The ledger of the OpenAI manuscript, version 1.1, October 1, 2026

Appendix C: Realizing the admissible stress cone (pp. 157 to 165)

MC.1: Target and obstruction: the admissible cone on a relaxed-only interval

Node ns-mc-1-target-and-obstruction-the-admissible-cone-on-a-relaxed, kind move, pp. 157-158.

• Statement: Appendix C target (pp. 157 to 158): with s = (a, −bs), a = 1 − 2DX log E, bs = 2DXU/E, ps = (XQs/L, XNs/(LE)), ts = −bs/a, vs = a + bs²/a, Pc = ps,1 + tsps,2, Jc = ps,2 − tsps,1, 𝒰 = Pc + Jc²/4 − |Jc|√((Pc − 2)/2 + Jc²/16), the relaxed cone (4.21) is Pc > 2, vs < 𝒰; the admissible cone adds vs > 2, equivalently (Lemma 4.5) (4.22): Pc > vs, (vs − 2)Jc² < 2(Pc − vs)². The input (Corollary B.10, Propositions A.4, A.7) has the strict relaxed cone throughout; a compact I = [X−, X+] ⋐ (Xa, Xb) contains every point where admissibility may fail. The only inequality to manufacture is vs > 2 on I.
• Obligation: This move fixes what the waves can realize, and therefore what the profile must satisfy. In section 7.1 (p. 74) the chart shear is g_0=F_0(-a,b_s), with N=g_0/|g_0| and K=N^\perp. The reference growth rate is \lambda_0^2=2aF_0^2(1-2/v_s), and c_0^2=(v_s-2)/2 with c_0<0, so pulses grow only if v_s>2. A growing pulse has polarization y/x=c_0\sqrt{1+s^2}+O(S_*^{-1}) (7.21). Its covariance column is therefore H_\sigma=h_\sigma(-A_cN-\sigma u_*K+e_\sigma) with A_c=-c_0\sqrt{1+u_*^2}>0 (7.28). Positive squared amplitudes y=H^{-1}T_{0,*} (Proposition 7.5) exist exactly for targets with T_N<0 and |T_K|<(u_*/A_c)(-T_N). Since u_*/A_c increases to 1/|c_0| as u_*\to\infty, the union of these cones is (7.1): T\cdot N<0 and |c_0\,T\cdot K|<|T\cdot N|. That is a cone about the shear direction (1,t_s) with half-opening \arctan\sqrt{2/(v_s-2)}, and by (4.23) it is exactly (4.22). The first inequality is also the sign of energy extraction from the shear, -g_0\cdot T=-|g_0|T_N>0 in (7.22). So the leading stress must lie in the cone for three reasons: amplitudes must be real (positive squared weights), only growing pulses carry the stress, and their polarization caps the ratio of transverse to along-shear flux at 1/|c_0|. Where v_s\le2, \lambda_0^2\le0: there is no growing direction, c_0 is undefined, and Proposition 7.5 has nothing to work with. Without this appendix, Theorem 4.6(iii) (v_s-2 bounded below on the closed annulus) is unproved.
• Mechanism: The appendix exploits a split in how the profile enters the cone test. The shear s uses logarithmic radial derivatives (4.11). The vector p_s and the moments m=(M,I,J,S,C_p) use only profile values, cumulative radial integrals, and \eta-derivatives ((4.15), (4.16)). So the shear can move at order one while p_s moves by O(N^{-1}). The construction freezes p_s pointwise and asks which shears are admissible for that p_s. It finds a loop of them averaging to the given shear (MC.2 to MC.4) and realizes the loop by fast radial modulation (MC.5 to MC.7). All changes live in I plus one reserved patch.
• Antecedent: None cited in Appendix C. The introduction (p. 2) cites Leibovich and Stewartson [15] and Billant and Gallaire [2, 3] as centrifugal-instability precedents for the wave dynamics. It cites Lifschitz and Hameiri [17] and Friedlander and Vishik [14] for wavevector and polarization evolution. (Digest's note, not in the paper: put \Omega=u_\theta/r, W=u_z, \Gamma=r^2\Omega. Then (4.11) gives a=-r\Omega'/\Omega and b_s=W'/\Omega, hence a(v_s-2)=(\Omega'\Gamma'+W'^2)/\Omega^2. For a>0 (\Omega'<0), the condition v_s>2 is exactly the Leibovich-Stewartson condition V\Omega'(\Omega'\Gamma'+W'^2)<0, and with W'=0 it is Rayleigh's \Gamma'<0.)
• Cost: The input must satisfy the relaxed cone strictly, with a positive minimum on compact sets, and admissibility on neighborhoods of both ends of I. One unused reserved patch must remain downstream. This move fixes X_\pm and X_{an}.
• Backward question: "The waves realize only stresses in a cone about the shear direction, with opening set by v_s, and only where v_s>2. My joined profile satisfies P_c>2 and v_s<\mathcal U but has v_s\le2 somewhere in the annulus. Which profile data enter the cone test through radial derivatives and which through integrals, and can I change the first kind at order one while freezing the second?"
• Checkable: This is a finite-dimensional check. For given (a,b_s,p_s), compute \Psi of (4.35) and test all four entries >0. Equivalently, set N=(-a,b_s)/|(-a,b_s)|, K=(-N_z,N_\theta), c_0=-\sqrt{(v_s-2)/2}, T=p_s-(a,-b_s), and test T\cdot N<0 and |c_0T\cdot K/T\cdot N|<1. Then pick u_* with u_*/\sqrt{1+u_*^2}>|c_0T\cdot K/T\cdot N|, set A_c=-c_0\sqrt{1+u_*^2} and H=[-A_cN-u_*K,\ -A_cN+u_*K], and check H^{-1}T>0 componentwise. Symbolically, check -2F_0N_\theta(2F_0N_\theta+|g_0|)=2aF_0^2(1-2/v_s) and c_0^2=(v_s-2)/2. The digest ran these checks. Among about 2\times10^5 random samples there were zero disagreements between (4.21) with v_s>2, (4.22), and (7.1). H^{-1}T>0 held in all 30,668 admissible samples. Sympy confirmed both identities and the Leibovich-Stewartson rewriting.
• Depends on: M4.7 (the target is Lemma 4.5's admissible cone, the relaxed inequalities (4.21) plus v_s > 2, equivalently (4.22).); M4.4 (the cone is written through the shear s = (a, -b_s), the integrated vector p_s, and T0 = F(p_s - s) of (4.11).); MB.14 (the input is Corollary B.10's joined profile, admissible on the inner collar and only relaxed out to the matching point.); MA.9 (Proposition A.4 gives admissibility only from the intermediate power law onward, which leaves the relaxed-only interval I.)
• Refs: pp. 157 to 158 (Appendix C preamble); p. 27, (4.11); pp. 30 to 32, (4.20) to (4.23), Lemma 4.5; p. 38, (4.35); p. 74, (7.1) and the frame; pp. 80 to 81, (7.21), (7.22); pp. 82 to 83, (7.23) to (7.29), Proposition 7.5; p. 2 (citations).
• Verification: statement completeness audit 2026-10-01: FRAGMENT; statement replaced from the digest; hypotheses not-checked; computation checked False

MC.2: Exponentially tilted ratio family: mean t_s, a floor on P_c, unbounded variance

Node ns-mc-2-exponentially-tilted-ratio-family-mean-t-s-a-floor-on-p-c, kind move, pp. 159.

• Statement: Lemma C.1, first part (p. 159). Let Pc(t) = ps,1 + ps,2t, 0 < d0 < ½ min(Pc(ts) − 2), Me(z) = ⟨e^{z sin θ'}⟩θ' (C.4) and t(θ'; μ) = ts + d0(e^{μps,2 sin θ'}/Me(μps,2) − 1)/ps,2 for μ ≥ 0 (C.5), equal to ts + d0μ sin θ' at ps,2 = 0. Then ⟨t⟩θ' = ts, Pc(t) ≥ Pc(ts) − d0 > 2, and the variance V(μ, ps,2) = ⟨(t − ts)²⟩θ' increases strictly in μ > 0 (C.6) and tends to ∞. A finite cover gives one μmax with V(μmax, ps,2) > 3/min a on I × [−1, 1] (C.7).
• Obligation: The loop must consist of shear directions t with mean t_s, so that the vectors can later average to (a,-b_s). It needs as much variance as required, because the variance is what lifts v above 2 (MC.3). Each member must also keep P_c(t)>2, since that is what makes the upper cone bound exceed 2 (2<\mathcal U\le P_c whenever P_c>2, proof of Lemma 4.5). A large symmetric oscillation of t would push P_c(t)=P_c(t_s)+p_{s,2}(t-t_s) below 2 on one side whenever p_{s,2}\ne0.
• Mechanism: The weight w=e^{\mu p_{s,2}\sin\theta'}/M_e(\mu p_{s,2}) is positive with mean one. Setting p_{s,2}(t-t_s)=d_0(w-1) gives mean zero and bounds the change in P_c below by -d_0, while excursions that increase P_c are unbounded. So t is bounded on one side and free on the other, and every large excursion goes in the helpful direction. The variance is (d_0/p_{s,2})^2(\langle w^2\rangle-1) with \langle w^2\rangle=M_e(2z)/M_e(z)^2. Strict monotonicity comes from log-convexity of the moment generating function: g'' is the variance of \sin\theta' under the tilted density. Growth comes from Laplace's method at the maximum of \sin. Joint smoothness through p_{s,2}=0 uses G(p)/p=\int_0^1G'(up)\,du.
• Antecedent: None cited. (Digest's note: M_e(z) is the modified Bessel function I_0(z), and (C.5) is an exponential tilt. The paper names neither.)
• Cost: New constants d_0 and \mu_{\max}. \mu_{\max} can be large, because V grows only like \sqrt{\mu|p_{s,2}|} when p_{s,2}\ne0 (digest: M_e(2z)/M_e(z)^2\sim\sqrt{\pi z}). The family is unbounded in t as \mu\to\infty, which is why the next margin \delta_L must be chosen after \mu_{\max}.
• Backward question: "How can I oscillate the shear ratio t about t_s with arbitrarily large variance, without ever letting the along-shear inviscid coefficient P_c(t)=p_s\cdot(1,t) fall to 2?"
• Checkable: Use scipy.special.i0 for M_e and quadrature in \theta', on sample (a,b_s,p_s) with P_c(t_s)>2. Check \langle t\rangle=t_s and \min_{\theta'}P_c(t)\ge P_c(t_s)-d_0. Check that \langle(t-t_s)^2\rangle equals the closed form V, that V increases in \mu, that V/(d_0^2\mu^2/2)\to1 as \mu\to0, and that [M_e(2z)/M_e(z)^2]/\sqrt{\pi z}\to1. The digest ran this at p_{s,2}=1,0,-1.5: all identities held to quadrature precision. The Bessel ratio was 0.980, 0.998, 0.9998 at z=10,100,1000. For p_{s,2}<0 the loop's t stayed below t_s+d_0/|p_{s,2}|, as predicted.
• Depends on: MC.1 (works on the relaxed-only interval I with p_s frozen, where P_c(t_s) > 2 has a positive minimum.); M4.7 (P_c(t), J_c(t) are the cone coordinates (4.20) along the direction (1, t); keeping P_c > 2 keeps U(P_c, J_c) > 2 (Lemma 4.5).)
• Refs: p. 159, (C.4) to (C.7); p. 31 (proof of Lemma 4.5: 2<v_-\le P_c).
• Verification: statement completeness audit 2026-10-01: FRAGMENT; statement replaced from the digest; hypotheses astra-spot-check-2026-10-01; computation checked False; Astra spot-check: correct

MC.3: Target instability level v and smooth tilt strength \mu(X,\eta)

Node ns-mc-3-target-instability-level-v-and-smooth-tilt-strength-mu-x, kind move, pp. 159-160.

• Statement: Pp. 159 to 160. Choose 0 < δL < 1 with 𝒰(Pc(t), Jc(t)) > 2 + δL for 0 ≤ μ ≤ μmax (C.8), and δL < min(vs − 2) near ∂I. With ζL smooth in vs, 0 ≤ ζL ≤ 1, ζL = 1 for vs ≤ 2 + δL/8, ζL = 0 for vs ≥ 2 + δL/4, set v∗ = 2 + δL/2, ρ = ζL(vs)²(v∗ − vs), v = vs + ρ (C.9), and solve V(μ, ps,2) = ρ/a, 0 ≤ μ < μmax (C.10). Then 0 ≤ ρ < 3; v = v∗ if vs ≤ 2 + δL/8; 2 < vs ≤ v ≤ v∗ in between; v = vs, μ = 0 if vs ≥ 2 + δL/4. In every case 2 < v < 𝒰(Pc(t), Jc(t)) for every θ'.
• Obligation: This fixes how unstable each loop member is. It must exceed 2 (growth) and stay below \mathcal U(P_c(t),J_c(t)) for every member (quadratic cone test). It must equal v_s, with no oscillation, wherever admissibility already holds, in particular near \partial I, which gives (C.3). And it must be chosen so that \mu depends smoothly on (X,\eta), including where \rho=0.
• Mechanism: The loop vectors will be v(1,t)/(1+t^2), and the averaging identity of MC.4 forces v=a\langle1+t^2\rangle=v_s+aV. So each loop member's excess instability over the mean shear is exactly a times the variance, and (C.10) sets that variance. The target v_* sits just above 2 because \mathcal U can approach 2 along the unbounded direction of the family (digest: for p_{s,2}=0, \mathcal U\to2 as |t|\to\infty). So \delta_L is fixed only after \mu_{\max} makes the family compact. Squaring the cutoff makes \sqrt\rho=\zeta_L(v_s)\sqrt{v_*-v_s} smooth, since v_*-v_s\ge\delta_L/4 on the support. V is even in \mu with V=\tfrac12d_0^2\mu^2+O(\mu^4p^2), so its signed square root is smooth and odd, with derivative d_0/\sqrt2 at \mu=0. The implicit function theorem applied to that square root and \sqrt\rho/\sqrt a gives smooth \mu through \rho=0; (C.6) handles \rho>0.
• Antecedent: None cited (implicit function theorem).
• Cost: New constants \delta_L and v_* and the cutoff \zeta_L, chosen in the order d_0\to\mu_{\max}\to\delta_L. On the modulated set the instability margin is only about \delta_L/2. Rewriting the p. 74 formula with a=v/(1+t^2) (digest's rewriting) gives \lambda_0^2=2F_0^2(v-2)/(1+t^2)\le F_0^2\delta_L there. So the positive lower bound for \lambda_0 used in section 7.1 is small, though fixed.
• Backward question: "If each loop vector is v(1,t)/(1+t^2) and their average must be (a,-b_s), what does that force on v? How do I place v strictly between 2 and the upper cone bound for every member, smoothly in (X,\eta), with no change where nothing is broken?"
• Checkable: Use sample data that violate only v_s>2. Take \delta_L as a fraction of \min_{\theta',\,\mu\le\mu_{\max}}[\mathcal U(P_c(t),J_c(t))-2], form \rho, and solve (C.10) by Brent's method. Check a\langle1+t^2\rangle=v and 2<v<\mathcal U(P_c(t),J_c(t)) for all \theta'. The digest ran this at three samples. For example, with a=1, b_s=0.3, p_s=(5,1) (v_s=1.09): \min(\mathcal U-2)=0.0564, \delta_L=0.0282, v=2.0141, \mu=1.274, and a\langle1+t^2\rangle=2.014093=v.
• Depends on: MC.2 (the variance ρ/a is reached by the tilted family, whose variance V increases strictly in μ and exceeds 3/min a at μ_max.); M4.7 (v must satisfy 2 < v < U(P_c(t), J_c(t)) for every loop member: the relaxed bound plus the viscous inequality.); MC.1 (δ_L is shrunk below min(v_s - 2) near ∂I, where the input is already admissible, so v = v_s there.)
• Refs: pp. 159 to 160, (C.8) to (C.10); p. 74 (\lambda_0^2).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

MC.4: The lift \varphi: exact mean (a,-b_s) and uniform admissible margins (Lemma C.1)

Node ns-mc-4-the-lift-varphi-exact-mean-a-b-s-and-uniform-admissible, kind move, pp. 158-160.

• Statement: Lemma C.1 (pp. 158 to 160). Reparametrize θ' by φ, φ(0) = 0, dφ/dθ' = a(1 + t²)/(2πv), and set (aL, −bL) = v(1, t)/(1 + t²). Since a⟨1 + t²⟩θ' = vs + aV = v, φ(θ' + 2π) = φ(θ') + 1, and ∫0^1 aL dφ = a, ∫0^1 (−bL) dφ = −bs (C.1). Each member has ratio t and level v, so is admissible with ps fixed; compactness gives Ψj(aL, bL, ps) ≥ κL, j = 1, ..., 4, on I × [−1, 1] × (ℝ/ℤ) (C.2), and (aL, −bL) = (a, −bs) within δ∂ of ∂I (C.3). Section 4 restates this as Lemma 4.11, (4.36) to (4.37).
• Obligation: Proposition C.2 needs three things from the loop. It needs a period-one family whose \varphi-mean is exactly the input shear, so that a_L-a and E(b_L-b_s) have zero mean and admit periodic antiderivatives (C.11). It needs a uniform margin \kappa_L, so that O(N^{-1}) perturbations stay admissible. And it needs constancy near \partial I, so that the modification is compactly supported inside I.
• Mechanism: Directions with the right mean ratio are not yet vectors with the right mean. Weighting each direction by the time the loop spends there fixes both components at once. With speed proportional to (1+t^2)/v, the vectors v(1,t)/(1+t^2) integrate to (a/2\pi)\int_0^{2\pi}(1,t)\,d\theta'=(a,at_s). Normalizing the period to one is exactly the identity a\langle1+t^2\rangle=v, which is why MC.3 set v=v_s+aV. The inverse reparametrization is smooth because d\varphi/d\theta' has a positive minimum on the compact family. (Digest's note: v_s=a+b_s^2/a is jointly convex in (a,b_s) on a>0, being the perspective of 1+t^2. By Jensen, a nonconstant loop with mean shear (a,-b_s) must contain shears with larger v_s; the construction puts every member at the same level v_s+a\,\mathrm{Var}(t).)
• Antecedent: None cited.
• Cost: Constants \kappa_L and \delta_\partial, depending on a,b_s,p_s,I.
• Backward question: "Given admissible directions t(\theta') with mean t_s at a common level v, how do I make the vectors themselves, not just their ratios, average to exactly (a,-b_s)?"
• Checkable: By quadrature in \theta', check \int_0^{2\pi}(d\varphi/d\theta')\,d\theta'=1, \int a_L\,(d\varphi/d\theta')\,d\theta'=a, \int(-b_L)(d\varphi/d\theta')\,d\theta'=-b_s, and \min_{\theta'}\Psi_j(a_L,b_L,p_s)>0 for j=1,\dots,4. The digest ran this at three samples: the period was 1.000000, both means were exact to six digits, and all four minimal gaps were positive (for a=1, b_s=0.3, p_s=(5,1): 0.630, 0.0141, 1.706, 5.045).
• Depends on: MC.3 (the level v = v_s + aV of (C.9), (C.10) makes the reparametrized period exactly 1 and every member admissible.); MC.2 (the ratio family has mean t_s, so the φ-weighted vectors v(1, t)/(1 + t²) average to (a, at_s) = (a, -b_s).); M4.7 (each member (t, v) with 2 < v < U(P_c(t), J_c(t)) is admissible by Lemma 4.5, giving the margin (C.2).)
• Refs: pp. 158 to 160, Lemma C.1, (C.1) to (C.3); pp. 38 to 39, Lemma 4.11, (4.36) to (4.37).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

MC.5: Fast radial modulation at phase N\log X with exact shear identities

Node ns-mc-5-fast-radial-modulation-at-phase-n-log-x-with-exact-shear, kind move, pp. 161.

• Statement: P. 161. Let \mathcal A,\mathcal B be the zero-mean periodic antiderivatives \partial_\varphi\mathcal A=-\tfrac12(a_L-a) and \partial_\varphi\mathcal B=\tfrac12E(b_L-b_s) (C.11). They exist by (C.1), vanish near \partial I, and are extended by zero. Set E_N=E\exp(\mathcal A(X,\eta,N\log X)/N) and U_N=U+\mathcal B(X,\eta,N\log X)/N (C.12). Then, exactly, a_N=a_L-2D_X\mathcal A/N and b_N=e^{-\mathcal A/N}(b_L+2D_X\mathcal B/(NE)) (C.13). Here D_X=X\partial_X at fixed \varphi, and all loop quantities are evaluated at \varphi=N\log X.
• Obligation: It turns the loop, a function of an extra variable \varphi, into actual radial profiles. Their shear (4.11) at radius X equals the loop shear at phase N\log X, up to O(N^{-1}).
• Mechanism: After substituting \varphi=N\log X, X\partial_X=D_X+N\partial_\varphi. The prefactor 1/N cancels the N from N\partial_\varphi, so the order-one part of the shear is \partial_\varphi\mathcal A and \partial_\varphi\mathcal B, which were defined to be the loop deviations. What remains is D_X(\cdot)/N. Modulating \log E additively makes a_N linear in \mathcal A. The axial shear picks up only the factor e^{-\mathcal A/N}, which is why \mathcal B carries the factor E. The phase is logarithmic because the shear is defined with D_X. Across I the shear traverses the loop N\log(X_+/X_-) times.
• Antecedent: None cited. (Digest's note: this is a one-dimensional fast-oscillation device, in which a small, rapidly varying function has a derivative that follows a prescribed loop with prescribed mean.)
• Cost: The large integer N. Mixed derivatives of the profile change with r\ge1 radial derivatives are only O(N^{r-1}). So X\partial_X(E_N-E) is order one, and the new profile is not C^1-close in X to the input. All later profile derivative bounds carry N-dependent constants.
• Backward question: "The shear is a logarithmic radial derivative of the profiles. Can I make that derivative trace a prescribed periodic loop by adding a small, rapidly oscillating term whose own derivative carries the deviation?"
• Checkable: Symbolic: differentiate (C.12) with \varphi=N\log X and substitute (C.11) to recover (C.13). The digest ran this in sympy and obtained a_N=a-2\partial_\varphi\mathcal A-2D_X\mathcal A/N and b_Ne^{\mathcal A/N}=2(XU_X+\partial_\varphi\mathcal B+X\partial_X\mathcal B/N)/E. Inserting (C.11) gives (C.13).
• Depends on: MC.4 (the zero-mean antiderivatives (C.11) exist because the loop's φ-mean is exactly (a, -b_s) (C.1), and vanish near ∂I by (C.3).); M4.4 (the shear formulas a = 1 - 2D_X log E and b_s = 2D_XU/E of (4.11) give (C.13) once φ = N log X.)
• Refs: p. 161, (C.11) to (C.13); duplicated in section 4.6 Step 2, p. 41, (4.38); outline p. 10, item 4.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

MC.6: Values, not derivatives: O(N^{-1}) control of moments, pressure, and p_s, and persistence of the cone

Node ns-mc-6-values-not-derivatives-o-n-1-control-of-moments-pressure, kind move, pp. 161-162.

• Statement: Pp. 161 to 162. On the range from X− through the first correction interval (X, E, H, L bounded below), the differences EN − E, UN − U (C.14), the shears (C.13) minus their loop values, mN − m and ΠN − Π (C.15) (axis pressure Π0 kept, ΠN = Π0 + Cp,N), and ps,N − ps (C.16) are ≤ CmN^{−1} through η-order m; mixed derivatives with r ≥ 1 radial derivatives are only Cr,mN^{r−1}. Hence, by (C.2) and (C.13), the admissible cone holds on I for large N, and between I and the correction patch by the input's margins.
• Obligation: The cone test (4.35) involves p_s as well as the shear, and the loop was built with p_s frozen. So the modulated profile's p_s must stay within the margin \kappa_L. The estimates also secure E_N>0 and the hypotheses of Lemma 4.4(ii).
• Mechanism: The phase N\log X does not depend on \eta, so \eta-derivatives never hit it and produce no powers of N. Every parameter derivative of the profile change therefore stays O(N^{-1}). The moments (4.15) integrate values. Q_s and N_s in (4.16) involve only values, moments, and first \eta-derivatives of moments, divided by XH and X. So p_s is Lipschitz in these data (Lemma 4.4(ii), (4.17), whose estimate contains no radial derivative of a difference). Between I and the patch, the profile values and shear are unchanged, but p_s still moves by O(N^{-1}) through the cumulative moments from the axis. The input's strict admissibility absorbs that.
• Antecedent: Lemma 4.4(ii) (internal). No external citation.
• Cost: It uses positive lower bounds for X,E,H,L on the fixed range and one extra \eta-derivative for p_s. N must exceed thresholds set by \kappa_L and the Lipschitz constant of \Psi on a compact neighborhood. Section 4.6 makes this explicit in (4.41): \Psi_i\ge\mu_L-C_\Psi(B_0+P_0)/N\ge\mu_L/2 on I, and \ge\mu_R/2 on [X_+,Y_1].
• Backward question: "Does the integrated inviscid vector p_s, which enters the cone test, see the fast oscillation at all? Which terms of (4.16) contain radial derivatives, and do \eta-derivatives of the oscillation cost powers of N?"
• Checkable: Take a smooth test profile on a compact X-interval and any smooth zero-mean periodic \mathcal A,\mathcal B, and build E_N,U_N for N=10,20,40,80. Compute the moments (4.15) by cumulative quadrature, Q_s,N_s by (4.16), and p_s by (4.11). Confirm that \sup|E_N-E|, \sup|\partial_\eta(E_N-E)|, \sup|m_N-m|, and \sup|p_{s,N}-p_s| scale like N^{-1}, while \sup|X\partial_X(E_N-E)| stays order one. (Not run by the digest.)
• Depends on: MC.5 (estimates the modulated profiles (C.12), whose values move by O(N^{-1}) because the phase N log X carries no η-dependence.); M4.6 (Lemma 4.4(ii) bounds Δp_s by value and moment changes with no radial derivative (4.17), so p_s moves only O(N^{-1}).); MC.4 (the loop's uniform margin κ_L (C.2) absorbs the O(N^{-1}) errors, so the admissible cone persists on I.)
• Refs: pp. 161 to 162, (C.14) to (C.16); pp. 28 to 30, (4.15) to (4.17), Lemma 4.4(ii) and the remark on p. 30; section 4.6, pp. 41 to 42, (4.39) to (4.41).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

MC.7: Exact restoration of the five cumulative moments on the first reserved patch

Node ns-mc-7-exact-restoration-of-the-five-cumulative-moments-on-the, kind move, pp. 162-163.

• Statement: Pp. 162 to 163. On the first reserved patch after (A.9), (Tw − 25, Tw − 20) with Tw = 60 log(1/λ), where U = 0 and E = K(η)X^{−1/2−λ}, add two fixed compact bumps to U and three to E with η-dependent coefficients. With moments ordered (M, J; I, S, Cp), the differential at zero is block diagonal with distinct power weights and uniformly invertible on [−1, 1] (Lemma A.1, Corollary A.3); the rest is exactly quadratic, so Lemma A.2 cancels the O_m(N^{−1}) discrepancy of (C.15) exactly, with coefficients O_m(N^{−1}). The five moments are restored and (C.16) and the cone persist inside the patch.
• Obligation: Every larger radius sees the cumulative integrals from the axis ((4.10), (4.16)). An uncorrected O(N^{-1}) moment error would propagate outward. The total moment identities (4.28) would fail, so the exterior stress would acquire the r^{-2} and r^{-1} tails described on p. 30 and would not vanish beyond X_b. The pressure normalization (4.25), the terminal compensation, and the heat exterior would all shift.
• Mechanism: Because the base has U=0 on the patch, the linearization decouples: U-bumps move only (M,J), and E-bumps move only (I,S,C_p). Each block is a moment matrix B_{ij}=\int X^{\alpha_i}\beta_j\,dX. By multilinearity, its determinant integrates the generalized Vandermonde determinant \det[x_j^{\alpha_i}] against the bumps. That determinant has constant sign on ordered points, because a nonzero combination of m distinct powers has at most m-1 positive zeros (Rolle induction, p. 127). The quadratic part is absorbed by the contraction c\mapsto B^{-1}(d-Q(c,c)) on the ball of radius 2\beta_0d_0, valid when 8\beta_0^2\kappa_0d_0\le1 (Lemma A.2). Taking N large makes d_0=O(N^{-1}) small enough.
• Antecedent: Lemmas A.1 and A.2 and Corollary A.3 (internal). The proof of Lemma A.1 is a Rolle's theorem induction; no external source is cited.
• Cost: It consumes the first reserved patch. The U-block inverse degenerates like \lambda^{-1} as the weights 1 and X^{-\lambda} coalesce (pp. 127 to 128; the paper claims no uniformity in that limit), so \lambda must be fixed before N. N is chosen after \lambda, the patch scales, and all input choices. Smallness is needed only for finitely many \eta-orders.
• Backward question: "The modulation leaves O(N^{-1}) errors in five integrals that control everything farther out. Is there a place where the base profile is a pure power law with U=0, so that five bumps give an invertible, nearly linear map onto those integrals without disturbing the cone?"
• Checkable: Compute B_{ij}=\int X^{\alpha_i}\beta_j(X)\,dX for \alpha=(0,-\lambda) and \alpha=(1/2,-1/2-\lambda,-3/2-\lambda), with rescaled \sigma' bumps ((A.5)) on ordered disjoint log-intervals. Confirm \det\ne0, that \lambda\,\mathrm{cond}(B_U) stays bounded as \lambda\to0, and that \mathrm{cond}(B_E) stays bounded. Then iterate c\mapsto B^{-1}(d-Q(c,c)) with |d|\sim N^{-1} and verify convergence with \|c\|\le2\beta_0\|d\|. The digest computed the conditioning: \lambda\,\mathrm{cond}(B_U)\approx5.9,6.1,6.1 at \lambda=0.1,0.01,0.001, and \mathrm{cond}(B_E)\approx2\times10^4, flat in \lambda, for its bump placement.
• Depends on: MC.6 (cancels the O_m(N^{-1}) discrepancy (C.15) that the modulation leaves in the five cumulative moments.); MA.3 (at α = -1/2 - λ, two U bumps and three E bumps give block-diagonal distinct-power Jacobians, invertible by Corollary A.3.); MA.4 (the bumps sit on the schedule's first reserved patch (T_w - 25, T_w - 20), where U = 0 and E = K(η)X^{-1/2-λ}.); MA.2 (the exactly quadratic system is solved by Lemma A.2 with O_m(N^{-1}) coefficients in each fixed η-order.)
• Refs: pp. 162 to 163; pp. 126 to 128, Lemma A.1, (A.1), Lemma A.2, (A.2) to (A.3), Corollary A.3, (A.4); p. 129, (A.5), (A.9); section 4.6 Step 3, pp. 42 to 43, (4.42).
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MC.8: Propagation of exact equality beyond X_{rep}, and the choice of N before q

Node ns-mc-8-propagation-of-exact-equality-beyond-x-rep-and-the-choice, kind move, pp. 163.

• Statement: P. 163 (Proposition C.2). (C.17): m̃(Xrep, η) = m(Xrep, η) and (Ẽ, Ũ) = (E, U) for X ≥ Xrep, so the five moments agree beyond the patch and Lemma 4.4(i) gives (m̃, Π̃, Q̃s, Ñs) = (m, Π, Qs, Ns), with η-derivatives, for X ≥ Xrep; terminal compensation, exterior moments and pressure vanishing at infinity are kept. The first and last collars and the two unused patches are unchanged. One finite N, chosen after all other finite profile choices (Remark B.9, p. 157), is fixed before q, the dyadic bands and the correction stages; all mixed derivatives are finite with N-dependent constants.
• Obligation: It keeps the exterior exactly as built in Appendix A (zero stress for X\ge X_b, heat exterior, canonical pressure), so Theorem 4.6(ii) and (v) survive. It also makes the profile a fixed object before any physical limit, so every constant in sections 5 to 10 is independent of q.
• Mechanism: Beyond X_{rep} the integrands of (4.15) coincide, so moment vectors that agree at X_{rep} agree for all larger X. With the same axis pressure datum, (4.7), (4.16), and (4.11) are identical formulas in identical inputs.
• Antecedent: Lemma 4.4(i) (internal).
• Cost: Every later constant may depend on N. The order of choices is: the profile parameters of (B.40), then N last within the profile construction, then q.
• Backward question: "If the five integrals agree at one radius and the profiles agree beyond it, is every derived quantity (pressure, radial velocity, Q_s, N_s, stress) automatically identical beyond it, so the exterior never learns about the modification?"
• Checkable: None: exact identity by integration.
• Depends on: MC.7 (starts from the exact equality of the five moments at X_rep produced by the five-bump restoration.); M4.6 (Lemma 4.4(i) turns equal fields and moments at X_rep, with the same Π0, into equal Π, Q_s, N_s for X ≥ X_rep.); MB.14 (Remark B.9 places the radial frequency N after every other profile choice in (B.40); it is then fixed before q.); MA.11 (the retained exterior is Appendix A's: the heat replacement with its compensation, the moment identities, and the canonical pressure.)
• Refs: p. 163, (C.17); pp. 28 to 29, Lemma 4.4(i); p. 157, Remark B.9, (B.40); section 4.6, p. 43, (4.43).
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MC.9: Nonvanishing of the stress in the open annulus, and positive lower bounds

Node ns-mc-9-nonvanishing-of-the-stress-in-the-open-annulus-and, kind move, pp. 163.

• Statement: Proposition C.3 (p. 163), part (ii) and first half of (iii). T0 = 0 for X ≤ Xa (Proposition B.5, (4.13)) and for X ≥ Xb (Lemma A.8), regions preserved by Proposition C.2; T0 ≠ 0 at interior points, since T0 = 0 would give ps = (a, −bs) by (4.11), so Pc = vs, contradicting the strict test Pc > vs. F, a > 0 and vs > 2 inside; at Xa, a = p1,r > 0, vs > 2 + c, F > 0; at Xb, bs = 0, a > 2 + h (A.56), heat factor positive. By compactness F, a and vs − 2 have positive minima on the closed annulus.
• Obligation: Theorem 4.6(ii): the support is exactly (X_a,X_b), so the unit direction n=T_0/|T_0| is defined and the amplitudes y_\sigma of Proposition 7.5 are positive throughout. Also the lower bounds of Theorem 4.6(iii), which section 7.1 uses to define N,K,\lambda_0>0,c_0<0 at every representative (p. 74 notes these follow from the profile bounds rather than being extra choices).
• Mechanism: Since T_0=F(p_s-s), (4.23) gives T_{0,\theta}+t_sT_{0,z}=F(P_c-v_s). The first admissible inequality therefore makes the along-shear component of the stress strictly positive, so the stress cannot vanish.
• Antecedent: None cited (internal identities (4.11), (4.23)).
• Cost: None new. (At the outer edge the instability margin is thin: 2+h<a\le2+2h with h<1/100, from (A.56).)
• Backward question: "After the modulation, could the stress vanish at an interior radius, leaving the direction undefined and forcing a zero wave amplitude inside the annulus?"
• Checkable: Symbolic identity (4.23): with T_0=F(p_s-(a,-b_s)) and t_s=-b_s/a, check T_{0,\theta}+t_sT_{0,z}=F(P_c-v_s) and T_{0,z}-t_sT_{0,\theta}=FJ_c. Then T_0=0 forces P_c=v_s.
• Depends on: M4.7 (T0 = 0 would force p_s = s, hence P_c = v_s and J_c = 0, contradicting Lemma 4.5's strict test P_c > v_s.); MC.6 (after the modulation the admissible cone holds at every interior point, so the strict test is available throughout.); MB.8 (T0 = 0 up to X_a from the stress-free axis profile, with a = p_1,r > 0 and v_s > 2 + c at X_a (Proposition B.5).); MA.12 (T0 = 0 for X ≥ X_b by Lemma A.8's backward stress formula.)
• Refs: p. 163; p. 27, (4.11), (4.13); p. 32, (4.23); p. 140, Lemma A.8; p. 143, (A.56); p. 151, Proposition B.5.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

MC.10: Edge directions and the uniform directional margin \kappa

Node ns-mc-10-edge-directions-and-the-uniform-directional-margin-kappa, kind move, pp. 163-164.

• Statement: Pp. 163 to 164. At Xa, by (B.30), T0 = eaB0 with B0(0, η) = Fps,r ≠ 0, so n extends smoothly and is a positive multiple of (1, ts): nz − tsnθ = 0, nθ + tsnz > 0. At Xb (Proposition A.10), n = (1, 0) and ts = 0. Inside, nθ + tsnz = (Pc − vs)/|ps − (a, −bs)| > 0 and nz − tsnθ = Jc/|ps − (a, −bs)|; 2 − (vs − 2)(nz − tsnθ)²/(nθ + tsnz)² is positive, extends continuously, and equals 2 at both edges. Compactness gives one 0 < κ < 2 with (4.26): nθ + tsnz ≥ κ and (vs − 2)(nz − tsnθ)² ≤ (2 − κ)(nθ + tsnz)².
• Obligation: Theorem 4.6(iii). The cone inequalities are homogeneous in T_0, and T_0\to0 at both edges, so the waves need a uniform margin for the direction on the closed annulus. Section 7.1 uses it to choose one u_* with u_*/\sqrt{1+u_*^2} above the supremum of |c_0(T_{0,*}\cdot K)/(T_{0,*}\cdot N)|, edges included (p. 74). Proposition 7.5 uses it for positivity of H^{-1}T_{0,*} up to the shell edges.
• Mechanism: At both edges the limiting direction lies on the axis of the cone (along the shear), where the directional quadratic expression takes its largest value, 2. So the margin cannot degenerate there. At X_a the stress is switched on by a flat reduction of the reference shear, so its leading part is parallel to the shear itself. At X_b the axial stress vanishes six powers of y_b faster than the angular stress ((A.50)).
• Antecedent: None cited (internal: (B.30), Proposition A.10).
• Cost: The constant \kappa, which fixes u_* in section 7.1.
• Backward question: "The stress vanishes at both edges, but the cone test is homogeneous. Does the unit direction have a limit, and does that limit sit strictly inside the cone, so that the margin is uniform up to the boundary?"
• Checkable: Given the profile, compute on a grid of the closed annulus A_n=n_\theta+t_sn_z and G_n=2-(v_s-2)B_n^2/A_n^2, with B_n=n_z-t_sn_\theta, using the edge factorizations for the limits. Check \min A_n>0 and \min G_n>0; check A_n=\sqrt{1+t_s^2}, B_n=0 at X_a and A_n=1, B_n=0 at X_b. Then \kappa=\min\{1,\min A_n,\min G_n\} (the recipe of section 4.6, p. 44).
• Depends on: MB.8 (the inner factorization (B.30), T0 = e_a B0 with B0(0, η) = F p_s,r ≠ 0, makes n extend to X_a parallel to the shear.); MA.14 (Proposition A.10's rates T_0,θ ~ e^{-4/δ²}δ^{-3}b_θ and T_0,z ~ e^{-4/δ²}δ³b_z give n(X_b) = (1, 0).); MC.9 (interior nonvanishing of T0 makes n = T0/|T0| defined, with n_θ + t_s n_z = (P_c - v_s)/|p_s - s| > 0.); M4.7 (the margin is the cone of (4.23) written for the unit direction, homogeneous in T0, with value 2 on the cone axis.)
• Refs: pp. 163 to 164; pp. 152 to 153, (B.30), (B.31); pp. 142 to 143, (A.48) to (A.50), (A.56); p. 33, (4.26); p. 44; p. 74.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

MC.11: The flat edge weight \zeta and the weighted stress bounds

Node ns-mc-11-the-flat-edge-weight-zeta-and-the-weighted-stress-bounds, kind move, pp. 163-164.

• Statement: Pp. 163 to 164. With ya = log(X/Xa), yb = log(Xb/X), δ = min{1, ya, yb} and ζ = exp(−t1²/ya² − 4/yb²) on (Xa, Xb), extended by zero (C.18), one has |T0| ≥ cζ and |∂^I T0| ≤ CIζδ^{−mI} for every fixed mixed profile derivative, constants independent of q (C.19). This follows from (B.31) on an inner collar, from the factor e^{−4/yb²}yb^{−3}bθ (bθ bounded below) and (A.51) on an outer collar, and from positive minima of ζ and |T0| on the compact middle; the zero extension of T0 is smooth and ζ is flat at both edges.
• Obligation: Theorem 4.6(iv), (4.27). Proposition 7.5 converts it into y_\sigma\ge c\sqrt{S_*}\zeta and |D^Iy_\sigma|\le C_IS_*^{b_I}\zeta\delta^{-a_I} (7.25). So the amplitudes a_\sigma=\sqrt{y_\sigma} obey \sqrt\zeta-weighted bounds and extend smoothly by zero at the shell edges. The coefficient classes of p. 18 carry the weights \zeta and \delta^{-d_{j,I}}.
• Mechanism: The two edges vanish at different flat rates. t_1 is the activation width in \log(X/X_a) from Proposition 4.10, and the 4 is inherited from the terminal smooth-step factor \psi_o=e^{-4/\delta^2}g(\delta) (p. 143). One product weight matches each edge up to smooth positive factors. Each derivative of an exponential factor costs finitely many inverse powers of y_a or y_b, hence \delta^{-m_I}.
• Antecedent: None cited (internal: the flat-integral lemma A.9 behind (B.30) and (A.48) to (A.51)).
• Cost: The weight \zeta and inverse powers \delta^{-m_I} of the capped logarithmic edge distance, which propagate into the weighted classes of sections 6 to 9.
• Backward question: "What single weight vanishes at the same flat rate as the stress at each edge, so that the stress is bounded below by it and each derivative is bounded by it times a finite inverse power of the edge distance?"
• Checkable: None: pure estimate from the edge factorizations (B.31) and (A.48) to (A.51).
• Depends on: MB.8 (the inner factor e^{-t1²/y_a²} and the bounds (B.31) come from the flat activation of width t1 in Proposition B.5.); MA.14 (the outer factor e^{-4/y_b²} and the bounds (A.51) come from Proposition A.10's terminal factorization.); MC.9 (on the compact middle of the annulus ζ and |T0| have positive minima because T0 does not vanish there.)
• Refs: pp. 163 to 164, (C.18), (C.19); p. 153, (B.31); p. 142, (A.48) to (A.51); p. 143 (\psi_o); p. 33, (4.27); p. 82, (7.25); p. 44; p. 18.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

MC.12: Carried-over conclusions: axis regularity, pressure normalization, exterior identities, reserved patches

Node ns-mc-12-carried-over-conclusions-axis-regularity-pressure, kind move, pp. 164-165.

• Statement: Proposition C.3 parts (i), (v), (vi), pp. 164 to 165. (i) The axis profiles of Proposition B.2 are smooth with F > 0, the analytic rectangle [0, Xan] × [−1, 1] of Corollary B.6 is kept, (4.4) to (4.5) give Cartesian smoothness, and (C.17) keeps the pressure normalization (4.25), ΠX = F². (v) The exterior moments (4.28) and heat exterior (4.29) are kept; beyond Xv = Xend, U = M = 0, so V0 = 0 by (4.7). (vi) The reserved Ipos, Imean ⊂ (Xa, Xv), sup Ipos < inf Imean, are unchanged with U = 0, E = cpatch(1 + η²)^{−1}X^{−1/2−λ} (4.30). All constants are independent of q.
• Obligation: It completes Theorem 4.6 for the modified profile. That means: a smooth Cartesian leading field at the axis; the canonical pressure; the exterior identities that make the stress vanish beyond X_b; the exact heat exterior (zero residual for X\ge X_{ext} in Theorem 3.1(iii)); and the two reserved patches used by Lemma 5.2 (positive-order background corrections) and Lemma 8.7 (mean corrections).
• Mechanism: Bookkeeping of supports. Every modification lies in (X_-,X_{rep}), strictly right of X_{an} and strictly left of the heat-compensation patch, the two later reserved patches, and X_v. Exact moment restoration makes all exterior formulas literally identical.
• Antecedent: None cited (internal).
• Cost: None new; it identifies X_v=X_{end}.
• Backward question: "Did any modification touch a region or an integral that an earlier or later stage of the construction depends on?"
• Checkable: None: bookkeeping of supports and exact identities.
• Depends on: MC.8 (exact equality beyond X_rep retains the canonical pressure (4.25), the moment identities (4.28), and the heat exterior (4.29).); MB.10 (Corollary B.6's analytic rectangle [0, X_an], carrying Proposition B.2's axis profile, gives part (i) untouched by the modification.); MA.10 (the heat profile of Lemma A.6 is smooth through η = ±1, which completes smoothness on every [0, R].); MA.4 (I_pos and I_mean are the schedule's third and fourth reserved patches, left with U = 0 and E = c_patch f X^{-1/2-λ} (4.30).)
• Refs: pp. 164 to 165; p. 25, (4.4) to (4.5); p. 26, (4.7); pp. 33 to 34, (4.25), (4.28) to (4.30); pp. 129 to 130, (A.9), (A.10).
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