Two Axioms
Vacuum self-similarity
A doubling-length recursion r(2L) = 1 + 1/r(L) on the closure ratio. The fixed point of this recursion is the golden ratio φ, which becomes the natural base of the cascade.
E8 maximality
The highest-rank exceptional lattice E8 is taken as the top of a structural sequence. The cascade descends from E8 through H4 and lower-dimensional structures to a single closure residue at the bottom.
The Seven-Rung Cascade
A1 and A2 together force a specific descent through seven structural rungs. The numbers along the cascade are the per-rung dimensional contributions; the depth N=583 is the sum of 24² rank-2 tensor shells at the D4 rung plus one scalar per rung.
- Base φ — emerges from axiom A1's recursion as the fixed point of r(2L) = 1 + 1/r(L)
- Depth N = 583 — explicitly 24² + 7 (576 rank-2 tensor shells at the D4 rung plus one scalar per rung)
- Factor 2 — the σ-orbit length of H4 inside E8 in the icosian construction (the same Galois twist that underpins the Schläfli decomposition)
- H4 rung — the 600-cell on S3, which serves as the discrete substrate
The Cosmological-Constant Formula
The same cascade ladder yields six further cosmological observables in closed form. Each has its own cascade expression and its own match against Planck 2018; all seven match simultaneously to within 0.5%.
| Observable | Cascade form | Predicted | Match | Grade |
|---|---|---|---|---|
| Λ·ℓP² | 2·φ−583·(1−δΛ) | 2.869 × 10−122 | +0.078% | Conditional Theorem |
| H0 | MP·φ−291.5·√(1−δH0) | 67.66 km/s/Mpc | +0.45% | Conditional Theorem |
| ΩΛ | (2/3)·(1−δΛ)/(1−δH0) | 0.6844 | −0.083% | Conditional Theorem |
| H0√ΩΛ | √(Λ/3) | 55.97 km/s/Mpc | +0.38% | Conditional Theorem |
| H0·t0 | closed-form Friedmann | 0.951 | −0.1% | Conditional Theorem |
| w (equation of state) | −1 exactly | −1 | <1σ | Conditional Theorem |
| t0 (age) | derived from H0, ΩΛ | 13.77 Gyr | −0.2% | Conditional Theorem |
The Conditional Theorem grading reflects the structure of the paper's argument: conditional on the two axioms the cascade form follows; the empirical match against Planck 2018 is then a separate statement testable against measurement. The headline result is the simultaneity of all seven matches inside the 0.5% band — not any single one in isolation.
Falsifiable Windows
| Prediction | Value | Falsifier window | Testable by |
|---|---|---|---|
| H0 | 67.66 km/s/Mpc | outside [66.5, 68.5] | Hubble tension toward Planck |
| w (equation of state) | −1 exactly | |w + 1| > 0.005 | DESI 2025+, Euclid, LSST |
| ΩΛ | 0.6844 | outside [0.680, 0.689] | CMB-S4, future surveys |
| H0√ΩΛ | 55.97 km/s/Mpc | outside [55.5, 56.3] | Same surveys |
| Substrate Ramanujan ratio | 230/240 | mathematical theorem | n/a — unconditional |
The paper describes the empirical entries as "open invitations" — explicit testable commitments that DESI, Euclid, LSST, CMB-S4 and successor cosmological surveys could decide against the framework. Falsifiability is built in, not appended. The final row is unconditional: an arithmetic identity on the substrate that survives independent of any measurement.
What Is — And Is Not — Claimed
What is claimed
- A two-axiom derivation of the cascade structure E8 → H4 → 40 → D4 → 16 → 8 → 0
- All seven cosmological observables (Λ, H0, ΩΛ, H0√ΩΛ, H0·t0, w, t0) match Planck 2018 simultaneously to within 0.5%
- The dimensionless cosmological constant matches Planck 2018 to 0.078% after a structural dipole correction
- 94 reproducible computational tests; full re-derivation runs from canonical generators alone in ≈10 minutes
- A synthetic positioning beneath established frameworks (FLRW, ΛCDM, dodecahedral topology, conformal cyclic cosmology) — bridge, not replacement
- Falsification windows that future cosmological surveys can decide against the construction, plus one unconditional substrate-arithmetic check (Ramanujan ratio 230/240)
What is not claimed
- A complete Theory of Everything — matter content remains at structural-correspondence grade, not derivation grade
- Replacement of quantum mechanics, general relativity, the Standard Model, or ΛCDM
- Replacement of Poincaré dodecahedral space or conformal cyclic cosmology — the framework is positioned as a synthetic layer beneath them
- Multiverse or anthropic-selection arguments
- Settled empirical bridge interpretation — the synthetic positioning is itself a claim under test
- Peer-reviewed status — preprint only
- The final word; the paper describes its falsifiers as "open invitations"
Verification — Reproducible From Scratch
The repository ships a Python verification suite at verify/verify_paper.py using NumPy, SciPy, and mpmath at 50-digit precision. The suite re-derives every numerical claim from canonical polytope constructions alone — no values are imported from the paper. The base φ, the depth N=583, the factor 2, and the structural correction δΛ are all computed by the suite from scratch, then compared against Planck 2018 measurements with bibliography pinned.
94 of 94 tests pass. The runtime bottleneck is an 8640-dimensional symmetric eigendecomposition. Total runtime on standard hardware is ≈ 10 minutes.
Where This Sits Beneath The Other VFD Work
The H4 rung of the cascade is the 600-cell — the same discrete substrate that underlies the closure-response operator Cφ, the V600 programme of finite-group papers, and the Schläfli decomposition + spectral bridge bundle. The factor of 2 in the Λ-formula is the same σ-Galois twist that characterises 2T ⊂ 2I in the Schläfli paper — the icosian construction is shared substrate.
Licensing — Commons of Knowledge, Stewardship of Power
The paper and supporting documents are released under CC BY-NC 4.0; the verification suite under PolyForm Noncommercial 1.0.0. The repository's stated intent: "Commons of knowledge. Stewardship of power." Open for verification and academic use; stewarded against commercial extraction or misrepresentation. Permission is required for paid products, SaaS offerings, commercial APIs, and branded packaging.
Two axioms, seven rungs, seven observables. All within 0.5% of Planck 2018, simultaneously. Preprint; falsifiers stated; not a Theory of Everything.