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At $x_0$ there is a collection of admissible abstract directions. Applying one direction gives a child object. The child is not forced to inherit the same direction list. It is queried again, and its own admissible directions create the next generation. This process continues without a preset depth.","Given any two objects $x$ and $y$, one can invent a one-step map that sends $x$ to $y$. Such a map does not reveal a generative structure. It hides the target inside the operation.","Let $\\mathsf T$ be a set of types and let\n\\[\n\\Xcal=\\coprod_{t\\in\\mathsf T}X_t\n\\]\nbe the disjoint union of the corresponding object spaces.","\\begin{definition}[Direction bundle]\nA direction bundle over $\\Xcal$ is a map\n\\[\n\\pi:\\Dcal\\longrightarrow\\Xcal.\n\\]\nThe fiber\n\\[\n\\Dcal_x=\\pi^{-1}(x)\n\\]\nis the complete set of directions admitted by the declared grammar at $x$.\n\\end{definition}","\\begin{definition}[Elementary deformation law]\nAn elementary deformation law is a map\n\\[\n\\Phi:\\Dcal\\longrightarrow\\Xcal\n\\]\nwith the understanding that $d\\in\\Dcal_x$ starts at $x$ and ends at $\\Phi(d)$. We also write\n\\[\nx\\star d=\\Phi(d).\n\\]\n\\end{definition}","The fiber $\\Dcal_x$ may depend on every property of $x$. A matrix near a spectral collision may expose different directions from a diagonal matrix. A function with zeros may forbid reciprocal deformation. A singular algebraic point may have a larger tangent cone than a smooth point.","\\begin{definition}[Typed generative deformation datum]\nA typed generative deformation datum is a tuple\n\\[\n\\boxed{\n\\Gcal=\n(\\mathsf T,\\Xcal,\\pi:\\Dcal\\to\\Xcal,\\Phi,\\mathcal R,\\tau,\\kappa,\\mathcal V),\n}\n\\]\nwhere $\\mathcal R$ is a path relation system, $\\tau$ is a completion structure, $\\kappa:\\Dcal\\to[0,\\infty)$ is a direction cost, and $\\mathcal V$ is a validity predicate.\n\\end{definition}","The data separate existence from computation. The full fiber $\\Dcal_x$ defines the mathematics. A finite explorer may inspect only a subset, a finite cover, or a best-first frontier.","\\begin{definition}[Direction-complete expansion]\nAn expansion rule is direction-complete relative to $\\Gcal$ at $x$ when it exposes every element of $\\Dcal_x$. It is direction-complete through depth $n$ when this holds at every endpoint of every admissible path of length less than $n$.\n\\end{definition}","Fix a root $x_0\\in\\Xcal$.","\\begin{definition}[Admissible path]\nAn admissible path of length $n$ from $x_0$ is a sequence\n\\[\n\\gamma=(d_1,\\ldots,d_n)\n\\]\nfor which\n\\[\nx_k=x_{k-1}\\star d_k,\n\\qquad\nd_k\\in\\Dcal_{x_{k-1}},\n\\qquad\n1\\le k\\le n.\n\\]\nIts endpoint is $\\ev(\\gamma)=x_n$. The empty path $\\varnothing$ has endpoint $x_0$.\n\\end{definition}","Write $\\Tcal_n(x_0)$ for the set of length-$n$ paths and\n\\[\n\\Tcal_{\\mathrm{fin}}(x_0)=\\coprod_{n\\ge0}\\Tcal_n(x_0).\n\\]\nEvery nonempty path has a unique parent obtained by deleting its final direction. Therefore $\\Tcal_{\\mathrm{fin}}(x_0)$ is a rooted tree even when different paths have the same endpoint.","\\begin{theorem}[Exact reachability by depth]\nFor every $n\\ge0$,\n\\[\n\\ev(\\Tcal_n(x_0))\n=\n\\Reach_n(x_0),\n\\]\nwhere $\\Reach_n(x_0)$ is the set obtained by applying exactly $n$ successive admissible directions.\n\\end{theorem}","\\begin{proof}\nThe statement is immediate at depth zero. Assume it holds at depth $n$. Every path of length $n+1$ is a path of length $n$ followed by one direction from the fiber over its endpoint. Thus its endpoint lies in the $(n+1)$-step reachable set. Conversely, every object in the $(n+1)$-step reachable set is produced by an $n$-step reachable object and one admissible direction. The induction closes.\n\\end{proof}","\\begin{theorem}[Free rooted path property]\nLet $Y$ be a rooted directed graph with root $y_0$. Suppose each admissible pair $(x,d)$ is assigned an edge operation on $Y$ in a way that depends only on the current image vertex and the label $d$. Then there is a unique root-preserving map\n\\[\nF:\\Tcal_{\\mathrm{fin}}(x_0)\\longrightarrow Y\n\\]\nthat follows those assigned edges along every path.\n\\end{theorem}","\\begin{proof}\nSet $F(\\varnothing)=y_0$. If $F(\\gamma)$ has been defined and $d$ is admissible at $\\ev(\\gamma)$, define $F(\\gamma d)$ by following the assigned $d$-edge from $F(\\gamma)$. Every path has one parent, so the definition is unambiguous. Any root-preserving map with the required edge behavior must obey the same recursion, which proves uniqueness.\n\\end{proof}","The cost of a finite path is\n\\[\n\\cost(\\gamma)=\\sum_{k=1}^n\\kappa(d_k).\n\\]\nA path record may also store error bounds, provenance, validity certificates, symbolic assumptions, and the first place at which an invariant changed. None of these fields alters reachability. They determine how a finite explorer prioritizes and audits the infinite object.","\\begin{definition}[Extension-compatible path relation]\nAn equivalence relation $\\sim$ on finite paths is extension-compatible when the following implication holds. If $\\gamma\\sim\\eta$ and a direction label $d$ is admissible after both paths with compatible meaning, then\n\\[\n\\gamma d\\sim\\eta d.\n\\]\nThe relation may also identify endpoints under a declared symmetry group, rewrite system, canonical form, gauge, or quotient law.\n\\end{definition}","\\begin{definition}[Quotient deformation graph]\nThe quotient deformation graph is\n\\[\n\\boxed{\n\\Ocal(x_0)=\\Tcal_{\\mathrm{fin}}(x_0)/\\sim.\n}\n\\]\nIts vertices are relation classes of paths. An edge is induced by one admissible extension.\n\\end{definition}","\\begin{theorem}[Well-defined quotient extension]\nIf $\\sim$ is extension-compatible, then the edge induced by an admissible direction does not depend on the representative of a path class.\n\\end{theorem}","\\begin{proof}\nChoose representatives $\\gamma$ and $\\eta$ of one class. By extension compatibility, every common admissible extension satisfies $\\gamma d\\sim\\eta d$. Therefore both representatives determine the same child class.\n\\end{proof}","A computational realization uses a canonical key\n\\[\nK:\\Xcal\\longrightarrow\\mathcal K\n\\]\nwith\n\\[\nK(x)=K(y)\n\\quad\\Longleftrightarrow\\quad\nx\\sim y\n\\]\nfor the declared finite example. Exact polynomial coefficients, reduced rational numbers, canonical symbolic expressions, and normalized matrices are suitable examples. Floating point rounding alone is not an exact quotient relation.","Let the root be $0\\in\\Z$. Every integer has two directions,\n\\[\n+1,\n\\qquad\n-1.\n\\]\nA path of length $n$ is a word in these two labels.","\\begin{theorem}[Raw and quotient growth of the integer walk]\nAt exact depth $n$, the raw path count is $2^n$. The distinct endpoint count is $n+1$.\n\\end{theorem}","\\begin{proof}\nThere are two choices at every step, giving $2^n$ words. If a path contains $k$ negative steps, its endpoint is\n\\[\n(n-k)-k=n-2k.\n\\]\nAs $k$ ranges from $0$ to $n$, these are $n+1$ distinct integers. Every path with the same value of $k$ has the same endpoint.\n\\end{proof}","Let each fiber $\\Dcal_x$ carry a metric $\\rho_x$. A finite set $N_\\epsilon(x)\\subseteq\\Dcal_x$ is an $\\epsilon$-net when every $d\\in\\Dcal_x$ lies within $\\epsilon$ of some $\\widehat d\\in N_\\epsilon(x)$.","Assume the object space carries a metric $d_X$.","\\begin{theorem}[Pathwise cover error]\nConsider an exact route\n\\[\nx_k=x_{k-1}\\star d_k\n\\]\nand a sampled route\n\\[\n\\widehat x_k=\\widehat x_{k-1}\\star\\widehat d_k.\n\\]\nSuppose at step $k$,\n\\[\nd_X(x\\star d,y\\star e)\n\\le\nL_k d_X(x,y)+M_k\\rho(d,e),\n\\]\nand\n\\[\n\\rho(d_k,\\widehat d_k)\\le\\epsilon_k.\n\\]\nThen\n\\[\n\\boxed{\n d_X(x_n,\\widehat x_n)\n \\le\n \\left(\\prod_{k=1}^nL_k\\right)d_X(x_0,\\widehat x_0)\n +\n \\sum_{j=1}^n\n M_j\\epsilon_j\n \\prod_{k=j+1}^nL_k.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nSet $e_k=d_X(x_k,\\widehat x_k)$. The hypothesis gives\n\\[\ne_k\\le L_ke_{k-1}+M_k\\epsilon_k.\n\\]\nRepeated substitution yields the displayed formula.\n\\end{proof}","\\begin{corollary}[Convergent direction covers]\nIf $L_k\\le1$, $M_k\\le M$, and\n\\[\n\\sum_{k\\ge1}\\epsilon_k<\\infty,\n\\]\nthen the total endpoint error of the sampled route is bounded uniformly in depth by\n\\[\nM\\sum_{k\\ge1}\\epsilon_k\n\\]\nwhen the roots coincide.\n\\end{corollary}","Sampling is an execution method, not the definition of the continuous forest. A continuous family is stored symbolically as an edge bundle\n\\[\n\\Ecal_x\n=\n\\{(d,x\\star d):d\\in\\Dcal_x\\}.\n\\]\nFinite nets approximate this bundle. Refinement replaces one coarse parameter cell by smaller cells. The cell itself remains a first-class object with a radius and a coverage certificate.","An infinite route is a sequence\n\\[\n\\omega=(d_1,d_2,\\ldots)\n\\]\nwhose every finite prefix is admissible. The set of infinite routes is the path boundary $\\partial\\Tcal(x_0)$.","An infinite route does not automatically determine an object of $\\Xcal$. A topology, metric, inverse limit, direct limit, or weak convergence rule is required.","\\begin{definition}[Convergent route]\nA route $\\omega$ is convergent in a metric realization when its endpoint sequence\n\\[\nx_0,x_1,x_2,\\ldots\n\\]\nconverges in the declared completion $\\widehat\\Xcal$.\n\\end{definition}","\\begin{theorem}[Summable-step convergence]\nSuppose $\\widehat\\Xcal$ is complete and\n\\[\nd_X(x_{n+1},x_n)\\le a_n\n\\]\nwith\n\\[\n\\sum_{n\\ge0}a_n<\\infty.\n\\]\nThen $(x_n)$ is Cauchy and the route has a unique limit in $\\widehat\\Xcal$.\n\\end{theorem}","\\begin{proof}\nFor $m>n$,\n\\[\nd_X(x_m,x_n)\n\\le\n\\sum_{k=n}^{m-1}d_X(x_{k+1},x_k)\n\\le\n\\sum_{k=n}^{m-1}a_k.\n\\]\nThe tail tends to zero. Completeness supplies a unique limit.\n\\end{proof}","Starting at $0$, let the directions at depth $n$ be\n\\[\n\\pm2^{-(n+1)}.\n\\]\nEvery infinite sign route converges because\n\\[\n\\sum_{n\\ge0}2^{-(n+1)}=1.\n\\]\nThe boundary therefore maps into $[-1,1]$. Different sign sequences can represent the same endpoint only at the usual binary-expansion ambiguity, which can be handled by a path relation.","Metric completion is not the only possibility. In arithmetic deformation, a branch may be a compatible tower\n\\[\nx_1\\leftarrow x_2\\leftarrow x_3\\leftarrow\\cdots\n\\]\nwhose object is an inverse limit. In nested function spaces, a branch may enter a direct limit. In weak probability or operator topologies, convergence may be defined only against a test class.","For every $r\\ge2$, let\n\\[\n\\Psi_r:\n\\Xcal^r\\times\\Dcal^{(r)}\n\\longrightarrow\n\\Xcal\n\\]\nbe a typed $r$-input deformation law.","\\begin{definition}[Operadic perturbation hyperforest]\nThe operadic perturbation hyperforest generated by a seed family is the set of all finite rooted typed trees whose leaves are seeds and whose internal vertices are labeled by admissible operations $\\Phi$ or $\\Psi_r$. Evaluation proceeds from leaves to root.\n\\end{definition}","Consider\n\\[\nu=s+\\lambda u^2.\n\\]\nRepeated substitution expands $u$ into rooted binary trees. If only shape is counted, the number of binary trees with $n$ internal vertices is the Catalan number $C_n$.","\\begin{proposition}[Catalan interaction growth]\nThe counts satisfy\n\\[\nC_0=1,\n\\qquad\nC_{n+1}=\\sum_{k=0}^nC_kC_{n-k}.\n\\]\n\\end{proposition}","\\begin{proof}\nAt the root of a binary tree with $n+1$ internal vertices, the left subtree has $k$ internal vertices and the right subtree has $n-k$. Summing over all $k$ gives the recurrence.\n\\end{proof}","The first values are\n\\[\n1,1,2,5,14,42,132,429,\\ldots.\n\\]\nThe rapid growth is not an implementation defect. It is the intrinsic size of the free nonlinear syntax before identities, symmetries, truncation rules, or cost weights are applied.","For a finite or countable depth layer, define\n\\[\n\\mathcal H_n=\\ell^2(\\Tcal_n(x_0)).\n\\]\nThe canonical basis vector associated with a path $\\gamma$ is written $\\lvert\\gamma\\rangle$.","For a prefix $\\alpha$ of length $k\\le n$, define the cylinder projector\n\\[\nP^{(n)}_\\alpha\n\\lvert\\gamma\\rangle\n=\n\\begin{cases}\n\\lvert\\gamma\\rangle,&\\alpha\\text{ is a prefix of }\\gamma,","For a prefix $\\alpha$ of length $k\\le n$, define the cylinder projector\n\\[\nP^{(n)}_\\alpha\n\\lvert\\gamma\\rangle\n=\n\\begin{cases}\n\\lvert\\gamma\\rangle,&\\alpha\\text{ is a prefix of }\\gamma,\\\\\n0,&\\text{otherwise}.\n\\end{cases}\n\\]","\\begin{theorem}[Exact cylinder refinement]\nFor every prefix $\\alpha$ and every depth $n>\\abs{\\alpha}$,\n\\[\n\\boxed{\nP^{(n)}_\\alpha\n=\n\\sum_{d\\in\\Dcal_{\\ev(\\alpha)}}P^{(n)}_{\\alpha d},\n}\n\\]\nwith strong convergence when the child set is countable.\n\\end{theorem}","\\begin{proof}\nThe descendant cylinder of $\\alpha$ is the disjoint union of the descendant cylinders of its one-step children. Orthogonal projection onto a disjoint union of basis supports is the sum of the corresponding orthogonal projectors.\n\\end{proof}","A complete cut is a prefix-free set $\\Gamma$ meeting every infinite route exactly once. Its projectors satisfy\n\\[\n\\sum_{\\alpha\\in\\Gamma}P_\\alpha=I.\n\\]\nThus one branch may be resolved deeply while another stops early. Adaptive exploration can be represented without discarding the unresolved region.","Let $A:\\mathcal H\\to\\mathcal K$ be a bounded operator and let $\\{P_\\alpha\\}$ and $\\{Q_\\beta\\}$ be complete route resolutions. Define\n\\[\nA_{\\beta\\alpha}=Q_\\beta A P_\\alpha.\n\\]\nThen\n\\[\nA=\\sum_{\\alpha,\\beta}A_{\\beta\\alpha}.\n\\]\nIf $B$ follows $A$, then\n\\[\n(BA)_{\\gamma\\alpha}\n=\n\\sum_\\beta B_{\\gamma\\beta}A_{\\beta\\alpha}.\n\\]\nEvery operator composition becomes a sum over compatible intermediate perturbation packets.","Define\n\\[\na_n=\\#\\Tcal_n(x_0)\n\\]\nfor raw paths and\n\\[\nq_n=\\#\\bigl(\\ev(\\Tcal_n(x_0))/\\sim\\bigr)\n\\]\nfor distinct quotient endpoints at exact depth $n$.","The raw branching entropy and object entropy are\n\\[\nh_{\\mathrm{raw}}\n=\n\\limsup_{n\\to\\infty}\\frac1n\\log a_n,\n\\qquad\nh_{\\mathrm{obj}}\n=\n\\limsup_{n\\to\\infty}\\frac1n\\log q_n.\n\\]\nAlways\n\\[\nh_{\\mathrm{obj}}\\le h_{\\mathrm{raw}}.\n\\]\nThe gap measures how much formal construction redundancy is removed by identities and symmetries.","Define\n\\[\nZ_{x_0}(s)\n=\n\\sum_{\\gamma\\ne\\varnothing}\n\\exp\\bigl(-s\\cost(\\gamma)\\bigr).\n\\]\nThis is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.","\\begin{theorem}[Geometric convergence bound]\nSuppose every vertex has at most $B$ children and every direction costs at least $c>0$. Then\n\\[\nZ_{x_0}(s)\n\\le\n\\sum_{n\\ge1}B^ne^{-scn}.\n\\]\nConsequently, $Z_{x_0}(s)$ converges whenever\n\\[\nsc>\\log B.\n\\]\nIn a full regular $B$-ary tree with constant cost $c$, it diverges whenever $sc\\le\\log B$.\n\\end{theorem}","\\begin{proof}\nThere are at most $B^n$ paths of length $n$, and each has cost at least $cn$. The upper bound is a geometric series with ratio $Be^{-sc}$. In the regular constant-cost case, equality holds at every depth, giving the divergence statement.\n\\end{proof}","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.98\\textwidth]{06_growth_phase_diagram.pdf}\n\\caption{The regular-tree phase diagram separates convergent and divergent weighted path sums. Branching and route cost compete through the threshold $sc=\\log B$.}\n\\label{fig:partition}\n\\end{figure}","A path zeta function measures construction complexity. A quotient zeta function measures distinct object complexity. Depending on the application, one may define\n\\[\nZ_{\\mathrm{obj}}(s)\n=\n\\sum_{[x]\\in\\Ocal(x_0)}e^{-sC([x])},\n\\]\nwhere $C([x])$ is the least path cost reaching the class. The two functions answer different questions and should be stored together.","\\begin{definition}[Generated family certificate]\nA generated family certificate is a tuple\n\\[\n\\Fcal=(\\Theta,F,\\{\\sigma_d\\}_{d\\in A},\\mathcal Q,\\mathcal I),\n\\]\nwhere\n\\begin{enumerate}[label=(\\alph*)]\n\\item $\\Theta$ is a parameter space\n\\item $F:\\Theta\\to\\Ocal(x_0)$ assigns an object class to every parameter\n\\item $A$ is a collection of closure directions\n\\item $\\sigma_d:\\Theta\\to\\Theta$ satisfies\n\\[\nF(\\sigma_d(\\theta))=d\\cdot F(\\theta)\n\\]\nfor every declared closure direction\n\\item $\\mathcal Q$ is a set of exact relations or recurrences\n\\item $\\mathcal I$ is a set of invariants shared or predictably transformed across the family\n\\end{enumerate}\n\\end{definition}","begin{definition}[Generated family certificate]\nA generated family certificate is a tuple\n\\[\n\\Fcal=(\\Theta,F,\\{\\sigma_d\\}_{d\\in A},\\mathcal Q,\\mathcal I),\n\\]\nwhere\n\\begin{enumerate}[label=(\\alph*)]\n\\item $\\Theta$ is a parameter space\n\\item $F:\\Theta\\to\\Ocal(x_0)$ assigns an object class to every parameter\n\\item $A$ is a collection of closure directions\n\\item $\\sigma_d:\\Theta\\to\\Theta$ satisfies\n\\[\nF(\\sigma_d(\\theta))=d\\cdot F(\\theta)\n\\]\nfor every declared closure direction\n\\item $\\mathcal Q$ is a set of exact relations or recurrences\n\\item $\\mathcal I$ is a set of invariants shared or predictably transformed across the family\n\\end{enumerate}\n\\end{definition}","uple\n\\[\n\\Fcal=(\\Theta,F,\\{\\sigma_d\\}_{d\\in A},\\mathcal Q,\\mathcal I),\n\\]\nwhere\n\\begin{enumerate}[label=(\\alph*)]\n\\item $\\Theta$ is a parameter space\n\\item $F:\\Theta\\to\\Ocal(x_0)$ assigns an object class to every parameter\n\\item $A$ is a collection of closure directions\n\\item $\\sigma_d:\\Theta\\to\\Theta$ satisfies\n\\[\nF(\\sigma_d(\\theta))=d\\cdot F(\\theta)\n\\]\nfor every declared closure direction\n\\item $\\mathcal Q$ is a set of exact relations or recurrences\n\\item $\\mathcal I$ is a set of invariants shared or predictably transformed across the family\n\\end{enumerate}\n\\end{definition}","cal I),\n\\]\nwhere\n\\begin{enumerate}[label=(\\alph*)]\n\\item $\\Theta$ is a parameter space\n\\item $F:\\Theta\\to\\Ocal(x_0)$ assigns an object class to every parameter\n\\item $A$ is a collection of closure directions\n\\item $\\sigma_d:\\Theta\\to\\Theta$ satisfies\n\\[\nF(\\sigma_d(\\theta))=d\\cdot F(\\theta)\n\\]\nfor every declared closure direction\n\\item $\\mathcal Q$ is a set of exact relations or recurrences\n\\item $\\mathcal I$ is a set of invariants shared or predictably transformed across the family\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Inductive soundness of a family certificate]\nSuppose the seed member $F(\\theta_0)$ is valid, every closure identity is proved, and every relation in $\\mathcal Q$ is preserved by the parameter actions. Then every finite parameter word in the maps $\\sigma_d$ produces a valid member of the certified family.\n\\end{theorem}","\\begin{proof}\nThe root parameter gives a valid member. If a valid member has parameter $\\theta$, applying a closure direction $d$ gives $F(\\sigma_d(\\theta))$ by the closure identity. The preserved relations remain true after the step. Induction over the parameter word proves the claim.\n\\end{proof}","For a sequence $a_0,a_1,\\ldots$, the implementation can search for a constant-coefficient relation\n\\[\na_n=c_1a_{n-1}+\\cdots+c_ra_{n-r}\n\\]\nusing exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers\n\\[\n(c_1,c_2)=(1,1).\n\\]\nThis is a small example of the certificate principle. Fitting alone is not enough. The relation must be verified and then proved or bounded on the intended domain.","\\begin{theorem}[Finite-depth completeness]\nIf every direction fiber reached before depth $N$ is finite and the breadth-first explorer has no active resource cap below the resulting path count, then it records every admissible path of length at most $N$.\n\\end{theorem}","\\begin{proof}\nBreadth-first exploration inserts every root. Assume every path of length $n<N$ is eventually removed from the frontier. The algorithm queries its complete finite direction fiber and inserts every one-step extension. Therefore every path of length $n+1$ is inserted. Induction proves the result.\n\\end{proof}","When a fiber is countably infinite, exhausting one vertex before visiting another may starve entire branches. A fair explorer dovetails local enumerators. For example, at global stage $m$ it may inspect the first $m$ directions of every path among the first $m$ frontier paths.","Let $\\Phi_k$ and $\\widetilde\\Phi_k$ be exact and perturbed deformation laws along corresponding routes. Suppose\n\\[\nd_X(\\Phi_k(x,d),\\widetilde\\Phi_k(y,e))\n\\le\nL_kd_X(x,y)+M_k\\rho(d,e)+\\eta_k,\n\\]\nwhere $\\eta_k$ measures model error.","\\begin{theorem}[Recursive deformation stability]\nIf the direction mismatch at step $k$ is at most $\\epsilon_k$, then the endpoint error satisfies\n\\[\ne_n\n\\le\n\\left(\\prod_{k=1}^nL_k\\right)e_0\n+\n\\sum_{j=1}^n\n(M_j\\epsilon_j+\\eta_j)\n\\prod_{k=j+1}^nL_k.\n\\]\n\\end{theorem}","\\begin{proof}\nThe one-step inequality gives\n\\[\ne_k\\le L_ke_{k-1}+M_k\\epsilon_k+\\eta_k.\n\\]\nIterate the recurrence.\n\\end{proof}","The calculus is useful only when the abstract layers survive contact with concrete objects. The examples below use different state spaces, direction grammars, quotient relations, and completion rules. They are not presented as interchangeable models. Their common feature is the same audited production sequence:\n\\[\n\\text{seed}\\longrightarrow\\text{directions}\\longrightarrow\\text{raw routes}\n\\longrightarrow\\text{quotient classes}\\longrightarrow\\text{certificate or completion}.\n\\]","Let\n\\[\np(x)=x^2+1,\n\\qquad\n(Tp)(x)=p(x+1),\n\\qquad\n(Dp)(x)=p'(x).\n\\]\nThe two primitive directions $T$ and $D$ satisfy\n\\[\nD(Tp)=T(Dp)=2x+2.\n\\]\nThe raw path tree therefore contains two length-two routes,\n\\[\np\\xrightarrow{T}Tp\\xrightarrow{D}DTp,\n\\qquad\np\\xrightarrow{D}Dp\\xrightarrow{T}TDp,\n\\]\nbut the quotient graph contains one endpoint class because $DTp=TDp$ exactly.","The complete finite grammar used by the package also contains reflection, multiplication by $x$, and the forward finite difference. Starting from $x^2+1$, the depth-four experiment records \\PolynomialRawPaths\\ raw paths and \\PolynomialQuotientNodes\\ exact polynomial classes under coefficient canonicalization.","Start from\n\\[\nf_0(x)=\\cosh x.\n\\]\nThe declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give\n\\[\nD(\\cosh x)=\\sinh x,\n\\qquad\nR(\\cosh x)=\\operatorname{sech}x,\n\\qquad\nL(\\cosh x)=\\tanh x.\n\\]\nApplying the same grammar again produces $\\operatorname{csch}x$, $\\coth x$, powers, parity components, and doubled arguments. Reciprocal and logarithmic derivative are rejected at the zero expression and the failed attempts remain in the audit.","The executable atlas starts from the eight-term Fibonacci prefix\n\\[\na=(0,1,1,2,3,5,8,13).\n\\]\nIts fixed grammar contains first difference, partial summation, the binomial transform, sign alternation, and reversal. For a finite sequence $a=(a_0,\\ldots,a_{N-1})$, define\n\\[\n(Da)_0=a_0,\n\\qquad\n(Da)_n=a_n-a_{n-1},\n\\]\n\\[\n(Ca)_n=\\sum_{j=0}^{n}a_j,\n\\qquad\n(Ba)_n=\\sum_{j=0}^{n}\\binom{n}{j}a_j.\n\\]\nThe two triangular operators satisfy\n\\[\nDC=CD=I\n\\]\non every finite sequence under the stated boundary convention. Hence the raw routes\n\\[\na\\xrightarrow{D}Da\\xrightarrow{C}CDa,\n\\qquad\na\\xrightarrow{C}Ca\\xrightarrow{D}DCa\n\\]\nreturn to the seed class exactly.","rom the eight-term Fibonacci prefix\n\\[\na=(0,1,1,2,3,5,8,13).\n\\]\nIts fixed grammar contains first difference, partial summation, the binomial transform, sign alternation, and reversal. For a finite sequence $a=(a_0,\\ldots,a_{N-1})$, define\n\\[\n(Da)_0=a_0,\n\\qquad\n(Da)_n=a_n-a_{n-1},\n\\]\n\\[\n(Ca)_n=\\sum_{j=0}^{n}a_j,\n\\qquad\n(Ba)_n=\\sum_{j=0}^{n}\\binom{n}{j}a_j.\n\\]\nThe two triangular operators satisfy\n\\[\nDC=CD=I\n\\]\non every finite sequence under the stated boundary convention. Hence the raw routes\n\\[\na\\xrightarrow{D}Da\\xrightarrow{C}CDa,\n\\qquad\na\\xrightarrow{C}Ca\\xrightarrow{D}DCa\n\\]\nreturn to the seed class exactly.","ifference, partial summation, the binomial transform, sign alternation, and reversal. For a finite sequence $a=(a_0,\\ldots,a_{N-1})$, define\n\\[\n(Da)_0=a_0,\n\\qquad\n(Da)_n=a_n-a_{n-1},\n\\]\n\\[\n(Ca)_n=\\sum_{j=0}^{n}a_j,\n\\qquad\n(Ba)_n=\\sum_{j=0}^{n}\\binom{n}{j}a_j.\n\\]\nThe two triangular operators satisfy\n\\[\nDC=CD=I\n\\]\non every finite sequence under the stated boundary convention. Hence the raw routes\n\\[\na\\xrightarrow{D}Da\\xrightarrow{C}CDa,\n\\qquad\na\\xrightarrow{C}Ca\\xrightarrow{D}DCa\n\\]\nreturn to the seed class exactly.","A second, closed subfamily starts from the impulse $\\delta=(1,0,0,\\ldots)$. Repeated summation gives\n\\[\n(C^k\\delta)_n=\\binom{n+k-1}{k-1}\n\\qquad(k\\ge1),\n\\]\nby induction and the hockey-stick identity. This provides a genuine family certificate: perturbation depth $k$ is the parameter, $C$ advances the parameter, $D$ reverses it away from the boundary, and the displayed coefficient formula is an exact invariant relation.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.99\\textwidth]{11_sequence_atlas.pdf}\n\\caption{The finite Fibonacci seed is transformed by difference, summation, binomial weighting, sign alternation, and reversal. A separate impulse route yields the closed binomial subfamily $C^k\\delta$.}\n\\label{fig:sequence-family}\n\\end{figure}","Represent a one-dimensional Gaussian law by the exact parameter pair $(\\mu,v)$, where $v=\\sigma^2>0$. The grammar contains shifts by $\\pm1$, reflection, convolution with $N(0,1)$, centering, scaling by two, and standardization. Their parameter actions are\n\\[\n\\begin{aligned}\n(\\mu,v)&\\mapsto(\\mu\\pm1,v),\n& (\\mu,v)&\\mapsto(-\\mu,v),","Represent a one-dimensional Gaussian law by the exact parameter pair $(\\mu,v)$, where $v=\\sigma^2>0$. The grammar contains shifts by $\\pm1$, reflection, convolution with $N(0,1)$, centering, scaling by two, and standardization. Their parameter actions are\n\\[\n\\begin{aligned}\n(\\mu,v)&\\mapsto(\\mu\\pm1,v),\n& (\\mu,v)&\\mapsto(-\\mu,v),\\\\\n(\\mu,v)&\\mapsto(\\mu,v+1),\n& (\\mu,v)&\\mapsto(0,v),\\\\\n(\\mu,v)&\\mapsto(2\\mu,4v),\n& (\\mu,v)&\\mapsto(0,1).\n\\end{aligned}\n\\]\nStandardization is state dependent: it is exposed only away from the canonical seed. Reflection is invisible at centered laws, while convolution and scaling change entropy through the variance coordinate.","One explicit route is\n\\[\n(0,1)\\xrightarrow{\\text{shift }+1}(1,1)\n\\xrightarrow{\\ast N(0,1)}(1,2)\n\\xrightarrow{\\text{scale }2}(2,8)\n\\xrightarrow{\\text{standardize}}(0,1).\n\\]\nThe final edge is a many-to-one canonicalization direction. It produces a strong quotient collapse without erasing the route that created the noncanonical law. The differential entropy\n\\[\nh(N(\\mu,v))=\\frac12\\log(2\\pi e v)\n\\]\nis unaffected by shifts and reflection, increases under unit convolution, and changes by $\\log 2$ under scaling by two.","1)}(1,2)\n\\xrightarrow{\\text{scale }2}(2,8)\n\\xrightarrow{\\text{standardize}}(0,1).\n\\]\nThe final edge is a many-to-one canonicalization direction. It produces a strong quotient collapse without erasing the route that created the noncanonical law. The differential entropy\n\\[\nh(N(\\mu,v))=\\frac12\\log(2\\pi e v)\n\\]\nis unaffected by shifts and reflection, increases under unit convolution, and changes by $\\log 2$ under scaling by two.","Begin with the path graph $P_3$ on three vertices. The finite grammar contains graph complement, line-graph formation, leaf attachment, edge subdivision, and a relabeling direction. Raw states are labeled adjacency matrices. For a graph with $n\\le6$ vertices, the quotient key is the lexicographically least adjacency bit string over all $n!$ vertex permutations.","For the seed $P_3$, the first descendants include\n\\[\n\\overline{P_3},\n\\qquad\nL(P_3)\\cong P_2,\n\\qquad\nP_3\\text{ with one attached leaf},\n\\qquad\nP_4\\text{ obtained by subdivision}.\n\\]\nIf $\\rho$ reverses the vertex labels, then\n\\[\n\\operatorname{key}(G)=\\operatorname{key}(\\rho G\\rho^{-1})\n\\]\nexactly, although the two raw paths and raw adjacency matrices remain distinct.","Let\n\\[\nA_0=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}.\n\\]\nThe declared matrix directions are transpose, negation, squaring, conjugation by the coordinate swap, symmetrization, commutation with\n\\[\nJ=\\begin{pmatrix}0&-1","0&1\\end{pmatrix}.\n\\]\nThe declared matrix directions are transpose, negation, squaring, conjugation by the coordinate swap, symmetrization, commutation with\n\\[\nJ=\\begin{pmatrix}0&-1\\\\1&0\\end{pmatrix},\n\\]\nand adjugation. Every result is stored as an exact integer matrix. The metadata include trace, determinant, rank, symmetry, and squared Frobenius norm.","The first calculations are explicit:\n\\[\nA_0^{\\mathsf T}=\\begin{pmatrix}1&0\\\\1&1\\end{pmatrix},\n\\qquad\nA_0^2=\\begin{pmatrix}1&2\\\\0&1\\end{pmatrix},\n\\qquad\n\\operatorname{adj}(A_0)=\\begin{pmatrix}1&-1\\\\0&1\\end{pmatrix}.\n\\]\nThey satisfy\n\\[\n(A^{\\mathsf T})^{\\mathsf T}=A,\n\\qquad\nA\\operatorname{adj}(A)=\\det(A)I,\n\\qquad\n\\det(SAS)=\\det(A)\n\\]\nfor the swap matrix $S$. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.","0&1\\end{pmatrix}.\n\\]\nThey satisfy\n\\[\n(A^{\\mathsf T})^{\\mathsf T}=A,\n\\qquad\nA\\operatorname{adj}(A)=\\det(A)I,\n\\qquad\n\\det(SAS)=\\det(A)\n\\]\nfor the swap matrix $S$. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.","The implementation works directly after quotienting global phase. A state is one of the six exact Bloch-axis vectors\n\\[\n(0,0,\\pm1),\n\\qquad\n(\\pm1,0,0),\n\\qquad\n(0,\\pm1,0),\n\\]\nrepresenting the Pauli eigenstates $|0\\rangle$, $|1\\rangle$, $|+\\rangle$, $|-\\rangle$, $|+i\\rangle$, and $|-i\\rangle$. The primitive grammar contains $H,S,X,Z$, acting as signed permutations of the Bloch coordinates:\n\\[\nH(x,y,z)=(z,-y,x),\n\\qquad\nS(x,y,z)=(-y,x,z),\n\\]\n\\[\nX(x,y,z)=(x,-y,-z),\n\\qquad\nZ(x,y,z)=(-x,-y,z).\n\\]\nThese rules are exact and avoid floating-point phase canonicalization.","tienting global phase. A state is one of the six exact Bloch-axis vectors\n\\[\n(0,0,\\pm1),\n\\qquad\n(\\pm1,0,0),\n\\qquad\n(0,\\pm1,0),\n\\]\nrepresenting the Pauli eigenstates $|0\\rangle$, $|1\\rangle$, $|+\\rangle$, $|-\\rangle$, $|+i\\rangle$, and $|-i\\rangle$. The primitive grammar contains $H,S,X,Z$, acting as signed permutations of the Bloch coordinates:\n\\[\nH(x,y,z)=(z,-y,x),\n\\qquad\nS(x,y,z)=(-y,x,z),\n\\]\n\\[\nX(x,y,z)=(x,-y,-z),\n\\qquad\nZ(x,y,z)=(-x,-y,z).\n\\]\nThese rules are exact and avoid floating-point phase canonicalization.","Starting from $|0\\rangle$, one has\n\\[\n|0\\rangle\\xrightarrow{H}|+\\rangle\n\\xrightarrow{S}|+i\\rangle\n\\xrightarrow{S}|-\\rangle,\n\\]\nwhile $H^2=X^2=Z^2=I$ on the projective orbit and $S^4=I$. The number of raw gate words grows as $4^n$, but the quotient orbit saturates at six objects.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.99\\textwidth]{13_qubit_atlas.pdf}\n\\caption{The left panel realizes the quotient states on the Bloch sphere. The right panel shows the relation-reduced gate graph for the displayed $H$ and $S$ generators.}\n\\label{fig:qubit-orbit}\n\\end{figure}","Take the rational triangle\n\\[\nK_0=\\left\\{(0,0),(2,0),\\left(1,\\frac32\\right)\\right\\}.\n\\]\nThe grammar contains quarter turns, reflection in the $x$ axis, horizontal shear, scaling by two, and unit translations in the two coordinate directions. Each descendant is again a finite rational point configuration, so the grammar is queried recursively at every child.","For example,\n\\[\nR(x,y)=(-y,x),\n\\qquad\nH(x,y)=(x+y,y),\n\\qquad\nS_2(x,y)=(2x,2y).\n\\]\nThe relation $R^4=I$ closes one route loop exactly. Reflection is an involution. Rotation preserves signed area, reflection reverses its sign, shear preserves it, and scaling by two multiplies it by four. These invariants separate visually similar but mathematically different descendants.","The executable completion example starts from $x_0=0$ and exposes the two depth-dependent directions\n\\[\nd_n^{\\pm}=\\pm2^{-(n+1)}.\n\\]\nA finite route with signs $\\varepsilon_n\\in\\{-1,+1\\}$ reaches\n\\[\nx_N=\\sum_{n=0}^{N-1}\\varepsilon_n2^{-(n+1)}.\n\\]\nEvery infinite sign route is Cauchy because\n\\[\n|x_m-x_n|\n\\le\n\\sum_{k=n}^{m-1}2^{-(k+1)}\n<2^{-n}.\n\\]\nThe completed object therefore contains a limit for every infinite route. The selected periodic sign pattern used in the figure converges numerically to $5/7$.","s\n\\[\nd_n^{\\pm}=\\pm2^{-(n+1)}.\n\\]\nA finite route with signs $\\varepsilon_n\\in\\{-1,+1\\}$ reaches\n\\[\nx_N=\\sum_{n=0}^{N-1}\\varepsilon_n2^{-(n+1)}.\n\\]\nEvery infinite sign route is Cauchy because\n\\[\n|x_m-x_n|\n\\le\n\\sum_{k=n}^{m-1}2^{-(k+1)}\n<2^{-n}.\n\\]\nThe completed object therefore contains a limit for every infinite route. The selected periodic sign pattern used in the figure converges numerically to $5/7$.","The same architecture covers iterated function systems. For the middle-third Cantor maps $L(x)=x/3$ and $R(x)=(x+2)/3$, finite words identify cylinder intervals and infinite words receive points only in the completed boundary. The dyadic example is used in the executable release because its exact tail certificate is especially transparent.","With four ordered leaves and one binary operation, the free operadic layer contains the five Catalan parenthesizations\n\\[\n((ab)c)d,\n\\quad\n(a(bc))d,\n\\quad\n(ab)(cd),\n\\quad\na((bc)d),\n\\quad\na(b(cd)).\n\\]\nThey are different syntax trees. If the operation is associative, all five evaluate to one quotient object. If it is also commutative, the $4!$ permutations of the leaves are identified as well. The raw tree and the quotient must therefore remain separate even in elementary algebra. The complete syntax-to-quotient visual appears in \\cref{fig:operadic}.","For a prime $p$, a state at level $m$ is a residue class modulo $p^m$ satisfying a declared congruence. Its direction fiber is the finite set of next base-$p$ digits that lift the current solution to level $m+1$.","For $x^2=1$ over powers of $3$, the two simple roots lift uniquely:\n\\[\n1\\pmod 3\\longleftarrow1\\pmod 9\\longleftarrow1\\pmod{27}\\longleftarrow\\cdots,\n\\]\n\\[\n-1\\pmod 3\\longleftarrow-1\\pmod 9\\longleftarrow-1\\pmod{27}\\longleftarrow\\cdots.\n\\]\nFor the singular equation $x^2=0$ over powers of $2$, some nodes have several valid next digits and some branches later die. Thus branching number is an object property detected by the local congruence geometry.","The \\BuiltInChecks\\ built-in checks verify:\n\\begin{itemize}\n\\item exact raw and quotient growth for the integer walk\n\\item the dyadic completion tail bound\n\\item monotone decay of the continuous-direction cover error\n\\item Catalan and Fibonacci identities\n\\item generation of $\\tanh x$ and $\\operatorname{sech}x$\n\\item inverse sequence transforms\n\\item Gaussian standardization\n\\item six-state projective qubit closure\n\\item purity preservation\n\\item matrix and affine involutions\n\\item graph relabeling invariance\n\\item unique simple-root lifting and singular-root branching\n\\end{itemize}\nAll \\BuiltInChecks\\ checks pass in the recorded run. The result file stores every Boolean check separately rather than recording only a final aggregate.","Symbolic examples & Polynomial route collapse, analytic generation of $\\tanh$ and $\\operatorname{sech}$, and explicit rejection of invalid reciprocal operations.","The proposed object can be summarized in one line:\n\\[\n\\boxed{\n\\begin{gathered}\n\\text{object}\n\\longrightarrow\n\\text{all declared directions at that object}\n\\longrightarrow\n\\text{all directions at every child}\n\\\\[0.25em]\n\\longrightarrow\n\\text{path forest}\n\\longrightarrow\n\\text{relation quotient}\n\\longrightarrow\n\\text{completed family universe}.\n\\end{gathered}\n}\n\\]","\\midrule\n\\endhead\nDirection fiber & $\\Dcal_x=\\pi^{-1}(x)$","One-step deformation & $x\\star d=\\Phi(d)$ for $d\\in\\Dcal_x$","Length-$n$ path layer & $\\Tcal_n(x_0)$","Endpoint map & $\\ev(d_1,\\ldots,d_n)=x_n$","Raw finite forest & $\\Tcal_{\\mathrm{fin}}(x_0)=\\coprod_{n\\ge0}\\Tcal_n(x_0)$","Path cost & $\\cost(\\gamma)=\\sum_k\\kappa(d_k)$","Quotient object graph & $\\Ocal(x_0)=\\Tcal_{\\mathrm{fin}}(x_0)/\\sim$","Infinite boundary & $\\partial\\Tcal(x_0)$","Cover recursion & $e_k\\le L_ke_{k-1}+M_k\\epsilon_k$","Cover closed bound & $e_n\\le (\\prod L_k)e_0+\\sum_jM_j\\epsilon_j\\prod_{k>j}L_k$","Summable branch condition & $d(x_{n+1},x_n)\\le a_n$, $\\sum a_n<\\infty$","Cylinder projector & $P^{(n)}_\\alpha\\lvert\\gamma\\rangle=1_{\\alpha\\preceq\\gamma}\\lvert\\gamma\\rangle$","Projector refinement & $P_\\alpha=\\sum_dP_{\\alpha d}$","Transfer block & $A_{\\beta\\alpha}=Q_\\beta A P_\\alpha$","Path product law & $(BA)_{\\gamma\\alpha}=\\sum_\\beta B_{\\gamma\\beta}A_{\\beta\\alpha}$","Raw entropy & $h_{\\mathrm{raw}}=\\limsup n^{-1}\\log a_n$","Object entropy & $h_{\\mathrm{obj}}=\\limsup n^{-1}\\log q_n$","Path partition function & $Z_{x_0}(s)=\\sum_{\\gamma\\ne\\varnothing}e^{-s\\cost(\\gamma)}$","Regular-tree threshold & $sc>\\log B$","Family closure & $F(\\sigma_d(\\theta))=d\\cdot F(\\theta)$","Integer walk has $2^n$ paths and $n+1$ endpoints & proved & exact counting","\\begin{table}[htbp]\n\\centering\n\\small\n\\caption{Selected analytic objects generated from the seed $\\cosh x$.}\n\\label{tab:analytic-objects}\n\\begin{tabular}{rL{0.26\\textwidth}L{0.48\\textwidth}}\n\\toprule\nDepth & Object & One shortest generating route","$p$-adic simple & 10 & 10 & 8 & 0","$p$-adic singular & 21 & 21 & 20 & 0","\\midrule\n\\endhead\nInteger walk & $0$ with $+1$ and $-1$ & $2^n$ routes collapse to $n+1$ endpoints, while exact binomial route multiplicities remain available.","Polynomials & $x^2+1$ with differentiation, translation, reflection, difference, and multiplication & Commuting and involutive operator routes become exact quotient relations between polynomial endpoints.","Analytic functions & $\\cosh x$ with derivative, reciprocal, logarithmic derivative, parity, square, and doubling & Hyperbolic families, poles, singular descendants, and invalid operations coexist in one audited graph.","Sequences & A Fibonacci prefix and the impulse $\\delta_0$ with difference, summation, binomial transform, sign alternation, and reversal & Exact transform relations generate structured integer families rather than one undifferentiated list of examples.","Gaussian laws & $N(0,1)$ with shift, scale, reflection, convolution, centering, and standardization & The relation-reduced atlas lives in exact $(\\mu,\\sigma^2)$ coordinates and exposes state-dependent canonicalization.","Matrices & $\\left(\\begin{smallmatrix}1&1\\\\0&1\\end{smallmatrix}\\right)$ with transpose, square, adjugate, conjugation, symmetrization, and commutator & Exact trace, determinant, rank, and symmetry metadata distinguish invariant-preserving routes from genuine deformation.","Graphs & $P_3$ with complement, line graph, subdivision, leaf addition, and relabeling & Different labeled adjacency matrices collapse to canonical unlabeled graph-isomorphism classes.","Quantum states & The projective seed $|0\\rangle$ with exact Bloch actions of $H,S,X,Z$ & Exponentially many gate words saturate at six Pauli eigenstates after global phase has been quotiented out.","Congruence lifts & Roots of $x^2=1$ over $3^m$ and $x^2=0$ over $2^m$ & Simple roots lift uniquely, while singular roots acquire state-dependent branching and dead ends."]}