Status legend
- 🟢 Mature — formal derivation exists, needs refinement/review
- 🟡 Promising — direction is clear, needs computational work
- 🔴 Speculative — intuition exists, formalization missing
Tier 1: Consolidate what we have
| # | Task | Status | Horizon |
|---|---|---|---|
| 1.1 | Effective action for the inflaton — formally show that the canonical inflaton φ = ln(τ₂) (the imaginary part of τ) on an FRW background reproduces an α-attractor Lagrangian with α = 2/3 | 🟢 | Weeks |
| 1.2 | LiteBIRD forecast — Fisher matrix for (nₛ, r), significance of α=2/3 vs α=1 | 🟢 | Weeks |
| 1.3 | Peer review — send formal document to 2–3 experts in inflation/modular forms | 🟢 | Parallel |
The weakest link is 1.1: without it, the inflaton = ln(τ₂) identification is heuristic. With it, the full chain A3 → shape of the metric → (postulate N) → α=2/3 → r=2(1−nₛ)² is formally closed (K=−1 is the scale fixed by N, not a consequence of the symmetry).
Tier 2: Extend predictions
| # | Task | Status | Horizon |
|---|---|---|---|
| 2.1 | Non-Gaussianity from Z₃ — bispectrum from the Z₃ symmetry of the ρ-point (τ = e^{iπ/3}) | 🟡 | Months |
| 2.2 | Running spectral index — precise prediction for dnₛ/d ln k from α = 2/3 | 🟡 | Weeks |
| 2.3 | Cosmological phase φ(z) and LRDs — test tan²(φ(z)) against JWST RUBIES/JADES catalogs | 🟡 | Month |
| 2.4 | Mass defect from observer projection (IR addendum, v5) — projection P onto the Stab(i)-invariant yields ~98% of hadron mass (gluonic contribution / QCD trace anomaly) as the size of the projection defect | 🔴 | Months |
Note on 2.3 / 2.4: both are independent speculative addenda outside the main r prediction. If they fail, the core result stands. (v5 shifts the emphasis of the second addendum from LRD/JWST to the QCD mass defect.)
Tier 3: Deep extensions
| # | Task | Status | Horizon |
|---|---|---|---|
| 3.1 | Nₑ from Γ(5) — physical reason for Nₑ = [PSL(2,ℤ):Γ(5)] = 60 | 🟢 done (2026-04-21) | — |
| 3.2 | Standard Model from octonions — rigorous derivation of U(1)×SU(2)×SU(3) from Cayley-Dickson tower | 🔴 | Year+ |
| 3.3 | Three generations from Z₃ — three fermion generations from Z₃ orbifold symmetry | 🔴 | Year+ |
| 3.4 | Dark energy from residual phase — Λ as φ₀ > 0 (φ never reaches zero exactly) | 🟡 | Year+ |
Theorem 4 closed (2026-04-21): the maximum-modulus principle forces Configuration A for the canonical potential V_A with no Hessian; A₅ combinatorics (20 order-3 elements → 10 axes → ×6 triangles) gives 60 = |A₅| with no open computation. Consequence: r = 2/900 ≈ 0.0022 is fully parameter-free for V_A — the first parameter-free TOE-level prediction in the framework. See theorem-4-m1-dyn-uzavreni and m1-dyn-b-closure for details.
What can kill the theory
| Fatal result | Consequence |
|---|---|
| LiteBIRD: r > 0.01 or r < 0.0005 | α = 2/3 ruled out |
| LiteBIRD: r ≈ 0.004 exactly | Starobinsky preferred, τ-geometry needs revision |
| Formal error in A3 → shape of the metric → α=2/3 (given N) | Core result collapses |
| Effective action doesn't reproduce α-attractors | Inflaton = τ identification fails |
What would NOT kill it:
- φ(z) fails vs JWST → cosmological branch falls, r prediction stands
- SM from octonions doesn't work → independent branch
- Nₑ ≠ 60 → standard Nₑ from reheating sufficient
Key timeline
Now → Effective action + peer review outreach
Weeks → LiteBIRD Fisher forecast
Month → JWST LRD data test
2032 → LiteBIRD launch and first results