TECHNICAL, Route-FUS v1 (leg 314): the finite-unstable-spectrum condition, classified

Scoping leg. Builds nothing. solver/ is read, never written, never run. Runner experiments/p2_route_fus_v1_scoping.py; curated data writeup/data/p2_route_fus_v1.json; evidence + figure experiments/p2_route_fus_v1_scoping_evidence.py → writeup/figures/fig77_route_fus_v1.png; novelty pass writeup/novelty/leg_314.md (committed first, 8c1f35f). Every number quoted below is in the JSON.

1. The gate, in its pre-committed wording

"Does the scoping produce a definite classification (finiteness for these profiles is (i) provable, with the argument sketched and its load-bearing step named; (ii) checkable only numerically, with a named certified-count/lower-bound route this repository's spectral tools bear on; or (iii) open, with the obstruction named?"

Answer: yes) a definite classification was produced. The classification is (iii) OPEN.

The gate text is immutable and was not edited. The classification is computed by the runner from five pre-committed discriminators, not asserted; the classifier's self-tests show it reaches all four of its outcomes (DISCHARGED, i_PROVABLE, ii_NUMERICAL, iii_OPEN), so this is a verdict that could have come out differently (lesson 90).

2. The object

arXiv:2509.14185 p.19:

"For a computer-assisted proof to be feasible, it is desirable that the spectrum in the right-half µ-plane consists of a finite number of eigenvalues."

Leg 175 banked that this is "called 'desirable' and assumed, which no residual reduction discharges", and that neither 2509.14185 nor 2511.22819's precision fix touches it. This pass re-verified the second half on the instrument: 2511.22819's abstract never mentions unstable spectrum, eigenvalues or spectral finiteness, instability appears only ordinally ("1st unstable", "4th unstable solution for IPM"). Leg 175's reading stands.

Two further restrictions accompany the paper's own mode count, and both matter here: it is a PINN eigensolve "under the assumption that there exist eigenvalues with non-negative real part that lie on the real axis", and "restricted to Ψ that lie within the same symmetry class".

3. The finding: the condition is not realization-invariant

The load-bearing observation is that the FUS condition, as stated, is not a property of the profile. It is a property of the (profile, realization) pair, and 2509.14185 names no realization.

This is not this leg's inference. It is Xu, arXiv:2607.19762 (22 Jul 2026, 41 pp), a paper this repository already holds and has audited (legs 70 / 171 / 173 / 249). For the a = 0 CLM profile Ω(y) = −y/(y²+¼):

• a realization dichotomy (the "in-strip smear" seen in generic discretizations is the spectrum of the maximal L² realization, which the origin-H² choice eliminates;
• in origin-H², the essential spectrum in {Re λ ≥ −½} is the single vertical line {Re λ = −½}, produced by a log-widening Weyl sequence, with a Hardy–Mellin resolvent bound clearing the rest of the half-plane;
• the point spectrum over all of ℂ is exactly {0, 1}) scaling and time-shift symmetry modes, with no embedded eigenvalues; quotienting gives a spectral gap of ½.

Same PDE, same profile, two spectral pictures. In one realization the unstable set is finite (and purely symmetry-induced); in the other it is smeared. So "the spectrum in the right-half µ-plane consists of a finite number of eigenvalues" has no truth value until the space is fixed, in particular the condition imposed at the profile's singular point. This is lesson 91 (name the realization) applied to somebody else's assumption.

4. The smear, made quantitative on banked numbers

solver/rescaled_spectrum.py's discretization is the maximal-L² side of Xu's dichotomy: leg 70's Route-RC established that from the code, finding it imposes no origin condition at X = 0. Route-I's banked ladder (writeup/data/p2_route_i_v1_driven.json, i5_stability) therefore measures the smear directly.

Realization, named in full (lesson 91): solver/rescaled_spectrum.py's compactified odd-sine basis (OddCompactBasis), gCLM at a = ½, p = 3, µ = 0 (inviscid), dilation mode excised by identity, unstable tolerance Re > 1e-6.

• n_unstable = K − 3 exactly. Least-squares slope dn/dK = 1.000000, max |residual| = 0. Verdict DIVERGENT on the pre-committed criterion (strictly increasing and slope ≥ 0.5).
• max Re is flat to 0.751% across the ladder while max |Im| grows ×3.267 as K grows ×3.000. The unstable set is not a fixed finite collection being resolved better; it is a curve on which Re rises with |Im|, whose visible extent is set by the truncation. solver/marginal_flow.py's own frequency_profile docstring states this independently.

4a. The tidy closed form is diagnosed, not banked

n = K − 3 is exactly the kind of suspiciously tidy closed form the standing discipline says to check rather than quote. The bookkeeping: the Jacobian is K × K (OddCompactBasis carries b_k, k = 1..K), unstable_count excises one eigenvalue by identity (the dilation mode, L(X Ω_X) = 0 exactly; its measured real part falls 0.2710 → 8.899e-05 → 4.156e-08 along the ladder), leaving K − 1 judged. So

offset 3 = 1 excised dilation + 2 eigenvalues with Re ≤ 1e-6.

min Re converges to −2 (−2.0814, −2.0073, −2.0017). That the two non-positive eigenvalues are isolated symmetry/edge modes is consistent with Route-E's banked finding that the only grid-converged isolated eigenvalues of this flow are its two exact symmetry modes, but this leg did not measure them and does not assert it (doing so needs a fresh gCLM eigendecomposition, which is banned). The load-bearing quantity is the slope, not the offset: the unstable count is proportional to the number of degrees of freedom.

4b. It is not an under-resolution artifact: the check runs the other way

The obvious objection is that a badly resolved profile manufactures spurious unstable directions. If so, the count would fall as resolution improves. It rises: across the ladder the profile residual improves by 10.17 decades and λ_truncation by 5.84 decades while n_unstable goes 45 → 93 → 141. Better resolution buys more unstable directions. That is the continuum signature and the opposite of the artifact signature.

4c. The positive control (lesson 90)

A control that cannot come out differently is not a control. Ask what would have to change for the counting code to report finite: only µ. On the identical ladder, the identical basis and the identical profile family, the viscous rows report n_unstable = 0 at every K (verdict K_STABLE, all_zero = True). The DIVERGENT verdict is therefore a statement about the inviscid operator, not about the instrument.

Figure fig77 shows all three panels: (a) the divergent inviscid count against the flat viscous control, (b) Re flat while |Im| grows, (c) resolution improving while the count rises.

5. Why this is (iii) OPEN and not (ii)

The tempting classification is (ii), checkable only numerically. It is wrong, and the reason is the whole content of this leg.

Every certified numerical route counts inside a bounded box. The finiteness claim is a statement about |Im µ| → ∞: a discrete eigenvalue set in a half-plane may still accumulate at infinite imaginary part, and forbidding that accumulation requires a resolvent bound, not a count. No finite computation, however rigorously enclosed, sees the region where the condition actually lives.

The literature confirms this split, and the sharpest confirmation is a paper that does the work. Guo–Hadžić–Jang–Schrecker, arXiv:2509.12435 (Nonlinear stability of the Larson–Penston collapse, 15 Sep 2025, 149 pp) prove maximal dissipativity of the linearised operator on arbitrarily large backward light cones, and then establish mode stability by

"a high-order energy method in low- and high-frequency regimes (relying on monotonicity) and rigorous computer-assisted techniques in the intermediate regime."

The computer does the bounded middle. The unbounded high-frequency end is done analytically. This repository already recorded the box at leg 265 / verify_265: Re λ ∈ [0,1], |Im λ| ≤ 8, with b₀ = 1/5, b₁ = 8 (VNODE-LP; code at github.com/mrischrecker/Larson-Penston-Stability).

The same structural step, under its other name, is BCG arXiv:2208.09445 (and non-radially CGSS arXiv:2310.05325): L = A₀ − δ_g + K with A₀ maximally dissipative and K compact on X = H₀^{2m}(B(0,2)), giving Λ = σ(L) ∩ {Re λ > −δ_g/2} finite, of finite algebraic multiplicity, a theorem, supported on an unbounded region, and the thing that makes the unstable spectrum finite. Under its third name it is Xu's Hardy–Mellin resolvent bound. The classical ancestor is Weyl's theorem plus a high-frequency estimate; the old fluid-adjacent template is the 2007/2010 wave-map mode-stability trio (math-ph/0702025, 1006.2172, 1003.0707).

None of these is available for CCF, IPM, 2D Boussinesq or 3D Euler with boundary, and no certified count has been published for any of them.

5a. Four independent literatures corroborate the same split

The bounded-box / unbounded-tail split is not an artefact of the blow-up literature. It recurs, with the same division of labour, in four places found this pass:

• Barker–Zumbrun, arXiv:1601.00837, the closest thing to a certified unstable count anywhere. Interval arithmetic plus rigorous ODE bounds plus an Evans-function winding number on ∂(B(0,R) ∩ {Re λ ≥ 0}). The decisive detail is the radius: R = (√γ + ½)², enclosing all possible unstable eigenvalues, is derived analytically. The computer works inside the disc; the disc itself is a theorem. Exactly this leg's classification, in a worked example.
• Gallay–Wayne, in the weighted spaces L²(m), σ_ess = {Re λ ≤ −(m−1)/2}. Finiteness of the unstable set is bought by raising the weight m, i.e. by changing the realization. Independent confirmation of obstruction (1): the answer is a property of the space.
• Weyl / Kato IV.5.35, and the Jörgens–Vidav–Voigt chain (with Chicone–Latushkin's evolution-semigroup version): the classical statement that the essential spectrum is invariant under relatively compact perturbation. This is the ancestor of "maximal dissipativity + compact K", and it is where the unbounded region is discharged by estimate.
• Chen–Hou, sidesteps the condition rather than discharging it: the computer-assisted 3D Euler-with-boundary proof is built on energy estimates with a finite-codimension stability argument, not on a certified enumeration of the unstable spectrum. That an existing computer-assisted blow-up proof avoids the FUS condition is evidence about its cost.
• Bricmont–Kupiainen: the old renormalisation-group template in which finiteness of the unstable directions is imposed by the choice of space, again not proved for a given profile.

5b. The one absence claim, and its audit (MF3)

The classifier's discriminator d4 (no certified unstable-mode count exists for CCF, IPM, 2D Boussinesq or 3D Euler with boundary) is this leg's only absence-based discriminator, and absence claims are the ones a broken search instrument can fake. It was audited against the orchestrator's defect report MF3 ("the arXiv endpoint returns zero for any two ANDed quoted phrases"); the audit is at writeup/novelty/leg_314.md §4a. Result, in short:

• MF3 did not reproduce on the instrument used here, ANDed quoted pairs returned 6, 1, 2 and 251 results. No zero from an ANDed quoted query is banked anywhere in this leg.
• The only all-zero return came from a curl/urllib instrument independently diagnosed as broken (301 → 429) and discarded before MF3 arrived; none of its output is cited.
• d4 rests on a multi-instrument sweep (WebSearch, WebFetch, direct PDF pulls of 2208.09445, 2310.05325, 2509.14185), and its strongest support is positive, not negative: 2509.14185 p. 19 calls finiteness "desirable": the authors state in their own words that they have not established it.

So the (iii)-OPEN branch does not rest on any query zero. Obstructions (1) and (2) below are presence claims resting on quoted text, and would stand even if d4 were withdrawn entirely.

6. The classification, stated

(iii) OPEN. The named obstruction, in two parts, both load-bearing:

1. The condition is not realization-invariant, and no realization is named. Until the space is fixed (in particular the condition at the profile's singular point) the FUS hypothesis has no truth value to prove, disprove or check. Xu's dichotomy shows one profile with two answers; §4 shows the smear side is not hypothetical but quantitative, n = K − 3.
2. Once a realization is fixed, the residual obligation is a high-frequency resolvent bound on the unbounded region |Im µ| → ∞: Hardy–Mellin, or maximal dissipativity plus relatively compact perturbation. It is a theorem, not a computation, and no certified count on a bounded box can supply it.

The corollary worth stating on its own: for this condition, numerics is a refuter, not a verifier. A truncation ladder can falsify finiteness (as it does in §4) and can count what sits inside a box. It can never establish finiteness.

7. What a construction leg would need: named, not built

The brief forbids building this, and it is not built. Recorded so the DM can draft or decline it.

Prerequisite before either half: name the realization.

Half A is buildable here. Half B is the load-bearing half and it is a theorem. Building half A alone would produce a certified count that does not discharge the condition, which is precisely the trap this leg exists to mark. External precedent for exactly this shape: GHJS arXiv:2509.12435.

8. Bans

Walked term by term in writeup/novelty/leg_314.md §3. The one that made contact is "another gCLM measurement leg", and it bound and redesigned this leg: following leg 70's Route-RC precedent, the runner performs arithmetic on counts already banked and carries an AST self-guard asserting zero banned calls (newton, continuation, spectrum, converged_spectrum, unstable_count, eigvals, …) and zero solver/numpy/scipy imports, both verified in the JSON and re-checked by the evidence script. One frequency_profile call was made as a runtime cost probe before the ban walk completed; it reproduced Route-I's banked max_re to ~3e-14 and no quantity from it is banked, quoted, or in the JSON. It is reported rather than quietly dropped. The ban is not lifted, and this leg does not lift it.

The near-miss on "closure is a property of the SPACE" is recorded explicitly in the novelty file: the banned move was explaining a certificate failure by appeal to the space, refuted because leg 49's gauge ablation put the 5604× in the border rows. Here the realization-dependence is a cited published theorem (Xu, Prop. 2), audited from this repo's own code by leg 70, not an explanatory habit.

9. Ceiling

Wall 1 and Wall 2 stand. No link of the L1→L4 chain moved. Clay odds remain ~0.05%, unchanged.

Classifying a condition is not discharging it. If anything this leg makes the FUS hypothesis harder to state (it shows the hypothesis is under-specified as written) not easier to prove.

Scope, stated as narrowly as it holds. The divergent count is measured in one realization of one model (gCLM, a = ½, p = 3, maximal-L² compactified odd-sine basis) and is not a claim about CCF, IPM, Boussinesq or 3D Euler with boundary. It demonstrates that the missing step can fail, not that it does fail for those four. What holds for those four is the statement of §6: open, obstruction named.