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Schrödinger as the Envelope Equation of a Gapped Electromagnetic Carrier

Year: 2026
Type: preprint
Abstract: 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).
DOI: 10.5281/zenodo.21534577
Language: English
Open Access status: green
OpenAlex ID: W7170846381
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