phi_upper
plain-language theorem explainer
The golden ratio φ, realized as an element of ℤ[φ], sits strictly below the rational 1618034/1000000. Anyone citing the certified sigma0 ladder-ratio brackets for the forced dimensionless ratio uses this upper bound. The proof is a one-line `decide` on a decidable positivity predicate built from integer arithmetic on ℤ[φ].
Claim. With $\varphi$ the golden-integer unit $1+\varphi_0$ in $\mathbb{Z}[\varphi]$, the strict rational upper bound $\varphi < 1618034/1000000$ holds, equivalently the integer witness for $0 < 1618034 - 1000000\cdot\varphi$ is positive under the sigma0 sign predicate.
background
Recognition Science forces a single primitive dimensionless ratio φ (T6 fixed point). Masses and other ladder observables then sit at integer powers φ^k of that ratio. This module certifies those powers without continuum analysis: phiPow k is an explicit element of ℤ[φ], and rational comparisons are integer predicates.
RatGt p q x asserts that the positive rational p/q lies strictly above x ∈ ℤ[φ]. It is defined by feeding the witness q·x − p (up to sign) into the already-decidable positivity predicate IsPos on golden integers. Because √5-irrationality is proved at sigma0, ties never occur and every concrete instance reduces by kernel decide inside {propext, Quot.sound}.
The local setting is the DeltaSpine ladder-ratio program: one forced ratio carried end-to-end with rational brackets that both the kernel and #eval certify, before any real embedding.
proof idea
One-line tactic proof: by decide. Unfolding RatGt 1618034 1000000 phi yields an IsPos goal on an explicit golden-integer linear form. The decidable instance (inherited from the sigma0 sign predicate on ℤ[φ], via the √5 square-comparison test) closes the goal with no further lemmas or classical choice.
why it matters
Supplies the upper half of the certified φ bracket inside ladder_ratio_brackets, the sigma0 bundle that pins φ, φ⁵, and φ⁸ each inside an explicit rational interval by pure integer arithmetic. That bundle is the arithmetic core quoted by ladder_ratio_real_brackets, which only translates the same inequalities into ℝ (sigma1).
In the forcing chain this is the concrete numerical face of T6: φ is not merely named; it is trapped between 1.618033 and 1.618034 with a machine-checkable proof that never leaves {propext, Quot.sound}. Downstream mass-law and octave (φ⁸) displays inherit the same certified precision. No open scaffold remains on this conjunct.
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