M06-v1: exact infinite-space geometric convergence
Frozen candidate for the unchanged original M06

Claim. The channel in original.tex satisfies, for every integer N>=0,
  (1/2)||Phi^N(tau_1)-tau_2||_1 <= 500001 exp[-N/(2*10^12)].
The proof below is infinite-dimensional. Its finite arithmetic controls are
checks of identities and constants; they are not an inference from a cutoff.

Notation q=e^-1, pi_r=(1-q^2)q^(2r), mu_r=(1-q)q^r. In the finite sector
K=2n_S+n_B=k use basis |r,k-2r>, 0<=r<=floor(k/2). The real symmetric
Jacobi matrix H_k has zero diagonal and edge r to r+1 equal to
sqrt[(r+1)(k-2r)(k-2r-1)]. Define U_k(s)=exp(-is H_k).
The exact population kernel, with rows as input populations, is
 P(r,j)=(1-q) sum_(l>=0) q^l 100 integral_.01^.02
          |[U_(2r+l)(s)]_(j,r)|^2 ds,
where a matrix entry outside its sector is zero. Each summand is nonnegative
and each row sums to one. On diagonal inputs, coherences between distinct
target occupations vanish after tracing the bath because the bath occupation
k-2j is different for different j. Thus the channel preserves diagonality,
and the population total variation distance is the full trace distance.

At the product pi tensor tau_1 the joint weight of |r,k-2r> is the constant
 c_k=(1-q^2)(1-q)q^k within sector k. Unitary conjugation therefore fixes this
product exactly. Since H_k is real symmetric, U_k(s) is symmetric and
|U_(j,r)|^2=|U_(r,j)|^2. Summing sector weights proves detailed balance
pi_r P(r,j)=pi_j P(j,r). Each fixed-time kernel separately has these properties.
The nonnegative sums and bounded unitaries justify all state identities by
trace-norm convergence of the direct sum; no occupation cutoff is assumed.

1. A scalar time-average bound.
Put h(omega)=100 integral_.01^.02 cos(s omega) ds, x=|omega|/100.
Thus h=int_1^2 cos(tx)dt, with h(0)=1 and
 |h| <= |sin(x/2)/(x/2)|.
For sqrt(2)/100<=x<=1, the elementary bound 1-cos y>=y^2/3 for |y|<=2
implies 1-h>=7x^2/9>=7/45000>1/10000.
For x>=1, set u=x/2. For1/2<=u<=2, sin(u)/u decreases, so |h|<=2sin(1/2)
<=1-1/24+1/1920=1841/1920<.96. For u>=2, |h|<=1/u<=1/2.
Consequently 1-h(omega)>=1/10000 whenever |omega|>=sqrt2.

We also have h(omega)>=-99/100 for every real omega. For0<=x<=1/2,
h>=1-(7/6)x^2>=17/24. For x>=1/2, u=x/2>=1/4. If u<=2, monotonicity
of sinc gives |h|<=4sin(1/4)<=1-1/96+1/30720=30401/30720<.99;
if u>=2, |h|<=1/2. Both elementary bounds are global, not grid bounds.

2. Uniform downward probability from the bath-vacuum branch.
Fix ANY integer r>=1. In sector K=2r start at |r,0>. Let Lambda have the
spectral probability measure of the finite-sector Hamiltonian at this vector.
Its first two off-diagonal squared couplings are2r and12(r-1), so
 E Lambda=0, E Lambda^2=2r, E Lambda^4=28r^2-24r.
For independent Lambda,Lambda', Omega=Lambda-Lambda' satisfies
 E Omega^2=4r, E Omega^4=80r^2-48r<=80r^2.
Paley-Zygmund applied to Omega^2 gives Pr[Omega^2>=2r]>=1/20.
The random-time survival probability is E h(Omega). Since 1-h>=0 everywhere
and >=1/10000 on this event, average escape probability is >=1/200000.
In this vacuum-bath sector every escaped state has strictly smaller target
occupation. Therefore for ALL r>=1,
 P(r,{0,...,r-1}) >= (1-q)/200000 =: d0.
This uses exact low moments in every sector, not a finite-cutoff estimate.

3. Infinite-chain conductance and positive gap.
For any nonempty A subset N with pi(A)<=1/2, zero is not in A because
pi0=1-q^2>1/2. Set m=min A>=1. Then
 Q(A,Abar)>=pi_m P(m,{0,...,m-1})>=d0 pi_m
 >=d0(1-q^2)pi(A), since pi(A)<=sum_(r>=m)pi_r=q^(2m).
So conductance Phi>=d0(1-q^2). Using q<2/5 and1-q^2>4/5 gives
Phi>12/(5*10^6)>2*10^-6. Cheeger's Poincare bound yields
 <f,(I-P)f> >= (Phi^2/2) Var_pi f >=10^-12 Var_pi f.
A self-contained proof: for nonnegative g vanishing at0, all level sets have
pi mass<1/2. Coarea yields sum pi_i Pij |g_i^2-g_j^2|>=2Phi||g||2^2.
Cauchy-Schwarz and sum pi_i Pij(g_i+g_j)^2<=4||g||2^2 give
E(g)>=Phi^2||g||2^2/2. Split f-f(0) into positive/negative parts; their
Dirichlet cross term has the favorable sign, and ||f-f(0)||2^2>=Var f.
Extend from finite-support real functions by density to L2(pi).

4. Excluding negative spectrum.
For real finite-support f, let F_k=diag(f(r)) in sector k. In an eigenbasis
of that sector's Hamiltonian, the stationary quadratic form is a positive
sum of |(F_k)ab|^2 h(lambda_a-lambda_b), with constant positive sector weight
(1-q^2)(1-q)q^k. The preceding bound h>=-.99 gives P>=-.99 I on L2(pi).
Absolute convergence and density justify the infinite sum/extension.
Therefore the centered L2 operator norm is <=1-10^-12.

5. Initial state is NOT in L2; exact tail conversion.
For mu^(K)=mu 1_(r<=K), the omitted mass is q^(K+1), and its density has
squared L2(pi) norm ((1-q)/(1+q))(K+1). Center at its own total mass.
Contraction, Cauchy-Schwarz and omitted-mass control give
 TV(mu P^N,pi)<=.5 sqrt[((1-q)/(1+q))(K+1)] (1-10^-12)^N+q^(K+1).
Take K=N and lambda=10^-12. Since
 sqrt(N+1) exp(-lambda N/2)<=lambda^-1/2,
 TV<=500001 exp[-N/(2*10^12)].
Thus certified, very conservative constants are C=500001,c=1/(2*10^12),
valid for EVERY integer N>=0 in the exact infinite Fock-space model.

6. Finite energy, heat and gross work.
The initial density f0=q^-r/(1+q) belongs to L^(3/2)(pi), and
 ||f0||_(3/2)=[(1-q^2)/((1+q)^(3/2)(1-sqrt q))]^(2/3)<2.
Indeed the cube of this norm is (1+sqrt q)^2/(1+q)<=2.
Reversibility/Markov contractivity gives the same bound for every f_N.
Holder and r^6<=6! binom(r+5,6) imply E_muN r^2 <=
2 [720/(1-q^2)^6]^(1/3)<30. Also E_pi r^2<4.
Cauchy-Schwarz with |mu_N-pi|<=mu_N+pi therefore gives
 |E_N-E_pi|<=sqrt[68 TV(mu_N,pi)]<6000 exp[-N/(4*10^12)].
Conservation of2n_S+n_B gives, at collision j,
 Q_j=2(E_(j-1)-E_j), w_j=[E_(j-1)-E_j]_+.
These are the discarded-mode heat and gross supplied mean work; no monotonic
energy assumption is made. Total signed heat through N is2(E0-E_N), and
net bare-energy work is E0-E_N. Gross W_N=sum_j w_j<=10^17 for all N,
using the energy-error geometric series and1-exp(-d)>=d/2 for d<=1,
d=1/(4*10^12). The infinite limits of signed heat and net work are
2q/(1-q^2) and q/(1-q^2), respectively. The gross upper bound is intentionally
loose; no claim that gross work equals net work.

For the stronger convention charging positive mean work separately given
every random time and its history, write e_j(h) for that branch target mean.
Every fixed-time channel separately preserves pi and contracts L^(3/2)(pi),
so the same second-moment bound<30 holds for EVERY history. The charge is
[e_(j-1)(h)-e_j(h,s)]_+<=e_(j-1)(h)<sqrt30<6. Consequently expected
conditional-history gross work is at most6N. The expected signed heat and
net-work telescoping formulas are unchanged. The constant10^17 estimate
is asserted only for the averaged-collision convention, not this stronger
branchwise convention. Both conventions are explicitly accounted for.

Each collision uses one
fresh thermal mode and time<=1/50, so N collisions use N modes and time<=N/50.
The interaction coefficient is one and its operator norm remains unbounded.

7. Original-scope and source comparison.
The constants apply to the original fixed uniform interval [.01,.02], the
original initial tau_1, the original target tau_2, and every integer N>=0.
No control, measurement, local rotation, postselection, retained bath mode,
or modified pulse distribution is introduced. Truncation is used only in
an exact estimate of the initial tail; the evolving channel is never truncated.

The original primary source is arXiv:2512.10703v2, Sections IV.2 and
Appendices B.2--B.3 (https://arxiv.org/html/2512.10703v2). Equations66--70
use a short-time expansion, and equations71--80 use low moments and a
Fano-type quantity. They do not state or prove the exact randomized-channel
geometric trace-distance bound established here. This is a narrow comparison
with that source, not a claim of an exhaustive novelty search.

Status at freeze: supported by the self-contained proof and exact arithmetic
controls; separate-agent review requested. External review and global novelty
assessment remain pending. No publication or external contact is authorized.

