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