# Modular and integer flow components for moduli four and five

Research draft, 4 September 2026. Candidate original result; not peer reviewed.

## Statement and provenance

Fix an orientation of a finite loopless multigraph G. An integer k-flow is an
integer circulation f with 0 < |f(e)| < k on every edge. A modular flow takes
nonzero values in Z/kZ and obeys conservation there. Two flows are adjacent if
their difference is supported on a simple cycle; parallel edges may form a
cycle of length two. Write I_k(G) and M_k(G) for these two reconfiguration
graphs. Empty graphs are regarded as connected in the vacuous sense of the
source problem.

**Theorem.** For k = 4 or k = 5, reduction modulo k induces a bijection
between the connected components of I_k(G) and M_k(G). In particular,

\[
 I_k(G)\text{ is connected}\quad\Longleftrightarrow\quad
 M_k(G)\text{ is connected}.
\]

This addresses the specified small-modulus cases of Problem 2.6 in
[Esperet–Hendrey–Lagoutte–Marseloo–Norin–Steiner,
*Nowhere-zero flow reconfiguration*, v4, 3 July 2026](https://arxiv.org/html/2512.17342v4#S2.SS2).
That version expressly states that k=4 is unknown. The equivalence for
arbitrary k is also Conjecture A in
[Cranston–Li–Su–Wang–Xu, *Reconfiguration of Nowhere-zero Flows*,
23 June 2026](https://arxiv.org/html/2606.24685v1#S1).
The theorem here does not assert that every graph has a 4-flow or 5-flow,
nor that either reconfiguration graph is always connected. The general
modulus conjecture remains outside its scope.

## Two elementary lifting facts

An oriented cycle has a signed unit vector c: its value is +1 on edges
traversed in their reference direction, -1 on oppositely traversed edges,
and zero elsewhere. This vector is an integer circulation. A cycle is
*conformal* to an integer circulation f when every nonzero c(e) has the
same sign as f(e).

**Cycle fact.** Every edge with nonzero integer flow lies on a conformal
simple cycle.

To see this, reverse the negative-flow edges and use positive magnitudes.
For an edge u→v, let R be the vertices reachable from v along positive-flow
arcs. If u is not in R, there is positive flow into R, while no positive
flow can leave R by its definition. Summing conservation over R is a
contradiction. Thus a shortest directed path from v to u, together with
u→v, gives the desired simple cycle. This works component by component
and permits parallel edges.

**Bounded lifting fact (including zeros).** Every modular circulation q,
with zeros allowed, has an integer circulation h congruent to q such that
|h(e)| < k for every edge.

First choose a spanning forest. Give each nonforest edge its residue as
an integer and solve conservation on forest edges by removing leaves.
Conservation modulo k shows, at each removal, that the assigned integer
is congruent to the required residue. Conservation at each final root
follows because the total divergence in its component is zero. This
produces an integer lift f, initially without a magnitude bound.

If |f(e)| ≥ k, choose a conformal simple cycle c through e and replace
f by f-kc. Congruence and conservation are preserved. Consider the
nonnegative integer potential

\[
 P(f)=\sum_e\max\{\lvert f(e)\rvert-k+1,0\}.
\]

On a changed edge with positive magnitude a, its new magnitude is |a-k|.
If a<k, both contributions to P are zero. If a≥k, its contribution
strictly decreases. Thus no contribution increases, and the chosen edge
strictly decreases P. Iteration terminates at the required bounded lift.
In such a lift, a zero residue has integer value exactly zero.

**Fiber fact.** Any two bounded nowhere-zero integer lifts f and g of
the same modular flow lie in the same component of I_k(G).

On each edge, g-f belongs to {-k,0,k}. Choose a conformal cycle c in g-f
and replace f by f+kc. Every changed edge immediately attains its final
value g(e); every other edge keeps its current value. Consequently all
intermediate flows stay nonzero and bounded. The number of edges on
which f differs from g decreases. Repetition gives a path from f to g.

## The midpoint argument

The following two lemmas hold for every integer k≥2.

**Unit move.** Suppose two nowhere-zero modular flows satisfy q'=q+c.
For any bounded integer lift f of q, f+c is a valid bounded nowhere-zero
integer lift of q'. Indeed, adding c(e) changes a coordinate by at most
one. The only possible violations of the interval (-k,k) are its
endpoints ±k, and both would have residue zero. An integer zero would
likewise contradict q'(e)≠0. Therefore this modular edge lifts directly.

**Double move.** Suppose q'=q+2c, with both q and q' nowhere zero.
Form the auxiliary modular circulation r=q+c, **allowing zero values**.
Choose a bounded integer lift h of r by the bounded lifting fact. Put

\[
 a=h-c,\qquad b=h+c.
\]

Both a and b are integer circulations; their residues are q and q',
respectively. Outside the cycle they equal h, whose residues there are
nonzero. On the cycle they differ from h by exactly one. Since |h(e)|<k,
each endpoint value lies in [-k,k]. It cannot equal -k, 0, or k because
its residue is nonzero. Hence a and b are valid integer k-flows, and
b-a=2c makes them adjacent. The auxiliary h is **not** asserted to be a
vertex of I_k(G); it serves only to construct this adjacent pair.

Given any prescribed integer lifts f of q and g of q', join f to a
within its fiber, take the edge a→b, and join b to g within its fiber.

## Proof of the theorem

If two modular flows differ on a simple cycle, conservation at its
vertices forces their difference to be a single constant a times its
signed unit vector c. For k=4 or 5, every nonzero residue a has a
representative in {1,-1,2,-2}. Reverse the cycle orientation when
necessary. Thus every modular edge is covered by the unit or double
move lemma. These lemmas and the fiber fact lift any modular path
between the residues of prescribed integer endpoints to an integer path.

Conversely, every integer cycle move reduces either to a modular cycle
move or to the same modular flow. Deleting repetitions projects an
integer path to a modular path. The bounded lifting fact gives
surjectivity on modular vertices. These facts prove the claimed
bijection on connected components, including the empty-state-space
case. ∎

## Formal and computational verification

`formal/FlowReconfiguration.lean` formalizes the argument over a finite
integer circulation lattice equipped with signed unit cycles and the
cycle fact above. That is an explicit structural hypothesis, not an
assumption about connectivity. The proof derives bounded lifting, fiber
connectivity, and the midpoint lemma from it. Modular states are
represented by their bounded integer lifts.

The graph-specific cycle fact and the spanning-forest construction
identifying all modular graph flows with these residue classes are
proved mathematically above. They are not yet instantiated from an
incidence-matrix definition inside Lean. Consequently this is a
kernel-checked abstract theorem and a self-contained graph proof, not
an end-to-end Lean formalization of the standard graph definitions.

`reconfigure.py` implements the graph constructions using exact integer
arithmetic, including loopless multigraphs with parallel edges.
Its path checker directly checks conservation, bounds, and connected
2-regular support of each step. Computational examples are sanity
checks, not substitutes for the proof.

The independent verifier enumerates all edge assignments on 12 small graphs
for each of k=4 and k=5. It checks 3,060 lifted modular moves, 4,092 pairs
of endpoints in common fibers, and all 12,276 returned path steps. It
also checks bounded lifting with zero residues and compares complete
component partitions using pairwise difference supports. The checked
classes include parallel edges, disconnected graphs, bridges, and empty
graphs. Separate exploratory enumeration found matching component counts
in 300 sampled graph/modulus cases. Reproduction instructions and exact
records accompany the note.

## Novelty check

Searches on 4 September 2026 covered the source paper's current v4,
the June follow-up, and queries for the cyclic/integer converse,
small moduli, lifts, and midpoint arguments. No prior proof of the
theorem above was found. This is evidence for a candidate new result,
not a certification of priority or a substitute for expert review.
No claim is made to resolve the arbitrary-k conjecture.