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Preprint . 2026
License: CC BY
Data sources: ZENODO
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
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VPSCF Paper 2: Deterministic Full Clifford Closure, Coefficient Spaces, and Grade Projections for Variable-Probability Signature Clifford Fields

Authors: Nguyen, Duy Hoang;

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

Abstract

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.

Keywords

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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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
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