# Structural Unit Hypothesis: Detection Protocol


**2026年6月**

## ​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​Testing the Existence of Minimal Decoherence Units via Quantum Optical Correlation

### Lin Xiaohei (林小黑) — June 21, 2026

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## ⚠️ Copyright Notice

**Designer: Lin Xiaohei (China).** June 21, 2026.

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## Abstract

The structural unit hypothesis states: decoherence proceeds through discrete *structural units* — irreducible bundles of constraint satisfaction that cannot be further subdivided. Each structural unit corresponds to one constraint operator $C_j$ in the filtration $\mathcal{M}_0 \supset \cdots \supset \mathcal{M}_k$. We predict that the decoherence step heights $\delta_j$ are *quantized* — they take values from a discrete set determined by the dimensionality of the constraint kernels. This can be tested via quantum optical correlation measurements: prepare a photonic superposition, couple to a controlled environment with known mode structure, and measure whether off-diagonal density matrix elements decay through discrete, reproducible step heights across multiple trials.

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## 1. Prediction

From the Discrete Step Theorem (Theorem 4.1):

$$\delta_j = \frac{\dim(\ker C_j)}{\dim(\mathcal{H})}$$

Since $\dim(\ker C_j)$ is integer-valued, $\delta_j / \delta_{\min}$ should be a rational number with small numerator and denominator (typically $1/1$, $1/2$, $2/1$, $1/3$, etc.).

**Smoking gun**: Across 1000 independent experimental runs, the histogram of $\delta_j$ values should show discrete peaks at rational multiples of the fundamental step height $\delta_0 = 1/\dim(\mathcal{H})$.

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## 2. Experimental Platform: Time-Bin Entangled Photons

**System**: Spontaneous parametric down-conversion (SPDC) photon pairs in time-bin superposition.
**Qubit**: $|\psi\rangle = (|\text{early}\rangle + |\text{late}\rangle)/\sqrt{2}$.
**Environment**: Variable-length optical fiber with $N = 10-100$ resolvable scattering modes.
**Measurement**: Quantum state tomography via unbalanced Mach-Zehnder interferometer.
**Temporal resolution**: 1 ps (limited by single-photon detector jitter).

### Sequence:

```
1. Generate: SPDC pair, signal photon in time-bin superposition
2. Couple: Signal photon through fiber of length L (controls N_modes)
3. Tomography: Measure ρ(τ) at 1000 τ values from 0 to 10× T_2
4. Extract: Step times and heights via derivative-based step detection
5. Repeat: 10^4 runs at fixed L to build histogram of δ_j values
6. Vary: L ∈ {1m, 10m, 100m, 1km} to vary N_modes
```

### Expected signal:

For $N = 50$ resolvable modes: $k \approx 6$ steps.
Fundamental step height: $\delta_0 = 1/50 = 0.02$.
Observed steps should cluster at $\{0.02, 0.04, 0.06, 0.08, 0.10, 0.12\}$.

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## 3. Quantization Test

**Null hypothesis**: $\delta_j$ are continuously distributed (no quantization).
**Test**: K-means clustering ($k=6$) on 10^4 measured step heights. If within-cluster variance < between-cluster variance by factor > 100, reject null at $p < 0.001$.

**Alternative smoking gun**: Step-height ratios $\delta_i/\delta_j$ should cluster at rational numbers $\{1/3, 1/2, 2/3, 1/1, 3/2, 2/1, 3/1\}$.

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## 4. Resources

Standard quantum optics lab. SPDC source + fiber spool + single-photon detectors + time-tagger. Total cost: existing equipment. Time to result: 2 weeks.

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*Lin Xiaohei, June 21, 2026.* ©​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌ 2026 Lin Xiaohei.

## §7.1 An Unresolved Open Question

The core of this framework contains a deliberate ambiguity that the author chooses not to resolve.

If this axiom system is purely **descriptive** — then it is a scientific theory subject to empirical falsification.
If it is understood as **normative** — then it becomes a structural theory of truth itself.

The author refuses to resolve this ambiguity. Not because it cannot be resolved. Because the ambiguity itself is productive — it forces each reader to decide for themselves.

The author's position on this question is not absent. It is withheld.

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© 2026 林小黑 (Lin Xiaohei). All rights reserved.
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https://gitee.com/samforce/structural-cognition