all_constants_from_phi
All named constants are functions of φ alone.
plain-language theorem explainer
Recognition Science fixes the speed of light, reduced Planck constant, gravitational constant and Planck length as algebraic expressions in the golden ratio phi within its native units. A physicist deriving the constant sector from the composition law would cite this result to eliminate free parameters. The proof is a one-line constructor that assembles five prior lemmas on each component.
Claim. In RS-native units, $c = 1$, there exists an integer $n$ such that $h = phi^n$, there exists an integer $m$ such that $G = phi^m$, $G h = 1$, and the Planck length equals 1.
background
The module derives physical constants from the Recognition Science foundation. The composition law forces the unique cost J(x) = (x + x^{-1})/2 - 1. Self-similarity fixes phi as the golden ratio. The eight-tick octave then requires three spatial dimensions, yielding fundamental scales tau_0 = 1 tick and ell_0 = 1 with coherence energy phi^{-5}. Constants appear as ratios of these scales. Upstream results include the definition of G_rs as phi^5 and the tick as the unit time quantum.
proof idea
The proof is a one-line wrapper that applies the conjunction of c_rs_eq_one, ℏ_algebraic_in_φ, G_algebraic_in_φ, G_ℏ_product and planck_length_eq_one. These lemmas are established earlier in the module from the phi-ladder and the forcing chain.
why it matters
This result closes the constant sector by showing every fundamental constant is algebraic in phi, which is forced by J-uniqueness and the self-similar fixed point. It supports the claim that the entire physics follows from the composition law with no free parameters. The module narrative links it to the eight-tick cycle forcing D = 3 and the mass formula on the phi-ladder.
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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}"