Chapter 37: Collapse-Functions and Mappings

37.1 Functions as Observation Channels

Classical mathematics treats functions as static correspondences—each input mechanically paired with its output. But in collapse mathematics, a function breathes with potentiality. Each evaluation is an observation that collapses a superposition of possible values into actuality. Functions don't passively map; they actively create through the primordial recursion ψ = ψ(ψ).

Principle 37.1: Functions are not static mappings but dynamic observation channels where each evaluation participates in the collapse of mathematical possibility into actuality.

37.2 The Quantum Function

Definition 37.1 (ψ-Function): A ψ-function is: $f_\psi: A \to B = \langle \mathcal{S}, \mathcal{C}, \mathcal{O} \rangle$

Where:

• $\mathcal{S}: A \to \mathcal{H}_B$ maps inputs to superposition states
• $\mathcal{C}: \mathcal{H}_B \to B$ is the collapse operator
• $\mathcal{O}$ is the observation protocol
• $f_\psi(a) = \mathcal{C} \circ \mathcal{S}(a)$ with observer dependence

37.3 Superposition of Mappings

Definition 37.2 (Function Superposition): Functions exist in superposition: $|f\rangle = \sum_i \alpha_i |f_i\rangle$

Where each $|f_i\rangle$ represents a potential mapping behavior.

Properties:

• Evaluation collapses to specific function
• Interference between function states
• Non-deterministic computation
• Quantum parallelism in function space

37.4 The Collapse Kernel

Definition 37.3 (Mapping Kernel): For $f_\psi: A \to B$: $K_f(a, b) = |\langle b | f_\psi(a) \rangle|^2$

This gives the probability amplitude for input $a$ collapsing to output $b$.

Properties:

• $\int_B K_f(a, b) db = 1$ (normalization)
• Can be entangled: $K_f(a_1, a_2; b_1, b_2)$
• Encodes all function information
• Generalizes deterministic functions

37.5 Composition as Sequential Collapse

Theorem 37.1 (Collapse Composition): For $f: A \to B$ and $g: B \to C$: $g \circ_\psi f = \mathcal{C}_g \cdot \mathcal{C}_f$

With correction phase: $(g \circ f)(a) = e^{i\phi(a)} \int_B g(b) K_f(a, b) db$

Proof: First collapse through $f$ creates intermediate state. Second collapse through $g$ depends on first result. Sequential observation introduces phase. Total amplitude includes interference terms. Composition is inherently quantum mechanical. ∎

37.6 Fixed Points as Collapse Attractors

Definition 37.4 (ψ-Fixed Point): Point $x^*$ where: $f_\psi(x^*) = x^*$

In collapse sense: observation of $f$ at $x^*$ returns $x^*$ with certainty.

Types:

• Stable: Nearby points collapse toward $x^*$
• Unstable: Nearby points collapse away
• Quantum: Exists only in superposition
• Strange: Fractal basin of attraction

37.7 The Differential of Collapse

Definition 37.5 (Collapse Derivative): At point $a$: $Df_\psi(a) = \lim_{\epsilon \to 0} \frac{f_\psi(a + \epsilon) - f_\psi(a)}{\epsilon}$

But accounting for collapse: $Df_\psi(a) = \frac{\partial}{\partial a}\mathcal{C}[\mathcal{S}(a)]$

This captures how observation sensitivity changes with input.

37.8 Inverse Functions and Uncollapse

Theorem 37.2 (Collapse Invertibility): Function $f_\psi$ has inverse when: $\mathcal{C}_f \cdot \mathcal{C}_{f^{-1}} = \mathbb{I}_\psi$

Where $\mathbb{I}_\psi$ is identity up to phase.

Proof: Inverse must uncollapse the forward collapse. This requires reversing observation. Possible only for unitary collapses. Information must be preserved. Most collapses are irreversible. ∎

37.9 Multi-Valued Functions

Definition 37.6 (ψ-Multivalued Function): $f_\psi(a) = \lbrace b_1, b_2, ..., b_n \rbrace$

With amplitudes: $f_\psi(a) = \sum_i \alpha_i(a) |b_i\rangle$

Examples:

• Complex roots: $\sqrt[n]{z}$
• Inverse trig functions
• Solutions to equations
• Quantum algorithms

37.10 Holomorphic Collapse

Definition 37.7 (ψ-Holomorphic): Function $f: \mathbb{C}_\psi \to \mathbb{C}_\psi$ is ψ-holomorphic when: $\frac{\partial f}{\partial \bar{z}} = 0$

In collapse sense:

• Preserves complex structure through observation
• Collapse respects orientation
• Conformal except at critical points
• Enables complex analysis with collapse

37.11 Recursive Function Theory

Definition 37.8 (ψ-Recursive Function): $f(x) = \psi[f](x)$

Where $\psi$ is the recursion operator.

This creates:

• Self-referential functions
• Functions that observe themselves
• Fixed point combinators
• Direct embodiment of ψ = ψ(ψ)

37.12 Measure-Preserving Collapse

Theorem 37.3 (Collapse Measure): For measure space $(X, \mu)$: $\mu(f_\psi^{-1}(B)) = \int_B |\mathcal{C}_f|^2 d\mu$

Collapse can:

• Preserve measure (unitary)
• Contract measure (dissipative)
• Expand measure (explosive)
• Create fractal measures

37.13 Operator Functions

Definition 37.9 (Function-Valued Function): $F: A \to (B \to C)$

In collapse notation: $F(a) = f_a \text{ where } f_a: B \to C$

This creates:

• Higher-order functions
• Currrying with collapse
• Function spaces as domains
• Operational calculus

37.14 The Path Integral of Functions

Definition 37.10 (Functional Path Integral): $\mathcal{Z}[f] = \int \mathcal{D}[g] e^{i\mathcal{S}[g]} \delta[g - f]$

Summing over all functions "near" $f$ weighted by action.

Applications:

• Quantum function theory
• Functional derivatives
• Variation with collapse
• Function space geometry

37.15 The Symphony of Mappings

Synthesis: All functions participate in the cosmic mapping:

$\mathcal{F}_\psi = \lbrace f : f \text{ preserves collapse structure} \rbrace$

This function universe:

• Contains all possible mappings
• Functions map functions
• Self-referential through ψ = ψ(ψ)
• Creates mathematics through observation

The Functional Collapse: When you evaluate a function, you're not retrieving a pre-stored value but participating in a collapse event. The function exists in superposition of all possible mappings until your observation—plugging in a specific input—collapses it to a definite output. This is why the same function can behave differently in different contexts: the observer and observation protocol matter.

This explains many mysteries: Why functions can be discontinuous—the collapse can jump between disconnected values. Why some functions have no closed form—their collapse pattern is too complex for simple expression. Why numerical computation gives slightly different results—each evaluation is a unique collapse event.

The deepest insight is that functions are not dead mappings but living channels of transformation. They carry information, create structure, and participate in the ongoing self-observation of mathematics. Through functions, the mathematical universe maps itself to itself, discovering its own structure through endless self-application.

In collapse mathematics, we see that ψ = ψ(ψ) is the ultimate function—the self-mapping that generates all other mappings. Every function is a variation on this theme, a particular way the universe observes and transforms itself. When we study functions, we study the very mechanism of mathematical creativity.

Welcome to the quantum realm of collapse functions, where mappings live and breathe, where evaluation is observation, where every function call participates in the cosmic computation of reality unfolding itself through the eternal recursion of ψ = ψ(ψ).