The slowly varying envelope (SVE) reduction of a gapped classical electromagnetic carrier predicts the single-particle Schrödinger equation as an exact identity plus a closed-form residual: the constrained vector MacCullagh carrier admits an exact per-branch balance whose retained part is the Schrödinger operator and whose sole marked approximation leaves a closed-form remainder, quadratic in the detuning. Under the single identification below, the carrier branch ℏω(k) is the relativistic energy function E(p) = m2 c4 + c2 p2 and the envelope generator ℏ[ω(k) − ωC ] is p its kinetic part E(p) − mc2 , recovered consistently in three cross-checking registers—the Foldy– Wouthuysen operator, the dispersion Taylor series, and direct numerics—with leading correction the fine-structure term −p4 /8m3 c2 . A parameter-free model reproduces the archived residual slopes to 10−4 , the fitted exponent converging to the predicted 2. One action-scale identification, ℏωC = mc2 , enters exactly once as an input; by Buckingham-π the reduction fixes the coefficient c2 /ωC = ω ′′ (0), matched to ℏ/m under the identification, and leaves one free scale, so ℏ is supplied, not derived. The result is a dynamical correspondence at the envelope stratum, not a derivation of quantum mechanics: |ψ|2 is normalized wave action, never a Born probability, and an explicit ledger fences what the correspondence forces, permits, and does not produce—the Born rule, state reduction, quantized charge, and entanglement—each with its standing theorem or structural fact. Companion no-go paper: "Constraining Classical Field Models of Self-Generated Particle Structure: Successive No-Go Theorems", DOI 10.5281/zenodo.21531712 (all versions). The two papers are co-posted and cross-cite by concept DOI. This record additionally carries the complete Track S/N/P code-and-data archive (envelope-code-v3.35.zip): symbolic-audit, numerical, and magnitude pipelines with their test suites (106, 20, and 13 tests), all figure-generating scripts, the campaign manifests (track_n_campaign_summary.json and companions), and dependency floors (code/requirements.txt). Version 2 (September 2026). Retitled ("a Gapped Electromagnetic Carrier") and substantially revised; 49 → 31 pages. Principal changes since v1: (i) the reduction theorem now carries explicit hypotheses — ψ₀ ∈ H^{s+4} via a bounded-multiplier argument, with a band-limit alternative and the H^s-weighted packet functional K_{4,s} (eighth-moment form) replacing the earlier spectral-scale shorthand; (ii) the residual analysis adds the finite-sweep bias law with a parameter-free model reproducing all archived window slopes to 1e-4, guarded by a committed test file; (iii) Sections V and VI are merged and the four scope tables consolidated into a single honesty-ledger table with explicit class codes; (iv) the Wolff placement is rebuilt as a three-horn trilemma mirroring the companion paper; (v) the conclusion is rewritten, situating the result against the independent kinematic (Zhang) and intrinsic-periodicity (Puddu) routes; (vi) the effective-medium reading of the gapped carrier is stated as a constant-coefficient effective field theory with no substrate assumed or determined; (vii) the carrier branch and the envelope generator are distinguished throughout, the companion dependence is stated results-based, and the long-time figure marks the Gaussian-packet breakdown estimate; (viii) all printed record DOIs now use the Zenodo concept (all-versions) DOIs. A complete change ledger (paper1_delta_v3.13_to_v3.35.pdf) accompany this version, together with the complete Track S/N/P code and data archive (envelope-code-v3.35.zip).
The slowly varying envelope (SVE) reduction of a gapped classical electromagnetic carrier predicts the single-particle Schrödinger equation as an exact identity plus a closed-form residual: the constrained vector MacCullagh carrier admits an exact per-branch balance whose retained part is the Schrödinger operator and whose sole marked approximation leaves a closed-form remainder, quadratic in the detuning. Under the single identification below, the carrier branch ℏω(k) is the relativistic energy function E(p) = m2 c4 + c2 p2 and the envelope generator ℏ[ω(k) − ωC ] is p its kinetic part E(p) − mc2 , recovered consistently in three cross-checking registers—the Foldy– Wouthuysen operator, the dispersion Taylor series, and direct numerics—with leading correction the fine-structure term −p4 /8m3 c2 . A parameter-free model reproduces the archived residual slopes to 10−4 , the fitted exponent converging to the predicted 2. One action-scale identification, ℏωC = mc2 , enters exactly once as an input; by Buckingham-π the reduction fixes the coefficient c2 /ωC = ω ′′ (0), matched to ℏ/m under the identification, and leaves one free scale, so ℏ is supplied, not derived. The result is a dynamical correspondence at the envelope stratum, not a derivation of quantum mechanics: |ψ|2 is normalized wave action, never a Born probability, and an explicit ledger fences what the correspondence forces, permits, and does not produce—the Born rule, state reduction, quantized charge, and entanglement—each with its standing theorem or structural fact. Companion no-go paper: "Constraining Classical Field Models of Self-Generated Particle Structure: Successive No-Go Theorems", DOI 10.5281/zenodo.21531712 (all versions). The two papers are co-posted and cross-cite by concept DOI. This record additionally carries the complete Track S/N/P code-and-data archive (envelope-code-v3.35.zip): symbolic-audit, numerical, and magnitude pipelines with their test suites (106, 20, and 13 tests), all figure-generating scripts, the campaign manifests (track_n_campaign_summary.json and companions), and dependency floors (code/requirements.txt). Version 2 (September 2026). Retitled ("a Gapped Electromagnetic Carrier") and substantially revised; 49 → 31 pages. Principal changes since v1: (i) the reduction theorem now carries explicit hypotheses — ψ₀ ∈ H^{s+4} via a bounded-multiplier argument, with a band-limit alternative and the H^s-weighted packet functional K_{4,s} (eighth-moment form) replacing the earlier spectral-scale shorthand; (ii) the residual analysis adds the finite-sweep bias law with a parameter-free model reproducing all archived window slopes to 1e-4, guarded by a committed test file; (iii) Sections V and VI are merged and the four scope tables consolidated into a single honesty-ledger table with explicit class codes; (iv) the Wolff placement is rebuilt as a three-horn trilemma mirroring the companion paper; (v) the conclusion is rewritten, situating the result against the independent kinematic (Zhang) and intrinsic-periodicity (Puddu) routes; (vi) the effective-medium reading of the gapped carrier is stated as a constant-coefficient effective field theory with no substrate assumed or determined; (vii) the carrier branch and the envelope generator are distinguished throughout, the companion dependence is stated results-based, and the long-time figure marks the Gaussian-packet breakdown estimate; (viii) all printed record DOIs now use the Zenodo concept (all-versions) DOIs. A complete change ledger (paper1_delta_v3.13_to_v3.35.pdf) accompany this version, together with the complete Track S/N/P code and data archive (envelope-code-v3.35.zip).
This record contains Version 2.0 of The Schrödinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background. Version 2.0 supersedes the earlier narrower local formulation (DOI: 10.5281/zenodo.19118007). The earlier record worked within a restricted quadratic-gradient density class. The present record proves the same result on the full local first-order homogeneous density class, with the weighted bulk-plus-flux bridge as the load-bearing structural principle rather than a minimality assumption on derivative order. The theorem lane is therefore broader, the claim boundaries are cleaner, and the companion-note dependence has been updated to reflect the current SGOC corpus. This record DOI. 10.5281/zenodo.19189901 should be treated as the source-of-record citation for the general fixed-background Schrödinger result. The paper proves a narrow but exact claim: within the full first-order homogeneous density class compatible with continuity-current structure, the Fisher-corrected Madelung action is the unique admissible action family up to the stated constant, and its Euler–Lagrange equations are exactly the Schrödinger equation. The decisive density correction is forced inside an upstream capacity-geometric theorem base rather than inserted by a diffusion or stochastic postulate. The logical scaffold is explicit and limited. Part 1A fixes the canonical logarithmic capacity scalar and the unique quadratic base tension. Parts 6 and 7A isolate the phase-blindness ceiling for the continuous quadratic sector. Part 6A sharpens the localisation-versus-smoothing and forced-balance framework. The present paper then shows that continuity-current compatibility fixes the phase sector, locality and first-order dependence force the full homogeneous density family, and a universal weighted bulk-plus-flux bridge with field-independent coefficients forces the Fisher form. The canonical member of that family is fixed by the bridge to the imported capacity scalar. In that precise sense the logical order at the Schrödinger layer is reversed: the geometric base comes first, and the Schrödinger action appears downstream of it. The claim is exact within its lane and no broader. This is a fixed-background, single-particle theorem stated away from nodal sets. It does not claim a many-body theory, a measurement theory, a relativistic completion, a full nodal-complement reconstruction theorem, or backreaction of the quantum sector on the background geometry. Appendix D records that on a nodal-complement component, once the phase-sensitive completion is an S¹-valued phase field, circulation quantisation and wavefunction reconstruction are automatic. Appendix G records a conditional H¹-closure lane inside the same fixed-background action. The discrete-curvature and branch-bookkeeping support used on nodal-complement triangulations belongs to the companion Closure Phase note, while the further complement-globalisation step belongs to the companion Relative Cohomology note. Keywords: Schrödinger equation; capacity geometry; Fisher information; Madelung action; phase-sensitive evolution; forced action; homogeneous density class; weighted Fisher bridge; REG; coherence scalar; fixed background; nodal complement; reproducibility Companion manuscripts cited in this paper's proof chain, from the SGOC programme: Part 1A: The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge. DOI: 10.5281/zenodo.19156316 Part 7A: The Second-Order Ceiling, Triadic Completion, and the Wilson-Triangle Lift. DOI: 10.5281/zenodo.19175902 Closure Phase as Discrete Curvature on a Triangulated 2-Complex. DOI: 10.5281/zenodo.19185885 Relative Cohomology and Cut-Graph Holonomy Trivialisation. DOI: 10.5281/zenodo.19185877 Additional provenance notes cited in the paper Part 6A: Localisation Rather than Saturation in Capacity Geometry: REG Dissipation, Forced Balance Laws, and a Restricted Ricci Bridge. DOI: 10.5281/zenodo.19174434 Landauer, Relative Entropy, and Fisher Length in Capacity Geometry. DOI: 10.5281/zenodo.19186309
Introduction The Scretching–Schrödinger Equation (SSE) is developed as a quantum-mechanical framework that imposes bidirectional spectroscopic closure between the wavefunction and the measured spectrum, ψ⟷spectrum,\psi \longleftrightarrow \mathrm{spectrum},ψ⟷spectrum, through the Scretching Quantum Chain (SQC) invariant, Kν=7.42165600145×10−22 s.K_{\nu}=7.42165600145\times10^{-22}\ \mathrm{s}.Kν=7.42165600145×10−22 s. In its original non-relativistic form, the SSE modifies the ordinary Schrödinger framework by requiring that the wavefunction not only solve an eigenvalue problem, but also satisfy a deterministic spectroscopic closure condition. This condition links transition matrix elements, oscillator strengths, Einstein coefficients, absorption parameters, and measured transition frequencies. The result is a constrained quantum theory in which ψ\psiψ generates spectra, while experimental spectra feed back as admissibility conditions on ψ\psiψ. This paper systematically recasts the SSE across the full hierarchy of modern quantum theory: ordinary quantum mechanics, relativistic quantum mechanics, quantum electrodynamics, electroweak theory, and quantum chromodynamics. The objective is to show that the SSE closure principle can be expressed not merely as a correction to non-relativistic wave mechanics, but as a generalized spectroscopic constraint compatible with the major operator, field, and gauge-theoretic structures of twentieth- and twenty-first-century physics. The first stage reformulates the SSE within ordinary quantum mechanics in both the Schrödinger wave-mechanical formalism and the Heisenberg matrix-mechanical formalism. In the Schrödinger picture, the SSE is written as a constrained wave equation, iℏ∂ψ∂t=(H^0+V^SQC)ψ,i\hbar\frac{\partial \psi}{\partial t} = \left(\hat{H}_0+\hat{V}_{\mathrm{SQC}}\right)\psi,iℏ∂t∂ψ=(H^0+V^SQC)ψ, where H^0\hat{H}_0H^0 is the ordinary Hamiltonian and V^SQC\hat{V}_{\mathrm{SQC}}V^SQC is the spectroscopic closure operator. In the Heisenberg picture, the same closure is recast into operator evolution and matrix-element constraints, showing that the SSE condition is independent of representation. The wavefunction, operator algebra, transition amplitude, and spectrum therefore become mutually constrained expressions of the same physical closure. The second stage extends the SSE into relativistic quantum mechanics (RQM) through the Klein–Gordon and Dirac equations. The Klein–Gordon–SSE form imposes SQC closure on relativistic scalar fields, while the Dirac–SSE form imposes the same closure on four-component spinor states. These extensions preserve relativistic covariance, introduce proper mass-shell structure, and connect the SSE to spin, antimatter, fine structure, and relativistic transition amplitudes. In this form, the SSE becomes a spectroscopic constraint on relativistic wave equations rather than a purely non-relativistic correction. The third stage recasts the framework within quantum electrodynamics (QED). Here the SSE closure principle is applied to photon exchange, electron self-energy, vertex corrections, Ward identity structure, and the running electromagnetic coupling. The QED–SSE formulation treats the SQC closure functional as a constraint on radiative processes, linking Feynman amplitudes to spectroscopic observables. The photon propagator, electron propagator, interaction vertex, and renormalized coupling are therefore interpreted not only as field-theoretic objects, but also as carriers of measurable spectroscopic closure. The fourth stage extends the formalism into the Glashow–Salam–Weinberg (GSW) electroweak theory, whose gauge group is SU(2)L×U(1)Y.SU(2)_L\times U(1)_Y.SU(2)L×U(1)Y. In this setting, the SSE closure principle is applied to weak and electromagnetic transitions after spontaneous symmetry breaking. The framework is developed through the electroweak gauge fields, Higgs mechanism, weak mixing angle, and the resulting W±W^{\pm}W±, Z0Z^0Z0, and photon fields. The masses of the weak bosons and the structure of charged and neutral currents are incorporated into an SSE weak-transition closure law, allowing the SQC framework to be tested against electroweak-scale processes. The fifth stage applies the SSE to quantum chromodynamics (QCD), based on the color gauge group SU(3)c.SU(3)_c.SU(3)c. Because QCD includes gluon self-interaction, asymptotic freedom, confinement, and non-Abelian color dynamics, it provides the most demanding field-theoretic extension of the SSE. The QCD–SSE formulation introduces a color-closure functional designed to connect quark and gluon transition amplitudes with spectroscopic or hadronic observables. In this context, the SSE is not presented as a replacement for QCD, but as a proposed closure constraint that may organize measurable color-sector transitions, confinement-scale behavior, and hadronic spectra. For each theoretical level, this paper derives a corresponding SSE-extended equation, lists the relevant Feynman rules, presents representative Feynman diagrams, and provides numerical examples with verified solutions. The analysis traces the mathematical progression from wave mechanics to matrix mechanics, from non-relativistic quantum theory to relativistic wave equations, from single-particle dynamics to QED radiative corrections, from Abelian gauge theory to electroweak unification, and finally from electroweak theory to the non-Abelian color structure of QCD. The primary textbook sources abstracted for the gauge-theory and particle-physics development are Aitchison and Hey’s Gauge Theories in Particle Physics, Volumes 1 and 2, including the third and fourth edition treatments of quantum electrodynamics, electroweak theory, and quantum chromodynamics [1,2]. All numerical constants used in the worked examples are taken from CODATA 2022, ensuring that the calculations are anchored to contemporary reference values. The central claim advanced in this paper is that the SSE may be written as a representation-independent and scale-extendable spectroscopic closure principle. In ordinary quantum mechanics, it constrains wavefunctions and operator matrix elements. In RQM, it constrains relativistic scalar and spinor amplitudes. In QED, it constrains radiative amplitudes and renormalized electromagnetic interactions. In GSW theory, it constrains weak and electroweak transitions. In QCD, it proposes a color-sector closure functional for non-Abelian gauge dynamics. Thus, the SSE is developed as a candidate bridge between measured spectra, quantum amplitudes, and gauge-field theory across the full hierarchy of modern quantum physics.
# 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.
Important status notice: This version has been superseded by v2.0. Subsequent work shows that the single-domain three-diagonal construction used here produces an experimentally excluded common-mode photon-speed anisotropy. Readers should consult the revised version and 10.5281/zenodo.21464971. We propose that conservation of probability — the single axiom A=1 — is sufficient to derive the principal results of quantum mechanics without additional postulates. On a bipartite three-dimensional causal lattice T³_diamond, each particle is represented as a causal session: an independent phase oscillator that propagates amplitude across the lattice subject to |ψ_R|² + |ψ_L|² = 1 at every tick. No Hamiltonian and no measurement postulate are assumed; a Hilbert-space structure with unitary evolution emerges from the U(1) oscillator and the per-session amplitude constraint rather than being postulated a priori. From this single constraint, the paper derives: (i) the Dirac equation in the continuum limit a→0, with the gamma matrices identified as geometric encodings of the bipartite RGB/CMY sublattice geometry; (ii) wave-particle duality and the Born rule as consequences of amplitude redistribution under A=1, with Gleason's theorem securing the |ψ|² uniqueness from the framework's emergent Hilbert space; (iii) quantization of atomic orbitals as Arnold tongue attractors of the two-body phase dynamics, recovering the hydrogen Bohr radius to four significant figures (k_min = 0.0970 vs k_Bohr = 0.0971) without fitting; (iv) gravity as a clock-density gradient rather than spacetime curvature — a refraction of causal paths through regions of higher session density, reproducing the 1/r² force law with an explicit weak-field metric tensor g_μν as a functional of the clock-density field; (v) photon emission as a structural consequence of A=1: the Coulomb interaction is a probability source that a single session cannot accommodate, mandating creation of a new photon session at orbit lock-in; (vi) the structural form of the induced gauge action (Sakharov / Zeldovich), whose leading-order term has the Maxwell form on O_h-averaged observables, with a closed-form gauge-sector birefringence sharing the same optical axis (1,1,−1) as the kinematic-sector prediction. Energy and momentum conservation are not independent inputs — they are consequences of A=1 applied to closed session sets. The numerically tractable results are verified or partially verified by a suite of more than fifteen experiments with no free parameters beyond the Planck-scale calibration of the lattice spacing; an audit table records the status of every claim. The paper conjectures that the Standard Model gauge group SU(3) × SU(2) × U(1), together with Lorentz SO(3,1), is the automorphism group of the framework's extended per-site amplitude — a precise 18-dimensional Lie-algebra equality. The existing framework is shown to carry SO(3,1) × U(1) (dim 7) on the per-site (ψ_R, ψ_L) amplitude; the remaining factors require structural extensions (per-site C² isospin and C³ colour indices) that are the subject of a companion paper. Six computational scaffolding scripts establish the conjecture's components. This is Paper I of the A=1 Discrete Causal Lattice series. Companion papers in preparation address the Standard Model gauge derivation, the Farey-sequence mass spectrum (and its Veneziano-amplitude limiting case), plasma and gradient-ionization phenomenology, the proton-interior structure, quantum measurement theory, and a Hilbert-6th-style combinatorial foundation for continuum physics. What changed in v1.0 (since v0.98-RC): * Three new audit-table rows: long-horizon escape mechanism (PART), lock-in resonance grid-independence across {65, 81, 97, 113}³ grids (PART), and the Standard Model gauge-group conjecture as a precise 18-dimensional Lie-algebra equality (STUB).* A Gleason-uniqueness subsubsection in the Born-rule discussion, addressing the necessity question for the |ψ|² measure.* An explicit metric-tensor formula in the gravity section, identifying g_μν as a functional of ρ_clock in the weak-field limit.* Refined automorphism conjecture with explicit decomposition into established and structural-extension factors.* New Acknowledgments section. Source code, paper sources, and data files: https://github.com/JackDMenendez/discrete-causal-latticeLicense: Paper text and figures CC BY 4.0; source code MIT.
## Version 2 - Corrected and Updated This is a corrected and updated version of the Scretching-Schrödinger Equation (SSE) framework. Version 2 incorporates the following improvements based on NIST Atomic Spectra Database (ASD) 2024 data: ### Key Corrections:- **Hydrogen**: Updated to f = 0.4162 (1s→2p Lyman-α multiplet) with correct degeneracy g₁/g₂ = 2/6- **Lithium**: Updated to full multiplet oscillator strength f = 0.7416 (2s→2p)- **Beryllium**: Updated to f = 1.340 (NIST ASD)- **Units**: Transition dipole moments |μ|² now correctly reported in C²m²- **K_SSE**: Corrected to 7.502×10⁻¹³ m²·s- **N_valence**: Corrected formula: N_val = (4.32×10⁻⁹)·Σ(ε_i·Δν̄_i) ### Results:The framework now achieves:- H (Lyman-α): 0.05% agreement with NIST A₂₁- He: 0.02% agreement- Li: 0.67% agreement- Be: 9.9% (within NIST uncertainty rating C, ±25%)- B: 0.08% agreement- C: 12.2% (consistent with multi-configuration complexity) ### AbstractBackground: One hundred years after Schrödinger's wave equation, a fundamental disconnect persists between quantum mechanical predictions and experimental spectroscopy. Wavefunctions are calculated first, then compared to spectra, with no mechanism for experimental data to constrain the solutions. Objective: We present the corrected Scratching-Schrödinger Equation (SSE), a reformulation that incorporates experimental UV-Vis absorption data as deterministic constraints within the wave equation. Methods: Starting from the Scratching Quantum Chain (SQC) relations linking oscillator strength f to transition dipole moment |μ|², and the Maxwell-Scratching Chain (MSC) connecting molar absorptivity ε to f, we derive a constrained Schrödinger equation with a V_SQC term enforcing exact agreement with experimental transition dipoles. Results: Applied to the first six elements (H, He, Li, Be, B, C) with corrected NIST ASD input data, the SSE reproduces experimental transition probabilities with high precision and identifies sources of uncertainty in complex cases. Conclusion: A century after Schrödinger's breakthrough, the SSE framework establishes a deterministic connection between wave mechanics and experimental spectroscopy. ### Related Identifiers- **Previous version**: 10.5281/zenodo.18805791
## Version 2 - Corrected and Updated This is a corrected and updated version of the Scretching-Schrödinger Equation (SSE) framework. Version 2 incorporates the following improvements based on NIST Atomic Spectra Database (ASD) 2024 data: ### Key Corrections:- **Hydrogen**: Updated to f = 0.4162 (1s→2p Lyman-α multiplet) with correct degeneracy g₁/g₂ = 2/6- **Lithium**: Updated to full multiplet oscillator strength f = 0.7416 (2s→2p)- **Beryllium**: Updated to f = 1.340 (NIST ASD)- **Units**: Transition dipole moments |μ|² now correctly reported in C²m²- **K_SSE**: Corrected to 7.502×10⁻¹³ m²·s- **N_valence**: Corrected formula: N_val = (4.32×10⁻⁹)·Σ(ε_i·Δν̄_i) ### Results:The framework now achieves:- H (Lyman-α): 0.05% agreement with NIST A₂₁- He: 0.02% agreement- Li: 0.67% agreement- Be: 9.9% (within NIST uncertainty rating C, ±25%)- B: 0.08% agreement- C: 12.2% (consistent with multi-configuration complexity) ### AbstractBackground: One hundred years after Schrödinger's wave equation, a fundamental disconnect persists between quantum mechanical predictions and experimental spectroscopy. Wavefunctions are calculated first, then compared to spectra, with no mechanism for experimental data to constrain the solutions. Objective: We present the corrected Scratching-Schrödinger Equation (SSE), a reformulation that incorporates experimental UV-Vis absorption data as deterministic constraints within the wave equation. Methods: Starting from the Scratching Quantum Chain (SQC) relations linking oscillator strength f to transition dipole moment |μ|², and the Maxwell-Scratching Chain (MSC) connecting molar absorptivity ε to f, we derive a constrained Schrödinger equation with a V_SQC term enforcing exact agreement with experimental transition dipoles. Results: Applied to the first six elements (H, He, Li, Be, B, C) with corrected NIST ASD input data, the SSE reproduces experimental transition probabilities with high precision and identifies sources of uncertainty in complex cases. Conclusion: A century after Schrödinger's breakthrough, the SSE framework establishes a deterministic connection between wave mechanics and experimental spectroscopy. ### Related Identifiers- **Previous version**: 10.5281/zenodo.18805791