We extract a rigid, reproducible two-threshold RG skeleton from the Mittermeier RG Attractor and state it in a form that makes an explicit particle–vacuum bridge visible. The central object is the General Mittermeier Flow Equation: an algebraically closed rational beta function for β_λ(u) := -∂_u ln λ\u0302(u) in a monotone, dimensionless flow coordinate u ≥ 0, whose parameters depend only on α_M, π, and e.
\nThe attractor fixes the plastic constant ρ by ρ³ = ρ + 1 and the Mittermeier Planck–Boundary normalization:
\nκ_M := λ\u0302(0) = e^-(2 + ln ρ) = e\u207b² / ρ
\nα_M := κ_M / 14 = e\u207b² / (14ρ)
\nDefining the refined thresholds by q_M = 2e\u207b¹ / ρ and p_M = q_M / √π, the flow admits the exact ρ-free rewrite:
\nβ_λ(u) = 15/16 - [ 2e\u207b¹ / √π ] / [ 1 + (28e / √π · α_M)u ] + [ e\u207b² / (98α_M) ] / [ 1 + (28e · α_M)u ]
\nHere, the integer 28 = 2 × 14 is forced by the threshold sector after substituting ρ = e\u207b² / (14α_M), and 98 = 7 × 14 is forced by the fixed residue ratio 1/7 in the IR pole. This \"particle bridge\" is a change of variables, not a fit: the paper verifies pointwise identity between the canonical refined normal form and the fine-structure constant α_M form.
\nWorking directly with the general α_M–π–e form, we derive a compact invariant set:
\nAn exact u-plateau of 15/16.
\nStrict monotonicity β'_λ(u) > 0 for all u ≥ 0 (no overshoot).
\nA universal deep-IR approach law: β_λ(u) = 15/16 - C · u\u207b¹ + O(u\u207b²) with a closed coefficient C(α_M, π, e).
\nA unique analytic inflection point u_inf.
\nNumerical values at the closed constant: β_λ(0) ≈ 0.711638, C ≈ 0.983986, and u_inf ≈ 0.629160.
\nWith the SSOT chart, the plateau β_λ(u) → 15/16 implies λ\u0302(u) ~ e^-(15/16)u and therefore λ\u0302(k) ~ (k / M_Pl)^((15/16)α_RG). For the QG-closed/FRW-closure choice α_RG = 32/15, this exponent equals 2, i.e., λ\u0302(k) ~ (k / M_Pl)² as k → 0 (infrared).
\nUsing CODATA-2022 α\u2080\u207b¹ = 137.035999177(21), the measured coupling implies: κ_M / (14α\u2080) - 1 = -13.245 ppm
\nThis ppm proximity is slightly larger than the CODATA uncertainty on α\u2080 and is reported as a deterministic numerical pattern; deriving α\u2080 dynamically would require an action-level completion beyond this companion.
", "languages": [ { "id": "eng", "title": { "en": "English" } } ], "publication_date": "2026-02-24", "publisher": "Zenodo", "resource_type": { "id": "publication-preprint", "title": { "de": "Preprint", "en": "Preprint" } }, "rights": [ { "description": { "en": "This work is protected by copyright law. \u00a9 2026 Rainer Andreas Mittermeier.\nPublished under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0)." }, "link": "https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode.de", "title": { "en": "CC BY-NC-ND 4.0" } } ], "subjects": [ { "subject": "Renormalization group (RG) flow, beta function, two-threshold normal form, vacuum coupling, vacuum sector, attractor dynamics, closed-form parametrization, boundary constant, boundary normalization, fine-structure constant, particle-gravity bridge, metrology, CODATA, asymptotic scaling, UV tail, limiting-curvature, curvature bridge, FRW calibration, cosmology-quantum gravity interface, reproducibility, audit-stable numerics" } ], "title": "The General Mittermeier Flow Equation: A Closed Two-Threshold RG Normal Form from the Planck\u2013Boundary Constant to an \u03b1\u2013\u03c0 Vacuum\u2013Particle Bridge" }, "parent": { "access": { "owned_by": { "user": "1316751" }, "settings": { "accept_conditions_text": null, "allow_guest_requests": false, "allow_user_requests": false, "secret_link_expiration": 0 } }, "communities": {}, "id": "18749267", "pids": { "doi": { "client": "datacite", "identifier": "10.5281/zenodo.18749267", "provider": "datacite" } } }, "pids": { "doi": { "client": "datacite", "identifier": "10.5281/zenodo.18749268", "provider": "datacite" }, "oai": { "identifier": "oai:zenodo.org:18749268", "provider": "oai" } }, "revision_id": 5, "stats": { "all_versions": { "data_volume": 95987160.0, "downloads": 94, "unique_downloads": 52, "unique_views": 79, "views": 110 }, "this_version": { "data_volume": 95987160.0, "downloads": 94, "unique_downloads": 52, "unique_views": 79, "views": 110 } }, "status": "published", "swh": {}, "updated": "2026-03-15T20:09:34.285616+00:00", "versions": { "index": 1, "is_latest": true } }