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38   pdftitle={Projector-Resolved Large Algebraic Signal Model},
39   pdfauthor={Komorebi P.},
40   pdfsubject={Exact multi-atlas coordinates, Bayesian operator programs, residual-driven refinement, memory, and active planning},
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107 \begin{titlepage}
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115 \vspace*{1.0in}
116 {\large\color{Blue}\bfseries PROJECTOR-RESOLVED SEED CALCULUS: CAPSTONE MODEL\par}
117 \vspace{0.25in}
118 {\fontsize{31}{35}\selectfont\bfseries Projector-Resolved\\[3pt]Large Algebraic Signal Model\par}
119 \vspace{0.18in}
120 {\fontsize{17}{21}\selectfont\color{Muted} Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning\par}
121 \vspace{0.55in}
122 \begin{minipage}{0.80\linewidth}
123 \large
124 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.
125 \end{minipage}
126 \vfill
127 {\Large\bfseries Komorebi P.\par}
128 \vspace{0.45in}
129 \end{titlepage}
130 
131 \begin{abstract}
132 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.
133 
134 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.
135 
136 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.
137 \end{abstract}
138 
139 \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.}
140 
141 \tableofcontents
142 \clearpage
143 
144 \section{The final construction problem}
145 
146 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.
147 
148 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.
149 
150 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.
151 
152 \begin{figure}[H]
153 \centering
154 \includegraphics[width=.98\linewidth]{01_pr_lasm_architecture.pdf}
155 \caption{The completed PR-LASM architecture.  Exact analysis, statistical inference, graph memory, active planning, and reconstruction form one loop.}
156 \end{figure}
157 
158 \subsection{What ``large'' means}
159 
160 The word \emph{large} does not refer only to parameter count.  The model is large because it maintains a structured family of possibilities:
161 \[
162 \boxed{
163 \text{atlas}\times\text{resolution}\times\text{operator program}
164 \times\text{memory}\times\text{packet refinement}\times\text{action}.}
165 \]
166 A small number of parameters inside many typed alternatives can express more structural uncertainty than one dense map with the same total coefficient count.  The model is algebraic because every alternative has an explicit coordinate map, feature map, operator design, probability, and reconstruction status.
167 
168 \subsection{The strongest defensible claim}
169 
170 The architecture is a complete candidate for a projector-resolved, exact-reconstruction, multi-atlas, fully algebraic large signal model.  ``Complete'' here means that the finite architecture closes its own declared loop: observation, exact analysis, compression, inference, memory, planning, action, residual measurement, refinement, and reconstruction.  It does not mean that the operator grammar contains every possible law of nature or that the planner solves every interactive environment.
171 
172 \section{The PR-LASM object}
173 
174 \begin{definition}[Projector-resolved large signal model]
175 A finite PR-LASM instance is a tuple
176 \[
177 \mathfrak L=
178 (X,\mathfrak U,\mathfrak E,\mathfrak P,\mathfrak Z,
179 \mathfrak G,\mathfrak B,\mathfrak H,\mathfrak R,\mathfrak C),
180 \]
181 where:
182 \begin{enumerate}
183 \item $X$ is the observation space;
184 \item $\mathfrak U=\{U_a\}_{a\in A}$ is a family of exact coordinate atlases;
185 \item $\mathfrak E=\{E_{a,\alpha}\}$ is the family of coordinate selectors;
186 \item $\mathfrak P=\{P_{a,\alpha}\}$ is the pulled-back projector family;
187 \item $\mathfrak Z$ is a nested family of statistical states;
188 \item $\mathfrak G$ is a typed operator-program grammar;
189 \item $\mathfrak B$ is the Bayesian posterior over programs and parameters;
190 \item $\mathfrak H$ is the explicit memory state;
191 \item $\mathfrak R$ is the state-action transition graph;
192 \item $\mathfrak C$ is the planning and reconstruction contract.
193 \end{enumerate}
194 \end{definition}
195 
196 The tuple separates exact coordinates from learned states.  This is not merely editorial.  It prevents a low prediction error in a compressed state from being reported as exact object reconstruction.
197 
198 \begin{figure}[H]
199 \centering
200 \includegraphics[width=.96\linewidth]{02_exact_archive_vs_statistical_state.pdf}
201 \caption{The exact coordinate archive and the statistical state serve different purposes.}
202 \end{figure}
203 
204 \section{Exact multi-atlas coordinates}
205 
206 Let $X$ be a finite-dimensional Hilbert space.  For every atlas $a\in A$, let
207 \[
208 U_a:X\to Y_a
209 \]
210 be unitary, and let $\{E_{a,\alpha}\}_{\alpha\in I_a}$ be orthogonal coordinate projectors satisfying
211 \[
212 E_{a,\alpha}E_{a,\beta}=\delta_{\alpha\beta}E_{a,\alpha},
213 \qquad
214 \sum_{\alpha\in I_a}E_{a,\alpha}=I_{Y_a}.
215 \]
216 Define the object-space packets
217 \[
218 \boxed{P_{a,\alpha}=U_a^*E_{a,\alpha}U_a.}
219 \]
220 
221 \begin{theorem}[Per-atlas packet reconstruction]
222 For every atlas $a$, the operators $P_{a,\alpha}$ are orthogonal projectors and
223 \[
224 \sum_{\alpha\in I_a}P_{a,\alpha}=I_X.
225 \]
226 Consequently,
227 \[
228 x=\sum_{\alpha}P_{a,\alpha}x,
229 \qquad
230 T=\sum_{\alpha,\beta}P_{a,\beta}TP_{a,\alpha}.
231 \]
232 \end{theorem}
233 
234 \begin{proof}
235 Conjugation by a unitary operator preserves multiplication, adjoints, and sums.  Therefore
236 \[
237 P_{a,\alpha}P_{a,\beta}
238 =U_a^*E_{a,\alpha}E_{a,\beta}U_a
239 =\delta_{\alpha\beta}P_{a,\alpha},
240 \]
241 and 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.
242 \end{proof}
243 
244 \subsection{A redundant tight-frame atlas}
245 
246 Choose weights $w_a>0$ and define
247 \[
248 \Acal x=(w_aU_ax)_{a\in A},
249 \]
250 with synthesis
251 \[
252 \Scal(c_a)_{a\in A}
253 =
254 \frac{1}{c}
255 \sum_{a\in A}w_aU_a^*c_a,
256 \qquad
257 c=\sum_{a\in A}w_a^2.
258 \]
259 
260 \begin{theorem}[Exact multi-atlas tight-frame reconstruction]
261 The analysis and synthesis maps satisfy
262 \[
263 \boxed{\Scal\Acal=I_X.}
264 \]
265 Moreover,
266 \[
267 \norm{\Acal x}^2=c\norm{x}^2.
268 \]
269 \end{theorem}
270 
271 \begin{proof}
272 By unitarity,
273 \[
274 \Scal\Acal x
275 =\frac{1}{c}\sum_a w_aU_a^*(w_aU_ax)
276 =\frac{1}{c}\sum_aw_a^2x=x.
277 \]
278 Similarly,
279 \[
280 \norm{\Acal x}^2
281 =\sum_aw_a^2\norm{U_ax}^2
282 =c\norm{x}^2.
283 \]
284 \end{proof}
285 
286 \begin{figure}[H]
287 \centering
288 \includegraphics[width=.96\linewidth]{03_multi_atlas_tight_frame.pdf}
289 \caption{The canonical dual averages exact reconstructions with the correct frame normalization.}
290 \end{figure}
291 
292 The normalized positive packets
293 \[
294 \widetilde P_{a,\alpha}
295 =\frac{w_a^2}{c}P_{a,\alpha}
296 \]
297 satisfy
298 \[
299 \sum_{a,\alpha}\widetilde P_{a,\alpha}=I_X.
300 \]
301 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.
302 
303 \section{Concrete coordinate atlases}
304 
305 \subsection{Spatial coordinates}
306 
307 The identity atlas stores the observation itself.  It is the natural chart for local geometry, connected components, boundaries, coordinates, and direct pixel or field values.
308 
309 \begin{figure}[H]
310 \centering
311 \includegraphics[width=.68\linewidth]{04_sample_signal.pdf}
312 \caption{A sample finite signal in its spatial chart.}
313 \end{figure}
314 
315 \subsection{Fourier coordinates}
316 
317 For an $H\times W$ array, the normalized two-dimensional transform is
318 \[
319 (Fx)_{k,\ell}
320 =\frac{1}{\sqrt{HW}}
321 \sum_{m=0}^{H-1}\sum_{n=0}^{W-1}
322  x_{m,n}e^{-2\pi i(km/H+\ell n/W)}.
323 \]
324 The inverse is $F^*$.  Disjoint frequency masks $E_B$ produce projectors
325 \[
326 P_B=F^*E_BF.
327 \]
328 Fourier space therefore appears inside the same packet language as any other unitary atlas.
329 
330 \begin{figure}[H]
331 \centering
332 \includegraphics[width=.68\linewidth]{05_fourier_coordinates.pdf}
333 \caption{Global oscillatory structure becomes visible in Fourier coordinates.}
334 \end{figure}
335 
336 \subsection{Haar-wavelet coordinates}
337 
338 The separable Haar transform repeatedly applies
339 \[
340 (a,b)\longmapsto
341 \left(\frac{a+b}{\sqrt2},\frac{a-b}{\sqrt2}\right)
342 \]
343 along columns and rows.  The inverse reverses those operations.  Scale-location selectors in the packed transform pull back to exact wavelet projectors.
344 
345 \begin{figure}[H]
346 \centering
347 \includegraphics[width=.68\linewidth]{06_haar_coordinates.pdf}
348 \caption{The same object in a multilevel wavelet chart.}
349 \end{figure}
350 
351 \subsection{Two-dimensional polyphase coordinates}
352 
353 For modulus $N$, define
354 \[
355 x_{r,s}[q,t]=x[Nq+r,Nt+s],
356 \qquad 0\le r,s<N.
357 \]
358 The complete channel family is exactly inverted by interleaving.  In object space, the phase projector is
359 \[
360 (P_{N,r,s}x)[m,n]
361 =
362 \mathbf 1_{m\equiv r\ (N)}
363 \mathbf 1_{n\equiv s\ (N)}x[m,n].
364 \]
365 
366 \begin{figure}[H]
367 \centering
368 \includegraphics[width=.82\linewidth]{07_polyphase_channels.pdf}
369 \caption{Polyphase coordinates expose address classes without dropping samples.}
370 \end{figure}
371 
372 \subsection{Categorical color packets}
373 
374 A finite-valued frame can be encoded by the one-hot coordinates
375 \[
376 C_j[m,n]=\mathbf 1_{x[m,n]=j}.
377 \]
378 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.
379 
380 \begin{figure}[H]
381 \centering
382 \includegraphics[width=.82\linewidth]{08_color_packets.pdf}
383 \caption{Color packets preserve categorical mass and exact label reconstruction.}
384 \end{figure}
385 
386 \subsection{Reconstruction certificates}
387 
388 \begin{certificate}[Exact atlas residuals]
389 Across the tested frames, the maximum observed residuals were
390 \[
391 \begin{array}{lr}
392 \toprule
393 \text{atlas}&\text{maximum absolute residual}\\
394 \midrule
395 \text{spatial}&0\\
396 \text{Fourier}&3.55\times10^{-15}\\
397 \text{Haar}&2.49\times10^{-14}\\
398 \text{polyphase-2}&0\\
399 \text{polyphase-4}&0\\
400 \text{categorical color}&0\\
401 \text{weighted multi-atlas frame}&5.33\times10^{-15}\\
402 \bottomrule
403 \end{array}
404 \]
405 The nonzero values are double-precision roundoff.
406 \end{certificate}
407 
408 \begin{figure}[H]
409 \centering
410 \includegraphics[width=.88\linewidth]{09_reconstruction_certificates.pdf}
411 \caption{The exact coordinate archive is accompanied by explicit numerical residuals.}
412 \end{figure}
413 
414 \section{From exact coordinates to nested statistical states}
415 
416 The exact archive is too large to place directly inside every candidate regression.  PR-LASM therefore computes interpretable summaries:
417 \begin{itemize}
418 \item pooled spatial values at several block sizes;
419 \item color masses and first spatial moments;
420 \item radial and angular Fourier energies plus low-frequency complex coefficients;
421 \item wavelet subband energies and coarse signed coefficients;
422 \item phase-channel means, deviations, and occupancy values;
423 \item connected-component, boundary, and change descriptors inherited from the interactive packetizer.
424 \end{itemize}
425 
426 \begin{figure}[H]
427 \centering
428 \includegraphics[width=.88\linewidth]{10_feature_portfolio.pdf}
429 \caption{The feature groups remain named before they are projected.}
430 \end{figure}
431 
432 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
433 \[
434 z^{\mathrm{macro}}=Q_{1:k_1}r(x),
435 \qquad
436 z^{\mathrm{meso}}=Q_{1:k_2}r(x),
437 \qquad
438 z^{\mathrm{micro}}=Q_{1:k_3}r(x),
439 \]
440 with
441 \[
442 k_1<k_2<k_3<d.
443 \]
444 The supplied dimensions are $48$, $96$, and $192$.
445 
446 \begin{boundary}[Projection status]
447 The nested states are deterministic compressions.  They are not claimed to reconstruct the original frame.  Exact reconstruction belongs to the coordinate archive, not to the statistical projections.
448 \end{boundary}
449 
450 \section{Adaptive exact packet refinement}
451 
452 A packet description should become finer only where the added structure improves the declared objective.  The release uses a complete Haar packet tree on a finite vector.
453 
454 \begin{definition}[Complete pruning]
455 A set $\Lambda$ of packet-tree nodes is a complete pruning when every root-to-leaf path meets $\Lambda$ exactly once.
456 \end{definition}
457 
458 \begin{theorem}[Exact mixed-depth synthesis]
459 Every complete pruning reconstructs the root signal exactly.
460 \end{theorem}
461 
462 \begin{proof}
463 Every selected node either is the root or has a unique sibling selected or reconstructed at the same depth.  Applying inverse Haar synthesis to sibling pairs moves the selected representation one level upward.  Repeating finitely many times produces the root.  The complete-pruning condition prevents both missing branches and overlapping ancestor-descendant selections.
464 \end{proof}
465 
466 Let $c(v)$ be any finite additive node cost.  Define recursively
467 \[
468 V(v)=
469 \min\left\{c(v),\sum_{u\in\operatorname{ch}(v)}V(u)\right\}.
470 \]
471 
472 \begin{theorem}[Finite best-basis dynamic program]
473 The recursion returns a globally minimum-cost complete pruning in time linear in the number of tree nodes.
474 \end{theorem}
475 
476 \begin{proof}
477 At a leaf, the only complete pruning is the leaf.  At an internal node, every complete pruning either keeps the node or uses complete prunings of both children.  Induction on depth proves optimality, and each node is evaluated once.
478 \end{proof}
479 
480 \begin{figure}[H]
481 \centering
482 \includegraphics[width=.98\linewidth]{11_adaptive_packet_tree.pdf}
483 \caption{A mixed-depth complete pruning selected by dynamic programming.}
484 \end{figure}
485 
486 \begin{figure}[H]
487 \centering
488 \includegraphics[width=.88\linewidth]{12_packet_basis_depths.pdf}
489 \caption{Selected packet counts and energy distribution across refinement depth.}
490 \end{figure}
491 
492 In twenty-four structured frame tests, the selected basis used between $11$ and $61$ packets, with mean $32.58$.  The maximum reconstruction residual was $1.95\times10^{-14}$.
493 
494 \section{The typed operator-program grammar}
495 
496 A large signal model should not hide all mechanism uncertainty inside one dense matrix.  PR-LASM uses a typed program
497 \[
498 g=(a,r,f,\kappa),
499 \]
500 where $a$ is an atlas, $r$ is a resolution, $f$ is an operator family, and $\kappa$ contains family-specific ranks or periods.
501 
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
503 \[
504 \phi_g(z_t,q_t,u_t,m_t,t)\in\R^{d_g}.
505 \]
506 The target is
507 \[
508 y_t^{(g)}=
509 \begin{bmatrix}
510  z_{t+1}^{(g)}-z_t^{(g)}\\
511  R_t/100\\
512  \Delta L_t\\
513  \mathbf 1_{\mathrm{win}}\\
514  \mathbf 1_{\mathrm{loss}}\\
515  \mathbf 1_{\mathrm{changed}}
516 \end{bmatrix}.
517 \]
518 
519 \begin{figure}[H]
520 \centering
521 \includegraphics[width=.98\linewidth]{13_operator_grammar.pdf}
522 \caption{The finite grammar keeps mechanism names and penalties explicit.}
523 \end{figure}
524 
525 \subsection{Linear and action-bilinear programs}
526 
527 The basic linear program uses
528 \[
529 \phi_{\mathrm{lin}}=[1,z_t,q_t,u_t].
530 \]
531 The action-bilinear extension adds products
532 \[
533 z_{t,i}u_{t,j},
534 \qquad
535 q_{t,i}u_{t,j},
536 \]
537 for a declared low interaction rank.
538 
539 \subsection{Quadratic packet interactions}
540 
541 The quadratic program adds
542 \[
543 z_{t,i}z_{t,j},
544 \qquad
545 q_{t,i}q_{t,j},
546 \qquad i\le j,
547 \]
548 within a controlled rank.  This is a finite packet jet, not an unrestricted polynomial expansion.
549 
550 \subsection{Delay and memory programs}
551 
552 The delayed program appends $z_{t-1}$.  The power-memory packet is
553 \[
554 m_t^{\mathrm{pow}}
555 =
556 \frac{\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}z_{t-\ell}}
557 {\displaystyle\sum_{\ell=1}^{H}\ell^{-\alpha}},
558 \]
559 while the exponential packet is
560 \[
561 m_t^{\mathrm{exp}}
562 =
563 \frac{\displaystyle\sum_{\ell=1}^{H}\rho^{\ell-1}z_{t-\ell}}
564 {\displaystyle\sum_{\ell=1}^{H}\rho^{\ell-1}}.
565 \]
566 The supplied defaults are $\alpha=0.68$ and $\rho=0.82$.
567 
568 \begin{figure}[H]
569 \centering
570 \includegraphics[width=.88\linewidth]{18_memory_kernels.pdf}
571 \caption{Power-law and exponential histories are retained as distinct memory mechanisms.}
572 \end{figure}
573 
574 \subsection{Phase-conditioned programs}
575 
576 For modulus $N$, the phase program appends the one-hot phase
577 \[
578 e_{t\bmod N}
579 \]
580 and interactions $z_{t,i}e_{t\bmod N}$.  Alternating, cyclic, or clocked dynamics therefore become linear in an expanded polyphase state.
581 
582 \begin{figure}[H]
583 \centering
584 \includegraphics[width=.90\linewidth]{19_phase_conditioned_dynamics.pdf}
585 \caption{A periodic mechanism is simpler in its natural phase chart.}
586 \end{figure}
587 
588 \subsection{Gated and hybrid programs}
589 
590 The gated program uses a smooth gate
591 \[
592 \gamma_t=\frac{1}{1+e^{-4\bar q_t}}
593 \]
594 and 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.
595 
596 \section{Bayesian operator selection}
597 
598 For one program $g$, write
599 \[
600 Y=X_gB_g^\top+\varepsilon,
601 \]
602 with independent output variances and a normal-inverse-gamma prior.  If the prior precision is $\Lambda_0=\lambda_0I$, then
603 \[
604 \Lambda_n=\Lambda_0+X_g^\top X_g,
605 \qquad
606 M_n=Y^\top X_g\Lambda_n^{-1}.
607 \]
608 The posterior coefficient covariance is $\Lambda_n^{-1}$, and each output variance has an inverse-gamma posterior.
609 
610 The implementation computes held-out Student predictive likelihood when enough rows are available.  The program log weight is
611 \[
612 \boxed{
613 W_g=
614 \ell_{\mathrm{pred}}(g)
615 -\lambda\,\operatorname{Complexity}(g)
616 -\eta\,\operatorname{Mechanisms}(g)
617 -\rho\,\operatorname{ResolutionPenalty}(g).}
618 \]
619 The posterior is
620 \[
621 p(g\mid D)=\frac{e^{W_g}}{\sum_he^{W_h}}.
622 \]
623 
624 \subsection{Posterior predictive mixture}
625 
626 If program $g$ predicts scalar vector $\mu_g$ with variance $V_g$, then
627 \[
628 \bar\mu=\sum_gp_g\mu_g,
629 \]
630 and the total predictive variance is
631 \[
632 \boxed{
633 V_{\mathrm{total}}
634 =
635 \underbrace{\sum_gp_gV_g}_{\text{within-program}}
636 +
637 \underbrace{\sum_gp_g(\mu_g-\bar\mu)^2}_{\text{between-program}}.}
638 \]
639 The second term is model disagreement and drives information-seeking behavior.
640 
641 \section{Exact transition memory}
642 
643 Each resolved frame receives a deterministic hash of its complete normalized array.  An observed public action defines a graph edge
644 \[
645 (s,a)\longmapsto s'.
646 \]
647 Repeated observations accumulate reward, progress, win, loss, and visit counts.
648 
649 \begin{definition}[Deterministic replay edge]
650 An edge is deterministic when repeated replay of the same action history and current action returns the same state signature.  If two targets are observed, the edge is marked nondeterministic and is excluded from exact shortest-path reasoning.
651 \end{definition}
652 
653 The graph supports:
654 \begin{itemize}
655 \item known winning and progressing actions;
656 \item untried actions at the current state;
657 \item frontier states with incomplete action coverage;
658 \item shortest known paths to a frontier;
659 \item loop and failure counts.
660 \end{itemize}
661 
662 \begin{figure}[H]
663 \centering
664 \includegraphics[width=.94\linewidth]{20_state_transition_graph.pdf}
665 \caption{Exact graph memory complements rather than replaces the learned world model.}
666 \end{figure}
667 
668 \begin{proposition}[Replay safety rule]
669 If replay of the committed history produces a signature different from the expected current signature, all later replay probes are disabled for that run.
670 \end{proposition}
671 
672 This rule sacrifices probe efficiency rather than treating a nondeterministic or hidden-state environment as deterministic.
673 
674 \section{Active planning}
675 
676 For candidate action $a$, the planner forms a utility
677 \[
678 \begin{aligned}
679 U(a)=&\ \widehat R(a)
680 +\lambda_L\widehat{\Delta L}(a)
681 +\lambda_W\widehat p_{\mathrm{win}}(a)
682 -\lambda_D\widehat p_{\mathrm{loss}}(a)\\
683 &+\lambda_N\operatorname{Novelty}(a)
684 +\lambda_I\operatorname{Information}(a)
685 -\lambda_V\operatorname{Variance}(a)\\
686 &+\lambda_H\operatorname{RoleHistory}(a)
687 -\lambda_Q\operatorname{RepeatCount}(a)^2
688 +\lambda_F\mathbf1_{\mathrm{untried}}.
689 \end{aligned}
690 \]
691 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.
692 
693 \begin{figure}[H]
694 \centering
695 \includegraphics[width=.94\linewidth]{23_active_planning_utility.pdf}
696 \caption{The planner combines exact and probabilistic evidence without merging their logical types.}
697 \end{figure}
698 
699 \section{Residual-driven model evolution}
700 
701 The model loop is
702 \[
703 \boxed{
704 \text{analyze exactly}
705 \to
706 \text{fit typed posterior}
707 \to
708 \text{predict and act}
709 \to
710 \text{measure residual}
711 \to
712 \text{refine selectively}.}
713 \]
714 A residual can trigger:
715 \begin{itemize}
716 \item a finer statistical projection;
717 \item a different atlas expert;
718 \item a packet-tree split;
719 \item a delay or memory mechanism;
720 \item an action interaction;
721 \item an explicit graph edge rather than a learned approximation.
722 \end{itemize}
723 
724 \begin{figure}[H]
725 \centering
726 \includegraphics[width=.92\linewidth]{24_residual_refinement_cycle.pdf}
727 \caption{The exact archive remains fixed while the predictive view changes.}
728 \end{figure}
729 
730 \section{Algorithms}
731 
732 \subsection{Exact multi-atlas packetization}
733 
734 \begin{lstlisting}[language=Python,caption={Core exact analysis contract.}]
735 analysis = analyzer.analyze(frame)
736 assert max(analysis.reconstruction_residuals().values()) < tolerance
737 exact = analysis.tight_frame_synthesis(weights)
738 assert max_abs(exact - frame) < tolerance
739 features = analyzer.feature_groups(analysis)
740 z_macro = Q[:48] @ concatenate(features)
741 z_meso  = Q[:96] @ concatenate(features)
742 z_micro = Q[:192] @ concatenate(features)
743 \end{lstlisting}
744 
745 \subsection{Program fitting}
746 
747 \begin{lstlisting}[language=Python,caption={Typed Bayesian program comparison.}]
748 for program in grammar:
749     X = stack(program.design_transition(row) for row in records)
750     Y = stack(program.targets(row) for row in records)
751     posterior = normal_inverse_gamma_fit(X_train, Y_train)
752     score = predictive_log_likelihood(posterior, X_valid, Y_valid)
753     weight[program] = score - complexity_penalty(program)
754 probability = softmax(weight)
755 \end{lstlisting}
756 
757 \clearpage
758 \subsection{Active agent loop}
759 
760 \begin{lstlisting}[language=Python,caption={PR-LASM planning loop.}]
761 while budget_remains and not terminal:
762     packets = exact_and_statistical_analysis(observation)
763     candidates = public_action_candidates(observation, packets)
764     probes = deterministic_replay_probes(history, candidates)
765     if enough_records:
766         world_model.fit(memory)
767     scores = utility(candidates, probes, graph, posterior, memory)
768     action = argmax(scores)
769     next_observation = environment.step(action)
770     record_transition(observation, action, next_observation)
771     update_graph_and_residuals()
772 \end{lstlisting}
773 
774 \section{Computational organization}
775 
776 Structured transforms are applied before dense learning.  The exact Fourier transform uses FFT structure, the Haar transform uses local butterfly operations, and polyphase extraction is a permutation of addresses.  Dense linear algebra is confined to projected states and program-specific design matrices.
777 
778 \begin{figure}[H]
779 \centering
780 \includegraphics[width=.98\linewidth]{25_computational_path.pdf}
781 \caption{The implementation keeps full-resolution transforms structured and small-state inference dense.}
782 \end{figure}
783 
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:
785 \[
786 \begin{array}{ll}
787 \text{FFT atlas}&O(n\log n),\\
788 \text{Haar atlas}&O(n),\\
789 \text{polyphase atlas}&O(n),\\
790 \text{feature projection}&O(dk),\\
791 \text{one regression fit}&O(md_g^2+d_g^3),\\
792 \text{all program fits}&\displaystyle\sum_{g=1}^GO(md_g^2+d_g^3).
793 \end{array}
794 \]
795 The architecture is designed so that $d_g$ is controlled by the selected rank and resolution rather than by the full signal dimension.
796 
797 \section{Controlled program-identification benchmark}
798 
799 The mechanism benchmark is deliberately algebraic.  It does not claim to reproduce the uncertainty of arbitrary physical or interactive data.  Its purpose is narrower: verify that when data are generated by one typed mechanism, the implementation can distinguish that mechanism from simpler alternatives on held-out records.
800 
801 Seven systems were generated:
802 \begin{enumerate}
803 \item Fourier-atlas linear transfer;
804 \item spatial quadratic interaction;
805 \item delayed portfolio dynamics;
806 \item Fourier power-memory dynamics;
807 \item portfolio exponential-memory dynamics;
808 \item phase-two polyphase dynamics;
809 \item color-gated dynamics.
810 \end{enumerate}
811 
812 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.
813 
814 \begin{certificate}[Mechanism and atlas identification]
815 The highest posterior family matched the declared generating family in all seven systems.  The highest posterior atlas matched the declared atlas in all seven systems.
816 \end{certificate}
817 
818 \begin{figure}[H]
819 \centering
820 \includegraphics[width=.96\linewidth]{14_mechanism_identification.pdf}
821 \caption{Every controlled mechanism was identified by its typed posterior.}
822 \end{figure}
823 
824 \begin{figure}[H]
825 \centering
826 \includegraphics[width=.96\linewidth]{15_atlas_identification.pdf}
827 \caption{Every controlled coordinate atlas was identified.}
828 \end{figure}
829 
830 \subsection{Linear ablation}
831 
832 The baseline is one linear macro-resolution program using the portfolio atlas.  PR-LASM uses the full typed posterior.  The test-state RMSE values are:
833 
834 \begin{table}[H]
835 \centering
836 \small
837 \begin{tabular}{lrrr}
838 \toprule
839 system&linear baseline&PR-LASM&ratio\\
840 \midrule
841 linear Fourier&0.2449&0.0221&0.0903\\
842 quadratic spatial&0.2708&0.1240&0.4580\\
843 delay portfolio&0.3112&0.0629&0.2020\\
844 fractional Fourier&0.2735&0.0654&0.2393\\
845 Volterra portfolio&0.2793&0.0642&0.2300\\
846 phase-two polyphase&0.1465&0.0436&0.2978\\
847 gated color&0.0502&0.0385&0.7674\\
848 \bottomrule
849 \end{tabular}
850 \caption{Held-out state-prediction RMSE.  The benchmark is controlled and mechanism-specific.}
851 \end{table}
852 
853 \begin{figure}[H]
854 \centering
855 \includegraphics[width=.96\linewidth]{16_operator_rmse.pdf}
856 \caption{The full typed model versus the single linear portfolio baseline.}
857 \end{figure}
858 
859 \begin{figure}[H]
860 \centering
861 \includegraphics[width=.92\linewidth]{17_rmse_ratios.pdf}
862 \caption{All seven state RMSE ratios are below one; the median is $0.2393$.}
863 \end{figure}
864 
865 \begin{boundary}[Interpretation of the controlled benchmark]
866 The result verifies program construction, posterior comparison, and held-out prediction when the generating family belongs to the supplied grammar.  It does not prove that the grammar contains the mechanism of an arbitrary environment.
867 \end{boundary}
868 
869 \section{Deterministic interactive benchmarks}
870 
871 Four source-blind environments test the integrated planning loop:
872 \begin{itemize}
873 \item a five-step action sequence;
874 \item a spatial click target;
875 \item a grid-navigation task with walls;
876 \item a six-step alternating phase switch.
877 \end{itemize}
878 All expose only observations and public actions.  Deterministic replay is available, so the graph and probe system can be tested exactly.
879 
880 \begin{certificate}[Interactive systems]
881 PR-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.
882 \end{certificate}
883 
884 \begin{figure}[H]
885 \centering
886 \includegraphics[width=.90\linewidth]{21_agent_actions.pdf}
887 \caption{Committed actions and replay probes in the four interactive systems.}
888 \end{figure}
889 
890 \begin{figure}[H]
891 \centering
892 \includegraphics[width=.90\linewidth]{22_agent_graph_sizes.pdf}
893 \caption{Exact graph memory remained compact in the finite smoke tasks.}
894 \end{figure}
895 
896 \section{The supplied twenty-five-environment suite}
897 
898 The release bundles twenty-five local environment trees.  Static validation checks metadata, source presence, source syntax, the expected game-class structure, and prohibited network or process-control imports.  All twenty-five trees pass.
899 
900 The supplied baseline-action totals vary substantially, so the runner retains per-environment budgets rather than using one small global cap.
901 
902 \begin{figure}[H]
903 \centering
904 \includegraphics[width=.90\linewidth]{26_arc25_action_scale.pdf}
905 \caption{Distribution of supplied baseline-action totals across the local suite.}
906 \end{figure}
907 
908 \begin{figure}[H]
909 \centering
910 \includegraphics[width=.90\linewidth]{27_arc25_suite_map.pdf}
911 \caption{Level counts and baseline-action totals for the twenty-five local environment trees.}
912 \end{figure}
913 
914 \begin{boundary}[No official execution claim]
915 The official local runtime was not available in the present build container.  Therefore this paper reports no solved count, score, or timing for the twenty-five environments.  The release includes offline run scripts for the target machine and preserves the original strict network guard.
916 \end{boundary}
917 
918 \section{Software architecture}
919 
920 \begin{longtable}{L{0.37\linewidth}L{0.55\linewidth}}
921 \toprule
922 module&role\\
923 \midrule
924 \endfirsthead
925 \toprule
926 module&role\\
927 \midrule
928 \endhead
929 \texttt{exact\_atlas.py}&unitary Fourier, orthonormal Haar, polyphase, categorical color, projector queries, tight-frame synthesis\\
930 \texttt{lasm\_packetizer.py}&exact archive, named feature groups, deterministic nested projections\\
931 \texttt{lasm\_refinement.py}&complete packet trees, mixed-depth synthesis, best-basis dynamic programming\\
932 \texttt{lasm\_programs.py}&typed design maps for linear, nonlinear, phase, delay, and memory mechanisms\\
933 \texttt{lasm\_model.py}&Bayesian fitting, posterior weights, uncertainty decomposition, action prediction\\
934 \texttt{lasm\_memory.py}&committed states, lag packets, power memory, exponential memory, role statistics\\
935 \texttt{lasm\_graph.py}&exact transition edges, nondeterminism marking, frontier and path queries\\
936 \texttt{lasm\_agent.py}&replay probes, utility scoring, graph updates, source-blind interactive loop\\
937 \texttt{lasm\_benchmark.py}&exactness, controlled operator, and interactive benchmarks\\
938 \texttt{lasm\_cli.py}&benchmark, smoke, single-environment, and suite commands\\
939 \bottomrule
940 \end{longtable}
941 
942 The command
943 \begin{lstlisting}
944 python -m komorebi_arc3_offline.lasm_cli benchmark \
945   --output results/pr_lasm_benchmark
946 \end{lstlisting}
947 regenerates the benchmark ledgers.  The PowerShell wrappers provide the same operation on Windows.
948 
949 \section{Verification}
950 
951 \begin{certificate}[Automated suite]
952 The complete source release passes
953 \[
954 \boxed{36/36\text{ automated tests}.}
955 \]
956 The 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.
957 \end{certificate}
958 
959 \begin{figure}[H]
960 \centering
961 \includegraphics[width=.92\linewidth]{29_verification_dashboard.pdf}
962 \caption{Summary of the finite verification state.}
963 \end{figure}
964 
965 \section{Claim ledger}
966 
967 \begin{longtable}{L{0.36\linewidth}L{0.18\linewidth}L{0.37\linewidth}}
968 \toprule
969 statement&status&boundary\\
970 \midrule
971 \endfirsthead
972 \toprule
973 statement&status&boundary\\
974 \midrule
975 \endhead
976 Per-atlas packet reconstruction&theorem&finite unitary atlas and complete coordinate selectors\\
977 Weighted multi-atlas synthesis&theorem&finite family of unitary atlases with positive weights\\
978 Categorical color reconstruction&exact construction&finite one-hot palette\\
979 Mixed-depth packet pruning&theorem&finite complete binary packet tree\\
980 Best-basis dynamic program&theorem&finite additive node cost\\
981 Bayesian regression formulas&exact finite formulas&declared normal-inverse-gamma model\\
982 Typed program selection&finite certificate&seven controlled systems generated from the supplied grammar\\
983 RMSE improvement&finite certificate&controlled held-out benchmark only\\
984 State-action graph&exact finite data structure&deterministic edges only\\
985 Interactive solutions&finite certificate&four bundled deterministic systems\\
986 Twenty-five environment integrity&static certificate&source trees, not executed games\\
987 Official environment score&not claimed&official local runtime absent from build container\\
988 Universal operator discovery&not claimed&finite declared grammar\\
989 AGI&not claimed&outside the mathematical and empirical scope\\
990 \bottomrule
991 \end{longtable}
992 
993 \begin{figure}[H]
994 \centering
995 \includegraphics[width=.94\linewidth]{28_claim_boundary.pdf}
996 \caption{The proof and benchmark boundary is part of the model specification.}
997 \end{figure}
998 
999 \section{What has been completed}
1000 
1001 The construction now contains every layer required for one coherent large signal model:
1002 \begin{enumerate}
1003 \item exact multi-atlas observation coordinates;
1004 \item Fourier, Haar-wavelet, polyphase, spatial, and categorical realizations;
1005 \item a normalized redundant reconstruction theorem;
1006 \item named feature groups and nested statistical states;
1007 \item exact adaptive packet-tree refinement;
1008 \item a typed nonlinear and memory-bearing operator grammar;
1009 \item Bayesian evidence and posterior uncertainty;
1010 \item exact transition-graph memory;
1011 \item replay-safe active planning;
1012 \item synthetic mechanism-identification and interactive certificates;
1013 \item a source-blind offline adapter and twenty-five local environment trees;
1014 \item a reproducible package, test suite, figures, tables, and run scripts.
1015 \end{enumerate}
1016 
1017 \begin{figure}[H]
1018 \centering
1019 \includegraphics[width=.72\linewidth]{30_closing_wheel.pdf}
1020 \caption{The completed model closes exact analysis, learning, exploration, planning, and reconstruction.}
1021 \end{figure}
1022 
1023 \section{Remaining research directions}
1024 
1025 The model is finished as a finite architecture, but several extensions are mathematically substantive.
1026 
1027 \subsection{Program grammar growth}
1028 
1029 A future compiler should be able to propose new typed operator programs while preserving an explicit description length and proof-status boundary.  Arbitrary feature generation would destroy the meaning of the posterior over mechanisms.
1030 
1031 \subsection{Learned invertible atlases}
1032 
1033 The present exact atlases are fixed.  One may add learned transforms only when their inverse, conditioning, and packet selectors remain explicit.  A learned bijection without stable synthesis is not an exact atlas in the present sense.
1034 
1035 \subsection{Continuous and stochastic worlds}
1036 
1037 The finite graph can be replaced by kernels or stochastic transition operators.  Exact replay edges then become probability distributions, and frontier planning requires uncertainty over both the world and the graph itself.
1038 
1039 \subsection{Scaling and hardware}
1040 
1041 Packet-sparse expert fitting, batched small regressions, structured transforms, and graph-local updates are natural GPU targets.  The present CPU certificates do not establish an optimal hardware design.
1042 
1043 \subsection{Empirical evaluation}
1044 
1045 The next empirical step is straightforward: install the official local runtime on the target machine, execute the bundled suite, preserve complete traces, and compare PR-LASM against the earlier Bayesian operator runner and graph-only ablations under identical budgets.  That is an experiment, not another missing architectural component.
1046 
1047 \section{Conclusion}
1048 
1049 PR-LASM treats a signal world as more than a stream of vectors.  An observation has several exact coordinate descriptions; an operator has packet transfer blocks; nonlinear behavior has interaction features; memory has typed kernels; uncertainty lives over mechanisms and representations; exploration creates an exact graph where possible; and every compression remains attached to an exact reconstruction archive.
1050 
1051 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:
1052 \begin{center}
1053 \fbox{\begin{minipage}{0.91\linewidth}
1054 \centering
1055 \textbf{signal object}
1056 $\to$
1057 \textbf{exact atlas portfolio}
1058 $\to$
1059 \textbf{packet features and nested states}
1060 $\to$
1061 \textbf{typed operator posterior}\\[4pt]
1062 $\to$
1063 \textbf{memory and transition graph}
1064 $\to$
1065 \textbf{active action}
1066 $\to$
1067 \textbf{residual}
1068 $\to$
1069 \textbf{selective refinement}
1070 $\to$
1071 \textbf{exact reconstruction}
1072 \end{minipage}}
1073 \end{center}
1074 Within its finite declared scope, that loop is complete.
1075 
1076 \appendix
1077 \clearpage
1078 \section{Exact transform formulas}
1079 
1080 \subsection{Haar analysis and synthesis}
1081 For one pair,
1082 \[
1083 \ell=\frac{a+b}{\sqrt2},\qquad h=\frac{a-b}{\sqrt2},
1084 \]
1085 and
1086 \[
1087 a=\frac{\ell+h}{\sqrt2},\qquad b=\frac{\ell-h}{\sqrt2}.
1088 \]
1089 Applying these identities separably and recursively proves the finite two-dimensional transform used in the code.
1090 
1091 \subsection{Polyphase analysis and synthesis}
1092 For modulus $N$,
1093 \[
1094 (\Acal_Nx)_{r,s}[q,t]=x[Nq+r,Nt+s].
1095 \]
1096 The synthesis operator places each channel value back at its unique original address.  Therefore
1097 \[
1098 \Scal_N\Acal_N=I.
1099 \]
1100 
1101 \subsection{Fourier selector projectors}
1102 For any diagonal zero-one mask $E_B$,
1103 \[
1104 P_B=F^*E_BF,
1105 \]
1106 so
1107 \[
1108 P_B^2=P_B,\qquad P_B^*=P_B.
1109 \]
1110 Disjoint masks annihilate, and a complete mask partition sums to the identity.
1111 
1112 \section{Bayesian regression formulas}
1113 
1114 For one output $y$ and design matrix $X$, take
1115 \[
1116 \beta\mid\sigma^2\sim N(0,\sigma^2\Lambda_0^{-1}),
1117 \qquad
1118 \sigma^2\sim\operatorname{InvGamma}(a_0,b_0).
1119 \]
1120 Then
1121 \[
1122 \Lambda_n=\Lambda_0+X^\top X,
1123 \qquad
1124 \mu_n=\Lambda_n^{-1}X^\top y,
1125 \]
1126 \[
1127 a_n=a_0+\frac n2,
1128 \qquad
1129 b_n=b_0+\frac12(y^\top y-\mu_n^\top\Lambda_n\mu_n).
1130 \]
1131 For a feature row $x_*$, the predictive mean is $x_*^\top\mu_n$ and the Student predictive scale contains
1132 \[
1133 1+x_*^\top\Lambda_n^{-1}x_*.
1134 \]
1135 The implementation applies this independently to every output coordinate while sharing the design covariance.
1136 
1137 \section{Reproducibility commands}
1138 
1139 \subsection{Linux or macOS shell}
1140 \begin{lstlisting}
1141 cd Komorebi_PR_LASM_Ultimate
1142 export PYTHONPATH="$PWD/src"
1143 pytest -q
1144 python -m komorebi_arc3_offline.lasm_cli benchmark \
1145   --output results/pr_lasm_benchmark
1146 \end{lstlisting}
1147 
1148 \subsection{PowerShell}
1149 \begin{lstlisting}
1150 $Work = "$env:USERPROFILE\Downloads\Komorebi_PR_LASM_Ultimate"
1151 Set-Location $Work
1152 $env:PYTHONPATH = "$Work\src"
1153 python -m pytest -q
1154 & "$Work\Run_PR_LASM_Benchmark.ps1"
1155 & "$Work\Run_PR_LASM_Smoke.ps1"
1156 \end{lstlisting}
1157 
1158 \subsection{Local twenty-five-environment run}
1159 \begin{lstlisting}
1160 & "$Work\Run_PR_LASM_5_Easiest.ps1" -EnvName "arc3-offline"
1161 & "$Work\Run_PR_LASM_All_25.ps1" -EnvName "arc3-offline"
1162 \end{lstlisting}
1163 
1164 \section{Machine-readable result files}
1165 
1166 The benchmark directory contains:
1167 \begin{itemize}
1168 \item \texttt{exactness\_benchmark.json};
1169 \item \texttt{operator\_benchmark.json} and \texttt{operator\_benchmark.csv};
1170 \item \texttt{agent\_benchmark.json} and \texttt{agent\_benchmark.csv};
1171 \item \texttt{benchmark\_summary.json};
1172 \item per-environment traces, posterior traces, graph summaries, and result files.
1173 \end{itemize}
1174 
1175 \section{Test map}
1176 
1177 \begin{longtable}{L{0.42\linewidth}L{0.49\linewidth}}
1178 \toprule
1179 test group&content\\
1180 \midrule
1181 \endfirsthead
1182 \toprule
1183 test group&content\\
1184 \midrule
1185 \endhead
1186 Inherited offline runner&actions, packetizer, Bayesian model, replay, offline guard, official-adapter contract, CLI, suite discovery, suite bundle\\
1187 Exact atlases&spatial, Fourier, Haar, polyphase, color, weighted tight-frame synthesis\\
1188 Packet projectors&Fourier band completeness, polyphase completeness, wavelet subband completeness\\
1189 Nested packetizer&determinism, dimensions, reconstruction ledger\\
1190 Adaptive basis&mixed-depth selection and exact synthesis\\
1191 Transition graph&frontier paths and winning edges\\
1192 Program posterior&seven parametrized mechanism and atlas identifications\\
1193 Interactive agent&sequence, click, navigation, and phase-switch environments\\
1194 \bottomrule
1195 \end{longtable}
1196 
1197 \end{document}

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