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Volume 3 — Collapse Algebra and Tensor Operators

Algebraic Structures in Golden-Base Tensor Space​

This volume develops abstract algebra entirely within the φ-constrained framework. Groups, rings, and fields emerge from trace composition operations that respect the no-11 constraint, creating a new algebraic foundation.

Chapter Index​

Chapter 048: GroupCollapse​

Invertible Trace Structures under Collapse Composition

Constructs groups where elements are traces and the operation preserves φ-constraint.

Chapter 049: RingCollapse​

Dual Operation System for Trace Addition and Folding

Develops rings with trace addition and multiplication maintaining structural integrity.

Chapter 050: FieldCollapse​

Irreducible Trace Quotients and φ-Divisibility

Fields emerge from irreducible trace structures with full arithmetic.

Chapter 051: TensorOps​

Algebraic Operations on Collapse Trace Tensors

Extends operations to full tensor spaces while preserving constraints.

Chapter 052: CollapseMatrix​

Trace Matrix Construction from Structured Block Tensors

Matrices built from φ-constrained trace blocks.

Chapter 053: EigenCollapse​

Spectral Stability through Trace Eigenstructure

Eigenvalues and eigenvectors in collapse tensor space.

Chapter 054: ClosureCollapse​

Algebraic Closure of Trace Groups under Collapse Semantics

Achieving closure while maintaining golden constraints.

Chapter 055: CollapseBracket​

Anti-Symmetric Collapse Commutator Systems

Lie bracket operations on trace structures.

Chapter 056: TraceIdentity​

Tensor Algebraic Identities across Collapse Transformations

Fundamental identities that hold in trace algebra.

Chapter 057: PolyCollapse​

Collapse Polynomial Systems over φ-Indexed Variables

Polynomials with coefficients and variables from trace space.

Chapter 058: CollapseIdeal​

Structural Ideals over φ-Rank Tensor Trace Space

Ideal theory in the context of collapse structures.

Chapter 059: FieldExtend​

Trace Field Extensions and Collapse Compatibility

Extending fields while preserving structural constraints.

Chapter 060: ModuleCollapse​

φ-Module Structures with Tensor Multiplicity

Modules over collapse rings with tensor action.

Chapter 061: CollapseBasis​

Basis Vector Fields for Collapse-Algebraic Structures

Finding basis elements in trace vector spaces.

Chapter 062: CollapseHomomorph​

Morphism Mapping between Collapse Tensor Algebras

Structure-preserving maps between algebraic objects.

Chapter 063: AlgebraOverObserver​

Algebraic Structures Varying over Observer Spectra

How algebraic properties change with observer perspective.

Key Concepts Introduced​

  1. Trace Groups: Invertible composition operations
  2. Collapse Rings: Dual operations maintaining φ-constraint
  3. Tensor Fields: Full field structure in trace space
  4. Spectral Algebra: Eigenstructure of collapse operators
  5. Polynomial Traces: Polynomial rings over traces
  6. Observer Algebra: Perspective-dependent structures

Dependencies​

  • Volume 0: Language and trace foundations
  • Volume 1: Arithmetic operations
  • Volume 2: Logic for algebraic properties

Next Steps​

  • Volume 4: Geometric structures from algebra
  • Volume 5: Spectral analysis and constants
  • Volume 15: Category theory perspective

"Algebra blooms where golden traces compose."