A Position-Weighted Asymmetric Embedding for DNA Sequence Representation: Mathematical Characterization and a BRCA1 Case Study

Compact numerical representations of DNA sequences can reduce the dimensionality of genomic data, but aggregation may result in the loss of positional information. This study investigates a position-weighted representation in which each nucleotide at position k contributes to one of four nucleotide channels according to a strictly increasing positional weighting function. The resulting sequence representation is a fixed four-dimensional vector, independently of sequence length. For a single nucleotide substitution at position j relative to a fixed reference, one nucleotide channel decreases and another increases by the same position-dependent magnitude, producing a zero-sum and antisymmetric change vector whose magnitude identifies the substitution position in exact arithmetic. A finite-precision analysis was additionally developed to derive a conservative sufficient precision condition for discrete position recovery. The approach was evaluated using the complete 126,033-nucleotide BRCA1 reference sequence. A deterministic single-nucleotide substitution was introduced independently at every position, and all 126,033 tested substitutions were correctly recovered using double-precision arithmetic under the evaluated pairwise-summation procedure, despite the conservative theoretical bound requiring 59 significand bits. The maximum continuous position error was 0.2227 positions. Computational benchmarking against one-hot encoding (OHE) and k-mer count representations showed that the proposed representation reduced a 5000-nucleotide sequence to four floating-point values (32 bytes), although its direct implementation required greater encoding time. In a complementary controlled Gradient Boosting experiment, the sequence-length-normalized weighted representation, padded OHE, and normalized 1-mer frequencies all produced classification performance close to chance after controlling for genomic window and sequence length, while the four-dimensional weighted representation substantially reduced memory requirements and downstream training time relative to padded OHE. These results characterize the proposed method as a compact, position-dependent DNA representation with analytically interpretable behavior for single nucleotide substitutions relative to a fixed reference. The findings distinguish positional identifiability and computational compactness from pathogenicity prediction and do not imply global sequence injectivity or diagnostic capability.