Higher-Order Collapse in Simplicial Complexes Is the Release of Stored Material Memory

Authors: Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Luo’s branch-consistent robustness formalism shows that a simplicial complex can lose a higher-order functional channel while its graph skeleton remains unchanged. Pudding Theory reads this phenomenon as Material Memory. The filled triangles of the complex are not decorative higher-order additions to a graph. They are the storage sites of repeated co-activation. Their coface constraints preserve a trace in the Hodge 1-Laplacian, and the first nonharmonic branch is the measurable carrier of that trace. Branch switching is therefore not a minor spectral inconvenience. It is a loss of identity in the memory-bearing channel. The source paper defines the correct observable by keeping the same branch fixed. Pudding Theory explains why that definition is physically forced: robustness belongs to the remembered channel, not to the smallest label available after damage. If the branch-conditioned sensitivity mass were measured to be uniformly distributed over triangles at fixed coface energy, this Postulate would be falsified.

Source Synopsis

Kaiming Luo studies robustness in higher-order networks represented as simplicial complexes. The central object is the Hodge \(1\)-Laplacian, which acts on edge-space variables and separates exact, harmonic, and coexact components of flow. In this setting, triangles are not merely visual fillings of a graph. They enter the operator through the upper term \(B_2 W_2 B_2^\top\), thereby constraining edge-space dynamics.

The paper argues that a common robustness observable, the instantaneous smallest positive eigenvalue of the Hodge \(1\)-Laplacian, is generically ill-defined under triangle deletion. As triangles are removed, the harmonic subspace can grow. When this occurs, eigenvalue branches may switch. The smallest positive eigenvalue after a deletion step may no longer represent the same nonharmonic mode that was being monitored before the step. The observable then changes physical identity while retaining a similar numerical label.

Luo resolves the problem by fixing the first nonharmonic branch of the intact complex and following that same branch throughout the damage process. This defines branch-consistent functional robustness. The normalized robustness \(R_{HO}(f)=\tilde{\mu}_1(f)/\tilde{\mu}_1(0)\) measures the decay of one chosen higher-order functional channel as a fraction \(f\) of triangles is removed.

The deletion rule called Mode Sensitivity follows from first-order perturbation theory. For a triangle \(\tau\), its sensitivity is the negative derivative of the tracked branch under removal of that triangle. Thus the ranking is not an external attack heuristic. It is the local response field of the observable itself.

The numerical results show rapid collapse. In synthetic and empirical clique complexes, removing a small fraction of high-sensitivity triangles can drive the tracked nonharmonic branch to zero. Graph-level observables, including algebraic connectivity and giant connected component size, remain unchanged because the \(1\)-skeleton is preserved. The paper also shows that critical simplices often localize along sparse bridge-like carriers rather than dense triangle-rich cores. Finally, the tracked eigenvalue predicts edge-space relaxation timescales, linking the spectral observable to dynamics.

Postulate Lens

This paper applies Material Memory: matter retains the trace of repeated signals, and the trace biases future probability.

The fit is direct. In Luo’s simplicial complexes, triangles encode repeated higher-order co-occurrence. The coface term in the Hodge \(1\)-Laplacian is the mathematical storage of that trace. The tracked nonharmonic branch is the active memory channel through which prior higher-order organization biases future edge-space relaxation.

Pudding Theory Reading

Pudding Theory reads Luo’s result as a theory of stored higher-order memory. The graph skeleton is the visible contact structure. The filled triangles are the memory substrate. They record that three pairwise relations have not merely existed side by side, but have repeatedly acted as a coherent unit. This distinction is exactly what the graph cannot see.

In the source framing, triangle deletion alters coface structure while preserving the \(1\)-skeleton. In Pudding Theory terms, it removes localized memory traces while leaving the pairwise carrier intact. The failure of graph observables is therefore expected. They measure the persistence of contact, not the persistence of remembered joint constraint. A graph can remain connected and still lose the stored pattern that made its edge-space dynamics functional.

The branch-consistent observable becomes physically necessary under this reading. A memory trace is not identified by being the smallest currently positive eigenvalue. It is identified by continuity of the channel that carries it. When the instantaneous minimum switches branches, it does not discover robustness. It changes the subject. Luo’s fixed-branch construction restores identity to the memory-bearing mode.

Mode Sensitivity also changes meaning. It is not just a perturbative score. It is a map of where the stored trace is loaded in simplex space. A triangle with high \(MS_\tau\) is not important because it is dense, central, or graph-prominent. It is important because the tracked memory channel passes through it. This explains why bridge-like simplices can dominate collapse. A sparse intercommunity conduit can carry more functional memory than a dense local cluster if the tracked branch is concentrated there.

The source paper treats localization as a spectral property of the eigenvector. Pudding Theory treats it as the spatial distribution of retained signal. The coexact energy \(\|W_2^{1/2}B_2^\top u\|^2\) is the measurable memory mass of the mode. Its concentration predicts vulnerability. If that mass is held by a few triangles, the remembered higher-order constraint can be erased by a small deletion budget. If it is diffuse, the memory channel resists localized damage.

The free parameter that changes status is the critical deletion fraction \(f_c\). In a purely algorithmic reading, \(f_c\) is an empirical outcome of a ranking strategy. In the Pudding Theory reading, \(f_c\) is structurally constrained by the concentration profile of the stored trace. The collapse threshold is set by how unevenly the memory mass is distributed across triangles.

Falsifiable Observable

The distinguishing observable is the concentration of branch-conditioned sensitivity mass at fixed coface energy, measured by the ranked cumulative curve \(\sum_{\tau \in R(f)}MS_\tau/\sum_\tau MS_\tau\). Pudding Theory predicts that rapid collapse requires concentrated memory support, not merely high triangle density or graph centrality. If the branch-conditioned sensitivity mass were measured to be uniformly distributed over triangles at fixed coface energy, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The reading risks renaming spectral localization as memory. Luo already gives the mathematics. The Hodge \(1\)-Laplacian has coexact structure. Branch consistency prevents eigenvalue relabeling. No new ontology is required.

Geisel: The mathematics is not in dispute. The question is what the mathematics says the system is. Luo shows that the graph skeleton can remain intact while function disappears. That is not ordinary structural robustness. It is the erasure of a higher-order trace.

Tanaka: But the trace is just the contribution of \(B_2W_2B_2^\top\). Calling it Material Memory may add metaphor without prediction.

Geisel: It adds a constraint. The relevant carrier is the continuous branch, not the instantaneous smallest eigenvalue. That is exactly the rule one would impose if the observable were a stored channel. It also identifies the sensitivity field as memory loading, not as centrality.

Tanaka: The source already rejects centrality explanations.

Geisel: Yes, and Pudding Theory explains why they fail. Centrality counts prominence in the visible skeleton. The collapse follows retained higher-order signal. A sparse bridge can carry the memory because the mode is stored there. The theory commits to concentration of branch-conditioned sensitivity as the condition for fast collapse.

Discussion

The Pudding Theory reading buys a sharper interpretation of Luo’s definitional correction. Branch consistency is not only a cleaner spectral convention. It is the rule required when the system stores functional history in higher-order constraints. The monitored object must be the same remembered channel across damage.

This reading also explains the exact separation between structural and functional robustness. The graph records pairwise availability. The simplicial cofaces record joint history. A system can preserve all pairwise contacts and still lose the memory that organizes edge-space response. That is why graph-level robustness remains flat while the tracked mode collapses.

The limitation is clear. The reading applies where simplices encode repeated or functional co-occurrence, as in clique complexes derived from interaction data or designed higher-order systems. It would be weakened in complexes whose triangles are arbitrary bookkeeping devices with no relation to repeated signal history. The decisive tests are therefore not visual. They are branch-based, dynamical, and sensitivity-weighted. The conclusion would change if collapse were consistently governed by graph centrality or local triangle density after controlling for branch-conditioned sensitivity.

References

1. K. Luo, “Hidden Higher-Order Vulnerabilities in Simplicial Complexes Revealed by Branch-Consistent Functional Robustness,” arXiv:2603.24286, DOI: doi:10.48550/arxiv.2603.24286, 2026.

2. S. Ochs, “Pudding Theory: A Topological Theory of Information Fields,” QBist Lab Working Paper, 2026.

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