# Two-Mode Jaynes–Cummings Model of Endogenous and Exogenous Phase Coherence **Complete Zenodo record with historical development** **DOI:** [10.5281/zenodo.22904994](https://doi.org/10.5281/zenodo.22904994) **Authors:** Miguel Angel Percudani (corresponding: miguel_percudani@yahoo.com.ar, ORCID [0009-0007-1748-3212](https://orcid.org/0009-0007-1748-3212)) & Jorge Iván Díaz **Affiliation:** Independent Research Collaboration **Publication date:** 22 September 2026 **Related records:**- Binary Universe Ontology: [10.5281/zenodo.18015439](https://doi.org/10.5281/zenodo.18015439) (2025)- UAT framework: [10.5281/zenodo.17886549](https://doi.org/10.5281/zenodo.17886549) (2025)- Antifrequency datasets: [10.5281/zenodo.17861265](https://doi.org/10.5281/zenodo.17861265) (2025) --- ## 1. Central paper — read this first > **`track_A_formal.pdf`** (746 kB, 7 pages) > *A Two-Mode Jaynes–Cummings Model of Endogenous and Exogenous Phase Coherence: Falsifiable Predictions for Neural Dynamics* This is the **formal, falsifiable Track A**. The model contains **no geometric constants** ($\Phi_{\text{UAT}}$, $k_{\text{UAT}}$, $\kappa_{\text{crit}}$). The coupling strength $g_0$ is a free parameter to be fitted to EEG/fMRI data. **This PDF is the one to cite.** All other files in this record are complementary or supplementary. --- ## 2. Two-track structure The record is organized in two independent tracks: ### Track A — Formal, falsifiable, testable now A two-mode Jaynes–Cummings model with Lindblad dissipation and Default Mode Network modulation. The model produces a two-dimensional phase diagram in the plane $(\alpha = g/\delta,\ \gamma_\phi)$ and four falsifiable predictions for EEG/fMRI experiments. All parameters are free and listed in the paper. **Track A remains valid even if Track B is refuted.** Documents:- **`track_A_formal.pdf`** (7 pages) — main paper, 4 falsifiable predictions, phase diagram- **`track_A_field_equations.pdf`** (15 pages) — technical appendix: complete field equations, Hilbert space, operator algebra, Lindblad dissipator, master equation, reduced dynamics, BDF numerical scheme- **`track_A_Extended_version_full_derivation.pdf`** (20 pages) — extended appendix that **supersedes** the previous one: it contains all the material of `track_A_field_equations.pdf` plus three new sections with full step-by-step derivations: - (i) mean-field closure of the reduced dynamics - (ii) expansion of the density matrix in the Jaynes–Cummings eigenbasis (dressed states) - (iii) linear stability analysis of the Liouvillian superoperator and spectral gap $\Delta$ Both appendix versions are provided for convenience. **The extended version is the recommended reference.** ### Track B — Speculative research program An ontological interpretation of Track A in terms of the Binary Universe Ontology (Bit 1 = analytic qubit, Bit 0 = substrate oscillator, causal friction = Jaynes–Cummings exchange, homeostasis = Lindblad dissipation, DMN = baryonic filter). Contains the constants $\Phi_{\text{UAT}} = \pi^2/8$, $k_{\text{UAT}} \approx 10.90677$, $\kappa_{\text{crit}} \approx 10^{-78}$ **explicitly labeled as speculative hypotheses, not derived results.** Document:- **`track_B_speculative.pdf`** (6 pages) — *Binary Universe Ontology as an Interpretation of Two-Mode Coupling: The Geometric Origin of $k_{\text{UAT}}$ as a Speculative Research Program* The two tracks are independent. Track A does not use any of the constants from Track B. Track B does not modify the numerical results of Track A. --- ## 3. Figures (stand-alone PNGs for reuse) - **`v5_run1_rabi.png`** (130 kB) — Rabi oscillation: $\langle \sigma_z \rangle$, $\langle n_0 \rangle$, $S_{vN}$, purity for $g = 17$ Hz, 10 periods- **`v6_scan2D.png`** (150 kB) — Phase diagram in $(\alpha = g/\delta,\ \gamma_\phi)$ plane, four steady-state observables- **`v6_cortes1D.png`** (173 kB) — One-dimensional cuts at $\gamma_\phi \in \{0.09, 0.44, 2.26, 11.60\}$ Hz --- ## 4. LaTeX sources (for reproducibility) - `LaTex_track_A_formal.tex` (16 kB) — source of the central paper- `LaTex_track_A_field_equations.tex` (31 kB) — source of the 15-page field-equations appendix- `LaTex_track_A_Extended_version_full_derivation.tex` (45 kB) — source of the 20-page extended appendix- `LaTex_track_B_speculative.tex` (17 kB) — source of the speculative interpretation All sources compile with `pdflatex` (two passes required for cross-references). No external dependencies beyond the standard TeX Live distribution. --- ## 5. Code archives (historical development) The numerical results were developed over six versions. Each `trackA_vN.zip` contains the scripts and data as they were at that stage. Historical versions are included for transparency; **only v5 and v6 are needed to reproduce the published results.** ### Historical / supplementary - **`trackA_v2.zip`** (326 kB) — v2: manual RK4 integrator, initial dark state $|0,0\rangle$ bug, crashes for $\gamma_\phi \gtrsim 10$ due to stiffness- **`trackA_v3.zip`** (426 kB) — v3: fix with BDF implicit solver (`scipy.integrate.solve_ivp`), first stable runs- **`trackA_v4.zip`** (630 kB) — v4: first sweep in $(\alpha, \gamma_\phi)$ plane- **`trackA_v4_1.zip`** (643 kB) — v4.1: JC coupling corrected (standard Jaynes–Cummings, not anti-JC) ### Final — use these to reproduce - **`trackA_v5.zip`** (1,626 kB) — **IMPORTANT**: Rabi runs and endogen/exogen contrast at equal Rabi period. Generated `v5_run1_rabi.png`.- **`trackA_v6.zip`** (510 kB) — **IMPORTANT**: Final 2D scan and 1D cuts, phase diagram, boundary $\alpha_{\text{cruce}}(\gamma_\phi)$, empirical scaling $g_{\text{cruce}} \cdot \gamma_\phi^{0.62} \approx 3.95$. **This is the version used in `track_A_formal.pdf`.** --- ## 6. How to reproduce the central result ### Numerical ```bash# Main result (phase diagram + Rabi)unzip trackA_v6.zip && cd trackA_v6python -m venv venv && source venv/bin/activatepip install numpy scipy pandas matplotlibpython main.py # BDF solver, rtol=1e-7, atol=1e-9# Outputs: v6_scan2D.png, v6_cortes1D.png ## 12. Update: intrinsic transition measures (v8.1) In September 2026, the numerical exploration was extended with v8.1, which replaces the threshold-based definition of the "coexistence window" used in earlier versions (v7.x) with threshold-free intrinsic measures based on the product P(t) = R(t) · I(t). Key result: two distinct coexistence regimes emerge: - Vigilance: slow, weak, sustained (P_max = 0.006, W_half = 46, eta = 0.87) - Flow / Endogenous / Sleep: fast, intense, brief (P_max ≈ 0.013-0.016, W_half ≈ 8-15, eta ≈ 0.24-0.38) The trade-off between peak intensity and effective duration is quantified. See trackA_v8_1.zip for code, data, and figures. Licencia: CC BY 4.0 para manuscritos, MIT para código.
Two-Mode Jaynes–Cummings Model of Endogenous and Exogenous Phase Coherence Complete Zenodo record with historical developmentDOI:10.5281/zenodo.22904994Authors: Miguel Angel Percudani (corresponding: miguel_percudani@yahoo.com.ar, ORCID 0009-0007-1748-3212) & Jorge Iván Díaz Affiliation: Independent Research Collaboration Publication date: 22 September 2026 Related records:- Binary Universe Ontology: 10.5281/zenodo.18015439 (2025)- UAT framework: 10.5281/zenodo.17886549 (2025)- Antifrequency datasets: 10.5281/zenodo.17861265 (2025) --- ## 1. Central paper — read this first > track_A_formal.pdf (746 kB, 7 pages) > A Two-Mode Jaynes–Cummings Model of Endogenous and Exogenous Phase Coherence: Falsifiable Predictions for Neural Dynamics This is the formal, falsifiable Track A. The model contains no geometric constants (ΦUAT, kUAT, κcrit). The coupling strength g0 is a free parameter to be fitted to EEG/fMRI data. This PDF is the one to cite. All other files in this record are complementary or supplementary. --- ## 2. Two-track structure The record is organized in two independent tracks: ### Track A — Formal, falsifiable, testable now A two-mode Jaynes–Cummings model with Lindblad dissipation and Default Mode Network modulation. The model produces a two-dimensional phase diagram in the plane (α=g/δ,γϕ) and four falsifiable predictions for EEG/fMRI experiments. All parameters are free and listed in the paper. Track A remains valid even if Track B is refuted. Documents:- track_A_formal.pdf (7 pages) — main paper, 4 falsifiable predictions, phase diagram- track_A_field_equations.pdf (15 pages) — technical appendix: complete field equations, Hilbert space, operator algebra, Lindblad dissipator, master equation, reduced dynamics, BDF numerical scheme- track_A_Extended_version_full_derivation.pdf (20 pages) — extended appendix that supersedes the previous one: it contains all the material of track_A_field_equations.pdf plus three new sections with full step-by-step derivations: - (i) mean-field closure of the reduced dynamics - (ii) expansion of the density matrix in the Jaynes–Cummings eigenbasis (dressed states) - (iii) linear stability analysis of the Liouvillian superoperator and spectral gap Δ Both appendix versions are provided for convenience. The extended version is the recommended reference. ### Track B — Speculative research program An ontological interpretation of Track A in terms of the Binary Universe Ontology (Bit 1 = analytic qubit, Bit 0 = substrate oscillator, causal friction = Jaynes–Cummings exchange, homeostasis = Lindblad dissipation, DMN = baryonic filter). Contains the constants ΦUAT=π2/8, kUAT≈10.90677, κcrit≈10−78explicitly labeled as speculative hypotheses, not derived results. Document:- track_B_speculative.pdf (6 pages) — Binary Universe Ontology as an Interpretation of Two-Mode Coupling: The Geometric Origin of kUAT as a Speculative Research Program The two tracks are independent. Track A does not use any of the constants from Track B. Track B does not modify the numerical results of Track A. --- ## 3. Figures (stand-alone PNGs for reuse) - v5_run1_rabi.png (130 kB) — Rabi oscillation: ⟨σz⟩, ⟨n0⟩, SvN, purity for g=17 Hz, 10 periods- v6_scan2D.png (150 kB) — Phase diagram in (α=g/δ,γϕ) plane, four steady-state observables- v6_cortes1D.png (173 kB) — One-dimensional cuts at γϕ∈{0.09,0.44,2.26,11.60} Hz --- ## 4. LaTeX sources (for reproducibility) - LaTex_track_A_formal.tex (16 kB) — source of the central paper- LaTex_track_A_field_equations.tex (31 kB) — source of the 15-page field-equations appendix- LaTex_track_A_Extended_version_full_derivation.tex (45 kB) — source of the 20-page extended appendix- LaTex_track_B_speculative.tex (17 kB) — source of the speculative interpretation All sources compile with pdflatex (two passes required for cross-references). No external dependencies beyond the standard TeX Live distribution. --- ## 5. Code archives (historical development) The numerical results were developed over six versions. Each trackA_vN.zip contains the scripts and data as they were at that stage. Historical versions are included for transparency; only v5 and v6 are needed to reproduce the published results. ### Historical / supplementary - trackA_v2.zip (326 kB) — v2: manual RK4 integrator, initial dark state ∣0,0⟩ bug, crashes for γϕ≳10 due to stiffness- trackA_v3.zip (426 kB) — v3: fix with BDF implicit solver (scipy.integrate.solve_ivp), first stable runs- trackA_v4.zip (630 kB) — v4: first sweep in (α,γϕ) plane- trackA_v4_1.zip (643 kB) — v4.1: JC coupling corrected (standard Jaynes–Cummings, not anti-JC) ### Final — use these to reproduce - trackA_v5.zip (1,626 kB) — IMPORTANT: Rabi runs and endogen/exogen contrast at equal Rabi period. Generated v5_run1_rabi.png.- trackA_v6.zip (510 kB) — IMPORTANT: Final 2D scan and 1D cuts, phase diagram, boundary αcruce(γϕ), empirical scaling gcruce⋅γϕ0.62≈3.95. This is the version used in track_A_formal.pdf. --- ## 6. How to reproduce the central result ### Numerical ```bash# Main result (phase diagram + Rabi)unzip trackA_v6.zip && cd trackA_v6python -m venv venv && source venv/bin/activatepip install numpy scipy pandas matplotlibpython main.py # BDF solver, rtol=1e-7, atol=1e-9# Outputs: v6_scan2D.png, v6_cortes1D.png ## 12. Update: intrinsic transition measures (v8.1) In September 2026, the numerical exploration was extended with v8.1, which replaces the threshold-based definition of the "coexistence window" used in earlier versions (v7.x) with threshold-free intrinsic measures based on the product P(t) = R(t) · I(t). Key result: two distinct coexistence regimes emerge: - Vigilance: slow, weak, sustained (P_max = 0.006, W_half = 46, eta = 0.87) - Flow / Endogenous / Sleep: fast, intense, brief (P_max ≈ 0.013-0.016, W_half ≈ 8-15, eta ≈ 0.24-0.38) The trade-off between peak intensity and effective duration is quantified. See trackA_v8_1.zip for code, data, and figures. Licencia: CC BY 4.0 para manuscritos, MIT para código.
This record contains the complete development of a two-mode Jaynes–Cummings model of endogenous and exogenous phase coherence, organized in two independent tracks. TRACK A — FORMAL AND FALSIFIABLE A two-mode quantum model in which an analytic sequential mode (a qubit) interacts with a non-analytic substrate mode (a truncated harmonic oscillator) through a time-dependent Jaynes–Cummings coupling with Lindblad dissipation. The coupling is modulated by an empirically accessible signal identified with Default Mode Network (DMN) activity. The model contains no speculative constants: all parameters are free and listed in the main paper. Numerical integration with an implicit BDF solver produces a two-dimensional phase diagram in the plane (α = g/δ, γφ), where the boundary between analytic-dominated and substrate-dominated steady states is a curve rather than a single threshold. The paper states four falsifiable predictions testable with current EEG/fMRI techniques. Track A documents: • track_A_formal.pdf — main paper, 7 pages • track_A_field_equations.pdf — technical appendix, 15 pages • track_A_Extended_version_full_derivation.pdf — extended appendix, 20 pages, supersedes the previous one and adds full derivations of the mean-field closure, the dressed-state expansion, and the linear stability of the Liouvillian superoperator TRACK B — SPECULATIVE RESEARCH PROGRAM An ontological interpretation of the same two-mode structure in terms of the Binary Universe Ontology: Bit 1 = analytic qubit (baryonic, finite, sequential), Bit 0 = substrate oscillator (non-baryonic, infinite, superposed); Jaynes–Cummings exchange = causal friction; Lindblad dissipation = entropic homeostasis; DMN = baryonic filter. The document presents the geometric constants Φ_UAT = π²/8, kUAT ≈ 10.90677, and κcrit ≈ 10⁻⁷⁸ explicitly as speculative hypotheses, not as derived results. Track B documents: • track_B_speculative.pdf — 6 pages INDEPENDENCE OF THE TWO TRACKS Track A does not use any constant from Track B. Track B does not modify the numerical results of Track A. Track A remains valid even if Track B is refuted. RECORD CONTENTS • 4 PDF documents (main paper, two technical appendices, speculative interpretation) • 4 LaTeX sources, compilable with pdflatex • 3 standalone figures in PNG format (Rabi oscillation, 2D phase diagram, 1D cuts) • 6 code archives (trackA_v2 through trackA_v6) documenting the full historical development, with v5 and v6 being the versions that generated the published results NOTES The 20 fMRI sessions of Agustina Vélez Picatto (led by Pablo Barttfeld, IIPsi — CONICET / Universidad Nacional de Córdoba) are cited as compatibility, not validation, as of 22 September 2026. No peer-reviewed DOI has been issued yet. Reproducibility: all results can be regenerated from trackA_v6.zip using standard Python scientific libraries (numpy, scipy, pandas, matplotlib) and the BDF implicit solver with rtol = 1e-7, atol = 1e-9. Compilation of the LaTeX sources requires two pdflatex passes for cross-references. License: CC BY 4.0 for manuscripts, MIT for code.
This internal working note documents the results of the Camino 1 program, an extension of the v8.1-3D analysis of the two-mode Jaynes-Cummings model with Lindblad dissipation. It is a **methodological and phenomenological** contribution: it uses the Jaynes-Cummings formalism as an effective language for two coexisting dynamical modes, and it does not claim that the underlying system is quantum. ## What is formal and verifiable The model is the same two-mode Jaynes-Cummings system with Lindblad dissipation used in v8.1-3D, integrated with BDF. Two coexistence proxies are analyzed: - Rmath(t)=∣r(t)∣, the Bloch vector length of the reduced qubit (coherence proxy) - Rbio(t)=Tr(ρ(t)∣1⟩⟨1∣), the qubit excited population (population proxy) ## What is phenomenological The proxies are phenomenological simplifications of coherence and population. No claim is made that they correspond to specific physical quantities in a real system. ## Main findings 1. **Single-exponential validity**: The single-exponential description of Rbio(t) is valid only at intermediate coupling (g≈4–8 Hz). Outside this region, the fit fails or is a lower bound. 2. **Two-exponential structure**: In the failed regime (g≳15 Hz), a two-exponential model is strongly preferred by AIC in 43 out of 45 cells. The two timescales are separated by a factor of ∼150. 3. **Negative result on naive scaling**: The predictions T1∼1/ΩR and T2∼1/γtotal are **not** confirmed. The two timescales depend on the full parameter triplet (g,γϕ,γa). 4. **Purcell formula**: Among four theoretical models (Fermi golden rule, Purcell, dressed-state, Bloch-Redfield), the **Purcell formula** is the best approximation for T1 (median ratio 0.68). 5. **No simple form for coherence**: No simple functional form (single exponential, stretched exponential, power law, double exponential) fits Rmath(t) well. All have mean R2 in the range 0.31–0.36. 6. **Proxies are complementary**: The two proxies are uncorrelated in value (mean corr≈+0.09) but almost perfectly correlated in their time derivatives (mean corr≈+0.91). The value correlation changes sign with the regime. Both proxies are needed for a complete characterization. 7. **Fock-space convergence**: The observables are identical for N0∈[12,20] (relative change 0.00%). The results are fully converged at N0=12. ## Contents of this release The LaTeX technical note and the associated CSV files with the fit results, regime classification, and proxy comparison. The generating scripts are provided in the companion release of v8.1-3D. ## Scope disclaimer No empirical dataset is analyzed or claimed in this note. The purpose is methodological: to document a reproducible numerical procedure and its regime of validity. The failure of the naive scaling and the lack of a simple functional form for Rmath are reported honestly as negative results, which motivate further theoretical analysis.
This v8.1 update of Track A presents a threshold-free characterization of coexistence dynamics in the two-mode Jaynes-Cummings model with Lindblad dissipation, solved using BDF. It replaces the earlier versions v3, v4, v4.1, v5, and v6, and supersedes the intermediate versions v7, v7.1, and v7.3. Formal content includes: the model (Hilbert space, operators, Hamiltonian, Lindblad dissipator, master equation), the BDF numerical scheme, and the definitions of the intrinsic measures of the coexistence product P(t)=R(t)·I(t): A_coex, P_max, t_peak, W_1/2, tau_eff, eta, and the peak_reached flag. Informative content includes the numerical values used for reproducibility: N0=15, t_max=50, n_steps=3000, omega0=6 Hz, omega1=40 Hz, gamma_a=0.1 Hz, n_th=0.05, and four parameter points labeled Vigilance, Flow, Endogenous, and Sleep. The speculative content (labeling R as Reason and I as Instinct, and mapping the four points to consciousness states) is explicitly presented as a heuristic ontology, not as a validated neuroscience claim. Main results: with t_max=50 all peaks are reached, fixing the v8 artifact where the Vigilance bio peak was not reached. Vigilance is characterized as slow access (t_peak,bio=22.9, W_1/2,bio=45.9, tau_bio=42.5). Flow shows the highest P_max,math=0.318. Endogenous and Sleep have similar P_max,bio (~0.0155) but differ in tail width (W_1/2,bio=9.15 vs 8.18). In the 2D scan, the optima of A_coex and tau_eff do not coincide, showing a trade-off between integrated action and duration. This update contains only the v8.1 files plus this technical note: v8.1_summary.csv, v8.1_traj_.csv (4 files), v8.1_products.png, v8.1_bars.png, v8.1_scan2D_.csv (optional), track_A_v8.1_intrinsic.py, and this note. Versions v7, v7.1, and v7.3 are intentionally omitted to avoid confusion. DOI: 10.5281/zenodo.22904994
Manuscript submitted to Atoms (MDPI) on 30 September 2026 (manuscript atoms-4632260), with all code and data needed to reproduce its numerical results. Files: the manuscript (PDF, version 3.9), its LaTeX source with the ten figures (pauli_exclusion_wall_atoms_submission.zip), the supplementary material File S1 (pauli_exclusion_wall_supplementary.zip: the two-dimensional Q-ball scripts P1, P1b/P1c, P2 and P4 with their machine-readable outputs; the three-dimensional isorotating-cell analysis scripts and run reports, including the free-amplitude finite-element solver; the fully three-dimensional fixed-isospin pair relaxation; and the two-triad relaxation solver and reports, in phase to 3200 steps and out of phase to 4000 steps), and README.md, which describes the contents. The isorotating-cell technical note that the paper cites is a separate record, doi:10.5281/zenodo.23036586. This version completes the file set of the record first published on 30 September 2026, whose first version lacked the supplementary archive.
我们为二纠缠的原子 A 和 B 局部地与独立的洞交往的一个双 Jaynes-Cummings 模型学习纠纷动力学 and b，分别地但是在地点 Aa 和 Bb 之间没有相互作用。我们指出那在那里存在在的一个参数范围同时的 pairwise 纠纷为遥远的聚会的突然的死亡(ESD ) ，即 AB， ab， Ab 和 Ba，可以发生。我们证明在这个同时的 ESD 时期期间，各种各样的多国参加的纠纷能被创造。我们的结果暗示在二个独立地点之间的 pairwise 纠纷可以被变成说明在二个独立地点之间的关联的另外的多国参加的形式。