Scale-Dependent Data-Conditioned Inference for the Inflationary Consistency Relation from Decoherence-Induced Occupancy

Author: Raman Marozau · ORCID: 0009-0000-0241-1135 · Independent Researcher

Date: 2026-03-28

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Abstract

We show that a single Mukhanov–Sasaki/Bogoliubov computation, applied to the Theory of Everything (ToE) decoherence mechanism, simultaneously yields three linked data-conditioned inferences from one parameter set: (i) an infrared consistency-ratio deviation $Q(k) = c_s^\ast/(1+2\bar{n}_k) < 1$ at scales $k \lesssim k_0$, with mean IR deviation 6–70% depending on $k_0$ (peak deviation up to 97% at the lowest $k$); (ii) ring-down oscillations in the scalar power spectrum with phase-stable behavior in the amplitude-relevant region (serving as an internal consistency check); and (iii) non-Gaussianity suppression by the same occupancy factor ($R_{f_{NL}}(k) = Q(k)$). The standard inflationary limit $Q = 1$ is recovered at the pivot scale $k = 0.05$ Mpc$^{-1}$ as a null test. Using public BICEP/Keck 2018 + Planck 2018 + BAO chains (1,948,224 samples, $r = 0.0163 \pm 0.0101$), we demonstrate that 75 out of 125 parameter combinations (60%) satisfy all three channels simultaneously, including the manuscript reference point ($k_0 = 0.002$, $\varepsilon_H = 0.01$, $\Gamma/H = 5$). This is a robust data-conditioned inference within the tested domain, not a detection, and is testable with future low-$\ell$ B-mode constraints on independently measured $n_t$.

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1. The Claim

In the tested ToE parameter domain ($k_0 \geq 0.002$ Mpc$^{-1}$, $\varepsilon_H = 0.001$–$0.05$, $\Gamma/H = 1$–$20$), a single Mukhanov–Sasaki/Bogoliubov computation simultaneously yields three linked data-conditioned inferences from one parameter set:

(i) An IR consistency-ratio deviation $Q(k) < 1$ (~6%–70%, scale-dependent);

(ii) Ring-down oscillations with phase-stable behavior in the amplitude-relevant region (internal consistency check — passes for all 125 points, confirming phase coherence but not independently constraining the parameter space);

(iii) Non-Gaussianity suppression by the same occupancy factor ($R_{f_{NL}}(k) = Q(k)$).

The standard limit $Q = 1$ is recovered at the pivot scale. This is falsifiable with future low-$\ell$ B-mode measurements.

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2. What Is New

• One solver, three channels. All three observables ($Q$, ring-down, $f_{NL}$) are computed from a single call to the Mukhanov–Sasaki solver with Bogoliubov matching at $\eta_0$. They are not independent fits — they are three consequences of one mechanism.

• Phase-metric correction. The ring-down phase $\phi_k = \arg(\alpha_k \beta_k^\ast)$ is evaluated only where observationally relevant (amplitude-weighted mask, $>1\%$ of peak). This removes branch-cut artifacts at high $k$ where $|\beta_k| \to 0$.

• Manuscript point passes all three channels. At $k_0 = 0.002$, $\varepsilon_H = 0.01$, $\Gamma/H = 5.0$: Channel 1 (Q) PASS, Channel 2 (ring-down) PASS, Channel 3 ($f_{NL}$) PASS.

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3. Data

The tensor spectral index $n_t$ is not free in these chains. This is precisely the assumption being tested: the ToE predicts $n_t \neq -r/8$ at low $k$.

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4. Method

4.1 ToE Parameters

These parameters are not present in BK18 chains. They define the decoherence mechanism:

4.2 Computation Pipeline

For each parameter point $(k_0, \varepsilon_H, \Gamma/H)$:

1. Instantiate ToETheoryErrorEval (from src/toe_decoherence_validation/toe_theory.py).
2. Call _compute_ms_on_sparse_grid(k_grid, eta_0, c_s, eps_H, eta_H, s, Gamma_over_H).
3. This single call returns: $\bar{n}_k$, $\phi_k$, $\theta_k$, $A_\text{ring}$, $r_k$ at each $k$.
4. From these, compute:
   • Channel 1: $Q(k) = c_s^\ast / (1 + 2\bar{n}_k)$
   • Channel 2: $A_\text{ring}(k)$, $\phi_k(k)$ (with amplitude-weighted phase mask)
   • Channel 3: $R_{f_{NL}}(k) = 1/(1 + 2\bar{n}_k) = Q(k)/c_s^\ast$

All three channels emerge from the same Bogoliubov coefficient $\beta_k$. No additional fitting.

4.3 Channel Pass Criteria

Phase weight: $W(k) = A_\text{ring}(k) \cdot (1 + 2\bar{n}_k)$. Points with $W < 1\%$ of maximum are masked (phase undefined where $|\beta_k| \to 0$).

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5. Results

5.1 Main Quantitative Result: IR Deviation from $Q = 1$

At the manuscript reference point ($k_0 = 0.002$, $\varepsilon_H = 0.01$, $\Gamma/H = 5.0$):

Mean IR deviation ($k \leq 0.002$): 12.3%. Pivot: $Q = 0.9999999975$ ($\approx 1$ to 9 decimal places).

5.2 Joint-Channel Consistency

Scan: 5 values of $k_0$ × 5 values of $\varepsilon_H$ × 5 values of $\Gamma/H$ = 125 points.

The 75 feasible points span $k_0 \in \{0.002, 0.005, 0.01\}$ at all tested $\varepsilon_H$ and $\Gamma/H$. The 50 non-feasible points have $k_0 \in \{0.0005, 0.001\}$ where the mean IR deviation is below the 5% threshold (individual $k$-point deviations reach 6–7%, but the mean across the IR window $k \leq 0.002$ does not meet the criterion due to the coarse $k$-grid in that region).

Note on Channel 2: Ring-down passes for all 125 points (100%). After implementing the gauge-robust complex-phase metric ($\Delta\phi_k = \arg(u_{k+1} u_k^\ast)$ with amplitude-weighted edges), the phase is smooth wherever observationally relevant (phase score 0.98 at manuscript point, weighted $|\Delta\phi| = 0.067$ rad, well below $\pi/2$ threshold). This channel serves as an internal consistency check confirming that the MS solver produces physically coherent phase behavior. The feasible region is determined by Channels 1 and 3; Channel 2 confirms phase coherence across the entire tested domain.

Phase metric robustness: at the manuscript point, 4 of 7 $k$-points are significant (obs_weight $> 1\%$ of max). The remaining 3 points ($k \geq 0.02$) have $|\beta_k| \to 0$ and are correctly masked as phase-undefined.

Manuscript point ($k_0 = 0.002$, $\varepsilon_H = 0.01$, $\Gamma/H = 5.0$): ALL THREE PASS.

5.3 Data-Conditioned Inference of ToE-Induced Deviation

This section reports a data-conditioned inference of ToE-induced deviation, not a standalone prediction and not a detection.

Operationally, we took the observed BK18 posterior values of $r$ (from chains with $n_t = -r/8$), propagated them through the Mukhanov–Sasaki solver with fixed ToE parameters, and computed $\Delta_Q(k) \equiv Q_\text{ToE}(k;\, r_\text{BK18}) - 1$.

Therefore, what is inferred here is the ToE-implied departure from the consistency baseline $Q = 1$ conditioned on real BK18 data for $r$. What is not done here is a direct data measurement of $Q(k)$ (or free-$n_t$ inference of $n_t$) from polarization data; in BK18, $n_t$ is fixed, so the chains encode $Q = 1$ by construction.

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6. Null Test: Pivot Scale

At $k = 0.05$ Mpc$^{-1}$ (Planck pivot), the mode is deep sub-horizon at $\eta_0$ ($k/k_0 = 25$ for manuscript $k_0$). The Bogoliubov coefficient $\beta_k \to 0$, giving $\bar{n}_k \approx 10^{-9}$ and $Q = 1.000000000$.

This is physically expected: decoherence at $\eta_0$ does not affect modes that are deep sub-horizon at that time. The ToE reduces to standard inflation at the pivot scale.

This null test is passed for all 125 parameter points in the scan.

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7. Robustness

7.1 Sensitivity to $\varepsilon_H$

The consistency ratio $Q(k)$ is nearly independent of $\varepsilon_H$ in the tested range (0.001–0.05). At fixed $k_0 = 0.002$:

Variation: $< 1$ percentage point across a factor-50 range in $\varepsilon_H$. The effect is controlled by $k_0$ (horizon geometry at $\eta_0$), not by slow-roll details.

7.2 Sensitivity to $\Gamma/H$

$\Gamma/H$ does not affect $Q(k)$ or $f_{NL}$ suppression (identical values for all $\Gamma/H$ at fixed $k_0$, $\varepsilon_H$). It affects only the ring-down damping rate: larger $\Gamma/H$ suppresses oscillation amplitude faster.

7.3 $k_0$ as the Key Parameter

$k_0$ determines the scale and amplitude of the ToE effect:

The ToE-implied deviation is a family of curves parameterized by $k_0$. Constraining $k_0$ from data is the key next step.

7.4 Quasi-Invariant $Q(k_0) \approx 0.94$

At $k = k_0$ (the decoherence scale), $Q(k_0) \approx 0.94$ across all tested $\varepsilon_H$ values. This is a candidate structural invariant of the matching prescription, arising because $\bar{n}_k(k_0)$ is determined by horizon geometry at $\eta_0$, not by slow-roll parameters.

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8. Falsification Criteria

8.1 Confirmation

The ToE receives support if, in an analysis with free $n_t$ and low-$\ell$ B-mode data:

1. The posterior prefers $Q(k) < 1$ at $k \lesssim k_0$ with $> 3\sigma$ significance;
2. The scale dependence matches the ToE-implied form (stronger in IR, vanishing at pivot);
3. The three channels ($Q$, ring-down, $f_{NL}$) are jointly consistent from one parameter set.

8.2 Refutation

The ToE is refuted if data with free $n_t$ yield:

1. $Q(k) = 1$ at low $k$ within errors incompatible with the ToE-implied 6–70%;
2. No scale-dependent enhancement toward the IR;
3. Robustness under marginalization over ToE parameters.

8.3 Inconclusive

If uncertainties on $n_t$ at low $k$ exceed the ToE-implied $\Delta n_t$, the result is non-discriminating.

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9. Limitations

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10. Reproducibility

All code and data are open:

# Extract BK18 chains
tar xzf modeling/chains_no_data_files.tar.gz -C /tmp/bk18_chains/

# Single-point evaluation (manuscript parameters)
python src/toe_decoherence_validation/evaluate_bk18.py

# Sensitivity map (k0 × eps_H scan)
python src/toe_decoherence_validation/evaluate_bk18_map.py

# Joint analysis (three channels, 125 points)
python src/toe_decoherence_validation/joint_analysis.py

Physics implementation: src/toe_decoherence_validation/toe_theory.py, imports MS solver from src/toe_decoherence_validation/mukhanov_sasaki.py.

All plots generated automatically in plots/.

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Appendix A: Phase Metric Definition

The ring-down phase $\phi_k = \arg(\alpha_k \beta_k^\ast)$ is physically meaningful only where the Bogoliubov coefficient $|\beta_k|$ is nonzero. At high $k$ ($k \gg k_0$), $|\beta_k| \to 0$ and the phase is numerically undefined.

The gauge-robust complex-phase weighted metric (implemented per companion recommendation):

1. Construct unit complex phase vector: $u_k = e^{i\phi_k}$
2. Compute local phase step: $\Delta\phi_k = \arg(u_{k+1} \cdot u_k^\ast)$ — this is gauge-invariant and removes branch-cut artifacts
3. Compute observational weight: $W(k) = A_\text{ring}(k) \cdot (1 + 2\bar{n}_k)$
4. Mask points where $W(k) < 0.01 \cdot \max(W)$ (phase observationally irrelevant)
5. Compute edge weights: $w_\text{edge} = \frac{1}{2}(W_{k} + W_{k+1})$, normalized to sum to 1
6. Weighted absolute phase variation: $\langle|\Delta\phi|\rangle_w = \sum w_\text{edge} \cdot |\Delta\phi_k|$
7. Phase score: $S_\phi = \text{clip}(1 - \langle|\Delta\phi|\rangle_w / \pi,\; 0,\; 1)$
8. Pass criterion: $\langle|\Delta\phi|\rangle_w < \pi/2$
9. If fewer than 2 significant points: status = undetermined (not pass)

At the manuscript point: $S_\phi = 0.98$, $\langle|\Delta\phi|\rangle_w = 0.067$ rad, 4 significant points.

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Figures

• Figure 1: Three channels at manuscript point ($k_0 = 0.002$, $\varepsilon_H = 0.01$, $\Gamma/H = 5$). Left: $Q(k)$ and $R_{f_{NL}}(k)$. Right: $A_\text{ring}(k)$ and $\phi_k$ (unwrapped, masked). → plots/joint_three_channels.png

• Figure 2: Sensitivity heatmap: $k_0 \times \varepsilon_H \to \max(1-Q)$. → plots/sensitivity_max_deviation.png

• Figure 3: Joint feasibility map: number of channels passing (0–3) at each $(k_0, \varepsilon_H)$. → plots/joint_feasibility_map.png

• Figure 4: Ring-down amplitude $A_\text{ring}(k)$ vs $\Gamma/H$. → plots/ringdown_vs_gamma.png

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References

1. R. Marozau, "A Theory of Everything from Internal Decoherence, Entanglement-Sourced Stress–Energy, Geometry as an Equation of State of Entanglement, and Emergent Gauge Symmetries from Branch Algebra" (manuscript).

2. P. A. R. Ade, Z. Ahmed, M. Amiri, D. Barkats, R. Basu Thakur, C. A. Bischoff, D. Beck, J. J. Bock, H. Boenish, E. Bullock et al. (BICEP/Keck Collaboration), "Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season," Phys. Rev. Lett. 127, 151301 (2021). doi:10.1103/PhysRevLett.127.151301

3. N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak et al. (Planck Collaboration), "Planck 2018 results. VI. Cosmological parameters," Astron. Astrophys. 641, A6 (2020). doi:10.1051/0004-6361/201833910

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Document generated from src/toe_decoherence_validation/ pipeline using BK18 public chains (NASA LAMBDA) and ToE physics from src/toe_decoherence_validation/toe_theory.py. All results reproducible via commands in Section 10.