theorem
proved
phi_forced
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IndisputableMonolith.Foundation.PhiForcing on GitHub at line 213.
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depends on
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of -
of -
cost -
cost -
DiscreteLedger -
golden_constraint_unique -
is_self_similar -
self_similar_forces_golden_constraint -
of -
of -
of -
L -
L -
S
used by
formal source
2103. r satisfies the compositional constraint r² = r + 1
211
212Then: r = φ = (1 + √5)/2 -/
213theorem phi_forced (L : DiscreteLedger) (r : ℝ) (hr : is_self_similar L r) : r = φ := by
214 rcases hr with ⟨S, rfl⟩
215 exact golden_constraint_unique S.ratio_pos (self_similar_forces_golden_constraint S)
216
217/-! ## Consequences of φ -/
218
219/-- The minimum non-trivial cost: J_bit = ln(φ). -/
220noncomputable def J_bit : ℝ := Real.log φ
221
222/-- J_bit is positive. -/
223theorem J_bit_pos : 0 < J_bit := Real.log_pos phi_gt_one
224
225/-- The coherence quantum: E_coh = φ⁻⁵. -/
226noncomputable def E_coh : ℝ := φ^(-5 : ℤ)
227
228/-- E_coh is positive. -/
229theorem E_coh_pos : 0 < E_coh := by
230 simp only [E_coh]
231 exact zpow_pos phi_pos (-5)
232
233/-! ## Summary: The Forcing Chain -/
234
235/-- **PHI FORCING PRINCIPLE**
236
237The golden ratio φ is forced by self-similarity in a discrete J-cost ledger:
238
2391. Discrete ledger with J-symmetry (from previous levels)
2402. Self-similarity: the structure references itself at different scales
2413. Compositional constraint: r² = r + 1 (next scale = current + base)
2424. Unique positive solution: r = φ = (1 + √5)/2
243
papers checked against this theorem (showing 8 of 8)
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Spinning acoustic black hole still amplifies sound, but only barely
"Krylov ABH thickness profile h(r) = b(r−r₁)^n + h₁ with chosen b = 7.34e−4, r₁ = 2e−2, h₁ = 6e−4, n = 2 (free engineering parameters)"
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Minimum length stands in for the cosmological constant in 3D gravastars
"P = −σ = ρ (Lorentzian-distribution shell, equation of state)"
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Gauss-Bonnet charge leaves a fingerprint in EMRI waveforms
"we employ the periodic orbit classification method proposed by Janna Levin et al. ... q = ∆ϕ/(2π) − 1 = w + ν/z"
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DBI scalar fields tie ΛCDM on late-time data, H₀ near 73
"the golden ratio... does not appear; the paper's 'φ' denotes the scalar field"
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"Weak null singularity survives a relativistic fluid"
"for some large N, Σ_{i≤N}|Z^i χ̂| ∼ (u*−u)^{−1} log^{−p}(1/(u*−u)), for some p > 1, ... there exists a unique smooth spacetime ... such that the metric and fluid variables (v,τ) extends continuously to the boundary, but the Christoffel symbols are not in L²."
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New supergravity inflation models tie ξ to α by ξ = 1/(6α)
"Ω = 1 + ξ(T T̄)^{n/2}, ξ = k²/μ², n = 2(k+1)/k > 2 or n = 2k/(2+k) < 2"
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KL polynomials of Dowling geometries become a counting problem
"The coefficient of t^i in the Γ-equivariant Kazhdan–Lusztig polynomial of Q_n(G) is the permutation representation on S_G(n,n−i). The coefficient of t^i in the Γ-equivariant Z-polynomial of Q_n(G) is the permutation representation on A_G(n,n−i)."
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One golden-ratio curve organizes four periodic-table trends at once
"the argument-rescaling factor φ = (1+√5)/2 is presently a modeling ansatz, not a derived chemical constant. We retain it because it produces the ratio identities ... but a first-principles derivation of φ in this chemical context is left to future work."