# When the observation window changes the result

**Graviton · Research Note 02 · 27 October 2026**

Computational research note. Finite quantum model. No experimental data. Not peer reviewed.

## For the curious reader

> A result can be correct and still depend on when you look.

Imagine comparing two recordings of a repeating signal. One looks quieter during the first interval, then louder during the next. Neither recording has changed; your observation window has. Our quantum calculation has this problem: adding an interaction can reduce a particular average in one interval and increase it in another. We publish the full comparison so readers can examine how much the conclusion depends on that choice.

Here, “when we look” means the interval we choose to analyze. Changing that interval does not alter the simulated state or its evolution.

This study concerns a small, specified quantum system. It does not simulate a physical wormhole, measure consciousness, or establish a new law of gravity.

## Abstract

Graviton's earlier finite-model calculation found a lower mean absolute Majorana autocorrelation after adding a specified quartic interaction, at three positive coupling strengths and within the observation window 12–24. We retain that diagnostic and centered RMS-energy matching while extending the calculation to all 30 quartic additions absent from the five-term baseline and constructed from its seven active Majorana operators. Six signed couplings and four observation windows produce 720 contrasts across 180 Hamiltonian settings. On the declared primary time grid, 130 settings have contrasts of both signs across the four windows. The original addition at coupling 0.2 has contrast −0.036111 in window 12–24 and +0.006218 in window 36–48; both signs persist on two finer grids. Six of the full set of 720 contrasts change sign under refinement, and this sensitivity is retained in the report. An independent matrix-exponential calculation agrees with the spectral calculation to within 1.5 × 10⁻¹⁴ at the specified check points. The earlier bounded result is reproduced, but its extension to an always-negative contrast fails within the tested family. These findings concern this particular diagnostic and do not determine thermalization, teleportation performance, or a gravitational interpretation.

## 1. Question and scope

The earlier calculation used the addition ψ₁ψ₂ψ₃ψ₅, couplings 0.1, 0.2 and 0.3, and the interval 12–24. An overall rescaling was removed by matching the centered RMS energies. That controls one possible explanation for the difference, but cannot establish that the same sign persists for different interactions or observation windows.

We ask whether the negative contrast extends throughout a stated finite family. A positive contrast is a counterexample to that extension, not a contradiction of the earlier restricted calculation.

The Hamiltonian is taken from the literature [1–3]. This is an exploratory extension of Graviton's existing calculation, not a blind validation, external preregistration, or priority claim. The protocol was saved before this sweep ran; the earlier reference result was already known. Every specified case is included.

## 2. Definitions

- **Hamiltonian:** the energy operator that generates time evolution. The matrices here are 16 × 16.
- **Majorana operator:** a Hermitian fermionic operator with the anticommutation normalization given below. These are mathematical matrices, not detected particles.
- **Quartic interaction:** a product of four distinct Majorana operators, ordered by increasing index.
- **Autocorrelation:** the normalized trace overlap between an operator and its later version. Here it starts at one.
- **Commuting operators:** operators whose product does not depend on order. Noncommutation alone does not establish thermalization.
- **Centered RMS energy:** the root-mean-square spread of energy eigenvalues about their mean. Matching it does not match an entire spectrum.
- **Observation window:** the interval used to compute the reported average. A finite system can revisit earlier patterns, making the window consequential.

## 3. Model and reproducible construction

Use four qubits, eight Majorana operators, and the conventions

\[
\{\psi_i,\psi_j\}=\delta_{ij}I_{16},\qquad
\rho=I_{16}/16,\qquad \beta=0,\qquad \hbar=1.
\]

For j = 1,…,4, with the first qubit as the leftmost factor,

\[
\psi_{2j-1}=\frac{Z^{\otimes(j-1)}\otimes X\otimes I^{\otimes(4-j)}}{\sqrt2},
\qquad
\psi_{2j}=\frac{Z^{\otimes(j-1)}\otimes Y\otimes I^{\otimes(4-j)}}{\sqrt2}.
\]

X, Y and Z are Pauli matrices. Define P_abcd = ψ_a ψ_b ψ_c ψ_d, with a < b < c < d. The five-term baseline printed in [2] is

\[
H_0=-0.36P_{1245}+0.19P_{1347}-0.71P_{1356}
     +0.22P_{2346}+0.49P_{2357}.
\]

This model is related to the experimental program in [1] and the discussion in [2,3]. Our state and observable do not reproduce that experiment's teleportation protocol.

Enumerate every increasing four-index subset of {1,…,7} except the five present in H₀: 30 additions. For each addition P, use

\[
\lambda\in\{-0.3,-0.2,-0.1,0.1,0.2,0.3\}.
\]

Set A = H₀ + λP and define

\[
\widetilde A=A-\frac{\operatorname{Tr}A}{16}I_{16},\qquad
r(A)=\sqrt{\frac{\operatorname{Tr}(\widetilde A^2)}{16}},\qquad
H_{P,\lambda}=\frac{r(H_0)}{r(A)}\widetilde A.
\]

All perturbed Hamiltonians therefore share H₀'s centered RMS energy. Spectral shape, conserved quantities and the relative interaction strengths can still differ. Neither the model's energy scale nor its time coordinate is calibrated to a laboratory system.

## 4. Observable and numerical method

For each Hamiltonian,

\[
G_i(t)=\frac{2}{16}\operatorname{Re}\operatorname{Tr}
\left[e^{+iHt}\psi_i e^{-iHt}\psi_i\right].
\]

The declared discrete statistic is

\[
D_W(H)=\frac{1}{7N_W}\sum_{i=1}^{7}\sum_{t_n\in W}|G_i(t_n)|,
\qquad
\Delta D_W=D_W(H_{P,\lambda})-D_W(H_0).
\]

The four windows are [0,12], [12,24], [24,36] and [36,48]. The primary step Δt = 0.1 gives 121 samples per window, including both endpoints. Shared boundary samples appear in each adjacent window. Refinements use Δt = 0.05 and 0.025. This is a discrete average, not a numerical claim about an exact continuous-time integral.

ψ₈ commutes with every Hamiltonian in this family and has G₈(t) = 1. It is checked as a conserved spectator and excluded from D. The seven active operators remain embedded in the full 16-dimensional representation.

For H = V diag(E) V†, the main calculation evaluates

\[
G_i(t)=\frac{2}{16}\sum_{a,b}
 |(V^\dagger\psi_iV)_{ab}|^2\cos[(E_a-E_b)t].
\]

A separate implementation of the matrix exponential uses a Taylor series with scaling and squaring. It checks every Hamiltonian at times 0, 6, 12, 18, 24, 36 and 48, without using the spectral expression for evolution. These checks test numerical implementation, not physical interpretation.

## 5. Results

All counts below refer to the declared primary grid.

| Window | Negative contrast | Positive contrast |
|---|---:|---:|
| 0–12 | 116 | 64 |
| 12–24 | 155 | 25 |
| 24–36 | 176 | 4 |
| 36–48 | 96 | 84 |

Of the 180 Hamiltonian settings, 130 have both positive and negative contrasts across the windows. On each finer grid this count is 132. The count is a description of the specified enumeration, not an estimated probability or a set of independent replications.

The original reference interaction P₁₂₃₅ at λ = 0.2 is particularly informative:

| Window | Baseline D | Perturbed D | Contrast ΔD | Finest-grid contrast |
|---|---:|---:|---:|---:|
| 0–12 | 0.372443 | 0.369375 | −0.003069 | −0.003159 |
| 12–24 | 0.188286 | 0.152175 | −0.036111 | −0.035806 |
| 24–36 | 0.415505 | 0.294993 | −0.120512 | −0.119893 |
| 36–48 | 0.249832 | 0.256050 | +0.006218 | +0.006914 |

Both signs survive refinement. Thus even this reference addition does not always reduce D.

Two additions, P₁₂₆₇ and P₄₅₆₇, commute with H₀. They remain as controls at all six couplings. The other 28 additions do not commute with H₀. Among the 168 noncommuting settings, the four windows respectively contain 52, 18, 4 and 74 positive contrasts. The counterexamples are therefore not confined to commuting controls.

## 6. Verification and grid sensitivity

The declared numerical verification tolerance is 10⁻¹⁰. Checks cover the Majorana algebra, baseline-term commutation, Hermiticity, initial normalization, RMS-energy matching, conservation of ψ₈, the unit bound on absolute correlations, and agreement of two evolution methods.

- Maximum spectral-versus-exponential difference at the specified points: 1.4988 × 10⁻¹⁴.
- Maximum difference from the three historical reference contrasts: 5.27 × 10⁻¹⁶.
- Maximum contrast shift between the primary and finest grids: 0.00198419885, or 0.001985 rounded upward.

Six of 720 contrasts change sign between primary and finest grids:

1. P₁₃₅₇ at λ = ±0.3, in windows 24–36 and 36–48: four contrasts.
2. P₂₃₅₆ at λ = ±0.3, in window 36–48: two contrasts.

The Δt = 0.05 and 0.025 grids agree on all contrast signs. The initial grid is retained, with the six sensitive results explicitly marked. Grid differences are not statistical error bars or solver failures, and do not provide a rigorous bound on a continuous-time average. The broad conclusion that the contrast need not be negative survives all three grids.

## 7. What the result does not establish

The models are related; some results coincide. No observations are randomly sampled and no sampling confidence intervals are appropriate. The exact finite enumeration is useful for testing the stated extension, but not for estimating how often a generic quantum system behaves this way.

D is not a teleportation fidelity, an out-of-time-order correlator, an entanglement entropy, a size-winding measure, or a thermalization rate. A smaller value within one interval does not uniquely identify any of those phenomena. RMS matching controls an overall energy scale but leaves many other differences between Hamiltonians intact. The calculation cannot settle the disagreement about a gravitational interpretation of the original experiment.

The historical Hamiltonian and normalization are credited to their sources. Graviton's contribution is this specified sweep, its numerical checks, and the interpretation of its limits. We make no priority claim for the diagnostic or for the general observation that conclusions in finite systems can depend on the observation window.

## 8. Next research step

For an equilibration study, first choose a subsystem, an initial-state family and a directly relevant local observable. Account for its stationary or conserved component, and select observation windows before inspecting results. A recent preprint [4] derives finite-window bounds for a different, stationary-subtracted mean-square diagnostic. Its assumptions would require a separate implementation; its bounds are not applied to D here.

For a gravitational interpretation, compare several independently specified observables against suitable nongravitational alternatives and investigate system-size dependence. The present note motivates that broader comparison but does not supply it.

## 9. Reproduction and downloadable artifacts

Keep `verify-robustness.py` beside `protocol.json` and run:

```sh
python3 -m pip install numpy
python3 verify-robustness.py
```

Python 3.10+ is recommended; the primary calculation used NumPy 2.3.5. The script writes `results.json` and `contrasts.csv`. To keep an existing output unchanged, use `--output-dir PATH`. If the earlier Graviton dataset is present at its repository location, the script additionally verifies the historical reference values; standalone numerical checks still run without that file.

For figures, keep `render-figures.py` beside `results.json`, install Matplotlib and run:

```sh
python3 -m pip install matplotlib
python3 render-figures.py
```

The figures are generated from the numerical output. The interactive website also reads that same JSON, rather than maintaining a separate numerical implementation.

Files include the protocol; the complete 180-case result with all three grids; a CSV of the 720 window comparisons; a reference trace and contrast figure in SVG and PNG; an SVG map of all 30 additions at λ = 0.2; and the plotting source. The JSON records the script and protocol hashes, software version and actual numerical errors. Those records identify the calculation behind this note.

## References

1. Jafferis, D. et al. (2022). *Traversable wormhole dynamics on a quantum processor.* Nature 612, 51–55. https://doi.org/10.1038/s41586-022-05424-3
2. Kobrin, B., Schuster, T. & Yao, N. Y. (2023). *Comment on “Traversable wormhole dynamics on a quantum processor”.* https://arxiv.org/abs/2302.07897
3. Jafferis, D. et al. (2023). *Response to the comment.* https://arxiv.org/abs/2303.15423
4. Ćwikliński, P. (2026). *Finite-time quantum equilibration from local correlations.* Preprint, version 2, 21 October 2026. https://arxiv.org/abs/2610.16351v2