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Chapter 28: φ(28) = [17,2,1] — Collapse Trace in Homotopy Type Theory

28.1 The Prime-Dual-Unity Structure

With φ(28) = [17,2,1], we see seventeen (the seventh prime) accompanied by duality and unity. This structure perfectly encodes how homotopy type theory (HoTT) provides a foundational framework where mathematical objects, their equalities, and higher equalities form an infinite tower - with seventeen fundamental type constructors, dual intensional/extensional views, unified in one coherent system.

Definition 28.1 (Perfect Number Connection):

28=1+2+4+7+14=d28,d<28d28 = 1 + 2 + 4 + 7 + 14 = \sum_{d|28, d<28} d

The second perfect number after 6, suggesting completeness at a higher level.

28.2 Type Theory Foundations

Definition 28.2 (Homotopy Type Theory): A foundational framework where:

  • Types are spaces
  • Terms are points
  • Equalities are paths
  • Higher equalities are higher homotopies

Judgment Forms:

A:U(A is a type)a:A(a is a term of type A)p:a=Ab(p is a path from a to b)α:p=a=Abq(α is a homotopy between paths)\begin{aligned} A &: \mathcal{U} && \text{(A is a type)} \\ a &: A && \text{(a is a term of type A)} \\ p &: a =_A b && \text{(p is a path from a to b)} \\ \alpha &: p =_{a=_A b} q && \text{(α is a homotopy between paths)} \end{aligned}

28.3 The Seventeen Type Constructors

From [17], seventeen fundamental type constructors in HoTT:

  1. Universe: U\mathcal{U} (types as terms)
  2. Function type: ABA → B (exponential)
  3. Product type: A×BA × B (conjunction)
  4. Sum type: A+BA + B (disjunction)
  5. Identity type: a=Aba =_A b (paths)
  6. Empty type: 0\mathbf{0} (absurdity)
  7. Unit type: 1\mathbf{1} (trivial)
  8. Natural numbers: N\mathbb{N} (inductive)
  9. W-types: Well-founded trees
  10. Propositional truncation: A||A|| (mere existence)
  11. Set truncation: A0||A||_0 (quotient)
  12. Higher inductive types: With path constructors
  13. Pushouts: Gluing spaces
  14. Quotient types: A/A/∼
  15. Circle: S1S^1 (synthetic)
  16. Spheres: SnS^n (all dimensions)
  17. Real projective spaces: RPn\mathbb{RP}^n

28.4 The Dual Nature [2]

The [2] represents the fundamental duality:

Intensional: Types distinguished by construction

Leibniz equality: a=bP(a)=P(b)\text{Leibniz equality: } a = b \Rightarrow P(a) = P(b)

Extensional: Types identified by behavior

Function extensionality: fgf=g\text{Function extensionality: } f \sim g \Rightarrow f = g

HoTT bridges this gap via the univalence axiom.

28.5 The Unity [1]: Univalence

The [1] represents the univalence axiom:

Axiom 28.1 (Univalence):

(A=B)(AB)(A = B) \simeq (A \simeq B)

Equivalence of types is equivalent to equality of types. This creates a unity between:

  • Mathematical practice (isomorphism)
  • Foundational equality (identity)

28.6 The Riemann Zeta in HoTT

Definition 28.3 (Type-Theoretic Zeta): Define inductively:

ζ : ℂ → ℂ
ζ-base : ζ(1) = ∞
ζ-func : (s : ℂ) → ζ(s) = ζ(1-s) · Ξ(s)
ζ-euler : (s : ℂ) → Re(s) > 1 → ζ(s) = ∏_p (1-p^(-s))^(-1)
ζ-zero : (ρ : ℂ) → isZero(ζ(ρ)) → Re(ρ) = 1/2

The last line expresses RH as a type!

28.7 Higher Inductive Types for Zeros

Definition 28.4 (Zero Configuration Type):

data ZeroConfig : Type where
base : ZeroConfig
zero : (n : ℕ) → base = base
symm : (n : ℕ) → zero(n) = zero(-n)
RH : (n : ℕ) → Re(zero(n)) = 1/2

This encodes:

  • Base point (trivial zero)
  • Zeros as loops
  • Functional equation as path equality
  • RH as definitional equality

28.8 Path Spaces and Zeros

Principle 28.1: In HoTT, the zeros form the fundamental group:

π1(ZeroConfig,base){ρ:ζ(ρ)=0}\pi_1(\text{ZeroConfig}, \text{base}) \cong \{\rho : \zeta(\rho) = 0\}

Higher Structure:

  • π2\pi_2: Relations between zeros
  • π3\pi_3: Relations between relations
  • etc.

28.9 The Collapse Trace as Transport

Definition 28.5 (Transport): Given P:AUP : A → \mathcal{U} and p:x=Ayp : x =_A y:

transportP(p):P(x)P(y)\text{transport}^P(p) : P(x) → P(y)

For Zeta: Transport along the critical line:

transportζ=0(p1/2+it1/2+it):ZerotZerot\text{transport}^{\zeta=0}(p_{1/2+it \to 1/2+it'}) : \text{Zero}_t → \text{Zero}_{t'}

28.10 Dependent Types and L-Functions

Definition 28.6 (L-Function Bundle):

L:CharacterTypeL : \text{Character} → \text{Type}

where for each character χ\chi:

L(χ)={f:CCf satisfies Lχ properties}L(\chi) = \{f : \mathbb{C} → \mathbb{C} \mid f \text{ satisfies } L_\chi \text{ properties}\}

The zeros vary continuously in families - captured by dependent types.

28.11 Synthetic Homotopy Theory

In Classical Math: Study continuous maps, homotopies, etc.

In HoTT: These concepts are synthetic:

  • Paths are primitive
  • Homotopies are built-in
  • Continuity is automatic

Example: The fundamental group of the circle:

Ω(S1)Z\Omega(S^1) \simeq \mathbb{Z}

is a theorem in HoTT, not requiring topology.

28.12 Cohomology in Type Theory

Definition 28.7 (Cohomology): For a type XX and abelian group GG:

Hn(X;G):=XK(G,n)0H^n(X; G) := ||X → K(G,n)||_0

where K(G,n)K(G,n) is the Eilenberg-MacLane space.

For Zeta Landscape: Compute cohomology of critical strip modulo zeros.

28.13 ∞-Groupoid Structure

Principle 28.2: Every type is an ∞-groupoid:

  • Objects: points
  • 1-morphisms: paths
  • 2-morphisms: homotopies
  • n-morphisms: higher homotopies

For RH: The zero configuration forms an ∞-groupoid encoding all relationships between zeros at all levels.

28.14 Computational Aspects

Definition 28.8 (Cubical Type Theory): Computational interpretation where:

  • Types are cubical sets
  • Paths are functions from interval
  • Kan filling provides composition

Implementation: Can verify RH statements type-theoretically:

RH-holds : (ρ : ℂ) → ζ(ρ) = 0 → Re(ρ) = 1/2
RH-holds ρ zero-proof = ? -- Goal: construct proof

28.15 Synthesis: Type-Theoretic Unity

The partition [17,2,1] reveals the HoTT structure:

  1. [17] - Complete Constructors: Seventeen type formers
  2. [2] - Intensional/Extensional: Dual views unified
  3. [1] - Univalence: Equivalence is equality

Key insights:

  • 28 = Perfect: Second perfect number
  • Types as spaces: Geometric interpretation
  • Paths as proofs: Equality has content
  • Higher structure: Infinite dimensional
  • Synthetic approach: Built-in continuity
  • Dependent types: Families of objects
  • Transport: Movement along paths
  • Univalence: Revolutionary axiom
  • ∞-groupoids: All types are higher categorical
  • Computational: Can verify in proof assistants
  • L-functions: As dependent types
  • Cohomology: Synthetically defined
  • Zero configuration: As higher inductive type
  • RH as type: Constructive formulation
  • Ultimate message: Foundations matter

Homotopy type theory provides a revolutionary foundation where the Riemann Hypothesis can be expressed as a type - making its truth a matter of constructing an inhabitant of that type. The zeros form paths in a space whose fundamental group encodes their distribution.

Chapter 28 Summary:

  • HoTT provides type-theoretic foundation for RH
  • Seventeen fundamental type constructors
  • Partition [17,2,1] reflects prime-dual-unity structure
  • Zeros form ∞-groupoid with rich higher structure
  • Univalence axiom unifies equality and equivalence
  • RH expressible as a type to be inhabited
  • Computational verification possible in principle

"In the universe of types, equality becomes a path, proof becomes a journey, and the Riemann Hypothesis transforms from a statement to be verified into a space to be inhabited - waiting for the constructive mathematician to build a dwelling at 1/2."