Independent audit of report M1 and M5 against the current manuscript
Date: 9 October 2026
Scope: selection.tex, receiver.tex, caps.tex, and the selected-source base
argument in source-history.tex. This is a local verification of the
uniform closure and protected-cap implications, not certification of
every spectral, core, cold-tail or compactness lemma in the paper.

SUMMARY

M5 is substantively addressed. The protected actual-cap corollary now
contains a quantitative early margin, late lower slope and upper-speed
bound; the final selection proof defines a continuity radius at each
sparse step and controls the entire freely variable remaining tail.
I checked those implications. They do not contain a remaining logical
gap conditional on the stated actual-stage bounds.

M1 is substantially but not completely addressed in presentation. The
current selection proof still displays exactly the criticized expression
M_input <= M_ref + C C_past delta^(1-1/beta) + o(1), without displaying
one explicit trial-dependent error. However, the newly added prepared-
child history lemma proves enough estimates to replace this o(1) by an
explicit expression. This is a repairable quantitative consolidation,
not a new transmission estimate. The output reset exponent is genuinely
independent of the incoming reset exponent in the stated finite-memory
argument: incoming error is used to enter a common forcing ball and to
obtain a nonzero coefficient, not as an additive error in the output
polynomial. I independently checked this dependency rather than merely
relying on the assertion in the final induction.

M1: EXACT CURRENT LOCATIONS AND DEPENDENCIES

* Current assembled source lines 8703 ff.: prop:next-uniform-induction.
  Module selection.tex lines 9-214.
* Current lines 8823-8831: the criticized o(1) is still present.
* repair:prepared-child-history, current lines 8008 ff., now specifies
  fixed trial constants, later q thresholds, depth/phase/homotopy
  uniformity, zero extension at the controlled endpoint, and a weighted
  absolute norm. Its eq:repair:prepared-advance-error bounds the
  cap-to-reference advance. The preparation ball and all fixed moments
  become independent of C_past after its integrated drift is smaller
  than the fixed radial margin. This is valid for each fixed trial
  C_past, not uniformly over unbounded trial constants.
* The future moment prefactor is independent of the density-stop trial
  C_s. The factor exp[C(1+C_s) delta u/beta] is at most two after q is
  chosen for that fixed C_s, since u <= C_U beta log(1/q) and
  delta log(1/q) = log(1/q)^(-3). The uncontrolled actual future is
  not used: it is zero extended at the controlled endpoint.
* eq:ar-weighted-input and lem:ffr-comparison establish a density
  envelope using only an upper input ball. C_a is computed before
  choosing C_s. The strict choice
      C_s > max{1, 2 C_a exp(sigma L_0)/|C_*|}
  gives ||w_i-W_kappa||_X < C_s delta/2 through min(H_+,T_s), and
  excludes a density stop. Only then is the actual signed coefficient
  C_q evaluated on the whole prospective interval. This order is valid.
* lem:ffr-memory, current lines 7734-7786, normalizes each pure
  coefficient and uses the shifted generator L_rot-j sigma. The j=1
  error consists of (i) a decaying complementary component; (ii) the
  missing future main input tail; and (iii) a small weighted remainder.
  For j>=2 every difference has one lower-order error and the reserved
  derivatives close its product estimate. The external main term is
  exponentially damped after normalization. Hence
      ||h_j(s)-F_j(C_q)|| <= C_j (1+s)^d_j exp(-d s)+C_j eta_q
  for fixed input ball and future envelope. No incoming reference-state
  error appears on this right side. Its effect on C_q is removed by
  exact coordinate matching (lem:ffr-gauge); matching only the linear
  amplitude would not suffice.
* At s=S_q+O(1), S_q=(6/sigma) log(1/q)+O(log log(1/q)), so the
  memory transient is q^(6d/sigma) times a fixed logarithmic power.
  All other remainders used for reset have their own positive q powers.
  Therefore the output intrinsic exponent chi' is independent of the
  incoming chi. One can take a common chi strictly below both the base
  and receiver intrinsic exponents. This makes coefficient constants
  absorbable by a later common q threshold; it is not successive
  degradation of chi with depth.

AN EXPLICIT QUANTITATIVE REPAIR FOR THE REMAINING M1 DISPLAY

Let r=C_W/sigma and ell_q=-s_q=O(log log(1/q)). The weighted advance
calculation in the prepared-child lemma gives

 E_adv(q;C_reset) :=
 C delta^(-r-1/beta)(1+ell_q)
   [C_reset delta^(M+1)+q^chi L^D+q(1+ell_q)].

This is an explicit upper bound on the noncanonical contribution on
[s_q,0], apart from Gaussian tails smaller than any fixed q power.
The negative preparation history adds
  C C_past delta^(1-1/beta).
The fixed canonical history and the future polynomial majorant supply
one constant M_ref independent of C_past, C_reset, C_s and depth.
Every remaining retained field, own seed and omitted leak has already
been bounded by a positive q power times a fixed logarithmic power.
Call their finite sum C q^eta L^d_in, choosing eta>0 below their
strict positive exponents. Then, for each fixed trial triple after
its later q threshold, one may replace lines 8823-8831 by

 M_input <= M_ref + C C_past delta^(1-1/beta)
                       + E_adv(q;C_reset) + C q^eta L^d_in.

For transparency, M_input should be explicitly defined as the maximum
over the finitely many required C^k weighted input norms (or just the
largest required order R). Higher stream differentiation acts only on
the fixed smooth spatial bases, so these norms use the same history
bound. Constants can be chosen once for that finite list.

The convergence is immediate and does not use an unspecified diagonal
in the stage depth: M+1>r+2 and beta>1 give
M+1-r-1/beta>1, so the first term of E_adv tends to zero even after
the factor 1+ell_q. All its q terms tend to zero against their finite
logarithmic factors. The C_past term tends to zero because 1-1/beta>0.

The invariant lemma can therefore be made explicit as follows. Compute
the finite output constants P_0,R_0 and intrinsic chi_0>0 in the ball
M_input<=M_ref+1. P_0 includes the receiver cap-relative envelope
constant and the selected-source envelope constant; R_0 includes both
finite reset constants. Choose C_past>2P_0, C_reset>2R_0 and
0<chi<chi_0. Choose the logarithmic exponent D large enough for both
base and receiver output errors. For these fixed trial constants choose
one q so that the preceding input errors are below one, the global
geometric/core/localization and strict clock inequalities hold, and the
constant times q^chi_0 errors are bounded by q^chi L^D. The stage maps
the class with constants (C_past,C_reset,chi,D) into itself, with the
same fixed-radius localization constants. Every cited estimate is
uniform in depth and phases, so the minimum of this finite set of q
thresholds is common to all N. A choice-order table would make this
more readily checkable; this is presently narrated rather than tabulated.

Important qualification: this repair consolidates existing estimates.
It cannot independently cure a failure in the spectral or actual core/
moment lemmas feeding the input ball. Those claims require their own
audits. I found no counterexample to the uniform-closure implication
itself.

FOLLOW-UP VERIFICATION OF THE IMPLEMENTED QUANTITATIVE CONSOLIDATION

The parent subsequently implemented the explicit input norm, E_adv,
E_in, eta_in, constant-order table and physical cap constants in module
selection.tex lines 115-300. I checked the new formulas independently.
The input norm is exactly the scaled seed norm and weighted C^R field
norm required by the finite receiver recursion. The five conservative
receiving powers
  2-epsilon_0, 4-epsilon_0, K-4-e_B-epsilon_0,
  2-e_P-epsilon_0, K-6-epsilon_0
are correct after division by q^6 and recentering of the discount.
They are positive under e_B<K-6, e_P<1, epsilon_0<1/2 and
epsilon_0<K-6. The half-minimum leaves slack for logarithmic losses.
E_adv has the correct delta comparison and discount powers. The trial
constants-before-q ordering and absorption of output intrinsic q errors
into one common smaller chi close the announced invariant class.

One concrete cross-reference typo was found and reported to the parent:
selection.tex line 166 must use {repair:prepared-advance-error}, without
an additional leading eq:. No mathematical error was found in the new
consolidation within this audit's scope. Suggested exposition details:
M_ref also depends on the already fixed ratios/clocks; the common
normalized all-time Lipschitz constant across j is justified by the
older fourfold origin-growth bound and bounded |y| |grad T_adj(y)|,
rather than rescaling the physical outer global-velocity norm; cite
thm:next-selected-source for the selected-source cap counterpart.

M5: QUANTITATIVE CAP PROTECTION IS ALREADY PRESENT

Current cor:ffr-cap, assembled lines 7896-7966, module receiver.tex
lines 599-669, establishes:
  d_cap := min{L_0/4,1/(8 C_speed)};
  d/dt |A(w)| >= sigma delta/2 a.e. on
        [t_q-d_cap,t_q+d_cap];
  sup_{0<=t<=t_q-d}|A(w(t))|
       <= delta-min{delta/4,sigma delta d/2}
        for each fixed d in (0,d_cap].

I checked the argument for these facts. The all-time upper envelope
excludes a cap before the late window. The exact-coordinate expansion
makes the window endpoints straddle the cap and puts its crossing in
the middle half. The upper-speed estimate prevents escape from the
[3delta/4,5delta/4] band during the common short interval. Within that
band the modulus derivative is sigma|A| plus an error o(delta), hence
the lower slope. An earlier excursion followed by return would require
a downward crossing of that band, impossible under the integrated
lower slope. The displayed precap margin follows. The selected-source
base has the analogous explicit linear-coordinate estimate and
instantaneous identity (source-history.tex lines 738-771); its common
constants can be decreased to the same ones. This remains uniform for
all allowed nonselected amplitude reductions because all error bounds
were uniform for those reductions.

To make the dependence on physical scale explicit, choose any common
normalized e in (0,d_cap], set
  h := min{delta/4,sigma delta e/2}, c := sigma delta/2.
If a_j(t_phys)=A(w_j(A_j t_phys)) denotes the normalized colour's
observable in its stipulated rotating frame, then
  e_j=e/A_j, h_j=h, c_j=A_j c.
Its fixed-scale global Lipschitz bound is A_j times the normalized one.
At the outer scale A_1=1 these are common physical constants independent
of N. If using an unnormalized physical colour/test, its observable also
has the amplitude-mass scaling A_j R_j^2 and the cap level must be
scaled accordingly; the paper uses the normalized version.

caps.tex, ph:first-cap, is an elementary correct stability lemma:
if ||a-b||_infty=epsilon<min(h/2,ce/4), no hit is possible before
tau(a)-2epsilon/c, while a hit has occurred by tau(a)+2epsilon/c.
The phase error is <=delta^(-1)(1+2 L_obs/c)epsilon. This bound requires
protection for a but no differentiability of b. Protected uniform
limits retain a unique first hit in their local windows.

The final selection proof now defines the positive continuity radius
r_k in eq:reader-continuity-radius (assembled lines 9020-9034) and
requires Delta_tail(N_{k+1})<min_{i<=k}r_i (lines 9035-9040).
The half-supremum definition is legitimate because the admissible-
radius property is monotone and contains a nonempty initial interval.
Every later change is in the whole phase tail beginning at N_{k+1},
so the final datum lies inside every earlier radius r_i without needing
a viscosity-uniform continuity radius. This addresses exactly the
distinction raised by the friend report. The cap functional is the same
outer colour, test and cap level throughout. The unchanged outer initial
colour is essential and is correctly fixed here.

RELIABILITY VERDICT WITHIN THIS AUDIT'S SCOPE

No demonstrated theorem-invalidating error was found in the M1/M5
closure or protected-phase arguments after the revision. M5 is resolved;
M1 merits the displayed quantitative consolidation above and an explicit
constant-choice table. It would be unjustified to infer from these local
checks that the entire paper is solid, particularly if an independent
audit finds a failure in an upstream spectral/core/history hypothesis.
