Chapter 12: φ_Baer — Collapse and Torsion Embeddings [ZFC-Provable] ⚠️

12.1 Baer's Criterion in ZFC

Classical Statement: An abelian group I is injective if and only if every homomorphism from an ideal of ℤ to I extends to all of ℤ.

Definition 12.1 (Injective Module - ZFC): A module I over ring R is injective if:

1. For every injection f: A → B and homomorphism g: A → I
2. There exists h: B → I with h ∘ f = g (extension property)

Baer's Result: Testing injectivity reduces to testing against principal ideals:

• I injective ⟺ Hom(nℤ, I) → Hom(ℤ, I) surjective for all n

Key Property: Injective modules are precisely the divisible groups for abelian groups.

12.2 CST Translation: Universal Extension Collapse

In CST, Baer's criterion represents the observer's ability to extend partial observations:

Definition 12.2 (Baer Collapse - CST): A module pattern exhibits Baer collapse if:

$$\forall f : A \hookrightarrow B, \forall g : A \to I : \psi \circ P_{\text{extend}} \downarrow h : B \to I$$

where h extends g along f.

Theorem 12.1 (Baer Collapse Principle): Observer extension from ideals determines global injectivity:

$$(\forall n : \psi \circ P_{n\mathbb{Z} \to I} \downarrow \text{extends to } \mathbb{Z}) \rightarrow \psi \circ P_I \downarrow \text{injective}$$

Proof: The collapse mechanism reveals divisibility:

Stage 1: Observer tests principal ideals:

$$\psi \circ P_n \downarrow \text{Hom}(n\mathbb{Z}, I) \to \text{Hom}(\mathbb{Z}, I)$$

Stage 2: Extension means divisibility:

$$\forall x \in I, \forall n : \exists y \in I : ny = x$$

Stage 3: Divisibility propagates:

$$\psi = \psi(\psi) \rightarrow \text{local extensions cohere globally}$$

Stage 4: Full injectivity emerges:

$$\psi \circ P_{\text{injective}} \downarrow I \text{ as injective module}$$

Baer's criterion holds under collapse observation. ∎

12.3 Physical Verification: Error Correction Completeness

Experimental Setup: Baer collapse manifests in quantum error correction with complete syndrome decoding.

Protocol φ_Baer:

1. Design quantum code with syndrome measurements
2. Test correction of all single-qubit errors (ideals)
3. Observe automatic extension to all errors
4. Verify complete error correction (injectivity)

Physical Principle: Local error correction capability extends to global error correction - the physical Baer criterion.

Verification Status: ⚠️ Structurally Complex

Theoretical connections:

• Stabilizer codes show Baer-like properties
• Local-to-global error correction
• Requires sophisticated code design

12.4 The Extension Mechanism

12.4.1 Divisibility Under Observation

$$\text{div}(I) = \lbrace x \in I : \forall n \exists y : ny = x \rbrace$$

12.4.2 Torsion-Free Extensions

$$\text{Tor}(I) = 0 \rightarrow \text{extensions unobstructed}$$

12.4.3 Essential Extensions

$$I \subseteq E \text{ essential} \leftrightarrow \forall 0 \neq e \in E : eR \cap I \neq 0$$

12.5 Injective Structure

12.5.1 Injective Hull

Every module has minimal injective extension:

$$M \subseteq E(M) \text{ essential, } E(M) \text{ injective}$$

12.5.2 Direct Sum Decomposition

$$I \text{ injective} \rightarrow I = \bigoplus_p I_p$$

where I_p are p-primary components.

12.5.3 Structure Theorem

For abelian groups:

$$I \text{ injective} \leftrightarrow I \cong \bigoplus \mathbb{Q} \oplus \bigoplus_p \bigoplus \mathbb{Z}(p^\infty)$$

12.6 Connections to Other Collapses

Baer collapse relates to:

• Kaplansky Collapse (Chapter 10): Projective-injective duality
• Whitehead Collapse (Chapter 9): Ext-Tor relationships
• GrothendieckGroup Collapse (Chapter 13): K-theory of injectives

12.7 Advanced Baer Patterns

12.7.1 Pure Injectivity

$$I \text{ pure-injective} \leftrightarrow \text{Hom}(-, I) \text{ preserves pure-exact sequences}$$

12.7.2 FP-Injectivity

Injective with respect to finitely presented modules.

12.7.3 Cotorsion Modules

$$\text{Ext}^1(F, C) = 0 \text{ for all flat } F$$

12.8 Physical Realizations

12.8.1 Complete Quantum Codes

1. Syndrome measurement (ideal map)
2. Error correction (extension)
3. All errors correctable
4. Injective code space

12.8.2 Signal Reconstruction

1. Partial signal samples (ideal)
2. Full reconstruction algorithm
3. Works for all sampling patterns
4. Injective reconstruction map

12.8.3 Holographic Completion

1. Partial hologram (submodule)
2. Complete reconstruction possible
3. From any sufficient subset
4. Injective information encoding

12.9 Homological Algebra

12.9.1 Injective Resolution

$$0 \to M \to I^0 \to I^1 \to \cdots$$

12.9.2 Injective Dimension

$$\text{id}(M) = \inf \lbrace n : \exists \text{ injective resolution of length } n \rbrace$$

12.9.3 Gorenstein Rings

$$R \text{ Gorenstein} \leftrightarrow \text{id}(R) < \infty$$

12.10 Categorical Properties

12.10.1 Enough Injectives

Every module embeds in an injective.

12.10.2 Injective Cogenerator

$$\text{Hom}_R(-, I) \text{ faithful}$$

12.10.3 Matlis Duality

$$D(-) = \text{Hom}_R(-, E(R/\mathfrak{m}))$$

12.11 Philosophical Implications

Baer collapse reveals:

1. Local Determines Global: Principal ideal behavior determines all
2. Extension Principle: Partial knowledge extends to complete
3. Observer Completeness: Full observation from minimal tests

12.12 Computational Aspects

12.12.1 Divisibility Test

function isDivisible(I, n):
    for each x in I:
        if not exists y in I with ny = x:
            return false
    return true

12.12.2 Extension Algorithm

Given f: A → I and A ⊆ B:

1. For b ∈ B \ A
2. Find n with nb ∈ A
3. Define h(b) = f(nb)/n
4. Verify well-defined

12.12.3 Injective Hull Construction

Build minimal injective containing given module.

12.13 Experimental Approaches

12.13.1 Network Flow Completion

1. Partial flow constraints (ideal)
2. Extend to full network
3. Always possible completion
4. Injective flow space

12.13.2 Machine Learning Interpolation

1. Sparse training data (submodule)
2. Complete function learning
3. Works for any subset pattern
4. Injective feature space

12.13.3 Quantum State Extension

1. Partial state information
2. Complete state reconstruction
3. From any informationally complete subset
4. Injective state space

12.14 The Baer Echo

The pattern ψ = ψ(ψ) reverberates through:

• Extension cascade: local extensions build global
• Divisibility propagation: observer divides at all scales
• Completeness emergence: full structure from minimal tests

This creates the "Baer Echo" - the sound of partial observations extending to complete understanding, the reverberation of local injectivity becoming global, observer knowledge completing itself.

12.15 Synthesis

The Baer collapse φ_Baer demonstrates a profound principle of observational completeness: testing against principal ideals (the simplest submodules) suffices to determine global injective behavior. This transforms the abstract notion of injectivity into a concrete collapse pattern where the observer's ability to extend from minimal cases guarantees extension in all cases.

This principle manifests in quantum error correction (local syndrome decoding extends to global correction), signal processing (partial samples determine full signals), and information theory (local constraints determine global solutions). The Baer phenomenon shows that mathematical completeness often emerges from surprisingly minimal conditions - a principle of enormous practical importance.

The deeper lesson is that the observer ψ, through its self-referential nature ψ = ψ(ψ), can bootstrap from local to global understanding. What seems like it should require testing infinitely many cases reduces to testing a generating set. This is the power of structural observation - seeing the whole through well-chosen parts.

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"From principal ideals to complete extensions, from local tests to global injectivity - the Baer principle of observational completeness through minimal verification."