TITLE:
An Angular Reformulation of the Kronecker Delta through a Jeno-Type Structural Nullity—Gram-Schmidt Orthogonalization and a Sliding-Ladder Application
AUTHORS:
Oscar Mauricio Jeno Soto
KEYWORDS:
Kronecker Delta, Gram Matrix, Gram-Schmidt Orthogonalization, Sliding Ladder, Unilateral Contact, Orthogonality, Structural Nullity
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.8,
August
24,
2026
ABSTRACT: A geometric reformulation of the Kronecker delta is proposed for finite-dimensional real inner-product spaces equipped with normalized basis vectors. The construction separates index identity, represented by
δ
ij
, from off-diagonal angular coupling, represented by
(
1−
δ
ij
)cos
θ
ij
. The resulting quantity coincides with the Gram matrix for normalized bases and reduces exactly to the classical Kronecker delta for orthonormal bases. The label structural nullity distinguishes an off-diagonal pair whose metric coupling vanishes by orthogonality; it does not introduce a new arithmetic zero. Two applications are developed. First, Gram-Schmidt orthogonalization is interpreted as the systematic elimination of off-diagonal Gram couplings, quantified by the sign-independent functional
N(
B
)=
∑
i