Route-DWM v1, a per-constant width ledger of BCG's r-dominance argument (leg 305)

Gate answer: YES. The re-derivation reproduces both window endpoints through an independent code path and yields a per-constant width ledger naming the costliest constant.

Data: writeup/data/p2_route_dwm_v1.json. Runner: experiments/p2_route_dwm_v1.py. Figure: fig82, writeup/figures/fig82_route_dwm_v1_ledger.png. Source: Buckmaster, Cao-Labora, Gómez-Serrano, arXiv:2208.09445, Smooth imploding solutions for 3D compressible fluids, Forum of Math. Pi 13 e6 (2025); e-print tarball md5 45ea63c45a1a199ecfb4dc4a15431600, file RadialImplosion31_FinalArxiv.tex. Every l.NNN below is a line number in that file.

1. What was open

Leg 266 (verified and corrected by leg 300) fixed the certificate obligation at the stability step with F_dis retained and r outside the dominance window (1.1666667, 1.1909830): a window 6.854× narrower than the target window (1, 7/6], closed form (7+3√5)/2. Nobody knew whether that width is sharp for BCG's argument or an artifact of one generous intermediate constant. This leg re-derives the window tracking every intermediate constant and measures which constant costs the most width.

Leg 240 had already banked the endpoints, the width, the δ_dis margin and the γ-ceiling. Those are used here as an internal reference to be reproduced through a different code path, never re-claimed as this leg's findings.

2. The independent path (pre-registered before measuring)

A reproduction through the same formula is not a reproduction. Both endpoints are re-derived from different displayed equations than legs 240 and 300 used.

\eqref{eq:rstar} is evaluated in exactly one place) check T4: purely as the cross-check target. Nothing else consumes it.

3. Reproduction: every gate-triggering tolerance passes

Measured window, 60 digits:

lower  1.16666666666666666666666666666666666666666666666666666666666
upper  1.19098300562505257589770658281718094113984541009711856893228
width  0.02431633895838590923103991615051427447317874343045190226562
ratio  6.85410196624968454461376050309691435316092753941728858640298
γ-ceil 2.15470053837925152901829756100391491129520350254025375203704

The ratio is (7+3√5)/2, agreeing with leg 300's re-derivation. It is not 6.855; that transcribed digit is not used anywhere in this leg's artifacts.

The γ-ceiling deserves a note. Solving r*(γ) = 2γ/(γ+1) requires r* on both of BCG's branches, and the root-solve selects the branch by measurement rather than by reading \eqref{eq:rstar}: for 1 < γ < 5/3, D_{Z,1} reaches zero only at the realness edge, while for γ ≥ 5/3 it crosses zero strictly inside the domain. That this branch structure falls out of the root-solve, and that T4 then agrees with the two-branch closed form to exactly zero over 12 γ values, is the strongest single piece of evidence that the transcription of R₁, R₂ and D_{Z,1} is faithful.

The r=1 conditioning trap, reproduced and diagnosed

BCG state at l.603 that R₂ = 0 at r = 1, hence k(1) = 1. The radicand is a sum of terms of magnitude up to 24.528 cancelling to exactly zero, so in IEEE double it evaluates to 1.2212453270876722e-14 of rounding dust and the square root costs half the digits:

• float: |k(1) − 1| = 1.381374710174299e-07, failing a 1e-9 tolerance, reproducing leg 302's measured 1.29e-07 to within the difference in probe arithmetic;
• 60-digit: R₂ radicand at r=1 is 0.000 exactly, and |k(1) − 1| = 0 exactly.

This was pre-registered as an expected float failure and it decides nothing: the float number is emitted under conditioning_trap, labelled diagnostic, and is not an input to any verdict. A conditioning artifact reported as a sharpness finding would have been this leg's serious failure mode; it was diagnosed before it was believed.

4. The ledger

Eleven named constants, each appearing in a displayed BCG equation, each perturbed one at a time multiplicatively with everything else held fixed, and the window recomputed at γ = 7/5. move_to_close is the relative change in that one constant for which the dominance window becomes as wide as the target window 1/6, i.e. the deficit falls to 1×.

Costliest constant (pre-registered rule: smallest |move_to_close|): C1_c_lap, the Laplacian's own scaling weight, at −42.705%.

Why c_lap is the costliest, and why it is not generous

Naming the constant is not the realization; naming why it is the one is. At γ = 7/5 the lower endpoint is the root of 2 − r + (1/α)(1−r), and c_lap = 2 is the 2 in that expression. It is not an estimate, a Sobolev constant, or a bookkeeping convenience. It is the number of spatial derivatives in νΔ, entering through the parabolic scaling of the Laplacian under BCG's self-similar change of variables. Its width elasticity is −13.708% of the window per 1% move, precisely because it is the additive constant term of the exponent rather than a coefficient of r.

[CORRECTED 2026-08-12, leg 336, measured against this leg's own ledger JSON (writeup/data/p2_route_dwm_v1.json, ledger[], field M2_pct_of_window_per_1pct).] The sentence originally read "Its width elasticity is the largest of any constant: −13.708% of the window per 1% move". That is false on the ledger's own metric: C9_a1's elasticity is M2_pct_of_window_per_1pct = −31.058%, 2.27× larger in magnitude than C1's −13.708%. The table above prints — for C5–C11 because their reachable move_to_close is undefined (their far side is capped), but the JSON still computes a one-sided M2 elasticity for every row from whichever perturbation direction is non-flat, and on that column C1 is not the largest; C9_a1 is (|−31.058|), ahead of C1 (|−13.708|), C8_a2's and C10_a0's and C6_c_lin's and C5_c4's and C7_R2scale's rows (all smaller in magnitude). C1_c_lap remains the costliest constant only under the pre-registered rule stated above (smallest |move_to_close|), which is a narrower and different metric than elasticity: see §6 for whether the SHARP verdict depends on either claim.

Its move_to_close has a closed form, and the runner's bisected value agrees with it to 1e-40:

c_lap*  = (9 − 3√5)/2 = 1.14589803375031545538623949690308564683907246058271141359365
s*      = (9 − 3√5)/4 = 0.572949016875157727693119748451542823419536230291355706796825

Since c_lap = 2s for the operator ν(−Δ)^s, closing the deficit by moving this constant means replacing νΔ with hypodissipation ν(−Δ)^s at s = 0.5729…, below the s = 1 of the actual Navier–Stokes viscosity and indeed below s = 1/2. That is a different PDE, not a sharper proof of the same one. This is what "sharp" means here in operational terms.

The upper endpoint cannot be moved at all, and that is a measurement

Seven of the eleven constants, every constant living in D_{Z,1} or R₂, report unreachable, and the reason is structural rather than numerical. At γ = 7/5:

R₁ radicand at r*        = 3E-59          (i.e. zero)
R₁ radicand roots        = 1.19098300562505257589770658281718094113984541009711856893228
                           2.30901699437494742410229341718281905886015458990288143106773
D_{Z,1} at realness edge = −1.639E-30     (i.e. zero)

r* is exactly the smaller root of BCG's R₁ radicand, that is, exactly the point where P_s and P̄_s merge, a saddle-node of the phase portrait. It is not an estimate of a threshold; it is a discriminant. Consequently the upper endpoint is pinned: perturbing any D_{Z,1}/R₂ constant can push D_{Z,1}'s zero below the edge, lowering r*, but can never push it above, because R₁ ceases to be real there and P_s ceases to exist. r* sits at a maximum of every one of these seven one-parameter families.

The measurement that establishes this, rather than asserting it, is the response exponent. The runner perturbs by ε = 1e-5 and 1e-6 and fits |ΔW| ∼ ε^p:

• six of the seven capped rows (C5–C10): p = 1.996…–1.9996, i.e. p = 2, a stationary maximum, not a kink;
• the four uncapped rows: p = 0.9999…–1.0, i.e. p = 1, as an ordinary derivative should be.

[CORRECTED 2026-08-12, leg 336, measured against ledger[] fields scaling_exponent_up / scaling_exponent_dn in writeup/data/p2_route_dwm_v1.json.] The bullet originally read "all seven capped rows". C11_aR1 is capped (capped_r_star_at_a_maximum: true) but reports scaling_exponent_up = scaling_exponent_dn = "flat": neither side shows the p ≈ 2 response; both are identically flat. C11_aR1 is not a seventh instance of the p = 2 phenomenon; see the T6 correction below for the adjudication (inert-by-construction, not missing data).

p = 2 is forced by the geometry: D_{Z,1} meets zero at the edge with a vertical tangent, because R₁ ∼ √(edge − r) there, so an O(ε) perturbation of any constant inside D_{Z,1} moves the endpoint only by O(ε²). C11_aR1's row multiplies R₁ itself, which is the quantity that vanishes at r* (R₁_radicand_at_r_star ≈ 0, structural_findings in the JSON); its term is R₁ × (polynomial), and R₁ = 0 at the evaluation point regardless of how the polynomial factor is perturbed, so the row measures a derivative of a term that is identically zero at r*: a different, degenerate case from the six genuine p = 2 rows, not a data gap.

A pre-registered tolerance that failed, reported rather than smoothed

Tolerance T6 (elasticity linearity: the central-difference dW/dln c at ε = 1e-5 versus ε = 1e-6, relative deviation ≤ 1e-4) FAILS, and it is reported as failing. T6 was pre-registered as a non-gate-triggering diagnostic, and the diagnosis is exact: for six of the seven capped rows, the central difference is itself O(ε) because one side is flat and the other is quadratic, so halving ε halves it and the relative deviation is exactly 0.9. Measured: 0.8991–0.8999 for those six. An additional check declared as additional, T6b, applies the same test to the four uncapped rows and passes at ~9e-15.

[CORRECTED 2026-08-12, leg 336, measured against ledger[] field M1_linearity_rel_dev in writeup/data/p2_route_dwm_v1.json.] The passage originally read "for the seven capped rows … Measured: 0.8991–0.8999 for all seven". C11_aR1's M1_linearity_rel_dev is 0E-54 (exactly zero, not ≈0.899) because both its one-sided derivatives (M1_one_sided_up, M1_one_sided_dn) are themselves exactly 0, not merely small. Adjudication: this is inert-by-construction, not missing or broken data. C11_aR1's row is R₁ × (9(γ−2)γ + ((2−3γ)γ+5)r + 5), and R₁ = 0 at r = r* by the structural finding above (R₁ radicand ≈ 3E-59, i.e. R₁ itself vanishes there: this is the same fact that defines r* as the saddle-node). Perturbing the polynomial factor by ±ε multiplies a zero by (1±ε), which stays exactly zero to the runner's precision; there is no missing measurement, no broken perturbation, and no undetected response, the term genuinely contributes nothing to dW at r*, unlike the other six capped rows, whose one-sided derivatives are small but nonzero (Lesson 90 tell: C11 is a control that structurally cannot come out differently, which is exactly why it does not belong to the "seven" count).

So T6's failure is not an arithmetic defect. It is the numerical signature of BCG's constants sitting at a stationary maximum of r*: the same finding as the previous subsection, arriving independently through a tolerance that was written down before the measurement.

5. The discrete sub-ledger

BCG's admissible speeds are not an interval. Lemma \ref{lemma:k} (l.583) makes k : [1, r*) → [1, ∞) a bijection and the profiles live at r_n := k⁻¹(n) for odd n ≥ 3 (l.367–369). So the width deficit also has a counting form:

k(7/6) = 16.3479210516613384873990168461313841556225924879068376948995

16 admissible speeds fall at or below the dominance threshold, 7 of them odd (n = 3,5,7,9,11,13,15). BCG's Euler profiles at those seven speeds have no Navier–Stokes counterpart under the dominance argument; the first odd speed that survives is n = 17. The continuum deficit 6.854× and the discrete deficit "the first seven odd profiles" are two readings of the same gap.

6. Verdict: sharp, on the pre-registered rule

The rule, fixed before measuring: SLACK iff at least one constant is classified ESTIMATE and its move_to_close is within a factor 2 of unity. Otherwise SHARP FOR BCG'S ARGUMENT AS STATED.

All eleven constants classify as EXACT_IDENTITY. Not one is an estimate, a Sobolev or interpolation constant, or a numerically-tuned bound; each is a derivative count, an ideal-gas exponent, a polynomial coefficient in a discriminant, or an algebraic combination of γ. There is therefore no constant that a sharper argument could sharpen. The verdict is SHARP_FOR_BCG_ARGUMENT_AS_STATED, and the named realization (lesson 91) is:

The 6.854× deficit is not the price of a generous estimate. It is the distance between two exact thresholds: a derivative count (c_lap = 2, the order of the Laplacian) and a discriminant (r*, the merge point of P_s and P̄_s). Both endpoints are rigid. Closing the deficit by moving the costliest constant means changing the operator to ν(−Δ)^{0.5729…}, i.e. solving a different equation.

What this does and does not say. It says the deficit cannot be reduced by sharpening any constant in this argument. It does not say the deficit is irreducible: a different argument (one that does not route the viscous term through pointwise domination by the self-similar profile at all) is untouched by this ledger. Measuring the shape of an obligation is not discharging it.

[ADDED 2026-08-12, leg 336.] The verdict above depends only on the M4_class field of all eleven rows (EXACT_IDENTITY, unanimous) and on move_to_close. Neither correction in §4 touches either input. The elasticity correction (C9_a1 at −31.058%, not C1 at −13.708%, is the largest-magnitude M2_pct_of_window_per_1pct) does not change any row's M4_class, and C1_c_lap is unaffected as costliest-by-move_to_close because that metric was never elasticity to begin with. The C11_aR1 all-zero-row adjudication (inert-by-construction, not missing data) removes one row from the "seven capped p ≈ 2 / T6 ≈0.899" count but changes no row's M4_class and adds no ESTIMATE. The SHARP verdict is UNMOVED by both corrections.

7. Relation to leg 315 (Route-TMS)

Leg 315 landed the complementary question. It asked what machinery could close this gap and identified the sonic-crossing r-tube of BCG's autonomous (W,Z) self-similar Euler ODE flow as reachable by Taylor-model stepping but not by this repository's Newton–Kantorovich apparatus, recording the same shortfall 6.8541019662496845446. This leg asks which constant makes the gap that wide. The two answers compose: the costliest constant named here, c_lap = 2, is the thing leg 315's proposed enclosure would have to beat, and since c_lap is a derivative count rather than an estimate, an enclosure cannot beat it by being tighter; it would have to certify the domination on a region where δ_dis < 0, by a different mechanism.

Leg 315 also found the cost blocker, which matters for reading this leg's verdict: Zgliczyński's ODE→PDE bridge assumes dissipativity, while BCG's rescaled system is quasilinear hyperbolic. So even had this ledger come out slack, "slack" would not have meant "reachable". It came out sharp, which if anything strengthens that reading.

8. Walls

Wall 1 and Wall 2 stand. Measuring the shape of an obligation is not discharging it; no link of the L1→L4 chain moves. The object is 3D compressible Navier–Stokes (BCG, γ = 7/5), which the source leg itself flagged as not the incompressible system the Clay problem asks about. Clay odds ~0.05%, unchanged.