Golden Germ Third-Order Ledger

Abstract

The explicit third-order golden ledger cancels every local mode below phi to the fifth power and leaves a prime-summable normalized deviation.

Theorem 1.1 (Third-order local cancellation reaches the phi-fifth boundary).

$$\begin{array}{r}\forall s\in \mathbb{C},\\ \frac{1}{{\phi }^5}<\Re (s)\Rightarrow \\ x(p):=p^{-s\times {\phi }^2},y(p):=p^{-s\times {\phi }^3},\mathrm{Summable}(p:\mathrm{Primes}(\mathbb{N})\mapsto |(1-y(p{)}^2{)}^{-1}\times (1-x(p{)}^2\times y(p))\times (1-y(p))\times (1+x(p){)}^{-1}\times \sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)}-1|).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/EulerGerm/GoldenGermThirdOrderLedger.golden_third_normalized_factor_deviation_norm_summable (✓ std3). ∎

Source. Repository-derived.

Commentary.

The proof computes beta at modes four and five: mode four has weight two phi-squared plus phi-cubed, while mode five has weight exactly phi to the fifth power. Splitting after six modes isolates the remaining tail.

The frozen golden_germ_second_order_factorization gives the global factorization and its unique continuation, but does not expose the local normalized remainder needed here. This ledger instead reuses the canonical definitions germLocalFactor and o5Beta, together with o5_beta_zero, o5_beta_power_law, o5_beta_closed_form, and o5_beta_growth, and proves the local identity independently from the six-mode expansion. Its displayed factors cancel minus y-squared and plus x-squared y; the first retained monomial x y-squared lies exactly on the threshold, and the tail starts there and grows linearly.

This is the next local extraction step on the golden Euler germ staircase used in OACTC parts 580 and 581 and on the RH-route O-5 control line. It advances the absolute-summability boundary to real part greater than one over phi to the fifth power. It does not assert O-5, a global continuation or nonvanishing theorem, or the Riemann Hypothesis.

References

• Truth anchor: D5/S3/Analytic/EulerGerm/GoldenGermThirdOrderLedger.golden_third_normalized_factor_deviation_norm_summable