# Global Geometry II: local rules, observable-relative universality, and obstruction-aware inverse design **Research manuscript draft — 2026-08-30** **Status:** alpha research manuscript; not peer reviewed; no publication, physical-validation, or literature-priority claim. The separately addressed U2 evaluation-v2 checkpoint is `UNRESOLVED`, not a confirmatory decision. ## Abstract How can a finite-radius rule, which sees only a bounded neighborhood, produce a global shape, dimension, or geometry? We develop a discrete framework in which the answer has three parts. First, local incidence and metric data glue into a global piecewise geometry only when local compatibility and global integral constraints agree. Second, equivariant local dynamics propagate state through finite causal cones while preserving or modifying only the structures they are authorized to change. Third, a refinement and scaling program can erase some microscopic distinctions while retaining others. The resulting notion of universality is therefore relative to a declared metric, operator, observable family, and convergence topology. The formal package contains sixteen proved-here finite or elementary propositions. Two results isolate the central phenomenon. Unit-edge square and triangular lattice graphs converge, in their graph-length metrics, to nonisometric normed planes with four- and six-vertex unit balls, respectively. Yet uniform random walks on the corresponding orthogonal and equilateral step sets have the same covariance-normalized Gaussian finite-dimensional limit. Thus a pair of local constructions can be universal as diffusion laws and nonuniversal as metric spaces. A third result proves a robust cylinder-law basin for fixed-dimensional, centered, bounded-range iid increments with uniformly nondegenerate covariance, after exact law-specific covariance whitening. We accompany the theory with deterministic finite artifacts: analytic spectral calibration on periodic lattices; a 5-by-5 anisotropy/sample-size resolution map inside the proved diffusion basin; four bounded filled-disk construction families; a deliberately nonconfirmatory preview atlas; and an inverse compiler that orders surface, curvature, circle-packing, and metric compatibility gates before synthesis. Representative certificates include admitted intrinsic targets on a tetrahedral sphere, triangular disk, annulus, and periodic torus; a target that passes Gauss--Bonnet but fails a Chow--Luo subset inequality by exactly \(-\pi/5\); a zero-curvature sphere rejected by Gauss--Bonnet; and a degenerate triangle rejected by an exact binary64-dyadic sign test. A programmable q-star sheet and hardware-in-the-loop emulator translate the intrinsic theory into a digital engineering instrument, without claiming physical validation. All finite numerical findings remain distinct from continuum theorems. Bounded digital Darwin/Windows replay status is delegated to the content-addressed [comparison record](artifacts/global-geometry-ii/certificates/macos-windows-comparison-v1.json), whose inputs are [the macOS local record](artifacts/global-geometry-ii/certificates/macos-local-v2.json) and [the Windows local record](artifacts/global-geometry-ii/certificates/windows-local-v1.json). That comparison closes only its declared digest-and-focused-test surface. The v1 records are now historical prior-release evidence and do not certify the enlarged v2 surface. A later strict-partial and screening checkpoint is documented in the [U2 evaluation report](GLOBAL_GEOMETRY_II_U2_EVALUATION_REPORT.md): the finite preview remains `NOT_EVALUATED`, the evaluation checkpoint is `UNRESOLVED`, and the full confirmatory U2 atlas and every physical, camera, device, or hardware experiment remain `NOT_RUN`. ## 1. The local-to-global question The phrase “local rules produce global geometry” can hide several different claims. A local formula may create one finite trajectory, a sequence of larger finite objects, a convergent refinement family, or a robust limiting law. These statements are not interchangeable. Global Geometry II keeps the chain explicit: \[ \text{local rule} \longrightarrow \text{finite trajectory} \longrightarrow \text{refinement family} \longrightarrow \text{declared macroscopic limit}. \] The programme has forward, inverse, and recognition forms: \[ \begin{array}{rcl} \text{forward} &:& \text{local programme}\to\text{global geometry},\\ \text{inverse} &:& \text{target geometry}\to\text{candidate local programme},\\ \text{recognition} &:& \text{local observations}\to\text{admissible global worlds}. \end{array} \] The constructive answer developed here is: \[ \boxed{ \begin{aligned} \text{global geometry} ={}&\text{locally compatible cells and measurements}\\ &+\text{global obstruction and conservation laws}\\ &+\text{dynamical selection from initial and boundary data}\\ &+\text{a declared refinement and observation scale}. \end{aligned}} \] Locality is sufficient to build global organization because neighboring patches share data, updates propagate, and repeated composition reaches the whole carrier. Locality is not sufficient to guarantee an arbitrary global world: topology, total curvature, conservation, and model-specific inequalities can obstruct it. Nor does a visually stable large object by itself establish a continuum limit. The observation and scaling contract is part of the geometry being claimed. ## 2. Evidence discipline Every scientific statement in the project has one primary label: | Label | Meaning in this work | |---|---| | `proved` | A complete proof is given here, or a cited theorem is used after its hypotheses are checked. | | `computational` | A deterministic or statistical finite calculation with a versioned method and falsifier. | | `conjectural` | A quantified mathematical statement for which the required proof is not present. | | `engineering` | A model, emulator, calibration target, or design approximation with a declared envelope. | | `speculative` | An interpretation that has no evidentiary force. | A proved record additionally declares `proved-here` or `proved-external`. Tests can verify that software implements a formula; test success alone never turns a numerical statement into a theorem. “Exact” is reserved for symbolic or combinatorial statements on the declared finite object. A binary64 evaluation of an exact identity remains a finite numerical audit unless an exact representation or justified enclosure is supplied. The implementation uses closed, versioned records. Each claim names its scope, method, assumptions, falsifier, dependencies, artifacts, tests, and limitations. Internal 32-bit canonical checksums bind replay records and detect accidental mutations; they are not security boundaries. Generated artifact containers add canonical SHA-256 content addresses. ## 3. Mathematical setting ### 3.1 Carrier, metric, measure, and operator A finite carrier is a typed incidence structure \[ K=(C_0,C_1,\ldots,C_D,\prec), \] with finite cell sets and codimension-one incidence. A simplicial or cellular carrier supplies boundary maps over \(\mathbb F_2\) satisfying \(\partial_{q-1}\partial_q=0\). The principal implemented surface class is a finite triangular complex \(K=(V,E,F)\). At time \(n\), the declared discrete world is \[ \mathfrak G_n=(K_n,\ell_n,\mu_n,W_n,U_n,A_n,B_n,\Pi_n). \] Here \(\ell\) assigns positive intrinsic edge lengths, \(\mu\) is a declared cell measure, \(W\) assigns transport conductances, \(U\) is local state, \(A\) is actuator state, and \(B\) records boundary data. The display or measured embedding \(\Pi\) is not authoritative intrinsic geometry unless a configuration explicitly compiles lengths from it. For a connected one-skeleton, \[ d_\ell(x,y)=\min_{\gamma:x\leadsto y}\sum_{e\in\gamma}\ell_e. \] For a valid triangulated surface, the project can instead use the piecewise-Euclidean length metric, which permits paths through face interiors. Those metrics need not share a limit. Likewise, the transport operator is declared rather than inferred from the metric. One implemented family is the symmetric normalized Laplacian \[ L=I-D^{-1/2}WD^{-1/2}. \] Graph, finite-element, cotangent, and random-walk operators are different objects and receive different identifiers. ### 3.2 Finite-radius rules For a cell \(c\), let \(\mathcal N_R(c;\mathfrak G)\) be its rooted, typed, decorated radius-\(R\) neighborhood in the declared adjacency. A synchronous deterministic local rule applies the same isomorphism-invariant kernel to every such neighborhood and commits all writes at once: \[ \mathfrak G_{n+1}=F_R(\mathfrak G_n;\theta,b_n). \] The locality classes are: - **L0:** a bounded neighborhood and fixed constants; - **L1:** L0 plus a disclosed boundary value, precompiled target, broadcast, or global scalar; and - **L2:** a global eigensolve, optimization, equilibrium solve, matching, or any computation that can depend on the whole state. An L2 compiler may create immutable target data used by an L1 runtime. This does not make the compiler local. Structural rewrites are also separate from state updates: a rule that never changes \(K\) cannot change the carrier's homology. ### 3.3 Generability and recognition The framework distinguishes four strengths of generability: 1. exact finite generability; 2. approximate finite generability in a declared target distance; 3. asymptotic generability in a declared convergence topology; and 4. robust generability under a declared perturbation class. “Programmable” means that an inverse compiler returns a rule programme and the strongest certificate supported at one of these levels. “Recognizable” means that the estimator is identifiable and stable under a declared observation model; merely calculating a descriptor is not recognition. ### 3.4 Refinement and observable profiles A refinement programme is a family \[ \mathscr P= \left\{(\mathfrak G_h,a_h,b_h,\tau_h,\iota_h,\mathbb P_h)\right\}_{h\downarrow0}, \] where distance, measure, and time are rescaled by \(a_h,b_h,\tau_h\), \(\iota_h\) is a comparison or coupling map, and \(\mathbb P_h\) is the law of any seeded disorder. A refinement must identify its resolution, boundary model, mesh-quality gates, and convergence topology. The observable profile is not one dimension number. It contains distinct clocks: \[ \Phi_h= \left(d_V(r),d_s(t),d_w(m),\kappa_h,\beta_k(r), \lambda_2,\operatorname{transport},\operatorname{stability}\right). \] Metric radius \(r\), heat time \(t\), walk time \(m\), and filtration scale are not silently identified. Agreement of a few components is profile equivalence, not isometry. ### 3.5 Universality is relative to retained structure Two families can agree at several distinct levels: replay, finite isomorphism, finite intrinsic isometry, finite profile equivalence, finite-dimensional or cylinder-law equivalence, common limit object, or path-space/object-law universality. Robustness modifies any one of these levels when its conclusion holds uniformly over a predeclared microscopic basin; it is not a stronger topology. For random rescaled objects \(\mathcal Z_h^A,\mathcal Z_h^B\) in a declared state space and topology \(\mathcal T\), object-level universality means \[ \mathcal Z_h^A\Rightarrow\mathbb Q_\infty, \qquad \mathcal Z_h^B\Rightarrow\mathbb Q_\infty \] for one limit law. The topology determines what is retained: metric-measure shape, rooted neighborhoods, Dirichlet form, curvature measure, topology, marks, or a combination. Universality in one projection does not imply universality in another. ## 4. Proved-here proposition package The machine-readable ledger contains sixteen proposition records. Their proofs are elementary or finite, and their scopes are deliberately narrow. | ID | Exact statement in its declared scope | Local-to-global role | |---|---|---| | P3.1 | A synchronous radius-\(R\) rule on fixed adjacency has a radius-\(RT\) causal cone after \(T\) steps. | Bounds propagation. | | P3.2 | Neighborhood-isomorphism kernels with equivariant commit commute with relabeling. | Prevents identifier-dependent geometry. | | P3.3 | State-only updates on a fixed carrier preserve homology and Euler characteristic. | Separates geometric evolution from topology rewrite. | | P3.4 | A deterministic radius-\(R\), \(T\)-round observer cannot distinguish roots with matching radius-\(RT\) decorated neighborhoods. | Gives a recognition obstruction. | | P4.1 | Positive shared edge lengths satisfying every strict face inequality glue to a piecewise-Euclidean complex; valid links make it a surface. | Turns compatible local cells into global intrinsic geometry. | | P4.2 | Positive tangency radii with \(\ell_{ij}=r_i+r_j\) satisfy all face inequalities. | Supplies an automatically compatible local metric family. | | P4.3 | With interior target angle \(2\pi\) and boundary target angle \(\pi\), finite angle defects sum to \(2\pi\chi(K)\). | Gives a global curvature obstruction. | | P4.4 | The normalized-Laplacian zero multiplicity equals the number of connected components. | Couples global topology to diffusion. | | P4.5 | Symmetric edge flux conserves the measure-weighted total. | Produces a global invariant from local cancellation. | | P5.1 | Solvable circle-radius edge sums are unique on a connected nonbipartite graph and retain a one-dimensional alternating gauge on a bipartite graph. | States inverse identifiability exactly. | | P5.2 | If every edge of a fixed connected \(N\)-vertex graph changes by at most \(\varepsilon\), every shortest-path distance changes by at most \((N-1)\varepsilon\). | Quantifies finite metric stability. | | P5.3 | Closed deterministic generation, seeds, broadcasts, and synchronous updates determine one finite trajectory. | Grounds replay. | | P6.1 | Straight inherited subdivision is isometric in the piecewise-Euclidean metric. | Supplies a null refinement control. | | P8.1 | Unit-edge square and triangular graph metrics have nonisometric normed-plane limits. | Proves metric nonuniversality. | | P8.2 | Square and equilateral-triangular simple walks have the same Gaussian finite-dimensional diffusion limit after common covariance scaling. | Proves a scoped universality relation. | | P8.3 | Centered bounded-range iid laws with a uniform covariance gap have one standard Gaussian finite-dimensional limit after exact law-specific whitening, uniformly over the declared basin. | Proves robust level-5/cylinder-law universality. | The most important structural consequences are already visible in P3.1, P4.3, and P6.1. A bounded rule reaches a global world only by repeated composition; total curvature can reject globally inconsistent local targets; and increasing cell count can be geometrically null when it merely subdivides an unchanged intrinsic surface. ### 4.1 P8.1: exact graph-metric nonuniversality Let the square coordinate graph have generators \(\{\pm e_1,\pm e_2\}\), and let the triangular coordinate graph have generators \(\{\pm e_1,\pm e_2,\pm(e_2-e_1)\}\). Give every edge length \(h\). The exact word distances from the origin are \[ d_\square(0,(m,n))=|m|+|n|, \] and \[ d_\triangle(0,(m,n))=\max\{|m|,|n|,|m+n|\}. \] Each formula follows from a matching lower bound on the coordinate changes of one allowed step and an explicit path attaining it. Multiplying by \(h\) and rounding continuum points to nearby lattice points changes distances by \(O(h)\). On bounded pointed sets the two graph-length spaces therefore converge to polyhedral normed planes. The square unit ball has four extreme points; the triangular unit ball has six. By affine rigidity of surjective isometries of real normed spaces, an isometry would preserve the number of extreme points. Hence the limiting graph metrics are not isometric. This is a negative control, not a failure of the programme. It demonstrates that a common dimension exponent or heat profile cannot identify the entire geometry. ### 4.2 P8.2: a common finite-dimensional diffusion limit Embed the square step set as \[ S_\square=\{\pm(1,0),\pm(0,1)\}, \] and the triangular step set as \[ S_\triangle=\{\pm u,\pm v,\pm(u-v)\}, \quad u=(1,0),\quad v=(1/2,\sqrt3/2). \] Choose every step independently and uniformly. Symmetry gives mean zero. A direct calculation gives the same one-step covariance for both laws: \[ \frac1{|S_A|}\sum_{z\in S_A}zz^{\mathsf T}=\frac12I_2, \qquad A\in\{\square,\triangle\}. \] Their characteristic functions consequently have the common expansion \[ \varphi_A(\xi)=1-\frac14\lVert\xi\rVert^2 +O(\lVert\xi\rVert^4). \] For \(Z_n^A(t)=n^{-1/2}X^A_{\lfloor nt\rfloor}\), \[ \varphi_A(\xi/\sqrt n)^{\lfloor nt\rfloor} \longrightarrow \exp(-t\lVert\xi\rVert^2/4). \] For finitely many times, every linear combination of positions decomposes into disjoint increment blocks. Independence makes their characteristic functions multiply, proving convergence to one centered Gaussian process in finite-dimensional distribution, with \[ \mathbb E[B_sB_t^{\mathsf T}]=\frac12\min(s,t)I_2. \] At spatial edge scale \(h\), accelerating time to \(2h^{-2}\) steps per unit physical time gives the standard planar Brownian finite-dimensional laws. The fourth moments of the two microscopic step laws differ, and P8.1 proves that their graph metrics differ. P8.2 therefore identifies a genuine but strictly scoped universality relation. It does not establish path-space tightness, filled-disk boundary behavior, a common graph metric, or robustness to disordered conductance. ### 4.3 P8.3: a robust whitened cylinder-law basin Fix dimension \(d<\infty\), support bound \(M<\infty\), and \(0<\kappa\le1\). Let \(\mathcal P_{M,\kappa}\) be a family of laws on \(\mathbb R^d\) such that, for \(X\sim P\), \[ \mathbb E_PX=0, \qquad \lVert X\rVert\le M\ \text{almost surely}, \qquad \kappa I\preceq\Sigma_P\preceq\kappa^{-1}I. \] For iid increments define \[ W_n^P(t)=n^{-1/2}\Sigma_P^{-1/2} \sum_{j=1}^{\lfloor nt\rfloor}X_j, \] where \(\Sigma_P^{-1/2}\) is the exact symmetric covariance whitening for that law. For every fixed schedule \(0=t_0