Signed Currents and Structural Matching in Möbius Shells
Year: 2026
Type: article
Abstract: Signed Currents and Structural Matching in Möbius Shells studies an exact arithmetic matching problem associated with the Möbius and Mertens functions. Odd squarefree integers in a dyadic shell are grouped according to their largest prime factor. The resulting prime-window capacities define a bipartite Hasse packet graph, with shores determined by the parity of the number of prime factors in a core. Complete structural matching asks whether a maximum matching always saturates the smaller shore. In that case, the number of unmatched shell vertices is exactly the absolute Mertens imbalance. The principal result establishes complete structural matching at every integer endpoint greater than or equal to the ceiling of the exponential of 2,148,007,937. The proof combines uniform positivity of large-core logarithmic deletion currents with a simultaneous repair of the negative currents that may occur at smaller cores. Every active core whose natural logarithm is at least 524,288 has positive current at every local scale. For the remaining cores, an explicit one-sided estimate bounds the total negative contribution available to each repair. Prime-lifted four-cycle corrections then remove all negative edge amounts while accounting for shared capacities. Root correction and bipartite integrality produce the required matching. A separate theorem proves eventual positivity of the individual smaller-core currents at a larger explicit local onset. The analytic development includes an exact rankwise evaluation of a continuous cofactor majorant, harmonic renewal inequalities, truncated-prime and required-prime positivity estimates, and a signed comparison with a continuum profile. Absolute clock moments and spatial-derivative bounds control the contribution of small clock parameters. Exact arithmetic prefixes and an adaptive terminal cutoff extend large-core positivity to every local scale. The comparison retains the unit cofactor, excluded core primes, prime-entry discontinuities, moving boundaries, and actual terminal-prime windows. An explicit logarithmic-integral replacement isolates the signed arithmetic trace without discarding the underlying cofactor restrictions. The paper also establishes Hall inequalities for substantial families of cuts. Every singleton is safe. Combining explicit prime allocations with the finite matching theorem in the preceding manuscript excludes all prime-source cuts on a smaller shore and every cut containing at most seven cores at every endpoint. An even-shore obstruction must contain at least forty-one cores, while an odd-shore obstruction without prime cores must contain at least four hundred and five. Odd cuts with arbitrarily many prime cores and at most one higher-rank core are also covered. Rankwise coprime-packing criteria, based on classical shadow theorems, apply to families of unbounded size. Further simultaneous allocations combine smooth higher cores with specified rough-core families under common capacity budgets. An exact signed-boundary identity expresses Hall slack in terms of edge currents and the root allowance. Signed-Laplacian and effective-resistance criteria provide sufficient matching certificates beyond pointwise current positivity. A first-event formulation shows that any initial failure of saturation must occur at a single adverse arithmetic event meeting an old tight Hall block. The resulting deficiency is exactly one. The event is safe precisely when its residual closure reaches unused capacity on the larger shore. An exact dual criterion characterizes feasibility within a prescribed family of source-dependent prime-diamond repairs, while slack augmentation distinguishes ordinary matching from the stronger requirement of preserving every current marginal. Explicit parent-capacity estimates sharpen this event analysis. Whenever the natural logarithm of a parent's local scale is at least 1,800, that parent absorbs the demands of any eight active children with a strict quantitative reserve. A universal nine-child assertion is false for actual arithmetic prime-window capacities. Consequently, each lower parent of the minimum-rank layer of a root-free tight block either lies below the stated scale or supports at least nine selected children. Restricted-prime examples and shifted complexes with assigned capacities delimit the conclusions available from local sharing and elementary Hall inequalities alone. These examples do not refute the full arithmetic matching conjecture. The all-endpoint conjecture remains open. The finite coverage through thirty-two billion is used from the preceding manuscript and is not recomputed here. The present eventual theorem leaves the interval between that finite range and its explicit exponential onset unresolved. Completing the structural matching theorem would remove the additional graph obstruction exactly; it would leave the separate Riemann-hypothesis-scale bound for the Mertens function unresolved. The accompanying materials comprise the paper, its LaTeX source, exact arithmetic verification programs, finite numerical certificates, and documentation of the supporting computations. Numerical certificates establish their stated finite inequalities and enumerations; the analytic arguments are presented in the manuscript.
Author: Fathi, Kevin
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213753780
FWCI:
Cites:
Cited by:
Related to: