Large Signal Model software

Search aliases: =

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: A

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 184

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  \mathfrak U=\{U_a\}_{a\in A}

Search aliases: a

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 184

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  \mathfrak U=\{U_a\}_{a\in A}

Search aliases: B mathfrak

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: C mathfrak

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: E

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 185

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  \mathfrak E=\{E_{a,\alpha}\}

Search aliases: E mathfrak

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: E_a,α alpha

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 185

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  \mathfrak E=\{E_{a,\alpha}\}

Search aliases: G mathfrak

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: H mathfrak

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: L mathfrak

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: P

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 186

  model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memory state; item R is the state-act…

  Exact mathematical passage

  \mathfrak P=\{P_{a,\alpha}\}

Search aliases: P mathfrak

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: P_a,α alpha

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 186

  model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memory state; item R is the state-act…

  Exact mathematical passage

  \mathfrak P=\{P_{a,\alpha}\}

Search aliases: R mathfrak

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: U

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 184

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  \mathfrak U=\{U_a\}_{a\in A}

Search aliases: U mathfrak

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: U_a

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 184

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  \mathfrak U=\{U_a\}_{a\in A}

Search aliases: X

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: Z mathfrak

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 176

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  
  \mathfrak L=
  (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
  \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
  

Search aliases: α alpha

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 185

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  \mathfrak E=\{E_{a,\alpha}\}

Search aliases: Λ Lambda

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• Definition passage: Complete pruning · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 455

  [Complete pruning] A set Λ of packet-tree nodes is a complete pruning when every root-to-leaf path meets Λ exactly once.

  Exact mathematical passage

  \Lambda

Search aliases: ∈ in

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• Definition passage: Projector-resolved large signal model · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 184

  [Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed operator-program grammar; item B is the Bayesian posterior over programs and parameters; item H is the explicit memo…

  Exact mathematical passage

  \mathfrak U=\{U_a\}_{a\in A}

Search aliases: *

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 217

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  \boxed{P_{a,\alpha}=U_a^*E_{a,\alpha}U_a.}
  

Search aliases: +

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 318

  For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.

  Exact mathematical passage

  
  (Fx)_{k,\ell}
  =\frac{1}{\sqrt{HW}}
  \sum_{m=0}^{H-1}\sum_{n=0}^{W-1}
   x_{m,n}e^{-2\pi i(km/H+\ell n/W)}.
  

Search aliases: 1 mathbf

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 359

  For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].

  Exact mathematical passage

  
  (P_{N,r,s}x)[m,n]
  =
  \mathbf 1_{m\equiv r\ (N)}
  \mathbf 1_{n\equiv s\ (N)}x[m,n].
  

Search aliases: 1_changed mathbf mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: 1_loss mathbf mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: 1_m≡ r (N) mathbf equiv

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 359

  For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].

  Exact mathematical passage

  
  (P_{N,r,s}x)[m,n]
  =
  \mathbf 1_{m\equiv r\ (N)}
  \mathbf 1_{n\equiv s\ (N)}x[m,n].
  

Search aliases: 1_n≡ s (N) mathbf equiv

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 359

  For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].

  Exact mathematical passage

  
  (P_{N,r,s}x)[m,n]
  =
  \mathbf 1_{m\equiv r\ (N)}
  \mathbf 1_{n\equiv s\ (N)}x[m,n].
  

Search aliases: 1_untried mathbf mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: 1_win mathbf mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: 1_x[m,n]=j mathbf

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 375

  A finite-valued frame can be encoded by the one-hot coordinates C_j[m,n]= 1_x[m,n]=j. The original frame is recovered by the unique active color at each address. This reconstruction is categorical rather than linear over the original scalar encoding, so it is kept as a separately typed exact atlas.

  Exact mathematical passage

  
  C_j[m,n]=\mathbf 1_{x[m,n]=j}.
  

Search aliases: a_0

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1115

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  \beta\mid\sigma^2\sim N(0,\sigma^2\Lambda_0^{-1}),
  \qquad
  \sigma^2\sim\operatorname{InvGamma}(a_0,b_0).
  

Search aliases: a_n

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1126

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  a_n=a_0+\frac n2,
  \qquad
  b_n=b_0+\frac12(y^\top y-\mu_n^\top\Lambda_n\mu_n).
  

Search aliases: Acal A mathcal

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 247

  Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.

  Exact mathematical passage

  
  \Acal x=(w_aU_ax)_{a\in A},
  

Search aliases: Acal_N A_N Acal mathcal

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1093

  Polyphase analysis and synthesis. For modulus N, ( A_Nx)_r,s[q,t]=x[Nq+r,Nt+s]. The synthesis operator places each channel value back at its unique original address. Therefore S_N A_N=I.

  Exact mathematical passage

  
  (\Acal_Nx)_{r,s}[q,t]=x[Nq+r,Nt+s].
  

Search aliases: appendix

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1061

  [4pt] → memory and transition graph → active action → residual → selective refinement → exact reconstruction Within its finite declared scope, that loop is complete. appendix Exact transform formulas. Haar analysis and synthesis. For one pair, ℓ=fraca+bsqrt2, h=fraca-bsqrt2, and a=fracℓ+hsqrt2, b=fracℓ-hsqrt2. Applying these identities separably and recursively proves the finite two-dimensional transform used in the code.

  Exact mathematical passage

  4pt]
  $\to$
  \textbf{memory and transition graph}
  $\to$
  \textbf{active action}
  $\to$
  \textbf{residual}
  $\to$
  \textbf{selective refinement}
  $\to$
  \textbf{exact reconstruction}
  \end{minipage}}
  \end{center}
  Within its finite declared scope, that loop is complete.
  
  \appendix
  \clearpage
  \section{Exact transform formulas}
  
  \subsection{Haar analysis and synthesis}
  For one pair,
  \[
  \ell=\frac{a+b}{\sqrt2},\qquad h=\frac{a-b}{\sqrt2},
  

Search aliases: B

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: b

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: b_0

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1115

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  \beta\mid\sigma^2\sim N(0,\sigma^2\Lambda_0^{-1}),
  \qquad
  \sigma^2\sim\operatorname{InvGamma}(a_0,b_0).
  

Search aliases: B_g^top top

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 599

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  
  Y=X_gB_g^\top+\varepsilon,
  

Search aliases: b_n

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1126

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  a_n=a_0+\frac n2,
  \qquad
  b_n=b_0+\frac12(y^\top y-\mu_n^\top\Lambda_n\mu_n).
  

Search aliases: bar q bar

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 591

  The gated program uses a smooth gate γ_t=frac11+e^-4bar q_t and appends γ_tz_t and (1-γ_t)z_t. The hybrid program contains a controlled combination of quadratic, delayed, power-memory, exponential-memory, and interaction terms. Its higher prior penalty prevents it from winning merely because it has more columns.

  Exact mathematical passage

  
  \gamma_t=\frac{1}{1+e^{-4\bar q_t}}
  

Search aliases: bar q_t bar

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 591

  The gated program uses a smooth gate γ_t=frac11+e^-4bar q_t and appends γ_tz_t and (1-γ_t)z_t. The hybrid program contains a controlled combination of quadratic, delayed, power-memory, exponential-memory, and interaction terms. Its higher prior penalty prevents it from winning merely because it has more columns.

  Exact mathematical passage

  
  \gamma_t=\frac{1}{1+e^{-4\bar q_t}}
  

Search aliases: barμ bar mu

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 627

  If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.

  Exact mathematical passage

  
  \bar\mu=\sum_gp_g\mu_g,
  

Search aliases: bfseries

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: bmod

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 577

  For modulus N, the phase program appends the one-hot phase e_tbmod N and interactions z_t,ie_tbmod N. Alternating, cyclic, or clocked dynamics therefore become linear in an expanded polyphase state.

  Exact mathematical passage

  
  e_{t\bmod N}
  

Search aliases: bottomrule

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 390

  [Exact atlas residuals] Across the tested frames, the maximum observed residuals were lr atlas · maximum absolute residual spatial · 0 Fourier · 3.55×10^-15 Haar · 2.49×10^-14 polyphase-2 · 0 polyphase-4 · 0 categorical color · 0 weighted multi-atlas frame · 5.33×10^-15 The nonzero values are double-precision roundoff.

  Exact mathematical passage

  
  \begin{array}{lr}
  \toprule
  \text{atlas}&\text{maximum absolute residual}\\
  \midrule
  \text{spatial}&0\\
  \text{Fourier}&3.55\times10^{-15}\\
  \text{Haar}&2.49\times10^{-14}\\
  \text{polyphase-2}&0\\
  \text{polyphase-4}&0\\
  \text{categorical color}&0\\
  \text{weighted multi-atlas frame}&5.33\times10^{-15}\\
  \bottomrule
  \end{array}
  

Search aliases: C

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: c

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: c_a

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 251

  Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.

  Exact mathematical passage

  
  \Scal(c_a)_{a\in A}
  =
  \frac{1}{c}
  \sum_{a\in A}w_aU_a^*c_a,
  \qquad
  c=\sum_{a\in A}w_a^2.
  

Search aliases: C_j

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 375

  A finite-valued frame can be encoded by the one-hot coordinates C_j[m,n]= 1_x[m,n]=j. The original frame is recovered by the unique active color at each address. This reconstruction is categorical rather than linear over the original scalar encoding, so it is kept as a separately typed exact atlas.

  Exact mathematical passage

  
  C_j[m,n]=\mathbf 1_{x[m,n]=j}.
  

Search aliases: caption

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: centering

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: ch operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 467

  Let c(v) be any finite additive node cost. Define recursively V(v)= min{c(v),∑_u∈ch(v)V(u)}.

  Exact mathematical passage

  
  V(v)=
  \min\left\{c(v),\sum_{u\in\operatorname{ch}(v)}V(u)\right\}.
  

Search aliases: changed mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: clearpage

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: Complexity operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 611

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  \boxed{
  W_g=
  \ell_{\mathrm{pred}}(g)
  -\lambda\,\operatorname{Complexity}(g)
  -\eta\,\operatorname{Mechanisms}(g)
  -\rho\,\operatorname{ResolutionPenalty}(g).}
  

Search aliases: D

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: d

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: d_g

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 503

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)

  Exact mathematical passage

  
  \phi_g(z_t,q_t,u_t,m_t,t)\in\R^{d_g}.
  

Search aliases: d_g^2

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 785

  For n signal coordinates, A exact unitary atlases, projected state dimension k, G candidate programs, and m observed transitions, the dominant modeled costs are: ll FFT atlas · O(nlog n), Haar atlas · O(n), polyphase atlas · O(n), feature projection · O(dk), one regression fit · O(md_g^2+d_g^3), all program fits · displaystyle∑_g=1^GO(md_g^2+d_g^3). The architecture is designed so that d_g is controlled by the selected rank and resolution rather than by the full signal dimension.

  Exact mathematical passage

  
  \begin{array}{ll}
  \text{FFT atlas}&O(n\log n),\\
  \text{Haar atlas}&O(n),\\
  \text{polyphase atlas}&O(n),\\
  \text{feature projection}&O(dk),\\
  \text{one regression fit}&O(md_g^2+d_g^3),\\
  \text{all program fits}&\displaystyle\sum_{g=1}^GO(md_g^2+d_g^3).
  \end{array}
  

Search aliases: d_g^3

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 785

  For n signal coordinates, A exact unitary atlases, projected state dimension k, G candidate programs, and m observed transitions, the dominant modeled costs are: ll FFT atlas · O(nlog n), Haar atlas · O(n), polyphase atlas · O(n), feature projection · O(dk), one regression fit · O(md_g^2+d_g^3), all program fits · displaystyle∑_g=1^GO(md_g^2+d_g^3). The architecture is designed so that d_g is controlled by the selected rank and resolution rather than by the full signal dimension.

  Exact mathematical passage

  
  \begin{array}{ll}
  \text{FFT atlas}&O(n\log n),\\
  \text{Haar atlas}&O(n),\\
  \text{polyphase atlas}&O(n),\\
  \text{feature projection}&O(dk),\\
  \text{one regression fit}&O(md_g^2+d_g^3),\\
  \text{all program fits}&\displaystyle\sum_{g=1}^GO(md_g^2+d_g^3).
  \end{array}
  

Search aliases: e

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: e''

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: e^-2π i(km/H+ℓ n/W) pi ell

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 318

  For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.

  Exact mathematical passage

  
  (Fx)_{k,\ell}
  =\frac{1}{\sqrt{HW}}
  \sum_{m=0}^{H-1}\sum_{n=0}^{W-1}
   x_{m,n}e^{-2\pi i(km/H+\ell n/W)}.
  

Search aliases: e^-4bar q_t bar

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 591

  The gated program uses a smooth gate γ_t=frac11+e^-4bar q_t and appends γ_tz_t and (1-γ_t)z_t. The hybrid program contains a controlled combination of quadratic, delayed, power-memory, exponential-memory, and interaction terms. Its higher prior penalty prevents it from winning merely because it has more columns.

  Exact mathematical passage

  
  \gamma_t=\frac{1}{1+e^{-4\bar q_t}}
  

Search aliases: e^W_g

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 620

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  p(g\mid D)=\frac{e^{W_g}}{\sum_he^{W_h}}.
  

Search aliases: e^W_h

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 620

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  p(g\mid D)=\frac{e^{W_g}}{\sum_he^{W_h}}.
  

Search aliases: E_a,β beta

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 211

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  E_{a,\alpha}E_{a,\beta}=\delta_{\alpha\beta}E_{a,\alpha},
  \qquad
  \sum_{\alpha\in I_a}E_{a,\alpha}=I_{Y_a}.
  

Search aliases: E_B

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 324

  For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.

  Exact mathematical passage

  E_B

Search aliases: e_tbmod N bmod

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 577

  For modulus N, the phase program appends the one-hot phase e_tbmod N and interactions z_t,ie_tbmod N. Alternating, cyclic, or clocked dynamics therefore become linear in an expanded polyphase state.

  Exact mathematical passage

  
  e_{t\bmod N}
  

Search aliases: exp mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 560

  The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.

  Exact mathematical passage

  
  m_t^{\mathrm{exp}}
  =
  \frac{\displaystyle\sum_{\ell=1}^{H}\rho^{\ell-1}z_{t-\ell}}
  {\displaystyle\sum_{\ell=1}^{H}\rho^{\ell-1}}.
  

Search aliases: F

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: f

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: F^*

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 324

  For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.

  Exact mathematical passage

  F^*

Search aliases: fontsize

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: G

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 784

  For n signal coordinates, A exact unitary atlases, projected state dimension k, G candidate programs, and m observed transitions, the dominant modeled costs are: ll FFT atlas · O(nlog n),

  Exact mathematical passage

  G

Search aliases: g

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: H

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: h

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: I

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: i

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: I_a

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 210

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  \{E_{a,\alpha}\}_{\alpha\in I_a}

Search aliases: I_X

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 223

  [Per-atlas packet reconstruction] For every atlas a, the operators P_a,α are orthogonal projectors and ∑_α∈ I_aP_a,α=I_X. Consequently, x=∑_αP_a,αx, T=∑_α,βP_a,βTP_a,α.

  Exact mathematical passage

  
  \sum_{\alpha\in I_a}P_{a,\alpha}=I_X.
  

Search aliases: I_Y_a

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 211

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  E_{a,\alpha}E_{a,\beta}=\delta_{\alpha\beta}E_{a,\alpha},
  \qquad
  \sum_{\alpha\in I_a}E_{a,\alpha}=I_{Y_a}.
  

Search aliases: includegraphics

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: Information operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: InvGamma operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1115

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  \beta\mid\sigma^2\sim N(0,\sigma^2\Lambda_0^{-1}),
  \qquad
  \sigma^2\sim\operatorname{InvGamma}(a_0,b_0).
  

Search aliases: j

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: K

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: k

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: k_1

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: k_2

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: k_3

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: L

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: l

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: L_t

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: Large

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: large

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: lin mathrm

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 528

  The basic linear program uses ϕ_lin=[1,z_t,q_t,u_t]. The action-bilinear extension adds products z_t,iu_t,j, q_t,iu_t,j, for a declared low interaction rank.

  Exact mathematical passage

  
  \phi_{\mathrm{lin}}=[1,z_t,q_t,u_t].
  

Search aliases: linewidth

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: log

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 785

  For n signal coordinates, A exact unitary atlases, projected state dimension k, G candidate programs, and m observed transitions, the dominant modeled costs are: ll FFT atlas · O(nlog n), Haar atlas · O(n), polyphase atlas · O(n), feature projection · O(dk), one regression fit · O(md_g^2+d_g^3), all program fits · displaystyle∑_g=1^GO(md_g^2+d_g^3). The architecture is designed so that d_g is controlled by the selected rank and resolution rather than by the full signal dimension.

  Exact mathematical passage

  
  \begin{array}{ll}
  \text{FFT atlas}&O(n\log n),\\
  \text{Haar atlas}&O(n),\\
  \text{polyphase atlas}&O(n),\\
  \text{feature projection}&O(dk),\\
  \text{one regression fit}&O(md_g^2+d_g^3),\\
  \text{all program fits}&\displaystyle\sum_{g=1}^GO(md_g^2+d_g^3).
  \end{array}
  

Search aliases: loss mathrm

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: M

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: m

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: m_a

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: M_n

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 603

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  
  \Lambda_n=\Lambda_0+X_g^\top X_g,
  \qquad
  M_n=Y^\top X_g\Lambda_n^{-1}.
  

Search aliases: m_t

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 502

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)

  Exact mathematical passage

  m_t

Search aliases: m_t^exp mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 560

  The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.

  Exact mathematical passage

  
  m_t^{\mathrm{exp}}
  =
  \frac{\displaystyle\sum_{\ell=1}^{H}\rho^{\ell-1}z_{t-\ell}}
  {\displaystyle\sum_{\ell=1}^{H}\rho^{\ell-1}}.
  

Search aliases: m_t^pow mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 553

  The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.

  Exact mathematical passage

  
  m_t^{\mathrm{pow}}
  =
  \frac{\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}z_{t-\ell}}
  {\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}},
  

Search aliases: macro mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: Mechanisms operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 611

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  \boxed{
  W_g=
  \ell_{\mathrm{pred}}(g)
  -\lambda\,\operatorname{Complexity}(g)
  -\eta\,\operatorname{Mechanisms}(g)
  -\rho\,\operatorname{ResolutionPenalty}(g).}
  

Search aliases: meso mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: micro mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: mid

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 620

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  p(g\mid D)=\frac{e^{W_g}}{\sum_he^{W_h}}.
  

Search aliases: midrule

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 390

  [Exact atlas residuals] Across the tested frames, the maximum observed residuals were lr atlas · maximum absolute residual spatial · 0 Fourier · 3.55×10^-15 Haar · 2.49×10^-14 polyphase-2 · 0 polyphase-4 · 0 categorical color · 0 weighted multi-atlas frame · 5.33×10^-15 The nonzero values are double-precision roundoff.

  Exact mathematical passage

  
  \begin{array}{lr}
  \toprule
  \text{atlas}&\text{maximum absolute residual}\\
  \midrule
  \text{spatial}&0\\
  \text{Fourier}&3.55\times10^{-15}\\
  \text{Haar}&2.49\times10^{-14}\\
  \text{polyphase-2}&0\\
  \text{polyphase-4}&0\\
  \text{categorical color}&0\\
  \text{weighted multi-atlas frame}&5.33\times10^{-15}\\
  \bottomrule
  \end{array}
  

Search aliases: min

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 467

  Let c(v) be any finite additive node cost. Define recursively V(v)= min{c(v),∑_u∈ch(v)V(u)}.

  Exact mathematical passage

  
  V(v)=
  \min\left\{c(v),\sum_{u\in\operatorname{ch}(v)}V(u)\right\}.
  

Search aliases: N

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: n

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: norm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 266

  [Exact multi-atlas tight-frame reconstruction] The analysis and synthesis maps satisfy S A=I_X. Moreover, norm A x^2=cnormx^2.

  Exact mathematical passage

  
  \norm{\Acal x}^2=c\norm{x}^2.
  

Search aliases: Novelty operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: O

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: o

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: p

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: P_a,β beta

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 227

  [Per-atlas packet reconstruction] For every atlas a, the operators P_a,α are orthogonal projectors and ∑_α∈ I_aP_a,α=I_X. Consequently, x=∑_αP_a,αx, T=∑_α,βP_a,βTP_a,α.

  Exact mathematical passage

  
  x=\sum_{\alpha}P_{a,\alpha}x,
  \qquad
  T=\sum_{\alpha,\beta}P_{a,\beta}TP_{a,\alpha}.
  

Search aliases: P_B

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 325

  For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.

  Exact mathematical passage

  
  P_B=F^*E_BF.
  

Search aliases: P_B^*

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1107

  Fourier selector projectors. For any diagonal zero-one mask E_B, P_B=F^*E_BF, so P_B^2=P_B, P_B^*=P_B. Disjoint masks annihilate, and a complete mask partition sums to the identity.

  Exact mathematical passage

  
  P_B^2=P_B,\qquad P_B^*=P_B.
  

Search aliases: P_B^2

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1107

  Fourier selector projectors. For any diagonal zero-one mask E_B, P_B=F^*E_BF, so P_B^2=P_B, P_B^*=P_B. Disjoint masks annihilate, and a complete mask partition sums to the identity.

  Exact mathematical passage

  
  P_B^2=P_B,\qquad P_B^*=P_B.
  

Search aliases: p_g

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 627

  If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.

  Exact mathematical passage

  
  \bar\mu=\sum_gp_g\mu_g,
  

Search aliases: P_N,r,s

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 359

  For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].

  Exact mathematical passage

  
  (P_{N,r,s}x)[m,n]
  =
  \mathbf 1_{m\equiv r\ (N)}
  \mathbf 1_{n\equiv s\ (N)}x[m,n].
  

Search aliases: par

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: pow mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 553

  The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.

  Exact mathematical passage

  
  m_t^{\mathrm{pow}}
  =
  \frac{\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}z_{t-\ell}}
  {\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}},
  

Search aliases: pred mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 611

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  \boxed{
  W_g=
  \ell_{\mathrm{pred}}(g)
  -\lambda\,\operatorname{Complexity}(g)
  -\eta\,\operatorname{Mechanisms}(g)
  -\rho\,\operatorname{ResolutionPenalty}(g).}
  

Search aliases: Q

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 432

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  Q

Search aliases: q

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: Q_1:k_1

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: Q_1:k_2

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: Q_1:k_3

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: q_t

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 502

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)

  Exact mathematical passage

  q_t

Search aliases: q_t,i

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 532

  The basic linear program uses ϕ_lin=[1,z_t,q_t,u_t]. The action-bilinear extension adds products z_t,iu_t,j, q_t,iu_t,j, for a declared low interaction rank.

  Exact mathematical passage

  
  z_{t,i}u_{t,j},
  \qquad
  q_{t,i}u_{t,j},
  

Search aliases: q_t,j

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 542

  The quadratic program adds z_t,iz_t,j, q_t,iq_t,j, i≤ j, within a controlled rank. This is a finite packet jet, not an unrestricted polynomial expansion.

  Exact mathematical passage

  
  z_{t,i}z_{t,j},
  \qquad
  q_{t,i}q_{t,j},
  \qquad i\le j,
  

Search aliases: R

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: R mathbb

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 432

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  r(x)\in\R^d

Search aliases: r

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: R^d R mathbb

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 432

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  r(x)\in\R^d

Search aliases: R^d_g R mathbb

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 503

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)

  Exact mathematical passage

  
  \phi_g(z_t,q_t,u_t,m_t,t)\in\R^{d_g}.
  

Search aliases: r_l

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: R_t

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: RepeatCount operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: ResolutionPenalty operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 611

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  \boxed{
  W_g=
  \ell_{\mathrm{pred}}(g)
  -\lambda\,\operatorname{Complexity}(g)
  -\eta\,\operatorname{Mechanisms}(g)
  -\rho\,\operatorname{ResolutionPenalty}(g).}
  

Search aliases: RoleHistory operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: S

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: s

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: s'

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 644

  Each resolved frame receives a deterministic hash of its complete normalized array. An observed public action defines a graph edge (s,a)⟼ s'. Repeated observations accumulate reward, progress, win, loss, and visit counts.

  Exact mathematical passage

  
  (s,a)\longmapsto s'.
  

Search aliases: Scal S mathcal

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 251

  Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.

  Exact mathematical passage

  
  \Scal(c_a)_{a\in A}
  =
  \frac{1}{c}
  \sum_{a\in A}w_aU_a^*c_a,
  \qquad
  c=\sum_{a\in A}w_a^2.
  

Search aliases: Scal_N S_N Scal mathcal

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1097

  Polyphase analysis and synthesis. For modulus N, ( A_Nx)_r,s[q,t]=x[Nq+r,Nt+s]. The synthesis operator places each channel value back at its unique original address. Therefore S_N A_N=I.

  Exact mathematical passage

  
  \Scal_N\Acal_N=I.
  

Search aliases: section

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: selectfont

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: StatusBox

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: subsection

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: T

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: t

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: tableofcontents

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: top

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 599

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  
  Y=X_gB_g^\top+\varepsilon,
  

Search aliases: toprule

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 390

  [Exact atlas residuals] Across the tested frames, the maximum observed residuals were lr atlas · maximum absolute residual spatial · 0 Fourier · 3.55×10^-15 Haar · 2.49×10^-14 polyphase-2 · 0 polyphase-4 · 0 categorical color · 0 weighted multi-atlas frame · 5.33×10^-15 The nonzero values are double-precision roundoff.

  Exact mathematical passage

  
  \begin{array}{lr}
  \toprule
  \text{atlas}&\text{maximum absolute residual}\\
  \midrule
  \text{spatial}&0\\
  \text{Fourier}&3.55\times10^{-15}\\
  \text{Haar}&2.49\times10^{-14}\\
  \text{polyphase-2}&0\\
  \text{polyphase-4}&0\\
  \text{categorical color}&0\\
  \text{weighted multi-atlas frame}&5.33\times10^{-15}\\
  \bottomrule
  \end{array}
  

Search aliases: total mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 631

  If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.

  Exact mathematical passage

  
  \boxed{
  V_{\mathrm{total}}
  =
  \underbrace{\sum_gp_gV_g}_{\text{within-program}}
  +
  \underbrace{\sum_gp_g(\mu_g-\bar\mu)^2}_{\text{between-program}}.}
  

Search aliases: u

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: U_a^*

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 217

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  \boxed{P_{a,\alpha}=U_a^*E_{a,\alpha}U_a.}
  

Search aliases: u_t

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 502

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)

  Exact mathematical passage

  u_t

Search aliases: u_t,j

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 532

  The basic linear program uses ϕ_lin=[1,z_t,q_t,u_t]. The action-bilinear extension adds products z_t,iu_t,j, q_t,iu_t,j, for a declared low interaction rank.

  Exact mathematical passage

  
  z_{t,i}u_{t,j},
  \qquad
  q_{t,i}u_{t,j},
  

Search aliases: untried mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: V

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 467

  Let c(v) be any finite additive node cost. Define recursively V(v)= min{c(v),∑_u∈ch(v)V(u)}.

  Exact mathematical passage

  
  V(v)=
  \min\left\{c(v),\sum_{u\in\operatorname{ch}(v)}V(u)\right\}.
  

Search aliases: v

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: V_g

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 626

  If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.

  Exact mathematical passage

  V_g

Search aliases: V_total mathrm

Link to this entry · Copy TeX

• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 631

  If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.

  Exact mathematical passage

  
  \boxed{
  V_{\mathrm{total}}
  =
  \underbrace{\sum_gp_gV_g}_{\text{within-program}}
  +
  \underbrace{\sum_gp_g(\mu_g-\bar\mu)^2}_{\text{between-program}}.}
  

Search aliases: Variance operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: vfill

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: W

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: w

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: w_a

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 246

  Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.

  Exact mathematical passage

  w_a>0

Search aliases: w_a^2

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 251

  Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.

  Exact mathematical passage

  
  \Scal(c_a)_{a\in A}
  =
  \frac{1}{c}
  \sum_{a\in A}w_aU_a^*c_a,
  \qquad
  c=\sum_{a\in A}w_a^2.
  

Search aliases: W_g

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 611

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  \boxed{
  W_g=
  \ell_{\mathrm{pred}}(g)
  -\lambda\,\operatorname{Complexity}(g)
  -\eta\,\operatorname{Mechanisms}(g)
  -\rho\,\operatorname{ResolutionPenalty}(g).}
  

Search aliases: W_h

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 620

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  p(g\mid D)=\frac{e^{W_g}}{\sum_he^{W_h}}.
  

Search aliases: widehat p widehat

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: widehat p_loss widehat mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: widehat p_win widehat mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: widehat R widehat

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: widehatΔ L widehat Delta

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: widetilde P widetilde

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 293

  The normalized positive packets widetilde P_a,α =fracw_a^2cP_a,α satisfy ∑_a,αwidetilde P_a,α=I_X. They need not be mutually annihilating across different atlases. They are frame packets, not one enlarged orthogonal projector family. This distinction prevents the simultaneous use of several atlases from being mistaken for a direct-sum decomposition inside the original space.

  Exact mathematical passage

  
  \widetilde P_{a,\alpha}
  =\frac{w_a^2}{c}P_{a,\alpha}
  

Search aliases: widetilde P_a,α widetilde alpha

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 293

  The normalized positive packets widetilde P_a,α =fracw_a^2cP_a,α satisfy ∑_a,αwidetilde P_a,α=I_X. They need not be mutually annihilating across different atlases. They are frame packets, not one enlarged orthogonal projector family. This distinction prevents the simultaneous use of several atlases from being mistaken for a direct-sum decomposition inside the original space.

  Exact mathematical passage

  
  \widetilde P_{a,\alpha}
  =\frac{w_a^2}{c}P_{a,\alpha}
  

Search aliases: win mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: x

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: X^top top

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1121

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  \Lambda_n=\Lambda_0+X^\top X,
  \qquad
  \mu_n=\Lambda_n^{-1}X^\top y,
  

Search aliases: x_*

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1131

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  x_*

Search aliases: x_*^top top

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1131

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  x_*^\top\mu_n

Search aliases: X_g

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 599

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  
  Y=X_gB_g^\top+\varepsilon,
  

Search aliases: X_g^top top

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 603

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  
  \Lambda_n=\Lambda_0+X_g^\top X_g,
  \qquad
  M_n=Y^\top X_g\Lambda_n^{-1}.
  

Search aliases: x_m,n

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 318

  For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.

  Exact mathematical passage

  
  (Fx)_{k,\ell}
  =\frac{1}{\sqrt{HW}}
  \sum_{m=0}^{H-1}\sum_{n=0}^{W-1}
   x_{m,n}e^{-2\pi i(km/H+\ell n/W)}.
  

Search aliases: x_r,s

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 354

  For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].

  Exact mathematical passage

  
  x_{r,s}[q,t]=x[Nq+r,Nt+s],
  \qquad 0\le r,s<N.
  

Search aliases: Y

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 207

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  U_a:X\to Y_a
  

Search aliases: y

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: Y^top top

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 603

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  
  \Lambda_n=\Lambda_0+X_g^\top X_g,
  \qquad
  M_n=Y^\top X_g\Lambda_n^{-1}.
  

Search aliases: y^top top

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1126

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  a_n=a_0+\frac n2,
  \qquad
  b_n=b_0+\frac12(y^\top y-\mu_n^\top\Lambda_n\mu_n).
  

Search aliases: Y_a

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 207

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  U_a:X\to Y_a
  

Search aliases: y_t^(g)

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: z

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: z^macro mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: z^meso mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: z^micro mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 433

  Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.

  Exact mathematical passage

  
  z^{\mathrm{macro}}=Q_{1:k_1}r(x),
  \qquad
  z^{\mathrm{meso}}=Q_{1:k_2}r(x),
  \qquad
  z^{\mathrm{micro}}=Q_{1:k_3}r(x),
  

Search aliases: z_t

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 502

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)

  Exact mathematical passage

  z_t

Search aliases: z_t+1^(g)

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: z_t,i

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 532

  The basic linear program uses ϕ_lin=[1,z_t,q_t,u_t]. The action-bilinear extension adds products z_t,iu_t,j, q_t,iu_t,j, for a declared low interaction rank.

  Exact mathematical passage

  
  z_{t,i}u_{t,j},
  \qquad
  q_{t,i}u_{t,j},
  

Search aliases: z_t,j

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 542

  The quadratic program adds z_t,iz_t,j, q_t,iq_t,j, i≤ j, within a controlled rank. This is a finite packet jet, not an unrestricted polynomial expansion.

  Exact mathematical passage

  
  z_{t,i}z_{t,j},
  \qquad
  q_{t,i}q_{t,j},
  \qquad i\le j,
  

Search aliases: z_t-1

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 552

  The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.

  Exact mathematical passage

  z_{t-1}

Search aliases: z_t-ℓ ell

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 553

  The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.

  Exact mathematical passage

  
  m_t^{\mathrm{pow}}
  =
  \frac{\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}z_{t-\ell}}
  {\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}},
  

Search aliases: z_t^(g)

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: × times

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 118

  [3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable. vfill Largebfseries Komorebi P. This paper presents the Projector-Resol…

  Exact mathematical passage

  3pt]Large Algebraic Signal Model\par}
  \vspace{0.18in}
  {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
  \vspace{0.55in}
  \begin{minipage}{0.80\linewidth}
  \large
  A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction system, while learned operator programs remain typed, probabilistic, and auditable.
  \end{minipage}
  \vfill
  {\Large\bfseries Komorebi P.\par}
  \vspace{0.45in}
  \end{titlepage}
  
  \begin{abstract}
  This paper presents the Projector-Resolved Large Algebraic Signal Model (PR-LASM), a finite, executable world-model architecture built from exact projector decompositions, coordinate transport, polyphase refinement, operator programs, Bayesian evidence, residual-driven adaptation, memory packets, and active planning.  Each observation is retained in a complete multi-atlas archive consisting of spatial, unitary Fourier, orthonormal Haar, two-dimensional polyphase, and categorical color coordinates.  These atlases are not concatenated informally: their weighted direct sum is proved to be a finite tight frame with an explicit canonical synthesis map.  The model then forms deterministic interpretable features and nested macro, meso, and micro states for statistical prediction without confusing those compressed states with the exact archive.
  
  The learned dynamics are a posterior over a typed finite grammar containing linear, quadratic, action-bilinear, delayed, power-memory, exponential-memory, phase-conditioned, gated, and hybrid programs.  A normal-inverse-gamma regression is fitted inside every program, and predictive evidence is combined with explicit complexity and mechanism penalties.  Exact deterministic replay transitions are stored in a state-action graph; the planner combines known graph outcomes, replay probes, posterior progress, win and loss probabilities, novelty, information value, role statistics, and loop penalties.
  
  The release contains exact reconstruction tests, adaptive mixed-depth packet-tree synthesis, controlled program-identification benchmarks, deterministic interactive planning tests, a static audit of twenty-five supplied local environments, a reproducible Python package, and offline run scripts.  All thirty-six automated tests pass.  On seven controlled algebraic systems, the intended operator family and intended atlas are identified in every case.  The median state-prediction RMSE relative to a single linear portfolio baseline is $0.2393$.  Four deterministic interactive systems are solved.  No official benchmark score or claim of general intelligence is made.
  \end{abstract}
  
  \StatusBox{\textbf{Status.}  The exact finite coordinate and reconstruction theorems, program definitions, Bayesian formulas, graph algorithms, and supplied tests are complete under the hypotheses written here.  The controlled benchmarks verify the implementation of the declared mechanisms.  The twenty-five bundled environment trees are statically validated, but they were not executed with the official local runtime in the present build environment.}
  
  \tableofcontents
  \clearpage
  
  \section{The final construction problem}
  
  The technical series developed seeds, projectors, transfer blocks, nonlinear interactions, polyphase refinement, special functions, stochastic and quantum realizations, computational packet algorithms, arithmetic packets, and local-to-global completion.  The remaining engineering question is whether those ingredients can be assembled into one coherent model of a changing signal world.
  
  A conventional predictor usually commits early to one representation.  A vector model receives coordinates that have already forgotten geometry.  A convolutional model privileges translation-local structure.  A Fourier model privileges global frequency.  A wavelet model privileges scale and location.  A graph model privileges relational neighborhoods.  None of these choices is wrong, but no single one is uniformly natural.
  
  PR-LASM keeps several exact coordinate atlases simultaneously.  It then lets the posterior decide which atlas, resolution, operator family, and memory mechanism best explain the observed transitions.  The exact frame remains recoverable regardless of that choice.
  
  \begin{figure}[H]
  \centering
  \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
  \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
  \end{figure}
  
  \subsection{What ``large'' means}
  
  The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
  \[
  \boxed{
  \text{atlas}\times\text{resolution}\times\text{operator program}
  \times\text{memory}\times\text{packet refinement}\times\text{action}.}
  

Search aliases: β beta

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 211

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  E_{a,\alpha}E_{a,\beta}=\delta_{\alpha\beta}E_{a,\alpha},
  \qquad
  \sum_{\alpha\in I_a}E_{a,\alpha}=I_{Y_a}.
  

Search aliases: γ gamma

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 591

  The gated program uses a smooth gate γ_t=frac11+e^-4bar q_t and appends γ_tz_t and (1-γ_t)z_t. The hybrid program contains a controlled combination of quadratic, delayed, power-memory, exponential-memory, and interaction terms. Its higher prior penalty prevents it from winning merely because it has more columns.

  Exact mathematical passage

  
  \gamma_t=\frac{1}{1+e^{-4\bar q_t}}
  

Search aliases: γ_t gamma

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 591

  The gated program uses a smooth gate γ_t=frac11+e^-4bar q_t and appends γ_tz_t and (1-γ_t)z_t. The hybrid program contains a controlled combination of quadratic, delayed, power-memory, exponential-memory, and interaction terms. Its higher prior penalty prevents it from winning merely because it has more columns.

  Exact mathematical passage

  
  \gamma_t=\frac{1}{1+e^{-4\bar q_t}}
  

Search aliases: Δ Delta

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 507

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .

  Exact mathematical passage

  
  y_t^{(g)}=
  \begin{bmatrix}
   z_{t+1}^{(g)}-z_t^{(g)}\\
   R_t/100\\
   \Delta L_t\\
   \mathbf 1_{\mathrm{win}}\\
   \mathbf 1_{\mathrm{loss}}\\
   \mathbf 1_{\mathrm{changed}}
  \end{bmatrix}.
  

Search aliases: δ delta

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 211

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  E_{a,\alpha}E_{a,\beta}=\delta_{\alpha\beta}E_{a,\alpha},
  \qquad
  \sum_{\alpha\in I_a}E_{a,\alpha}=I_{Y_a}.
  

Search aliases: δ_αβ delta alpha beta

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 211

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  E_{a,\alpha}E_{a,\beta}=\delta_{\alpha\beta}E_{a,\alpha},
  \qquad
  \sum_{\alpha\in I_a}E_{a,\alpha}=I_{Y_a}.
  

Search aliases: ε varepsilon

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 599

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  
  Y=X_gB_g^\top+\varepsilon,
  

Search aliases: η eta

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 611

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  \boxed{
  W_g=
  \ell_{\mathrm{pred}}(g)
  -\lambda\,\operatorname{Complexity}(g)
  -\eta\,\operatorname{Mechanisms}(g)
  -\rho\,\operatorname{ResolutionPenalty}(g).}
  

Search aliases: κ kappa

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 497

  A large signal model should not hide all mechanism uncertainty inside one dense matrix. PR-LASM uses a typed program g=(a,r,f,κ), where a is an atlas, r is a resolution, f is an operator family, and κ contains family-specific ranks or periods.

  Exact mathematical passage

  
  g=(a,r,f,\kappa),
  

Search aliases: λ lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 602

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  \Lambda_0=\lambda_0I

Search aliases: Λ_0 Lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 602

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  \Lambda_0=\lambda_0I

Search aliases: λ_0 lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 602

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  \Lambda_0=\lambda_0I

Search aliases: Λ_0^-1 Lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1115

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  \beta\mid\sigma^2\sim N(0,\sigma^2\Lambda_0^{-1}),
  \qquad
  \sigma^2\sim\operatorname{InvGamma}(a_0,b_0).
  

Search aliases: λ_D lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: λ_F lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: λ_H lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: λ_I lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: λ_L lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: Λ_n Lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 603

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  
  \Lambda_n=\Lambda_0+X_g^\top X_g,
  \qquad
  M_n=Y^\top X_g\Lambda_n^{-1}.
  

Search aliases: λ_N lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: Λ_n^-1 Lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 603

  For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.

  Exact mathematical passage

  
  \Lambda_n=\Lambda_0+X_g^\top X_g,
  \qquad
  M_n=Y^\top X_g\Lambda_n^{-1}.
  

Search aliases: λ_Q lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: λ_V lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: λ_W lambda

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 677

  For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive dominant bonuses. This ordering ensures that an already observed winning action is not ignored because a statistical model has not yet fit it.

  Exact mathematical passage

  
  \begin{aligned}
  U(a)=&\ \widehat R(a)
  +\lambda_L\widehat{\Delta L}(a)
  +\lambda_W\widehat p_{\mathrm{win}}(a)
  -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
  &+\lambda_N\operatorname{Novelty}(a)
  +\lambda_I\operatorname{Information}(a)
  -\lambda_V\operatorname{Variance}(a)\\
  &+\lambda_H\operatorname{RoleHistory}(a)
  -\lambda_Q\operatorname{RepeatCount}(a)^2
  +\lambda_F\mathbf1_{\mathrm{untried}}.
  \end{aligned}
  

Search aliases: μ mu

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 626

  If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.

  Exact mathematical passage

  \mu_g

Search aliases: μ_g mu

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 626

  If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.

  Exact mathematical passage

  \mu_g

Search aliases: μ_n mu

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1121

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  \Lambda_n=\Lambda_0+X^\top X,
  \qquad
  \mu_n=\Lambda_n^{-1}X^\top y,
  

Search aliases: μ_n^top mu top

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1126

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  a_n=a_0+\frac n2,
  \qquad
  b_n=b_0+\frac12(y^\top y-\mu_n^\top\Lambda_n\mu_n).
  

Search aliases: π pi

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 318

  For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.

  Exact mathematical passage

  
  (Fx)_{k,\ell}
  =\frac{1}{\sqrt{HW}}
  \sum_{m=0}^{H-1}\sum_{n=0}^{W-1}
   x_{m,n}e^{-2\pi i(km/H+\ell n/W)}.
  

Search aliases: ρ rho

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 560

  The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.

  Exact mathematical passage

  
  m_t^{\mathrm{exp}}
  =
  \frac{\displaystyle\sum_{\ell=1}^{H}\rho^{\ell-1}z_{t-\ell}}
  {\displaystyle\sum_{\ell=1}^{H}\rho^{\ell-1}}.
  

Search aliases: ρ^ℓ-1 rho ell

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 560

  The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.

  Exact mathematical passage

  
  m_t^{\mathrm{exp}}
  =
  \frac{\displaystyle\sum_{\ell=1}^{H}\rho^{\ell-1}z_{t-\ell}}
  {\displaystyle\sum_{\ell=1}^{H}\rho^{\ell-1}}.
  

Search aliases: σ sigma

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1115

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  \beta\mid\sigma^2\sim N(0,\sigma^2\Lambda_0^{-1}),
  \qquad
  \sigma^2\sim\operatorname{InvGamma}(a_0,b_0).
  

Search aliases: σ^2 sigma

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1115

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  \beta\mid\sigma^2\sim N(0,\sigma^2\Lambda_0^{-1}),
  \qquad
  \sigma^2\sim\operatorname{InvGamma}(a_0,b_0).
  

Search aliases: ϕ phi

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 503

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)

  Exact mathematical passage

  
  \phi_g(z_t,q_t,u_t,m_t,t)\in\R^{d_g}.
  

Search aliases: ϕ_g phi

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 503

  For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)

  Exact mathematical passage

  
  \phi_g(z_t,q_t,u_t,m_t,t)\in\R^{d_g}.
  

Search aliases: ϕ_lin phi mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 528

  The basic linear program uses ϕ_lin=[1,z_t,q_t,u_t]. The action-bilinear extension adds products z_t,iu_t,j, q_t,iu_t,j, for a declared low interaction rank.

  Exact mathematical passage

  
  \phi_{\mathrm{lin}}=[1,z_t,q_t,u_t].
  

Search aliases: ℓ ell

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 318

  For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.

  Exact mathematical passage

  
  (Fx)_{k,\ell}
  =\frac{1}{\sqrt{HW}}
  \sum_{m=0}^{H-1}\sum_{n=0}^{W-1}
   x_{m,n}e^{-2\pi i(km/H+\ell n/W)}.
  

Search aliases: ℓ^-α ell alpha

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 553

  The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.

  Exact mathematical passage

  
  m_t^{\mathrm{pow}}
  =
  \frac{\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}z_{t-\ell}}
  {\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}},
  

Search aliases: ℓ_pred ell mathrm

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 611

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  \boxed{
  W_g=
  \ell_{\mathrm{pred}}(g)
  -\lambda\,\operatorname{Complexity}(g)
  -\eta\,\operatorname{Mechanisms}(g)
  -\rho\,\operatorname{ResolutionPenalty}(g).}
  

Search aliases: → to

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 207

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  U_a:X\to Y_a
  

Search aliases: ∑ sum

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 211

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  E_{a,\alpha}E_{a,\beta}=\delta_{\alpha\beta}E_{a,\alpha},
  \qquad
  \sum_{\alpha\in I_a}E_{a,\alpha}=I_{Y_a}.
  

Search aliases: ∑_a sum

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 273

  By unitarity, S A x =frac1c∑_a w_aU_a^*(w_aU_ax) =frac1c∑_aw_a^2x=x. Similarly, norm A x^2 =∑_aw_a^2normU_ax^2 =cnormx^2.

  Exact mathematical passage

  
  \Scal\Acal x
  =\frac{1}{c}\sum_a w_aU_a^*(w_aU_ax)
  =\frac{1}{c}\sum_aw_a^2x=x.
  

Search aliases: ∑_a,α sum alpha

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 298

  The normalized positive packets widetilde P_a,α =fracw_a^2cP_a,α satisfy ∑_a,αwidetilde P_a,α=I_X. They need not be mutually annihilating across different atlases. They are frame packets, not one enlarged orthogonal projector family. This distinction prevents the simultaneous use of several atlases from being mistaken for a direct-sum decomposition inside the original space.

  Exact mathematical passage

  
  \sum_{a,\alpha}\widetilde P_{a,\alpha}=I_X.
  

Search aliases: ∑_a∈ A sum in

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 251

  Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.

  Exact mathematical passage

  
  \Scal(c_a)_{a\in A}
  =
  \frac{1}{c}
  \sum_{a\in A}w_aU_a^*c_a,
  \qquad
  c=\sum_{a\in A}w_a^2.
  

Search aliases: ∑_g sum

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 627

  If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.

  Exact mathematical passage

  
  \bar\mu=\sum_gp_g\mu_g,
  

Search aliases: ∑_g=1^G sum

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 785

  For n signal coordinates, A exact unitary atlases, projected state dimension k, G candidate programs, and m observed transitions, the dominant modeled costs are: ll FFT atlas · O(nlog n), Haar atlas · O(n), polyphase atlas · O(n), feature projection · O(dk), one regression fit · O(md_g^2+d_g^3), all program fits · displaystyle∑_g=1^GO(md_g^2+d_g^3). The architecture is designed so that d_g is controlled by the selected rank and resolution rather than by the full signal dimension.

  Exact mathematical passage

  
  \begin{array}{ll}
  \text{FFT atlas}&O(n\log n),\\
  \text{Haar atlas}&O(n),\\
  \text{polyphase atlas}&O(n),\\
  \text{feature projection}&O(dk),\\
  \text{one regression fit}&O(md_g^2+d_g^3),\\
  \text{all program fits}&\displaystyle\sum_{g=1}^GO(md_g^2+d_g^3).
  \end{array}
  

Search aliases: ∑_h sum

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 620

  The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.

  Exact mathematical passage

  
  p(g\mid D)=\frac{e^{W_g}}{\sum_he^{W_h}}.
  

Search aliases: ∑_m=0^H-1 sum

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 318

  For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.

  Exact mathematical passage

  
  (Fx)_{k,\ell}
  =\frac{1}{\sqrt{HW}}
  \sum_{m=0}^{H-1}\sum_{n=0}^{W-1}
   x_{m,n}e^{-2\pi i(km/H+\ell n/W)}.
  

Search aliases: ∑_n=0^W-1 sum

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 318

  For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.

  Exact mathematical passage

  
  (Fx)_{k,\ell}
  =\frac{1}{\sqrt{HW}}
  \sum_{m=0}^{H-1}\sum_{n=0}^{W-1}
   x_{m,n}e^{-2\pi i(km/H+\ell n/W)}.
  

Search aliases: ∑_u∈ch(v) sum in operatorname

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 467

  Let c(v) be any finite additive node cost. Define recursively V(v)= min{c(v),∑_u∈ch(v)V(u)}.

  Exact mathematical passage

  
  V(v)=
  \min\left\{c(v),\sum_{u\in\operatorname{ch}(v)}V(u)\right\}.
  

Search aliases: ∑_α sum alpha

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 227

  [Per-atlas packet reconstruction] For every atlas a, the operators P_a,α are orthogonal projectors and ∑_α∈ I_aP_a,α=I_X. Consequently, x=∑_αP_a,αx, T=∑_α,βP_a,βTP_a,α.

  Exact mathematical passage

  
  x=\sum_{\alpha}P_{a,\alpha}x,
  \qquad
  T=\sum_{\alpha,\beta}P_{a,\beta}TP_{a,\alpha}.
  

Search aliases: ∑_α,β sum alpha beta

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 227

  [Per-atlas packet reconstruction] For every atlas a, the operators P_a,α are orthogonal projectors and ∑_α∈ I_aP_a,α=I_X. Consequently, x=∑_αP_a,αx, T=∑_α,βP_a,βTP_a,α.

  Exact mathematical passage

  
  x=\sum_{\alpha}P_{a,\alpha}x,
  \qquad
  T=\sum_{\alpha,\beta}P_{a,\beta}TP_{a,\alpha}.
  

Search aliases: ∑_α∈ I_a sum alpha in

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 211

  Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.

  Exact mathematical passage

  
  E_{a,\alpha}E_{a,\beta}=\delta_{\alpha\beta}E_{a,\alpha},
  \qquad
  \sum_{\alpha\in I_a}E_{a,\alpha}=I_{Y_a}.
  

Search aliases: ∑_ℓ=1^H sum ell

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 553

  The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.

  Exact mathematical passage

  
  m_t^{\mathrm{pow}}
  =
  \frac{\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}z_{t-\ell}}
  {\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}},
  

Search aliases: ∼ sim

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 1115

  For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independently to every output coordinate while sharing the design covariance.

  Exact mathematical passage

  
  \beta\mid\sigma^2\sim N(0,\sigma^2\Lambda_0^{-1}),
  \qquad
  \sigma^2\sim\operatorname{InvGamma}(a_0,b_0).
  

Search aliases: ≡ equiv

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 359

  For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].

  Exact mathematical passage

  
  (P_{N,r,s}x)[m,n]
  =
  \mathbf 1_{m\equiv r\ (N)}
  \mathbf 1_{n\equiv s\ (N)}x[m,n].
  

Search aliases: ≤ le

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 354

  For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].

  Exact mathematical passage

  
  x_{r,s}[q,t]=x[Nq+r,Nt+s],
  \qquad 0\le r,s<N.
  

Search aliases: ⟼ longmapsto

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• Use in source · Projector_Resolved_Large_Algebraic_Signal_Model.tex, line 339

  The separable Haar transform repeatedly applies (a,b)⟼ (fraca+bsqrt2,fraca-bsqrt2) along columns and rows. The inverse reverses those operations. Scale-location selectors in the packed transform pull back to exact wavelet projectors.

  Exact mathematical passage

  
  (a,b)\longmapsto
  \left(\frac{a+b}{\sqrt2},\frac{a-b}{\sqrt2}\right)