G-001 — Multiple-zero density bounds as extra chain constraint ("Idea C")

Date: 2026-08-17 | Session: 001 | Track: A | Tag: [FAILED-ATTEMPT(obstruction below)] Opened: D-009 §4 (Idea C), thread T5. Killed: same day, before literature investment — the obstruction is unconditional on any s₂ bound.

Attempt

Feed an unconditional upper bound on on-line multiple zeros (s₂ ≤ κN, Levinson/CGG-style input) into the chain as an extra constraint, hoping to beat 2 − R(ψ). Rationale at opening: the 0.682 ceiling theorem constrains only 2-moment certificates; multiplicity bounds are a different data class.

Obstruction (the s₂/p substitution symmetry)

On-line doubles and off-line pairs play IDENTICAL roles in the chain: both cost 2 in N(I′), both occupy the Q′ positive-index budget at the flat charge 4 (D-005). Replace each on-line double at ordinate γ by an off-line pair {½+δ+iγ, ½−δ+iγ}. In every bandwidth-one two-moment datum: e^{2πij(γ+iδ)/N} + e^{2πij(γ−iδ)/N} = 2e^{2πijγ/N} + O(δ) (δ→0), which equals the double's contribution m=2. Hence the adversary moves all multiple mass into depth→0 off-line pairs leaving every usable measurement unchanged; any s₂-bound is vacuous against this configuration.

• For A(i): extremal realizable with s₂ = 0 ⟹ no gain from bounding s₂.
• For A(ii): pairs are distinct zeros ⟹ substitution only RAISES N_d. Same invisibility that killed zero-density input (D-009 Idea A); paper already flags it (§7.2(b): "off-line pairs of depth→0, spectrally the same"; §1.4 insensitivity to o(N) off-line zeros).

Lesson

The binding wall is not moments, windows, or multiplicities — it is that OFF-LINE-PAIR MASS at shallow depth is invisible to every unconditional data class in the method. Any certificate improvement must either (1) see depth directly (needs prime-pair/support>1 data — walls S1/S2, D-006 §6), or (2) constrain p by new unconditional input — and p-bounds are essentially RH-strength (the D–H/Epstein gate). Multiplicity-side inputs (s₂) are provably useless within this framework.

Salvage

• The substitution argument itself is a clean, quotable lemma for the paper's obstruction-map section ("no multiplicity-side input can improve the certificate").
• Cross-track value: confirms that the ξ′-analogue (Remark 7.1, thread T4) works BECAUSE ξ′-zeros behave differently — worth understanding precisely why (the extra derivative breaks the s₂/p symmetry there?). → keep thread T4 open with this sharper question. → ANSWERED (D-022, session 004): the symmetry does NOT break — it is measurement-side and function-independent. The ξ′ gain is data-side (perfect-square node in the diagonal law D₁ ≈ s(1−2s)² from the −Λ + (2/l)(Λ·log) cancellation); not portable back to ξ. The salvage hypothesis above is hereby corrected.