ND0158

GaussianSignedNorm(ap,an,bp,bn,N)

The actual sum (ap−an)²+(bp−bn)² of two witnessed signed-coordinate squares.

Conservative notation; not a theorem, primitive, or axiom.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Definition in prerequisite notation

∃ ge_real_square_lowerlayer. ∃ ge_imaginary_square_lowerlayer. SignedDifferenceSquare(ap,an,ge_real_square_lowerlayer) ∧ (SignedDifferenceSquare(bp,bn,ge_imaginary_square_lowerlayer) ∧ N = ge_real_square_lowerlayer + ge_imaginary_square_lowerlayer)

Only definitions earlier in this acyclic notation graph are used here.

Hygienic expanded first-order definition 
exists ge_real_square_lowerlayer ge_imaginary_square_lowerlayer. ((((((ap) * (ap))) + (((an) * (an)))) = ((ge_real_square_lowerlayer) + (((((ap) * (an))) + (((an) * (ap))))))) /\ ((((((bp) * (bp))) + (((bn) * (bn)))) = ((ge_imaginary_square_lowerlayer) + (((((bp) * (bn))) + (((bn) * (bp))))))) /\ ((N) = ge_real_square_lowerlayer + ge_imaginary_square_lowerlayer)))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

SignedDifferenceSquare(p,n,s)

Definitions depending on this notation

GaussianSignedDivisionRemainder(ap,an,bp,bn,cp,cn,dp,dn,qp,qn,up,un,rp,rn,sp,sn,U,V)GNorm(z,n)

Checked theorems using this definition

GI0012 · gaussian_signed_norm_existsGI0013 · gaussian_signed_norm_functionalGI001A · gaussian_signed_norm_productGI001B · gaussian_signed_norm_integer_transportGI001C · gaussian_signed_norm_conjugateGI001E · gaussian_signed_norm_nonzeroGI002F · gaussian_signed_norm_balanceGI0030 · gaussian_conjugate_product_is_normGI0032 · gaussian_adjoint_product_is_norm_scaleGI0033 · gaussian_residual_conjugate_identityGI003A · gaussian_signed_euclidean_division_existsGI004D · gaussian_norm_of_representationGI004E · gaussian_norm_for_representationGI004F · gaussian_norm_exists