Iteration-25 prep — LEM-A1‴-T-ℂ: the complex-c extension (DRAFT)

Status: [DRAFT — UNREFEREED]. Integrator's hand computation, written 2026-07-12 while the iteration-25 fleet waits on a rate-limit window. The iteration-25 scope agent identified this as the single possible upstream gap for the Feshbach route: if route (i) uses boundary values of V_eff(z) for z off the real axis near κ, the Riesz-basis machinery is invoked at complex c(z). This note drafts that extension. It must be refereed (default REFUTE) before anything consumes it.

Statement

LEM-A1‴-T-ℂ (draft). Fix c₀ > 1. There is an open neighborhood N ⊂ ℂ of c₀ and a constant family of bounds such that for every c ∈ N the pencil family {a_n(c) = cos(s_n η) + i·tanh(πτ(c))·sin(s_n η), b_n(c) = cos(s_n η)}, s_n = n + iτ(c), is a Riesz basis of L²((0,π); ℂ²), with frame bounds jointly continuous in c and bracketing the real-c bounds of LEM-A1‴-T as c → c₀.

The five steps (each elementary)

Holomorphy of the corner frequency. τ(c) = arccosh(√c)/π is holomorphic on a neighborhood of (1, ∞) in ℂ (√· is holomorphic off (−∞,0]; arccosh is holomorphic off (−∞,1]; √c₀ > 1). Write τ(c) = ρ(c) + iσ(c); on a small enough N, ρ > 0 and |σ| < ½ (the ½ matters only for step 2's uniformity, any bound < ∞ works).
Weighted Fourier survives complex weights. E_n(c) = e^{i s_n x} = e^{−τ(c)x}e^{inx} on (0, 2π). Multiplication by e^{−τ(c)x} factors as [multiplication by the unimodular e^{−iσ(c)x}] ∘ [multiplication by the positive e^{−ρ(c)x}]. The first factor is unitary; the second is bounded invertible with bounds [e^{−2πρ}, 1]. So {E_n(c)} is a Riesz basis of L²(0, 2π) with frame bounds [2π e^{−4πρ(c)}, 2π] — the real-c proof verbatim, with |·| taken through. (No use is made of σ = 0.)
The folding miracle is τ-agnostic. (Uf)(η) = (f(η), f(2π−η)) is unitary exactly as before, and (U E_n)(η) = (e^{i s_n η}, q(c)·e^{−i s_n η}) with q(c) = e^{2πi s_n} = e^{2πin}·e^{−2πτ(c)} = e^{−2πτ(c)} — independent of n for any complex τ. The n-independence used only two facts: integer real parts of s_n, and an n-independent imaginary part. Neither requires τ real.
The intertwiner persists by continuity. The coefficient data v±(c), t(c) = tanh(πτ(c)) are holomorphic in τ hence in c on N (tanh has poles only at πτ ∈ iπ(ℤ+½), i.e. τ ∈ i(ℤ+½) — away from τ(c₀) > 0 real). det M(c) = (t(c)/2)e^{2πτ(c)} is holomorphic and nonzero at c₀ (t(c₀) ∈ (0,1)); shrink N so |det M(c)| ≥ ½|det M(c₀)| on N̄ compact. M(c) is constant in η, so ‖M(c)‖, ‖M(c)⁻¹‖ are continuous on N̄, uniformly bounded.
Assembly. F_n(c) = M(c)(U E_n(c)) exactly, for all n, by the same algebra as the real case (the identity is rational-exponential in s_n; it holds as an identity of holomorphic functions of τ, having been verified on the real τ interval — uniqueness of analytic continuation). A bounded invertible operator carries a Riesz basis to a Riesz basis; the frame bounds multiply by [‖M⁻¹‖⁻², ‖M‖²]. ∎ (draft)

What still needs care (referee targets)

(R1) The eigenvector interpretation at complex c. Steps 1–5 make {a_n(c), b_n(c)} a Riesz basis of defined formulas. That they are the eigenvector family of the analytically-continued transverse pencil needs the Birkhoff data continued: the characteristic quadratic z² − 2(2c−1)z + 1 has discriminant 16c(c−1) ≠ 0 for all c ∈ N (shrinking N off {0,1}), so the root system continues holomorphically with no collision. Verify the continued lattice is exactly {n + iτ(c)} (it is, by the same cos²(πs) = c computation — but state it).
(R2) Step 5's analytic-continuation appeal. ~~The identity F_n = M(UE_n) was machine-verified at real c only.~~ Done (2026-07-12): machine-verified at four complex points c ∈ {1.5+0.1i, 4−0.25i, 1.02+0.01i, 25+i}: max relative intertwiner error 4.4×10⁻¹⁶ over n = −6..6; det M matches (t/2)e^{2πτ} to 7×10⁻¹⁵; Re τ(c) > 0 at all four points as step 1 requires. (Also true symbolically: writing cos and sin as exponentials, F_n = M(UE_n) is finite trig algebra valid for every complex s.)
(R3) What the Feshbach route actually needs. If route (i) only ever evaluates on the real axis (limiting absorption from above), this whole note may be unnecessary — the scope agent's caveat was conditional. The consumer should say which.

Appendix — referee-support data for the Bari supplement (2026-07-12)

The iteration-25 Bari prover (output cached, unrefereed) claims the Lipschitz constant L(τ₀) = π·√(5π/3)·cosh(π(τ₀ + ½)) for the eigenvector map Φ(s) on the strip |Im s − τ₀| ≤ ½. Integrator's numerical check (sup of ‖∂ₛΦ‖_{L²} over Re s ∈ [0,40], Im s spanning the strip; Lipschitz constant on a convex set = sup of derivative norm):

τ₀ (c ≈)	L claimed	measured sup‖∂ₛΦ‖	ratio	verdict
0.0462 (1.02)	20.64	8.79	0.43	HOLDS
0.2108 (1.5)	33.92	11.03	0.33	HOLDS
0.4196 (4)	64.81	17.87	0.28	HOLDS
0.7298 (25)	171.30	42.07	0.25	HOLDS

The claimed constant is valid with 2.4–4.3× headroom (an over-estimate is harmless for the Bari argument — only finiteness and explicitness are load-bearing). Worst point sits at Re s = ½, Im s at the top of the strip, consistent with the cosh envelope. [measurement — supports, does not referee, the claimed proof]

Addendum 2 (2026-07-12) — two new structural observations for the iteration-25 fleet

(N1) The compact-core exclusion program [NEW ROUTE — DRAFT, unrefereed]. Never tried in 24 iterations: computer-assisted spectral exclusion. The pieces now exist: (i) the scope verdict (compact-window architecture is forced AND sufficient; the gate follows from exhausting (1,∞) by compacts, with the four asymptotic corners covered separately); (ii) iteration 24's explicit constants (frame bounds, closed-form Gram, exact lattice) — the precondition rigorous numerics always needs; (iii) the extended Birman–Schwinger principle for nonlocal kinetic parts (Ishida–Lőrinczi–Sasaki, arXiv:2109.01564). The criterion: with A = A₀ + K, A₀ = coth²(πD), for κ in a window W ⊂ (1,∞), κ ∈ σ_p(A) requires 1 ∈ σ(BS(κ)) for the Birman–Schwinger operator of the pair (A₀, K); a rigorous interval-arithmetic bound ‖BS(κ)‖ < 1 uniformly on W × [mR-box] is an unconditional no-eigenvalue theorem on that box — no decay-class reasoning (so no BIC-no-go conflict: the bound is computed for THE operator, not a class). Architecture: compact (mR, κ) core by machine; corners mR → 0 (perturbative Mourre, iteration 22), mR → ∞ (large-mass asymptotics — open, name it), κ → 1⁺ (FG-1989/LEM-K0 threshold ledger), κ → ∞ (scope's vacuity note). Honest caveats: needs (H1)-type relative compactness to ground BS; interval-arithmetic discretization error must be bounded by the explicit frame constants — that bound is the real work. Status: route minted, nothing proved.

(N1a) First feasibility scan — the NAIVE splitting fails; the bounded-transform form is the viable one [measurement, 2026-07-12]. Lattice scan at L = 192, m ∈ {1, 2}: with raw K = A_sym − coth²(πD_lat), ‖K‖ ≈ 9.7 — as large as the lattice A₀'s whole spectral range [1, 10.9] (the two regularizations disagree at the spectral top: box quantization vs Peschel — an artifact concentrated at small ε), and ‖|K|^{1/2}(A₀−κ−iη)^{-1}|K|^{1/2}‖ > 1 at every tested κ ∈ [1.5, 20]. So the compact-core program must be built on the bounded-transform (Konno–Kuroda-type) Birman–Schwinger, not the raw splitting — and there the feasibility indicator already exists: the iteration-23 resolvent probe measured ‖(A_sym+1)^{-1} − (coth²(πD)+1)^{-1}‖ ≈ 0.377–0.397 (refinement-stable) with fast singular-value decay (sv[12] ~ 4e-4). A resolvent-difference norm < 1 with rapid decay is exactly the shape a Konno–Kuroda exclusion needs. Route refined, still nothing proved; the interval-arithmetic-with-explicit-constants step is untouched.

(N1b) The mR → ∞ corner does NOT close by norm bounds — own conjecture refuted, trap documented [measurement + retraction, 2026-07-12]. The integrator conjectured the fourth corner of the compact-core program (mR → ∞) closes by norm collapse (‖A(mR)‖ → 1). A two-L lattice scan refutes the design: the apparent growth of ε_min with mass is a pure lattice-mass artifact — the signature is exact row-shift ε_min(L=128, m) = ε_min(L=256, 2m) (e.g. 2.0339 vs 2.0340), i.e. ε_min is a function of m_lat = 8m/L alone in the scanned range. In the continuum, the iteration-18 law (ε_min·ln L → π² at fixed mR) means σ(A) = [1, ∞) a.c. for every mR: A is unbounded, no norm-collapse corner closure exists. The mR → ∞ corner of the compact-core program is genuinely OPEN (what varies with mR is spectral weight, not spectral support). TRAP for future probes: any large-mass lattice scan must include the two-L row-shift check before reading mass dependence as physical.

(N2) Exact parity split [elementary — verify and consume]. Q(−ξ, η) = Q(ξ, η) and the η-edge conditions are ξ-independent, so interval reflection z ↦ −z (⟺ ξ ↦ −ξ) commutes with the full coupled system and with A. Every candidate eigenfunction splits into even/odd sectors; LEM-A1⁗'s transversality statement decouples into two half-multiplicity statements, one per parity — simplifying the local Pontryagin index count. (Consistent with the reflection-degenerate ladder observed numerically in iteration 18.)