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: - $R_{\text{math}}(t) = |\mathbf{r}(t)|$, the Bloch vector length of the reduced qubit (coherence proxy) - $R_{\text{bio}}(t) = \text{Tr}(\rho(t)\,|1\rangle\langle 1|)$, 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 $R_{\text{bio}}(t)$ is valid only at intermediate coupling ($g \approx 4$–$8$ Hz). Outside this region, the fit fails or is a lower bound. 2. **Two-exponential structure**: In the failed regime ($g \gtrsim 15$ 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 $\sim 150$. 3. **Negative result on naive scaling**: The predictions $T_1 \sim 1/\Omega_R$ and $T_2 \sim 1/\gamma_{\text{total}}$ are **not** confirmed. The two timescales depend on the full parameter triplet $(g, \gamma_\phi, \gamma_a)$. 4. **Purcell formula**: Among four theoretical models (Fermi golden rule, Purcell, dressed-state, Bloch-Redfield), the **Purcell formula** is the best approximation for $T_1$ (median ratio 0.68). 5. **No simple form for coherence**: No simple functional form (single exponential, stretched exponential, power law, double exponential) fits $R_{\text{math}}(t)$ well. All have mean $R^2$ in the range 0.31–0.36. 6. **Proxies are complementary**: The two proxies are uncorrelated in value (mean $\text{corr} \approx +0.09$) but almost perfectly correlated in their time derivatives (mean $\text{corr} \approx +0.91$). 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 $N_0 \in [12, 20]$ (relative change 0.00%). The results are fully converged at $N_0 = 12$. ## 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 $R_{\text{math}}$ are reported honestly as negative results, which motivate further theoretical analysis.
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 work presents a systematic numerical study of the dissipative Jaynes-Cummings model, consisting of a two-level system (qubit) coupled to a truncated harmonic oscillator, both subject to Markovian dissipation. The model is standard in quantum optics and open quantum systems. The master equation is solved using the Lindblad formalism with BDF integration. The study focuses on two standard observables: the qubit excited-state population P_exc(t) and the qubit Bloch vector length |r(t)|. A three-dimensional parameter scan is performed over the coupling g, the qubit dephasing rate gamma_phi, and the oscillator relaxation rate gamma_a, giving 216 cells in total. The parameters used are omega_0 = 6 Hz, omega_1 = 40 Hz, giving a detuning delta = 34 Hz, and thermal occupation n_th = 0.05. MAIN FINDINGS: The single-exponential description of the excited-state population is valid only at intermediate coupling (g approximately 4 to 8 Hz). Outside this region, the dynamics exhibits a multi-timescale structure. The decay time T is non-monotonic in g, with a minimum at g approximately 7 Hz. The location of the minimum depends only weakly on gamma_a, following a power law with exponent 0.127. At strong coupling (g greater than 15 Hz), a two-exponential model is strongly preferred by the Akaike Information Criterion in 43 out of 45 cells. The two timescales are separated by a factor of approximately 150. The fast timescale T_1 is best approximated by the Purcell formula (median ratio of numerical to theoretical value 0.68). The Fermi golden rule overestimates the decay rate by a factor of approximately 4.5. Neither the dressed-state correction nor the Bloch-Redfield calculation improves on the Purcell formula. The Bloch vector length does not follow any simple functional form (single exponential, stretched exponential, power law, or double exponential). All models have mean R-squared in the range 0.31 to 0.36. Its time derivative is, however, almost perfectly correlated with that of the population (mean correlation 0.91). The results are converged with respect to the Fock-space truncation at N_0 = 12, with identical observables for N_0 between 12 and 20. SCOPE: This is a purely numerical study of a well-defined quantum-optical model. It contains no empirical data, no claims about biological or cognitive systems, and no speculative ontology. The model, the numerical method, and the observables are standard. The results include several negative findings (the failure of perturbative formulas, the lack of a simple functional form for the Bloch vector) which are reported honestly and which identify the regime where non-perturbative methods are required. The purpose of this release is to provide reproducible numerical data and a benchmark for future analytical work on the dissipative Jaynes-Cummings model in the strong-coupling regime. CONTENTS: The LaTeX technical note (approximately 20 pages), the Python scripts for the numerical simulations, and the CSV files with the fit results, regime classification, and theoretical comparison. REFERENCES: E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).G. Lindblad, Commun. Math. Phys. 48, 119 (1976).H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).E. M. Purcell, Phys. Rev. 69, 681 (1946).C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Wiley (1992).W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH (2001).
# 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** (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≈10−78 **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 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 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
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 dataset and code repository accompanies the article of the same title (version 5, May 2026) and provides all simulation data, source code, and raw terminal output necessary to reproduce and verify the results. Study summary We investigate the critical dynamics of the two-dimensional Ising model at the exact critical point (Tc ≈ 2.269185) using the local Metropolis Monte Carlo algorithm. The central focus is the characterization and comparison of two widely used correlation time estimators: the exponential decay criterion τ₁/e and the integrated autocorrelation time τ_int, following the convention of Sokal (1997). Simulations were performed for seven system sizes L ∈ {16, 24, 32, 48, 64, 96, 128} with up to 50 independent replicas per system size, using both Python (NumPy/PCG64) and C++ (std::mt19937, compiled with -O2) implementations. The two implementations were developed independently, enabling cross-validation of results across languages. Main findings The coefficient of variation (CV = std/mean) of τ_int converges to a physical floor of approximately 0.44–0.52 regardless of the number of replicas (8 or 50) or programming language, confirming that this floor is determined by the physics of the estimator under local Metropolis dynamics at criticality — not by statistical sampling limitations. The distribution of τ₁/e is genuinely bimodal at the critical point: individual replicas alternate between low (~200–500 MC steps) and high (~3000–5000 MC steps) correlation times for the same system size, with mean-to-median ratios of up to 2.6× and bimodal ratios (mean of upper half / mean of lower half) of up to 23.5×. This bimodality does not disappear with more replicas, confirming it is a physical property of the estimator. No detectable power-law scaling τ ~ L^z is found for L ≤ 128 in any of the six independent runs (R² < 0.14 in all log-log fits using all 7 system sizes). The accepted asymptotic value z ≈ 2.17 (Li & Sokal, 1989) implies an expected τ_int of approximately 73,600 MC steps at L = 128 — 76× larger than the observed value of 972. This quantifies the dominance of finite-size corrections in the range studied, and justifies treating the effective dynamic exponent z_eff as a continuously varying quantity rather than a fixed constant. The C++ implementation achieves speedups of 98× to 782× relative to Python depending on system size, enabling the full 50-replica run across all seven system sizes to complete in approximately 33 minutes. Cross-validation caveat Cross-validation between Python and C++ is fully valid for τ₁/e. For τ_int, the Python scripts used a fixed equilibration of n_equil = 3000 steps for all L — documentably insufficient for L ≥ 24 where τ_int > 600 (n_equil/τ < 3–5τ). This generates systematic underestimation of up to 86% at L = 64. The valid cross-validation for τ_int is between the C++-8 and C++-50 runs, where CV agreement (~0.52 in both) confirms the physical floor is robust. Connection to the EIE/ZIE framework This study provides the empirical foundation for the EIE/ZIE adaptive meta-control framework (Patent BR 10 2025 024459 4), justifying three architectural decisions: (1) τ_int as the preferred estimator of the correlation time τ_corr; (2) the effective dynamic exponent z_eff treated as a continuously estimated dynamic variable rather than a fixed constant; and (3) the dynamic freezing time τ_freeze introduced as an observable independent of τ_corr, motivated by the bimodal behavior of τ₁/e which reveals that the system alternates between regimes of high and low correlation persistence — a phenomenon invisible to conventional autocorrelation-based diagnostics. Repository contents Cpp/ — C++ source code (3 scripts) and simulation data (6 CSV files + raw terminal output) Python/ — Python source code (2 scripts) and simulation data (4 CSV files + raw screen output) Paper/ — Article in English (PDF) and Portuguese (PDF) README.md — Full repository documentation including file descriptions, column definitions, reproduction instructions, and physical parameters LICENSE.txt — CC BY 4.0 for data; MIT for code; patent notice Joemerson da Silva Lima — Independent Researcher, Cuiabá, Mato Grosso, Brazil joemerson93lima@gmail.com Patent BR 10 2025 024459 4
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。