M14 complete-original candidate at T=4/5.
The retained-memory witness uses three collisions with bath word(1,2,1),
times(217/1000,404/1000,179/1000), and all phases0. Three is permitted by
'at most four'. The certificate gives
 E_mem(4/5) <=0.259093772014,
 E_reset(4/5) >=0.259300152798,
so the strict gap exceeds0.000206380784. These are conservative bounds,
not the optimized numerical gap. All energies use the original unit scale.

1. Exact reset reduction and a positive finite subensemble.
Fresh thermal baths erase the phase dependence and keep target populations
diagonal. Thus every reset strategy is four ordered scalar kernels P_s,
s>=0,sum s<=4/5; zero pulses include fewer collisions. The kernel on conserved
K=2r+l is exp(-is H_K), with tridiagonal off-diagonal
sqrt((r+1)(K-2r)(K-2r-1)). Restrict initial target r<=10 and each of the four
initial bath occupations l<=10 WITHOUT renormalizing. Omitted contributions
to final energy are nonnegative, so the resulting minimum is a lower bound
for every original reset strategy; no tail subtraction is needed on this side.
After j pulses every retained path has r<=10+5j, so target states0..30 retain
every reachable trajectory. Each truncated-bath row has exact mass1-exp(-11).

2. Uniform interval envelopes for a pulse time.
For s in[c-h,c+h], Taylor's theorem gives entrywise
 P_s >=P_c-h|P'_c|-h^2 M2/2,
 P_s <=P_c+h|P'_c|+h^2 M2/2,
where M2 bounds |P''| throughout0<=s<=4/5. For a spectral proposal V diag(w)V^T,
the absolute Fourier bound for a transition probability is
2(A0*A2-A1*A1), A_j=|V|diag(w^j)|V|^T, with entrywise products. This follows
by summing |V_ra V_ja V_rb V_jb|(w_a-w_b)^2; it holds for every time.
The exact-Hamiltonian defect and all arithmetic errors are added below.

3. No trust in a numerical eigenvalue solver.
Quantize proposed V,w to rational multiples of2^-60. Enclose every H_K
off-diagonal by integer square-root endpoints with that denominator. Exact
integer products bound ||V^T V-I||<=e_o and ||HV-V diag(w)||<=e_r via n times
the largest entry enclosure. These bounds are saved for every K<=70.
Let J>=||H||, A>=||diag(w)||, and e_o<.01. Polar-correct V to an exact orthogonal
Q. One has ||V-Q||<=e_o; a conservative bound for0<=s<=.8 is
 u=.8[2e_r+(J+4A)e_o]+4e_o
on ||exp(-isH)-V exp(-is diag(w))V^T||. This follows by comparing H with
V diag(w)V^T and then Q diag(w)Q^T, using unitary Duhamel and polar bounds.
The first and second time-derivative errors are at most
 d=Ju+2e_r, e=J^2u+2e_r(J+A).
Using proposal amplitude norms<=2,2A,2A^2 gives probability error
u(4+u), first-derivative error2(2d+2Au+ud), and second-derivative error
2(4Ad+d^2)+2(2e+2A^2u+ue). These are computed as exact fractions, then enlarged.

4. Arithmetic budget, stated explicitly.
MPFR evaluates rational phases at160 bits, with |phase argument|<2^14.
Input rounding is <2^-145 and trigonometric rounding <2^-159; conversion to
binary64 is <2^-53. An absolute1e-14 bound for each proposed sin/cos is ample.
All spectral matrices have dimension<=36, J,A<1000, and ||V||<2. The standard
IEEE binary64 dot-product bound gamma_m=m*2^-53/(1-m*2^-53), with at most8n
real operations per complex dot, bounds each proposed U entry error by1e-12
and each U' entry error by1e-9. For A0,A1,A2 the errors are bounded by1e-12,
1e-9,1e-6 respectively using the same formula; propagating them through
2(A0*A2-A1^2), with A0<=2,A1<=2000,A2<=2e6, costs <1e-3.
Thermal weights are checked against exact rational exp(-1) enclosures to
absolute1e-14. Summing at most11 bath weights, including all scalar arithmetic,
fits within the extra budgets: P1e-8, P'1e-5, P''0.01. The last interval
evaluation and float-to-grid conversion receive another1e-7 per entry.
There is no underflow-sensitive negative cancellation used for a final sign.
These margins depend on the stated IEEE/MPFR arithmetic contract, unlike the
pure-Fraction M10 certificate. They should be independently audited before release.

5. Exact integer branch bounds.
Quantize kernel lower/upper envelopes outward on the grid10^-8; initial
probabilities and row mass are rounded down using exact fractions. Given
a nonnegative vector of later costs, minimize each row's cost over the
interval probabilities with at least that row mass. Fill the remaining
mass greedily at the smallest costs. This is an exact linear-program minimum.
It permits different times for different target occupations, a relaxation,
so it only lowers the original common-time optimum. Perform four backward
steps and the initial average with integer arithmetic, rounding down after
each step. All products/sums are <=30*10^16<2^63; no integer overflow is possible.

The four time coordinates are rational dyadic fractions of4/5. Bisect boxes,
discard only boxes whose lower endpoints already exceed the budget, and clip
each upper endpoint by the other lower endpoints. This covers every point
of the original simplex. At termination all14049 surviving boxes have an
exact integer lower bound greater than0.2593. The cover is saved in NPZ and
can be regenerated by the checker. The discarded boxes are exactly infeasible.

6. Memory upper bound and infinite-dimensional tail.
For fixed bath parities e1,e2 and L=r+floor(l1/2)+floor(l2/2), the full invariant
sector has basis(r,u,v),r+u+v=L. Each pulse is a direct sum of the same certified
one-bath tridiagonal exponentials, with the other bath a spectator. Retain all
sectors L<=18, all initial states in those sectors, and their original thermal
weights. No projection is inserted between pulses. The computed energy is
0.2590829355357667. Missing L>18 implies initial total occupation S>18.
Conservation of2N_S+N_b1+N_b2 bounds final target energy by initial S, hence
the missing energy is at most the explicit negative-binomial tail B_3(18)
from R04, evaluated with outward rational arithmetic (about9.84e-6).

The sector dimensions are<=190. The exact one-pulse spectral error is<1e-9;
its proposed entry arithmetic is<1e-12 in blocks of size<=19. Three products
therefore have operator error<3.2e-9, up to dense-product roundoff<1e-10 by
the gamma_n bound and Frobenius norms. Since target occupation<=18, the
total energy error is<2e-7. Thermal-weight and summation errors are also below
1e-8; the checker reserves a full1e-6. Adding this and B_3(18) gives the
stated upper bound, separated from the global reset lower bound.

This is a memory-versus-reset result only. It is not a claim of a coherence
advantage, collective-target advantage, or globally optimized memory control.
