M05: binary inverse-CDF encoding of the exact coherent correction
22 October 2026. Qualification by /root/qualification_backlog.
One scoped implementation theorem; not a complete-original verdict.

1. Statement and exact source interface

Let H=H* be the rounded local Hamiltonian on m sites, with fixed local
dimension/range, H=A+D where A is the sum of onsite terms and the interaction
normalization alpha_D is O(m J_int). Let beta>0 and S=S*. Take known rational
bounds E>=max(1,||H||), B>=max(1,||S||) and 0<eta<1/4. Local matrices and beta
have binary descriptions. The letter b below includes their bit lengths,
the supplied scalar-description lengths, log E and the bits needed to
read/aggregate the local input list; it is not the source's notation b for
block size. B and eta are also binary inputs. Bounds E and B may always be
padded to powers of two, without changing the asymptotic estimates.

Assume an actual unitary encoding S/B to normalized error rho is available
at gate and classical construction cost C_S(rho). The R44 frozen theorem
provides controlled real-time encodings for binary t in [-T,T], with error
xi and cost polynomial of fixed degree in
  m, b, 1+T alpha_D, log(1/xi), and the time-register bit length.
It compiles onsite fast forwarding and the interaction-picture oracle from
the local matrices; neither free spectral access nor a table of all t is
assumed. This is a pinned dependency, not a new simulation theorem here.

There is a unitary U_G that is EXACTLY Hermitian and involutive and whose
zero-ancilla block K_G satisfies
  ||alpha_G K_G - G_H(S)|| <= eta,
  G_H(S) = beta^(-1) integral_0^infinity
       [exp(itH) S exp(-itH) - exp(-itH) S exp(itH)]/sinh(2 pi t/beta) dt,
  alpha_G = O(B[1+log(1+beta E B/eta)]).
It uses O(1) calls to the S encoding and its inverse at
rho=eta/(128 alpha_G), and otherwise gate and classical cost polynomial
of fixed degree in
  m, b, 1+beta alpha_D, log B, log(1/eta).
Bit lengths of beta, B, eta, and scale bounds are included. In particular
there is no factor E or alpha_H in the normalization or gate count; log E
is explicitly charged. The dependence on B itself remains in alpha_G and
in the supplied S-encoding cost. No uniform O(1) operator-norm bound for the
tanh Schur multiplier is assumed.

The target is exactly Rouze--Stilck Franca 2610.03578v1 Eq. (3.11), with
tau_t(X)=exp(itH)Xexp(-itH). It is Hermitian for Hermitian S. For their local
generator S=V*V is the loss operator of the dressed RESET stack, not an
arbitrary list of uncharged jumps. Section 5 below supplies this interface.

2. Integrable cutoff and normalization

Put u=2 pi t/beta and define rational/integer parameters
  nu=1+beta E B/eta, a=1/(64 nu),
  U=2+ceil(log_2(128 B/eta)),
  r=ceil(log_2(1/a))+2, alpha_G=2 B(r+1).
The ceilings are computed by comparisons of binary integers/rationals.
Thus a<1/64 and U>1. The correction is
  (1/(2 pi)) integral_0^infinity
       [tau_{beta u/(2 pi)}(S)-tau_{-beta u/(2 pi)}(S)] csch(u) du.
For C=||[H,S]||<=2EB, Duhamel gives
  ||tau_t(S)-tau_-t(S)||<=2|t|C<=4EB|t|.
As csch(u)<=1/u, the omitted part below a has norm at most
  beta E B a/pi^2 <= eta/(64 pi^2).
For u>=U, use the bound 2B on the numerator and
  integral_U^infinity csch(u)du=-log(tanh(U/2))<=4 exp(-U).
The upper tail has norm at most 4B exp(-U)/pi < eta/64.
These are norm bounds, not a scalar cancellation assumption.

Let
  I=integral_a^U csch(u)du=log(tanh(U/2))-log(tanh(a/2)).
Since tanh(a/2)>=a/4, I<=log(4/a)<=r log 2. Hence
  w=B I/pi <= alpha_G.
The truncated correction is w times the average of
  sigma tau_{sigma t}(S/B), sigma in {+1,-1} uniformly,
with u density csch(u)/I and t=beta u/(2 pi).
The scalar c=w/alpha_G lies in [0,1]. A single scalar rotation with top
entry c pads the normalization to the exactly specified alpha_G. This
logarithmic mass is an inherited singular-kernel idea, not a novelty claim.

3. Coherent inverse CDF; no exponentially large weight table

Write L_a=log(tanh(a/2)), L_U=log(tanh(U/2)). For v in [0,1], set
  w(v)=(1-v)L_a+v L_U,
  u(v)=log((1+exp(w(v)))/(1-exp(w(v)))),
  t(v)=beta u(v)/(2 pi).
Direct differentiation, or the antiderivative log(tanh(u/2)), proves
that this is the increasing inverse CDF of csch(u)/I on [a,U].
Prepare n uniform qubits, Q=2^n, and reversibly compute t(v_j) for
  v_j=(j+1/2)/Q, 0<=j<Q.
No amplitude preparation for Q unequal weights, Q-entry table, or loop over
Q is used. The large Q only supplies an n-bit index. The sign is one qubit.

The binary computation is uniform and polynomial in its bit budget. For
completeness, elementary log/exp evaluations can be implemented by binary
range reduction, rational Taylor series with explicit remainder, and
Newton refinement, followed by reversible simulation of the resulting
finite Boolean arithmetic circuit. The following bounds identify all
required guard bits and rule out hidden dependence on 1/a or exp(U).
Compute log(tanh(a/2)) as log(a/2)+log(tanh(a/2)/(a/2)); the second factor
is analytic near zero and bounded away from zero, so the exponentially
small a is handled by its exponent and a relative mantissa. Evaluate the
inverse as log1p(2 exp(w)/[-expm1(w)]), using range-reduced series. In
particular do not separately subtract logs that discard the second copy
of exp(w) near the tiny lower endpoint. At the upper endpoint expm1
retains the small denominator.
Throughout the interval,
  1-exp(w)>=1-tanh(U/2)=2/(exp(U)+1)>=exp(-U),
  |d u/d w|=2 exp(w)/(1-exp(2w))<=C exp(U).
Consequently O(U+|log beta|+log(1/zeta_t)+log(1/a)) bits suffice for
absolute time error zeta_t. This quantity is polynomial in the listed
input sizes. Guard bits also cover multiplication by beta, pi evaluation,
and endpoint logs. Subtractions use the indicated stable expressions.
If beta is very small, absolute precision is easier; |log beta| is still
charged. Compute, use as a control register, and uncompute all arithmetic.

4. Discretization, oracle errors, and exact Hermiticity

The normalized map t -> tau_t(S/B) is 2E-Lipschitz in operator norm.
For any increasing t(v) with range length at most T=beta U/(2 pi), the
midpoint-quantile discretization has Wasserstein-1 error <=T/Q: partition
[0,1] into Q equal cells, couple each cell to its midpoint, and sum the
within-cell variations of the monotone function. Thus the operator error
after multiplying by w is <=2 alpha_G E T/Q.

Use a power of two Q>=max(1,128 alpha_G E beta U/eta), and allocate
  zeta_t <= eta/(128 alpha_G E),
  xi <= eta/(128 alpha_G), rho <= eta/(128 alpha_G).
Time rounding changes the output by <=2 alpha_G E zeta_t. Each simulated
time evolution is a contraction block P_t with ||P_t-exp(itH)||<=xi.
Using the same compiled oracle adjoint for the inverse, and fresh product
ancillas, the projected conjugation block differs from tau_t(S/B) by at
most 2xi+rho. One can first Hermitize the S oracle without increasing rho.
Approximate the scalar c and its rotation to projected error at most
eta/(128 alpha_G), with endpoint clamping to [0,1]. Its target stays far
from 1, so no ill-conditioned rotation evaluation is hidden.
The sign phase and uniform Hadamards are exact. Summing the two tails,
grid, time arithmetic, two propagators, S block, and scalar errors is
strictly less than eta (each displayed bound has substantial slack).
Finite-gate synthesis is included in xi, rho and the scalar budget.

Even if a compiled LCU unitary W is not Hermitian, the unitary
  U_G = (H_flag tensor I) [[0,W],[W*,0]] (H_flag tensor I)
is exactly Hermitian, squares to I, and its all-zero block is
(K+K*)/2, where K is the all-zero block of W. This preserves the norm
error to the Hermitian target. W* means reverse the ACTUAL compiled gates,
not approximate an independent inverse. Its controlled implementation has
only a constant oracle overhead. This supplies the involutive Hamiltonian
oracle required by the stacked-jump simulator, not merely a Hermitian
target matrix with an arbitrary unitary completion.

All oracle ancillas are included in m-independent bookkeeping registers.
The simulation time bound is T=O(beta[1+log B+log(1/eta)]). The bit sizes
of Q, a, t and all synthesis accuracies are polynomial in
b+log m+log B+log(1/eta). Substitution in the frozen R44 interaction-picture
construction proves the cost claim. In particular the enormous numerical
grid required by a high onsite frequency is not enumerated.

5. Actual reset loss interface and charged parameter consequence

Use the R45 frozen construction on reset block B0 and larger localized
region R. It supplies the collective dressed map
  V=(I_env tensor O_R) V_0 O_R^(-1), alpha_V=exp(O(x)), x=1+beta J_int,
where V_0 is implemented by swapping the old B0 into the environment and
preparing its Gibbs purification. O_R and its separately constructed
inverse use the R44 coefficient oracle and ordered Dyson LCU. All recursive
children, inverse circuits and amplification calls are already charged in
that dependency. The reset is on B0, not a Gibbs state on all of R.

If Pi selects successful output ancillas and the jump-label subspace of
the unitary encoding of V, then a block encoding of
  V*V/alpha_V^2
is obtained by U_V*, a block encoding of the known projector Pi, and U_V.
The projector is (I+(2Pi-I))/2 and is implementable by a success-bit check;
it does not enumerate jump labels. Alternatively use the standard product
rule on the rectangular stack with distinct ancillas. Set B=alpha_V^2.
If V is encoded to absolute error e_V, its loss error is at most
  (2 alpha_V+e_V)e_V.
Choose e_V<=B rho/[4(1+alpha_V)] so this normalized loss error is <=rho.
Hence C_S(rho) is O(1) calls to the actual R45 V construction at that
accuracy, plus polynomial ancilla checks. It is not a free S oracle.

For the source reset length ell<=C x^8 exp(8 beta J_int), R45 gives
  C_V(e)<=poly(m,b,x,log(1/e))
           + exp(C x^3)(1+b)^c [1+log(1/e)]^(C x+c),
up to factors exp(poly(x)) from dressing and normalization; c is fixed.
Thus this correction inherits a temperature-dependent precision exponent
O(x), and only polynomial binary-input overhead. Its normalization is
  alpha_G <= exp(Cx)[1+b+log m+x+log(1/eta)].
When precisions involve log alpha_G, factors (1+log(1+b))^(Cx) can be
absorbed into exp(O(x log(2+x))) times a fixed-degree polynomial in (1+b):
for y>=1,k>=1, y^k<= (k/e)^k exp(y). No temperature-dependent polynomial
in the numerical onsite energy is introduced.

This closes the exact coherent-correction encoding gap left explicitly by
R45. The separate composition assessment checks the local simulator and
lists what still requires global review. The original M05 bound also asks
for total classical preprocessing, global preparation, input precision and
the temperature dependence of the log(N/epsilon) exponent. This theorem
alone is not that conclusion, and does not change an original-card status.
