Explicit Local-Unit Cancellation for $389.a1$ at $p=11$

Following up on v0.7 (which was not separated into its own suite; this document explains its own preamble) which compressed the problem down to the local descent gates P5-DESC11 and P5-VAL011, the goal of this document is to individually isolate every explicitly computable 11-local factor in the Burns-Kurihara-Sano rank-2 element to see if there are still invisible denominators hiding. Using the minimal model, it precisely calculates #E(F₁₁)=16, a₁₁=-4, and the good prime truncation factor L₁₁(E,1)⁻¹=16/11 (valuation -1); the split multiplicative truncation factor for the bad prime 389 is 388/389 (valuation 0). The true technical core is precisely calculating the valuation of the 11-adic logarithm: using standard parameters of the formal group, it precisely calculates that 16P' falls into the formal group and v₁₁(t(16P'))=1. Combined with the property that the formal logarithm on t∈11ℤ₁₁ agrees with t itself modulo 11², it deduces v₁₁(log_ω(P))=1 with a residue of 4 mod 11. Multiplying the three factors: v₁₁(u_loc)=(-1)+0+1=0, and u_loc≡4 (mod 11)—which is exactly an 11-adic unit. This means that under the already fixed normalization, multiplying by u_loc changes neither the domain barrier nor the valuation, hence β_ξ·u_loc∈ℚ₁₁ ⟺ β_ξ∈ℚ₁₁, and their valuations are equal. The document explicitly states: this does not prove β_ξ∈ℚ₁₁ nor does it prove its valuation is zero; the only confirmed positive result is that "there are no hidden 11-local denominators needing explanation"—eliminating an entire erroneous direction that could have otherwise consumed research time.