M06-v1 — independent strict review, 22 October 2026

Judgments
Validity: supported internally; no proof gap or refutation found.
Original coverage: complete original for the fixed bosonic channel, uniform
times in [1/100,1/50], initial tau1 and target tau2, on the full Fock space.
The result does not claim uniform trace mixing from arbitrary initial states,
fixed-time collisions, a general pulse distribution, or bounded interaction
operator norm. Its rate constants are deliberately conservative.
Source comparison: difference supported in a bounded current primary-source
comparison; no specific unresolved covering theorem found.
External status: no external expert acceptance established by this review;
no submission or publication performed.

Target and proof reconstruction
Original campaign/originals/M06.tex:7–23; candidate
campaign/continuation-2026-10-21/candidates/M06-v1/proof.txt is frozen in the
input hash manifest. I read all of it and the full-state/charging manuscript.
1. Opening sector construction: each K=2nS+nB sector is finite and the
operator is the prescribed direct sum. Bath tracing kills target coherences
on diagonal inputs. The joint tau2 tensor tau1 weight is constant in a sector,
so every time kernel fixes it. Real symmetry of the sector Hamiltonian gives
reversibility. Thus population total variation equals the actual trace
distance in the specified trajectory, not a proxy from a few moments.
2. Sections 1–2: the scalar Fourier average h has a global floor -.99, and
1-h>=1/10000 above frequency sqrt(2). The endpoint of the bath-vacuum sector
has moments 2r and 28r^2-24r for every r>=1. Independent spectral differences
have moments 4r and 80r^2-48r; Paley–Zygmund gives mass >=1/20 above 2r.
Every nontrivial transition from that endpoint lowers the target occupation.
The resulting downward probability (1-e^-1)/200000 is uniform in ALL r.
This is the decisive infinite-space bridge; it is not extrapolated from a
cutoff or a short-time series.
3. Sections 3–4: pi0>1/2 ensures a small stationary set has minimum m>=1.
Its downward exit at m and the geometric stationary tail give positive
conductance. The supplied coarea/Cauchy–Schwarz argument gives the Poincare
gap, while the sector spectral quadratic form and global lower h bound
exclude eigenvalues near -1. The centered L2 contraction is <=1-10^-12.
4. Section 5 explicitly recognizes that the initial density is not in
L2(pi). Its truncation at K has known omitted mass and squared L2 norm
proportional to K+1. Taking K=N and absorbing sqrt(N+1) gives
TV(Phi^N(tau1),tau2)<=500001 exp[-N/(2*10^12)] for all integer N>=0.
5. Section 6: the initial density lies in L^(3/2)(pi), and all fixed-time
kernels are contractions there. The uniform second moment and Cauchy–Schwarz
turn trace convergence into the stated energy bound. Conservation gives
Q_j=2(E_{j-1}-E_j) and gross supplied mean work [E_{j-1}-E_j]_+; no energy
monotonicity is assumed. The manuscript distinguishes the averaged-state
charging convention (uniform bound 10^17) from conditioning on sampled time
history before charging (expected gross work <=6N). Both give finite values
at every N and report heat/net work consistently with the original.

Primary sources and equivalent-result search
• Png–Hu–Scarani, 2512.10703v2, §IV.2 and Appendices B.2/B.3, Eqs. (66)–(80):
the iterative cooling and low-moment analysis uses short-pulse expansions.
It does not prove the exact randomized channel's full-state geometric rate.
The source's Fano-factor discussion cannot substitute for trace convergence;
the candidate does not use that inference. https://arxiv.org/html/2512.10703v2
• 2505.10054, §II Eq. (1), Theorems 2/3 and §IV: finite-dimensional
energy-preserving molecule collisions and modified HBAC protocols. These
hypotheses and control operations differ from the unbounded cubic oscillator
channel here; the thermalization/cooling statements do not cover its exact
infinite-space mixing inequality. https://arxiv.org/html/2505.10054
• Recent cooling searches also found 2607.23641 (multi-qubit dynamic cooling)
and 2610.27948v1 (slow-driving thermodynamic geometry). Their stated models
are finite-dimensional optimized controls, not this randomized channel. The
latter's Eqs. (2), (5) and propositions explicitly use the slow-driving GKSL
setting; no covering channel theorem was identified.
https://arxiv.org/abs/2607.23641
https://arxiv.org/html/2610.27948v1

Checks and manuscript comparison
Both clean candidate replays pass exact constant and moment checks, including
rejection of a wrong fourth moment. Newly written independent_controls.py,
also run twice in clean copies, reconstructs the two endpoint Jacobi edges
symbolically and derives m4 and the spectral-difference moment without
importing candidate code. This supports the algebra; the conductance and
tail arguments were reviewed analytically, not inferred numerically.
Manuscript M06-randomized-cooling/main.tex:58–77, 79–127, 128–206, 207–305,
and 306–368 follow the same argument and preserve the initial-state and
charging restrictions. No unsupported manuscript strengthening found.
Next action: retain as complete internally supported result with its exact
channel/initial-state scope and conservative rate explicit.
