Physics Predictions

The Numerical Prediction Catalogue compiles 67 quantitative predictions derived from the single master constant ι_τ = 2/(π+e), with zero free continuous parameters. Each prediction compares a τ-derived value to experimental measurement under explicit source, unit-context, and verification boundaries. Fifteen predictions achieve sub-10 ppm precision, including the electron mass (0.025 ppm), weak mixing angle (−0.65 ppm), and Higgs mass (+8.0 ppm).

The catalogue sits on top of the Calibration Cascade. Two inputs — the algebraic posit ι_τ = 2/(π+e) and a single SI anchor, the neutron mass m_n — feed a five-layer dependency overlay:

1. L0 Algebraic→
2. L1 Dimensionless→
3. L2 SI Anchor→
4. L3 SI-Derived→
5. L4 Verification

Most entries live at L1 (dimensionless ratios) or L3 (SI readout routes), but the cascade also records unresolved mapping and unit-context review status.

Three kinds of prediction — honest a-priori accounting

“67 zero-parameter predictions” is a load-bearing phrase and deserves to be unpacked here, at the catalogue’s front door, not only in an appendix. Not all catalogue entries are the same kind of thing:

• ~50 are retro-consistency post-dictions — predictions for constants that were already measured, decades ago, before the τ-framework was developed. The electron mass, fine-structure constant, Koide relation, weak mixing angle, and Higgs self-coupling live here. The τ-value is computed algebraically from ι_τ; the experimental value existed first. These demonstrate that ι_τ captures the relevant scale without tuning, but they are not novel forward tests in the philosopher-of-science sense.
• ~10 are tension accounts — predictions that commit the framework to a specific side of an open empirical discrepancy. The Hubble constant h = 2/3 + ι_τ²/17 ≈ 0.6735 (early-universe-consistent), the W-boson mass vs the CDF II anomaly, muon g−2, and S₈ live here. Each is a forward commitment, strongest where the tension pre-existed the derivation.
• ~7 are genuine forward tests — predictions for quantities not yet measured at the precision required to confirm or falsify. The framework lives or dies on these.

These ratios are the canonical accounting reported in the Results Overview and developed in full in the Prediction Timing Ledger. They are structural estimates at precision-tier granularity, not exact per-prediction counts. A reader landing directly on this browse page should not take “67 zero-parameter predictions” as uniformly forward-test weight.

The strongest forward tests

The genuinely forward Category-C tests are concentrated in the Falsification Pack (N1–N30). The flagship is:

• N9 — CMB-S4 tensor-to-scalar ratio r = ι_τ⁴ ≈ 0.01363, falsifiable at roughly 14σ on CMB-S4’s design sensitivity σ(r) ≈ 0.001 over 2028–2035. This is the sharpest single empirical test the framework voluntarily stakes itself on.

Other Category-C tests include the neutrino mass sum Σm_ν = 0.089 eV (DESI, Euclid, KATRIN, Project 8), neutrinoless double-beta decay with a τ-specified half-life (nEXO, LEGEND, CUPID), the normal mass ordering (JUNO, DUNE, Hyper-Kamiokande), exact proton stability, and the structural prediction of no supersymmetric partners at any scale (LHC, FCC, CEPC). Any one of these, if falsified, refutes the corresponding τ-derivation.

How to read these predictions — scope labels

Every prediction in the catalogue below carries a scope label that states its epistemic status within the framework. The four-tier discipline is canonical to the program (see Scope Labels):

• Established — classical mathematics independently verified in the literature; used as a building block inside τ, not a τ-consequence. (No predictions in this catalogue; catalogue entries are τ-derivations.)
• τ-effective — the default scope for a numerical prediction entry: a quantitative τ-derivation with a specific number and an experimental target. Most prediction records sit here.
• Conjectural — the derivation depends on an axiom that is computationally verified at all tested finite bounds but whose infinite-limit extension is asserted rather than proven. TauLib marks these with explicit axiom declarations, and the catalogue flags those entries on the card.
• Metaphorical — philosophical or analogical extensions beyond formal mathematics. Not used in the numerical prediction catalogue; reserved for the metaphysics world readout.

The precision tier chip on each card is orthogonal to the scope label: it states how tightly the τ-derived value matches the current measurement. Sub-10 ppm means agreement at the sixth decimal place; 10–1000 ppm is the next tier; 1–5% covers known-imperfect sector matches; structural covers derivations whose content is categorical (e.g., “proton is stable”, “no fourth generation”) rather than a specific decimal.

How to read precision — the Tier A / B / C taxonomy

Precision tiers on a card answer “how close is τ to the measured number today?” The Calibration Cascade adds an orthogonal question: which route through the cascade produced this prediction, and what is the best precision that route can in principle reach? The N1–N30 falsifiers are tagged by three structural tiers:

• Tier A (~0.025 ppm) — mass-ratio route. The prediction is a ratio of two masses (or two dimensionless L1 quantities). Because both numerator and denominator carry the same SI anchor, the SI-calibration error cancels exactly. Tier A predictions are falsifiable at the precision of PDG/CODATA mass-ratio data — already at the 10⁻⁸ level for leptons today. Most sub-10-ppm entries in the catalogue are Tier A.
• Tier B (~3 ppm) — closing-identity route. The prediction is an SI-anchored observable routed through the G–α closing identity (V.T20). These entries inherit the residual uncertainty of the SI anchor m_n and the closing identity’s loop order. Achievable precision is ~3 ppm; examples include αG, several electroweak L3 observables, and the ppm-tier cosmological predictions.
• Tier C (~0.8%) — fine-structure leading-order route. The prediction uses α as a derived input at leading order without the full closing-identity correction. Tier C is the coarsest route and governs the 1–5% entries and most “structural” binary predictions.

Read the card precision chip and the cascade tier together: a “sub-10 ppm” Tier A entry is a closed book today; a “sub-10 ppm” Tier B entry is still waiting on the SI anchor to tighten; a “structural” Tier C entry is a yes/no falsifier, not a decimal comparison.

Using this catalogue

Use the filters below to narrow by physics domain, precision tier, scope, or canonical book. The filter row surfaces ppm precision; the Tier A / B / C calibration-tier distinction is not yet a UI chip on each card (it is a per-prediction structural classification flagged for a follow-up data uplift). For the 30 sharpest falsifiable predictions with named experiments and explicit timelines, see the Falsification Pack. For the first-order response to “could 15 sub-10-ppm hits just be numerology?” see the Fit-Space Argument. The companion publication artifact remains available as a free download: Numerical Prediction Supplement (PDF, 1.11 MB).