:latent}
   z_{t+1}=F_\theta(z_t,u_t),\qquad y_t=O_\theta(z_t,q_t),
  

Search aliases: π pi

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• Use in source · Neuralese_Final_Research_Edition.tex, line 162

  rror accumulates] Let the intended semantic operation be the identity and the learned implementation be a planar rotation R_ε. Every step preserves Euclidean norm exactly. After k steps the state has rotated by kε. With a readout determined by the sign of the first coordinate, a state starting at (1,0) eventually changes class for any fixed 0<ε<π/2. This counterexample separates numerical stability from correct composition without involving underflow, overflow, or a defective normalization imple…

  Exact mathematical passage

  0<\varepsilon<\pi/2

Search aliases: σ sigma

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• Use in source · OSL_Final_Research_Edition.tex, line 179

  Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The 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 …

  Exact mathematical passage

  
   \mathcal P_{W,d}=\bigcup_{j=0}^{d}G_W^j,
   \qquad \sigma_W(p)=\bigl(p(s)\bigr)_{s\in S_W}.
  

Search aliases: σ_W sigma

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• Use in source · OSL_Final_Research_Edition.tex, line 179

  Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The 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 …

  Exact mathematical passage

  
   \mathcal P_{W,d}=\bigcup_{j=0}^{d}G_W^j,
   \qquad \sigma_W(p)=\bigl(p(s)\bigr)_{s\in S_W}.
  

Search aliases: τ tau

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• Use in source · OSL_Final_Research_Edition.tex, line 109

  A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates 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.

  Exact mathematical passage

  \Gamma\vdash e:\tau

Search aliases: ℓ ell

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• Use in source · Neuralese_Final_Research_Edition.tex, line 106

  Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_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.

  Exact mathematical passage

  O_h(z)=\arg\max_j \ell_j(z)

Search aliases: ℓ_i ell

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• Use in source · Neuralese_Final_Research_Edition.tex, line 168

  Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A 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 unt…

  Exact mathematical passage

  m=\ell_i-\max_{j\ne i}\ell_j>0

Search aliases: ℓ_j ell

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• Use in source · Neuralese_Final_Research_Edition.tex, line 106

  Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_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.

  Exact mathematical passage

  O_h(z)=\arg\max_j \ell_j(z)

Search aliases: ↔ leftrightarrow

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• Use in source · Neuralese_Final_Research_Edition.tex, line 355

  From latent computation to an explicit language. The intended interface. The working architecture separates three timescales: fast latent computation ↔ typed semantic checkpoint ↔ persistent evidence ledger. A 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, n…

  Exact mathematical passage

  
  \text{fast latent computation}\ \leftrightarrow\ \text{typed semantic checkpoint}\ \leftrightarrow\ \text{persistent evidence ledger}.
  

Search aliases: ↦ mapsto

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• Use in source · Neuralese_Final_Research_Edition.tex, line 156

  [Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A 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.

  Exact mathematical passage

  x\mapsto x/\norm{x}

Search aliases: ∅ varnothing

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• Use in source · Neuralese_Final_Research_Edition.tex, line 113

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  
   N_H=\bigcap_{|w|\le H}\ker(CM_w),\qquad M_\varnothing=I.
  

Search aliases: ∑ sum

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• Use in source · Neuralese_Final_Research_Edition.tex, line 197

  Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When 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 o…

  Exact mathematical passage

  
   e_t\le a^t e_0+\varepsilon R\sum_{j=0}^{t-1}a^j.
  

Search aliases: ∑_i=1^k sum

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• Use in source · OSL_Final_Research_Edition.tex, line 79

  Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert 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. For a=b=1 and ε_i≤ε, the bound is kε, not ε. A small o…

  Exact mathematical passage

  
   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}.
  

Search aliases: ∑_j=0^t-1 sum

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• Use in source · Neuralese_Final_Research_Edition.tex, line 197

  Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When 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 o…

  Exact mathematical passage

  
   e_t\le a^t e_0+\varepsilon R\sum_{j=0}^{t-1}a^j.
  

Search aliases: ∘ circ

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• Use in source · Neuralese_Final_Research_Edition.tex, line 46

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  
   F_w=F_{a_k}\circ\cdots\circ F_{a_1}.
  

Search aliases: ∞ infty

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• Use in source · Neuralese_Final_Research_Edition.tex, line 123

  [Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility 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_…

  Exact mathematical passage

  N_\infty

Search aliases: ∩ cap

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• Use in source · Neuralese_Final_Research_Edition.tex, line 120

  [Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility 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_…

  Exact mathematical passage

  
   N_{H+1}=N_H\cap\bigcap_{a\in\mathcal A}M_a^{-1}N_H.
  

Search aliases: ∼ sim

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• Use in source · Neuralese_Final_Research_Edition.tex, line 80

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  
   z\sim_H z'\iff O(F_w(z))=O(F_w(z'))\text{ for every }|w|\le H.
  

Search aliases: ∼_H sim

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• Use in source · Neuralese_Final_Research_Edition.tex, line 80

  The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The 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…

  Exact mathematical passage

  
   z\sim_H z'\iff O(F_w(z))=O(F_w(z'))\text{ for every }|w|\le H.
  

Search aliases: ∼_H+1 sim

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• Use in source · Neuralese_Final_Research_Edition.tex, line 96

  nproposition[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ 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)∼_HF_a(z'). Both successor states are in R_≤ L+1. The horizon d…

  Exact mathematical passage

  z\sim_{H+1}z'

Search aliases: ≈ approx

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• Use in source · Neuralese_Final_Research_Edition.tex, line 269

  Norm ratio at 2048 · 1.0000 · 1.0000 · ≈0 · ≈0

  Exact mathematical passage

  \approx0

Search aliases: ≠ ne

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• Use in source · Neuralese_Final_Research_Edition.tex, line 147

  Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact.

  Exact mathematical passage

  Mz\ne0

Search aliases: ≤ le

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• Use in source · Neuralese_Final_Research_Edition.tex, line 71

  An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, 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.

  Exact mathematical passage

  d(z,z')\le\varepsilon

Search aliases: ⊆ subseteq

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• Use in source · Neuralese_Final_Research_Edition.tex, line 57

  [Semantic continuation] If [source reference] holds on a forward-invariant domain D⊆ Z, then O(F_w(z))=bar O(G_w(q(z))) for every z∈ D and every finite w. Forward 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. Without forward invariance, a one-step certificate may leave its own scope. Without observation factorization, the reduced state may preserve transitions but omi…

  Exact mathematical passage

  D\subseteq Z

Search aliases: ⊕ oplus

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• Use in source · Neuralese_Final_Research_Edition.tex, line 206

  [Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality. If K is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-k…

  Exact mathematical passage

  V=K^\perp\oplus K

Search aliases: ⊥ perp

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• Use in source · Neuralese_Final_Research_Edition.tex, line 206

  [Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality. If K is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-k…

  Exact mathematical passage

  U(K^\perp)=K^\perp

Search aliases: ⋂ bigcap

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• Use in source · Neuralese_Final_Research_Edition.tex, line 113

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  
   N_H=\bigcap_{|w|\le H}\ker(CM_w),\qquad M_\varnothing=I.
  

Search aliases: ⋂_a∈ A bigcap in mathcal

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• Use in source · Neuralese_Final_Research_Edition.tex, line 120

  [Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility 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_…

  Exact mathematical passage

  
   N_{H+1}=N_H\cap\bigcap_{a\in\mathcal A}M_a^{-1}N_H.
  

Search aliases: ⋂_|w|≤ H bigcap le

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• Use in source · Neuralese_Final_Research_Edition.tex, line 113

  Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.

  Exact mathematical passage

  
   N_H=\bigcap_{|w|\le H}\ker(CM_w),\qquad M_\varnothing=I.
  

Search aliases: ⋃ bigcup

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• Use in source · OSL_Final_Research_Edition.tex, line 179

  Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The 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 …

  Exact mathematical passage

  
   \mathcal P_{W,d}=\bigcup_{j=0}^{d}G_W^j,
   \qquad \sigma_W(p)=\bigl(p(s)\bigr)_{s\in S_W}.
  

Search aliases: ⋃_j=0^d bigcup

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• Use in source · OSL_Final_Research_Edition.tex, line 179

  Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The 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 …

  Exact mathematical passage

  
   \mathcal P_{W,d}=\bigcup_{j=0}^{d}G_W^j,
   \qquad \sigma_W(p)=\bigl(p(s)\bigr)_{s\in S_W}.
  

Search aliases: ⋯ cdots

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• Use in source · Neuralese_Final_Research_Edition.tex, line 45

  Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that semantics is both operation-compatible and sufficient for the declared outputs. A decoder that predicts only the pre…

  Exact mathematical passage

  w=a_1\cdots a_k

Search aliases: ⟨ langle

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• Use in source · Neuralese_Final_Research_Edition.tex, line 210

  inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality. If K is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block in this orthogonal decomposition. A nonorthogonal coor…

  Exact mathematical passage

  \langle Ux,k\rangle=\langle x,U^{-1}k\rangle=0

Search aliases: ⟩ rangle

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• Use in source · Neuralese_Final_Research_Edition.tex, line 210

  inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality. If K is the fixed kernel of a quotient, the theorem excludes a nonzero visible-to-kernel block in this orthogonal decomposition. A nonorthogonal coor…

  Exact mathematical passage

  \langle Ux,k\rangle=\langle x,U^{-1}k\rangle=0

Search aliases: ⟶ longrightarrow

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• Use in source · Neuralese_Final_Research_Edition.tex, line 91

  [Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ 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)∼_HF_a(z'). Both successor states are in R_≤ L+1. The horizon decreases alo…

  Exact mathematical passage

  
   Q_{L,H+1}\longrightarrow Q_{L+1,H},\qquad [z]\longmapsto[F_a(z)].
  

Search aliases: ⟼ longmapsto

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• Use in source · Neuralese_Final_Research_Edition.tex, line 91

  [Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ 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)∼_HF_a(z'). Both successor states are in R_≤ L+1. The horizon decreases alo…

  Exact mathematical passage

  
   Q_{L,H+1}\longrightarrow Q_{L+1,H},\qquad [z]\longmapsto[F_a(z)].