Measurement Operators on Scutoid Spacetime#
A measurement needs three decisions: which cells to inspect, what number to attach to each cell, and how to combine those numbers. Geometry answers the first question. A distance and an amplitude rule answer the second. If the cell values have local internal frames, parallel transport answers the third.
The cell reconstruction is developed in Scutoid Spacetime: Moving Cells and Neighbor Changes. The comparison links used here follow Direct Field Observables and Lattice QFT on the Fractal Set and Discrete Yang–Mills Actions, Noether Identities, and Quantum Reconstruction.
A line or hyperplane selects the Voronoi cells it intersects. Each selected swept cell supplies an amplitude and a geometric phase. Transport brings its field value to a common reference before addition. The norm of this sum is independent of local frame choices for any fixed set of paths. Independence from the paths themselves requires a further condition: the relevant loop transports must act trivially. A spanning tree makes that condition finite and directly testable.
1. Probes and the cells they select#
Imagine passing a sheet of paper through a collection of cells. A cell is selected when some part of it touches the sheet; its center need not lie on the sheet. The same rule works for a thin straight probe. This distinction matters computationally: a line can pass through the middle of a square while missing all four vertices.
Hyperplanes and straight lines refer to a specified Euclidean coordinate chart. The Voronoi cells may use either Euclidean distance or the declared spatial metric. Those choices are separate parts of the measurement.
Definition 804 (Measurement schedule and observation geometry)
Fix \(T>0\) and measurement times
At each time, specify the included sites, spatial domain \(M\), metric used for the Voronoi cells, and any observation clipping region. The cells \(\operatorname{Vor}_i(t_k)\) are those of Definition 674, with the clipping convention applied when present. In coordinate formulas below, the observation region lies in a fixed chart \(x\in\mathbb R^d\).
When the recorded history is discrete, select available frames or declare the interpolation used to evaluate an intervening measurement time. The measurement interval \(T\) need not equal the algorithm’s time step.
Definition 805 (Hyperplane probe)
For a Euclidean unit normal \(n\in\mathbb R^d\) and offset \(b\), define the ambient hyperplane and its signed coordinate distance by
The probe inside the observation domain is \(H\cap M\). The displayed formula measures signed distance to the ambient hyperplane; distance to a clipped portion of that hyperplane can differ.
Definition 806 (Line probe)
For an anchor \(p\in\mathbb R^d\) and Euclidean unit direction \(u\in\mathbb R^d\), define
This is the unsigned distance to the ambient line. A segment probe is obtained by specifying a closed parameter interval for \(\lambda\); its distance and intersection tests use that interval.
Cell selection and adjacency#
Definition 807 (Pierced Voronoi cell: hyperplane)
A cell is pierced when its intersection with the probe is nonempty. For a hyperplane, the selected indices are
This convention includes tangencies and contact along a cell boundary.
Definition 808 (Pierced Voronoi cell: line)
For a line probe, the selected indices are
For a segment, replace \(L\) by the specified segment.
Definition 809 (Pierced neighbor graph)
Let \(E_{\mathrm{DT}}(t_k)\) be the edges of the specified Delaunay or cell-adjacency construction on the slice. For either selected set \(P_{t_k}\), define the induced graph
Adjacency uses the geometric convention in Definition 675 and Definition 676. For a general metric it is supplied by that construction; the Euclidean Delaunay predicates apply under Lemma 279. The induced graph can have several connected components.
Lemma 323 (Exact piercing tests for convex polyhedral cells)
Let \(C\ne\varnothing\) be a convex polyhedron. If \(C\) is bounded with vertex set \(V\), then
For a half-space description \(C=\{x:a_\ell\cdot x\le c_\ell\}\), line piercing is equivalent to feasibility of
Thus positive coefficients give upper bounds on \(\lambda\), negative coefficients give lower bounds, and zero coefficients require a nonnegative right-hand side. The intersection is nonempty precisely when all zero constraints hold and the largest lower bound is at most the smallest upper bound. Segment bounds are included in this interval intersection.
Proof
A linear functional on a bounded polyhedron takes its minimum and maximum at vertices. Its image on a convex set is an interval, so that image contains zero exactly under the stated inequalities. For a line, substitute \(x=p+\lambda u\) into every defining half-space. Dividing each nonzero coefficient with its corresponding inequality direction gives exactly the listed bounds. These substitutions are reversible, proving sufficiency as well as necessity. For an unbounded cell, the hyperplane test can instead be performed as affine feasibility with the same half-space constraints.
The vertex sign test succeeds because a linear function takes every value between its extrema on a convex cell. Distance to a line has a different minimum: that minimum may occur inside the cell. For example, the horizontal axis pierces \([-1,1]^2\), although every vertex has distance one from it. The interval test keeps the geometry of the whole cell.
2. Swept cells, representatives, and Euclidean phases#
The slice gives us a selected label. To attach a quantity to its evolution, we need the cell swept out during a time interval. The endpoint positions alone do not determine that region. In particular, a cloning replacement can jump: the recorded slot continues, while its parent identifier records a separate genealogical relation.
A midpoint is often enough for a phase feature. Its appeal is that it is cheap and reproducible. Its meaning remains an average of the chosen points; the center of mass of a curved swept region generally requires an integral over the region.
Definition 810 (Scutoid indexing on a measurement slice)
Use the interpolation and event policy of Definition 680. Write \(S_{i,k}\) for the swept cell of recorded slot \(i\) on \([t_k,t_{k+1}]\), with explicit one-sided pieces at replacement jumps when required. Bottom association selects the cells whose bottom index lies in \(P_{t_k}\); top association uses the corresponding top cells on the terminal slice. The set selected by bottom association is denoted \(\mathcal S_{t_k}\).
An association across a birth, death, or missing endpoint must be supplied by the reconstruction. A parent identifier can define a separate lineage observable, but does not by itself identify that parent’s swept cell with the replaced slot’s cell. A measurement interval containing several events retains those events or records an explicit coarsening rule.
Definition 811 (Center-only representative)
For supplied bottom and top positions \(x^-_{i,k},x^+_{i,k}\), define
These are barycenters of the two specified endpoint points. They need not be volume barycenters of \(S_{i,k}\) or lie inside a nonconvex swept cell. Coordinates use one chart and, on a periodic domain, a declared consistent lift before averaging.
Definition 812 (Vertex-augmented representative)
If finite bottom and top vertex sets \(V^-_{i,k},V^+_{i,k}\) are supplied in the same coordinates, define
This averages endpoint generators and endpoint vertices with equal weights. For a cell of finite positive measure \(\nu\), its spatial volume barycenter is instead \(\nu(S)^{-1}\int_S x\,d\nu(t,x)\), when that integral exists. The vertex formula requires a finite vertex representation; it does not supply one for a curved or unbounded cell.
Definition 813 (Euclidean hyperplane phase)
For either chosen representative \(\bar x(S)\), define
The scale \(k_H\) has inverse-length units in the specified coordinates, so the phase is dimensionless. Reversing the probe orientation reverses this phase.
Definition 814 (Euclidean line phase)
Define the unsigned line-distance phase
where \(k_L\) has inverse-length units. Reversing \(u\) leaves this phase unchanged.
3. Intrinsic distances and graph measurements#
A path along graph edges is a permitted route between sites. Its length can exceed the shortest route through the surrounding space. If the metric also changes the cost of moving in different directions, Euclidean edge lengths introduce another difference. Both effects can be bounded, but the bounds need geometric information.
There is also a change of endpoint to watch. Distance from a cell representative to its walker measures displacement within the reconstructed cell. Distance from a walker to the set of pierced walkers measures access to the probe. A walker already in that set has access distance zero even when its cell representative has moved.
Definition 815 (Intrinsic probe phases and their graph alternatives)
On a specified measurement slice, let \(g\) be a positive-definite spatial metric and \(d_g\) its length distance. If \(g=D_x^2V+\epsilon_\Sigma I\), require the spectral bounds of Lemma 277; positivity of \(\epsilon_\Sigma\) alone is insufficient. For a nonempty probe portion \(B\subset M\), set
These intrinsic phases are unsigned. An oriented hyperplane variant is \(\operatorname{sgn}(n\cdot\bar x-b)\phi_g^H(S)\). Intrinsic evaluation requires \(\bar x(S)\in M\), with the slice and metric specified even when \(\bar x\) comes from a time interval.
For a spatial graph with sites \(x_i\), define \(d_{\mathrm{DT}}\) using Euclidean edge lengths. If each edge has a specified admissible spatial curve, define \(d_{G,g}\) using its \(g\)-length instead. A shortest-path calculation approximates an intrinsic phase only with a distance comparison and a rule attaching the representative and the probe to the graph.
Lemma 324 (Metric comparison and endpoint errors for graph distances)
For a graph whose edges are admissible rectifiable curves in \(M\),
If the edges are straight segments in the chart and \(\lambda I\preceq g\preceq\Lambda I\) along them, then
These comparisons do not assert that the graph contains routes close to minimizing geodesics. Suppose, in addition, that for the required vertices
For a nonempty set of attached probe sites \(X_P=\{x_j:j\in P\}\), assume their \(d_g\)-Hausdorff distance from the chosen probe portion \(B\) is at most \(h\), and \(d_g(\bar x,x_i)\le a\). Then
Proof
Every graph path concatenates admissible curves. Its \(g\)-length is at least the infimum over all such curves, proving the first assertion. The spectral bounds give \(\sqrt\lambda|\dot\gamma|\le |\dot\gamma|_g\le\sqrt\Lambda|\dot\gamma|\) on each edge. Integrating, summing, and taking infima over graph paths proves the second.
The additional graph estimate survives taking the minimum over \(j\in P\). Distance to a set is a 1-Lipschitz function of its starting point. Replacing one target set by another at Hausdorff distance at most \(h\) changes its distance function by at most \(h\), as follows by the triangle inequality and an arbitrarily close point to each infimum. Applying these two facts and the graph estimate gives the last bound.
Definition 816 (Euclidean walker-relative phases)
For the specified endpoint positions and representative, define
Here \(k_W\) has inverse-length units. For the center-only representative, both phases equal \(k_W\|x^+_{i,k}-x^-_{i,k}\|/2\).
Definition 817 (Intrinsic endpoint phases and graph probe-access phases)
With a specified spatial metric \(g_-\) or \(g_+\) for each endpoint comparison, the intrinsic endpoint phases are
All compared points must lie in the domain of the respective distance. Separately, the computable graph probe-access phases are
where \(d_G(i,P)=\min_{j\in P}d_G(i,j)\). A missing endpoint or unreachable set gives an undefined measurement, or an explicitly flagged infinite distance, rather than a finite phase. For a reachable pierced vertex these access phases vanish. They describe access to selected sites; approximating the intrinsic endpoint phases instead requires graph attachments to \(\bar x\) and the endpoints, with errors controlled as above.
4. Amplitudes and local complex fields#
An amplitude determines how strongly each selected cell contributes. Volume weighting asks how much reconstructed space-time the cell occupies. Probability weighting asks how much mass a specified sampling law assigns to a region. Inverse-density weighting has a third role: it compensates for uneven sampling when estimating volume.
After choosing an amplitude, multiplying it by a complex phase gives a signal feature. Interference in the resulting sum is an algebraic property of complex numbers. A probabilistic or physical interpretation of its squared magnitude requires a separately specified measurement model.
Definition 818 (Scutoid amplitude and its reference measure)
Choose a nonnegative finite amplitude for each included swept cell. Two geometric choices are
The first uses a declared volume measure \(\nu\) on the reconstruction, such as coordinate measure \(dt\,dx\) or \(dt\,d\operatorname{vol}_{g_t}\). Its units follow that measure. The notation \(w_{\mathrm{geo}}(S)\) in the second formula denotes a specified nonnegative volume quadrature. For example, under the sampling law \(q\,d\nu\) with \(q>0\), the inverse-density estimator from Definition 689 is
For a deterministic finite-volume \(S\) and samples with that marginal law, its expectation is \(\nu(S)\), since \(\int\mathbf1_S q^{-1}q\,d\nu=\nu(S)\). If \(S\) is selected from the same samples, this expectation needs the corresponding conditional sampling argument.
Here \(A_Q\) estimates geometric volume by inverse-density quadrature. A probability amplitude such as \(A_\mu(S)=\mu(B_S)\) instead requires a specified probability law \(\mu\) on the slice and a measurable footprint \(B_S\). Using a QSD for \(\mu\) requires the appropriate marginal of the identified QSD; its mass is dimensionless and differs from inverse-density weighting.
Definition 819 (Measurement functional in a common frame)
For a declared common trivialization of the field values, define the raw complex signal
In the scalar case, \(\psi(S)=A(S)e^{i\phi(S)}\) in that trivialization. In a \(q\)-dimensional unitary representation, the field lies in \(\mathbb C^q\). Choose one of the probe phases above, optionally adding a specified endpoint or access phase. Every summand uses the same units and normalization convention.
When each cell has its own local frame, this raw sum depends on those frames. The intrinsic comparison uses the reference-transported sum of Definition 823.
5. Comparing fields through a connection#
Suppose two observers express vectors using different axes. Adding their coordinate lists before aligning the axes gives an answer that changes when either observer rotates a notebook. A connection records how to make that alignment along an edge. A chain of edges transports all vectors into one observer’s frame, where their sum has a definite meaning.
The geometric phase remains a scalar under this change of internal axes. The internal direction of the field changes with the frame. Keeping these two transformations separate fixes the orientation signs in the formulas.
Definition 820 (Edge parallel transport on the measurement graph)
Use a unitary representation of a group \(G\subset U(q)\) and a fiber \(V_i\simeq\mathbb C^q\) at each measurement vertex. The comparison link
maps coordinates at \(j\) into the frame at \(i\), following Definition 699 and Definition 731. Under local frame changes,
The scalar case uses \(G=U(1)\); \(SU(q)\) gives a nonabelian example. For a directed path \(\gamma=(i_0,i_1,\ldots,i_m)\) traversed from \(i_0\) to \(i_m\), the forward transport is
The rightmost factor acts first. Reverse paths have inverse transport.
Definition 821 (Edge-type transport assignment)
Specify the graph on which the measurement transports its fields and assign a comparison link to every edge used. A recorded Fractal Set edge can inherit a supplied link of the same representation from Definition 658:
An IG comparison uses its spatial link.
A CST edge included across slices uses its temporal link, with the inverse for reverse traversal.
An IA edge uses its attribution link when that link is part of the selected field construction.
A Delaunay neighbor pair need not be a recorded IG pair. Such an edge requires a declared connection reconstruction or another path through available links. Geometry alone does not assign its gauge matrix. All links in a product must have matching source and target fibers. Interaction-pair doublets use a graph of the corresponding pair states, as in Definition 700; passing from vertex fields to pair fields requires that explicit representation choice.
Definition 822 (Scutoid field and transport to a reference)
Attach the bottom-associated cell \(S_{i,k}\) to the fiber at its recorded vertex \(i\) on slice \(t_k\). Choose a unit vector \(\chi_i\in V_i\) and set
The amplitude and geometric phase are gauge scalars; \(\chi_i\) transforms as \(G_i\chi_i\), so \(\psi_i\) transforms as \(G_i\psi_i\). No fiber at the geometric representative is required for this definition.
Within a connected measurement component choose a reference vertex \(r\) and a path \(\gamma_i:i\to r\) for each included field. The paths are fixed as part of the measurement, independently of local frame coordinates. Define
For disconnected components use one reference per component, or supply additional comparison paths in a specified larger graph.
Definition 823 (Reference-transported measurement)
For the cells attached to one reference component, define
The scalar amplitude statistic and nonabelian intensity statistic retain the conventions
Thus \(\mathcal J_r=\mathcal I_r^2\) in the first convention. Use one normalization consistently when comparing measurements. For disconnected components, \(\sum_c\mathcal J_{r_c}\) is a scalar statistic without requiring an identification of their fibers.
Lemma 325 (Reference-transport covariance)
For any fixed paths and any unitary connection,
Consequently \(\mathcal I_r\) and \(\mathcal J_r\) are gauge invariant. Moreover,
These conclusions require no flatness assumption.
Proof
Under the frame change, the factors in \(T_\gamma=U_{i_mi_{m-1}}\cdots U_{i_1i_0}\) become
Each adjacent pair \(G_j^{-1}G_j\) cancels. The surviving factors are \(G_{i_m}T_\gamma G_{i_0}^{-1}\). Multiplication by \(G_{i_0}\psi_{i_0}\) proves field covariance, and summation gives the same transformation for \(\mathcal M_r\). Unitarity preserves its norm; equivalently, \(\mathcal M_r\mathcal M_r^\dagger\) transforms by conjugation and its trace is unchanged. Finally every \(T_{\gamma_i}\) is unitary and \(\|\psi_i\|=A(S_{i,k})\). The triangle inequality proves both bounds.
Gauge invariance has now been proved, even if different routes rotate a vector differently. The routes are part of this measurement, just as the position of the probe is part of it. Changing a route is a change of the measurement protocol. The next result identifies precisely when that change has no effect.
6. Path independence and loop observables#
Start with a spanning tree: it reaches every vertex using one route, with no loops. Every extra edge closes a loop against that tree. These extra edges provide a finite list of tests for whether all possible routes agree. Checking small triangles suffices only when those triangles also account for the larger loops. A hole in the selected graph can leave an additional loop to check.
Definition 824 (Gauge-compatible probe field)
A unit section \(s_i\in V_i\) is parallel on the measurement graph when
One may impose this as an additional compatibility requirement on \(s_i=e^{i\phi(S_{i,k})}\chi_i\). In the scalar case, writing \(s_i=e^{i\theta_i}\) in a chosen trivialization gives
The total phase \(\theta_i\) includes the frame-dependent phase of \(\chi_i\); the geometric probe phase \(\phi\) by itself remains gauge invariant. The same equation for full fields, \(U_{ij}\psi_j=\psi_i\), also requires equal amplitudes along each edge, by unitarity. Parallel-section compatibility is optional for the measurement functional.
Theorem 464 (Path-independent reference transport on the pierced graph)
Let \(G_0=(V,E)\) be a finite connected component of the measurement graph, with unitary links as above. Fix a root \(r\), a spanning tree, and its forward transports \(B_i=T_{r\to i}^{\mathrm{tree}}\), with \(B_r=I\). The following conditions are equivalent:
Transport between any two vertices is independent of the chosen path.
Every closed path has identity transport.
For each of the \(|E|-|V|+1\) edges outside the tree, in either chosen orientation, \(B_i^{-1}U_{ij}B_j=I\).
Every link has the form \(U_{ij}=B_iB_j^{-1}\).
Under these conditions the reference-transported fields and their sum are path independent for every field assignment, and their norm statistics are gauge invariant. No compatibility between the geometric phases and the connection is needed.
Proof
If all paths with the same endpoints agree, a loop agrees with the constant path, proving condition 2. For an edge carrying \(U_{ij}:V_j\to V_i\), start at \(r\), follow the tree to \(j\), traverse that edge to \(i\), and return along the tree to \(r\). Its transport is \(B_i^{-1}U_{ij}B_j\). Thus condition 2 implies condition 3.
For a tree edge, the defining tree paths already give \(U_{ij}=B_iB_j^{-1}\): one endpoint extends the path to the other, and reverse traversal uses the inverse. For every edge outside the tree, the same identity follows by multiplying condition 3 on the left by \(B_i\) and on the right by \(B_j^{-1}\). Hence condition 3 implies condition 4.
Finally, on a path \(i_0\to\cdots\to i_m\), condition 4 gives
This depends only on the endpoints and proves condition 1. The non-tree edges are exactly \(|E|-(|V|-1)\) in number. Path independence of each transport gives path independence of the sum; gauge invariance is Lemma 325.
Corollary 142 (Reference changes and field-specific path independence)
For any connection, changing reference from \(r\) to \(r'\) by appending the same fixed path \(\delta:r\to r'\) to every transport gives
For a particular vector \(v\in V_i\), transport from \(i\) is independent of paths precisely when every loop based at \(i\) fixes \(v\). This is weaker than identity holonomy on the whole fiber. For example,
fixes \((1,0,0)^T\) while being nonidentity when \(\alpha\notin2\pi\mathbb Z\).
Proof
Appending a common path multiplies every summand by the same unitary \(T_\delta\), proving the first assertion. For two paths \(\gamma,\gamma'\) from \(i\) to the same endpoint,
The matrix on the right is the transport around the loop obtained by following \(\gamma\) and returning along \(\gamma'\). Conversely, compare a fixed outgoing path with that path preceded by any loop at \(i\). This proves necessity for every loop as well as sufficiency. The displayed matrix verifies the example directly.
Lemma 326 (When triangle tests establish global flatness)
Suppose the measurement graph carries specified triangular faces, and assume every closed edge path can be reduced to the constant path by a finite sequence of backtrack cancellations and replacements of two sides of a face by its third side, or the reverse replacements. If every face has identity holonomy, all loop transports are identity.
Without this reduction property, checking the triangles alone is insufficient. A square cycle with no triangular faces has no triangle tests, but assigning \(U(1)\) phase \(e^{i\alpha}\ne1\) to one forward edge and phase one to the other three gives nonidentity loop transport.
Proof
A backtrack contributes \(U_{ij}U_{ji}=I\). Identity transport around a triangle gives \(U_{cb}U_{ba}=U_{ca}\) for its two-side path \(a\to b\to c\). Thus each permitted replacement preserves the full ordered path product. After the finite reduction, the product is that of the constant path, namely \(I\). The square example has forward loop product \(e^{i\alpha}\), which proves the second assertion.
For nonabelian matrices, even the order of loop comparisons matters. A collection of triangles that spans cycles only after treating their edges as commuting symbols does not supply the reduction used in the proof. The spanning-tree test avoids that issue: it checks an explicit generating set of based loops in the graph. Taking an induced subgraph of a spatial tessellation can create holes, so the selected graph needs its own test.
Definition 825 (Gauge-invariant measurement through holonomy)
For a specified oriented closed path \(\mathcal C\) based at \(i\), define its Wilson statistic
This is Definition 701 with the path orientation and comparison-link convention made explicit. Reversing a path conjugates its Wilson statistic: \(W(\mathcal C^{-1})=\overline{W(\mathcal C)}\). Real statistics may use its real part or modulus. Loop statistics and the norms of reference-transported fields may be combined using a declared scalar function to define a gauge-invariant probe observable.
Proof
The path transformation proved in Lemma 325 gives \(T_{\mathcal C}\mapsto G_iT_{\mathcal C}G_i^{-1}\). Cyclicity of the trace proves invariance. Unitarity gives \(T_{\mathcal C^{-1}}=T_{\mathcal C}^\dagger\), proving the conjugation identity. A scalar function of invariant arguments remains invariant.
7. Recording a reproducible measurement#
The resulting observable is reproducible when its geometric and transport choices travel with the output. Two signals may disagree because one uses a different clipping window, a different distance, or a different path through a curved connection. Recording those choices makes the difference interpretable.
A neighbor-radius heuristic can help select candidate cells, but an exact piercing result requires a cell intersection test. Likewise, a graph distance is a well-defined graph observable even before a geodesic approximation theorem applies. Naming the quantity computed is enough to keep these measurements usable without assigning them an unsupported interpretation.
Algorithm 13 (Measurement specification)
For each requested measurement time:
Record the frame, included sites, observation window, coordinate chart, metric, probe parameters, and cell reconstruction. Use the corresponding cell-intersection rule; polyhedral Euclidean cells admit Lemma 323.
Select bottom or top associations, retaining slot identifiers and event policies. Record the representative rule, including periodic lifts and any interpolation across the measurement interval.
Record the phase formula and its scale. Distinguish ambient Euclidean distance, intrinsic distance, endpoint displacement, and graph access distance. For a claimed intrinsic approximation, retain its attachment and distance error bounds.
Record the amplitude measure or quadrature, its units, and any normalization. Identify the sampling law used for probabilistic weighting or inverse-density estimates.
Specify the representation and every link used in transport. Choose references and paths separately for connected components. A deterministic spanning forest supplies such paths. If path independence is claimed, check the non-tree loop products in Theorem 464.
Return the selected raw signal or transported statistic, with the amplitude-versus-intensity convention and any loop observables. Empty selections give zero sums; missing associations and inaccessible required paths receive an explicit undefined status.
This specifies an observable of the recorded history and the supplied geometric reconstruction. It leaves the Fractal Gas transition rule unchanged.