RECEIVER AUDIT OF REPORT M3 (9 October 2026)

Scope
Checked the current publication source, receiver module in full through the density closure and first-transform definition, finite residual/comparison proof, the prepared-child history proof, cap-to-reference inputs, inactive closure and packet force. This is a local audit; the spectral theorem, cold-source construction and global phase selection require their separate audits.

Verdict on M3
The current revision contains the substantive quantitative continuation required by M3. No circular use of the actual signed growing-coefficient lower bound was found in the receiver density closure. The requested standalone lemma is absent: the continuation is a carefully ordered subsection of the receiving-stage proof. This is an organizational request that can be satisfied by extracting the verified argument, not currently a demonstrated mathematical obstruction.

Current locations
Publication source lines 8367-8397: T_s and the deterministic prospective H_+; actual core hypotheses are imposed on density, not the protected coordinate.
Lines 8399-8423: shorter-stop upper source-history bounds and independence of the trial C_s.
Lines 8425-8499: all retained field coefficients and arbitrary-order collar mismatch.
Lines 8501-8549: fixed weighted derivative input ball and finite density comparison.
Lines 8551-8565: explicit strict improvement and completion of H_+.
Lines 8567 onward: only afterwards, entire controlled source history and full actual C_q.

Verification details
1. Completion of the child's fixed reference time does not assume the receiving coordinate cap. The unconditional inactive closure is applied with the child c as current colour, so the receiver i<c is an older colour controlled to <delta/4 on that interval. The child's cap-to-reference lemma uses current-child inactive closure and individual descendant contractions; no active-child small-density continuation is assumed on that advance. The strict C_0<C_2 clock slack allows its O(log L) advance.

2. Beyond the child reference time and up to min(H_+,T_s), older colours i'<i are controlled by unconditional inactive closure with i as current. The receiver i itself has ||(I-P_rad)w_i||_1<=2 C_s delta under the provisional density stop. The constant-core theorem then applies because C_2 L sqrt(C_s delta)->0 and C_2 L beta q->0 for every fixed C_s. Its hypotheses involve only actual density and radial displacement; no coordinate cap is used. Thus the child radial drift after its reference state has local coefficient C(1+C_s)delta/beta and unrestricted origin growth C|y|/beta through the whole stopped interval, including any portion beyond a hypothetical coordinate cap.

3. The source future upper bound has a fixed prefactor:
|B_c(u)| <= C(1+u)^4 exp[C(1+C_s)delta u/beta] + G_q,
0<=u<=C_2 beta L.
For fixed C_s, choose q so C(1+C_s)C_2 delta L<=log 2 and C(1+C_s)delta<sigma/2. Hence the discounted absolute input is bounded by 2C integral_0^infty e^(-sigma u/beta)(1+u)^4 du plus Gaussian errors, independently of C_s. Negative-history input is a fixed canonical majorant plus C C_past delta^(1-1/beta) and the vanishing reset term. Fix trial constants, decrease q to make those latter terms <=1. This yields one input ball before C_s is chosen, with only its later q threshold depending on the chosen trials.

4. Field input is measured in spatial C^R, not in time derivatives. Differentiating a retained stream harmonic changes its fixed spatial basis only. The moment coefficients can be merely measurable in time. In receiver units b_J=q^6 B and D=e^(-sigma(t-t_0))||B||_(C^R); the weighted absolute bound is exactly integral D<=B_0. This is the sole forcing bound used by the finite recursion for density closure.

5. After recentering, the own seed ratio is q^(K-6-epsilon_0) L^D ->0. The higher inner harmonics have weighted ratio at most q^(4-epsilon_0)L^D ->0. Actual older harmonic ratios have positive powers K-4-e_B-epsilon_0 and 2-e_P-epsilon_0, since e_B<K-6, e_P<1 and epsilon_0<1/2. Descendant fourth moments contribute q^(2-epsilon_0)L^D. Every exponent is uniform in depth. The polynomial q^-epsilon_0 loss comes from the finite negative source endpoint; the strict clock slack absorbs the O(log L) correction in sigma t_0.

6. The finite recursion is triangular in j+ell with derivative reserve r_(j,ell)=k_*+2(M-j)+2(N-ell), R=k_*+2M+2N+3. Diffusion consumes two derivatives from ell-1; transport/quadratic terms consume one from smaller j; heat interactions use smaller ell. Each finite coefficient is supported on the fixed derivative collars and has zero integral. Its sector set is finite. Nothing in this argument invokes a uniform all-angular-mode spectral estimate.

7. The residual formula includes every positive viscosity corrector and the exact radial heat base. Kato comparison with the ACTUAL divergence-free foreign field puts that field into the transport of z=w-w_app; it does not differentiate it. The remaining mismatch is controlled on a fixed broad collar by m_q and against the exact radial heat gradient off that collar by a Gaussian and the global linear radial growth bound. The fixed exponent C_1=C_p(||grad W||_X+1) is chosen before the orders; (M+1)sigma>C_1 absorbs the pure nonlinear residual. N and P can then satisfy
3(N+1)>C_2(C_1-sigma)_+ + chi,
P>6+epsilon_0+C_2(C_1-sigma)_+ + chi.
Thus ||w-w_app||_X<=C a^(M+1)+q^chi L^D a, and ||w-W_kappa||_X<=C_a a, with C_a dependent on the fixed finite input ball and orders, not the trial C_s. No nonzero C_q is used in the comparison lemma or its application.

8. The strict improvement is explicitly correct. Let E_0=exp(sigma L_0)/|C_*|. Then a(H_+)=E_0 delta. Choose C_s>max(1,2 C_a E_0), AFTER computing C_a in the fixed input ball, and then choose the common q threshold for this fixed trial. On [0,min(H_+,T_s)], a(t)<=a(H_+), so
||w_i-W_kappa_i||_X <= C_a E_0 delta < C_s delta/2.
Continuity at a hypothetical T_s<=H_+ contradicts its defining equality. Hence T_s>H_+. The small-amplitude condition a(H_+)<=a_* also follows from delta=L^-4 and fixed E_0. Using the fixed canonical number |C_*| to SET H_+ is harmless; no actual-transform lower bound is assumed.

9. Only after that contradiction is proved does the proof zero-extend the controlled moment history to [-beta t_0,beta(H_+-t_0)], invoke weighted absolute convergence, and define the integral C_q through H_+. Its actual nonzero limit is subsequently used for the first coordinate cap and intrinsic state reset. This order is correct.

Suggested precise standalone addition (does not change scope)

Lemma (Upper continuation to the prospective endpoint).
Fix the profile, sigma, C_1, finite M,N,J and input ball A_0,B_0. Let delta=L^-4, epsilon=q^6, a(t)=epsilon exp(sigma(t-t_0)), 0<=sigma t_0<=epsilon_0 L, H_+<=C_2 L, kappa<=q^3, and a(H_+)=E_0 delta with fixed E_0>0. Assume the strict finite-order inequalities above and fixed global radial-growth constant. For a trial C_s>=1 define T_s by ||w-W_kappa||_X=C_s delta. Suppose that on every H<=min(T_s,H_+) the actual retained field and mismatch satisfy the finite comparison hypotheses with the SAME A_0,B_0 and m_q<=q^P L^D. These hypotheses may require a later q threshold depending on fixed C_s, but the input constants do not. Then there is C_a depending only on the fixed profile, orders and input constants such that if C_s>2 C_a E_0, subsequently small q gives T_s>H_+ and ||w-W_kappa||_X<=C_a a(t) on [0,H_+].

Proof.
For q sufficiently small, a(H_+)<=a_* and the finite viscosity smallness conditions hold. The finite comparison lemma on every stopped horizon gives the bound with this fixed C_a. Its bound is <C_s delta/2 throughout the stopped interval, contradicting the equality at any T_s<=H_+. The comparison has no signed-coefficient hypothesis. End proof.

The receiving-stage proof verifies the hypotheses of this extracted lemma using its child-history lemma, unconditional inactive closure, actual constant-core theorem, and multipole/positive-envelope estimates. Listing those inputs next to the new lemma would satisfy the report's organizational request. This addition should not be represented as a new estimate or as an independent verification of the whole manuscript.

FOLLOW-UP AUDIT: newly extracted cor:ffr-upper-continuation

The proof is valid under all hypotheses of the finite comparison lemma. C_a is independent of C_s because profile, finite orders, A_0/B_0 and radial-growth constant are fixed; q thresholds may depend on the fixed trial, as stated. The endpoint relation makes a(H_+)->0, while kappa<=q^3 and H_+<=C_2 L imply every fixed finite viscosity-smallness condition. Strict error exponents match the comparison lemma exactly. Initial strict inequality is enough to ensure the first stop is a genuine crossing from below, but is NOT a replacement for the weighted initial-seed assumption.

One ambiguity in the standalone statement should be corrected before publication: it currently says only that 'the actual retained field and collar mismatch satisfy the finite comparison hypotheses.' For a literal standalone reading this need not impose w(0)=W+alpha Phi and (alpha/q^6)e^(sigma t_0)||Phi||_(H^r)<=A_0. Those initial data are essential to the conclusion at t=0. An O(delta) initial perturbation can obey ||w(0)-W||<C_s delta while exceeding C_a a(0) by an arbitrarily large ratio. Zero foreign field does not cure that. Add explicitly that the corollary works in the full setting of Lemma ffr-comparison, including its colour equation, initial seed, regularity and support assumptions, and eq ffr-input with the same A_0,B_0 on every stopped interval. This is a minor statement-clarification repair; the actual receiving-stage application already verifies the missing weighted seed inequality.
