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Quantum Mechanics from Finite Graded Equality

This paper proposes that standard quantum mechanics emerges uniquely from the hypothesis that equality possesses finite resolution, formalized through a graded distinguishability kernel that enforces finite state capacity, relational completeness, and reversible dynamics, thereby deriving complex coefficients, the Born rule, and local tomography as necessary consequences.

Original authors: Julian G. Zilly

Published 2026-03-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Julian G. Zilly

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to describe a picture to a friend, but you only have a very limited amount of memory to store the details. You can't describe every single pixel with infinite precision; you have to round things off.

This paper, "Quantum Mechanics from Finite Graded Equality," by Julian G. Zilly, argues that the strange, probabilistic, and "fuzzy" nature of the quantum world isn't a fundamental mystery of the universe. Instead, it is the inevitable result of one simple fact: Reality has a finite resolution.

Here is the breakdown of the paper's big ideas using everyday analogies.

1. The Core Idea: "Pixelated" Equality

In our everyday math, two things are either exactly the same or different. It's a binary switch: $1$ or $0$.

The Paper's Hypothesis: What if "sameness" isn't a switch, but a dimmer dial? What if you can only tell things apart up to a certain limit?

The author replaces the strict "Is it A or B?" with a "How different are A and B?" scale from 0 to 1. This is called Graded Equality.

Analogy: Imagine trying to distinguish between two shades of blue paint. If you have infinite eyes, you can see the difference between every drop. But if your eyes are "pixelated" (finite resolution), you can only tell apart a limited number of distinct shades. Once you run out of distinct shades, you can't make more.

2. The Three Big Consequences

When you accept that equality has a limit (finite resolution), three things must happen to make the system work logically:

A. Finite Capacity (The "Bucket" Limit)

Because you can only distinguish a finite number of states, the universe (or any system) has a maximum "bucket size" for information.

Analogy: Think of a digital camera with a fixed number of pixels. You can't take a photo with infinite detail. There is a hard limit on how many distinct images you can store. In quantum mechanics, this limit is called $N$ .

B. Relational Completeness (The "Map" is the Territory)

If you can only measure differences, then a "thing" isn't defined by what it is inside, but by how it relates to everything else.

Analogy: Imagine a city map where the only thing that exists is the distance between intersections. There are no "streets" or "buildings" in the map's code, only the relationships (distances) between points. If two points have the exact same distances to all other points, they are the same point. The paper calls this Saturation: the map contains all the information there is; there are no hidden "secret streets."

C. Reversible Dynamics (The "Dance")

If you can't store infinite detail, and you can't lose information (because the map is complete), the system must move in a perfect, reversible loop.

Analogy: Imagine a dancer spinning. If they had infinite memory, they could stop at any exact angle. But with finite resolution, they can only stop at specific "ticks" of a clock. To keep moving without losing information or getting stuck, they must spin in a perfect circle, visiting every "tick" exactly once before returning to the start. This is Cyclic Dynamics.

3. Why is the World Probabilistic? (The "Storage Crash")

This is the paper's most exciting part. Why can't we predict exactly what will happen next? Why do we need probabilities?

The author proves a "Capacity Deficit."

The Problem: To know the outcome of every possible measurement a particle could undergo, you would need to store a massive amount of data.
The Math: If a system has a capacity of $N$ (like a qubit), it can store about $\log_2(N)$ bits of information. But to be deterministic (knowing the answer to every possible question in advance), you would need roughly $N^2$ bits of storage.
The Result: The system simply doesn't have enough hard drive space to hold all the answers.
Analogy: Imagine a library with only 100 books (capacity). But the librarian is asked to write down the answer to every possible question about every book in the universe (determinism). The library runs out of paper.
The Solution: Since the library can't store the answers, it must guess (probabilistically). The "guessing" isn't random chaos; it's the most efficient way to use the limited space. This forces the universe to be probabilistic.

4. Where Do Complex Numbers and the Born Rule Come From?

Once you accept the system is a "finite-resolution loop," the math forces the rest of quantum mechanics to appear naturally.

Complex Numbers: To keep the "dance" (cyclic dynamics) smooth and reversible without breaking the rules of the loop, the math requires imaginary numbers (complex numbers). You can't do this dance with just real numbers if the loop is big enough ( $N \ge 3$ ).
The Born Rule ( $P = |c|^2$ ): This is the rule that tells us the probability of an outcome. The paper shows that because the system has limited space, the only way to measure "distance" between states that fits the geometry of the loop is to square the numbers.
- Analogy: If you are trying to fit a round peg (the quantum state) into a square hole (the measurement), the only way to make the fit perfect without wasting space is to use a specific formula. The paper proves that $P = |c|^2$ is the only formula that fits the geometry of a finite-resolution world.

5. Time and the "Unfolding"

The paper suggests that Time is just the system unfolding its limited information step-by-step.

Analogy: Imagine a movie reel. The whole movie exists at once (the "state"), but your projector (the "observer") can only show one frame at a time. Because the projector has limited memory, it can't show the whole movie instantly. It has to play the frames in order. That sequence of frames is time.

Summary: The "Big Reveal"

The paper argues that we don't need to invent weird quantum rules. We just need to admit that equality is fuzzy.

Equality is fuzzy (Finite Resolution).
Therefore, there is a limit to how much info a system can hold (Finite Capacity).
Because the system can't hold all the answers, it must guess (Probabilistic).
To keep the system consistent while guessing, it must follow complex, reversible loops (Unitary Evolution).
The specific way it guesses is the Born Rule.

The Takeaway: Quantum mechanics isn't a weird exception to reality; it's the natural consequence of a universe that has a finite amount of "pixelation." We are living in a high-resolution simulation where the resolution is just high enough to look smooth to us, but low enough to force the universe to be probabilistic.

Technical Summary: Quantum Mechanics from Finite Graded Equality

1. Problem Statement

Standard quantum mechanics (QM) is typically formulated using structures that are postulated rather than derived: complex Hilbert spaces, unitary evolution, and the Born rule ( $p = |\psi|^2$ ). While various reconstruction programs exist (e.g., Hardy, CDP, CBH), they often assume operational probabilities as primitives or rely on infinite-dimensional frameworks.

This paper addresses the foundational question: Can the entire structure of finite-dimensional quantum mechanics be derived from a single, minimal hypothesis regarding the nature of identity and distinguishability? Specifically, the author challenges the assumption of binary equality ( $x=y$ or $x \neq y$ ), which implies infinite precision, and proposes that physical reality operates under finite resolution.

2. Methodology

The paper employs an axiomatic reconstruction approach, starting from a single primitive mathematical structure and deriving the rest of the theory through logical necessity.

Core Hypothesis: Graded Equality

The author replaces the binary equality predicate with a graded distinguishability kernel $K: X \times X \to [0, 1]$ .

$K(x, x) = 0$ (Self-equality/indistinguishability).
$K(x, y) = 1$ (Perfect distinguishability).
$K(x, y) \in (0, 1)$ (Partial distinguishability).
$K$ is symmetric but not necessarily a metric (no triangle inequality).

The Two Axioms

From this primitive, two axioms are formalized:

Axiom 1: Finite Capacity with Informational Saturation.
- Finite Capacity: There exists a maximum number $N$ of mutually perfectly distinguishable states (a basis).
- Saturation: The kernel $K$ $K$ is the complete arbiter of physical structure.
  - Injectivity: States with identical $K$ -profiles are identical (Leibniz's Principle).
  - Surjectivity: Every consistent $K$ -profile corresponds to a physical state (no "missing" states or hidden variables).
  - Structural Leibniz: Any symmetry of a finite configuration (preserving $K$ ) extends to a global automorphism.
Axiom 2: Universal Relationality.
- Operational Completeness: All physical distinctions are registered by $K$ .
- Transport Consistency: Degrees of freedom must be comparable via $K$ .
- Basis Isotropy: The symmetry group acts transitively on the set of all bases (no measurement orientation is privileged).

Derivation Strategy

The paper traces the logical consequences of these axioms:

Self-Resolution: A system cannot determine its full relational identity (K-profile) using a single basis due to the "self-resolution problem." It requires evaluations against complementary bases.
Dynamics: To access these complementary bases, the system must undergo a cyclic transformation.
Algebraic Structure: Representation theory on the derived manifold forces the coefficient field to be complex numbers ( $\mathbb{C}$ ) for $N \geq 3$ .
Probability: Information-theoretic constraints (Capacity Deficit) and geometric constraints (Metric Compatibility) uniquely select the Born rule.

3. Key Contributions and Results

A. The Capacity Deficit and Indeterminism

The paper proves a "Capacity Halting Principle" (Theorem 86).

Available Storage: A system with capacity $N$ stores $\log_2 N$ bits.
Required Storage: A deterministic hidden-variable model assigning definite outcomes to all measurement contexts requires $\Omega(N^2)$ bits (specifically, $(M-1)\log_2 N$ for $M$ mutually unbiased bases).
Result: For $N \geq 3$ , the required storage exceeds available capacity. Determinism is mathematically impossible; the system must behave probabilistically. This reinterprets the Kochen-Specker theorem as an information-theoretic overflow rather than a logical contradiction.

B. Emergence of Complex Numbers and Hilbert Space

Field Selection: The cyclic dynamics required for self-resolution (an $N$ -cycle automorphism) forces the coefficient field to be $\mathbb{C}$ for $N \geq 3$ . Real numbers ( $\mathbb{R}$ ) are insufficient for $N \geq 3$ , and Quaternions ( $\mathbb{H}$ ) are excluded due to non-commutativity and spectral non-uniqueness (Theorem 44).
State Space: The state space is identified as the complex projective space $\mathbb{C}P^{N-1}$ .
Dynamics: The discrete cyclic symmetry ( $\mathbb{Z}_N$ ) extends to a continuous unitary group $U(N)$ via band-limited interpolation (Nyquist-Shannon theorem applied to the state manifold), yielding the Schrödinger equation.

C. Derivation of the Born Rule

The paper provides a novel derivation of the Born rule ( $p_k = |c_k|^2$ ) that works even for $N=2$ (where Gleason's theorem fails).

Metric Compatibility: The paper argues that the statistical distinguishability (Fisher-Rao metric) must be proportional to the geometric distinguishability (Fubini-Study metric) to avoid maintaining two independent notions of distance.
Uniqueness: The only probability assignment $p_k = f(|c_k|^2)$ that makes these metrics isometric is $f(x) = x$ , i.e., $p_k = |c_k|^2$ .

D. Tensor Products and Local Tomography

Composition: For spatially separated systems, the kernel composition rule is uniquely determined by associativity, leading to the tensor product structure ( $H_{AB} \cong H_A \otimes H_B$ ).
Local Tomography: The paper proves that local tomography (joint states determined by local measurements) is a consequence of the complex field and tensor product structure, not an independent postulate.

E. Time and Phase Granularity

Time as Unfolding: Time is defined as the sequential revelation of relational information as the system cycles through complementary contexts.
Phase Granularity: In a finite- $N$ system, phase is band-limited. The minimum distinguishable phase shift is $\delta \theta = 2\pi/N$ . Standard continuous phase is the $N \to \infty$ limit. This introduces a natural UV cutoff.

4. Significance and Implications

Foundational Rigidity: The paper demonstrates that quantum mechanics is not a collection of arbitrary postulates but a rigid structure forced by the hypothesis that "equality has finite resolution." Removing the idealization of infinite precision leads inevitably to QM.
Resolution of the Measurement Problem (Partial): It explains why probabilities arise (capacity deficit) and which probabilities (metric compatibility), though it remains agnostic on the "hard problem" of specific outcome selection, treating measurement as correlation formation.
Finite-N as Fundamental: Standard infinite-dimensional QM is reinterpreted as an effective theory ( $N \to \infty$ ) of a fundamentally finite system. This offers a natural resolution to UV divergences in Quantum Field Theory, suggesting they are artifacts of extrapolating a finite theory beyond its domain.
Refutability: The framework makes specific predictions:
- Deterministic, context-independent outcomes are impossible for $N \geq 3$ .
- The exponent in the Born rule must be exactly 2.
- Phase granularity exists at the scale of $2\pi/N$ (though unobservable internally due to the observer's own finite capacity).

5. Conclusion

Julian G. Zilly's paper successfully derives the core pillars of quantum mechanics—complex Hilbert spaces, unitary evolution, the Born rule, and tensor products—from a single primitive: graded equality. By formalizing the concept that physical systems have finite resolution and that states are exhaustively defined by their relational distinguishability profiles, the paper shows that indeterminism, complex numbers, and the specific form of quantum probability are not choices but necessary consequences of information-theoretic and geometric constraints. This work positions finite capacity not as a limitation, but as the generative principle of quantum structure.