Jump and Gradient Invariants in the Partition Graph

Abstract: We introduce edgewise jump invariants and gradient-type structures for the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts. Previous work on $G_n$ has focused mainly on vertex-level invariants such as degree, local simplex dimension, and support size. Here we study how such invariants change along edges. For an oriented edge $e=(λ,μ)$ and a vertex invariant $F$, we define the signed jump $Δ_e F=F(μ)-F(λ)$ and focus on the basic jump signature [ J(e)=(Δ_e d,Δ_eδ,Δ_eσ), ] where $d$ is degree, $δ$ is local simplex dimension, and $σ$ is support size. We prove that support jumps are universally bounded by $2$ and describe them in terms of local multiplicity data. We also develop a taxonomy of active, neutral, pure, and mixed transitions, relate nonzero jumps of integer-valued invariants to threshold-layer crossings, and discuss strict gradient orientations associated with real-valued vertex invariants. Finally, we formulate a reproducible protocol for a computational atlas of jump spectra, transition ranks, large-jump edges, and localization patterns. No large-scale computations are carried out here; the atlas is presented as a framework for subsequent work.

• The paper introduces edgewise jump signatures for degree, local simplex dimension, and support size, along with acyclic invariant-gradient orientations and cone-compatible path corridors.
• The paper proves that every support-size jump satisfies |Δσ|≤2, with sharpness already realized in G₈, and shows that support changes depend only on local transfer sizes and multiplicities.
• The paper frames degree and dimension jumps as neighbor-system and threshold-layer changes, while presenting a reproducible computational atlas for testing open questions about signature spectra and graph morphology.

Overview and motivation

This paper, by Fedor B. Lyudogovskiy (2605.28981), shifts attention in the partition-graph program from vertex-level to edgewise morphology. The partition graph $G_n$ has as vertices the integer partitions of $n$, with adjacency given by an elementary transfer: moving one unit from a donor part of size $p$ to a recipient part of size $q$ (with $q=0$ allowed, meaning creation of a new part), followed by reordering. Prior work in the same series developed vertex invariants—degree $d(\lambda)$, local simplex dimension $\delta(\lambda)$ in the clique complex $K_n=\mathrm{Cl}(G_n)$, and support size $\sigma(\lambda)=|\mathrm{supp}(\lambda)|$. The present paper asks how these quantities change along an edge rather than what values they take at a vertex.

The central object is the signed jump $\Delta_eF = F(\mu)-F(\lambda)$ for an oriented edge $n$0, and in particular the basic jump signature

$n$1

The paper's stated role is deliberately structural: it establishes a common edgewise language and isolates a small set of elementary structural facts, while deferring classification of realized signatures to future computation. No large-scale computations are carried out; the computational atlas is presented explicitly as a reproducible protocol, not as completed work.

Edgewise language and gradient orientations

Signed jumps are attached to oriented edges and change sign under orientation reversal ($n$2), whereas absolute signatures $n$3 are invariants of the unoriented edge. This signed/absolute distinction runs through the whole paper: signed data serve gradient-type questions, absolute data serve unoriented atlases.

For any real-valued invariant $n$4, the strict $n$5-gradient orientation sends each edge toward the endpoint with larger $n$6-value. The paper proves this orientation is acyclic (a directed cycle would force $n$7 to strictly increase around a loop), and derives a length bound: every strictly $n$8-increasing path has length at most $n$9, where $p$0 is the set of attained values. The author is careful to note that "gradient" here means only an orientation induced by a scalar invariant, not a vector field, and that there is no canonical gradient on $p$1—different invariants may induce opposed directions of drift.

Support jumps: the main exact result

The strongest theorem-type content concerns support size. Because an elementary transfer changes multiplicities only at the affected sizes $p$2 (size 0 omitted), the paper proves:

• Universal bound: for every edge, $p$3, so the support component of every signature lies in $p$4 independently of $p$5. The bound is sharp: in $p$6, the transfer $p$7 realizes $p$8.
• Local determination: writing $p$9 for the number of support sizes that disappear and $q$0 for those newly introduced, $q$1, so the support jump is completely determined by $q$2 together with the multiplicities at the affected sizes. Support activity is thus a strictly local phenomenon.

An immediate corollary constrains all realized signatures: no third component can exceed 2 in absolute value, which also serves as a consistency check in any computation.

Degree and dimension jumps: contrast with support

Degree and local-dimension jumps do not admit analogous universal bounds from the transfer argument alone. The degree jump decomposes as a neighbor-system imbalance:

$q$3

bounded above by the reorganization size $q$4. Equivalently, it equals new transfer outputs minus lost transfer outputs. Local-dimension jumps compare maximal clique structures through the two endpoints and record whether a transfer enters or exits regions of higher simplicial thickness.

A useful general principle connects jumps to layer structure: for any integer-valued invariant $q$5 with threshold layers $q$6, nonzero jumps are exactly threshold-layer crossings, and $q$7 counts precisely how many integer thresholds the edge crosses. Large jumps are therefore edges crossing many layers simultaneously.

Transition taxonomy and numerical versus structural neutrality

The active component set $q$8 yields a transition rank $q$9, classifying edges as fully neutral, pure, two-component mixed, or fully mixed, refined further by sign patterns (coherent versus opposed). The rank counts how many of the three threshold-layer systems an edge crosses.

A key caveat is established by example: in $q=0$0, the edge $q=0$1 has $q=0$2—both endpoints have degree 3, support 2, and local simplex dimension 2—yet their neighbor sets differ. Fully neutral edges therefore need not be structurally trivial, and the jump signature should be read as a numerical shadow of transition morphology, not a complete invariant. The paper is explicit about which claims are proven (the six structural facts listed above) and which remain computational or conjectural: realization of signatures for large $q=0$3, growth of transition-rank distributions, concentration of fully mixed transitions, correlation between large degree and support jumps, and asymptotic stabilization of the jump spectrum.

Corridors and positional drift

Pathwise behavior is formalized via cone-compatible transitions: for a vector invariant $q=0$4 and a sign region $q=0$5, a corridor is a path whose every edge satisfies $q=0$6. Weak simultaneous increase corresponds to $q=0$7. The author restricts the formal notion to path corridors, noting that naive subgraph-level corridors would be essentially vacuous since any subgraph's edges can be oriented by $q=0$8-value. Positional gradients based on axial distance $q=0$9, where $d(\lambda)$0 and $d(\lambda)$1, give a separate drift language that may disagree with invariant-gradient drift.

Computational atlas

The final technical section specifies a reproducible protocol: for fixed $d(\lambda)$2, compute signed and absolute spectra $d(\lambda)$3 and $d(\lambda)$4, symmetric signed histograms (symmetry follows from the orientation-reversal involution), transition-rank counts $d(\lambda)$5, eight-class active-set decompositions, joint signature counts sliced by the bounded support component, and localization statistics via $d(\lambda)$6-stack midpoints, axial distance, simplex-layer boundaries, and $d(\lambda)$7 or $d(\lambda)$8 total activity norms. A minimal worked instance is given: $d(\lambda)$9 has five edges with distribution $\delta(\lambda)$0, $\delta(\lambda)$1, $\delta(\lambda)$2, $\delta(\lambda)$3. Six guiding problems are posed, including sharpness of the support bound across $\delta(\lambda)$4 and growth rates of $\delta(\lambda)$5 and $\delta(\lambda)$6.

Limitations and open questions

The paper concedes its limits plainly. It proves only elementary structural facts; all statements about where large jumps occur, whether mixed transitions dominate asymptotically, whether jump spectra stabilize, and whether corridors persist across $\delta(\lambda)$7 are atlas-level observations awaiting proof. The degree and local-dimension components have no known universal bounds, and their growth with $\delta(\lambda)$8 is open. Whether edgewise statistics can reconstruct global morphology—or explain why $\delta(\lambda)$9 has comparatively simple homotopy type despite the graph's rich local structure—is posed as a question rather than answered.

Conclusion

The paper contributes a precise edgewise vocabulary for the partition graph: jump signatures, transition ranks, threshold-crossing equivalences, acyclic gradient orientations, and cone-compatible corridors. Its exact results—the universal support-jump bound with sharpness, the local multiplicity determination of support jumps, and the identification of jump magnitude with crossed thresholds—are modest but firm anchors. The bulk of the framework is deliberately prospective, providing a reproducible protocol whose empirical output remains to be generated and, where patterns recur, proved.