Independent audit of report.tex comments M2 and M4
Date: 2026-10-09
Scope: Current output/euler-publication/euler-plane-revised.tex and corresponding source-history/receiver/core-cold/selection modules. No manuscript edits.

CONCLUSION
The mathematical requests in M2 and M4 are substantively implemented in the current revision. I did not find a counterexample or an additional essential nonlinear gap in the specific history/localization chain audited. This is a local conclusion about that chain, not a certification of the entire manuscript.

M2: EXPLICIT WHOLE-HISTORY HANDOFF
Current locations:
- Lemma “Prepared-child reference history”, label repair:prepared-child-history, current lines 8008–8274.
- Reset hypothesis repair:prepared-weighted-reset, line 8039 onward.
- Weighted absolute convergence repair:prepared-history-convergence, line 8079 onward.
- Cap-advance estimate repair:prepared-advance-error, line 8132 onward.
- One-sided comparison eq:cr-integrated-error at lines 6671–6678; source-history module lemma lem:cr-actual-reference.
- Controlled-future proposition prop:next-actual-postcap-tail, lines 5974–6107.
- Uniform future tail cor:next-postcap-scale-margin, lines 6109–6146.
- Receiver completes stop first and only then uses the signed transform: eq:ar-history after the prospective interval has been reached. This is separately audited by another reader.

Exact objects already present:
L=log(1/q), δ=L^{-4}, r=C_W/σ, L_q=-s_q=O(log L), t_ref=τ_c-s_q.
B_q(u)=B_4(T_{-φ}w_c(t_ref+u)) for -t_ref≤u≤U_q and 0 otherwise.
The actual history is explicitly truncated at the controlled prospective endpoint, not an assertion about the unprotected infinite future.
The norm is ∫_R e^{-σu/β}|B_q(u)-B_4(Ω_ref(u))|du.
Complex convergence follows directly from |e^{-μ_βu}|=e^{-σu/β}; no cancellation is used for the norm bound.

Three interval check:
1) Preparation u≤s_q:
|B_q(u)|≤C C_past δ e^{σ(u-s_q)}+G_q(u).
The radial heat base has zero fourth complex moment. Inside the fixed preparation ball one uses L1 perturbation, outside one uses actual positive-envelope Gaussian tails.
Integrating the main term gives
  C C_past δ e^{-σs_q/β}/[σ(1-1/β)]
  ≤ C C_past δ^{1-1/β}.
Since β>1 this tends to zero. The canonical negative history is O(e^{σu}) and has the same vanishing tail on (-∞,s_q]. This includes the missing actual interval (-∞,-t_ref). Gaussian errors absorb the q^{-C} discount at the earliest negative endpoint, since t_ref=O(L) and κ_c≤q^{3/2}.

2) Advance s_q≤u≤0:
The exact difference equation keeps the actual foreign drift in divergence-free transport. Its forcing is bounded by C(E+q), using fixed canonical derivative/support bounds and actual older/descendant estimates independent of an active child's derivative.
E'(v)≤C_WE(v)+Cq yields
  sup E+∫E≤Cδ^{-r}(E_q+q),
which is stronger than the conservative bound used in the prepared-child lemma.
Whole moments follow by a fixed cutoff and superalgebraic tails. The displayed sufficient inherited-error condition is
  δ^{-r-1/β}(1+L_q)[C_reset δ^{M+1}+q^χ L^D+q(1+L_q)]→0.
For the proposed strict choice M+1>r+2, the reset part is
  C_reset L^{-4(M+1-r-1/β)}O(log L)→0,
because β>1. The q^χ and q terms also tend to zero after any fixed power of L and log L. Thus the amplification and discount requested in M2 are both explicitly controlled.

3) Controlled future 0≤u≤U_q:
Reference mass and fourth moment have fixed majorants. The own/descendant velocity is uniformly bounded; stopped older radial drift inside the parent core has coefficient a=C(1+C_s)δ/β.
The future moment bound is
  |B_q(u)|≤C(1+u)^4 exp[C(1+C_s)δu/β]+o(q^P).
Its prefactor does not multiply C_s; C_s only enters the weak exponential and later q threshold. Since U_q≤C_UβL and δL=L^{-3}, for any fixed C_s the exponential is at most 2 after a later threshold. Moreover C(1+C_s)δ<σ/2.
Therefore the weighted tail from J is at most
  C_{β,σ}(1+J)^4exp[-σJ/(2β)]+o(q^P).
For each fixed J, actual/canonical X convergence and envelope tightness give whole fourth-moment convergence on [0,J]. Taking q→0 for fixed J, then J→∞, proves the absolute convergence without exchange of unproved limits. U_q→∞ is explicitly required when invoking full prospective-interval convergence. Shorter density stops use only the upper input norm.

Uniformity:
All comparisons use normalized geometric mass/amplitude/velocity sums and fixed clocks. Constants are depth/phase/homotopy independent for the fixed trial class. The q threshold may depend on the finite C_past,C_reset,C_s. No uniformity over unbounded trial constants is claimed. This distinction is correct and sufficient for a one-time invariant-class selection.

M4: TWO-SCALE POSITIVE-ENVELOPE RESET
Current locations:
- Lemma lem:cr-actual-reference, lines 6629–6850.
- Integrated comparison eq:cr-integrated-error at lines 6671–6678.
- Parent-radius Gaussian eq:cr-parent-radius-tail at lines 6695–6700.
- Fixed reference corollary cor:next-source-reference-reset, lines 6861 onward.
- Independent parent-radius lemma lem:next-source-parent-handoff, lines 6930–6983.
- Actual future proposition prop:next-actual-postcap-tail and scale-margin corollary above.
- Receiver eq:ar-packet-moment and the subsequent descendant estimate distinguish aggregate packet moments from the individual child's signed fourth moment.

Objects:
The revision explicitly asserts both envelopes:
  P_child(t)=S_v(t,0)|w_child(0)|,
  P_packet(t)=S_v(t,0)∑_{l≥child}|w_l(0)|,
where the sum is in child coordinates and S_v is the actual common scalar propagator. The packet initially has bounded mass and fixed support even though it need not have a finite summed L∞ norm.
Both solve the same positive scalar equation. Individual colour L1,L∞ contractions bound each signed velocity and the convergent geometric velocity sum. They do not require an L∞ bound for the positive aggregate.

No log-log displacement loss:
Before the cap, own nonradial speed integrates to C C_past δ.
On the cap-to-reference segment,
  ∫_{s_q}^0 ||K*(Ω_ref(u)-W)||∞du≤Ca_ref=O(δ_ref).
This follows from the canonical negative exponential envelope, rather than multiplying a fixed δ_ref speed by the advance length O(log L).
The comparison contribution is ∫E≤Cδ^{-r}(E_q+q)=o(1).
Descendants contribute CβqC_2L=o(1), older nonrotational fields smaller.
Thus the radial displacement is O(C_pastδ)+O(δ_ref)+o(1).
Choosing δ_ref inside the fixed margin first and q afterward yields the fixed-radius envelope reset with constants independent of the fixed trial constants.

Uniform moment tightness:
The current lemma already provides each required higher fixed moment. For any one fixed m>4 this directly implies the requested fourth-moment tightness:
  sup_{j,allowed seeds} ∫_{|y|>R}|y|^4P_j(t_ref,y)dy
    ≤ C_m M_sup R^{4-m}→0 as R→∞,
for either P_child or P_packet.
This precise displayed consequence could be added as a minor editorial improvement. Taking m=6 is sufficient, so no simultaneous choice of infinitely many moment thresholds is required.
Strong L1 convergence is used only after this tail control; it is not treated as implying moment convergence.

Parent-radius reset is independently proved:
Set R_parent=c_R/q. On this ball older harmonic/core-leak estimates yield a_q|y| with a_q≤Cδ/β+o(q^P), while child/descendants contribute bounded additive speed D.
For H≤C_2L:
  a_qH=o(1), DH=O(L)≪R_parent.
The initial packet support is fixed. Exit therefore requires Brownian displacement of order R_parent/√κ, and its probability is ≤C exp[-cR_parent²/(κH)].
Conditional global moment growth after exit costs a fixed power q^{-C} times a fixed logarithm; since κ≤q^{3/2},
  R_parent²/(κC_2L)≥c q^{-7/2}/L,
so it is absorbed in the Gaussian exponent. The stated weighted parent-radius tail follows for every needed finite moment order.
This estimate is not inferred from the fixed-radius tail. It meets exactly the stronger hypothesis needed in the controlled-future proposition.

Additional stress tests:
- Signed zero circulation is irrelevant to the positive-envelope mass; envelope mass is positive and fixed/bounded. No cancellation is assumed in the tail estimates.
- The infinite packet lacks summed L∞, but positive initial truncations and monotone passage suffice; velocity estimates concern individual signed colours.
- Rotation does not change radial moments. The predictable-rotation argument can be justified by smooth approximation or the radial test function; no angular velocity bound enters the constants.
- κ may be much smaller than q^{3/2}; all Gaussian estimates improve and uniformity is preserved.
- The canonical positive history has polynomial fourth-moment growth because its smooth Euler velocity is bounded globally by conserved L1,L∞ norms; exponential discount is therefore integrable.
- The future moment lemma is explicitly conditional on actual stopped parent-core control and is not used to prove that control through a signed nonzero transform.
- At j=1 the older field is empty and only the fixed-radius reset is needed; the parent-radius statement is explicitly restricted to j≥2.

MINOR AUTONOMOUS ADDITIONS RECOMMENDED (NO THEOREM CHANGE)
1) Add the displayed R^{4-m} tightness inequality immediately after the two-envelope reset, naming both P_child and P_packet.
2) In the uniform-induction proof replace its inherited reset o(1) by the explicit quantity
  η_reset(q;C_reset):=
  Cδ^{-r-1/β}(1+|s_q|)
   [C_resetδ^{M+1}+q^χL^D+q(1+|s_q|)],
plus any already recorded intrinsic q-power error. It tends to zero for each fixed C_reset by M+1>r+2.
These additions answer the report more visibly but do not repair a newly identified essential gap; the quantitative arguments already appear in the current lemmas.

Limitations of this audit:
I did not rederive the spectral/finite nonlinear expansion or the global degree construction; their separate correctness remains outside this specific M2/M4 audit. No claim of complete-paper correctness is made here.

FOLLOW-UP CHECK OF NEW eq:cr-reference-tightness
The added sixth-to-fourth moment inequality is valid for either the child or source/deep envelope, uniformly over the fixed trial class and q below the finite sixth-moment threshold. The pointwise inequality |y|^4 1_{|y|>ρ}≤ρ^{-2}|y|^6 proves it. The common normalized mass supremum is finite: if C_seed bounds ||W||1+α||Φθ||1, then packet mass is at most C_seed/(1−βq²)≤2C_seed under the already imposed geometric smallness. Positive initial truncations justify it without summed L∞; p>2 and the existing NS setting already supply all assumptions used elsewhere.
One minor notation issue: the preceding parent-radius assertion already assigns R=c_R/q. The independent tail radius in the new tightness statement should therefore be renamed ρ (and ρ→∞), rather than reusing R. Explicit indices for M_sup can be copied from the displayed supremum, although the intended scope is already clear. The supremum does not range over unbounded trial constants; their q thresholds correctly remain class-dependent.

