Where a certificate for this operator can live, and where it cannot: the space axis, mapped and closed

Route PUB2, leg 186, v1. DRAFT, for the user's review. This leg's landing does not approve it.

This is the SECOND of two publication-scoping drafts, and it is a different document from the first. TECHNICAL_P2_PUB1_V1.md (leg 179) asks why the method failed in weighted ℓ¹: four separable causes, of which the A₂₁ inequality is one. This note asks a different question: given that it failed there, where else could a certificate for this operator possibly live? The two notes share exactly one table (leg 127's σ_min ladder), which PUB1 uses as the proof of a theorem and this note uses as the measurement of one endpoint of an axis. They are not to be merged without a deliberate editorial decision.

Status. Assembled entirely from banked results; nothing in it is new to the world, and its only contribution is arrangement. Every number is quoted from the JSON of the leg that produced it: the provenance audit is writeup/novelty/leg_186.md, which lists each number against the file and key it was read from.

What this note is not. It is not a certificate, not a theorem about the Navier–Stokes equations, and not a claim that any link of this project's L1 → L4 chain has moved. None has, in 275 legs (count current as of leg 276). The object throughout is the a = 0 Constantin–Lax–Majda (CLM) steady linearisation (an already-solved, already-published model) and every magnitude below bounds the difficulty of the real target from below, not above.

The sentence this note may not contain, and does not. "Xu proves the operator invertible, therefore a certificate is possible." That phrase was banned by the leg that read Xu at primary source, for the reason given in §3.3, and the ban is inherited here.

0. The question, and why three results answer it and not one

A radii-polynomial / Newton–Kantorovich certificate is not built on an operator. It is built on an operator in a space. Seven successive legs of this project failed to close such a certificate for a 1D fluid transport model, and PUB1 anatomises the failure. This note asks the question that survives the anatomy: the failure happened in one space: is the space the problem, and is there a better one?

That question has an axis, and the axis now has four points on it, all measured, all banked:

The convention note. The σ_min digits printed in this note are convention-relative; the property they are used for is not. Recorded by leg 249 §10 (W5) and carried into p2_route_h2c_v1_construction_correction_leg268.json as convention_caveat_recorded_by_249: the X ⊕ ℂ Gram puts unit weight on the border amplitude while the X block carries the bare Laguerre normalization (no 2π, no half-line ½). That is a choice. Sweeping that weight over 10⁻² … 10² moves σ_min across 0.01086 … 0.13580, a factor of 12.5, and adopting Xu's own y-space normalization for the X block (i.e. carrying the 2π) gives 0.0420 instead of 0.0908. [CORRECTED 2026-08-11, leg 280, per leg 277's verified finding.] 0.0420 is Xu's Definition 4.1 equivalent full-line norm (κ = 2π); Definition 4.1's own displayed half-line definition (4.2) carries π, not 2π, and gives 0.057643 instead (‖R‖_X = 17.348, not 23.792): 1.37× larger than 0.0420. Both are Xu-normalization readings; which one "Xu's own normalization" means was previously unresolved and is now named. In the same breath, because it is what keeps this from touching any conclusion below: the property leg 176's gate turns on (σ_min bounded away from zero uniformly in the truncation) is invariant under that weight, since a positive weight cannot send a positive limit to zero; and the only thing §§3–5 ever argue from is the contrast in ladder behaviour (flattens here, decays like M^{−(1−s)} there), which is likewise convention-invariant. Leg 249's own sentence was "the digits are not invariant, and PUB2 quotes the digits without the convention." This note now quotes the convention, and every later printing of 0.0908 / 0.090804 (in §3.2, §3.5, §4.5, §5(3) and §7) points back to this paragraph rather than repeating it.

Read separately these are three leg reports. Read together they are one statement, and §5 is that statement. The order below is the order a practitioner meets the decisions in: pick the space, discover it is dead, look for a better one, ask whether anything in between helps.

The fourth point was commissioned, and it has since landed. A construction leg on the origin-H² formulation (leg 176) was authorised and in flight at the time this note was drafted; it had then committed only its novelty pass (no runner, no data, no gate answer) and nothing was attributed to it or predicted for it. It has now landed (bb0f184), and its outcome is folded in at §3.5 as the fourth data point, quoted from its own report. §3.4 preserves, unaltered, what was quotable at drafting; §3.5 is an addition on top of it, not a rewrite of it.

Two limits on that fold-in, stated here rather than discovered later. (1) §§5–6 below were written when the axis had three points and are left as drafted; they are the three-result synthesis, and §3.5 says explicitly which of their sentences leg 176 makes more precise and which it leaves standing. (2) The verification leg originally commissioned against leg 176 (leg 192) committed only its own pre-registered novelty / prior-art pass (no verification runner, no re-derivation, no verdict) but the independent re-derivation has since been done and has reported: leg 249 re-derived leg 176's construction in exact rational arithmetic (its journal, novelty log and p2_route_h2cv_v1_postconstruction.json are on main, and byte-identical to the originating branch leg/249-h2cv2-v2 at e9db984, which is not itself an ancestor of main), confirming both conjuncts and correcting two banked values below the digits quoted here; a subsequent review pass re-pulled leg 249's figures from that branch and matched them character for character. Leg 176's numbers are therefore carried here as leg 176's float64 measurements, with leg 249's exact tier as the independent check on them: the corrections are recorded at §3.5 and in writeup/data/p2_route_h2c_v1_construction_correction_leg268.json.

1. Naming the axis honestly first, because the obvious framing is wrong

It is natural to picture ℓ¹_w and origin-H² as two points on one scale, with an interpolation family running between them. They are not. The two spaces differ in three independent coordinates, and this was the first finding of the leg that went looking for the interpolant:

So "interpolate between them" is not one move; it is at least two, and the third coordinate is not a space parameter at all. §4 checks the two that can be interpolated, separately, and finds them separately dead. A note that reported "no interpolant works" without saying this would be reporting a confusion rather than a result, and this is stated first for that reason.

2. Endpoint one: weighted ℓ¹, where the operator is not bounded below

The result. In ℓ¹_w with w_k = (1+k)^s and s < 1, for the assembled bordered a = 0 CLM steady linearisation L at μ = 0, every bounded operator A on the space (with all four blocks arbitrary, including A₂₁) satisfies

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.

The two ingredients, and only one is this project's. The inequality Z₁ ≥ 1 − ‖A‖ σ_min(L) follows from ‖x‖ ≤ ‖(I−AL)x‖ + ‖A‖‖Lx‖ and is the contrapositive-with-remainder of the Z₁ < 1 ⟹ invertible hypothesis every radii-polynomial paper states. It is folklore and it is not claimed here. What is this project's is the other half: that σ_min(L) = 0 on this operator in this space at every s < 1.

The measurement. σ_min ∼ M^{−p} fitted over M − K = 128 … 2048 at K = 4:

Max deviation from 1 − s over all s < 1 and all K ∈ {2, 4, 8}: 0.0219. Relative spread of σ_min across K for s < 1: 0.29 %, so the divergence is a property of the tail, not of where the finite/tail split is placed, which is why no choice of split escapes it. (At s = 1.5 that spread is 2.14, because there the obstruction is the cokernel entering the dual, a second and separate mechanism, reported separately rather than folded in.)

The witness is constructed, not found. An explicit sequence v_M = (z_M ; h^{(M)}) built from the tail block's analytic kernel matches the numerically-optimal direction to a ratio of 1.0000000000045 and a cosine of 0.9999999999999998, with z_K = 0.0 exactly and the entire residual of Lv carried by one row, the truncation edge.

The bound is attained, so nothing leaks. At A = L⁻¹ the inequality is an equality: max slack 1.2366e−08. Across the previously banked shape battery it holds 196 / 196 with minimum slack 7.7343e−10.

Two controls that can report the other answer. (i) Switch on dissipation: σ_min saturates, fitted exponent ≤ 2.63e−03 in absolute value at every μ > 0, against 0.6985 at μ = 0 in the same code path. (ii) A truncation-artifact audit, zero-pad the M-optimal direction into 2M and 4M, degrades by at most 1.3543×, bounded, not a return to O(1).

The repair that was the whole point of the assembled object does nothing. Bordering with the far-field amplitude changes σ_min by 5.7e−15 relative at μ = 0. The mechanism is not that the singular sequence has no far-field-amplitude component: that component is 6.5–6.8 % of ‖v‖₁ and growing with M. It is that its coupling column is supported on a single row, the truncation edge, so bordering has nowhere else to reach. (The same two code arms differ by 6.1e−02 at μ = 0.1, so the comparison is live in both directions.)

The consequence, as a magnitude. Any A reaching Z₁ ≤ 1 − δ needs ‖A‖_w ≥ δ/σ_min(L_M) ∼ δ·M^{1−s}; the measured floor grows 1.99× per doubling of M at s = 0 and 1.62× at s = 0.3, i.e. exactly 2^{1−s}. At any fixed M the floor is finite, so a finite-M counterexample is not excluded: one is already banked, and was ruled inadmissible for exactly that reason. What is excluded is a single bounded A working uniformly in M, which is the only sense the method has.

The scope line, which is load-bearing. This is a statement about the ℓ¹_w realization at s < 1 and about nothing else. No sentence here says the operator has no bounded approximate inverse: §3 is the reason it may not.

3. Endpoint two: origin-H², viable in form, capped in value

3.1 What was checked, and what passed

The published fact this endpoint rests on is not this project's: Xu (arXiv:2607.19762) proves the same a = 0 CLM linearisation is invertible on origin-H² after the standard modulation, with a spectral gap of 1/2. The question a scoping leg could actually answer is narrower and structural: does that realization supply a split, a shape for the approximate inverse, and freedom from §2's obstruction?

All three, yes, each re-derived from primary source rather than cited:

The supporting identities, re-derived numerically with every derivative analytic (three independent quadratures, no finite differences): the resolvent identity holds to a worst relative residual of 2.807e−14 over 18 (datum, z) combinations including complex z; the two symmetry modes L₀⁺(b⁻²) = b⁻² and L₀⁺(y b⁻²) = 0 hold to 1.790e−15 and 5.266e−16, and both also hold exactly by hand.

The bordered formulation is explicit, not gestured at. At z = 0 the only singular object in the kernel is a rank-one functional of the data, so

[ L₀⁺ m ] [ u ] [ f ] m(y) = y b(y)⁻² (the z = 0 null mode) [ ℓ 0 ] [ κ ] = [ 0 ] ℓ(f) = i f(0) − f′(0)/4 (the only z = 0 pole): one border column and one matching row per Hardy block, rank two on the real odd space. That is the same shape the ℓ¹_w assembly already used. Bordered-solve residuals are ≤ 2.221e−14; ℓ(f) after projection onto the solvability subspace is exactly 0.0 in all three test data.

3.2 The contrast with §2, as magnitudes

(Convention. The 0.0908 in this table, and every σ_min digit below it in this note, is convention-relative by a factor of 12.5: §0's convention note states the sweep and Xu's own 0.0420. What this table is here for is its truncation-dependence row (decays there, flattens here) and that contrast is convention-invariant.)

(Provenance of the two numbers, and why they were never in conflict. 0.71465 is the five-decimal rounding of leg 163's own banked implied_sigma_min_lower_witness = 0.7146549471256172, which is the reciprocal of its largest sampled ratio in full precision (1/1.39927667753796). No intermediate rounding of the ratio enters: the chain is one step, not two. 0.71465 is the display form; the witness bound itself is σ_min ≤ 0.71465495, from which the five-decimal figure differs by 4.9e−06: far below any digit this note uses it at, but recorded here so the printed form is not mistaken for the bound itself. Its producing leg wrote it as a lower witness σ_min ≥ 0.7147; that inequality is inverted. With f = L u, each datum gives ‖f‖_X/‖u‖_X = 1/r ≥ σ_min, so a finite family of sampled ratios bounds the infimum only from above, a sample can miss the worst direction and here it did, by a lot: leg 163's best ratio reached 12.71 % of the true ‖R‖_X = 11.0127, and leg 176's own four-datum re-run of the same construction reached only 7.88 % of it. Once the sign is read correctly the two legs agree: 0.0908 ≤ 0.71465. Under the erroneous ≥ reading they would not even be self-consistent, since the reciprocal of leg 176's re-run family is 1.1519, which cannot also be a lower bound on the same quantity. 0.0908 is the figure used throughout §3–§5; 0.71465 is retained here only as the (true, and very loose) upper bound leg 163's data actually establish. See §3.5.)

3.3 The ceiling, stated at full strength: this is the half that matters

The scoping gate answered YES, and the leg that answered it escalated rather than built, main untouched. Its own summary of the split was "a genuine yes/no split, not a clean win." The five-item obstruction census is reproduced here with its own severity gradings intact, because the third item is what closes the value of the YES:

• O1: the α^{−3/2} resolvent blow-up. Not a §2-class obstruction. Engaged only as Re z → −½. The certificate sits at z = 0, α = ½, where α^{−3/2} = 2.8284. No severity for a static certificate.
• O2: non-normality: the gap is not a decay rate in the X norm. Not a §2-class obstruction. No severity for a static invertibility certificate; HIGH severity downstream. Xu bounds his own result in his own abstract: the closed-form decay is obtained "on a weighted space of the conjugated variable reached from X by a bounded transfer map; we keep the two separate, since L₀ is non-normal and a spectral gap does not by itself give a decay rate in the X norm." So an origin-H² certificate would certify invertibility, not nonlinear stability, and nonlinear stability is what the chain's first link needs. This is why "invertible there, therefore a certificate is possible" is a forbidden sentence, not a shortcut.
• O3: a = 0 exactness dependence. FATAL for transfer, and that grading is the producing leg's own word, not a paraphrase. Every usable object in §3.1 is a consequence of the profile being the exact a = 0 CLM profile Ω(y) = −y/(y² + ¼): the single-simple-pole identity H(Ω) − iΩ = i/(y + i/2), the collapse of the nonlocal linearisation to a scalar first-order ODE on each Hardy block, the closed-form kernel, the exact Mellin norm 1/α. For a > 0 Xu proves only a conditional two-line inclusion under a hypothesis, no resolvent, no invertibility, no gap. This project's nominal target is not a CLM profile and inherits none of it. The corollary, in the census's own words: a certificate at a = 0 in origin-H² would certify an object Xu already inverts in closed form.
• O4: realization coupling for a > 0. Xu §4.6: "One cannot use the H² metric to empty the strip and the L² metric to close the origin channel." The origin-regularity index and the positions of the essential lines are coupled; a stronger realization shifts the origin line off instead. At a = 0 the two lines coincide, which is why the picture is clean there and only there. (§4 shows this is also what kills one half of the interpolation question.)
• O5: the §2 obstruction itself: checked for, and it does not recur. The contrast table of §3.2 is that check.

So the honest form of endpoint two is a conjunction, and both halves must travel together. The formulation is structurally viable: split, shape and bordered rows all explicit, verified against Xu at worst residual 2.8e−14. And it is capped at a = 0 exactness, where what it would certify is already known in closed form, with no transfer to the real target. Quoting either half alone misrepresents the result.

3.4 One finding from that leg that is not about certificates at all

The sharpest in-repo consequence is a realization-discipline finding. On the maximal L² realization, every point of the open strip −½ < Re λ < 3/2 is a genuine eigenvalue (Xu §4.6, with an explicit odd eigenfunction, confirmed numerically against the full nonlocal operator), so there is no L² spectral gap at all. The one feature separating those modes from the physical ones is the second derivative at the origin: requiring φ″ ∈ L² near zero is the whole difference. Every numerical object this repository owns for this operator sits in the realization without the gap: its own capability ledger already recorded that its grid imposes no origin condition. That is actionable independently of whether any certificate is ever built, and it is the transferable thing here: the realization discipline, not the certificate.

On the commissioned construction leg (leg 176): as drafted, it had not landed. (It has since landed; this paragraph is kept exactly as written, and the outcome is at §3.5.) At drafting it had committed only its novelty pass. Two things from that pass are quotable, both pre-registrations rather than results: (a) if it lands, the only thing it could claim as new is the discrete realization (Xu's own numerics use compactified-grid Newton continuation, log-Mellin discretization and FFT grids, with no Laguerre basis, no matrix representation and no finite-section truncation) "a claim about the method, not the theorem; the result is Xu's"; (b) Xu's three recorded gaps toward a computer-assisted proof (uniform large-imaginary-part bound, trace-ideal membership, quadrature-error bounds in trace norm) sit in a different strand of his paper and "this leg closes none of the three." That pass also states that leg 163's O3 is confirmed rather than weakened by it. No outcome is attributed or predicted.

3.5 The fourth data point: the construction leg landed, and its answer has two halves

Leg 176 built the formulation §3.1 scoped. Its own gate headline, verbatim: "YES on both conjuncts: with one magnitude that says NO and is reported in the same breath." Both halves are reproduced here at the strength leg 176 states them, and neither is quotable without the other. Float64 throughout, nothing interval-enclosed; 42 evidence checks, 0 failing.

The YES, half one: Xu's closed form reproduces. Xu eq. (4.23) at z = 0, bordered, checked pointwise against Xu's own ODE in y-space (a check that never mentions the discrete realization): max relative residual 4.767e−15, worst over six independent data 5.135e−15, against leg 163's established 2.8e−14 class. Every derivative analytic; no finite differences.

The YES, half two: it closes on leg 163's own diagnostic. σ_min of the bordered operator in the X metric, on a domain-only truncation (range untruncated), which gives an upper bound closing down onto σ_min rather than a Galerkin section:

(The N = 512 cell is quoted to the six figures the computation supports. Leg 176 banked 0.09080465147034879 there; an independent exact-rational re-derivation (leg 249, whose artifacts are on main; originating branch leg/249-h2cv2-v2 at e9db984) proves that value wrong from the 7th significant figure (the pencil AᵀG_cA − λG_d is not positive definite at that λ, so σ_min is strictly below it) and certifies σ_min ∈ (0.090804094, 0.090804194). The cause is the whitening leg 176 uses, whose error grows with the Gram's condition number; a Cholesky whitening of the same matrices agrees with the exact tier at every rung, and under it the ladder is monotone decreasing through N = 2048, so the N = 1024 rise below is a property of leg 176's whitening rather than of the Gram: the reliable window is wider than leg 176 claimed, not narrower. Full record: writeup/data/p2_route_h2c_v1_construction_correction_leg268.json.)

(Every digit in that ladder, and the headline below it, is stated in the X ⊕ ℂ weight convention of §0's convention note, a factor of 12.5 of freedom in the digits, none in the flatten-versus-decay behaviour they are read for.)

σ_min = 0.0908, ‖R‖_X = 11.0127, monotone decreasing through N = 512, varying by 0.139 % over a 16-fold truncation range. The N = 1024 row rises instead of falling; leg 176 attributes that to the float floor of an X Gram whose entries reach ~1e12 and reports it as the reason the reliable window stops at 512: that attributed cause is superseded by the footnote above, which carries leg 249's finding that the rise is a property of leg 176's eigh whitening rather than of the Gram, and that under Cholesky whitening the ladder is monotone through N = 2048. Leg 176's sentence is quoted here as leg 176's; the operative explanation is leg 249's, and the reliable window is wider than leg 176 claimed rather than narrower. In leg 176's own words the ladder is "float64 evidence of a positive limit, not a proof of one." (That quoted sentence is the last sentence of leg 176's C1_bordered_sigma_min_X.reading field. Its opening clause, "bounded away from zero and truncation-independent", is superseded and is deliberately not quoted here; the governing instruction is A1 of p2_route_h2c_v1_construction_annotation_leg274.json, which requires any quotation of that field to carry the corrected framing or cite the annotation beside it. This is the citation.) The tail block closes too, and it earns exactly that same reading and no stronger one: ‖T⁻¹‖_X rises monotonically across the ladder (3.994032 → 4.012071 → 4.021340 → 4.026241 → 4.028864 at N = 64 … 1024, 0.865 % in relative terms) with its own increments shrinking geometrically (1.804e−2 → 9.269e−3 → 4.901e−3 → 2.623e−3, ratios 0.514 / 0.529 / 0.535) (equivalently, the reciprocal tail σ_min = 1/‖T⁻¹‖_X falls with decrements 1.126e−3 → 5.745e−4 → 3.027e−4 → 1.617e−4, ratios 0.510 / 0.527 / 0.534, which are the smaller (≈16×) set and belong to σ_min, not to ‖T⁻¹‖_X) float64 evidence of a finite limit near 4.032, not a proof of one, and not a truncation-independent value. Earlier drafts of this note quoted 4.026, truncation-independent; that is the value at one rung of a sequence still climbing at N = 1024, and it understates the limit by 0.14 %. What survives untouched is the comparison the number is here for: this sequence converges, where in ℓ¹_w (legs 51/53) the tail inverse norm diverged with M. (At N = 512 leg 249's exact arithmetic certifies ‖T⁻¹‖_X ∈ [4.02623993, 4.02624155]; leg 176's banked 4.02614534796022 lies outside that bracket by 6.0e−5 relative, same whitening mechanism as above and recorded in the same correction artifact. Leg 176's own JSON reading field already read 4.03.) And the two controls report the other answer: unbordered σ_min collapses to the float floor (1.55e−15 → 6.97e−14), and the loose L² realization decays like N^{−1.49} with no gap at all, so the origin condition is now a measured magnitude rather than a citation.

The NO, in leg 54's own shape, and it is the same magnitude class this note's §2 reports. With A = blockdiag(finite bordered inverse, tail inverse) and Z₁ = ‖I − A𝕃‖_X:

Best cell 140.72 where < 1 is needed; growth ~K². So in leg 54's shape the X realization fails too (but, in leg 176's own reading, for a different reason than ℓ¹_w did. There, leg 127 showed the operator itself had no truncation-independent σ_min) a theorem, σ_min(L_M) = c_s M^{−(1−s)} → 0, so Z₁ ≥ 1 for every bounded A. Here that mechanism is measured absent rather than proved absent: the ladder flattens instead of decaying (§3.5), float64 evidence of a positive limit and not a proved truncation-independent value. On that reading, the strongest the data support, the failure is of the block-diagonal shape of A, not of the operator and not of the space. Leg 176 calls that distinction its most useful output and the reason both numbers are reported with neither standing for the other. This note adopts that framing and adds nothing to it: no claim is made here that some other, non-block-diagonal A closes it, that was not tested, by leg 176 or by anyone.

One correction this construction makes to §3.2's, §4.5's and §5(3)'s witness: now applied at all three sites, and it turns out to be a sign, not a magnitude. Earlier drafts quoted σ_min ≥ 0.71465 in §3.2, §4.5 and §5(3). That is leg 163's three-datum witness, and leg 176 states plainly that it was optimistic by 7.9× [FLAGGED 2026-08-11, leg 280, per leg 281's finding E6: this factor is convention-relative, ranging 5.265 … 656.95 over the same weight sweep §0 names, a 124.8× swing: the factor 7.9× holds only in the repo's own fixed weight convention]: "That is a witness, not a bound. The measured value is 0.0908." Reproducing leg 163's own quantity from the closed form gives 0.8681539 over four data while the operator norm is 11.0, i.e. random low-mode data does not find the worst direction. Tracing it further: 0.71465 is the five-decimal rounding of leg 163's banked implied_sigma_min_lower_witness = 0.7146549471256172 (§3.2 records the same provenance, in one step and with the 4.9e−06 display offset named), the reciprocal in full precision of leg 163's own largest sampled ratio 1.39927667753796, and since each datum gives ‖f‖_X/‖u‖_X = 1/r ≥ σ_min, a finite family of such ratios bounds σ_min only from above. So leg 163's data support σ_min ≤ 0.71465, the ≥ was inverted, and the two legs never actually disagreed, 0.0908 ≤ 0.71465. All three call-sites now carry 0.0908 as the measured figure, with 0.71465 retained only in §3.2 with its correct direction and its provenance named.

What the correction does and does not change. It changes no conclusion in §3–§5. But it does sharpen how one property may be stated. The ≥ sign was carrying the claim "bounded away from zero", i.e. a lower bound (and neither leg proves one: leg 176's own domain-only truncation is, in its words, "an upper bound closing down onto σ_min", and its ladder is "float64 evidence of a positive limit, not a proof of one." What is established is a monotone-decreasing ladder that flattens (0.0908878 → 0.090804 over N = 64 … 512, 0.0920 %; 0.139 % over the full 16-fold range) together with two controls that report the other answer) against ℓ¹_w, where the analogous ladder decays like M^{−(1−s)}. That contrast is a contrast of ladder behaviour in both spaces, which is all §3–§5 ever use it for, and it survives the correction intact. What does not survive, and is not written anywhere in this note, is any claim to a proved floor. Nor, per §0's convention note, any claim that the digit 0.0908 is convention-independent: it moves by a factor of 12.5 under the border weight and becomes 0.0420 in Xu's own y-space normalization (0.057643 under Definition 4.1's own displayed norm, §0), quoted here in one fixed convention. [CORRECTED 2026-08-11, leg 280, per leg 281's finding E5/E4: 0.71465, by contrast, IS convention-free, to 1.87e−16: it is 1/max(‖u‖_X/‖f‖_X) over a finite family, a ratio of X-norms with no border coordinate, so the weight cancels between numerator and denominator. The convention-dependence claim above applies to 0.0908 only, not to 0.71465.] The flatten-versus-decay contrast just described is what survives that freedom, and it is the only thing §3–§5 use.

What this does and does not change in §3.3's ceiling: nothing is lifted. Leg 176 restates O3 itself: a = 0 only, and what is certified is "an object Xu already inverts in closed form." It closes none of Xu's three recorded gaps toward a computer-assisted proof (uniform large-imaginary-part bound, trace-ideal membership, quadrature-error bounds in trace norm), forms no Y₀ and no Z₂, is float64 with nothing interval-enclosed, and moves no link of the L1 → L4 chain. Its own summary of what it is: "the first constructed (not merely scoped) certificate outside the ℓ¹_w lane this repository has built, and a new exact discrete realization of Xu's operator. Infrastructure, not a theorem." The claim grade of this fourth point is therefore construction, gate YES with a reported NO magnitude: strictly weaker than §2's theorem and not to be levelled with it; and §3.3's "escalated rather than built" now describes leg 163 specifically, not the state of the axis.

Completing §0's promise: the two sentences in §§5–6 leg 176 bears on, named rather than left to inference. (i) §5's opening "Separately they are: a theorem, an escalated scoping YES, and a scoping NO" counts the three points the synthesis was written from; with leg 176 the axis has four, as §0's own table already shows, and the fourth is the construction grade just stated. That sentence is left standing as the three-result synthesis it is, not silently upgraded. (ii) §6's claim table likewise has three rows and no leg-176 row; the grade that row would carry is the one in the preceding paragraph: construction, gate YES with a reported NO magnitude, strictly below §2's theorem grade and strictly below nothing else. Both are drafting-time scope, now disclosed here rather than discovered by a reader.

4. The segment between them is empty

Given a dead endpoint and a capped endpoint, the natural next move is to look for something in between. It was looked for. There is nothing there, and the two halves of the search fail for two different reasons.

4.1 MOVE A, the coefficient scale. The ℓ¹_w obstruction does not weaken at all.

Hold the domain and the realization; vary the index across ℓ^p_w, 1 ≤ p ≤ 2: the Fourier–Lebesgue / Besov-coefficient scale, which contains ℓ¹_w at p = 1. Measured.

First, the instrument was calibrated against the banked endpoint. Local slopes of §2's exponent converge monotonically from above, so the endpoint of a ladder is not the answer; the Aitken limits are 1.0000 / 0.7002 / 0.3090 at s = 0 / 0.3 / 0.7, deviating from 1 − s by at most 0.00905. A negative control on the same code path (the μ = 0 kernel fed to the dissipative operator at μ = 1) returns −0.2904, i.e. the ratio grows, so the μ = 0 decay is a property of the operator and not of the arithmetic. (This is explicitly not the same control as §2's own μ > 0 saturation check, and is not conflated with it.)

Then the mechanism, which is the whole of the check. Because the kernel solves the tail recursion exactly, its image vanishes in every row but the truncation edge: the single-row share of the residual norm is 0.9999999999998843, with 1 row above 1e−12 relative. The negative control that could have failed, the same counter fed a perturbed direction, reports 4088 rows, so "one row" is a property of the witness and not a tautology of the code.

And a one-entry vector has the same norm in every ℓ^p. At fixed (K, M, s):

Numerator relative spread across p: 5.653e−14. Denominator relative spread: 1.5707, i.e. 157 %. Moving along the interpolation scale changes the quantity the proof divides by by 157 %, and the quantity it divides by 6e−12 %. The index cannot reach the numerator, because the numerator is one entry.

Swept over the whole grid, the exponent tracks 1 − s and is p-blind where it is O(1):

Max deviation from 1 − s for s ≤ 0.7: 0.00905. Max spread across p at fixed s ≤ 0.7: 0.00904. Rows where p > 1 rescues an exponent that is dead at p = 1: 0. The technique weakens only as s → 1, and s was already available at p = 1 and already swept there. (The p-spread grows with s (0.0000 at s = 0, 0.0836 at s = 1.0, 0.3936 at s = 1.4) but that is the Aitken limit becoming ill-conditioned exactly where the exponent it extrapolates passes through zero, which is why the conclusion rests on the s ≤ 0.7 block. It is reported rather than trimmed.)

4.2 The invariant: the two-parameter scale is a one-parameter picture

Three membership thresholds decide the whole scale, and with the kernel decaying like m^{−2} (measured −2.00236), the cokernel functional growing like m^{+1} (measured +1.00118), and the target's coefficients like k^{−(1+α)} with α = 0.39735311167782:

All three depend on the single combination σ = s + 1/p, the Sobolev/Besov scaling index, which is precisely the invariant of interpolation. Measured as growth exponents of partial-sum norms rather than as convergent/divergent booleans, over 45 (p, s) points × 3 quantities: 114 quantities scored, 21 in the marginal band |σ − threshold| ≤ 0.15 (excluded from scoring, and reported rather than hidden, because at a threshold the true behaviour is logarithmic and a power-law fit correctly reads a small spurious exponent); worst deviation from the σ-prediction outside the band 0.01605; mismatches above 0.02 outside the band 0. The marginal band's log behaviour is demonstrated rather than asserted: at p=1, s=1 (σ = 2) the partial sums add a constant per decade over N = 10³…10⁶, max/min increment ratio 1.01114. And the single row nearest the band edge was chased rather than tolerated: its deviation falls monotonically 0.02955 → 0.02167 → 0.01605 → 0.01213 along a truncation ladder, so it is the partial sum's truncation and not a failure of the prediction.

4.3 The window's width is exactly zero, and the target is outside it anyway

The kernel leaves the space at σ ≥ 2; the cokernel enters the dual at σ ≤ 2. The only σ at which neither obstruction is present is σ = 2 exactly: a single point, at every p. At p = 1 that is s = 1, which is exactly the exponent this project's own ban list already calls "the ONE exponent at which bordering cannot help." The interpolation scale does not widen that point; it translates it.

And at that point the target is out of the space, by the same margin everywhere:

Both walls translate by exactly −1/p, so the gap between them is an invariant of the whole scale, equal to α − 1 = −0.602647 exponent units. Cross-checked against the directly measured p = 1 margin banked earlier by a different leg: −0.6062554687114012 against this asymptotic prediction of −0.602647, difference 0.003609, agreeing to 0.6 % (the first is a measured norm, the second an exponent-level asymptote). This is the p-independent generalisation of an earlier leg's own sentence: "the class where the operator is least bad is the class where the target has infinite norm." It holds on the whole scale, with the same 0.60-exponent-unit gap.

4.4 MOVE B, the origin-regularity index. No scale can even ask the question.

The other interpolation is the Sobolev index between L² and H² on the line. It fails, and for a reason of a different kind. This half was not re-measured; it is read out of endpoint two's own census, and that attribution is part of the result, not a footnote.

What forces a = 0 exactness is the single-simple-pole identity H(Ω) − iΩ = i/(y + i/2), and that is an equation satisfied by the PROFILE. It contains no norm, no weight and no index. Every space (interpolated or not, on the line or on the circle) inherits it if and only if the profile is the exact a = 0 one. No choice of scale can supply it, and none can remove it. So check (b) is not a question an interpolation scale is able to answer, which is a stronger and more honest statement than "the candidate scales fail it."

And the one interpolation that could even be attempted is ruled out in Xu's own text, quoted above as O4: "One cannot use the H² metric to empty the strip and the L² metric to close the origin channel." On the maximal L² realization there is no spectral gap at all; a stronger realization shifts the origin line off; an intermediate index gets neither. This sentence was named as the candidate obstruction in the leg's novelty pass before the check was run, so the NO is not a discovery its own construction conveniently arrived at.

4.5 The scale-by-scale answer, and one reason that is explicitly not the reason

Every scale fails at least one check, and no scale passes both. (The 0.0908 and 0.71465 in the last row are convention-relative digits) §0's convention note; the FAILS/passes verdicts in the table are not, since each rests on a decay exponent or on O3/O4.)

One constraint that is emphatically not the reason, recorded so it cannot later be mistaken for it. The CAP literature is bimodal (weighted ℓ¹ or Hilbert H^l, nothing between) because ℓ¹_ν is a Banach algebra under convolution (which is what the quadratic constant needs) and ℓ^p is not an algebra for p > 1. On the ℓ^p_w scale the algebra property is recovered whenever the weight embeds the space in ℓ¹, i.e. σ > 1, which holds throughout the region of interest, since the crossing is at σ = 2. So the Banach-algebra constraint does not bind here. The reasons are the ones in §4.1–§4.4.

5. What the three results say together

Separately they are: a theorem, an escalated scoping YES, and a scoping NO.

Together they are one statement about where a certificate for this operator can live, and it has four parts:

1. The obstruction that consumed roughly seventy legs of this project is a property of the certificate's SPACE, not of the operator. The same a = 0 CLM linearisation is not bounded below in ℓ¹_w at any s < 1 (§2, proved) and is invertible with an explicit closed-form inverse on origin-H² (§3, published by Xu, re-derived here at worst residual 2.8e−14). Those two facts do not conflict; they separate two realizations of one operator. The organising concept (that an obstruction can belong to the realization rather than to the operator) is itself published and named on this exact operator, and is used here, not claimed.

2. The ℓ¹_w endpoint is dead for a reason that is structural and not tunable. It is not the weight (the exponent is 1 − s at every s < 1, with 0.29 % spread across the split), not the split (the tail owns the divergence), not the shape of A (the argument never decomposes A), and not the bordering repair (5.7e−15). The method needs weighted ℓ¹ of Fourier coefficients to control its tail, and in that space this operator is not bounded below at all.

3. The origin-H² endpoint is structurally viable and simultaneously worthless for the real target, and both halves are load-bearing. Split, shape and bordered rows are explicit and verified; σ_min measured at 0.0908 (§3.5; the 0.71465 of earlier drafts was an upper bound from three data, not a floor) on a ladder that flattens rather than decaying: 0.139 % over a 16-fold truncation range, with a quadrature-window spread of 1.44e−04, and no proved floor anywhere in it; the ℓ¹_w obstruction does not recur. And every one of those objects is a consequence of a = 0 exactness (O3, FATAL for transfer), the operator is non-normal so invertibility is not stability (O2, HIGH downstream), and what a certificate there would certify is a closed form its author already wrote down. (The 0.0908 in this paragraph is a convention-relative digit (§0's convention note, a factor of 12.5) while the flattens-rather-than-decays statement it appears in is not.) The right-sized claim is: the space axis has a live point, and the live point is at the wrong object.

4. And there is nothing between the two endpoints. Not because the candidates were tried and found wanting one by one, but for two structural reasons: on the coefficient side the entire two-parameter family collapses to the one parameter σ = s + 1/p that legs 51/55/127 had already swept, and the window where both obstructions vanish is a single point whose distance from the target's own ceiling is the p-invariant −0.602647; on the origin-regularity side the requirement that kills transfer is an identity about the profile, which no space can supply or remove.

So the space axis is mapped and closed. Everything left is off it: a different object (the a > 0 profiles, where none of Xu's machinery applies), a different question (nonlinear stability rather than invertibility, which O2 says needs more than a gap), or a different discipline (enforce the origin condition in the discretization, the one genuinely transferable item, per §3.4).

6. What is claimable, and what is not

The three results sit at three different grades and must not be levelled:

Four things this note must never be read as saying. (1) That the operator has no bounded approximate inverse: it is invertible on origin-H², published. (2) That the Newton–Kantorovich or radii-polynomial method is obstructed in general. (3) That origin-H² offers a route to anything this project needs: O3 says the opposite, and §3.3 leads with it. (4) That any of this bears on the target problem. The object is the a = 0 CLM linearisation, whose first certificate constant is exactly zero for a degenerate reason (the anchor is one basis mode), so every magnitude here bounds the real target's difficulty from below.

7. Ceiling

The object throughout is the a = 0 CLM steady linearisation. No dynamics were run. Nothing is claimed about this project's nominal target profile beyond its banked coefficient exponent α = 0.39735311167782. Every σ_min digit in this note is stated in one fixed weight convention and is relative to it by a factor of 12.5 (§0's convention note gives the sweep (0.01086 … 0.13580) and Xu's own y-space figure (0.0420 rather than 0.0908 under Xu's equivalent full-line norm; 0.057643 under Definition 4.1's own displayed half-line norm) see §0, corrected 2026-08-11 per leg 277); what is invariant under that freedom, and what §§3–5 argue from, is the ladder's behaviour, not its digits. Every number is float64 at a stated truncation; nothing here is interval-enclosed or rigorous in the computer-assisted-proof sense, including §2, whose proof is exact but whose verifying measurements are floating-point. §3 built no certificate and formed no certificate constant. §4's σ-threshold algebra is asymptotic at the exponent level and agrees with the directly measured p = 1 margin to 0.6 %, not exactly; its MOVE-B half is read from §3's census rather than independently re-derived, and the leg that wrote it states plainly that it did not re-read Xu at primary source. The commissioned origin-H² construction leg had not landed at drafting and nothing in §§1–4.5 or §§5–6 is attributed to it; it has since landed and is folded in at §3.5 only, where it is float64 with nothing interval-enclosed, forms no Y₀ and no Z₂, and is carried as that leg's own float64 measurements. The independent check on them has since reported: the originally commissioned verification leg (leg 192) committed only a novelty pass, but leg 249 re-derived the construction in exact rational arithmetic, confirmed both conjuncts, and corrected two banked values below the digits quoted here (§0, §3.5); a later review pass re-pulled those figures from leg 249's own branch and matched them exactly, and those three artifacts are now on main, byte-identical to that branch. That check is exact where leg 176 is float64, and it does not convert any ladder here into a proved bound. No link of the L1 → L4 chain moved. None has moved in 275 legs (count current as of leg 276). Clay odds remain ~0.05 %.

This draft is for the user's review. Its landing records that the bundle reproduces from its sources; it does not approve the bundle for publication. It is one of two such drafts: see TECHNICAL_P2_PUB1_V1.md for the other, which answers a different question.