Independent upstream core / cold-tail / inactive-density audit
2026-10-09

Scope and method
----------------
I read all of core-cold.tex (1119 lines) and inactive.tex (745 lines),
checked the displayed estimates independently, and traced the selected-
cutoff cold-tail application in source-history.tex, lines 450--602. I also
checked the weighted off-diagonal Nash lemma in source-history.tex,
lines 319--380, because it is used to turn the cold aggregate's localization
into a collar velocity error. I did not verify the full spectral profile,
all coefficient recursions, the complete history reset, or the final phase
diagonalization in this audit. Consequently this is not a correctness
certificate for the main theorem.

Snapshots checked:
core-cold.tex SHA256
882116309abad4453bf3affc82107997b6d8b59e6bd23867a9eac4481cf16b66
inactive.tex SHA256
c5f7d1e3c24cb56ca1fc6faef24396da634214cdc242b01b7f5da9a1d4702f4e

Findings
--------
No central mathematical obstruction was identified in these upstream
arguments under their explicitly stated hypotheses. In particular:

1. Actual constant cores (core-cold.tex, lem:cont-C4-origin,
   lem:cont-radial-exit, prop:cont-outer-core).

   The fourfold far-field expansion contains only powers 3,7,11,...;
   its integral bound is C ||f||_infinity |z|. The near integral has
   the same order. This uses individual colour L-infinity bounds,
   never an L-infinity bound on the entire stack.

   The cosh radial-exit estimate has generator bound
   Delta cosh(lambda r) <= 2 lambda^2 cosh(lambda r); optimizing gives
   denominator 8 nu T in the exponent. The core bootstrap drift budget
   is a0 R_l / 8. Thus starting at radius a0 R_l / 2 leaves distance
   3 a0 R_l / 8 before exit, yielding c_* = 9/512 exactly.

   Older constant cores give tangential solid rotation. Older exterior
   nonradial densities start at cubic order, giving the geometric factor
   q^2/(beta-q^2). Unrestricted younger velocities are controlled by
   sum A_k R_k, which converges. The maximum over l <= i is finite and
   introduces no depth-dependent prefactor. The result is conditional
   on the actual nonradial density hypothesis; it does not prove that
   hypothesis. This distinction is respected in the manuscript.

2. Gaussian-energy signed cold moment (prop:resume-gaussian).

   With beta_G = 3/10, conjugation g = exp(beta_G |y|^2/2) w gives
   <w,Lw> = (1/2)||g||_2^2 - ||grad g||_2^2
              - (3/50)||y g||_2^2,
   D = ||grad g||_2^2 + (9/100)||y g||_2^2 + (3/5)||g||_2^2.
   The oscillator lower bound is sqrt(6)/5, and D <= 3 H_osc.
   Splitting H_osc as 3/50 plus 47/50 proves the stated
   -D/50 + F^2/25 estimate: the numerical remainder is less than 1/25.

   The bounded rational Gaussian weights avoid assuming membership in
   the full Gaussian space. Their Hessian terms indeed cancel after
   conjugation; the remaining confining term is negative uniformly.
   Young's inequality absorbs both the bounded self velocity and the
   weak linear foreign drift. Thus the preliminary finite-window entry
   argument does not invoke an unjustified oscillator gap for truncated
   weights.

   Fourfold nonradial h has all ordinary polynomial moments through
   degree three equal to zero. In the natural Gaussian weight this
   excludes Hermite degrees 0--3, giving <h,Lh> <= -2||h||^2 and
   D_h >= 3||h||^2. The cubic source EG is controlled by the strict
   1/20 Gaussian margin. The stated decay rate 19/10 for ||h|| follows
   after choosing the displayed small constants. Combining it with
   F <= C m exp(tau/25) gives physical-moment exponent
   2 + 1/25 - 19/10 = 7/50. The largest foreign contribution has
   exponent 2 + 1/25 + 51/25 = 102/25. These exponents are correct.

3. Local harmonic field / rough leak comparison
   (prop:resume-local-transfer).

   The cutoff is applied to a streamfunction, so it preserves divergence
   freedom and fourfold covariance and needs no derivative of the leak.
   The claimed derivative-of-Biot--Savart heat kernels have the correct
   scaling: nonlinear differentiated kernel norms are respectively
   t^(-3/2), t^(-1/2), and t^(-3/4) in L-infinity, L1, and L^(4/3).
   The late-time heat-mild integral is integrable. The early/late splits
   yield ||grad u_g||_infinity <= C (M/nu)/t.

   The H-minus-one comparison cancels the v.advection terms because
   v = J grad psi and v dot grad psi = 0. The common initial trace in
   velocity L2 is independently established using zero circulation,
   uniform first moments, and convergence of the vorticity in L2.
   The actual coefficient is integrable at time zero by p > 2 and the
   t^(-1/p), t^(-1/2-1/p) kernel bounds. Hence the subsequent singular
   (t/s)^(CM/nu) estimate does not discard an unknown initial term.
   With CM/nu < 1/4 its source integral is finite. The cutoff fourth
   moment test has gradient L2 norm O(R_*^4), as claimed.

4. Physical cold-tail powers and scope.

   At the selected cutoff, M = O(beta A_i q^2 R_i^2), r = q R_i,
   kappa = q^(3/2), S = O(q^(-1/2) H). The two physical powers are
   2+4+1/2-7/100 = 643/100 and
   2+4+5/2-51/25 = 323/50. Both exceed 6. The smallness budget has
   m S^(1/25) = O(beta q^(12/25) H^(1/25)). The leak terms are
   superalgebraic because their Gaussian exponent defeats exp(C H)
   for H = O(log(1/q)).

   The selected-source application does not incorrectly impose a fixed
   small density for O(log(1/q)) in the source-core bootstrap. Instead
   its provisional growing envelope has integral O(delta_*), deriving
   the source core directly, after which the cold estimate needs only
   C_3 R_*^2 H = O(delta_* q^(3/4) log(1/q)). That tends to zero.
   The manuscript explicitly restricts this erasure budget to the
   selected cutoff; it is not used to erase the intended active child
   at larger receiving levels.

5. Inactive rows and actual packet closure (inactive.tex).

   The self error satisfies the scaled L1/Lp Kato estimate with a fixed
   profile rate gamma A_i, without differentiating the error colour.
   The older field row q^(2(i-l)) follows from the cubic interior
   inverse-moment expansion. The inner row q^(6(l-i)) follows from
   the fourth exterior moment divided by R_i^6. The remaining density
   leak uses the receiver radius R_i in its Gaussian exponent rather
   than the small source radius; this distinction is essential and is
   explicitly present.

   The four error-row sums have the stated beta and q weights. Dividing
   them by (D q^6)^k gives the displayed bounded geometric sums, with
   eta(q) superalgebraically small. The positive comparison weights
   give inner ratio beta/D and outer ratio D q^8/beta exactly.

   The positive packet aggregate is used in L1/Lp and moments only.
   Its actual velocity is controlled by the separately bounded signed
   colours. For older colours the origin-growth sum is O(A_j |x|/beta),
   whereas the unrestricted inner packet has bounded velocity
   O(A_j R_j). The moment calculation yields
   A_j R_j^(n+2) (1+theta+sqrt(kappa_j theta))^n
   exp(C_n theta/beta), uniformly in depth.

   At an older collar the fourth packet moment contributes
   velocity/(A_j R_i) = (R_j/R_i)^6 = q^(6k). The tail split with
   Lp gives the other term q^k exp(-c_p q^(-2k)/(kappa_j S)).
   Thus the actual-packet force, not a presumed signed active impulse,
   closes the older inactive density stop. The choices
   C_2 lambda_B < K-6 and C_2(lambda_P+C_4/beta) < 1 make the
   field powers strictly smaller than the main q^6 scale, as stated.

6. Weighted off-diagonal Nash lemma (source-history.tex).

   Conjugation by exp(ell phi) yields drift u+2 kappa ell grad phi
   and potential ell u.grad phi + kappa ell^2 |grad phi|^2
   - kappa ell Delta phi. Integration gives the L1 coefficient with
   + kappa ell Delta phi; in L2 that term cancels against half the
   drift divergence. The resulting growth bounds, their adjoints,
   and the midpoint Nash composition produce exactly the claimed
   L1-to-L-infinity prefactor C/[kappa(t-s)] and exponential loss.
   Optimizing ell proves the collar Gaussian density bound without
   a summed L-infinity hypothesis or tangential-velocity derivative.

Limitations / wording
---------------------
No theorem alteration is justified by this audit. No counterexample was
found to the checked upstream statements under their hypotheses. All
uniformities above are local conditional estimates: using them in the
global fixed-data construction still depends on the downstream preparation,
history convergence, finite expansion, and invariant reset arguments.
The fact that these upstream estimates survive scrutiny cannot by itself
justify calling the whole manuscript's main proof complete.
