M05: compiled recursive block Gibbs purification and boundary dressing
Author: /root/qualification_backlog. 22 October 2026.
Status: scoped constructive implementation theorem; M05 remains open.

DELTA AND EXACT SCOPE
R02/R04 already gave a CONDITIONAL amplified recurrence; R03 already gave
the collective reset isometry. Those are inherited, not new claims here.
R44 supplied the missing coherent filtered-boundary coefficient oracle for
interacting Hamiltonians with arbitrary on-site scales. This packet supplies
an explicit ordered-product circuit using that oracle, its inverse dressing,
an actual recursive purification algorithm, and charged error/cost bounds.
No free block Gibbs, spectral, interpolation, partition-function, or successful
postselection oracle is assumed. This is one qualification, not several seats.

We use fixed local dimension and range; group a fixed number of sites to
reduce to nearest-neighbor form. Local terms are ordinary binary rational
matrices supplied to requested absolute precision. Let b bound their per-term
bit length, the temperature/bounds' bit lengths and integer magnitude bits.
If an input lists repeated local supports, first aggregate them while charging
their entire description length; include the raw description bits per fixed
local neighborhood in b. Thus reading/aggregation costs O(L poly(b)), rather
than silently assuming an arbitrarily long labelled list has O(L) read cost.
Let x=1+beta J_int. Constants below depend only on fixed local parameters.
There is a uniform integer kappa>=1, kappa<=C x, such that at every merge
the QBP coefficient below has norm <=kappa and Lipschitz constant <=C kappa^2.
We charge b explicitly; an arbitrarily long input cannot be read for free.

For an interval of L sites, h=ceil(log_2 L), and 0<delta<1/4, a circuit
prepares a purification whose system marginal has half trace-distance <=delta
from its Gibbs state. Its elementary-gate count AND classical construction
work obey, for fixed constants C,c,

 C_pur(L,delta) <= L^c exp[C kappa h]
  [1+b+kappa+h+z]^c [1+z+(kappa+1)h]^h,       z=log(1/delta).       (1)

The same upper bound pays for all inverse preparation calls, work-register
phase tests, and expanded circuit output. Polynomial-degree slack in c is
intentional. No claim of optimal beta dependence is made.

For the source's RESET block B, Eq(1.5) gives
 ell<=C_d x^8 exp(8 beta J_int), h=O(x), independent of N and delta.
Consequently (1) implies the conservative bound

 C_pur(ell,delta) <= exp[C x^3] (1+b)^c [1+log(1/delta)]^(C x+c).   (2)

This closes the previously conditional block-purification implementation in
the stated arbitrary-on-site binary-input model. The temperature-dependent
precision exponent is displayed. It is not a temperature-uniform exponent.

On ANY localized region R of m sites, the source's boundary dressing O_R
and O_R^{-1} have block encodings with normalization <=exp(Cx), absolute
error eta, and gate/classical cost poly(m,b,x,log(1/eta)), of fixed degree.
Using (2), these encodings implement R03's collective dressed reset and loss
with exp(O(x)) normalization, without enumerating d^(2 ell) jumps.
We do NOT implement the source coherent correction G_H(S) with a bound
uniform in the on-site energy value. Therefore this packet does NOT establish
the full global nearly-linear M05 algorithm or a full update simulation.

1. EXACT QBP TRANSPORT, TEMPERATURE AND ORACLE
At a binary merge H(s)=H_left+H_right+s V, let J bound both ||V|| and its
constant-size local-term block-encoding normalization. In one dimension
J<=C J_int. Use the source construction with parameter 2 beta:

 A(s)=-(beta/2) integral p_(2 beta)(t) e^(itH(s)) V e^(-itH(s)) dt,
 O'(s)=A(s)O(s), O(0)=I,
 O(1)e^(-beta(H_left+H_right))O(1)^*=e^(-beta H(1)).             (3)

Here p_theta(t)=4/(pi theta) log coth(pi |t|/theta), of unit mass and
E|t|<=theta. Equation (3), positivity of the kernel and the transport identity
are inherited from source Section3.1. Using parameter beta instead would
transport the HALF-temperature Gibbs operator and would be the wrong merge.
Writing gamma=beta J/2, norm and Duhamel estimates give
 ||A(s)||<=gamma,
 ||A(s)-A(s')||<= beta J^2 E_(p_2beta)|t| |s-s'|
                 <=2 beta^2 J^2 |s-s'|=8 gamma^2 |s-s'|,
 ||O(1)||,||O(1)^(-1)||<=e^gamma.                              (4)
For the last bound integrate the coefficient norm; the inverse satisfies
its usual right multiplication equation, with the same Gronwall bound.
Choose a known integer kappa>=max(1,gamma) uniformly for every merge.

The frozen R44 theorem supplies, coherently for binary s, an encoding of
A(s)/kappa with normalized error rho. Its gate and classical construction
cost is poly(L,b,x,log(1/rho),number of s bits), with fixed degree. Indeed
its interaction normalization is alpha_B=O(L J_int), so beta alpha_B=O(Lx).
Large on-site norms enter through their logarithms/input bits only. The
generator is NOT obtained by diagonalizing the entire block. The oracle
includes its real-time simulation, quadrature, arithmetic and input errors.

2. FINITE DYSON LCU WITHOUT A TIME-NODE TABLE
This paragraph constructs a circuit directly, so a unit-cost propagator
lemma is unnecessary. The sorted-time LCU mechanism is standard: in particular
Kim2610.01190v1 AppendixC.2 Eqs(203)--(207) explicitly includes repeated time
indices and retained reversible sorting data. We instantiate that mechanism
for the smoothed QBP coefficient with its R44 oracle and explicit temperature
bounds; we do not claim to originate the construction. For a coefficient A
with ||A||<=kappa and Lipschitz
constant D, expand the time-ordered solution at time1 as

 O(1)=sum_(k>=0) kappa^k/k! E[ B(t_(k)) ... B(t_(1)) ],
 B=A/kappa, t_1,...,t_k independent uniform[0,1],
 t_(1)<=...<=t_(k).                                           (5)

The k! chambers of the cube give the ordered simplex; collisions have zero
Lebesgue measure. The order in (5) is essential for noncommuting coefficients.
Choose K+1>=2e kappa and K+1>=log_2(16/eta). The omitted tail is at most
2 kappa^(K+1)/(K+1)!<=2^(1-(K+1))<=eta/8, using the factorial bound and
the geometric ratio <=1/2. Thus K=O(kappa+log(1/eta)) suffices.

Take M=2^n>=max(1,8D e^kappa/eta) and replace each uniform time by the
midpoint of its M-cell. Sorting is nonexpansive for the maximum coordinate
distance, so corresponding ordered times differ by at most1/(2M). Product
telescoping and summing the exponential series bound the total grid error by
 D e^kappa/(2M)<=eta/16. Repeated discrete indices cause no omitted terms:
their ordered products are precisely the rounded-cube approximation to (5).
The number of time bits is O(kappa+log(1/eta)+log(1+D)), not M.

Let alpha=sum_(k=0)^K kappa^k/k!<=e^kappa. Prepare the length register with
probabilities (kappa^k/k!)/alpha, and K independent n-bit uniform registers.
For branch k, copy/sort the first k indices reversibly, retaining comparison
bits so sorting can be undone. Apply the k ordered coefficient encodings
with SEPARATE block-encoding work registers, then undo sorting and time/length
preparation. The projected block is the finite sum in (5) divided by alpha.
Unused time registers and work registers are unprepared to zero. A comparison
network, controlled products and their inverse work cost poly(K,n); no M^k
list of ordered tuples is made or charged as a free data structure.

If each normalized coefficient block differs from B by at most rho, both
the implemented block and B are contractions. The product error is at most
k rho, so the unnormalized sum error is at most kappa e^kappa rho. Choose
rho<=eta/(16 kappa e^kappa). The length weights are rational because kappa
is integer; computing K factorial weights and their sum uses polynomially
many bits. Their square-root state preparation takes polynomial cost in
K and the requested bits. Give its unitary error and other synthesis errors
total <=eta/(16alpha). Including the tail and grid leaves ample slack for
absolute operator error eta. All extra precision is O(kappa+log(1/eta)).

Thus a complete unitary U has alpha<0|U|0>=K_O with ||K_O-O(1)||<=eta,
normalization alpha<=e^kappa, and fixed-degree polynomial gate/classical
cost in L,b,x,log(1/eta). The comparison registers, coefficient-oracle time
bits and all scalar arithmetic are included.

The SAME construction for -A(1-s) encodes O(1)^(-1): its solution is
Y(s)=O(1-s)O(1)^(-1), with Y(0)=I and Y(1)=O(1)^(-1).
Reversing the unitary U alone would encode O^*, NOT O^(-1); we explicitly
build this second coefficient oracle and circuit. Bounds (4) are unchanged.

3. COHERENT SUCCESS AMPLIFICATION WITH ACTUAL CHILD CIRCUITS
Let the two child preparation unitaries output a normalized vector phi_tilde,
including all retained work registers, within Euclidean distance e of the
chosen exact product purification phi, up to a global phase. Apply U from
Section2 on the system. The fresh LCU-zero success projection gives
K_O phi_tilde/alpha. If eta<=e^(-kappa)/2, then for EVERY normalized input
 ||K_O phi_tilde||>=e^(-kappa)/2,
 p_success>=p0=(1/4)e^(-4kappa).                              (6)
No partition-function ratio is computed and no successful branch is free.

For explicit accounting use Grover's pi/3 fixed-point scheme, not an assumed
exact state preparation. If a unitary W maps zero to sqrt(p)g+sqrt(1-p)b,
let R_0 and R_good multiply respectively the zero-input line and the good
subspace by omega=exp(i pi/3). Define W_0=W and
 W_(j+1)=W_j R_0 W_j^* R_good W_j.
The two-dimensional span of g,b is preserved. Put q=1-p. The bad amplitude
is multiplied by 1+(omega-1)(omega p+q)=-(omega-1)^2 q; its modulus is q.
Thus failure probability becomes q^3, and after j levels it is q^(3^j).
The good direction remains g up to a scalar phase. This is the standard
Grover2005 mechanism, reproduced here to make every call explicit.

Choosing the first 3^j>=p0^(-1)log(128/delta^2) gives failure amplitude small
enough for vector error <=delta/4, and uses
 q_amp=3^j<=3p0^(-1)log(128/delta^2)=exp(O(kappa)) O(1+z)         (7)
calls to W or W^*, including BOTH child circuits and their inverses on every
call. There are O(q_amp) selective phases. Multi-controlled zero/good tests
use reversible AND computation on all relevant work bits, a phase, and its
uncomputation. Synthesize their phase rotations to operator error
<=delta/(100q_amp); telescoping charges the total phase error <=delta/4.
Their arithmetic/gate overhead is polynomial in the register count and
log(q_amp/delta). This is coherent bounded-time preparation, with no measured
restart, hidden expected-time tail, or leftover failure probability omitted.

The inverse W^* is the exact reversal of its ALREADY COMPILED gate sequence.
Therefore the fixed-point calculation applies to the actual child outputs
and actual block K_O. Child approximation errors are not unexplained oracle
errors reused q_amp times: the good direction is exactly the normalized
K_O phi_tilde. For this direction use the elementary normalization inequality
 ||a/||a||-b/||b||||<=2||a-b||/||b||, giving

 distance(g, O phi/||O phi||)
 <=2e^kappa eta+2e^(2kappa)e.                                (8)

For a target node error delta, prepare EACH child to error
delta/(32e^(2kappa)) and take eta<=delta e^(-kappa)/32. Then e is at most
the sum of the two child errors. Equations(7),(8), plus phase synthesis,
give total vector error <delta. All old child work remains as purification
environment; the ideal reference has zero work registers. Global phases
do not affect either the system marginal or the reflections.

4. LEAVES, RECURRENCE AND ALL PRECISION COSTS
For a single fixed-dimensional site, diagonalize its rational Hermitian
input to the needed absolute accuracy classically. Fixed-degree algebraic
root isolation and linear algebra have polynomial bit cost; near-degenerate
clusters may also be grouped. Shift the minimum energy before exponentiating,
and omit Boltzmann amplitudes below the allocated error. Only polynomially
many bits in b, |log beta| and log(1/delta) are required; no operation count
proportional to the energy value or inverse Boltzmann probability is used.
Prepare sum_a sqrt(p_a)|a>|a> with fixed-size rotations and the eigenbasis
change. This supplies a true unitary and its inverse, with no leaf oracle.
The beta=0 case is the maximally mixed purification; zero interactions can
be handled as a product directly.

Use a balanced binary tree, padding its analysis by height h rather than
adding physical interactions. On every path the required logarithmic
accuracy increases by at most log32+2kappa at each edge. Thus
 z_leaf<=z+(log32+2kappa)h.                                   (9)
Every amplitude-amplification multiplier is bounded by
 exp(Ckappa)[1+z+(kappa+1)h]. The cost of a single nonrecursive merge,
including all work-bit phase tests, is bounded by a fixed-degree polynomial
in L,b,kappa,h,z. Its ancilla count is polynomial too: amplification reuses
its registers, and the tree contributes only O(L) nodes.

Equivalently, with P a fixed-degree polynomial, the safe circuit recurrence is
 C_h(delta)<=exp(Ckappa)(1+log(1/delta))
 [2 C_(h-1)(delta/(32e^(2kappa)))+P(2^h,b,kappa,log(1/delta))].  (10)
Increasing P covers phase-synthesis logs and register counts. Replacing each
factor by its maximum allowed by (9), and summing at most2L tree nodes,
proves (1). Each inverse has the same gate count. Classical compilation can
reuse subcircuit descriptions, but we conservatively charge expansion to the
full gate list; hence it satisfies the same bound. No cost saving from
unaccounted caching is needed. Polynomial local arithmetic does not acquire
an additional depth-dependent exponent: only the explicitly shown success
factors are multiplied over levels.

Absolute input precision is separate from circuit approximation. For rounded
H_hat, inherited R30 proves ||rho_beta(H)-rho_beta(H_hat)||_1
<=beta||H-H_hat||. With O(L) terms it suffices to supply operator error
O(delta/(beta L)) per term; allocate half the final trace budget to this and
run the circuit at half the desired accuracy. For temperature rounding use
the same bound for the dimensionless Gibbs generators: |Delta beta| alpha_H
must be O(delta). This costs log(1+alpha_H) extra bits, already part of b.
This equilibrium perturbation accounts for the input once, rather than
silently granting exact real matrices. All further R44 coefficient errors
use the rounded input and the budgets of Section2. Leaf and merge precisions
in (9), weight precision O(kappa+z_leaf), and phase precision
O(kappa+z_leaf+log(1+z_leaf)) are explicitly charged.

5. TEMPERATURE-SIZED RESET BLOCK VERSUS LOCALIZED DRESSING REGION
The source B.1 retains R_B(X)=sigma_beta^B tensor Tr_B X while replacing H
by H_R, R=B^{<=u}. The Gibbs reset is on B, not R. Its temperature-sized
ell has log ell=O(x). Substitution into (1), absorbing polynomial x factors
using 1+z+x^2<=C x^2(1+z), proves (2). At delta comparable to a global
update accuracy, z=O(log(N/epsilon)+poly(x)), the precision exponent is O(x).

By contrast, if one CHOOSES to prepare a Gibbs state recursively on a region
L=e^(P(x)) Q, Q=1+log(N/epsilon), then h=O(P(x)+log Q). The normalization
factor alone gives L^O(x)=exp(O(xP(x))) Q^O(x). It cannot be presented as a
temperature-only constant. The additional precision factor in (1), when
z=Theta(Q), permits Q^O(P(x)+log Q). Thus this particular upper-bound audit
does NOT establish a polylogarithmic global cost for that larger recursion;
it may be quasipolylogarithmic. This is an accounting limitation, not a lower
bound on all algorithms or a proof that the actual cost must attain it.
The source's architecture avoids that unnecessary large purification.

For the required O_R and inverse use Section2 directly with coefficient
-(beta/4)F_(beta,H_R(s))(W_boundary B). There are at most two interval ends,
so its kappa=O(x), independently of |B| and |R|. Its coefficient evaluation
cost is poly(|R|,b,x,log(1/eta)); it has no recursive child preparation.
The source localization can therefore enlarge R without adding purification
tree levels. Boundaries, full temperature in the reset, and half temperature
in the dressing are all distinct and kept as in the source.

6. COLLECTIVE RESET/LOSS INTERFACE AND THE REMAINING GAP
Let U_B be the circuit just proved, preparing a purification psi_B with
vector error delta_B. Swap the old B input into a fresh environment and
prepare psi_B on B and another environment. This gives R03's isometry V0
for the reset; its operator-norm error is <=delta_B because it tensors the
same prepared vector with every old input. It uses one U_B, no enumeration
of eigenvectors of the entire block or of an operator basis. Define
 V=(I_env tensor O_R) V0 O_R^(-1).
Its CP map is Ad_(O_R) R_B Ad_(O_R^(-1)), exactly the source reset jump map.
Any purification Kraus family may replace its displayed Hermitian-basis
family without changing that map, S=V^*V, or its required coherent completion.

Independent-ancilla block products encode V with normalization
alpha_V=alpha_O alpha_inverse<=e^(C x), and S with alpha_V^2<=e^(C x).
Operator error propagation costs at most e^(C x) times the sum of input
errors; for a requested eta choose delta_B and both dressing errors at most
eta e^(-C x)/C. The isometry and all inverse preparation calls are charged
by (2), while the two dressing calls use the fixed-degree regional bound.
This is inherited R03 algebra with its formerly missing implementations
now supplied, not an additional discovery claim.

The source generator is T(X)-{S,X}/2-i[G_H(S),X], with
G_H(S)=(i/2)tanh[(beta/4)ad_H](S).
Appendix B.4 encodes it through (beta/8)F_H(i[H,S]), whose straightforward
normalization contains alpha_H, potentially as large as the on-site ENERGY
VALUE. Neither (1) nor R44 removes that factor for this different input.
No uniform coherent-correction encoding or complete stacked-jump simulation
analysis is claimed here. In particular, implementing only the CP jump map
and loss is not implementing the source Gibbs-preserving update.

Full M05 remains open. The concrete route decision is to retain recursive
reset purification and regional inverse/dressing as compiled, and move the
remaining bottleneck to the exact coherent correction and final simulator
composition. The completed block theorem is stronger than another filtered
coefficient corollary, while narrower than the original global claim.

SOURCES AND VERIFICATION
Source2610.03578v1 Section3.1, Eq(1.5), Theorem3.11, Remark3.12 and FULL
AppendixB were read, with immutable local snapshots in inputs.json.
R44's frozen oracle, including its imported Low--Wiebe1805.00675v2 simulation
interface and actual local oracle construction, is an explicit dependency.
Grover quant-ph/0503205v1 Sections2--4 supplies the attributed pi/3 scheme;
its complete two-dimensional identity and query recurrence are reproved above.
Kim2610.01190v1 Theorem6.1 and AppendixC.2 were compared at theorem/circuit
level; its fixed-temperature Petz implementation is not the uniform bound
for the present smoothed QBP recursion. Its Dyson construction is credited.
Source/history comparison is in the companion file. No worldwide novelty
claim is made for Dyson LCU, QBP, fixed-point amplification or block encoding.
The checker tests the noncommuting ordered-product construction, exact pi/3
identity, approximation budgets and depth accounting. Floating ODE controls
are diagnostics; the uniform analytic proof carries the claim. Actual usage
is unknown. Freeze this version before any owner adoption.
