The SSD Method and the Algorithmic Economy of Reality: A Universal Interpreter, Latent Twin, and Experimental Test of Deviations from the Born Rule

Zlatko Pangarić

doi:10.20944/preprints202603.2375.v1

2. Common Ontology: The Space of Variations

2.1. Combinatorial Completeness—Postulate P1

Let A be a finite alphabet (for numerical sequences

A = {0, 1, \dots, 9}

; for images

A = {0, 1}

),

S = A^{*}

the set of all finite sequences, and

Ω = A^{N}

the space of infinite histories. Every history

H \in Ω

is an address in a mathematically complete space of all possible descriptions. For images of resolution

n \times n

:

I_{n} = {0, 1}^{n \times n} ≅ {0, 1}^{n^{2}}, | I_{n} | = 2^{n^{2}} .

(1)

Hypothesis 1

(Combinatorial Completeness (P1)). Every

X \in I_{n}

is a point in the space of variations. This is not a trivial mathematical tautology but a methodological assumption: no external agent prohibits any configurationa priori. The choice among possible histories is made exclusively through the selection mechanism of Section 3, not through ontological prohibition. P1 yields no direct empirical consequences but enables the formal definition of the space over which selection operates.

3. Universal Interpreter: The Principle of Reality Selection

3.1. Three-Layer Structure

The UI is organized through three conceptual layers:

Ontological layer: $Ω$ —the space of all formal configurations (P1).
Selective layer: filters and algorithmic economy (P3, P4*).
Epistemological layer: attribution of meaning to selected configurations (P2).

3.2. Selection Measure

The measure quantifying preference for compressible histories is:

d ν (H) = \frac{1}{Z} \cdot 2^{- λ K (H)} \cdot d μ (H),

(2)

where

K (H)

is the Kolmogorov complexity,

μ

is a reference (uniform) measure,

λ > 0

is the selection stringency parameter, and Z is a normalization constant. Histories with smaller K receive greater selection weight—economy of description, not ontological necessity, determines what becomes reality. Z is finite because

\sum 2^{- K (H)}

converges by Kraft’s inequality [5, Thm. 2.1.1].

3.3. Algorithmic Action

The algorithmic action of history H is defined as

S_{alg} (H) = K (H)

, i.e., the minimum description length. Observed reality is that history whose description requires the least informational effort, consistent with available observables. This principle is a heuristic framework—not an independently confirmed physical law—motivated by Solomonoff’s inductive theory [6] and Occam’s razor. Experimental verification is possible only indirectly, through testable consequences such as P5.

3.4. Postulates P2 and P3

Hypothesis 2

(Semantic Labeling (P2)). There exists a function

σ : I_{n} \to Σ

mapping to a finite label set Σ. Operationally, σ is approximated by the SSD descriptor and/or expert annotation.

Hypothesis 3

(Bounded Diameter of Classes (P3)). For each label

i \in Σ

there exists

L_{i} (n)

such that

diam (σ^{- 1} (i)) \leq L_{i} (n)

in the metric space

(I_{n}, d)

.

P2 and P3 are assumptions enabling the formal definition of semantic classes. They are necessary for Lemma G and the latent twin construction but yield no direct empirical predictions outside this framework.

3.5. Postulate P5—Algorithmic Correction to the Born Rule

In standard quantum mechanics, the probability of outcome i is

{| 〈 i | ψ 〉 |}^{2}

. Within the UI framework, every quantum outcome i (a bitstring of length n) is a history in

Ω

; applying the selection filter from Section 3.2 gives:

P_{UI} (i) \propto P_{Born} (i) \cdot 2^{- λ K (i)} .

(3)

Because K is uncomputable, we introduce the operational approximation

s^{*} (i)

(Section 4):

Hypothesis 4

(Algorithmic Correction to the Born Rule (P5)). Let

P_{Born} (i)

be the Born-rule probability and

s^{*} (i)

the SSD complexity index of outcome i. Then:

P_{UI} (i) = Z \cdot P_{Born} (i) \cdot 2^{- λ s^{*} (i)},

(4)

where

λ > 0

is the informational stringency parameter and Z is a normalization constant.

P5 is a testable hypothesis extending standard quantum mechanics; its status is empirical, not a priori. States with lower

s^{*}

have slightly increased probability relative to the Born rule; more complex states have slightly decreased probability. The effect is

O (λ)

and requires high statistical precision. Key limitations: (1) replacing K with

s^{*}

is inexact (Section 10); (2) P5 must remain consistent with existing high-precision Born-rule tests (Section 7.7).

4. SSD Method: Formal Framework

4.1. Definition of Symbolic Structures

For a numerical sequence

X = (x_{0}, x_{1}, \dots, x_{N - 1})

, define sliding triplets

S_{k} = (x_{k}, x_{k + 1}, x_{k + 2})

,

k = 0, \dots, N - 3

, with differences:

\begin{matrix} Δ_{1}^{1} & = x_{k} - x_{k + 1} (first transition), \\ Δ_{2}^{1} & = x_{k + 1} - x_{k + 2} (\sec ond transition), \\ Δ_{1}^{2} & = | Δ_{1}^{1} | - | Δ_{2}^{1} | (geometric acceleration) . \end{matrix}

(5)

The symbolic structure of a triplet is determined by the signs of these three quantities:

σ_{k} = (sgn (Δ_{1}^{1}), sgn (Δ_{2}^{1}), sgn (Δ_{1}^{2})), sgn \in {<, =, >},

(6)

generating

3^{3} = 27

possible local geometries. Each maps to a code

c \in [0, 26]

via base-3 indexing:

c (σ_{k}) = val (s_{1}) \cdot 9 + val (s_{2}) \cdot 3 + val (s_{3}), val (<) = 0, val (>) = 1, val (=) = 2 .

(7)

4.2. SSD Metrics

From the empirical distribution

p_{s} = N_{s} / (N - 2)

:

SSD entropy: $E_{SSD} = - \sum_{s} p_{s} {log}_{2} p_{s}$ (diversity of local geometries).
Symbolic space activity: $κ = | {s : p_{s} > 0} | / 27$ (fraction of activated symbolic states).
Transition entropy $ε$ : normalized entropy of the transition matrix between consecutive symbolic states, $ε \in [0, 1]$ .
Composite index: $s^{*} (H) = [κ (H) + ε (H)] / 2$ . Lower values indicate more structured (compressible) configurations; higher values indicate randomness.

s^{*}

is a computable proxy for K, exhibiting monotonic correlation with K on standard sequence classes. Its limitations are discussed in Section 10.

4.3. SSD Descriptor and Pseudometric

The SSD descriptor of configuration X is the frequency vector of the 27 symbols:

S_{SSD} (X) = (f_{0}, f_{1}, \dots, f_{26}) \in Δ^{26} .

(8)

The SSD pseudometric is the

L^{1}

norm of the difference of descriptors:

d_{SSD} (X, Y) = {∥ S_{SSD} (X) - S_{SSD} (Y) ∥}_{1} .

(9)

5. Latent Twin: Formal Framework

5.1. Definition and Acceptance Criteria

Let

X \in I

be a real configuration, Q a candidate set, and

Φ : I \to L

a coding map (SSD descriptor, AE/VAE/CLIP encoder, or their fusion). The latent twin is:

Q^{*} = arg min_{Z \in Q} D (Φ (X), Φ (Z)) .

(10)

Candidate

Q^{*}

is accepted as the latent twin of X if there exists a symmetry transformation

g \in G

such that:

\begin{matrix} d_{SSD} (S_{SSD} (X), S_{SSD} (g \cdot Q^{*})) & \leq ε (structural condition), \end{matrix}

(11)

\begin{matrix} d_{perc} (X, g \cdot Q^{*}) & < τ (perceptual condition), \end{matrix}

(12)

where

ε

and

τ

are calibratable thresholds and G is the group of meaning-preserving symmetries (flips,

90^{\circ}

rotations, mild photometric corrections).

5.2. SSD Isomorphism and Equivalence Classes

X and Q are SSD-isomorphic to tolerance

ε

(

X \sim_{SSD}^{ε} Q

) if

\exists g \in G

such that

d_{SSD} (S_{SSD} (X), S_{SSD} (g \cdot Q)) \leq ε

. This is an equivalence relation for fixed

ε

and closed G; its classes define structural types in the space of variations.

5.3. Postulate P4*—Local Density

Hypothesis 5

(Local Density (P4*)). There exist

ρ_{i} \in (0, 1]

and

r^{*} \geq 1 / N_{n}

such that for every

X \in σ^{- 1} (i)

:

\frac{| B_{r^{*}} (X) \cap σ^{- 1} (i) |}{| B_{r^{*}} (X) |} \geq ρ_{i} .

(13)

Near every configuration of class i there are sufficiently many other configurations of the same class—this assumption of local cohesion of semantic classes is necessary for finding a latent twin.

P4* is a technical assumption enabling Lemma G. No subsequent experimental test depends directly on it; it serves solely for internal consistency.

6. Local Clustering Lemma (Lemma G)

Theorem 1

(Lemma G—Discrete Version). Under Postulates P1–P4*(i), for every

X \in σ^{- 1} (i)

there exists

Y \in σ^{- 1} (i)

,

Y \neq X

, such that

0 < d (X, Y) \leq r^{*}

.

Theorem 2

(Lemma G—Asymptotic Version). Under Postulates P1–P3 and P4*(ii), for every

ε > 0

there exists

n_{0}

such that for all

n \geq n_{0}

and every

X \in σ^{- 1} (i)

there exists

Y \neq X

with

{\bar{d}}_{n} (X, Y) < ε

.

Proof of Lemma G (Discrete Version).

Suppose the contrary: all

Y \in B_{r^{*}} (X)

satisfy

σ (Y) \neq i

except X itself, giving

| B_{r^{*}} (X) \cap σ^{- 1} (i) | = 1

. Since

r^{*} \geq 1 / N_{n} > 0

, the ball contains at least one Hamming neighbor, so

| B_{r^{*}} (X) | \geq 2

, and therefore

1 / | B_{r^{*}} (X) | \leq 1 / 2 < ρ_{i}

(for

ρ_{i} > 1 / 2

), which contradicts P4*(i). □

Interpretation. In the space of variations, no configuration is structurally isolated within its semantic class. A structurally close alternative realization always exists, guaranteeing the existence of latent twins. The role of Lemma G in this paper is purely structural—ensuring internal consistency of the latent twin construction—with no direct empirical implications.

7. Experimental Test of Postulate P5: Design and Methodology

7.1. Objective

P5 predicts

P_{UI} (i) = Z \cdot P_{Born} (i) \cdot 2^{- λ s^{*} (i)}

. The experiment tests whether the distribution of quantum measurement outcomes correlates with their algorithmic complexity—a deviation from standard quantum mechanics, which predicts that probability depends solely on amplitude.

7.2. Operational Formulation of the Test

P5 is tested as follows:

1.: Prepare quantum state $| ψ 〉$ .
2.: Perform $N_{shots}$ measurements; collect outcomes $i \in {0, 1}^{n}$ .
3.: Compute $s^{*} (i)$ for each distinct outcome.
4.: Test the correlation between empirical frequency $P (i)$ and complexity $s^{*} (i)$ .

Key test (for a uniform state):

P (i) \propto 2^{- λ s^{*} (i)}

.

7.3. Choice of Quantum State

The minimal test uses a uniform superposition:

| ψ 〉 = \frac{1}{\sqrt{2^{n}}} \sum_{i} | i 〉,

(14)

with

P_{Born} (i) = 1 / 2^{n}

for all i. Standard theory predicts a uniform distribution; any systematic deviation is in principle detectable. For

n \leq 3

, all outcomes have similar complexity and the effect is undetectable;

n \geq 10

(preferably 20–50) is needed for meaningful

s^{*}

variation among outcomes.

7.4. Measurement and Data Collection

Required:

N_{shots} \sim 10^{6}

–

10^{9}

. For each outcome i: occurrence count

O_{i}

and empirical probability

P (i) = O_{i} / N_{shots}

.

7.5. Complexity Proxies

Three proxies are applied in parallel, all normalized to

[0, 1]

:

1.: SSD index $s^{*} (i) = [κ (i) + ε (i)] / 2$ .
2.: Lempel–Ziv complexity.
3.: zlib compression ratio.

Consistency across proxies controls for artefacts specific to any single approximant.

7.6. Statistical Test

Model tested:

log P (i) = - λ s^{*} (i) + C

.

Hypotheses:

H_{0} : λ = 0

(standard quantum mechanics);

H_{1} : λ \neq 0

(algorithmic correction).

Procedure: group outcomes by complexity bins, compute mean probability per bin, perform linear regression with

log P (i)

as the dependent variable and

s^{*} (i)

as the independent variable.

7.7. Consistency with Existing Experiments

The Born rule has been confirmed to precision

≲ 10^{- 6}

(spectroscopy) and

≲ 10^{- 4}

(direct quantum optical tests). For P5 to be consistent, we need:

λ \cdot Δ s_{max}^{*} ≲ 10^{- 4},

(15)

where

Δ s_{max}^{*}

is the maximum complexity difference between outcomes. For a 10-qubit system with

s^{*} \in [0.02, 0.42]

, this gives

λ ≲ 2.5 \times 10^{- 4}

.

7.8. Control of Systematic Errors

Key error sources: quantum hardware noise, readout bias, imperfect superposition. Controls:

1.: Readout matrix calibration.
2.: Ideal-case simulation (Qiskit Aer) as a reference.
3.: Cross-device comparison (IBM Q, IonQ, Rigetti).
4.: Randomization of measurement order.

7.9. Methodological Verification of the Statistical Test

Note on status: The simulation below confirms only that the test correctly recovers injected

λ

from synthetic data. It does not confirm that nature follows P5.

A simulation was run on all 1,024 possible 10-bit outcomes of a uniform superposition (Aer ideal simulator), with injected

λ = 0.01

:

P_{sim} (i) \propto 2^{- 10} \cdot 2^{- 0.01 \cdot s^{*} (i)}

. The SSD index

s^{*}

ranges from 0.0185 (periodic/all-zeros) to 0.4188 (most random). Linear regression returned: slope

= - 0.010000

; recovered

λ \approx 0.010000

;

r^{2} = 1.000000

. The value

r^{2} = 1

is expected because data were generated exactly without noise. In a real experiment,

r^{2}

will be substantially lower due to statistical noise and hardware imperfections.

Statistical power estimate. For realistic parameters (

n = 10

–20,

λ \sim 10^{- 4}

,

N_{shots} = 10^{8}

–

10^{9}

), expected statistical power is approximately 0.6–0.8, requiring careful experimental planning.

7.10. Interpretation of Possible Outcomes

$λ \approx 0$ : consistent with the Born rule; sets an upper bound on $λ$ .
Weak correlation: indication of an effect; requires independent replication.
Clear correlation: potential fundamental deviation from standard quantum mechanics.

8. Empirical Validation of the SSD Method

8.1. Decimal Sequences: $π$ , PRNG, QRNG

Table 1 reports SSD metrics for three classes of numerical sequences.

Finding 1. The SSD method is completely blind to the ontological origin of the sequence—deterministic (

π

, PRNG) and quantum (QRNG) sources yield virtually identical structural complexity metrics. SSD measures informational structure, not the data-generation mechanism.

8.2. Images and Textual Data

Table 2 summarizes SSD performance across domains (binary

30 \times 30

images; corpora of Python code, Shakespeare, and random text).

Finding 2. SSD is an effective measure of global structural complexity but not of fine semantic distinctions. This precisely defines its domain of application within the integrated model.

9. Theoretical Implications

9.1. Motivating the Born Rule in the UI Framework

The goal is not to claim a complete derivation of the Born rule—that would require resolving the quantum measurement problem, which remains open. The goal is to show that squaring the amplitude is not a priori unexpected within algorithmic economy, and that P5 represents a natural extension.

Explicit acknowledgment. The UI framework does not derive the Born rule; it is consistent with it at leading order. The following argument provides motivation, not deduction.

Decomposition of informational channels. Within the UI framework, the wave function

| ψ 〉 = \sum_{i} c_{i} | i 〉

is interpreted as a compressed record of the pair

(G, Φ)

, where G represents structural density (geometric pattern in the space of variations) and

Φ

the set of phase relations among components. This interpretation is non-standard; it is an additional assumption of the UI framework. Assuming G and

Φ

are informationally independent channels:

K (| ψ 〉) = K (G) + K (Φ) + O (1), ν (i) \propto 2^{- K (G_{i})} \cdot 2^{- K (Φ_{i})} .

(16)

If

| c_{i} |^{2} \sim 2^{- (K (G_{i}) + K (Φ_{i}))}

, then squaring the amplitude follows from algorithmic economy rather than being an arbitrary axiom. However, this relation is heuristic, not exact; alternative forms

| c_{i} |^{α}

for

α \neq 2

are not excluded by this argument.

Conclusion on motivation. This argument provides a consistent motivation for squaring the amplitude within the UI framework, but not a unique derivation. The Born rule appears as the leading-order term of algorithmic economy; P5 is the corrective term that is in principle testable.

Open problem. A formal proof that

α = 2

is the only value consistent with the UI framework remains explicitly open.

9.2. Categorical Framework

A functor

U : Img \to Sem

maps structural signatures to semantic classes, where Img has configurations (or SSD-equivalence quotients) as objects and Sem has semantic classes

σ^{- 1} (i)

as objects. Empirically, U is partially faithful: it distinguishes a portrait from noise but not `3’ from `5’ in MNIST, precisely defining where U is sufficient.

10. Limitations and Open Problems

10.1. The Epistemic Gap Between K and $s^{*}$ —Central Methodological Problem

P5 is formulated in terms of

K (i)

but the experiment measures

s^{*} (i)

. Formal reasons why

s^{*}

cannot uniformly approximate K:

1.: K is uncomputable; any computable approximation has non-uniform limitations.
2.: Pathological sequences exist with low K but high $s^{*}$ (globally simple, locally chaotic) and vice versa; $| K (X) - s^{*} (X) |$ can be arbitrarily large.
3.: SSD analyzes only triplets; structures requiring longer windows are invisible.

Epistemic consequence. If the experiment finds

λ = 0

using

s^{*}

, a non-zero

λ

in terms of K is not excluded. Conversely, finding

λ \neq 0

using

s^{*}

is a strong signal because pathological sequences do not form systematic patterns in outcomes from a uniform superposition.

Recommended control. Consistency across

s^{*}

, Lempel–Ziv, and zlib substantially reduces the probability that any observed effect is a method-specific artefact.

10.2. SSD Performance Profile

Table 3. SSD performance profile by application domain.

Task	Performance	Explanation
Structured vs. noise (images)	$> 95 %$	Global entropy and compressibility
Geometrically simple vs. complex	70–90%	Activity sum as transition measure
Digits (MNIST)	30–50%	Meta-structure partially helpful
Fine semantic distinctions	$< 30 %$	Requires spatially aware representations

10.3. Open Problems

1.: Formal proof that $α = 2$ is the unique value consistent with the UI framework.
2.: An upper bound on $λ$ derived from a comprehensive analysis of existing high-precision quantum experiments.
3.: Extension of SSD to windows of length $k > 3$ for detection of long-range correlations.
4.: Relationship between selection parameter $λ$ and empirically measurable constants (e.g., Planck’s constant or the dimensionality of the state space).

11. Conclusions

This paper develops an integrated theoretical framework connecting the space of variations, the Universal Interpreter, the algorithmic economy of reality, the concept of the latent twin, and the SSD measure into a coherent, operationally defined system. The central thesis—that reality is not determined by ontological necessity but by minimal informational cost—provides an alternative interpretive basis for understanding physical laws. Standard physical laws emerge as consistent with the principle of algorithmic economy; the Born rule receives motivation (though not a complete derivation) within this framework.

The SSD method is the key operational contribution of this work. Its applicability has been demonstrated on numerical sequences, images, and textual data, with clearly defined limitations: excellent for global structuredness vs. noise; insufficient for fine semantic distinctions.

Postulate P5 introduces a testable hypothesis about a correction to the Born rule based on the algorithmic complexity of outcomes. The experimental framework has been formulated with all methodological safeguards: the epistemic gap between K and

s^{*}

is explicitly addressed as the central problem (Section 10); the upper bound on

λ

from existing experiments is estimated (Section 7.7); and the methodological verification (Section 7) is clearly identified as a procedural check, not a confirmation of the hypothesis.

Lemma G formally ensures that no configuration is isolated within its semantic class, thereby guaranteeing the existence of latent twins and the stability of structural types.

The paper does not claim empirical confirmation of the algorithmic correction to quantum probabilities. It establishes a theoretical and experimental framework in which such a hypothesis is precisely formulated and testable. Although P5 is the central empirically testable consequence addressed here, the UI framework in principle generates a broader class of testable predictions through the introduction of algorithmic selection bias, the systematic investigation of which remains the subject of future work.

Author Contributions

Conceptualization, Z.P.; methodology, Z.P.; formal analysis, Z.P.; writing—original draft preparation, Z.P.; writing—review and editing, Z.P. The author has read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No empirical datasets were generated or analyzed in this study. The methodological verification (Section 7) is based on a deterministic simulation using the Qiskit Aer ideal simulator; the simulation code is available from the author upon reasonable request.

Acknowledgments

The author thanks the open-source communities behind Qiskit [14], NumPy, and SciPy for the computational infrastructure used in the methodological verification.

Conflicts of Interest

The author declares no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

SSD	Symbolic Structures of Differences
UI	Universal Interpreter
LT	Latent Twin
PRNG	Pseudo-random number generator
QRNG	Quantum random number generator
MNIST	Modified National Institute of Standards and Technology (digit database)

Appendix A. Formal Motivation of Postulate P5

Appendix A.1. Why Correct the Born Rule?

In standard quantum mechanics,

P_{Born} (i) = {| 〈 i | ψ 〉 |}^{2}

is an axiomatic postulate. Within the UI framework, every quantum outcome i (a bitstring of length n) is a point in

Ω

, and the same selection measure applies:

d ν (H) = Z^{- 1} \cdot 2^{- λ K (H)} \cdot d μ (H)

.

Appendix A.2. Derivation from the Selection Measure

Applying the selection filter to outcome i:

P_{UI} (i) \propto P_{Born} (i) \cdot 2^{- λ K (i)}

. The Born amplitude is retained because it is consistent with the leading-order of algorithmic economy—it encodes the informational cost of phase relations and structural density already embedded in

c_{i}

. The additional factor

2^{- λ K (i)}

penalizes configurations requiring a longer description. Replacing K with

s^{*}

:

P_{UI} (i) = Z \cdot P_{Born} (i) \cdot 2^{- λ s^{*} (i)} .

(A1)

Appendix A.3. Intuitive Interpretation

Low

s^{*} (i)

(periodic sequences, low Hamming weight) ⇒ factor

\approx 1

⇒ slightly increased probability. High

s^{*} (i)

(random bitstrings) ⇒ factor

< 1

⇒ slightly decreased probability. The effect is

O (λ)

, but systematic and cumulative. For

λ \approx 0.001

–

0.01

(consistent with the upper bound from Section 7.7), detection requires

10^{8}

–

10^{9}

measurements on 10–20 qubits.

Appendix A.4. Experimental Testability

The test reduces to linear regression:

log P_{emp} (i) = - λ s^{*} (i) + C

, with

H_{0} : λ = 0

vs.

H_{1} : λ \neq 0

. Section 7 confirms the algebraic correctness of the procedure. A real quantum hardware experiment is required for a meaningful test of the hypothesis.

References

Pangarić, Z. Symbolic Geometry of the Number π: Structures, Statistics, and Security. Preprints.org 2026. [Google Scholar] [CrossRef]
Pangarić, Z. Latentni dvojnik. Preprints.org 2026. [Google Scholar] [CrossRef]
Pangarić, Z. Symbolic Structures of Differences (SSD): A Geometrical Approach to Quantifying Complexity in Time Series. Preprints.org 2026. [Google Scholar] [CrossRef]
Bandt, C.; Pompe, B. Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett. 2002, 88, 174102. [Google Scholar] [CrossRef] [PubMed]
Li, M.; Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications, 4th ed.; Springer: New York, NY, USA, 2008. [Google Scholar]
Solomonoff, R.J. A formal theory of inductive inference. Inf. Control 1964, 7, 1–22. [Google Scholar] [CrossRef]
Chaitin, G.J. A theory of program size formally identical to information theory. J. ACM 1975, 22, 329–340. [Google Scholar] [CrossRef]
Tegmark, M. The Mathematical Universe Hypothesis. Found. Phys. 2008, 38, 101–150. [Google Scholar] [CrossRef]
Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J. 1948, 27, 379–423. [Google Scholar] [CrossRef]
Cover, T.M.; Thomas, J.A. Elements of Information Theory, 2nd ed.; Wiley: Hoboken, NJ, USA, 2006. [Google Scholar]
LeCun, Y.; Cortes, C.; Burges, C. MNIST handwritten digit database. ATT Labs 2010. [Google Scholar]
Carroll, S.M. Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime; Dutton: New York, NY, USA, 2019. [Google Scholar]
Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information, 10th ed.; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
Qiskit Contributors. Qiskit: An Open-source Framework for Quantum Computing, 2023. Available online: https://qiskit.org (accessed on 1 March 2026).

Table 1. SSD metrics for numerical sequences.

Source	Active Codes	$E_{SSD}$	$κ$	$ε$	$s^{*}$
Theory	17	3.83281	—	—	—
$π$	17	3.83233	0.512	1.000	0.756
PRNG	17	3.83190	0.512	0.999	0.756
QRNG	17	3.83479	0.518	1.001	0.759

Table 2. SSD performance across application domains.

Domain	Performance	Conclusion
Structured vs. noise	$> 95 %$	Reliably distinguishes order from randomness
Textual structure	$s^{*} \in [0.55, 0.75]$	Lower $s^{*}$ = more structured (code < language < random)
Fine semantic differences	$< 30 %$	Does not substitute for deep learning

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

The SSD Method and the Algorithmic Economy of Reality: A Universal Interpreter, Latent Twin, and Experimental Test of Deviations from the Born Rule

Abstract

Keywords:

Subject:

1. Introduction

1.1. Three Pillars of the Integrated Framework

1.2. Common Thesis

2. Common Ontology: The Space of Variations

2.1. Combinatorial Completeness—Postulate P1

3. Universal Interpreter: The Principle of Reality Selection

3.1. Three-Layer Structure

3.2. Selection Measure

3.3. Algorithmic Action

3.4. Postulates P2 and P3

3.5. Postulate P5—Algorithmic Correction to the Born Rule

4. SSD Method: Formal Framework

4.1. Definition of Symbolic Structures

4.2. SSD Metrics

4.3. SSD Descriptor and Pseudometric

5. Latent Twin: Formal Framework

5.1. Definition and Acceptance Criteria

5.2. SSD Isomorphism and Equivalence Classes

5.3. Postulate P4*—Local Density

6. Local Clustering Lemma (Lemma G)

7. Experimental Test of Postulate P5: Design and Methodology

7.1. Objective

7.2. Operational Formulation of the Test

7.3. Choice of Quantum State

7.4. Measurement and Data Collection

7.5. Complexity Proxies

7.6. Statistical Test

7.7. Consistency with Existing Experiments

7.8. Control of Systematic Errors

7.9. Methodological Verification of the Statistical Test

7.10. Interpretation of Possible Outcomes

8. Empirical Validation of the SSD Method

8.1. Decimal Sequences: π , PRNG, QRNG

8.2. Images and Textual Data

9. Theoretical Implications

9.1. Motivating the Born Rule in the UI Framework

9.2. Categorical Framework

10. Limitations and Open Problems

10.1. The Epistemic Gap Between K and s * —Central Methodological Problem

10.2. SSD Performance Profile

10.3. Open Problems

11. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

Abbreviations

Appendix A. Formal Motivation of Postulate P5

Appendix A.1. Why Correct the Born Rule?

Appendix A.2. Derivation from the Selection Measure

Appendix A.3. Intuitive Interpretation

Appendix A.4. Experimental Testability

References

MDPI Initiatives

Important Links

Subscribe

8.1. Decimal Sequences: $π$ , PRNG, QRNG

10.1. The Epistemic Gap Between K and $s^{*}$ —Central Methodological Problem