VPSCF Paper 2: Deterministic Full Clifford Closure, Coefficient Spaces, and Grade Projections for Variable-Probability Signature Clifford Fields

This paper develops the deterministic carrier, closure, and coefficient-semantics layer
underlying Variable-Probability Signature Clifford Fields. The variable-probability structure
was introduced in VPSCF1 through signature laws, local kernels, and averaged multiplication
tensors; the present paper does not introduce a new stochastic dynamics. Instead, it proves
the deterministic carrier, closure, and coefficient-semantics firewall that every pathwise or
averaged signature law must respect before probability-dependent signed forms, projected
dynamics, or analytic field equations can be stated without ambiguity. VPSCF1 [1] fixed a
common real blade carrier
A=spanR{1,i,j,k,ij,ik,jk,ijk}
and placed all signature dependence in frozen multiplication tensors
mσ : A⊗A→A, σ=(εi,εj,εk)∈{±1}3.
The present paper studies what VPSCF1 deliberately left open: full Clifford closure generated
by the scalar-vector sector, coefficient-valued products in spaces of the form V ⊗ A, and
grade projections back to truncated sectors. The first structural firewall is not a new
Clifford-generation fact, but its consequence for VPSCF semantics: the scalar-vector sector
A≤1 =spanR{1,i,j,k}
is not a closed algebraic carrier: repeated homogeneous grade-one generator products collapse
to scalar signature signs, whereas exact frozen products of distinct homogeneous grade-one
generators produce ordered bivector labels. The second point is that V ⊗A is only a coefficient
carrier until one specifies coefficient semantics. VPSCF2 isolates three baseline coefficient
regimes—pure metric or topological data, associative coefficient multiplication, and free
tensor-hierarchy bookkeeping—without claiming that they exhaust all possible coefficient
semantics. The third point is that projected products
U ⋆σ,≤r W =Π≤r(U ·σ W)
create closed effective truncated operations only by discarding higher grades, and those
discarded grades generate associator defects. Thus exact multiplication remains the frozen
full-grade product in A, while projected and averaged products are effective operations.