M05: local stacked-jump composition and original-target ledger
22 October 2026. Companion to M05-singular-correction-v1.txt.
One mechanism; no separate test or round credit for this consequence.

A. A valid simulator interface, including approximation errors

The imported simulator is Chen--Gao--Wang--Zhou, 2609.18757v1,
Definition 1.1 and Theorem 1.2. Its Hamiltonian oracle is an EXACT Hermitian
involution; its jump oracle encodes the rectangular stack
  V=sum_j |j> tensor L_j
in the nonzero output-label subspace. Normalizations enter through
  alpha=alpha_H+alpha_V^2, tau=alpha T.
This is not a theorem with a cost polynomial in the number of jumps.

The R45 stack and the correction theorem meet these interfaces. Include
an extra output flag equal to one on every successful stack output, so that
all successful environment labels are nonzero; the initial flag is zero.
Success ancillas remain separate. This is a constant-cost relabeling, not
an enumeration of labels. All workspace registers and log(number of labels)
are charged; they include the purification and arithmetic workspace and
are not assumed to be just the physical block size. The correction theorem's
one-flag Hermitian wrapper gives precisely the required involution.

Let Vtilde and Gtilde be the EXACT projected matrices of the compiled
unitaries, scaled by their stated normalizations. Gtilde is Hermitian.
These matrices define an exact Lindbladian
  Ltilde(X)=Tr_env(Vtilde X Vtilde*)
       -1/2 {Vtilde*Vtilde,X} - i[Gtilde,X].
There is no requirement that its Gibbs state be exact, or that the full
unitary completion be close to some preferred ideal completion. The
target L is the same expression with exact V,G. For e_V=||Vtilde-V||
and e_G=||Gtilde-G||, multiplication/partial-trace bounds on the Stinespring
maps, including arbitrary reference systems, give
  ||L-Ltilde||_diamond
       <=2 e_G+2(||V||+||Vtilde||) e_V
       <=2 e_G+2(2 alpha_V+e_V)e_V.
Indeed the completely positive terms differ by at most
(||V||+||Vtilde||)e_V; the loss matrices have the same bound and their
half-anticommutator map has diamond norm at most that loss norm.
Both generated semigroups are CPTP. Duhamel therefore gives
  ||exp(TL)-exp(TLtilde)||_diamond <= T||L-Ltilde||_diamond.
There is no exp(T||L||) stability factor.

For desired local channel error delta<1/4 take, for example,
  e_G<=delta/[32(1+T)],
  e_V<=delta/[64(1+T)(1+alpha_V)].
These put the perturbation contribution below delta/4. Independently build
the higher-precision V used inside S for the correction at the accuracy
specified in that theorem; using a different compiled stack there is
harmless. Allocate delta/2 to the exact-oracle simulator and its remaining
finite-gate synthesis. The whole local channel is within delta.

B. Gate and classical compilation accounting

For exact input oracles, Theorem 1.2 has query count
  O(tau+log(1/delta)/log(e+log(1/delta)/tau)),
and additional gates polynomial in that count, the oracle ancilla count,
log(number of jumps), and log((1+tau)/delta). A coarser fixed polynomial
bound in 1+tau+log(1/delta) suffices here. Use alpha_H=alpha_G from the
correction theorem, NOT a normalization of the physical Hamiltonian H.

Classical compilation cannot be concluded solely from a query theorem.
The source construction was therefore inspected at Sections 6.2, 7.1--7.2,
9.3--9.4. It has explicit binomial/filter coefficients with support O(q),
and their finite sums can be computed in polynomial(q,p) bit operations
at p-bit precision. It is enough here to choose the coarse admissible
q=ceil(C[1+tau+log(1/delta)]) for a sufficiently large universal C:
then q>=2 C_0 tau and sqrt(tau)(C_0 tau/q)^q<=delta/20, the explicit
Section 7.2 criterion. Optimal query scaling is not needed. The large
discretization parameter can be a power of two
  J>=max(2 tau,8q,20 tau^2/delta),
so log J=O(log(2+tau)+log(1/delta)). The construction represents the time
position in binary. Section 9.3 constructs the two-by-two rotations by
binary powers/rotations, and Algorithm 2 in Section 9.4 traverses O(log J)
binary decisions, with O(q) retained labels. There is no J-entry classical
table. Computing those scalar rotations, the O(q) coefficient state and
their gate decompositions to inverse-polynomial-in-call-count error uses
polynomial(q,log J,log(1/delta),oracle ancillas) classical work.
Thus this local use supplies both a gate bound and a polynomial classical
construction bound. The exact source circuit, not just its query headline,
is the imported interface. All called R44/R45 circuits and their inverses
are included. The polynomial degree is fixed.

C. Explicit chain parameters and what this packet does not certify

For the source's chain reset blocks put x=1+beta J_int and z=1+log(1/delta).
The reset length satisfies ell<=C x^8 exp(8 beta J_int). The source gives
T<=exp(O(x)) times a fixed polynomial/logarithm in x. Its Proposition 5.7
and Appendix B.1 give, with fixed-parameter constants,
  lambda=c/x,
  q_beta=C_d exp(2 beta J_int)x^4 log(2+beta J_int),
  mu=lambda/2, C_loc=exp(lambda/2)q_beta,
for radius u>=max(r,ell). Taking
  u=max(r,ell,ceil(mu^(-1)log(4 T C_loc/delta)))
gives localization error <=delta/4 and
  m=|B0^{<=u}|<=exp(Cx)(1+z).
This specifically uses the chain constants; treating the generic B.1
dependence on block size as an unspecified constant would not suffice.

The correction and R45 stack then satisfy
  alpha_V<=exp(Cx),
  alpha_G<=exp(Cx)(1+b+x+z),
and the local update compilation, including both gate and classical costs,
has the conservative form
  exp(C x^3)(1+b)^c (1+z)^(C x+c),
after increasing fixed constants C,c. The logarithms of b arising inside
high-precision child calls are absorbed as shown in the correction proof.
The exponent of precision is explicitly temperature dependent; c need
not be 2 in the temperature exponential. This is a substantive local
completion of the correction/stack interface left open in R45.

Substituting delta proportional to epsilon/K and the source's
K=O(N log(N/epsilon)) Poisson truncation is consistent with a total
  exp(C x^3) N (1+b)^c [1+log(N/epsilon)]^(C x+c+1)
ledger. That is a composition candidate, not a complete-original verdict
from this packet. A full review must still consolidate the entire global
algorithm: its prescribed rates/durations and initial-state mixing bound,
uniform finite-bit sampling of the truncated Poisson and block choices,
source-input rounding over all local terms, and the total classical cost
of reading, indexing, compiling and reusing the N local instances.
No free block-Gibbs oracle or jump enumeration may reappear in that review.

In particular the exact original displays no arbitrary-input-bit factor.
Our explicit bound contains (1+b)^c, and reading N b input bits already
costs that much in general. If b is polynomial in x and log(N/epsilon),
the displayed target shape follows after adjusting constants/exponents;
if b is unrestricted, this factor must be stated separately as required
by the original's instruction to specify all input-precision costs. It
must not silently be hidden in a temperature-only constant. Full-original
status requires an explicit interpretation of this input convention and
independent verification of the consolidated theorem. Neither a uniform
temperature-independent polylog exponent nor a complete M05 solution is
claimed here.
