Route-J v1: the primary-source pass

Phase 2 / P2, leg 42. Figure: writeup/figures/fig39_route_j_v1_literature.png. Code: solver/literature_gates.py, test_literature_gates.py (9/9), experiments/p2_route_j_v1_literature.py → writeup/data/p2_route_j_v1_literature.json. Working notes: PHASE2_P2_NOTES.md §31. Literature record: LITERATURE_CHECK.md, sixth pass (the first primary-source pass).

No new science. This leg reads the papers that six previous legs were blocked from reading, and retracts what they pre-empt. Seven standing claims are pre-empted, one is partial, two remain unsearched, and one result arrives from the literature that this project did not have. No link of the L1→L4 chain moved. Clay odds unchanged at ~0.05%.

0. What this leg is, in one paragraph

For six legs LITERATURE_CHECK.md grew, five passes, while Papers/ stayed empty, because the environment's egress allowlist did not include arxiv.org. Every novelty claim this project has ever made was search-level: assembled from search-engine summaries of papers nobody here had opened. This session egress worked. bash Papers/fetch.sh pulled all fourteen manifest entries on the first try; Tier 1 is read; and this leg is what the reading leaves behind. The deliverable is not a paragraph saying "we checked": that is exactly as durable as a figure looked at once, which Route-I established is not durable at all. It is a module of executable gates: each one re-derives a published number from the published equations and compares it to ours, so the check survives the session that made it.

The headline is a subtraction. Route-F v1's s_c = α/2 (the leg the project called "the only item that probes the actual obstruction between a toy certificate and NS") is in the literature, as s*(a) = 1/c_l(a), posted eleven days before we did it.

1. The sources, and the exponent dictionary that makes them comparable

Four papers do the work.

Before any comparison there is a translation, and getting it wrong is the only way to misread this literature. ALS and XU write

omega ~ tau^{-beta} f( x / tau^{c_l} ) ,     tau = t_c - t

so their c_l (ALS call it alpha, XU call it c_l) is the length exponent: what this project calls beta. Our alpha is the profile's far-field decay exponent, and §26's rescaling ODEs give beta = 1/alpha. Therefore

alpha_ours  ==  1 / c_l ,

and because their dissipation is Lambda^sigma while ours is (-Delta)^s with symbol |k|^{2s}, we have sigma = 2s. Gate 1 of the suite is nothing but this dictionary, checked to be self-inverse on both anchors, and it goes first because every other verdict in this leg inverts if it is wrong while still passing its own test.

2. J1, the constant ALS corrected in Schochet (1986), settled from our side

ALS §5.1 reproduce Schochet's exact a = 0, sigma = 2 pole-dynamics solution and note, parenthetically, that they are correcting the 1986 constant: K_pm = 24(3 ± √6), where Schochet printed 12(6 ± √6).

That is a checkable claim, and checking it is free. Substituting the ansatz (37)–(41) into ALS (36), the complex Burgers equation omega_+t = -i omega_+^2 + nu omega_+xx, with analytic ∂_t and ∂_xx (no differencing anywhere, so the residual tests the solution rather than a discretization of it):

Three (nu, t, pole) configurations each, worst quoted. 13.66 decades of separation. ALS are right, and we now know that rather than assume it. As a second check on the same solution, ALS (44) for the blow-up time agrees to 1e-13 with the time the pole trajectory (40)–(41) actually crosses the real axis, which for poles on the imaginary axis is available in closed form (√(1 + (5/3)Kνt) = 3).

This matters beyond bookkeeping: Route-H's Papers/MANIFEST.md lists Schochet CPAM 1986 as gating its closed form and as unavailable (publisher PDF, not on arXiv). It is still unavailable, but its content is now pinned through ALS, and the version of the constant that would have come from the 1986 paper is the wrong one.

3. J2: Route-H's closed form (E) is ALS equation (57)–(58)

Route-H v1 introduced, as the known-answer gate its whole leg hangs from,

omega(x, t) = -2 (1 + mu_0) kappa x / ( kappa^2 (T-t)^2 + x^2 ) ,    kappa = nu / mu_0    (E)

and recorded it as "at high risk of being known", declining to claim it. That was the right call, and it is now settled rather than suspected. ALS §5.3 solves a = 0, sigma = 1 by pole dynamics:

omega(x,t) = omega_{-1}(t)/(x - i v_c(t)) + conj ,
omega_{-1}(t) = omega_{-1}(0) ,   v_c(t) = (omega_{-1}(0) + nu) t + v_c(0)      (57)-(58)

Splitting (E) into partial fractions gives a conjugate pair of simple poles at x = ± i kappa (T-t), i.e. exactly (57), under the parameter map

omega_{-1}(0) = -(1 + mu_0) kappa ,      v_c(0) = kappa T .

Three independent confirmations, all in the gate suite:

1. Pointwise agreement. Three parameter sets (mu_0, nu, T) × four times including t = 0.99 T: worst relative difference 6.5e-15.
2. The evolution law, from the other direction. ALS's dv_c/dt = omega_{-1}(0) + nu evaluates to -(1+mu_0)ν/μ_0 + ν = -ν/μ_0 = -kappa, which is (E)'s own dv_c/dt. Agreement to 1e-13, and this is a derivation, not a fit.
3. ALS's blow-up time formula (59) returns our T exactly: absolute error 0.0, not "small". And omega_{-1}(0) < -nu, so (E) sits on ALS's blow-up branch by their own criterion rather than merely near it.

Verdict: PRE-EMPTED. What survives is (E)'s use as a known-answer gate, which is what Route-H built it for, and which is strictly better now that it carries a citation.

A second consequence, and a more interesting one. ALS's self-similar form (61) for this family carries nu inside the profile with the exponents fixed at c_l = beta = 1. That is a one-parameter family of viscous self-similar blow-ups at a = 0, which is precisely Route-H's alpha_1 = 0, the "line of viscous self-similar blow-ups". Route-H measured, numerically, the existence of an exactly-known family. Correct; not new.

4. J3, alpha(1/2) = 3, from the published a = 1/2 system, integrated cold

ALS §5.2 gives the a = 1/2, sigma = 1 pole system (double poles this time, and ALS note the leading 1/(x∓iv_c)^4 poles cancel only at a = 1/2, which is what selects that value):

dv_c/dt = -( omega_{-2}/(4 v_c) - nu ) ,      d omega_{-2}/dt = omega_{-2}^2 / (4 v_c^2)   (49)-(50)

This shares no grid, no basis, no formulation and no line of code with our compactified Newton solve. Integrating it into the collapse and measuring the local exponent:

c_l = 0.3333076 against the exact 1/3, relative error 7.7e-5, so alpha(1/2) = 3.000232. The independent structural check on the same trajectory: Omega ~ v_c^{-2.000144} against the exact -2.

So Route-E's twelve-digit alpha(1/2) = 3 is right, exact, and known: it is c_l(1/2) = 1/3 from the exact pole-dynamics solution, attributed by both ALS and XU to Lushnikov–Silantyev–Siegel and to J. Chen.

And this answers a question Route-E left open and flagged as unexplained. Route-E found alpha passes through odd integers at isolated a (alpha = 1 at a = 0, alpha = 3 at a = 1/2 (twelve digits), alpha = 5 at a = 0.5821792673) and recorded that "what stays unexplained is why alpha = 3 lands on a round rational while alpha = 5 does not." ALS §1 states it plainly: exact solutions as superpositions of pole singularities exist at a = 0 and a = 1/2 and, per Lushnikov et al., nowhere else. alpha = 3 is a round rational because there is an exact solution behind it. alpha = 5 is not because there is not. The alpha = 5 "resonance" is a property of our instrument, Lambda^5 happens to be a finite matrix there, and not of the problem.

4.1 The refusal predicate, and the loose first version of it

The first version of this measurement had a gate on tau: refuse a rung once tau falls below 1e3 · eps · max|S|, S being the accumulated quadrature. It passed, and it read c_l = 0.222 at its deepest "resolved" rung while reading 0.33333 in the middle.

The gate was on the wrong quantity. The local exponent is a difference quotient, so what must stay resolvable is d tau, not tau, and tau at the bad rungs is 1e-11, enormous compared to underflow, so no threshold on tau could ever have caught it. Gating d tau > 1e3 · eps · max|S| instead cuts the ladder at d tau ≈ 3.5e-14 and refuses the entire tail below it (157 161 of 200 001 samples in the driver's run, which is most of them because the reparametrised integration spends most of its steps there); the deepest refused rungs read c_l off by ≥ 0.32. Panel C draws the refused branch in red rather than truncating the axis, because a gate you cannot see is a gate you cannot audit.

This is banked lesson 57 ("is the signal above the solve error?") one level down: gate the quantity the measurement divides by.

5. J4, what lies above criticality: the one thing the literature gave us

Routes F, H and I all stop at s ≤ s_c. Route-F's sentence was "the scaling says which term dominates given the self-similar form", and it had no second half. ALS §5.1 has the second half, in the one case where it is exactly solvable.

Substituting omega ~ tau^{-beta} f(x/tau^{c_l}) into the equation term by term:

omega_t                ~ tau^{-beta - 1}
omega H(omega)         ~ tau^{-2 beta}
nu Lambda^sigma omega  ~ tau^{-beta - sigma c_l}

Below criticality omega_t balances stretching, forcing beta = 1, and dissipation is a correction, that is the regime every leg of this project has lived in. Above criticality the balance switches: dissipation balances stretching, giving

beta = sigma * c_l ,

and the time derivative becomes subdominant. Schochet is the witness: a = 0 gives c_l = 1, so sigma_c = 1/c_l = 1 and sigma = 2 is supercritical; the balance predicts beta = 2; and ALS (45) reports exactly omega ~ tau^{-2} f(x/tau).

We measure it rather than quote it. Rescaling the exact solution on a tau-ladder {1e-3, 3e-4, 1e-4, 3e-5, 1e-5} and reporting the spread of tau^beta omega across the ladder:

beta = 2 wins by 49×. But a minimum over four guesses is a fit, not a measurement, so the check that makes it one: ALS state (45) carries an O(tau^{-1}) correction, so the beta = 2 residual must shrink on deeper sub-ladders. It does: 0.0202 → 0.0060 → 0.00187 as the ladder deepens, falling like tau^1 relative to tau^{-2}, which is the stated correction behaving as stated.

The mechanism is the part worth carrying. The tau^{-2} is carried by a double pole whose residue is B = -12 i nu: proportional to nu, hence absent inviscidly. Above criticality the blow-up is not the inviscid one surviving viscosity; it is a different singularity that viscosity itself creates. Nothing in Routes F/H/I could have seen that, because all three are built on the assumption beta = 1.

And it sharpens what s_c means. XU are explicit, and we should be equally explicit: s* is not the sharp blow-up/regularity threshold, "which for this family remains unknown". At a = 0, s* = 1 and Schochet blows up at s = 2 > s*; XU also cite Sakajo, that at a = 0 blow-up persists at small viscosity regardless of the derivative order of the dissipation, with global solutions at large viscosity. So s* is not even nu-independent as a threshold. It is a relevance exponent for a fixed self-similar form, and Route-F's map is orientation, exactly as Route-F said, but the reason is now citeable rather than cautious.

6. J5: our alpha(a) branch is XU's s*(a), row for row

XU §6.1 eq. (6.3) defines the scaling-critical dissipation exponent by exactly our argument, dissipation enters the rescaled equation with coefficient e^{-gamma tau}, gamma = 1 - s c_l, and s* is where gamma = 0:

s*(a) = 1 / c_l(a) .

In our gauge that is alpha(a), and s_c = alpha/2 = s*/2. Route-F v1's headline. Against XU's Table 1, read straight out of Route-F's committed JSON so the comparison cannot drift from the data:

Worst row 3.1e-3, mean 1.2e-3, and XU state their own branch is good to "two or three significant figures at nonzero a", so this is agreement at the accuracy either side claims, not a discrepancy. The two exactly-known anchors agree to machine precision.

The endpoint is where we come off worst. a_c published (Lushnikov–Silantyev–Siegel) is 0.6890665; XU's recompute is 0.6888, 0.04% off; ours is 0.693493, 0.64% off. We are the least accurate of the three sources and should quote theirs. There is a gate asserting exactly that, which will fail if we ever become the better source, at which point the right response is to say so, not to delete the gate.

Also pre-empted: the ordinary-Laplacian crossing. Ours a ≈ 0.38281, XU a ≈ 0.39.

7. J6, the ledger

Twelve standing claims, with the verdict vocabulary fixed and gated:

Two entries need more than a row.

7.1 The re-classification, which has the largest forward consequence

XU's Proposition 2, the realization dichotomy, is the item to actually carry forward, and it is not a pre-emption of a claim so much as a re-description of a measurement.

Route-E measured the essential spectrum of the rescaled linearization as a continuum filling [0, +1] at a = 0 and [-2, +5] at a = 1/2, and shut the DSS lane with "a continuum has no eigenvalue to move". XU prove that on the origin-H^2 realization there is no such continuum in the strip (the point spectrum is exactly {0, 1} and the open strip is empty) while on the maximal L^2 realization the whole strip is spectrum, filled by an explicit family u_lambda(y) = y^{1-lambda}/(y + i/2)^2 that is L^2 but fails u'' ∈ L^2 at the origin. And they say, in as many words, that "the persistent essential smear that the grids without an origin condition place inside the strip" is the faithful spectrum of that maximal realization.

Our discretization has no origin condition. We were rendering the loose realization. So "the non-symmetry spectrum is continuous" is not a property of the operator; it is a property of which operator we discretized. The DSS conclusion survives, in the tight realization there is no continuum and no complex pair, so there is still nothing to bifurcate, and the lane stays shut on better evidence than before.

But Route-I's headline inherits the problem in full. Route-I reported that the inviscid rescaled fixed point at a = 1/2 has 141 of 144 unstable directions, and read that as "§26's essential spectrum as a count". If the essential spectrum in question is the maximal realization's strip, then the count is a count of that realization's spectrum: a realization whose eigenfunctions are precisely the ones failing the regularity condition at the collapse point. This does not make Route-I wrong, and it does not by itself change the mu > 0 half of the inversion (that half is about dissipation collapsing the spectrum onto a discrete negative ladder, and a discrete ladder is not realization-smear). It does mean the mu = 0 count is quoted in a realization nobody would choose deliberately, and the honest statement of Route-I's inversion needs the realization named in it. That is now the top-ranked correction item, and §8 says so.

7.2 The partial: finite support

HTW Proposition 2.3 (any H^1 solution of the profile equation with c_omega/c_l > 0 must be compactly supported) is proved for the De Gregorio model, a = 1, where non-degeneracy at the origin forces c_l = c_omega. The mechanism is the same one Route-D v12/v13 found: the profile is locally proportional to u + c_omega x, and the support ends where that vanishes. So the phenomenon and the mechanism are published.

What is not in HTW is the a-dependence: X_c as the zero of c + aU across a ∈ (0, 1/2), and the algebraic order 1/a of that zero. And the use we actually made of it (that a compactly supported profile in a global spectral basis puts its worst representation error exactly where the decay-graded codomain weight is largest) is a certification-space statement that no PDE paper would make. PARTIAL, with the surviving part narrower than "the profile ends".

8. What this changes, and what it does not

Does not change: any number in any committed artifact. Every measurement this project has made stands; nothing was found to be wrong. What changed is who found it first.

Changes, in priority order:

1. Route-I's stability inversion must name its realization. The mu = 0 count of 141 unstable directions out of 144 is a count in the maximal L^2 realization. Flagged in the ledger and in PHASE2_P2_NOTES.md §31; the writeup correction is the first item of the next leg, and it may require re-running I5 with an origin condition to say what the count is in the realization one would actually choose.
2. Route-F v1's writeups must lead with the pre-emption, not bury it. Done in LITERATURE_CHECK.md; the TECHNICAL_P2_ROUTEF_V1.md header now carries it.
3. Route-E's open question is closed (by ALS, not by us) and its a_c should quote the published value.
4. Route-H's (E) has a citation and stops being "at high risk".

Still unsearched at primary source: the Route-D methodological claims (the discrete-ball trap, the weighted-ℓ¹ no-go, the elasticity discipline), which remain the only claims in this project with a real chance of being new. Tier 2 (2302.12877, 2312.01702, 1908.09385) is fetched and text-extracted but not read closely. That is now a cheap spend and it is the obvious next literature action, but it is a literature action, and Directive 2 (the L1→L2 certification port) has now been deferred six times.

Clay: nothing here moves a link of the L1→L4 chain. The realistic prize is still a novel result on a model where blow-up is provable, and this leg made the "novel" half materially harder by deleting the candidates that were not.

9. Reproduce

bash Papers/fetch.sh                                          # 14 PDFs, gitignored
.venv/bin/python -u experiments/p2_route_j_v1_literature.py   # ~5 s
.venv/bin/python writeup/4_p2_lottery/p2_route_j_v1_evidence.py
.venv/bin/python test_literature_gates.py                     # 9/9