ArXe Theory: Falsifiable Predictions

A theory that cannot be wrong is not a theory. ArXe’s central claim — that physical reality emerges from the recursive resolution of a primordial logical contradiction, expressed through a hierarchy of levels T^k indexed by prime numbers (“arities”) — is only interesting to the extent that it can be checked against the world. This article collects the predictions that do exactly that: claims calculable from the ArXe structure before the experimental result is known, each with an explicit condition under which it would be refuted.

Three housekeeping notes before the list. First, ArXe distinguishes predictions from postdictions. A postdiction explains a number already known — useful for internal consistency, but not evidence on its own. Everything below is a prediction in the strict sense: derivable independently of the measured value, with a stated falsification condition. Second, several early ArXe predictions did not survive later scrutiny and were formally retired; they are listed at the end, because a theory that quietly drops its failures is not being honest about its track record. Third, each prediction below is marked with the date it was first stated in writing, alongside the date of the test that would resolve it. The gap between the two is what makes the prediction/postdiction distinction verifiable rather than a matter of trust: a claim dated December 2025 about a Belle-II measurement due in 2025–2027 could not have been reverse-engineered from a result that does not exist yet.

1. Structural predictions

These follow directly from the boundary-condition (BC) structure of the T^k hierarchy and the assignment of physical phenomena to specific arities. They are the most robust class in the corpus because none of them depends on a numerical formula being correct — only on the qualitative structure holding.

No gauge group between electromagnetism and the weak force. (Stated: March 2026 — open)
Between the electromagnetic level (T⁻⁵, arity 11) and the weak level (T⁻⁶, arity 13) there is no arity number available. ArXe predicts, therefore, that no gauge boson can exist with a coupling strength or symmetry structure intermediate between U(1) and SU(2).
Falsified by: discovery of a BSM gauge group with structure between U(1) and SU(2).

No gauge boson at a composite arity. (Stated: 2026 — open)
Every confirmed gauge force in ArXe sits at a prime arity — 7 (strong), 11 (electromagnetic), 13 (weak). The theory predicts this is not coincidental: a gauge force at a composite arity (9, 15, 21…) should be structurally impossible.
Falsified by: discovery of a gauge force whose carrier corresponds to a composite arity.

A fifth force, if it exists, sits at arity 17. (Stated: August 2026 — open)
If a new fundamental gauge force is ever discovered, ArXe predicts its carrier corresponds to level T⁻⁸ (arity 17) — the next prime past the weak force in the hierarchy — with a coupling strongly suppressed relative to the known three forces.

Normal neutrino mass hierarchy. (Stated: March 2026)
The ordering of T^k levels implies lighter neutrinos correspond to lower levels, which requires the normal (not inverted) mass hierarchy.
Test: JUNO, results expected from the late 2020s. Falsified by: confirmation of the inverted hierarchy.

Zero free parameters. (Standing claim since the earliest V4 core document, reaffirmed March 2026)
This is closer to a meta-prediction, but it is falsifiable in the strongest sense: if any specific ArXe derivation requires a fitted constant with no structural origin, the local derivation — and the broader claim that ArXe is parameter-free — fails at that point.

If dark matter is detected, its physical integer contains the factor 19. (Stated: March 2026 — open, awaiting detection)
Dark matter is assigned to level T⁻⁹ (arity 19) in the ArXe hierarchy. If a dark matter particle is directly detected, ArXe predicts its mass in Planck units, expressed as a physical integer P, will contain 19 as a factor.
Falsified by: a confirmed DM detection whose mass integer has no factor of 19.

2. Precision predictions from digit-level analysis

(Stated: March 2026, corrected and re-dated June 2026 after ALO + Grammar scrutiny — d for α⁻¹ and αₛ were both revised at that point, from an earlier and looser estimate)*

ArXe distinguishes, within any measured constant, the digits that are structurally fixed (“ArXe-pure,” up to a threshold d) from the digits that are set by measurement-protocol convention. This produces a genuinely unusual class of prediction: not a value, but a stability boundary* — a claim about which digits should agree across independent experiments and which should not.

Constant	d*	Structural basis	Test	Test window	Falsified by
α⁻¹ (fine structure)	3 digits	P = 137, pure at 3 digits	Penning trap vs. recoil interferometry	Testable now	Rb and Cs measurements differ in digit 1 or 2
αₛ (strong coupling)	2 digits	P = 3, pure at 2 digits	Lattice QCD vs. dijets vs. τ decays	Testable now	All methods converge to better than 0.5% precision
Higgs mass uncertainty	—	δ_ont = π/6 × mH (n=6)	HL-LHC	2028–2040	Uncertainty in mH falls below ±50 MeV
Critical exponent (phase transitions)	3 digits	δ_ont, n=6	Comparison across distinct experimental systems	Testable now	Variation in the first 3 digits
mₚ/mₑ	4 digits (estimated)	Base 1836 = 2²×3³×17	HD⁺ spectroscopy vs. antiproton measurements	Testable now	(estimate, pending formal digit-by-digit verification)

The αₛ case is worth stating explicitly because it predicts a persistent disagreement, which is an unusual thing for a theory to predict: ArXe holds that the ~1–2% tension between lattice, dijet, and τ-decay determinations of αₛ reflects a genuine 7-ary structure at the T⁻³ level, and will not shrink to 0.01% no matter how much computational precision improves. Convergence below 0.5% would be evidence against this.

3. The TDSL asymptotic prediction

(Stated: December 2025)

The Temporal Dimensional Structure Limit (TDSL) framework produces perhaps the cleanest single numerical prediction in the corpus:

At infinite energy, virtual screening vanishes and the bare electromagnetic coupling is revealed:
lim(E→∞) α⁻¹ = 4π × 11 = 138.230 (exact)

Test: a Future Circular Collider (~100 TeV), ultra-high-energy cosmic ray data, or precision measurements of the running of α at the highest accessible energies.
Falsified by: α⁻¹(E→∞) measured to a value other than 138.230.

A companion structural rule (the “Δn=0 test”) states that two observables belonging to the same T^k level should never diverge in their joint limit — over 60 retrospective cases are consistent with this, and any newly identified pair with Δn=0 is a live test.
Falsified by: a single Δn=0 pair whose joint limit diverges.

4. Particle physics: the tau lepton anomalous magnetic moment

(First stated: December 2024, in the initial 8/π mass-ratio derivation; formalized as a standalone falsifiable prediction December 2025)

The most experimentally imminent prediction in the corpus concerns the tau lepton. A geometric factor, 8/π ≈ 2.546, reproduces the tau/muon mass ratio (m_τ/m_μ = 16.817) to 0.08% precision. ArXe extends the same factor to the tau’s anomalous magnetic moment:

a_τ (ArXe) = (1.268 ± 0.018) × 10⁻³, an excess of ~7.6% over the Standard Model value (1.178 × 10⁻³).

Test: Belle-II, 2025–2027 — this is the prediction in the corpus with the tightest gap between the date it was put in writing (December 2025) and the date it can be checked.
Strong falsification: Belle-II measures a_τ consistent with the SM value within 1σ — the extension of the 8/π factor to magnetic moments is wrong.
Weak falsification: an intermediate value (~1.22 × 10⁻³) — the effect exists but is more suppressed than predicted, and the calculation needs refinement rather than rejection.

A related, coarser prediction: particles carrying a T⁻¹ component together with mass should be unstable, while those without should be stable — consistent with the electron, muon, and tau, though the photon required the rule to be refined (T⁻¹ alone is not sufficient; mass must also be present).

5. QCD

(Stated: 2026)

The QCD sector produces six distinct testable predictions, several with data already partially in hand:

Non-logarithmic structure in α_s(Q²) near the confinement scale Λ² — a “smoothed step” rather than the smooth logarithmic running of standard perturbative QCD. Test: deep inelastic scattering, τ decay, lattice QCD in the range Q² = (0.1–1 GeV)².
Λ_eff ∝ 1/r_RMS across hadron species — standard QCD treats Λ_QCD as universal; ArXe predicts it scales inversely with the hadron’s RMS radius, with slope ≈ 197 MeV·fm. Test: spectroscopy of ρ, ω, J/ψ, Υ.
Quark-gluon plasma viscosity saturating near the quantum bound η/s ≈ ℏ/(4πk_B), with a specific functional approach as T/T_c → ∞.
Jet quenching scaling as T³, rather than T² as in an ordinary plasma, reasoned from the topological (T⁻¹) nature of the interaction.
Confinement scale universality: Λ = 197 ± 50 MeV should recur across otherwise unrelated hadronic phenomena.

6. Chemistry and the periodic table

(Periodic table predictions stated: March 2026. Bonding predictions — linewidth, stability, bond-angle — stated: May 2026)

ArXe assigns each nucleus a coupling relationship to specific orbital angular momenta via arity, which produces several predictions about elements that have not yet been synthesized or fully characterized:

The g orbital (l=4) is structurally, not just nuclearly, unstable. It requires coupling to arity 9 = 3² — composite, with no irreducible ArXe level available. Superheavy elements with g-orbital electrons (testable near Z=121) should show instability beyond what nuclear physics alone predicts.
Z=119 should show no Madelung exception (119 = 7×17, composite — no phase commensurability with any ArXe level; expected configuration 8s¹, normal).
Z=127 should show a Madelung exception (127 is prime — phase commensurability predicts an orbital filling anomaly).
Z=137 should show anomalous electromagnetic sensitivity, since Z ≈ α⁻¹ places it at numerical resonance with the electromagnetic level operator.
Anomalous hyperfine interactions in the f-block, peaking near f⁷ and f¹⁴ configurations, testable by precision hyperfine spectroscopy on heavy lanthanides (Dy, Ho, Er, Tm).
A Kramers-doublet counting rule (stated: June 2026, priority-dated in that document): for lanthanide ions with more than 7 electrons in the 4f shell undergoing intra-level (4f→4f) transitions, the number of Kramers doublets in the principal emitting multiplet equals n − 4. Verified in two cases (Dy³⁺: 9→5; Er³⁺: 11→7); falsified by any Kramers ion with n>7 showing a different count.
Emission linewidth as a transition-type indicator (stated: June 2026): inter-level transitions (5d→4f) should show FWHM roughly two orders of magnitude broader than intra-level transitions (4f→4f) — verified in Ce³⁺ vs. Yb³⁺ in YAG, pending confirmation across other host crystals.

On bonding specifically: covalent intramolecular transitions are predicted to show FWHM below 10 nm against ionic charge-transfer transitions above 50 nm; molecular stability is predicted to require prime factors of combined atomic arity no greater than 13; and bond angles in novel hypervalent molecules are predicted to follow T² phase-ordering minima rather than electron-counting rules alone.

7. Fermion mixing (CKM/PMNS)

(Stated: July 2026)

A separate strand of the theory assigns the mixing angles between quark and lepton generations to mathematical attractors (φ, γ, e) determined by the boundary-condition structure of their recursion. This produces one prediction with a near-term experimental deadline:

The CP-violating phase δ_PMNS should carry the golden ratio φ, or no closed-form mathematical constant at all — not e or γ.
Current uncertainty is large (±52°/−25°, PDG 2023); the prediction becomes sharply testable once the error falls below 10°.
Test: DUNE (first results expected ~2028), Hyper-Kamiokande (~2027).

8. Gravity and the Planck scale

(Planck-scale unification predictions stated: August 2026. Graviton properties stated: December 2024 — the oldest dated prediction in this article, still awaiting a detection event to test it against)

The correction terms needed to unify the three Standard Model couplings at the Planck scale should factor entirely within the arity lexicon {7, 11, 13} — checkable against precision coupling-constant running data.
Gravity has no dedicated level in the T^k hierarchy of its own — it is an emergent scale, not a fourth fundamental interaction with its own arity. If a gravitational gauge boson with its own hierarchy level is ever discovered, this claim is refuted directly.
If the graviton is detected: exactly three independent observable properties (energy, momentum, helicity), helicity ±2, and zero mass — the last already consistent with LIGO/Virgo’s measurement of gravitational wave speed equal to c.

9. Retired and refuted predictions

Listed here deliberately, because a falsifiable theory has to show its failures alongside its successes:

The spectral index n_s formula with arities {23, 19, 17} — retired. Later analysis found the correct structural arities are {3, 5, 7, 11, 13, 17}; 23 and 19 do not appear.
A ~710 GeV resonance — retired. No clean factorization with the arity lexicon; no reproducible derivation method.
A ~534 GeV dark matter mass — retired. Derivation method not reproducible under the current framework.
The 7/11 lanthanide emission-ratio prediction (Er³⁺) — refuted outright; the observed ratio (0.983) is far from the predicted 0.636. Root cause: the wrong observable was targeted (photon frequency rather than the emitting level’s discrete doublet structure). Reformulated as the Kramers-doublet rule in Section 6.
The Mg-porphyrin FRM hypothesis — refuted by a discriminating test (Al-porphyrin fluorescence). The empirical trend it tried to explain (Mg outperforming Zn) still holds; the causal mechanism does not — it is the ordinary heavy-atom effect.

Live status check — August 2026

(For the four predictions whose test window has already opened, added at the time of publication)

Neutrino mass ordering (JUNO) — leaning toward confirmation, not yet decisive.
JUNO published its first physics result in Nature in June 2026, based on 59 days of data collected in autumn 2025. A follow-up analysis combining that data with the rest of the global oscillation dataset finds a preference for the normal ordering, with a p-value against the inverted ordering of 2–2.6% (2.2–2.3σ). This is not a discovery-level result, and JUNO’s own mass-ordering-dedicated analysis with larger statistics is still to come, but the first real data point available since this prediction was stated points in the direction ArXe predicts, not against it.

αₛ tension between methods — still present, prediction not falsified.
ArXe predicts the ~1–2% disagreement between lattice, τ-decay, and DIS determinations of αₛ will persist rather than converge. As of 2026 the world-average uncertainty on αₛ(M_Z) remains at the O(1%) level, and the specific tension between the τ-decay value and the lattice value is still measured at roughly 2.7 standard deviations. The tension has not gone away — the prediction remains live and unfalsified. It would be falsified the day all methods agree to better than 0.5%, which has not happened.

Tau anomalous magnetic moment (Belle-II) — window open, but no data precise enough yet.
This is the case to be most careful about. The 2025–2027 test window has technically begun, but no published Belle-II measurement currently has the precision needed to compare against the predicted 7.6% excess (a_τ = 1.268×10⁻³ vs. SM’s 1.178×10⁻³). Current collider-based bounds on a_τ are still an order of magnitude too wide — spanning roughly ±(5–8)×10⁻³ — to say anything about a 7.6% effect. Belle-II’s projected sensitivity of O(10⁻⁵) is expected only once its full dataset is analyzed. This prediction should be reported as open, not yet testable in practice, despite sitting inside its nominal test window.

Superheavy elements Z=119, 127, 137 — untested; element not yet synthesized.
As of 2026, element 119 has not been synthesized or confirmed by any laboratory (GSI, RIKEN, JINR, and LBNL all have ongoing or proposed attempts, none successful yet). The predictions tied to Z=119 (no Madelung exception), Z=127 (Madelung exception expected), and Z=137 (anomalous EM sensitivity) remain entirely untested — there is no element there yet to check them against.

Takeaway: of the four predictions whose clock has started, one shows early evidence consistent with ArXe (neutrino ordering), one remains unfalsified because the tension it predicted is still measured (αₛ), and two are simply not yet checkable despite being nominally “in window” — the tau moment for lack of precision, the periodic-table items for lack of the element itself. None of the four counts as a confirmation in the strong sense; none has failed either.

Priority timeline

The dates above matter individually, but the pattern across all of them is the actual argument for taking these as predictions rather than retrofitted explanations:

Prediction	Stated	Resolves	Status as of Aug 2026
Graviton properties (helicity, masslessness)	Dec 2024	Detection-dependent	Untested — no detection event
a_τ excess via 8/π	Dec 2024 → formalized Dec 2025	2025–2027 (Belle-II)	Open — window active, precision not yet sufficient
TDSL asymptotic α⁻¹ = 4π×11	Dec 2025	FCC-scale energies	Untested — energy scale not yet reached
Structural predictions (gauge gaps, DM factor 19, neutrino hierarchy)	Mar 2026	JUNO / DM detection / open	Neutrino hierarchy: early data leans consistent (2.2–2.3σ, not decisive); DM/gauge gap: untested
Periodic table anomalies (Z=119, 127, 137, g-orbital)	Mar 2026	Superheavy-element synthesis	Untested — Z=119 not yet synthesized
Chemical bonding rules	May 2026	Ongoing (databases, crystallography)	Not yet checked against corpus
Lanthanide Kramers/linewidth rules	Jun 2026	Ongoing (spectroscopy)	2 verified cases each; broader matrix survey pending
CKM/PMNS mixing attractors	Jul 2026	~2027–2028 (Hyper-K, DUNE)	Untested — current δ_PMNS error too large
Fifth-force arity, Planck-scale unification structure	Aug 2026	Open / future colliders	Untested
αₛ tension persists at 1–2%	Mar 2026	Testable now	Unfalsified — ~2.7σ τ-decay/lattice tension still measured

None of these have a test date earlier than the date they were stated — which is the only thing that actually distinguishes a prediction from an after-the-fact fit. Where a test date has already passed without resolution, that is noted as “open” rather than silently dropped. See the Live status check above for the four predictions with real data to compare against as of this writing.

Closing note

Most of physical theory-building produces retrodictions dressed up as predictions — formulas fit to known values, presented after the fact as if derived independently. The test above is whether a claim was calculable before the number was known, and whether there is a stated number or pattern that would make the theory wrong. By that standard, the strongest items here are the TDSL asymptotic limit for α⁻¹, the tau anomalous magnetic moment (testable at Belle-II within the next two years), the stability of α⁻¹ and αₛ to a fixed number of digits across measurement protocols, and the structural absence of any gauge group between U(1) and SU(2). The rest range from actively testable to genuinely speculative — flagged as such throughout — and the retired section exists precisely so the theory’s failures are not quietly dropped from the record.

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