Independent parent review of M06-v1 — 2026-10-21

Decision: supported internally; complete with respect to the exact original
M06 task. No blocking proof gap was found. The final candidate gives
  (1/2)||Phi^N(tau_1)-tau_2||_1 <= 500001 exp[-N/(2*10^12)]
for every integer N>=0, with explicit infinite-space and resource accounting.
The constants are deliberately conservative, not an estimate of the practical
cooling rate. This is a separate-agent review within the same model family,
not an external referee decision or an exhaustive novelty assessment.

Pinned version and reproduction
Candidate: campaign/continuation-2026-10-21/candidates/M06-v1.
Manifest SHA256:
7a4d481598604b774e1a1a1e7e4fde1851a227a352f05ffe4319ef40334acb63.
Proof SHA256:
6041825a345b020a2dfa9cb395b6774071958f97e2cac9232de9c909d295d555.
inputs.json records every payload hash. The original.tex hash agrees with
the unchanged campaign/originals/M06.tex. The candidate was copied into
replay-input/ and its self-contained checker replayed successfully there.
replay.json records the result. A separate 200-bit Arb/exact-fraction check
of the proof constants also passes; see check_bounds.py and bounds-check.json.
The frozen candidate has not been modified.

1. Exact quantum-to-population interface
The stated Jacobi matrix has the correct squared edge coupling
(r+1)(k-2r)(k-2r-1) in sector k=2n_S+n_B. Its direct sum defines the exact
unitary model in the original. On a diagonal input, different target output
occupations have different bath occupations within each sector, so tracing
the bath removes the corresponding off-diagonal elements. The original
initial tau_1 therefore remains diagonal under every collision, and population
total variation is exactly the full-state trace distance requested.

The product tau_2 tensor tau_1 has constant weight within each sector. It is
fixed by every collision time. Real symmetry of the sector Hamiltonian makes
the unitary matrix symmetric, giving detailed balance after summing the
thermal bath weights. These properties also hold before averaging over time.
All sums use nonnegative probabilities or convergent trace-class direct sums.
No finite occupation cutoff is used to define the evolving channel.

2. Uniform escape probability for every nonzero level
For input |r,0>, the first two squared Jacobi couplings are 2r and 12(r-1).
The endpoint spectral measure consequently has second moment 2r and fourth
moment 28r^2-24r. For two independent copies, their difference has second
moment 4r and fourth moment 80r^2-48r. Paley-Zygmund then gives probability
at least 1/20 of a frequency difference of magnitude at least sqrt(2r).

The scalar time average has the exact sinc representation used in the proof.
The inequalities on the two frequency ranges establish 1-h>=1/10000 for
|omega|>=sqrt(2), and 1-h>=0 everywhere. Hence escape from the bath-vacuum
endpoint has probability at least 1/200000 for every integer r>=1. Every
escaped state in that sector has smaller target occupation. Multiplication
by the bath-vacuum weight gives the uniform downward transition bound. The
finite walk controls check this moment interface; uniformity in r is supplied
by the analytic identities, not by the number of tested sectors.

3. Infinite-space positive and negative spectral bounds
The key conductance step is valid because the stationary atom pi_0 exceeds
1/2. A set of mass at most 1/2 cannot contain zero. Its least element m is
at least one and has mass at least (1-q^2) times the entire set's mass. The
uniform downward transition from m exits the set, yielding the claimed
conductance floor. The supplied coarea/Cauchy-Schwarz proof gives the
Poincare inequality on all L2(pi), including infinite sets, by nonnegative
integration and density. Splitting f-f(0) has the required favorable sign
for the Dirichlet cross term.

Separately, h>=-99/100 bounds each sector's quadratic form below by the
same multiple of its diagonal Hilbert-Schmidt norm. The weighted sector sum
is exactly the L2(pi) norm. Absolute convergence and density justify the
infinite sum. Combining this floor with the positive gap gives centered
operator norm at most 1-10^-12. The independent Arb conductance evaluation
gives a Cheeger gap floor greater than 3.7342e-12, safely above the value used.

4. The non-L2 initial state and the exact tail estimate
The untruncated initial density tau_1/pi is not in L2(pi). The proof explicitly
handles this issue. Its subprobability truncation at level K has omitted
mass q^(K+1) and squared L2 norm ((1-q)/(1+q))(K+1). Centering at that
subprobability's own mass, applying contraction, and bounding the omitted
mass gives the stated total-variation inequality. Taking K=N and absorbing
sqrt(N+1) into half the exponential rate yields the displayed C and c for
every N>=0. The tail estimate approximates only the initial distribution;
all subsequent transitions still use the full infinite kernel.

5. Energy and both work-accounting conventions
The initial density is in L^(3/2)(pi). Its norm cubed simplifies to
(1+sqrt(q))^2/(1+q)<=2. Markov contractivity and Holder's inequality give
the uniform second-moment bound below 30. Together with the trace-distance
estimate this implies the stated exponential energy convergence, with
prefactor below 6000. The numerical safety margins in these steps passed
the separate exact-fraction/Arb check.

Sector conservation gives signed discarded-bath heat
Q_j=2(E_(j-1)-E_j) and net bare-energy work E_(j-1)-E_j. The proof correctly
does not assume that target energy is monotone. With the positive part taken
after averaging each collision, the energy-error series bounds gross work
by 9.6e16, below the advertised 1e17. If the positive part is instead charged
conditional on every random time and its history, the same Lp and second-
moment bounds hold for every history because each fixed-time channel fixes
pi. Every branch charge is then below sqrt(30)<6, giving expected conditional
gross work at most 6N. The constant 1e17 bound is expressly limited to the
averaged convention. Both interpretations are therefore covered without
silently relaxing the original resource condition.

The signed heat and net-work telescoping limits follow from the energy
convergence. The resource statement retains N fresh bath modes, interaction
time at most N/50, coefficient one, and an unbounded interaction operator.
No additional controls, retained baths, or measurements have been added.

6. Primary-source comparison and remaining limits
The parent inspected the archived arXiv:2512.10703v2 text. Appendix B.2,
equations 66–70, uses a short-time expansion; Appendix B.3, equations 71–80,
uses low moments and a Fano-type quantity. These do not give the exact
randomized-channel trace-distance rate in the original M06 task. The narrow
comparison in the frozen proof is supported. Primary source:
https://arxiv.org/html/2512.10703v2 . A global novelty search and external
expert review remain pending. No external submission or communication occurred.
