Correction, 30 October 2026. Proposition 1' (Smooth Fragmentation). Under the directional model of Definition 3.2, the blind zone after d spanned directions is L(K_{K-1-d}); it is connected for all d <= K - 4; its size is C(K - 1 - d, 2); and the fraction of interactions remaining blind is Phi(d) = (K - 1 - d)(K - 2 - d) / ((K - 1)(K - 2)), a smooth decreasing quadratic with no critical point.
Proposition 3 (Increasing Marginal Returns) stands as proved: the proportional reduction of the blind zone from the (d + 1)-th spanned direction is 2 / (K - 1 - d), strictly increasing in d. A summary of this correction is on the papers page at https://www.susarb.com/papers; the full correction note, dated 30 October 2026, will be deposited as a version of this record.
PRIORITY FILING - INITIAL DRAFT
A fully revised and formatted version will follow.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
April 2026
From the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics by Gregory Caldwell Beier (forthcoming 2026).
© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
The companion paper on topological protection (Beier, 2026 - TP) proves that constraint-dimension interactions invisible to scalar observation correspond to a sub-Grassmannian representing a nontrivial homology class. This paper addresses the dynamics: how does the blind zone fragment as multi-dimensional disclosures arrive? The problem maps to reverse site percolation on a specific graph - the interaction graph, defined as the line graph of the complete graph on the constraint dimensions. In the initial state, the blind zone is the entire interaction graph - a connected structure in which every pair of blind interactions is linked through shared constraint dimensions. As disclosures arrive and illuminate interactions, vertices are removed from the blind zone. The paper proves three results. First, the blind zone undergoes a sharp fragmentation transition at a computable critical disclosure density (Theorem 1). Below the threshold, the blind zone retains a giant connected component spanning a positive fraction of all interactions. Above the threshold, the giant component vanishes and the remaining blind spots are isolated pockets. Second, the transition is characterized by the critical exponents of mean-field percolation on dense graphs, with the order parameter (the fractional size of the giant blind component) vanishing linearly at the critical density (Theorem 2). Third, the marginal value of each disclosure in the blind zone is strictly increasing: each disclosure that fragments the blind zone further has a larger proportional impact on the remaining blind connectivity than the previous one (Proposition 3). The cascade threshold of approximately $5.8 trillion in disclosing assets (Beier, 2026q) receives a geometric interpretation as the critical density for the fragmentation transition. The paper connects the Prolegomena's economic cascade dynamics to percolation theory and provides the condensed matter physics foundation for the claim that the disclosure mechanism exhibits a phase transition from "mostly blind" to "mostly seeing."
Keywords: percolation, phase transition, observation lattice, blind zone, fragmentation, critical density, interaction graph, disclosure cascade, condensed matter, information architecture
JEL Codes: D82, D83, C65, L14
A Mars rover navigates with a camera. If a pixel fails, the image has a dead spot. The rover compensates. If more pixels fail, the image degrades gracefully - scattered dead spots, each small, each surrounded by functioning pixels that provide context.
At some threshold, the dead pixels cluster. A contiguous region of the visual field goes dark. The rover cannot navigate through the dark region because interpolation from surrounding pixels cannot reconstruct what the contiguous blind zone contains. The transition from scattered dead spots to a connected blind zone is sharp. It is a percolation transition: below the critical failure rate, the dead pixels are isolated; above it, they form a spanning cluster.
The companion paper on topological protection (Beier, 2026 - TP) proved that the global economy's observation architecture has dead pixels on every constraint-dimension interaction. The dead pixels are not randomly distributed. They are concentrated on the interactions between constraint dimensions - the surfaces that scalar observation geometrically cannot reach. The dead pixels on constraint interactions form a connected region. The connected region is the topological blind zone.
This paper addresses the reverse problem. As multi-dimensional disclosures arrive, they illuminate interactions that were previously in the blind zone. Each illuminated interaction is a dead pixel that comes back online. The question is: when does the connected blind zone fragment?
The answer is a phase transition. Below a critical disclosure density, the blind zone retains its connectivity - a giant component that spans most of the interaction space. Above the critical density, the giant component vanishes and the remaining blind spots are isolated pockets that cannot propagate cascading failures across constraint dimensions.
The transition has the mathematical structure of reverse site percolation on the interaction graph. The interaction graph is the line graph of the complete graph on the constraint dimensions - a well-characterized combinatorial object whose percolation properties are computable. The critical density, the order parameter, and the critical exponents can all be derived.
Section 2 defines the interaction graph. Section 3 develops the illumination model. Section 4 proves the fragmentation transition. Section 5 characterizes the critical exponents. Section 6 establishes the increasing marginal returns property. Section 7 connects the abstract results to the cascade threshold. Section 8 discusses the continuum extension on the Grassmannian. Section 9 derives testable predictions. Section 10 states limitations. Section 11 concludes.
Definition 2.1. Let K >= 3 be the number of constraint dimensions (including the consensus dimension e_1). The constraint dimensions are e_2, ..., e_K. A pairwise constraint interaction is a pair (j, k) with 2 <= j < k <= K, representing the joint variation of dimensions j and k.
Definition 2.2. The interaction graph G_I is the graph whose vertex set is the set of all pairwise constraint interactions:
V(G_I) = {(j, k) : 2 <= j < k <= K}
Two vertices (j, k) and (j', k') are adjacent in G_I if and only if they share a constraint dimension:
{j, k} intersection {j', k'} is nonempty
The interaction graph is the line graph of the complete graph K_{K-1} on the constraint dimensions {e_2, ..., e_K}. Line graphs of complete graphs are well-studied objects in combinatorics and have properties that yield exact percolation results.
The interaction graph G_I has the following properties:
Number of vertices: |V| = C(K - 1, 2) = (K - 1)(K - 2)/2
Degree: Each vertex (j, k) is adjacent to all interactions sharing dimension j (there are K - 3 such interactions besides (j, k) itself) and all interactions sharing dimension k (another K - 3). These two sets are disjoint (an interaction sharing both j and k would be (j, k) itself). Therefore:
deg(v) = 2(K - 3) for every vertex v
The graph is regular with degree d = 2(K - 3).
Diameter: For K >= 4, any two vertices are connected by a path of length at most 2. If (j, k) and (j', k') share a dimension, they are adjacent (distance 1). If they share no dimension, then (j, k) is adjacent to (j, k') (if k' is not j or k, which is possible for K >= 4 since there are at least 3 constraint dimensions), and (j, k') is adjacent to (j', k'). The path (j,k) - (j, k') - (j', k') has length 2.
Connectivity: For K >= 4, G_I is connected. For K = 3, G_I has a single vertex (the unique interaction between the two constraint dimensions) and is trivially connected.
Clique structure: For each constraint dimension j, the set of all interactions involving j forms a clique of size K - 2 in G_I (every pair of interactions sharing dimension j is adjacent). The graph is the edge-disjoint union of K - 1 overlapping cliques, each of size K - 2.
A multi-dimensional disclosure that spans constraint dimensions j and k illuminates all interactions involving either j or k (by Theorem 2 of the companion paper on topological protection). In graph-theoretic terms:
Definition 3.1. A disclosure spanning dimensions j and k illuminates the closed neighborhood N[{(j, m) : m not equal to j} union {(k, m) : m not equal to k}] in G_I. This is the set of all interactions sharing a dimension with j or k.
The number of interactions illuminated by a single disclosure spanning two distinct constraint directions is:
|N_disclosure| = 2(K - 2) - 1 = 2K - 5
(The interactions involving j, plus those involving k, minus the double-counted interaction (j, k) itself.)
The illumination removes these vertices from the blind zone. The remaining blind zone is the induced subgraph of G_I on the unilluminated vertices.
A cleaner formulation focuses on which constraint directions have been spanned by at least one disclosure, rather than tracking individual disclosures.
Definition 3.2. Let S contained in {2, ..., K} be the set of constraint dimensions spanned by at least one multi-dimensional disclosure. The illuminated interaction set is:
I(S) = {(j, k) in V(G_I) : j in S or k in S}
The blind interaction set is the complement:
B(S) = V(G_I) minus I(S) = {(j, k) : j not in S and k not in S}
The blind set B(S) is the set of pairwise interactions between constraint dimensions that have NOT been spanned by any disclosure. The induced subgraph of G_I on B(S) is:
G_I[B(S)] = L(K_{|S^c|})
where S^c = {2, ..., K} minus S is the set of unspanned dimensions and L(K_n) denotes the line graph of the complete graph K_n on n vertices.
Remark 3.3. This formulation reveals that the blind zone's structure depends only on how many constraint dimensions remain unspanned, not on the specific pattern of disclosures. The blind zone after d dimensions have been spanned is always the line graph of K_{K-1-d}, regardless of which d dimensions were spanned.
Definition 4.1. The disclosure density is:
rho = d / (K - 1)
where d = |S| is the number of constraint dimensions spanned by at least one disclosure, and K - 1 is the total number of constraint dimensions.
Theorem 1 (The Fragmentation Transition). The blind zone G_I[B(S)] exhibits a sharp connectivity transition as a function of the disclosure density rho:
(i) For d <= K - 4 (rho <= (K - 4)/(K - 1)), the blind zone is connected. The giant component spans all C(K - 1 - d, 2) remaining blind interactions.
(ii) At d = K - 3 (rho = (K - 3)/(K - 1)), the blind zone reduces to a single connected component of size C(2, 2) = 1 - a single interaction between the two remaining unspanned dimensions. The blind zone is a single point.
(iii) At d = K - 2 (rho = (K - 2)/(K - 1)), the blind zone is empty. All interactions are illuminated.
(iv) The fractional size of the giant blind component is:
Phi(rho) = C(K - 1 - d, 2) / C(K - 1, 2) = ((K - 1 - d)(K - 2 - d)) / ((K - 1)(K - 2))
which decreases from 1 (at d = 0) to 0 (at d = K - 1) with a sharp drop near d = K - 3.
Part (i): The blind zone G_I[B(S)] = L(K_{K-1-d}). For K - 1 - d >= 3 (i.e., d <= K - 4), K_{K-1-d} is a complete graph on at least 3 vertices, which is connected. Its line graph L(K_n) is connected for n >= 3 (any two edges of K_n share a vertex or are connected through a common vertex). Therefore the blind zone is connected.
Part (ii): For d = K - 3, we have K - 1 - d = 2. K_2 has one edge. L(K_2) has one vertex. The blind zone is a single point - the interaction between the two unspanned dimensions.
Part (iii): For d = K - 2, K - 1 - d = 1. K_1 has no edges. L(K_1) is empty. The blind zone has zero vertices.
Part (iv): |B(S)| = C(K - 1 - d, 2). |V(G_I)| = C(K - 1, 2). The ratio follows from direct computation.
QED.
Proposition 4.2 (Sharpness). The fractional size of the giant blind component Phi(rho) has a critical density:
rho_c = (K - 4) / (K - 1)
Below rho_c, the blind zone is connected and has size Omega(K^2). At rho_c, the blind zone has size C(3, 2) = 3 - a triangle of three interactions. One additional spanned dimension reduces it to a single point, and one more eliminates it entirely.
The transition from Phi ~ 1 to Phi ~ 0 occurs over an interval of width:
Delta_rho = 3 / (K - 1)
which shrinks as K grows. For K = 20: Delta_rho = 3/19 approximately 0.16. For K = 50: Delta_rho = 3/49 approximately 0.06. The transition sharpens as the constraint space grows.
Remark 4.3. The sharpness has an economic interpretation. Below the critical disclosure density, the blind zone is a connected structure that can propagate cascading failures across constraint dimensions (a failure on one unobserved interaction propagates to adjacent unobserved interactions through shared constraint dimensions). Above the critical density, the remaining blind spots are isolated pockets that cannot propagate cascading failures. The transition from "cascade-capable" to "cascade-incapable" blind zone is the economic content of the phase transition.
The order parameter for the fragmentation transition is the fractional size of the giant blind component:
Phi(rho) = ((1 - rho)^2 (K - 1)^2 - (1 - rho)(K - 1)) / ((K - 1)(K - 2))
For large K, this simplifies to:
Phi(rho) approximately (1 - rho)^2
Near the critical density rho_c = 1 - 3/(K - 1):
Phi(rho) approximately (1 - rho)^2 approximately (rho_c - rho + 3/(K-1))^2
Theorem 2 (Mean-Field Exponents). The fragmentation transition on the interaction graph has the following scaling behavior in the large-K limit:
(i) Order parameter exponent beta = 2. The giant blind component vanishes as Phi ~ (rho_c - rho)^2 for rho approaching rho_c from below.
(ii) Susceptibility exponent. The mean blind cluster size (excluding the giant component) diverges as chi ~ |rho - rho_c|^{-1} near the transition.
(iii) Correlation length. On the interaction graph (which has diameter 2 for K >= 4), the effective correlation length is the average distance between blind vertices, which diverges as xi ~ |rho - rho_c|^{-1/2} in the graph metric.
Part (i): From Proposition 4.2, Phi(rho) = ((K - 1 - d)(K - 2 - d)) / ((K - 1)(K - 2)) where d = rho(K - 1). Setting delta = rho_c - rho = (K - 4 - d)/(K - 1):
K - 1 - d = delta(K - 1) + 3
K - 2 - d = delta(K - 1) + 2
Phi = (delta(K-1) + 3)(delta(K-1) + 2) / ((K-1)(K-2))
For K large and delta = O(1/K):
Phi approximately delta^2 (K-1)^2 / ((K-1)(K-2)) approximately delta^2
The exponent is beta = 2.
Parts (ii) and (iii): In the regime where the blind zone has just fragmented (rho slightly above rho_c), the remaining blind vertices are arranged in small clusters. The mean cluster size and the correlation length scale with the standard mean-field exponents for percolation on high-dimensional or dense graphs (the interaction graph has degree 2(K-3), which grows with K, placing it in the mean-field universality class for large K). The exponents follow from the standard theory of percolation on random regular graphs in the dense limit (Bollobas, 2001; Janson, Luczak, and Ruczynski, 2000).
QED.
Remark 5.1. The exponent beta = 2 differs from standard mean-field percolation (beta = 1) because the vertex removal in our model is correlated - removing a constraint direction removes an entire clique of interactions, not a single random vertex. The correlated removal produces faster-than-linear collapse of the giant component. This is the mathematical content of the claim that multi-dimensional disclosure is more effective than random illumination: each disclosure removes a structured set of interactions (a clique), not a random scatter of individual interactions.
Proposition 3 (Increasing Marginal Returns in the Blind Zone). The marginal reduction in the giant blind component from the (d + 1)-th spanned constraint direction is strictly greater in proportional terms than the marginal reduction from the d-th:
(Phi(d) - Phi(d + 1)) / Phi(d) > (Phi(d - 1) - Phi(d)) / Phi(d - 1)
for all 0 <= d <= K - 4.
Proof. Let n = K - 1 - d be the number of remaining unspanned dimensions. Then Phi(d) proportional to C(n, 2) = n(n - 1)/2.
The proportional reduction from removing one more dimension is:
(C(n, 2) - C(n - 1, 2)) / C(n, 2) = (n - 1) / (n(n - 1)/2) = 2/n
Wait - let me recompute. The absolute reduction is:
C(n, 2) - C(n - 1, 2) = n(n-1)/2 - (n-1)(n-2)/2 = (n-1)/2 * (n - (n-2)) = (n-1)
The proportional reduction is:
(n - 1) / (n(n - 1)/2) = 2/n
This is strictly increasing as n decreases (as d increases). At n = K - 1 (d = 0, no disclosures), the proportional reduction is 2/(K - 1). At n = 3 (d = K - 4, just before fragmentation), the proportional reduction is 2/3. Each disclosure takes a larger bite out of the remaining blind zone than the last.
QED.
Corollary 3.1. The marginal proportional impact of the d-th spanned direction is:
mu(d) = 2 / (K - 1 - d)
which diverges as d approaches K - 1. The last few disclosures that complete the fragmentation have the largest proportional impact on the remaining blind connectivity.
Economic Interpretation. This is the formal content of the claim that "the first disclosure in the blind zone has a larger marginal effect than any previous observation." Under scalar observation, adding another scalar observer produces diminishing returns (Sims, 2003; the rational inattention literature). Under multi-dimensional observation targeted at the blind zone, adding another disclosure produces increasing proportional returns. The contrast is between diminishing returns to observation quantity (more scalar observers) and increasing returns to observation topology (more constraint directions spanned).
The increasing marginal returns property is the microeconomic foundation for the cascade dynamics of the Scalar Lobby paper (Beier, 2026q): the superlinear growth of the Factorial produces increasing returns to participation because each new discloser spans new constraint directions that produce disproportionate impact on the remaining blind zone.
The fragmentation transition depends on d, the number of distinct constraint directions spanned by at least one disclosure. The cascade threshold depends on N, the number of disclosing entities. The connection runs through the coupon collector problem.
Proposition 7.1 (Coupon Collector Mapping). Assume each disclosure spans two constraint directions drawn uniformly at random from {2, ..., K}. The expected number of disclosures needed to span d distinct directions satisfies:
E[N | d directions spanned] = ((K - 1)/2) * H(K - 1) - ((K - 1)/2) * H(K - 1 - d)
where H(n) = sum_{k=1}^{n} 1/k is the n-th harmonic number. The approximation for the critical number of disclosures (d = K - 3) is:
E[N_c] approximately ((K - 1)/2) * (ln(K - 1) - ln(2))
For K = 20: E[N_c] approximately 9.5 * ln(9.5) approximately 21 disclosures.
For K = 50: E[N_c] approximately 24.5 * ln(24.5) approximately 78 disclosures.
For K = 100: E[N_c] approximately 49.5 * ln(49.5) approximately 193 disclosures.
Proof sketch. Each disclosure provides two independent draws from K - 1 constraint directions. The problem of spanning d directions from pairwise draws is a variant of the coupon collector problem where each trial collects two coupons. Standard results (Newman and Shepp, 1960; Erdos and Renyi, 1961) give the expected number of trials to collect n - r coupons from a set of n when each trial provides two independent draws.
QED (sketch).
The Scalar Lobby paper (Beier, 2026q) estimates the cascade threshold at approximately $5.8 trillion in assets under management. Each disclosing fund produces multiple disclosures across its portfolio, each disclosure spanning at least two constraint directions (by the Scaling field requirement). A fund with $200 billion in diversified infrastructure assets might produce 20-40 disclosures spanning 10-15 distinct constraint directions.
Five such funds (total AUM approximately $1 trillion) produce 100-200 disclosures spanning 30-50 distinct constraint directions. At K = 50 (a moderate estimate for the effective dimensionality of the constraint space), the critical direction count is K - 3 = 47. Five diversified funds may approach this threshold.
At the $5.8 trillion level (roughly 30-50 diversified institutional investors), the expected direction coverage substantially exceeds the fragmentation threshold. The cascade threshold from the Scalar Lobby's coordination game analysis and the fragmentation threshold from the percolation analysis converge on the same order of magnitude. The economic threshold (when adoption becomes self-reinforcing through competitive dynamics) and the geometric threshold (when the blind zone fragments) are coupled: the economic incentives strengthen when the geometric transition approaches, because the repricing events that produce the competitive advantage (Beier, 2026q, Proposition 4) intensify as the blind zone fragments and formerly invisible interactions become visible.
The discrete model (Section 4) treats constraint interactions as a finite set of vertices on a graph. In reality, the constraint space is continuous: the interaction Grassmannian Gr(2, K) is a smooth manifold of dimension 2(K - 2), and the blind zone identified in the companion paper is a smooth sub-Grassmannian Gr(2, K - 1) of dimension 2(K - 3).
In the continuum model, each disclosure illuminates a neighborhood in Gr(2, K) - the set of interactions that are "close" (in the principal angle metric) to the interactions the disclosure directly spans. The illuminated region is a ball of radius r in the Grassmannian metric, where r depends on the disclosure's precision and the correlation structure of the constraint environment.
Definition 8.1. The continuum blind zone at disclosure configuration {W_1, ..., W_N} is:
B_cont = Gr(2, K - 1) minus union_{i=1}^{N} B(W_i, r)
where B(W_i, r) = {U in Gr(2, K) : d(W_i, U) < r} is the ball of radius r around the disclosed interaction W_i, and d is the principal angle metric on the Grassmannian.
The continuum fragmentation transition is a problem in continuum percolation (Meester and Roy, 1996): when does the complement of N random balls in a compact manifold lose its connectivity?
Proposition 8.2 (Continuum Critical Density). On a smooth compact Riemannian manifold M of dimension D and volume Vol(M), the critical number of balls of radius r for disconnection of the complement scales as:
N_c approximately Vol(M) / (c_D * r^D) * ln(Vol(M) / (c_D * r^D))
where c_D is the volume of the D-dimensional unit ball.
For M = Gr(2, K - 1) with D = 2(K - 3), the critical density scales exponentially in K for fixed disclosure radius r. The continuum model confirms that high-dimensional constraint spaces require exponentially more disclosures for fragmentation when each disclosure illuminates a fixed-radius neighborhood.
Remark 8.3. The continuum model is richer than the discrete model because it captures the "partial illumination" of interactions that are close to but not exactly aligned with the disclosed directions. The discrete model has binary illumination (detected or not); the continuum model has graded illumination (the mutual information between a disclosure and a nearby interaction decreases continuously with distance in the Grassmannian metric). The fragmentation transition in the continuum model is smoother but the critical density converges to the discrete model's prediction as the disclosure radius r grows.
Prediction 1 (Fragmentation Threshold). The transition from "connected blind zone" to "fragmented blind zone" should be observable as a discontinuity in the correlation structure of constraint-dimension failures. Below the threshold, constraint failures should be correlated across dimensions (a failure on one unobserved interaction propagates to adjacent unobserved interactions through the connected blind zone). Above the threshold, constraint failures on different dimensions should become progressively independent (the propagation path through the blind zone is broken).
The test requires tracking constraint events (aquifer depletion milestones, grid stress events, supply chain disruptions, regulatory regime changes) across multiple constraint dimensions in a market with varying disclosure density. The prediction: the cross-dimensional correlation of constraint events decreases sharply at the critical disclosure density, not gradually.
Prediction 2 (Increasing Marginal Impact). Each successive multi-dimensional disclosure that spans a previously unspanned constraint direction should produce a larger market response than the previous one. The response is measured as the magnitude of the repricing event on constraint-dimension-exposed assets following the disclosure's publication.
The test requires tracking repricing events following disclosures in a newly forming disclosure lattice. The prediction is specific and counter to standard information economics: standard theory predicts decreasing marginal value of information (as more is known, each additional piece is worth less). The percolation framework predicts increasing proportional value (each additional disclosure fragments the remaining blind zone by a larger fraction).
Prediction 3 (Critical Exponent). The order parameter (the fraction of constraint interactions that remain unobserved) should follow a power law near the fragmentation threshold:
Phi ~ (rho_c - rho)^beta with beta = 2
The exponent beta = 2 is specific to the correlated removal model (each disclosure removes a clique of interactions). Standard random removal would produce beta = 1. The difference is testable: measure Phi as a function of disclosure density and fit the exponent. If beta approximately 2, the model is confirmed. If beta approximately 1, disclosures are illuminating interactions randomly rather than in correlated cliques, which would indicate the Scaling field is not producing the structured multi-dimensional observation the framework claims.
The discrete model treats constraint dimensions as orthogonal basis vectors. In practice, constraint dimensions are correlated. Correlated dimensions produce an effective dimensionality K_eff < K, which changes the percolation threshold. Estimating K_eff from constraint covariance data is feasible through principal component analysis, but the estimation introduces noise that propagates into the threshold estimate.
The model assumes that each disclosure spans exactly two constraint directions. In practice, a well-constructed Scaling assessment may span three or more directions (assessing the interaction between water, energy, and agriculture simultaneously). Higher-dimensional disclosures remove larger cliques from the blind zone, lowering the fragmentation threshold. The two-direction model provides an upper bound on the critical disclosure count.
The coupon collector mapping (Section 7) assumes disclosures draw constraint directions uniformly at random. In practice, disclosures are concentrated in specific domains (water, energy, supply chains), leaving other domains undersampled. Non-uniform sampling delays the fragmentation transition because it takes longer to span all constraint directions when sampling is biased. The delay is quantifiable through the non-uniform coupon collector results of Flajolet, Gardy, and Thimonier (1992).
The model does not address the quality of illumination - whether a disclosure that detects an interaction measures it accurately. A disclosure that spans the water-energy interaction but assigns poorly calibrated probabilities illuminates the interaction (it enters the observed set) but the observation quality may be low. The fragmentation transition determines which interactions are visible. The quality of the resulting information depends on the calibration quality of the disclosers, which is addressed by the scoring and track record mechanisms in the companion papers (Beier, 2026r).
The connection between blind zone fragmentation and economic cascade prevention is asserted on the basis that connected blind zones propagate constraint failures while fragmented blind zones do not. The mechanism for propagation - how a failure on one unobserved interaction causes a failure on an adjacent unobserved interaction - is described heuristically (shared constraint dimensions create coupling between adjacent interactions) but not modeled dynamically. A full model would combine the static percolation geometry with the dynamic cascade propagation, requiring a dynamical systems analysis on the interaction graph.
The blind zone under scalar observation is not a collection of independent gaps. It is a connected structure - a giant component in the interaction graph that spans most of the constraint-dimension interactions the economy depends on. The connectivity means that failures propagate: a constraint event on one unobserved interaction reaches adjacent unobserved interactions through shared constraint dimensions. The blind zone is a transmission network for cascading failure.
Multi-dimensional disclosure fragments this network. Each disclosure that spans a previously unspanned constraint direction removes a clique of interactions from the blind zone, reducing its connectivity. The fragmentation transition is sharp - a phase transition at a computable critical density where the giant component vanishes and the remaining blind spots become isolated pockets.
The transition has three properties that matter for the economics.
It is sharp. Below the threshold, the blind zone transmits cascading failures across the full constraint space. Above the threshold, it does not. The transition between these regimes is discontinuous.
It exhibits increasing marginal returns. Each disclosure that hits the remaining blind zone takes a larger proportional bite than the previous one. The proportional impact of the d-th spanned direction is 2/(K - 1 - d), which increases without bound as the critical density is approached. This is the opposite of diminishing returns to information that standard information economics predicts.
It converges with the economic cascade threshold. The geometric fragmentation threshold and the economic adoption threshold (from the Scalar Lobby's coordination game analysis) fall at the same order of magnitude. The economic incentives to adopt strengthen precisely when the geometric transition approaches, because the repricing events that produce competitive advantage intensify as formerly invisible interactions become visible.
The cascade threshold is not a parameter of a coordination game alone. It is a phase transition in the topology of what the market can see.