The Unequal Multiplicities

All-Fields programme paper (All-Fields Run, Batch 19 (AF)), November 2026. Written November 2026; publicly deposited 9 November 2026. The compositions of the two bounded readings at unequal multiplicity — the symmetric reading taken once, twice or three times against the one-sided reading taken a different number of times, six compositions in all — are graded at their own fronts, at the two grains the readings' ladders share. No unequal composition's kernel has a front-exact zero: the smallest front share among the twelve rows graded is 5.184×10⁻¹, eleven orders and more above the line at which a zero would be declared, and the largest is 9.859×10⁻¹. The shares are set first by the assignment of the multiplicities: at each grain the six order strictly by the one-sided reading's fraction of the total multiplicity, from 5.184×10⁻¹ where the symmetric reading carries three of the four factors to 9.859×10⁻¹ where the one-sided reading carries three of the four, and at fixed total depth four the equal fourfold's share stands strictly between its two unequal neighbours' at both grains. With this grading the composition family stands complete — the readings singly, their composition, its powers at equal multiplicity, and

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-09
DOI
https://doi.org/10.5281/zenodo.24086673
Primary Topic
Advanced Mathematical Theories and Applications
Type
preprint
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preprint

The Unequal Multiplicities

William Alexander Aldred
Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Theories and Applications
preprint

The Unequal Multiplicities

William Alexander Aldred
preprint en

Abstract

All-Fields programme paper (All-Fields Run, Batch 19 (AF)), November 2026. Written November 2026; publicly deposited 9 November 2026. The compositions of the two bounded readings at unequal multiplicity — the symmetric reading taken once, twice or three times against the one-sided reading taken a different number of times, six compositions in all — are graded at their own fronts, at the two grains the readings' ladders share. No unequal composition's kernel has a front-exact zero: the smallest front share among the twelve rows graded is 5.184×10⁻¹, eleven orders and more above the line at which a zero would be declared, and the largest is 9.859×10⁻¹. The shares are set first by the assignment of the multiplicities: at each grain the six order strictly by the one-sided reading's fraction of the total multiplicity, from 5.184×10⁻¹ where the symmetric reading carries three of the four factors to 9.859×10⁻¹ where the one-sided reading carries three of the four, and at fixed total depth four the equal fourfold's share stands strictly between its two unequal neighbours' at both grains. With this grading the composition family stands complete — the readings singly, their composition, its powers at equal multiplicity, and

Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Theories and Applications
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