Symmetric-family calibration with explicit confidence and drift

Every null is counted and retained. The positive square roots with a neutral pointer give the finite
isometry V = P
j
pEj ⊗|j⟩, retaining output source and any inert reference. For degenerate modes
a balanced rotation and a complete basis absorber realize γ = 1; known unsharpness permits other
values. Nondegenerate energy modes require the phase-reference stock from the finite-reference model.
No calibration phase resource is free.
A hit has probability wηc, independent of setting. Its conditional Bernoulli parameters are p and
q = (1 + γt)/2. This conditioning is justified by independent preparations and state-independent loss,
not by a reset wave. Let nZ, nX be hit counts and KZ, KX the true counts of label 0, +. An adversary
may change at most kZ, kX accepted labels, even after seeing all data, producing f
KZ, f
KX. Nulls cannot
be forged, suppressed or selected. This is robustness of the nominated reported-count readout; no
optimum over hidden corruption-controller records is asserted.

Let Iω be the ordinary CP interval for total hits nZ + nX in NZ + NX trials. For trusted ηc ∈[ηL, ηU],
use
Iw = [IL
ω /ηU, IU
ω /ηL] ∩[0, 1],

Proof.
Conditional on a hit count the true label count is binomial; CP coverage remains valid after
averaging over that random count. Monotonic endpoints and |K −f
K| ≤kf make the envelope contain
the true count’s interval on every corrupted transcript. Total hits are binomial with parameter w0ηc.
A union bound proves coverage; independence of the three intervals is unnecessary. CP coverage itself
follows by binomial-tail inversion, each excluded tail having probability at most β/2. Thus this is
finite-sample coverage for each fixed underlying state, not a posterior probability on a parameter. □
Hits alone identify wηc, not its factors. Without efficiency information the safe bound w ≥IL
ω still
follows from ηc ≤1. One event constrains the source for every reader, allowing subsequent reader or
sign-orientation choice without another source-confidence penalty. A transverse interval straddling
zero does not become a positive-direction certificate merely by clipping its lower endpoint.

Inputs may otherwise be adversarial and correlated through history. These are future admission
conditions, not consequences of finite calibration. Trace-distance qubit drift at most δ suffices for
dp = δ, dt = 2δ, but the componentwise class below may be larger.
On GS, enlarge the population interval to I′
p, put l = IL
t −dt ≥0, wL = (IL
w −dw)+, and define

Keep the original sign/equality check to reject extraneous roots.
This follows by differentiating
the ratio. Cases b = 0 or R = 0 reduce directly. Endpoint-only checking fails, for example, at
I = [−0.8, 0.8], l = 0.6, ∆= 0.3, (x, y, z) = (0.9, 0.3,
√

C3: sharper coherence interface.
Let u∗be the point of I closest to zero, rmax =
p

1 −u2, hence is minimized at u∗, proving
the claim. Supply this directly to the robust frontier; it already includes differential phase and must
not be multiplied by another cosine. An independent p, m box is valid but can be looser than Eq. (43),
which retains their correlation. If rmax = 0, the admitted states have zero transverse coherence; their
conditional error is 1/2 for every positive conclusive output. Use C2 directly and do not divide by rmax.
The positive-contrast robust frontier uses this interface only when mcal ≥0. If it is negative, use
the general admissible-error certificate C2 or another appropriately enlarged problem; do not insert it
into a theorem assuming nonnegative contrast.
At ∆= 0, direct calibration gives m ≥q. Separately inferring visibility at least q and phase
cosine at least q, then multiplying, gives only q2. Those extrema occur at different states: visibility q
with zero phase, or unit visibility with phase arccos q. Keeping the measured product removes this
unnecessary loss. No source Y assay is needed for this objective: the unmeasured component is retained
adversarially within the physical Bloch disk.
Divide Xmin(u) by
√
1 −u2. The result increases with l/
√

This structural family follows from the actual two-tap model: a balanced flag matrix times diag(a0, a1)
gives x = |a0|2, y = |a1|2, c = a∗
0a1. Input-independent mixtures preserve it by Cauchy–Schwarz.
Unknown path loss and phase change the coefficients but preserve the family. This is a physical-model
admission; finite probes cannot prove exact symmetry or dark-free behavior for an unrestricted device.
Use independent identically prepared copies within each trusted-probe batch |0⟩, |1⟩, |+⟩, | + i⟩, with
a fixed input-independent reader preparation and stationary pointer probabilities during calibration.
Every plus, minus and null is counted:

The fourth probe prevents an imaginary-offdiagonal blind spot. Six marginal CP intervals, with failures
summing to αD, give Ix, Iy, Ir, Ii. Corruption envelopes apply to bounded accepted-label errors. For
probe preparation trace error δprep, widen each event interval by δprep, since effects are trace-distance
contractive. That probe-preparation accuracy is independently trusted.
If IL
r > 0, infer