The Cascade Framework derives the complete observable content of the Standard Model — coupling constants, fermion mass ratios, quark mixing parameters, neutrino masses, and the cosmological constant — from the single cubic polynomial x³ = x² + 1, with zero free parameters. Its unique real root ψs ≈ 1. 4656 (the plastic constant) generates a quasi-crystalline orbit with quasi-period K = 44 and winding number W = 13, whose Sturmian structure encodes particle physics (inner K=13 word) and cosmology (outer K=44 word) as nested scales of the same algebraic object. Approximately 64 exact theorems are proved, with 30+ physical observables predicted at sub-0. 1% accuracy. The polynomial’s three roots carry physical roles: ψs encodes Being (spacelike anchor), ω⁻ encodes Gravity (amplitude-silent Frobenius), and ω⁺ encodes Becoming (the comma generator and arrow of time). Version 10 expands the suite from 10 to 15 papers and introduces five major structural results. Paper 11 derives the complete cosmic energy budget (ΩΛ, ΩDM, Ωb) from zero free parameters via the orbit’s Fibonacci/Tartini winding partition, resolves the Hubble tension algebraically, and shows 41–98% suppression of the σ8 tension through quasi-crystalline oscillations. Paper 12 proves that the class number h (−31) = 3 of the discriminant field Q (√−31) simultaneously forces the direction of spontaneous symmetry breaking (Theorem T59), determines the baryon fraction Ωb/Ωm = 2/13, and predicts the neutrino mass sum Σmν ≈ 0. 100 eV — all as consequences of the same number-theoretic identity. Paper 13 (“The Tartini Fugue”) establishes the complete musical structure of x³ = x² + 1: a 44-TET tuning system in which dark energy corresponds to an exact tritone, dark matter to an exact minor third, and the quasi-closure comma C is identified as the Tartini difference tone between the 4-stroke i-cycle and the Möbius quasi-2-stroke running in permanent stretto. Paper 14 explores the number theory of the framework, proving that the discriminant −31 = −M₅ (the fifth Mersenne prime) generates a 31, 79, 127 arithmetic progression whose members simultaneously encode the cascade’s winding numbers, Galois group order, and prime-counting identities — with the Chebotarev density theorem explaining the framework’s prediction accuracy tier structure. Paper 15 (“The Cascade Hologram”) closes the suite with three theorems: TUC proves ΩΛ = 1/ψs in two lines from Vieta’s formulas; THolo assigns the 13 bright orbit stations to the CFT boundary, the 31 shadow stations to the AdS bulk, and the comma C to the holographic entropy residual; and TAdS proves the AdS radius equals the ψ-anyon quantum dimension D = ψs^ (3/2) exactly — the geometry of the cosmos and the topology of quantum computation are the same algebraic number. Across the updated papers, v10 also establishes: the complete K=44 Sturmian word (triple-verified, 44 letters, 31 a’s, 13 b’s) ; the tribonacci-Fibonacci number structure of every cascade counting constant (K=44=T (9), W=13=F (7) =T (7), N₀=55=F (10) ) ; the nested self-similar hierarchy Sturmian (13, 31, 44) ⊃ Sturmian (5, 8, 13) as the orbit-level statement of holographic structure; a complete beat-frequency harmonic index table mapping every physical sector to an orbit mode m; and the ψ-anyon fusion category with exact F-matrix, S-matrix S₁₂ = ±i√ψs/D, and a proof of non-modularity (Verlinde fails). The two-line Unit Circle Theorem TUC is identified as the most fundamental proof in the framework: the cosmological constant is a Vieta product, not a free parameter. MANIFEST. md is a machine-readable index included at the root of the archive. It maps each file’s numeric prefix to its internal citation number (which diverge for several papers due to the framework’s historical development order), provides the canonical title and paper number for every file in the suite, records MD5 checksums for all 15 PDFs for integrity verification, and includes a compact framework summary with the concept DOI, theorem count, and observable tally. It is the single authoritative reference for anyone automating ingestion, citation, or verification of the paper suite.
Joshua Breault (Tue,) studied this question.