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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.\n\nThe 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.\n\nThe 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.\n\\end{abstract}\n\n\\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.}\n\n\\tableofcontents\n\\clearpage\n\n\\section{The final construction problem}\n\nThe 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.\n\nA 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.\n\nPR-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.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.98\\linewidth]{01_pr_lasm_architecture.pdf}\n\\caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}\n\\end{figure}\n\n\\subsection{What ``large'' means}\n\nThe word \\emph{large} does not refer only to parameter count.  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I_a}","kind":"use","occurrences":[16,20]},{"id":"symbol-71e2b91c820bc153","tex":"\\sum_{\\ell=1}^{H}","aliases":"∑_ℓ=1^H sum ell","display":"\\sum_{\\ell=1}^{H}","kind":"use","occurrences":[74,75]},{"id":"symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","occurrences":[128]},{"id":"symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","occurrences":[41]},{"id":"symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","occurrences":[40,72]},{"id":"symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","occurrences":[38,96]}],"contexts":["[3pt]Large Algebraic Signal Model\\par}\n\\vspace{0.18in}\n{\\fontsize{17}{21}\\selectfont\\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\\par}\n\\vspace{0.55in}\n\\begin{minipage}{0.80\\linewidth}\n\\large\nA 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.\n\\end{minipage}\n\\vfill\n{\\Large\\bfseries Komorebi P.\\par}\n\\vspace{0.45in}\n\\end{titlepage}\n\n\\begin{abstract}\nThis 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 coo…","\\begin{definition}[Projector-resolved large signal model]\nA finite PR-LASM instance is a tuple\n\\[\n\\mathfrak L=\n(X,\\mathfrak U,\\mathfrak E,\\mathfrak P,\\mathfrak Z,\n\\mathfrak G,\\mathfrak B,\\mathfrak H,\\mathfrak R,\\mathfrak C),\n\\]\nwhere:\n\\begin{enumerate}\n\\item $X$ is the observation space;\n\\item $\\mathfrak U=\\{U_a\\}_{a\\in A}$ is a family of exact coordinate atlases;\n\\item $\\mathfrak E=\\{E_{a,\\alpha}\\}$ is the family of coordinate selectors;\n\\item $\\mathfrak P=\\{P_{a,\\alpha}\\}$ is the pulled-back projector family;\n\\item $\\mathfrak Z$ is a nested family of statistical states;\n\\item $\\mathfrak G$ is a typed operator-program grammar;\n\\item $\\mathfrak B$ is the Bayesian posterior over programs and parameters;\n\\item $\\mathfrak H$ is the explicit memory state;\n\\item $\\mathfrak R$ is the state-action transition graph;\n\\item $\\mathfrak C$ is the planning and reconstruction contract.\n\\end{enumerate}\n\\end{definition}","model]\nA finite PR-LASM instance is a tuple\n\\[\n\\mathfrak L=\n(X,\\mathfrak U,\\mathfrak E,\\mathfrak P,\\mathfrak Z,\n\\mathfrak G,\\mathfrak B,\\mathfrak H,\\mathfrak R,\\mathfrak C),\n\\]\nwhere:\n\\begin{enumerate}\n\\item $X$ is the observation space;\n\\item $\\mathfrak U=\\{U_a\\}_{a\\in A}$ is a family of exact coordinate atlases;\n\\item $\\mathfrak E=\\{E_{a,\\alpha}\\}$ is the family of coordinate selectors;\n\\item $\\mathfrak P=\\{P_{a,\\alpha}\\}$ is the pulled-back projector family;\n\\item $\\mathfrak Z$ is a nested family of statistical states;\n\\item $\\mathfrak G$ is a typed operator-program grammar;\n\\item $\\mathfrak B$ is the Bayesian posterior over programs and parameters;\n\\item $\\mathfrak H$ is the explicit memory state;\n\\item $\\mathfrak R$ is the state-action transition graph;\n\\item $\\mathfrak C$ is the planning and reconstruction contract.\n\\end{enumerate}\n\\end{definition}","U,\\mathfrak E,\\mathfrak P,\\mathfrak Z,\n\\mathfrak G,\\mathfrak B,\\mathfrak H,\\mathfrak R,\\mathfrak C),\n\\]\nwhere:\n\\begin{enumerate}\n\\item $X$ is the observation space;\n\\item $\\mathfrak U=\\{U_a\\}_{a\\in A}$ is a family of exact coordinate atlases;\n\\item $\\mathfrak E=\\{E_{a,\\alpha}\\}$ is the family of coordinate selectors;\n\\item $\\mathfrak P=\\{P_{a,\\alpha}\\}$ is the pulled-back projector family;\n\\item $\\mathfrak Z$ is a nested family of statistical states;\n\\item $\\mathfrak G$ is a typed operator-program grammar;\n\\item $\\mathfrak B$ is the Bayesian posterior over programs and parameters;\n\\item $\\mathfrak H$ is the explicit memory state;\n\\item $\\mathfrak R$ is the state-action transition graph;\n\\item $\\mathfrak C$ is the planning and reconstruction contract.\n\\end{enumerate}\n\\end{definition}",",\\mathfrak H,\\mathfrak R,\\mathfrak C),\n\\]\nwhere:\n\\begin{enumerate}\n\\item $X$ is the observation space;\n\\item $\\mathfrak U=\\{U_a\\}_{a\\in A}$ is a family of exact coordinate atlases;\n\\item $\\mathfrak E=\\{E_{a,\\alpha}\\}$ is the family of coordinate selectors;\n\\item $\\mathfrak P=\\{P_{a,\\alpha}\\}$ is the pulled-back projector family;\n\\item $\\mathfrak Z$ is a nested family of statistical states;\n\\item $\\mathfrak G$ is a typed operator-program grammar;\n\\item $\\mathfrak B$ is the Bayesian posterior over programs and parameters;\n\\item $\\mathfrak H$ is the explicit memory state;\n\\item $\\mathfrak R$ is the state-action transition graph;\n\\item $\\mathfrak C$ is the planning and reconstruction contract.\n\\end{enumerate}\n\\end{definition}","numerate}\n\\item $X$ is the observation space;\n\\item $\\mathfrak U=\\{U_a\\}_{a\\in A}$ is a family of exact coordinate atlases;\n\\item $\\mathfrak E=\\{E_{a,\\alpha}\\}$ is the family of coordinate selectors;\n\\item $\\mathfrak P=\\{P_{a,\\alpha}\\}$ is the pulled-back projector family;\n\\item $\\mathfrak Z$ is a nested family of statistical states;\n\\item $\\mathfrak G$ is a typed operator-program grammar;\n\\item $\\mathfrak B$ is the Bayesian posterior over programs and parameters;\n\\item $\\mathfrak H$ is the explicit memory state;\n\\item $\\mathfrak R$ is the state-action transition graph;\n\\item $\\mathfrak C$ is the planning and reconstruction contract.\n\\end{enumerate}\n\\end{definition}","in A}$ is a family of exact coordinate atlases;\n\\item $\\mathfrak E=\\{E_{a,\\alpha}\\}$ is the family of coordinate selectors;\n\\item $\\mathfrak P=\\{P_{a,\\alpha}\\}$ is the pulled-back projector family;\n\\item $\\mathfrak Z$ is a nested family of statistical states;\n\\item $\\mathfrak G$ is a typed operator-program grammar;\n\\item $\\mathfrak B$ is the Bayesian posterior over programs and parameters;\n\\item $\\mathfrak H$ is the explicit memory state;\n\\item $\\mathfrak R$ is the state-action transition graph;\n\\item $\\mathfrak C$ is the planning and reconstruction contract.\n\\end{enumerate}\n\\end{definition}","tem $\\mathfrak E=\\{E_{a,\\alpha}\\}$ is the family of coordinate selectors;\n\\item $\\mathfrak P=\\{P_{a,\\alpha}\\}$ is the pulled-back projector family;\n\\item $\\mathfrak Z$ is a nested family of statistical states;\n\\item $\\mathfrak G$ is a typed operator-program grammar;\n\\item $\\mathfrak B$ is the Bayesian posterior over programs and parameters;\n\\item $\\mathfrak H$ is the explicit memory state;\n\\item $\\mathfrak R$ is the state-action transition graph;\n\\item $\\mathfrak C$ is the planning and reconstruction contract.\n\\end{enumerate}\n\\end{definition}","nate selectors;\n\\item $\\mathfrak P=\\{P_{a,\\alpha}\\}$ is the pulled-back projector family;\n\\item $\\mathfrak Z$ is a nested family of statistical states;\n\\item $\\mathfrak G$ is a typed operator-program grammar;\n\\item $\\mathfrak B$ is the Bayesian posterior over programs and parameters;\n\\item $\\mathfrak H$ is the explicit memory state;\n\\item $\\mathfrak R$ is the state-action transition graph;\n\\item $\\mathfrak C$ is the planning and reconstruction contract.\n\\end{enumerate}\n\\end{definition}","Let $X$ be a finite-dimensional Hilbert space.  For every atlas $a\\in A$, let\n\\[\nU_a:X\\to Y_a\n\\]\nbe unitary, and let $\\{E_{a,\\alpha}\\}_{\\alpha\\in I_a}$ be orthogonal coordinate projectors satisfying\n\\[\nE_{a,\\alpha}E_{a,\\beta}=\\delta_{\\alpha\\beta}E_{a,\\alpha},\n\\qquad\n\\sum_{\\alpha\\in I_a}E_{a,\\alpha}=I_{Y_a}.\n\\]\nDefine the object-space packets\n\\[\n\\boxed{P_{a,\\alpha}=U_a^*E_{a,\\alpha}U_a.}\n\\]","\\begin{theorem}[Per-atlas packet reconstruction]\nFor every atlas $a$, the operators $P_{a,\\alpha}$ are orthogonal projectors and\n\\[\n\\sum_{\\alpha\\in I_a}P_{a,\\alpha}=I_X.\n\\]\nConsequently,\n\\[\nx=\\sum_{\\alpha}P_{a,\\alpha}x,\n\\qquad\nT=\\sum_{\\alpha,\\beta}P_{a,\\beta}TP_{a,\\alpha}.\n\\]\n\\end{theorem}","\\begin{proof}\nConjugation by a unitary operator preserves multiplication, adjoints, and sums.  Therefore\n\\[\nP_{a,\\alpha}P_{a,\\beta}\n=U_a^*E_{a,\\alpha}E_{a,\\beta}U_a\n=\\delta_{\\alpha\\beta}P_{a,\\alpha},\n\\]\nand the complete sum is $U_a^*I_{Y_a}U_a=I_X$.  Inserting that identity on one or both sides gives object and operator reconstruction.\n\\end{proof}","Choose weights $w_a>0$ and define\n\\[\n\\Acal x=(w_aU_ax)_{a\\in A},\n\\]\nwith synthesis\n\\[\n\\Scal(c_a)_{a\\in A}\n=\n\\frac{1}{c}\n\\sum_{a\\in A}w_aU_a^*c_a,\n\\qquad\nc=\\sum_{a\\in A}w_a^2.\n\\]","\\begin{theorem}[Exact multi-atlas tight-frame reconstruction]\nThe analysis and synthesis maps satisfy\n\\[\n\\boxed{\\Scal\\Acal=I_X.}\n\\]\nMoreover,\n\\[\n\\norm{\\Acal x}^2=c\\norm{x}^2.\n\\]\n\\end{theorem}","\\begin{proof}\nBy unitarity,\n\\[\n\\Scal\\Acal x\n=\\frac{1}{c}\\sum_a w_aU_a^*(w_aU_ax)\n=\\frac{1}{c}\\sum_aw_a^2x=x.\n\\]\nSimilarly,\n\\[\n\\norm{\\Acal x}^2\n=\\sum_aw_a^2\\norm{U_ax}^2\n=c\\norm{x}^2.\n\\]\n\\end{proof}","The normalized positive packets\n\\[\n\\widetilde P_{a,\\alpha}\n=\\frac{w_a^2}{c}P_{a,\\alpha}\n\\]\nsatisfy\n\\[\n\\sum_{a,\\alpha}\\widetilde P_{a,\\alpha}=I_X.\n\\]\nThey 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.","For an $H\\times W$ array, the normalized two-dimensional transform is\n\\[\n(Fx)_{k,\\ell}\n=\\frac{1}{\\sqrt{HW}}\n\\sum_{m=0}^{H-1}\\sum_{n=0}^{W-1}\n x_{m,n}e^{-2\\pi i(km/H+\\ell n/W)}.\n\\]\nThe inverse is $F^*$.  Disjoint frequency masks $E_B$ produce projectors\n\\[\nP_B=F^*E_BF.\n\\]\nFourier space therefore appears inside the same packet language as any other unitary atlas.","The separable Haar transform repeatedly applies\n\\[\n(a,b)\\longmapsto\n\\left(\\frac{a+b}{\\sqrt2},\\frac{a-b}{\\sqrt2}\\right)\n\\]\nalong columns and rows.  The inverse reverses those operations.  Scale-location selectors in the packed transform pull back to exact wavelet projectors.","For modulus $N$, define\n\\[\nx_{r,s}[q,t]=x[Nq+r,Nt+s],\n\\qquad 0\\le r,s<N.\n\\]\nThe complete channel family is exactly inverted by interleaving.  In object space, the phase projector is\n\\[\n(P_{N,r,s}x)[m,n]\n=\n\\mathbf 1_{m\\equiv r\\ (N)}\n\\mathbf 1_{n\\equiv s\\ (N)}x[m,n].\n\\]","A finite-valued frame can be encoded by the one-hot coordinates\n\\[\nC_j[m,n]=\\mathbf 1_{x[m,n]=j}.\n\\]\nThe 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.","\\begin{certificate}[Exact atlas residuals]\nAcross the tested frames, the maximum observed residuals were\n\\[\n\\begin{array}{lr}\n\\toprule\n\\text{atlas}&\\text{maximum absolute residual}\\\\\n\\midrule\n\\text{spatial}&0\\\\\n\\text{Fourier}&3.55\\times10^{-15}\\\\\n\\text{Haar}&2.49\\times10^{-14}\\\\\n\\text{polyphase-2}&0\\\\\n\\text{polyphase-4}&0\\\\\n\\text{categorical color}&0\\\\\n\\text{weighted multi-atlas frame}&5.33\\times10^{-15}\\\\\n\\bottomrule\n\\end{array}\n\\]\nThe nonzero values are double-precision roundoff.\n\\end{certificate}","Let $r(x)\\in\\R^d$ be the concatenated feature vector.  A deterministic seeded QR factorization produces an orthonormal row system $Q$.  The nested states are\n\\[\nz^{\\mathrm{macro}}=Q_{1:k_1}r(x),\n\\qquad\nz^{\\mathrm{meso}}=Q_{1:k_2}r(x),\n\\qquad\nz^{\\mathrm{micro}}=Q_{1:k_3}r(x),\n\\]\nwith\n\\[\nk_1<k_2<k_3<d.\n\\]\nThe supplied dimensions are $48$, $96$, and $192$.","\\begin{definition}[Complete pruning]\nA set $\\Lambda$ of packet-tree nodes is a complete pruning when every root-to-leaf path meets $\\Lambda$ exactly once.\n\\end{definition}","Let $c(v)$ be any finite additive node cost.  Define recursively\n\\[\nV(v)=\n\\min\\left\\{c(v),\\sum_{u\\in\\operatorname{ch}(v)}V(u)\\right\\}.\n\\]","In twenty-four structured frame tests, the selected basis used between $11$ and $61$ packets, with mean $32.58$.  The maximum reconstruction residual was $1.95\\times10^{-14}$.","A large signal model should not hide all mechanism uncertainty inside one dense matrix.  PR-LASM uses a typed program\n\\[\ng=(a,r,f,\\kappa),\n\\]\nwhere $a$ is an atlas, $r$ is a resolution, $f$ is an operator family, and $\\kappa$ contains family-specific ranks or periods.","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\n\\[\n\\phi_g(z_t,q_t,u_t,m_t,t)\\in\\R^{d_g}.\n\\]\nThe target is\n\\[\ny_t^{(g)}=\n\\begin{bmatrix}\n z_{t+1}^{(g)}-z_t^{(g)}","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\n\\[\n\\phi_g(z_t,q_t,u_t,m_t,t)\\in\\R^{d_g}.\n\\]\nThe target is\n\\[\ny_t^{(g)}=\n\\begin{bmatrix}\n z_{t+1}^{(g)}-z_t^{(g)}\\\\\n R_t/100\\\\\n \\Delta L_t\\\\\n \\mathbf 1_{\\mathrm{win}}\\\\\n \\mathbf 1_{\\mathrm{loss}}\\\\\n \\mathbf 1_{\\mathrm{changed}}\n\\end{bmatrix}.\n\\]","The basic linear program uses\n\\[\n\\phi_{\\mathrm{lin}}=[1,z_t,q_t,u_t].\n\\]\nThe action-bilinear extension adds products\n\\[\nz_{t,i}u_{t,j},\n\\qquad\nq_{t,i}u_{t,j},\n\\]\nfor a declared low interaction rank.","The quadratic program adds\n\\[\nz_{t,i}z_{t,j},\n\\qquad\nq_{t,i}q_{t,j},\n\\qquad i\\le j,\n\\]\nwithin a controlled rank.  This is a finite packet jet, not an unrestricted polynomial expansion.","The delayed program appends $z_{t-1}$.  The power-memory packet is\n\\[\nm_t^{\\mathrm{pow}}\n=\n\\frac{\\displaystyle\\sum_{\\ell=1}^{H}\\ell^{-\\alpha}z_{t-\\ell}}\n{\\displaystyle\\sum_{\\ell=1}^{H}\\ell^{-\\alpha}},\n\\]\nwhile the exponential packet is\n\\[\nm_t^{\\mathrm{exp}}\n=\n\\frac{\\displaystyle\\sum_{\\ell=1}^{H}\\rho^{\\ell-1}z_{t-\\ell}}\n{\\displaystyle\\sum_{\\ell=1}^{H}\\rho^{\\ell-1}}.\n\\]\nThe supplied defaults are $\\alpha=0.68$ and $\\rho=0.82$.","ram appends $z_{t-1}$.  The power-memory packet is\n\\[\nm_t^{\\mathrm{pow}}\n=\n\\frac{\\displaystyle\\sum_{\\ell=1}^{H}\\ell^{-\\alpha}z_{t-\\ell}}\n{\\displaystyle\\sum_{\\ell=1}^{H}\\ell^{-\\alpha}},\n\\]\nwhile the exponential packet is\n\\[\nm_t^{\\mathrm{exp}}\n=\n\\frac{\\displaystyle\\sum_{\\ell=1}^{H}\\rho^{\\ell-1}z_{t-\\ell}}\n{\\displaystyle\\sum_{\\ell=1}^{H}\\rho^{\\ell-1}}.\n\\]\nThe supplied defaults are $\\alpha=0.68$ and $\\rho=0.82$.","For modulus $N$, the phase program appends the one-hot phase\n\\[\ne_{t\\bmod N}\n\\]\nand interactions $z_{t,i}e_{t\\bmod N}$.  Alternating, cyclic, or clocked dynamics therefore become linear in an expanded polyphase state.","The gated program uses a smooth gate\n\\[\n\\gamma_t=\\frac{1}{1+e^{-4\\bar q_t}}\n\\]\nand appends $\\gamma_tz_t$ and $(1-\\gamma_t)z_t$.  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.","For one program $g$, write\n\\[\nY=X_gB_g^\\top+\\varepsilon,\n\\]\nwith independent output variances and a normal-inverse-gamma prior.  If the prior precision is $\\Lambda_0=\\lambda_0I$, then\n\\[\n\\Lambda_n=\\Lambda_0+X_g^\\top X_g,\n\\qquad\nM_n=Y^\\top X_g\\Lambda_n^{-1}.\n\\]\nThe posterior coefficient covariance is $\\Lambda_n^{-1}$, and each output variance has an inverse-gamma posterior.","The implementation computes held-out Student predictive likelihood when enough rows are available.  The program log weight is\n\\[\n\\boxed{\nW_g=\n\\ell_{\\mathrm{pred}}(g)\n-\\lambda\\,\\operatorname{Complexity}(g)\n-\\eta\\,\\operatorname{Mechanisms}(g)\n-\\rho\\,\\operatorname{ResolutionPenalty}(g).}\n\\]\nThe posterior is\n\\[\np(g\\mid D)=\\frac{e^{W_g}}{\\sum_he^{W_h}}.\n\\]","If program $g$ predicts scalar vector $\\mu_g$ with variance $V_g$, then\n\\[\n\\bar\\mu=\\sum_gp_g\\mu_g,\n\\]\nand the total predictive variance is\n\\[\n\\boxed{\nV_{\\mathrm{total}}\n=\n\\underbrace{\\sum_gp_gV_g}_{\\text{within-program}}\n+\n\\underbrace{\\sum_gp_g(\\mu_g-\\bar\\mu)^2}_{\\text{between-program}}.}\n\\]\nThe second term is model disagreement and drives information-seeking behavior.","Each resolved frame receives a deterministic hash of its complete normalized array.  An observed public action defines a graph edge\n\\[\n(s,a)\\longmapsto s'.\n\\]\nRepeated observations accumulate reward, progress, win, loss, and visit counts.","For candidate action $a$, the planner forms a utility\n\\[\n\\begin{aligned}\nU(a)=&\\ \\widehat R(a)\n+\\lambda_L\\widehat{\\Delta L}(a)\n+\\lambda_W\\widehat p_{\\mathrm{win}}(a)\n-\\lambda_D\\widehat p_{\\mathrm{loss}}(a)","For candidate action $a$, the planner forms a utility\n\\[\n\\begin{aligned}\nU(a)=&\\ \\widehat R(a)\n+\\lambda_L\\widehat{\\Delta L}(a)\n+\\lambda_W\\widehat p_{\\mathrm{win}}(a)\n-\\lambda_D\\widehat p_{\\mathrm{loss}}(a)\\\\\n&+\\lambda_N\\operatorname{Novelty}(a)\n+\\lambda_I\\operatorname{Information}(a)\n-\\lambda_V\\operatorname{Variance}(a)\\\\\n&+\\lambda_H\\operatorname{RoleHistory}(a)\n-\\lambda_Q\\operatorname{RepeatCount}(a)^2\n+\\lambda_F\\mathbf1_{\\mathrm{untried}}.\n\\end{aligned}\n\\]\nExact 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.","The model loop is\n\\[\n\\boxed{\n\\text{analyze exactly}\n\\to\n\\text{fit typed posterior}\n\\to\n\\text{predict and act}\n\\to\n\\text{measure residual}\n\\to\n\\text{refine selectively}.}\n\\]\nA residual can trigger:\n\\begin{itemize}\n\\item a finer statistical projection;\n\\item a different atlas expert;\n\\item a packet-tree split;\n\\item a delay or memory mechanism;\n\\item an action interaction;\n\\item an explicit graph edge rather than a learned approximation.\n\\end{itemize}","For $n$ signal coordinates, $A$ exact unitary atlases, projected state dimension $k$, $G$ candidate programs, and $m$ observed transitions, the dominant modeled costs are:\n\\[\n\\begin{array}{ll}\n\\text{FFT atlas}&O(n\\log n),","For $n$ signal coordinates, $A$ exact unitary atlases, projected state dimension $k$, $G$ candidate programs, and $m$ observed transitions, the dominant modeled costs are:\n\\[\n\\begin{array}{ll}\n\\text{FFT atlas}&O(n\\log n),\\\\\n\\text{Haar atlas}&O(n),\\\\\n\\text{polyphase atlas}&O(n),\\\\\n\\text{feature projection}&O(dk),\\\\\n\\text{one regression fit}&O(md_g^2+d_g^3),\\\\\n\\text{all program fits}&\\displaystyle\\sum_{g=1}^GO(md_g^2+d_g^3).\n\\end{array}\n\\]\nThe architecture is designed so that $d_g$ is controlled by the selected rank and resolution rather than by the full signal dimension.","\\text{all program fits}&\\displaystyle\\sum_{g=1}^GO(md_g^2+d_g^3).\n\\end{array}\n\\]\nThe architecture is designed so that $d_g$ is controlled by the selected rank and resolution rather than by the full signal dimension.","For each system, $84$ records were used for fitting and $36$ independent records for testing.  The target-driving variables absent from simpler candidate programs were independently randomized.  This prevents the benchmark from being solved by time-index memorization.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.92\\linewidth]{17_rmse_ratios.pdf}\n\\caption{All seven state RMSE ratios are below one; the median is $0.2393$.}\n\\end{figure}","\\begin{certificate}[Interactive systems]\nPR-LASM solved all four systems.  The recorded committed-action counts were $38$, $1$, $29$, and $14$.  The corresponding graph sizes ranged from one to twenty states and from twelve to thirty-two transition edges.\n\\end{certificate}","\\begin{certificate}[Automated suite]\nThe complete source release passes\n\\[\n\\boxed{36/36\\text{ automated tests}.}\n\\]\nThe tests include all seventeen inherited offline-runner tests and nineteen PR-LASM tests covering exact atlas reconstruction, Fourier packet completeness, polyphase synthesis, wavelet subband synthesis, tight-frame reconstruction, deterministic packetization, mixed-depth best-basis reconstruction, graph frontiers, seven typed program-identification cases, and four interactive systems.\n\\end{certificate}","The result is not a language model disguised as a signal model, and it is not a dense predictor decorated with wavelets after the fact.  It is an algebraic production system:\n\\begin{center}\n\\fbox{\\begin{minipage}{0.91\\linewidth}\n\\centering\n\\textbf{signal object}\n$\\to$\n\\textbf{exact atlas portfolio}\n$\\to$\n\\textbf{packet features and nested states}\n$\\to$\n\\textbf{typed operator posterior}","[4pt]\n$\\to$\n\\textbf{memory and transition graph}\n$\\to$\n\\textbf{active action}\n$\\to$\n\\textbf{residual}\n$\\to$\n\\textbf{selective refinement}\n$\\to$\n\\textbf{exact reconstruction}\n\\end{minipage}}\n\\end{center}\nWithin its finite declared scope, that loop is complete.\n\n\\appendix\n\\clearpage\n\\section{Exact transform formulas}\n\n\\subsection{Haar analysis and synthesis}\nFor one pair,\n\\[\n\\ell=\\frac{a+b}{\\sqrt2},\\qquad h=\\frac{a-b}{\\sqrt2},\n\\]\nand\n\\[\na=\\frac{\\ell+h}{\\sqrt2},\\qquad b=\\frac{\\ell-h}{\\sqrt2}.\n\\]\nApplying these identities separably and recursively proves the finite two-dimensional transform used in the code.","\\subsection{Haar analysis and synthesis}\nFor one pair,\n\\[\n\\ell=\\frac{a+b}{\\sqrt2},\\qquad h=\\frac{a-b}{\\sqrt2},\n\\]\nand\n\\[\na=\\frac{\\ell+h}{\\sqrt2},\\qquad b=\\frac{\\ell-h}{\\sqrt2}.\n\\]\nApplying these identities separably and recursively proves the finite two-dimensional transform used in the code.","\\subsection{Polyphase analysis and synthesis}\nFor modulus $N$,\n\\[\n(\\Acal_Nx)_{r,s}[q,t]=x[Nq+r,Nt+s].\n\\]\nThe synthesis operator places each channel value back at its unique original address.  Therefore\n\\[\n\\Scal_N\\Acal_N=I.\n\\]","\\subsection{Fourier selector projectors}\nFor any diagonal zero-one mask $E_B$,\n\\[\nP_B=F^*E_BF,\n\\]\nso\n\\[\nP_B^2=P_B,\\qquad P_B^*=P_B.\n\\]\nDisjoint masks annihilate, and a complete mask partition sums to the identity.","For one output $y$ and design matrix $X$, take\n\\[\n\\beta\\mid\\sigma^2\\sim N(0,\\sigma^2\\Lambda_0^{-1}),\n\\qquad\n\\sigma^2\\sim\\operatorname{InvGamma}(a_0,b_0).\n\\]\nThen\n\\[\n\\Lambda_n=\\Lambda_0+X^\\top X,\n\\qquad\n\\mu_n=\\Lambda_n^{-1}X^\\top y,\n\\]\n\\[\na_n=a_0+\\frac n2,\n\\qquad\nb_n=b_0+\\frac12(y^\\top y-\\mu_n^\\top\\Lambda_n\\mu_n).\n\\]\nFor a feature row $x_*$, the predictive mean is $x_*^\\top\\mu_n$ and the Student predictive scale contains\n\\[\n1+x_*^\\top\\Lambda_n^{-1}x_*.\n\\]\nThe implementation applies this independently to every output coordinate while sharing the design covariance.","ign matrix $X$, take\n\\[\n\\beta\\mid\\sigma^2\\sim N(0,\\sigma^2\\Lambda_0^{-1}),\n\\qquad\n\\sigma^2\\sim\\operatorname{InvGamma}(a_0,b_0).\n\\]\nThen\n\\[\n\\Lambda_n=\\Lambda_0+X^\\top X,\n\\qquad\n\\mu_n=\\Lambda_n^{-1}X^\\top y,\n\\]\n\\[\na_n=a_0+\\frac n2,\n\\qquad\nb_n=b_0+\\frac12(y^\\top y-\\mu_n^\\top\\Lambda_n\\mu_n).\n\\]\nFor a feature row $x_*$, the predictive mean is $x_*^\\top\\mu_n$ and the Student predictive scale contains\n\\[\n1+x_*^\\top\\Lambda_n^{-1}x_*.\n\\]\nThe implementation applies this independently to every output coordinate while sharing the design covariance."]}