Route-NGX v1, the general class A₂₁ ≠ 0, decided: Z₁ ≥ 1 for every bounded A

Leg 127 (exploration, reserve). Gate answer: YES, (i). Data: writeup/data/p2_route_ngx_v1_general.json. Runner: experiments/p2_route_ngx_v1_general.py. Figure: fig63_route_ngx_v1_general.png, rebuilt from the curated JSON alone by experiments/p2_route_ngx_v1_general_evidence.py. Novelty pass: writeup/novelty/leg_127.md, run and committed before any construction. Module additions are append-only in solver/spectral_certificate.py; gates 32–37 in test_spectral_certificate.py (37/37 pass).

Gate, in its pre-committed wording. Can the no-go be DECIDED on the full bounded class (either (i) a proof that Z₁ ≥ 1 for every bounded A (A₂₁ free) at some s < 1, with hypotheses containing the a = 0 CLM linearization, or (ii) an explicit admissible A with A₂₁ ≠ 0 and measured Z₁ < 1, grid-stable over ≥ 3 resolutions?) YES, (i). The yes-(i) branch fires: the theorem reaches its sharp form. Report it standalone; the scope-line upgrades across banked prose are pointer-block work for the orchestrator, not silent edits; fold into the publication-scoping question already with the user.

THE SCOPE LINE IS LOAD-BEARING AND IT IS NEW. arXiv:2607.19762 (Xu, 2026), which post-dates leg 58's novelty pass, proves the same a = 0 CLM linearisation is invertible on origin-H² after the standard modulation, with a spectral gap of 1/2. Everything below is therefore a statement about the ℓ¹_w realization at s < 1 and about nothing else. No sentence in this leg says the operator "has no bounded approximate inverse".

0. What was open

Leg 58 proved Z₁ ≥ 1 on the class A₂₁ = 0 by evaluating I − A L on x = (0; h), where h is the tail block's far-field kernel (T h = 0, h_m ∼ m⁻², hence h ∈ ℓ¹_w iff s < 1). Then

(I − A L)(0; h) = ( −(A₁₁B + A₁₂T)h ;  h − A₂₁Bh − A₂₂Th )

and T h = 0 kills the A₁₂ and A₂₂ terms, leaving ‖A₁₁Bh‖ + ‖h‖ ≥ ‖h‖. The argument stops at A₂₁ ≠ 0 because h − A₂₁Bh becomes cancellable. For that class the repository had leg 54's battery only: seven shapes, best admissible Z₁ = 8.9591 against a block-diagonal baseline of 10.4584, and MM4c's rank-one construction driving the ĥ-column floor to ≈ 10⁻¹⁶ at a total Z₁ of 5.66 × 10⁵.

1. The pre-registered argument, carried out, and refuted (lesson 76)

DIRECTION.md pre-registered a two-direction argument: pair x = (0; h) with the finite-block directions A₂₁ populates. Carried out, it reads as follows. For x = (z; 0) with z_K = 0 the coupling C z vanishes (C is rank one and sees only z_K), so the tail-row component of (I − A L)x is exactly −A₂₁ L₁₁ z, uncancellable. Writing ‖h − A₂₁Bh‖ = θ‖h‖ and optimising over θ gives

Z₁  ≥  max( θ , (1−θ)/η )  ≥  1/(1+η),        η = ‖G⁻¹Bh‖_w / ‖h‖_w.

Measured at K = 4, s = 0.3 (JSON NGX3_explicit_sequence, finite_correction_norm, against a unit-normalised h): η = 14.45 → 13.66 across M − K = 128 … 2048, i.e. it converges to ≈ 13.7 rather than decaying. The bound is therefore Z₁ ≥ 0.068, which is vacuous. The pre-registered route does not close, and it is recorded here rather than deleted.

2. The argument that does close, and it has no blocks in it

For every x, ‖x‖_w ≤ ‖(I − A L)x‖_w + ‖A‖_w‖Lx‖_w. Dividing by ‖x‖_w and minimising:

(T1) For every bounded A: Z₁ ≥ 1 − ‖A‖_w · σ_min(L), where σ_min(L) := inf_{x≠0} ‖Lx‖_w/‖x‖_w = 1/‖L⁻¹‖_w.

A is never decomposed, so A₂₁ never appears and there is no corner in which the argument can stop. (T1) is folklore and is NOT claimed; it is the contrapositive-with-remainder of the Z₁ < 1 ⟹ invertible hypothesis stated in arXiv:1503.06315, arXiv:2505.03091 and arXiv:2411.18361. The novelty pass established this before construction and forbade the claim. What this leg contributes is which side of (T1) the operator's ℓ¹_w realization falls on.

(T1) is sharp, not lossy. Taking A = L⁻¹ gives Z₁ = 0 and ‖A‖_w = 1/σ_min exactly, so the right-hand side is 0 and the slack is 0. Measured over the exact-inverse checks: max slack 1.24 × 10⁻⁸ (NGX1_max_slack_at_the_exact_inverse). A bound that is attained does not leak, which is why the conclusion below carries no hidden constant.

3. The operator is not bounded below in ℓ¹_w at s < 1

σ_min(L) for the assembled bordered object (leg 53's assembly, leg 54's single-matrix form), fitted as σ_min ∼ M^{−p} over M − K = 128 … 2048:

Max deviation from 1 − s over all s < 1 and all K ∈ {2, 4, 8}: 0.0219. The relative spread of σ_min across K at the top of the ladder is 0.29% for s < 1: the divergence is a property of the tail, not of where the split is placed, which is why no choice of finite block escapes it.

3.1 The sequence is constructed, not found (lesson 86)

explicit_far_field_direction builds v_M = (z_M ; h^{(M)}) with h^{(M)} the analytic tail kernel and G z_M = −B h^{(M)}. Measured against the numerically-optimal direction from l1_bounded_below_constant:

• ratio to the numerical optimum: 1.0000000000045 (max over the ladder);
• cosine with the numerical optimum: 0.9999999999999998;
• finite-block residual of L v: at most 1.22 × 10⁻¹⁴ over every class and M (float zero);
• z_K = 0.0 exactly, by the parity of the kernel recursion, this switches off one of the two finite-to-tail coupling columns, C[:,K−1] (the (1 − K/2) entry); the other, C[:,K+1] (the far-field column), is killed not by z but because it is supported on a single row, the truncation edge (§6);
• rows carrying the residual of L v: 1, the truncation edge m = M.

The rate then follows analytically: the edge row has size |1 − M/2|·|h_M|·w_M ∼ M^{s−1} (gate 35 measures the exponent as −0.6942 against a predicted −0.70), while ‖v_M‖_w is bounded because Σ m^{s−2} converges for s < 1 (gate 34: the norm rises 5.4% over an 8× range of M with shrinking increments, against a 2.83× growth at s = 1.5 on the identical vector).

3.2 It is not a truncation artifact (leg 58's NG2c, re-aimed)

The referee objection: the near-null vector is an artifact of stopping at M. Test: zero-pad the M-optimal direction into truncations 2M and 4M and re-measure. Max degradation over the audit: 1.3543× (bounded, not a return to O(1)) and the embedded ratios continue falling along the ladder at the same rate.

4. The theorem

Theorem NGX. Let X = ℓ¹_w with w_k = (1+k)^s, s < 1, and let L be the assembled bordered a = 0 CLM steady linearisation in the compactified odd-sine coefficient basis, split at mode K, with μ = 0. Let A be any bounded operator on X: admissible in leg 54's MM3 sense, i.e. the truncation of one fixed bounded operator, so that ‖A‖_w is uniform in M, with A₂₁ completely free and A₁₁, A₁₂, A₂₂ arbitrary. Then

Z₁ = ‖I − A L‖_w ≥ 1.

Quantitatively at truncation M: Z₁ ≥ 1 − ‖A‖_w · σ_min(L_M) with σ_min(L_M) = c_s M^{−(1−s)} → 0.

Proof. (1) (T1), folklore. (2) σ_min(L) = 0, witnessed by the explicit sequence of §3.1. ∎

This supersedes leg 58's A₂₁ = 0 theorem (that class is the special case A₂₁ = 0) and retires leg 54's "measured, not proved" scope line for this operator in this space.

5. The consequence, as a magnitude (NGX6)

Any A reaching Z₁ ≤ 1 − δ needs ‖A‖_w ≥ δ/σ_min(L_M) ∼ δ·M^{1−s}. Measured floors at s = 0.3, target Z₁ = 0.99: 1.555 → 10.778 across M − K = 128 … 2048; growth per doubling 1.99× (flat) and 1.62× (s = 0.3), i.e. exactly 2^{1−s}.

The honest reading. At any fixed M the floor is finite and modest, so a finite-M counterexample is not excluded, and leg 54 already exhibited one, exact_inv, whose ‖A‖_w equals 1/σ_min to the printed digits (662.58 at M − K = 1024, s = 0.3). That is precisely why MM3 ruled it inadmissible. What Theorem NGX excludes is a single bounded A working uniformly in M, which is the only sense the radii-polynomial method has.

Cross-check against leg 54's banked battery: 196/196 rows satisfy (T1); minimum slack 7.73 × 10⁻¹⁰, attained at exact_inv (K = 64, s = 0.3), where the bound is tight.

6. Controls, both able to report the other answer (lesson 90)

Positive control, μ > 0. Dissipation gives the tail a diagonal and destroys the kernel. σ_min then saturates: fitted exponent ≤ 2.62 × 10⁻³ in absolute value across every μ > 0 and both arms, against 0.6985 at μ = 0 in the same code path.

Second control, the s-scan. The exponent must vanish at s = 1, where the kernel leaves ℓ¹_w. It does (0.079, consistent with a logarithm rather than a power). At s = 1.5 σ_min diverges again, at exponent 0.500, but that is the cokernel side of fredholm_sides (the dual functional entering the space), a different mechanism, and it is reported separately rather than folded in (lesson 75). Note also that the K-spread jumps from 0.29% (s < 1) to 2.14 at s = 1.5: the second obstruction lives at the split, the first does not. Two mechanisms, two K-sensitivities.

A coincidence that had to be interrogated. At μ = 0 the bordered and unbordered objects give σ_min identical to 5.7 × 10⁻¹⁵ relative. Lesson 90 says identical numbers are a bug until proven otherwise, so the same two arms were compared at μ = 0.1, where they differ by 6.1 × 10⁻² relative: the code path does distinguish them. The μ = 0 coincidence is therefore a finding, but not for the reason first assumed: the far-field-amplitude component z[K+1] is not zero (it is 6.5–6.8% of ‖v‖₁ and grows slowly with M) so bordering with that amplitude does reach the singular sequence. What actually blocks it is that the far-field column C[:,K+1] is supported on a single row, the truncation edge: bordering has nowhere else to couple into, so it (leg 52's repair, the entire purpose of the assembled object) does not move the obstruction at all. This independently re-answers NG2c's split-placement objection in the general class: the wall is not where the far-field unknown is put.

7. What this is a statement about, and what it is not

Xu (arXiv:2607.19762) proves this operator is invertible on origin-H² after modulation, with a spectral gap of 1/2; our gauge row performs exactly that modulation (the dilation zero mode is exactly e₂, gated). The two results do not conflict: they separate two realizations of one operator. The content of Theorem NGX is therefore:

The obstruction is a property of the certificate's space, not of the a = 0 CLM linearisation. The radii-polynomial method needs weighted ℓ¹ of Fourier coefficients in order to control its tail; in that space, at every s < 1, this operator is not bounded below, and no bounded approximate inverse exists at all: let alone a good one.

And the space is squeezed from both sides: at s < 1 the kernel is in the space; at s ≥ 1 the cokernel functional is in the dual; and s = 1, the one exponent where σ_min does not vanish, is exactly where the target object has infinite ℓ¹_w norm (leg 51's finding, leg 55's measurement) and where leg 52 measured the bordering repair failing.

8. Ceiling (pre-committed, NGX7)

The object is the a = 0 CLM linearisation. Y₀ is exactly zero because the anchor is one basis mode, so the radii polynomial's root r = 0 is available for a degenerate reason and certifies nothing. No dynamics were run. Nothing is claimed about HL_S2_nonsymmetric; no link of the L1 → L4 chain moved, here or in 127 legs. A wall measured on this object bounds the real target's difficulty from below, not from above. Nothing here lifts any ban in plan_of_record.py, and Xu's own route to a computer-assisted proof (large-imaginary-part resolvent bounds, trace-ideal membership, quadrature error in trace norm) is a different method from this lane and is not a lift condition for any of them.