Exact Response Monoids and Sharp Renormalization-Group Bounds
An effective axion coupling may satisfy global quantization and still have no source in a proposed heavy-particle sector. This paper determines which responses exist, how much gauge running their realization requires, and what further information is needed to distinguish their sources.
For , the three matched heavy gauge coefficients form an additive monoid: every allowed response is a sum of basic integer vectors with nonnegative integer multiplicities.
The classification covers arbitrary finite sums of coherent complex Dirac representations. An analytic reduction and a finite integer certificate establish completeness without a cutoff on representation dimension. Unrelated bare counterterms are not part of the matched heavy response.
Allowing coherently wound Majorana masses in real representations gives six irredundant generators. The globally allowed vector (3, 0, 0) has no coherent Dirac realization, but a real adjoint Majorana field supplies it. The vector (2, 1, 6) remains excluded in both coherent classes.
Four times any nonnegative globally allowed vector has a coherent Dirac realization; twice the vector suffices with real Majorana masses. These uniform factors are optimal. Coupling ratios alone therefore cannot reproduce the absolute-response exclusions.
Opposite winding signs can cancel anomaly coefficients, but their contributions to one-loop gauge running add. The joint criterion accounts exactly for that cost.
Here k is the matched anomaly triple, and S contains the unsigned heavy-matter indices. The symbol denotes the Dirac or Dirac–Majorana monoid; membership includes integrality and nonnegativity. The criterion assumes primitive mass windings +1 or −1, active thresholds, ordinary hypercharge normalization, and subtraction of other charged-field contributions.
For any anomaly vector, at most 70 Dirac candidates—or nine with real Majorana masses—give every minimal nonnegative weighted running cost and every simultaneous upper-budget test. Each frontier point is attained by a finite heavy sector.
For k = (3, 0, 0), the minimal Dirac indices are (3, 0, 72), (7, 0, 48), and (27, 0, 0). The middle point can minimize a combined colour–hypercharge cost even when neither endpoint does.
Two explicit low-dimensional sectors have the same anomaly and one-loop running but different two-loop gauge matrices. Replacing particles by response generators does not preserve higher-order evolution, decays, or cosmology.
For the matched response (17, −15, −354), at least one heavy threshold must lie above the bound below, under the specified ultraviolet validity and correction assumptions.
The comparison requires gauge couplings αi ≤ 0.12 through 1018 GeV, the stated heavy-Yukawa matrix-norm bounds, and continuous threshold matching without heavy–light mixing. Under the same assumptions, the componentwise triangle test gives approximately 3.10 × 107 GeV. The enlarged Standard Model input rectangle is a sensitivity set, not a statistical confidence region.
A favorable endpoint shift of one in each inverse coupling weakens the bound to 2.09 × 108 GeV. These allowances quantify conditional robustness; they do not calculate unknown matching corrections or higher orders.
At a heavy-mass ceiling of 108 GeV, 11,861 oriented responses pass global quantization and the triangle test. The exact condition removes a further 112 for Dirac matter and 54 when real Majorana masses are allowed. The improvement is limited but nonzero. The axion response is hypothetical; no observed axion signal is assumed.
Existence is the first test, not a complete viability judgment.
A representation-resolved catalogue contains 19,612 one- and two-multiplet spectra. Their distinct two-loop matrices are retained. Gauge-only data bound the coherent charged content; an additional gravitational coefficient permits the stated irreducible-source and finite-spectrum reconstruction results.
Specified examples connect particle production and reheating to entropy dilution, baryon preservation, primordial nuclei, and scalar stability. A radiative-relic calculation includes endpoint-resolved cascades and conditional nuclear constraints. Full neutrino collision/flavour transport, a joint CMB likelihood, and the complete quantum effective action remain outside those calculations.
The central response theorems do not require CHC. The compact-CHC construction is a separate application with additional assumptions. Neither a unique fundamental spectrum nor complete cosmological survival follows from the classification.
The source package contains the finite proof certificate, exact arithmetic tools, numerical programs, pinned inputs and original licenses, and five executed notebooks.
After extracting the source archive, run these commands from Axion_Completions_Source with Python 3.10 or later. These exact-response examples use only the standard library.
python3 scripts/anomaly_tomography.py gauge 3 0 0
python3 scripts/anomaly_tomography.py gauge 3 0 0 --majorana
python3 scripts/anomaly_tomography.py frontier 3 0 0 --weights 1 0 1/4The first command rejects a coherent Dirac source. The second permits a real adjoint Majorana source. The third gives the complete Dirac frontier and its minimum for the selected weights. RESULTS_GUIDE.md explains input conventions; README.md lists the separate dependencies and commands for the cosmological calculations.
The main text develops classification, running, and reconstruction. Appendix A gives the proof certificate; Appendices B–H contain spectrum and cosmology calculations; Appendix I treats compact CHC; Appendix J describes computational checks. PDF bookmarks provide direct navigation.