theorem
proved
all_constants_from_phi
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All named constants are functions of φ alone.
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IndisputableMonolith.Foundation.ConstantDerivations on GitHub at line 270.
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267 These are not free parameters. They are algebraic in φ,
268 and φ is forced by the self-similarity equation from the
269 unique cost J. -/
270theorem all_constants_from_phi :
271 -- c = 1
272 c_rs = 1 ∧
273 -- ℏ = φ^(-5)
274 (∃ n : ℤ, ℏ_rs = φ_val^n) ∧
275 -- G = φ^5
276 (∃ n : ℤ, G_rs = φ_val^n) ∧
277 -- G × ℏ = 1
278 G_rs * ℏ_rs = 1 ∧
279 -- Planck length = 1
280 planck_length_rs = 1 :=
281 ⟨c_rs_eq_one, ℏ_algebraic_in_φ, G_algebraic_in_φ, G_ℏ_product, planck_length_eq_one⟩
282
283/-! ## The Derivation Narrative -/
284
285/-- **THE CONSTANT DERIVATION NARRATIVE**
286
287 1. The composition law (d'Alembert) is the foundation.
288 2. It uniquely determines J(x) = ½(x + 1/x) - 1.
289 3. Self-similarity under J forces φ = (1+√5)/2.
290 4. The eight-tick cycle (2^D = 8) forces D = 3.
291 5. These determine the fundamental scales:
292 - τ₀ = 1 (fundamental tick)
293 - ℓ₀ = 1 (fundamental length)
294 - E_coh = φ^(-5) (coherence quantum)
295 6. The constants follow:
296 - c = ℓ₀/τ₀ = 1
297 - ℏ = E_coh · τ₀ = φ^(-5)
298 - G = φ^5 (curvature extremum)
299 - α ≈ 1/137 (geometric + gap-45)
300
papers checked against this theorem (showing 15 of 15)
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Caption-matching at web scale matches supervised ImageNet zero-shot
"We train a series of 5 ResNets and 3 Vision Transformers... We use the Adam optimizer with decoupled weight decay regularization... We use a very large minibatch size of 32,768... The learnable temperature parameter τ was initialized to the equivalent of 0.07."
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Image-only pretraining catches up to text-aligned vision features
"we train a ViT model with 1B parameters and distill it into a series of smaller models"
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Spinning acoustic black hole still amplifies sound, but only barely
"Effective sonic-horizon analogy: superradiance uses event horizon r_H, surface gravity κ_H, axial velocity v_φ; acoustic c_s, ρ_0 set as free SI parameters (343 m/s, 1.2 kg/m³)"
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Heavy jet mass yields α_s = 0.1148, matching thrust at last
"α_s(m_Z) = 0.1148^{+0.0015}_{-0.0022}, compatible with similar determinations from thrust and C-parameter."
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"Majorana neutrino propagators in matter, written out in full"
"E± = √(m² + (p∓g)²) ... the canonical momentum is π(x,t) = iη*(x,t)"
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Minimum length stands in for the cosmological constant in 3D gravastars
"β ≈ 1.15×10⁻²⁰ m = [0.583×10 TeV]⁻²; energy scale Λ_ml = 1/√β ∼ 10 TeV"
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String inflation without non-perturbative tricks
"ξ = -χ(CY)ζ[3]/(2(2π)^3), σ = -χ(CY)ζ[2]/(2(2π)^3) = -η (Eq. 2.9): tunable corrections depending on CY Euler characteristic"
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Quantum-well microcavities can squeeze light across tens of GHz
"We have employed the following experimentally accessible parameters: ω_b/2π=20 GHz, ω_d/2π=360 THz, κ_a/2π=20 GHz, κ_b/2π=1 MHz, κ_d/2π=2 GHz, g_ad/2π=20 GHz, g_db/2π=20 MHz, T=4 K."
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Gravity as a broken gauge symmetry, with no Big Bang singularity
"M_P^2 ≡ −8 k_W v^3 m^2, Λ ≡ ±6 m^2 (Eq. 24); Λ ≡ ±3 m^2 (Eq. 31)"
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A 38 TeV gauge operator could seed the universe's matter excess
"We introduce a dimension-six operator constructed from weak gauge fields: Q_{W̃} ∼ Λ⁻² ε_{ijk} W̃^i_{μν} W^{jνρ} W^{kμ}_ρ. We find that the BAU can be explained if Λ ≃ 38 TeV."
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DBI scalar fields tie ΛCDM on late-time data, H₀ near 73
"we adopt the specific form of the tension scalar f(ϕ) = λϕ⁴ and potential V(ϕ) = m²ϕ²/2, where λ and m are free, positive, and real parameters"
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Higgs–dilaton inflation in Weyl geometry hits a unitarity wall
"L = -(1/12)(ξ_0 φ_0² + ξ_1 φ_1²)R̃ - (1/4)F̃² + (1/2)D̃μφ_0 D̃μφ_0 + (1/2)D̃μφ_1 D̃μφ_1 - V(φ_0,φ_1), with V = λ_0 φ_0^4 + λ_1 φ_0² φ_1² + λ_2 φ_1^4 and ξ_1 = p·ξ_0"
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A four-strange-quark spectrum from 2.07 to 3.12 GeV
"input parameters ... m_s = (95±5) MeV, ⟨s̄s⟩ = (0.8±0.1)⟨q̄q⟩, ⟨g_s² G²⟩ = (0.88±0.25) GeV^4, m₀² = (0.8±0.2) GeV²; M = 2.23 ± 0.15 GeV ..."
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One Transformer matches keypoints across images and point clouds
"build our correspondence model with 600M parameters... trained jointly on diverse data combining pseudo point clouds from depth maps with real 3D correspondence annotations."
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One golden-ratio curve organizes four periodic-table trends at once
"The dimensionless landscape J_chem is converted to eV-scale ionization energies via a per-period scale E_p ... the values of E_p^(G) for Ne, Ar, Kr, Xe, Rn are matched to within ~5% by the two-parameter empirical pilot fit E_p^pilot = E_0 L_p^{2.1} p^{−0.72}"