Number theory · covering systems · finite probability

Odd Distinct Covering-System Conjecture

Collaboration beta

Must every distinct covering system include at least one even modulus?

C distinct cover, m_{i} > 1 \Rightarrow \exists i : 2 ∣ m_{i}

Known results and sources

Must every distinct covering system include at least one even modulus?

Research problem

Exact mathematical statement

A finite covering system is a family of congruence classes covering every integer. The Odd Distinct Covering-System Conjecture asks whether every covering system with distinct moduli greater than one must contain an even modulus:

\{a_i\pmod{m_i}\}\text{ covers }\mathbb Z,\ m_i>1,\ m_i\ne m_j \quad\Longrightarrow\quad 2\mid m_i\text{ for some }i.

Equivalently, no cover should exist with all moduli odd, distinct, and greater than one.

Problem infographic

Problem at a glance

Four text-free zones show distinct nontrivial odd-branching residue trees, a product grid retaining full shallow geometry, one source-reported local depth-two witness, and two unresolved continuations: the universal six-prime minimax and arbitrary-rank propagation. — The corrected full-p depth-two route is open: the source gives an exact rational witness for one hard placement, but every placement still needs a universal certificate family, independently of the normalized arbitrary-rank theorem.

Current mathematical picture

Where work on Odd Distinct Covering-System Conjecture stands

Partially resolved

The cumulative v4 source reports an internal omega(L)>=6 lower bound, exact six-prime restrictions, a corrected full-p finite interface and local certificate, while leaving both the universal finite minimax and the arbitrary-rank continuation open.

Leading routeExact root-recursive full-p H=2 route

Branch on root residues, apply live-root and tight-cover cuts, solve exact local profile oracles, prune deferred boxes, and export rational certificates for the universal finite minimax theorem.

Route status · Active route

Useful failureDepth-two scalar boundary functional

An exact integer dual certificate gives Φ₂ at least 50,213, above the allowed 44,449. Increase the cutoff or enrich the profile state, keeping the exact dual failure as a hard no-revisit boundary.

Route status · Narrowed route

Main reductionCorrected full-p finite boundary interface

The source reports eight delete-or-absorb sufficient full-p depth-two boundary criteria, with exponent padding and correlated shallow measures retained; the universal minimax inequality remains open.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the universal full-p depth-two minimax inequality

Prove or exactly refute the universal full-p H=2 minimax inequality for all placements of the twenty mixed shallow depth types.

Task status · Ready to work on

Later mathematical updatev4 separates the corrected full-p and arbitrary-rank blockers

The cumulative v4 source reports stronger internal six-prime results and exact local interfaces while preserving two independent open closing theorems.

v4 source revision order; not occurrence time or public priority

Work mapped so far

Odd Distinct Covering-System Conjecture in numbers

2.1kretained lines of mathematical investigation1,145 in the current working snapshot

Argument development: 1,750 · 83%
Explored or eliminated routes: 87 · 4%
Computational analysis: 115 · 5%
Open obligations: 50 · 2%
Definitions and setup: 98 · 5%

13selected mapped statements4routes investigated6open questions6contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 28, 2026

Source recordedAug 30, 2026

Recommended next task

Prove the universal full-p depth-two minimax inequality

Prove or exactly refute the universal full-p H=2 minimax inequality for all placements of the twenty mixed shallow depth types.

Suggested move: Implement the root-residue master with exact local value tables, deferred-box pruning, and rational primal or dual exports.

What would count as progress

Cover every shallow placement without silently assuming a product measure or omitted exponent padding.
Produce a replayable exact certificate family or an exact countercertificate identifying the missing state.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

23 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeExact root-recursive full-p H=2 route

Branch on root residues, apply live-root and tight-cover cuts, solve exact local profile oracles, prune deferred boxes, and export rational certificates for the universal finite minimax theorem.

Route status · Active route

Active routeNormalized arbitrary-rank propagation

Develop a normalized or multi-component potential that satisfies the exact O(1/p) block budget or records an explicit compensating decrease and retains quantified terminal margin.

Route status · Active route

Explored alternatives

Other routes

2 recorded

Narrowed routeDepth-two scalar boundary functional

An exact integer dual certificate gives Φ₂ at least 50,213, above the allowed 44,449. Increase the cutoff or enrich the profile state, keeping the exact dual failure as a hard no-revisit boundary.

Route status · Narrowed route

Narrowed routeIndependent recursive desingularization

The source states that only a genuinely coupled desingularization can remain viable. Any construction attempt must couple replacements across blocks rather than recurse independently.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementCorrected full-p finite boundary interface

The source reports eight delete-or-absorb sufficient full-p depth-two boundary criteria, with exponent padding and correlated shallow measures retained; the universal minimax inequality remains open.