phi_forced
plain-language theorem explainer
A self-similar discrete ledger forces its scale ratio to equal the golden ratio φ. Researchers deriving the Recognition Science forcing chain from T0 to T8 cite this result to establish the unique fixed point of self-similarity. The proof is a one-line term that extracts the ratio from the self-similarity witness and applies the uniqueness theorem for the golden constraint.
Claim. Let $L$ be a discrete ledger and let $r > 0$ satisfy the self-similarity condition on $L$. Then $r = φ$, where $φ = (1 + √5)/2$ is the unique positive root of $x^2 = x + 1$.
background
A DiscreteLedger packages a LedgerForcing.Ledger with a DiscretenessForcing.DiscreteConfigSpace to enforce finite-step structure. The module establishes that self-similarity on such a ledger, combined with the J-cost function, requires the scale ratio to obey the compositional identity $r^2 = r + 1$. Upstream, golden_constraint_unique states that any positive real satisfying the golden constraint equals φ, while self_similar_forces_golden_constraint shows that the self-similarity hypothesis on a ledger produces exactly that constraint.
proof idea
The term proof performs rcases on the self-similarity hypothesis to obtain the underlying scale structure S and a reflexivity equality. It then feeds S.ratio_pos together with the output of self_similar_forces_golden_constraint S directly into golden_constraint_unique.
why it matters
The declaration supplies the T6 step that φ is the self-similar fixed point inside the UnifiedForcingChain. It is invoked by origin_question_resolved to close the geometric-sequence closure clause and by fibonacci_char_poly_unique_pos_root to identify the unique positive root of the Fibonacci characteristic polynomial. The result therefore anchors the eight-tick octave and the subsequent derivation of D = 3.
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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."