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arxiv: 2606.19832 · v2 · pith:IB6BFCAJnew · submitted 2026-06-18 · 💻 cs.DC · cs.IT· cs.NI· math.IT

Ratio-Independent Three-Cycle Decomposition with Optimal Ordered Local-Switch Cost in Six-Regular Non-Axis Eisenstein--Jacobi Networks

Bader Albader This is my paper

Pith reviewed 2026-06-30 10:53 UTC · model grok-4.3

classification 💻 cs.DC cs.ITcs.NImath.IT
keywords Eisenstein-Jacobi networksHamiltonian cycle decompositionlocal-switch costsix-regular networksratio-independent decompositionunit-parallelogram exchangesinterconnection networksedge-disjoint cycles
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The pith

Every six-regular simple non-axis EJ network decomposes into three edge-disjoint Hamiltonian cycles with ratio-independent optimal local-switch cost.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a decomposition of every six-regular simple non-axis Eisenstein-Jacobi network into three edge-disjoint Hamiltonian cycles. It relies on a canonical ordered local-switch model that performs exchanges of unit parallelograms to rearrange edges into the cycles. The resulting costs are independent of the network ratio for d greater than or equal to four, hit explicit lower bounds for all listed d, and come with an O(d)-seed certificate that expands in linear time. A reader would care because the construction supplies explicit, verifiable cycle factors for these lattice interconnection networks.

Core claim

The paper claims that every six-regular simple non-axis EJ network admits a decomposition into three edge-disjoint Hamiltonian cycles via a canonical ordered local-switch model using unit-parallelogram exchanges. The d=1 case requires no switches, d=2 has total cost four, and d=3 and d greater than or equal to four achieve the d-1 lower bound. An equal-coordinate alternating lift removes reduced-ratio dependence for d greater than or equal to four. A block-chain invariant, exhaustive interior-template lemma, and parity-specific successor permutations together certify that the unused complement is empty and that coverage is complete.

What carries the argument

The ordered local-switch model based on unit-parallelogram exchanges, together with the block-chain invariant that advances rank by one modulo 4d-6.

If this is right

  • The d=1 branch needs no switches.
  • The d=2 case has optimal total cost four.
  • For d=3 and d greater than or equal to four both modified factors attain the component-counting lower bound d-1.
  • Factor-local switches commute, so the final factors and total cost are independent of interleaving order.
  • The O(d) seed records expand to the full edge lists in O(N) time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The commuting property of the switches may allow the same technique to produce disjoint cycle covers in other regular quotient-lattice graphs.
  • The ratio-independent lift for d greater than or equal to four suggests the method could be adapted to families of networks whose parameters vary continuously.
  • The parity-specific successor permutations supply an explicit rule that might be turned into a distributed construction algorithm.
  • Because the certificate is deterministic and dictionary-free, it could serve as a template for automated verification of cycle decompositions in larger interconnection networks.

Load-bearing premise

The unit-parallelogram exchange is the correct canonical model for counting and minimizing ordered local-switch cost in these networks.

What would settle it

A single six-regular simple non-axis EJ network of any norm d in which the three cycles either leave an edge uncovered or require more than the stated switch cost under the unit-parallelogram model.

Figures

Figures reproduced from arXiv: 2606.19832 by Bader Albader.

Figure 1
Figure 1. Figure 1: Positioning. Prior work proves non-coprime EJ EDHC existence through rectangu [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: The three EJ directions in the {1, τ} coordinate system and the fundamental quotient rhombus. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The three EJ direction factors in the {1, τ} coordinate system. Lemma 1 (Direction-component labels). Let d = gcd(a, b). The components of the natural direction factors are labeled by H : y (mod d), (8) V : x (mod d), (9) D : x − y (mod d). (10) Each direction factor consists of d cycles, each of length r = N/d. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: The three geometrically valid switch parallelograms. In particular, the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A local two-edge rhombus switch. The switch preserves degree two and changes [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Quotient-level minimal-alignment skeleton. The first two direction factors collapse [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Quotient-level minimal-alignment skeleton. The first two direction factors collapse [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Anchor-lock mechanism. The two designated [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Anchor-lock mechanism. Before the lock, the first [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Why complement incidence must be fine-grained. Lift phases preserve the residue [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Connector-label worked example for d = 6. The released connectors realize consec￾utive pairs in the alternating word W6 = (0, 5, 1, 4, 2, 3), so the complement-incidence graph is a path rather than a closed subcycle [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Early closure and seam-phase correction. The reduced connector tree is unchanged, [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Illustrative EDHC decomposition for α = 4 + 4ρ, N = 48, and d = 4. Panels (a)–(c) show the three Hamiltonian cycles H1, H2, H3. Solid edges are internal edges and dashed edges are wraparound quotient edges. The compact table reports edge counts and direction counts for this example only; the theorem is proved algebraically, not by this figure. Example 1 (Full lift-phase computation for a non-multiplier fam… view at source ↗
read the original abstract

Six-regular simple Eisenstein--Jacobi (EJ) networks are degree-six quotient-lattice interconnection networks. This paper gives a ratio-independent decomposition of every six-regular simple non-axis EJ network into three edge-disjoint Hamiltonian cycles using a canonical ordered local-switch model based on unit-parallelogram exchanges. The admitted $d=1$ branch needs no switches; $d=2$ has optimal total cost four; and for $d=3$ and $d\ge4$ both modified factors attain the component-counting lower bound $d-1$. Factor-local switches commute, so chronological interleaving does not alter the final factors or cost within the model. Orbit normalization identifies the exact domain and excludes the unique normalized non-axis norm-three degeneration. For $d\ge4$, an equal-coordinate alternating lift removes reduced-ratio dependence from the fine diagonal coordinate. A block-chain invariant, exhaustive interior-template lemma, and parity-specific successor permutations certify the unused complement: rank advances by one modulo $4d-6$, and arc and connector bijections prove complete coverage. The certificate uses $O(d)$ seed records and expands to the full edge lists in $O(N)$ time. Deterministic symbolic and full-quotient audits, including a dictionary-free fine-incidence check for every $4\le d\le201$, are provided in the accompanying reproducibility package and are not proof premises.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims a ratio-independent decomposition of every six-regular simple non-axis Eisenstein-Jacobi network into three edge-disjoint Hamiltonian cycles via a canonical ordered local-switch model based on unit-parallelogram exchanges. For d=1 no switches are needed, d=2 has total cost four, and for d=3 and d≥4 both modified factors attain the component-counting lower bound d-1. The construction uses an equal-coordinate alternating lift for d≥4, with completeness certified by a block-chain invariant, exhaustive interior-template lemma, and parity-specific successor permutations (rank advances by one modulo 4d-6, with arc/connector bijections); the certificate expands in O(N) time from O(d) seeds. Deterministic audits up to d=201 are supplied in a reproducibility package but are stated not to be proof premises.

Significance. If the claimed decomposition and its optimality hold for all d, the result would supply an explicit, ratio-independent construction of three Hamiltonian cycles in these degree-6 quotient-lattice networks together with a local-switch cost that meets the component-counting lower bound. The reproducibility package containing exhaustive dictionary-free incidence checks for 4≤d≤201 constitutes a concrete strength that allows independent verification of the finite cases.

major comments (2)
  1. [Abstract] Abstract: the central claim that the block-chain invariant together with parity-specific successor permutations and arc/connector bijections certifies complete coverage for arbitrary d≥4 rests on an asserted mathematical argument whose inductive step or explicit derivation ruling out additional modular obstructions is not visible in the manuscript; only computational audits to d=201 are supplied.
  2. [Abstract] Abstract: the component-counting lower bound d-1 is invoked to establish optimality of the modified factors, yet the independence of this bound from the particular construction (i.e., that it is not tautological with the switch model) cannot be verified without the explicit equations defining the bound and the factors.
minor comments (1)
  1. [Abstract] The abstract refers to “orbit normalization” and “the unique normalized non-axis norm-three degeneration” without defining the normalization map or the precise exclusion criterion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address each major comment below with clarifications drawn directly from the paper's arguments and indicate where revisions will improve visibility of the derivations.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the block-chain invariant together with parity-specific successor permutations and arc/connector bijections certifies complete coverage for arbitrary d≥4 rests on an asserted mathematical argument whose inductive step or explicit derivation ruling out additional modular obstructions is not visible in the manuscript; only computational audits to d=201 are supplied.

    Authors: The block-chain invariant is defined in Section 4.2 as the preservation of connectivity under the equal-coordinate alternating lift for d≥4. Lemma 4.5 provides the exhaustive interior-template lemma that enumerates all admissible local configurations. Lemma 5.3 establishes the parity-specific successor permutations, proving that rank advances by one modulo 4d-6 with no further modular obstructions possible due to the bijections between arcs and connectors (Theorem 6.1). The inductive step is embedded in the successor permutation construction, which holds for arbitrary d by the modulo arithmetic and the O(d) seed expansion. We agree that an explicit inductive paragraph would make this derivation more immediately visible and will add it in the revision. revision: yes

  2. Referee: [Abstract] Abstract: the component-counting lower bound d-1 is invoked to establish optimality of the modified factors, yet the independence of this bound from the particular construction (i.e., that it is not tautological with the switch model) cannot be verified without the explicit equations defining the bound and the factors.

    Authors: The component-counting lower bound d-1 is independent of the local-switch model and follows from the structure of the EJ network: in a 6-regular graph with N vertices, each Hamiltonian cycle covers N/2 edges, but the initial 2-factorization into three matchings yields d connected components per factor (from the lattice quotient). Connecting these requires at least d-1 switches, as each switch reduces the component count by at most one. The explicit equations are: let C_d denote the number of components in the d-parameterized matching (C_d = d), then lower bound on switches = C_d - 1 = d-1. This counting holds prior to any specific switch construction and is verified by the orbit normalization excluding the norm-three case. We will insert these equations and the component-counting derivation into the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: construction and invariants are independent of inputs

full rationale

The abstract and description present an explicit construction of the three-cycle decomposition via unit-parallelogram local switches, equal-coordinate alternating lift, block-chain invariants, interior-template lemma, and parity-specific successor permutations, with arc/connector bijections for coverage. The component-counting lower bound d-1 is invoked as an external counting argument to which the construction attains optimality; no equation or step is shown reducing a claimed prediction or result back to a fitted parameter, self-defined quantity, or self-citation chain. Computational audits up to d=201 are explicitly stated as non-premise supplements. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

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