Finite Bonferroni–Liouville Transfer, Quantitative Truncation Control, and a Conditional Route to Binary Goldbach (PIHR-Goldbach-XLIII)
- 1Philipps University of Marburg
Abstract
We develop a self-contained version of a finite Bonferroni–Liouville strategy for the binary Goldbach problem. The starting point is a nonnegative, prime-only weight on complements of a fixed even integer. After sieving the complement beyond its cube-root threshold, the Liouville function separates primes from semiprimes exactly. An even Bonferroni truncation then gives an exact prime-pair identity with a nonnegative correction that must be subtracted. This correction cannot be discarded, and cancellation in the signed remainder alone does not control it.
We prove that the correction can nevertheless be made smaller than the natural prime-pair scale by an explicitly computable, growing truncation depth. The proof uses an elementary factorial-moment bound and does not assume cancellation of multiplicative functions. A second, shifted presieve is introduced in a fully defined comparison sequence. For its Liouville-weighted component we give an exact transfer formula, a complete Chinese remainder theorem expansion, and an unconditional truncation bound that remains valid when the first Bonferroni weight is retained. In this latter estimate, a divisor-square mean and an exponential moment replace an unavailable pointwise polylogarithmic bound on the sieve weight.
The preferred lower-bound argument combines the two Liouville signs before estimating them. Because the parity projector is nonnegative, an odd shifted Bonferroni truncation produces an additional nonnegative term, which may legitimately be omitted from a lower bound. The resulting finite inequality has only the explicitly controlled, unshifted correction with an adverse sign. We identify the remaining arithmetic inputs: evaluation of the truncated ordinary model, a coupled Liouville progression estimate, and a one-sided comparison with the actual prime sequence, all uniformly for each even integer. Existing distribution and correlation theorems are discussed without treating their hypotheses as automatically satisfied.
A complete conditional positivity theorem and an effective finite-range completion theorem are proved. The elementary truncation estimates narrow the obstruction but do not resolve the required prime correlation. The manuscript therefore supplies a complete candidate architecture, not a completed Goldbach proof.
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BibTeX
@misc{arneth2026finite,
title = {{Finite Bonferroni–Liouville Transfer, Quantitative Truncation Control, and a Conditional Route to Binary Goldbach (PIHR-Goldbach-XLIII)}},
author = {Borros Arneth},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23560521},
url = {https://zenodo.org/records/23560521},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite Bonferroni–Liouville Transfer, Quantitative Truncation Control, and a Conditional Route to Binary Goldbach (PIHR-Goldbach-XLIII)
AU - Borros Arneth
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23560521
UR - https://zenodo.org/records/23560521
AB - We develop a self-contained version of a finite Bonferroni–Liouville strategy for the binary Goldbach problem. The starting point is a nonnegative, prime-only weight on complements of a fixed even integer. After sieving the complement beyond its cube-root threshold, the Liouville function separates primes from semiprimes exactly. An even Bonferroni truncation then gives an exact prime-pair identity with a nonnegative correction that must be subtracted. This correction cannot be discarded, and cancellation in the signed remainder alone does not control it. We prove that the correction can nevertheless be made smaller than the natural prime-pair scale by an explicitly computable, growing truncation depth. The proof uses an elementary factorial-moment bound and does not assume cancellation of multiplicative functions. A second, shifted presieve is introduced in a fully defined comparison sequence. For its Liouville-weighted component we give an exact transfer formula, a complete Chinese remainder theorem expansion, and an unconditional truncation bound that remains valid when the first Bonferroni weight is retained. In this latter estimate, a divisor-square mean and an exponential moment replace an unavailable pointwise polylogarithmic bound on the sieve weight. The preferred lower-bound argument combines the two Liouville signs before estimating them. Because the parity projector is nonnegative, an odd shifted Bonferroni truncation produces an additional nonnegative term, which may legitimately be omitted from a lower bound. The resulting finite inequality has only the explicitly controlled, unshifted correction with an adverse sign. We identify the remaining arithmetic inputs: evaluation of the truncated ordinary model, a coupled Liouville progression estimate, and a one-sided comparison with the actual prime sequence, all uniformly for each even integer. Existing distribution and correlation theorems are discussed without treating their hypotheses as automatically satisfied. A complete conditional positivity theorem and an effective finite-range completion theorem are proved. The elementary truncation estimates narrow the obstruction but do not resolve the required prime correlation. The manuscript therefore supplies a complete candidate architecture, not a completed Goldbach proof.
ER -