# Technical Companion: Detector Interfaces and Retained Outputs

Jeremy Rodgers. 27 October 2026.

Technical companion to the associated publication.

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## Source page 1

Technical Interfaces for Quantum Readers: Detectors, Finite
References and Calibration
Jeremy Rodgers
Independent Researcher
www.everythingequation.com
27 October 2026
Abstract
This technical companion preserves distinct operational interfaces that should not be forced into
one physical apparatus theorem: absorbing matched-template instruments; complete detector-output
comparison and finite microscopic closure repairs; finite charge-reference measurement; preparation
costs of nonlinear product-response interpretations; and task-specific versus general-effect calibration
and online monitoring. Results are conditional mathematical constructions using ordinary quantum
probabilities. Model scope, output omissions and preparation/reference costs remain explicit. The
note is comprehensive technical support, not a new Born-rule or consciousness paper.
Contents
1
Operational contracts and scope
2
1.1
Complete absorbing readers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
Complete retained-output comparisons and microscopic detector closure
4
2.1
A complete one-transition model and a reference-uniform certificate
. . . . . . . . . .
4
2.2
Exact scalar kernel formula: what intensity matching pays for . . . . . . . . . . . . . .
5
2.3
A coherent-field discriminator despite identical count records
. . . . . . . . . . . . . .
6
2.4
When is the binary reduction microscopically closed? . . . . . . . . . . . . . . . . . . .
7
3
Finite charge references and complete heralded phase measurement
8
3.1
Physical encoding and admitted experiment . . . . . . . . . . . . . . . . . . . . . . . .
8
3.2
Optimal exact heralding from finite reference support
. . . . . . . . . . . . . . . . . .
9
3.3
Full retained instrument: coherent versus classical charge record
. . . . . . . . . . . .
10
3.4
Reference dephasing: exact error–erasure boundary . . . . . . . . . . . . . . . . . . . .
11
3.5
Different objective: best no-erasure reference
. . . . . . . . . . . . . . . . . . . . . . .
13
4
Preparation cost of a product-response reconstruction
14
5
Symmetric-family calibration with explicit confidence and drift
15
5.1
Source, trusted assays and all nulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
5.2
One global finite confidence event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
5.3
Exact source-class minimax certificate . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
5.4
Actual-reader probes and a measured phase-axis bound
. . . . . . . . . . . . . . . . .
18
6
Continuum validation of a phase-grid design
18
1


## Source page 2

7
One calibration event and a finite adaptive record
19
8
General qutrit effect monitoring under a conditional drift contract
20
8.1
Operational contract and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
8.2
Nine probes and a general confidence body
. . . . . . . . . . . . . . . . . . . . . . . .
21
8.3
An exact finite reconstruction constant . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
8.4
Sharp conditional error for general asymmetric effects
. . . . . . . . . . . . . . . . . .
23
8.5
Finite-budget deployment and certificate expiry . . . . . . . . . . . . . . . . . . . . . .
25
8.6
Online calibration events and adaptive record guarantees . . . . . . . . . . . . . . . . .
25
9
Two-template monitoring and task-relative conditional-error certification
26
9.1
The same conditional-device contract, a smaller task . . . . . . . . . . . . . . . . . . .
26
9.2
One event certifies all thresholds and all inspected prefixes . . . . . . . . . . . . . . . .
27
9.3
Budget, expiry and a quantified comparison . . . . . . . . . . . . . . . . . . . . . . . .
28
9.4
What two probes can and cannot identify . . . . . . . . . . . . . . . . . . . . . . . . .
29
9.5
Adaptive monitoring and retained records . . . . . . . . . . . . . . . . . . . . . . . . .
29
10 Preservation and scientific status
30
1
Operational contracts and scope
This companion develops detector, reference and calibration interfaces within Shadow Theory’s pre-
existing source–readout programme [9].
A complete record law, a quantum output channel and
a conditional pointer statistic are different interfaces. Every section states which it covers. The
optical mode models use introduced interactions, and finite-reference costs apply to a separately
declared nondegenerate charge encoding. Established antecedents include Davies–Lewis instruments,
mode-selective conversion, Stinespring continuity, WAY/asymmetry constraints, symmetric-state
discrimination, polynomial multi-copy estimation and martingale calibration. They are credited in
the source text and bibliography [1, 3–6]. The source’s scalar event labels do not supply perspective-
admission or conscious-incorporation conditions.
1.1
Complete absorbing readers
M3: one reader, null output and retained reference.
The normalized matched-template
prescription in [7, pp. 18–21] and [8, Section 32, pp. 42–43] informs the template choice; the absorbing
dynamics and complete instrument are specified here. Let |r⟩be a normalized template in a finite
source space and Π = |r⟩⟨r|.
Fix a duration T ≥0 in seconds and a rate κ ≥0 in s−1, and
set the dimensionless α = κT, η = 1 −e−α. Adjoin an orthogonal absorbing flag f, choose zero
interaction-picture Hamiltonian, and a fixed jump J = √κ|f⟩⟨r|. The source-input instrument has
K0 = (I −Π) + e−α/2Π,
K1 = √η|f⟩⟨r|,
K†
0K0 + K†
1K1 = I.
(1)
Thus
Eclick = ηΠ,
pclick = (1 −e−α) tr(Πρ),
(2)
which differs from 1 −e−αc. For arbitrary inert reference R and joint input ρSR,
I1(ρSR) = η|f⟩⟨f| ⊗⟨r|ρSR|r⟩,
(3)
I0(ρSR) = (K0 ⊗IR)ρSR(K†
0 ⊗IR).
(4)
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## Source page 3

The inert reference is not traced out: its correct conditional assemblage is retained. The measurement
back-action can nevertheless change or destroy original source–reference correlations; retention is not
preservation of the input joint state.
Proof and time record.
The no-jump propagator solves  ̇K0(t) = −(κ/2)ΠK0(t). A jump at t
has operator density Lt = √κe−κt/2|f⟩⟨r|. Integrating LtρL†
t dt over [0, T] yields the click branch; an
absorbing flag prevents a second click. Completeness follows by integration or Eq. (1). Tensoring
Kraus operators with IR proves the reference formula. Retaining exact click time keeps LtρL†
t dt as a
separate classical record. Time bins integrate over each declared bin and retain its label. Coarsening
that record is an explicit forgetting operation.
□
M4: finite bank and tight simplification.
For normalized templates rj, fixed rates κj ≥0,
orthogonal absorbing flags fj and S = P
j κj|rj⟩⟨rj|, choose Jj = √κj|fj⟩⟨rj|. Then
K0(T) = e−ST/2,
Lj,t = √κj|fj⟩⟨rj|e−St/2,
(5)
E0(T) = e−ST ,
Ej(T) =
Z T
0
e−St/2κj|rj⟩⟨rj|e−St/2 dt.
(6)
Each coarse click map integrates Lj,tρL†
j,t and is CP even without commutation. Differentiating e−St
proves E0 + P
j Ej = I. In general the effects are not independent saturated projectors.
For P
j wj|rj⟩⟨rj| = I and κj = κwj, the exact instrument is
K0 = e−α/2I,
Kj = √ηwj|fj⟩⟨rj|,
Ej = ηwj|rj⟩⟨rj|,
E0 = e−αI.
(7)
The null probability is state independent and recorded. There is no separate irreversible loss sink in
this bank model. Tensoring the operators with IR retains any inert reference. Conditional click time
has density κe−κt/η, independent of source in this tight case; this need not hold for a non-tight bank.
Physical contract and experimental antecedents.
This ordinary Markovian one-excitation
model assumes preparable modes, fixed calibrated couplings, initially unexcited absorber/bath, a
specified interaction picture and clock, persistent orthogonal flag records, and retained nominated
source/controller and null/loss records.
These are introduced physical assumptions.
An optical
Hamiltonian alternative, Hint = ħg(arb† + a†
rb) with ar = P
x r∗
xax and vacuum output b, gives
Econverted = sin2(gT)Π on the one-photon sector. Coherent conversion and irreversible exponential
absorption are different controlled models.
Ansari et al. [2], Sec. II Eqs. (1)–(7), supply an actual shaped-pump conversion Hamiltonian
and measurement operator P
k ηk|k⟩⟨k|. Serino et al. [10], Secs. II–III and V–VI, demonstrate a
five-dimensional multi-output decoder and six MUB settings. This is a calibration platform, not
evidence for a microtubule or ontological identification. No new fundamental dynamics is proposed.
The instrument here declares registered classical time/label outputs and the residual source and
nominated inert reference. It does not identify the microscopic coherent bath outputs of two different
implementations. A stronger retained-output comparison must include the bath in both laws and is
not established here.
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2
Complete retained-output comparisons and microscopic detector
closure
2.1
A complete one-transition model and a reference-uniform certificate
Fix a finite input Hilbert space H0, horizon T, and common terminal output spaces. All changes of
basis used for comparison are declared in advance; an arbitrary environment unitary chosen afterwards
would change the meaning of retaining the physical output. For detector X = 0, 1, let HX(t) = HX(t)∗
and a jump-amplitude map BX(t) : H0 →O be bounded, measurable coefficients. The post-transition
propagation and every retained mark are included in BX; this propagation must be an isometry, or its
lost channels must be included. Set
QX(t) = BX(t)∗BX(t),
GX(t) = −iHX(t)/ħ−QX(t)/2,
 ̇NX(t) = GX(t)NX(t),
NX(0) = I.
Here HX is in joules, QX and GX in s−1, BX in s−1/2, and NX is dimensionless. This is an absorbing,
one-transition model: there are no later transitions back into the untriggered sector. A finite resonance
or continuum response may be used, but its amplitude map must actually be supplied.
Define the complete amplitude isometry
VXψ = NX(T)ψ ⊕
t 7→BX(t)NX(t)ψ

in
H0 ⊕L2([0, T]; O).
(8)
The first summand is the retained null/no-transition outcome, not a discarded population. The
corresponding time-resolved instrument has Kraus density KX(t) = BX(t)NX(t) and null operator
KX(∅) = NX(T). It may retain the output in O, coarsen time into bins, or apply a fixed common
decoder. Its normalized law is the ordinary quantum law of this specified instrument. It is not a
derivation of that law from a source theory. For a registered exact time, cq outputs are integrable
trace-class conditional kernels over the null atom and time interval, with integrated trace norm. This
classical output space is distinct from the coherent temporal L2 field in (8); no trace-class diagonal
density operator on that field is assumed. Coarsening to finite time bins gives ordinary cq blocks.
Proposition 2.1 (Full-output amplitude and kernel mismatch certificate). Let
e(t) =
Z t
0
∥(G1(s) −G0(s))N0(s)∥ds,
A2
T = e(T)2 +
Z T
0
(∥(B1(t) −B0(t))N0(t)∥+ ∥B1(t)∥e(t))2 dt.
(9)
Then V ∗
XVX = I and ∥V1 −V0∥≤AT . For every input state on H0 ⊗R, with arbitrary finite inert
reference R,
1
2∥(V1 ⊗IR)ρ(V ∗
1 ⊗IR) −(V0 ⊗IR)ρ(V ∗
0 ⊗IR)∥1 ≤min{1, AT }.
(10)
The same bound applies to the complete time/null/mark cq instrument after a common readout. With
a fixed input-independent original stock this is also a half-diamond upper bound for the two declared
output channels.
Proof. G∗
X + GX = −QX gives
d
dt(N∗
XNX) = −N∗
XQXNX.
Integration proves NX(T)∗NX(T) +
R KX(t)∗KX(t)dt = I, hence isometry and conservation of the null
population. The two-parameter propagator generated by G1 is a contraction. Variation of constants
gives
N1(t) −N0(t) =
Z t
0
N1(t, s)(G1(s) −G0(s))N0(s)ds,
∥N1(t) −N0(t)∥≤e(t).
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Since
K1 −K0 = (B1 −B0)N0 + B1(N1 −N0),
the direct-sum norm in (8), followed by the triangle inequality, proves (9). For normalized pure inputs,
the trace distance of two output vectors is at most their vector distance. Tensoring an operator with IR
does not change its operator norm; mixtures follow by convexity or purification and partial trace. Any
common time measurement, mark decoder or CPTP readout contracts trace distance. This proves the
reference-uniform statement without dividing by a branch probability or charging a reference dimension.
The fixed-stock condition supplies actual channels on an unknown input; arbitrary input-correlated
preparations supply only their stated statewise comparisons.
This certificate includes a term absent from a rate-only comparison: (B1 −B0)N0. Even Q1 = Q0
does not make it vanish. Conversely, matching the declared complex amplitude maps, null dynamics
and terminal propagation makes the full output equal. Under exact H, Q matching, the computable
cross-Gram operator
S = V ∗
0 V1 = N(T)∗N(T) +
Z T
0
N(t)∗B0(t)∗B1(t)N(t)dt
(11)
retains the omitted phase and mark data. For the coherent complete channels their exact half-diamond
distance is
q
1 −ν(S)2,
ν(S) =
min
ρ≥0, Tr ρ=1 | Tr(ρS)|.
(12)
Indeed the overlap of the two pure outputs of any reference-extended input is Tr(ρS), where ρ is its
input marginal; every such ρ has a finite purification. Mixed inputs cannot improve the optimum, by
purification and contraction. Equation (12) is the standard numerical-range description of isometric-
channel discrimination, here made explicit for the proposed detector lift. Time measurement can reduce
this distance, so a time-record comparison cannot be substituted for it.
2.2
Exact scalar kernel formula: what intensity matching pays for
For a prepared, untriggered scalar detector write
ΛX(t) =
Z t
0
ΓX(s)ds,
pX(∅) = e−ΛX(T),
pX(t) = ΓX(t)e−ΛX(t).
Let |bX(t)⟩be the normalized retained quantum mark following a transition at time t; assume the null
ray is common. On the common time/null space the positive output kernel is
σX(∅) = pX(∅)|0⟩⟨0|,
σX(t) = pX(t)|bX(t)⟩⟨bX(t)|.
The mark may include outgoing electron energy, polarization, vibrational state, source recoil and other
specified outputs; it is not merely a rendered color. Put c(t) = |⟨b0(t)|b1(t)⟩|.
Proposition 2.2 (Exact full time/mark discrepancy). The retained cq half trace distance is exactly
Dcq = 1
2|p1(∅) −p0(∅)|
+ 1
2
Z T
0
q
[p1(t) −p0(t)]2 + 4p1(t)p0(t)[1 −c(t)2] dt.
(13)
For equal rates it reduces to
Dcq =
Z T
0
Γ(t)e−Λ(t)q
1 −c(t)2 dt.
(14)
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For unequal rates one has the constructive calibration budget
Dcq ≤TV(P1, P0) +
Z T
0
min(p1(t), p0(t))
q
1 −c(t)2 dt,
(15)
where PX is the complete time-plus-null probability law.
Proof. For unit vectors u, v and nonnegative p, q, the nonzero eigenvalues of p|u⟩⟨u|−q|v⟩⟨v| lie in their
span, have sum p−q and product −pq(1−|⟨u, v⟩|2). Their absolute sum is
p
(p −q)2 + 4pq(1 −|⟨u, v⟩|2).
Integrate this identity on the time/null space. For the last bound split off |p−q| times the larger-weight
ray and compare the remaining equal weights. Common unchanged output factors, including a product
inert reference in this scalar-input comparison, may be tensored on without altering the result. For
input-correlated quantum outputs the corresponding full mark states must instead be included explicitly
in the compared kernels.
Thus exact intensity/rate matching really does fix all waiting-time statistics in this one-transition
model, not only one terminal probability. Its extension cost is determined by output-state overlaps.
For constant Γ, orthogonal marks give
Dbit = 0,
Dcq = 1 −e−ΓT .
At ΓT = 1, this is 0.6321205588. The reduced detector equivalence in [7, pp. 10–15] concerns a
smaller output interface. The calculation here quantifies the additional condition required for complete
retained-output comparison.
2.3
A coherent-field discriminator despite identical count records
A time-resolved measurement has already discarded coherence between different emission times.
Retaining the unmeasured complete field is a stronger promise. A declared, broadband absorbing
jump coupling can carry a time-dependent phase while leaving its reduced dissipator unchanged. With
constant rate Γ > 0, compare the two normalized complete vectors
|ΨX⟩= e−ΓT/2|0, vac⟩+
Z T
0
√
Γe−Γt/2eiθX(t)|1, 1t⟩dt.
(16)
Here |1t⟩denotes the continuum single-quantum temporal-mode basis; the integral is an L2 wavepacket,
not a distribution-valued state evaluated at a point. This is a signal-marker dilation of the abstract
absorbing detector, not a microscopic model of photoelectron emission or a microtubule. The symbols
0, 1 label ready/triggered states, not their energy ordering. An explicit Markov dilation uses the
coupling LX(t) =
√
ΓeiθX(t)|1⟩⟨0| with vacuum input; once triggered it cannot jump again. The phase
schedule and its source are externally prescribed resources. An autonomous realization would have to
include and retain that controller and clock and reprove the comparison.
For θ1(t) −θ0(t) = ∆ωt,
zT = ⟨Ψ0|Ψ1⟩= e−ΓT +
Γ
Γ −i∆ω[1 −e−(Γ−i∆ω)T ],
Dcoherent =
q
1 −|zT |2.
(17)
The reduced detector evolution and the entire count-time distribution agree exactly. Nevertheless, at
T = 1/Γ, ∆ω = 2Γ, Dcoherent = 0.5551876958. In the long-time limit,
Dcoherent =
|∆ω|
√
Γ2 + ∆ω2 .
(18)
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Consequently a claimed complete coherent-output error at most ε < 1 requires, for this family,
|∆ω| ≤Γε/
p
1 −ε2.
(19)
For one percent error the permissible detuning is approximately 0.0100005Γ. This is a usable tolerance,
not a proposed new physical law. Interference of the outgoing temporal mode with a calibrated reference
mode, or a phase-sensitive field measurement, can probe the missing phase; ideal equal-prior Helstrom
discrimination succeeds with probability (1+Dcoherent)/2. That optimal measurement is an in-principle
statement. Its apparatus is not included for free in the detector model. Counting and coherent-field
measurements are alternative output experiments; one cannot first measure an exact click time and
then claim its erased temporal coherence was retained.
2.4
When is the binary reduction microscopically closed?
Let the true finite detector have an untriggered subspace P0H, a triggered subspace P1H, a block-
preserving Hamiltonian, and absorbing jumps Ja = P1JaP0.
Define Q = P
a J∗
aJa.
The binary
population readout is C(ρ) = (Tr P0ρ, Tr P1ρ).
Proposition 2.3 (Exact closure and finite spectral repair). For a time-independent generator, the
binary populations obey the same absorbing scalar rate law  ̇p0 = −Γp0 for every microscopic input if
and only if
Q = ΓP0.
(20)
This statement concerns populations; fixed output marks and a two-dimensional quantum-coherence law
require additional assumptions.
If instead Q = Pk
i=1 γi|i⟩⟨i| on P0H, [H, Q] = 0, consider the declared controlled-rate model whose
complete jump preserves the label and is
J =
k
X
i=1
√γi |1, i⟩⟨0, i|
(21)
A k-label register is an exact finite repair of this specified model, including label coherence and its inert
purifying reference; additional output marks or phases, if present, must be retained in the repaired jump
as well. The register must be prepared once as part of the original stock, not redrawn after each use.
Proof. The dual generator gives L∗(P0) = −Q and L∗(P1) = Q. Equality of the scalar population
derivative for all states forces Q = ΓP0 by polarization; this condition also makes the two-dimensional
population observable space invariant under the dual generator, proving sufficiency for all times.
Under the second hypotheses, each label has no-jump amplitude e−γit/2, with its specified commuting
Hamiltonian phase, and jump amplitude √γie−γit/2. Equation (21) and its common no-jump dynamics
give exactly these amplitudes, also when the label is entangled with an inert reference. Full comparison
still retains any additional microscopic channels; none are deleted by naming a label.
For a diagonal original label stock with probabilities wi > 0, survival is S(t) = P
i wie−γit, not
generally an exponential. Its no-transition posterior is wi(t) = wie−γit/S(t). Therefore a reset wave or a
reset binary flag does not reset the actual label stock. An arbitrarily prescribed time-dependent hazard
−S′(t)/S(t) can imitate the survival of this one specified mixture; it does not establish preparation-
independent microscopic closure, restore the retained label correlations, or license independent successive
samples.
For k distinct rates, the number of labels in a time-homogeneous linear classical realization cannot
in general be reduced below k. Indeed its k × k derivative Hankel matrix is
Hab = (−1)a+bS(a+b)(0) =
X
i
wiγa+b
i
,
0 ≤a, b < k,
det H = (
Y
i
wi)
Y
i<j
(γi −γj)2 > 0.
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## Source page 8

Any n-state time-homogeneous linear realization S(t) = c∗eAtb has Hankel rank at most n, by factoring
each derivative as c∗AaAbb. Thus n ≥k. This is a standard realization-rank argument applied
constructively here, not a lower bound on every possible quantum or time-dependent surrogate.
An approximate repair can cluster rates while retaining their original label. For a rate γ replaced
by κ > 0, the exact affinity of its complete time/null law is
aT (γ, κ) = 2√γκ
γ + κ +

1 −2√γκ
γ + κ

e−(γ+κ)T/2.
(22)
This follows by adding the null overlap e−(γ+κ)T/2 to the integral of √γκe−(γ+κ)t/2. Cauchy–Schwarz
applied to |p −q| = |√p −√q|(√p + √q) gives TV(Pγ, Pκ) ≤
q
1 −a2
T . The same bound holds for their
coherent positive-amplitude wavepackets by the pure-state identity. With original label weights wi the
cq cost is at most P
i wi
q
1 −aT (γi, κg(i))2. For an unknown label distribution use the maximum. More
strongly, for the fully specified controlled model with the same retained labels, Hamiltonian phases
and marks on both sides, the coherent cross-Gram matrix is diagonal with entries aT (γi, κg(i)) > 0.
Equation (12) then gives the exact reference-uniform coherent-channel distance
max
i
q
1 −aT (γi, κg(i))2.
This bound admits label superpositions and arbitrary inert references; it does not replace their
amplitudes by a classical mixture. Different retained marks or phase schedules must instead be charged
with the full amplitude certificate. Thus a finite approximate repair has an explicit worst-case tolerance,
beyond a match to one mean rate.
3
Finite charge references and complete heralded phase measurement
3.1
Physical encoding and admitted experiment
Let the source S have basis |x⟩, x = 0, . . . , N −1, N ≥2, and dimensionless charge QS = P
x x|x⟩⟨x|.
For an energy implementation take HS = ħωQS, with ω > 0 in radians per second. A fixed, known
chirp setting a defines the orthonormal basis
|a, b⟩=
1
√
N
N−1
X
x=0
exp[2πi(ax2 + bx)/N]|x⟩,
b = 0, . . . , N −1.
(23)
No primality assumption is required for this single basis. Its diagonal quadratic mask commutes with
QS; the noncommuting resource is the Fourier mixing/readout, not the diagonal phase notation.
The active reference C has integer charges n and a fixed prepared state τC, independent of the
unknown source and its arbitrary inert reference R. At most M charge sectors are populated. The
ideal pure reference is
|uM⟩= M−1/2
M−1
X
n=0
|n⟩.
Admissible readout effects on SC commute with Q = QS + QC. All other accessible apparatus states
are invariant under the same U(1) symmetry and pointer records carry zero charge. Equivalently one
may use charge-preserving joint unitaries with invariant ancillas and a neutral pointer satisfying the
Yanase condition. A neutral-pointer Naimark dilation realizes every invariant POVM used here.
This is an explicit conditional resource model. It applies, for example, to coherent mixing of
equally spaced energy modes when no additional phase-bearing drive is admitted. It does not impose
a phase-reference cost on arbitrary degenerate spatial modes: if QS is proportional to I, their basis
8


## Source page 9

measurement commutes with charge and can be exact with no such token. An accessible macroscopic
oscillator, pump, hidden phase controller or purifying asymmetry register must be added to C; its
resource cannot be silently treated as free. A clock and fixed setting record are common to the
compared experiments; their allowed couplings must obey the declared symmetry. No speed, bandwidth
or interaction-strength bound is inferred from this resource count.
The task is an exact state-independent heralded measurement: outcomes b have induced effects
η|a, b⟩⟨a, b|, and the retained null effect is (1 −η)I. The theorem below optimizes η in this class, not
every possible approximate measurement with errors.
3.2
Optimal exact heralding from finite reference support
Sharp support/loss theorem.
For any reference supported on at most M integer charge sectors,
the largest possible exact heralding probability is
ηmax(N, M) = (M −N + 1)+
M
.
(24)
A uniform pure reference on M consecutive charges attains it. For M ≥N, equality among pure
reference states requires this uniform probability profile on a consecutive interval, up to charge
translation and arbitrary known phases compensated by the readout. For M < N, every reference has
zero exact success; no uniqueness assertion applies.
Proof.
For a pure reference |r⟩= P
n rn|n⟩set pn = |rn|2, with missing charges assigned pn = 0. An
invariant measurement sees independently the total-charge blocks
|vbq⟩=
1
√
N
N−1
X
x=0
rq−xeN(ax2 + bx)|x, q −x⟩,
eN(t) = e2πit/N.
Twirling is only a device for calculating invariant record probabilities; it does not authorize discarding
an actually retained coherent charge archive.
Within a block containing all N terms the reciprocal vectors
|wbq⟩=
1
√
N
X
x
eN(ax2 + bx)
r∗
q−x
|x, q −x⟩
satisfy ⟨wbq|vb′q⟩= δbb′. Averaging a measurement over the cyclic relabelings b preserves its mean success,
invariance and zero-error property. Every averaged zero-error effect therefore has form tq|wbq⟩⟨wbq|.
Its normalization requires tq
P
x p−1
q−x|x, q −x⟩⟨x, q −x| ≤I, so tq ≤minx pq−x, which is attainable. A
block with fewer than N terms cannot give positive symmetric zero-error success: otherwise all N
vectors would be linearly independent, since applying their zero-error effects to a linear relation isolates
each coefficient. There are fewer than N available dimensions. Thus the exact optimal average success
is
η(r) =
X
q
min
0≤x<N pq−x.
(25)
An exact state-independent heralded measurement must in particular discriminate this basis with mean
success η, so this is an upper bound for the nominated task.
For a finite subset A of integer charges, the number of contained length-N consecutive windows is
P
runs(|run| −N + 1)+ ≤(|A| −N + 1)+. Apply the layer-cake identity to At = {n : pn > t}:
X
q
min
x pq−x =
Z ∞
0
#{length-N windows in At} dt ≤(M −N + 1)+
M
Z ∞
0
|At| dt = ηmax.
9


## Source page 10

For M ≥N, the inequality is strict at every nonempty superlevel set of size below M, and at a
nonconsecutive set of size M. Equality therefore forces the stated uniform consecutive profile. Uniform
pn = 1/M has M −N + 1 complete windows and attains the bound. Mixed references cannot improve
it: decompose the reference into pure states within the same charge support. Zero-error probabilities
are nonnegative, so a zero-error mixture measurement is zero-error on each component, and its success
is bounded by the same number on every component. An invariant ancillary state decomposes into
pure fixed-charge states. On each component its addition merely shifts the reference charge labels, with
degeneracy vectors orthogonal between sectors; the same bound applies. Such ancillas therefore do
not improve it. The construction below attains the full state-independent effects, not just basis-state
average success.
□
For any allowed loss ceiling 0 < ε < 1, exact heralding requires and permits
M ≥
N −1
ε

.
(26)
At the hypothetical N = 180 energy-mode encoding, one-percent null probability requires at least
17,900 populated reference charges. This is a conditional resource requirement for the declared charge
encoding; angular divisions alone do not specify physical reference states. An attaining reference on
charges 0, . . . , M −1 has energy span (M −1)ħω and mean energy (M −1)ħω/2. No numerical energy
is inferred without a physical ω.
3.3
Full retained instrument: coherent versus classical charge record
Take M ≥N, K = M −N + 1, η = K/M, and I = {N −1, . . . , M −1}. Define normalized vectors
within total-charge sectors
|φbq⟩=
1
√
N
X
x
eN(ax2 + bx)|x, q −x⟩,
q ∈I.
The mutually orthogonal invariant projectors Fb = P
q∈I |φbq⟩⟨φbq| and F? = I −P
b Fb form a projective
measurement on SC. Outside the occupied input sector the identity may be extended arbitrarily
without affecting this experiment.
Retain a charge archive A by the invariant coherent copying isometry CQ|ξq⟩= |ξq⟩|q⟩A. Its basis
states are neutral; this records q without changing the charge of SC. Prepare C in |uM⟩and let
Jψ = ψ ⊗uM. The full operators from the logical input S to SCA are
Kb = CQFbJ = √η|χb⟩⟨a, b|,
|χb⟩= K−1/2 X
q∈I
|φbq⟩|q⟩A,
K? = CQF?J.
(27)
Direct substitution gives
K†
?K? = (1 −η)I,
X
b
K†
bKb + K†
?K? = I.
(28)
Consequently, for every joint input ρSR,
Ib(ρSR) = η|χb⟩⟨χb| ⊗⟨a, b|ρSR|a, b⟩,
I?(ρSR) = (K? ⊗IR)ρSR(K†
? ⊗IR).
(29)
This retains the active phase reference, source residual, coherent charge archive, neutral outcome
pointer and arbitrary inert reference. The normalized null operator is an isometry into the joint output.
It preserves the logical source information there; it does not restore the initial hardware source or
reference state, provide a free symmetry-preserving inverse, or make the reference reusable.
10


## Source page 11

A genuinely classical charge record is a different instrument: apply charge dephasing to A and
retain the value q. Then the conclusive output state becomes the fixed mixed state
σb = K−1 X
q∈I
|φbq⟩⟨φbq| ⊗|q⟩⟨q|A,
still multiplying η⟨a, b|ρSR|a, b⟩. The null channel is now a sum of charge-resolved Kraus branches and
is not generally an isometry. Indeed their K†
?,qK?,q are different partial diagonal projectors divided by
M; the Knill–Laflamme scalar condition fails. Thus the coherent-null conclusion is not applied to a
classically measured charge archive.
Exact complete-output comparison.
Declare the completed ideal basis-measurement channel J
to have output state |χb⟩⟨χb| in outcome b and zero null branch. For the classical-charge version use σb
instead. These are specified completions of the ideal readout, not a microscopic identification with an
arbitrary detector. The actual channel I has exactly
1
2∥I −J ∥⋄= 1 −η = N −1
M
.
(30)
In fact every normalized input, including every inert-reference extension, attains this distance.
Proof.
Each actual conclusive branch is precisely η times its ideal branch on every ρSR by Eq. (29).
The conclusive difference is negative with total trace 1 −η. The orthogonal actual null block is positive
with the same trace. Adding the block trace norms gives 2(1 −η). The same argument survives
classical charge recording, since the conclusive factorization and null trace remain exact. This is an
exact full-output identity, not a scalar probability bound substituted for an instrument error.
□
The isometry P
y Fy ⊗|y⟩with a neutral outcome pointer commutes with total charge and extends
to a blockwise charge-preserving unitary. The coherent charge-copy map has the same property.
Hence the model has a finite unitary realization; it has not hidden a noninvariant Fourier gate in the
apparatus. A common fixed readout of the outcome pointer supplies the declared instrument. Reading
additional microscopic pointer/environment degrees of freedom requires declaring their common states
and couplings as well; no universal detector equivalence follows.
The finite resource ledger of this explicit dilation is: source dimension N, active reference dimension
M, neutral charge-archive dimension N + M −1, and neutral outcome-pointer dimension N + 1, plus
the nominated fixed controller/clock and arbitrary inert R. A coherent charge archive is quantum
memory; a classical one still retains its entire record. The optimum in Eq. (24) counts populated
phase-reference charge sectors only. It is not a bound on total apparatus or complete-archive dimension,
and it does not identify a single-use Choi rank with retained history capacity.
3.4
Reference dephasing: exact error–erasure boundary
The exact-success boundary is fragile under a specified physical noise model. Let the accessible reference
state be
τν = (1 −ν)|uM⟩⟨uM| + νIM/M,
0 ≤ν ≤1.
(31)
This is partial dephasing in charge, not an unspecified decoherence rate. Its environment can be
included and retained as a fixed passive purification, but is unavailable to the nominated decoder
for measurement, conditioning, postselection or feedback. Retaining that quantum output does not
grant this access. If a later reader can use a classical noise flag or measure the purification, that is
extra accessible side information or a phase resource and the optimization must be recomputed. The
following frontier is not asserted optimal for that enlarged complete-archive readout.
11


## Source page 12

Give the N basis letters equal priors and retain every result, including nulls and charge records.
For total charge q let
Xq = {x : 0 ≤x < N, 0 ≤q −x < M},
mq = |Xq|,
wq = mq
NM ,
cq = ν + (1 −ν)mq
N
.
The weights wq sum to one. The subnormalized input block is
ρbq = (1 −ν)|ubq⟩⟨ubq| + νImq
NM
,
|ubq⟩=
X
x∈Xq
eN(ax2 + bx)|x, q −x⟩.
(32)
Its prior average is Imq/(NM).
Exact fixed-token frontier.
For a specified conclusive probability P ∈[0, 1], the maximum
unconditional correct-letter probability is exactly
Cmax(P) =
max
0≤tq≤1
P
q wqtq=P
X
q
wqcqtq.
(33)
It is obtained by accepting the highest mq blocks first, with fractional acceptance of the final tied block
if needed. The minimum conditional label error is 1 −Cmax(P)/P for P > 0. This is globally optimal
over all invariant POVMs at this fixed reference state; it is not claimed optimal over all reference
profiles at intermediate P.
Proof with a full POVM upper certificate.
For arbitrary conclusive effects Ebq with P
b Ebq ≤I,
the largest eigenvalue of ρbq is [ν + (1 −ν)mq]/(NM). Thus correct probability in that block is at most
cq times its conclusive probability. The latter is at most wq because the prior average is Imq/(NM).
This proves the linear-program upper bound without assuming a covariant candidate.
It is attained by the invariant effects
Ebq = tq
N |ubq⟩⟨ubq|,
X
b
Ebq = tqImq,
and the retained complementary null effect. Each effect occupies a maximal-eigenvalue direction of its
matching input block. The linear objective therefore sorts the cq. For a dual certificate at multiplier λ,
choose Yq = max(0, cq −λ)Imq/(NM). It obeys Yq ≥ρbq/N −λImq/(NM) for all b, and its trace sum
equals the attained Lagrange objective. This independently certifies the boundary against arbitrary
effects.
□
For M ≥N, there are M −N +1 blocks of size N, total weight η0 = (M −N +1)/M; every smaller
size m = 1, . . . , N −1 occurs twice, weight 2m/(NM). Therefore for 0 < P ≤η0 the sharp error floor
is
1 −Cmax(P)/P = ν(1 −1/N).
(34)
Beyond η0, the additional slopes are [ν + (1 −ν)m]/N for m = N −1, . . . , 1.
This is a closed
piecewise-linear correct-probability boundary.
At the interior-block operating point P = η0, the effects on every unknown logical input, not only
the basis test states, are
Eb = η0[(1 −ν)|a, b⟩⟨a, b| + νI/N],
E? = (1 −η0)I.
Thus for the declared classical pointer channel, retaining an arbitrary inert reference, the exact half-
diamond discrepancy from the pure-token heralded pointer channel is η0ν(1 −1/N). It is a uniform
12


## Source page 13

randomization of the ideal letter with probability ν; the maximum column total variation of that
classical postprocessing is 1 −1/N, attained by a basis input. This pointer certificate is not a full
active-token output bound.
Any ν > 0 makes exact zero-error heralding impossible at every positive conclusive probability: all
letter states have the same full support within each block. More generally, any strictly positive reference
density matrix on its populated charge space has the same obstruction: every relevant principal
submatrix is strictly positive, so each letter’s total-charge block has the same full support. This is an
exact-zero-error obstruction, not a claim that a small noise produces a large ordinary statistical error.
The nearby faithful repair is not to discard the reader but to keep the complete null/error archive
and use the exact controlled error floor. For example N = 5, M = 9, ν = 0.03 permits P = 5/9 with
minimum conditional error 0.024; no lower-confidence filtering can beat that value. At all-conclusive
readout,
Cmax(1) = (1 −ν)
"
1 −N2 −1
3NM
#
+ ν
N ,
M ≥N.
All these formulas concern declared label statistics. A full noisy retained-output comparison additionally
uses the explicit Kraus maps and retained purification; it is not inferred from label error alone. The
effects and their positive square roots already give such a normalized finite instrument, tensored with
IR for an arbitrary inert reference.
3.5
Different objective: best no-erasure reference
Exact finite eigenvalue optimization.
For a pure reference on charges 0, . . . , M −1 with amplitude
magnitudes rn ≥0, the maximum equal-prior correct-letter probability with no inconclusive output is
Pcorrect(r) =
1
N2
X
q
 N−1
X
x=0
rq−x
!2
= rTTN,Mr
N2
,
(TN,M)nm = (N −|n −m|)+.
(35)
Consequently the best reference has P opt
correct = λmax(TN,M)/N2, with its nonnegative principal eigen-
vector as amplitude profile.
Proof.
Within each total-charge block, cyclic symmetrization makes the seed effect have diagonal
entries 1/N and hence off-diagonal magnitudes at most 1/N by positivity. This upper-bounds correct
probability by the first expression, attained by a phase-aligned rank-one seed. Summing q counts
N −|n −m| pairs when |n −m| < N, proving the matrix formula. The variational principle and Perron
positivity give the optimum. Mixed references cannot improve a maximization over all pure states and
measurements. Compressing any nonconsecutive set of M integer charges decreases every pairwise
distance, hence increases this nonnegative kernel entrywise; consecutive charges are sufficient for the
global optimum at that cardinality.
□
At N = 2 this gives the known sine-state result
rn =
s
2
M + 1 sin (n + 1)π
M + 1 ,
P opt
error = 1 −cos[π/(M + 1)]
2
.
For the binary classical outcome channel, this is also the exact minimum half-diamond error to the ideal
basis measurement, uniform over inert references: the achieved effects are (I ± vX)/2 in target-basis
coordinates after the known diagonal chirp mask, with v = cos[π/(M +1)], and any channel comparison
is lower-bounded by its equal-prior basis discrimination error. This assertion does not compare retained
active-token outputs. The quadratic M−2 label-error scaling and the M−1 exact-heralding loss law
answer different questions.
13


## Source page 14

4
Preparation cost of a product-response reconstruction
Take a finite mode register or a finite spatial bin, a normalized state ρ, a dimensionless intensity effect
0 ≤A ≤I, and the normalized template projector B = |R⟩⟨R|. For a pure field Ψ =
√
Nψ with fixed
known total intensity N > 0, the normalized template coherence used in [7, 8] is ⟨ψ|B|ψ⟩and the bin
intensity is a fixed intensity scale times N⟨ψ|A|ψ⟩. Absorb these fixed scales and the susceptibility
into κ ≥0 with units s−1. This gives the precise proposed quantum-state interpretation
x = Tr(Aρ),
y = Tr(Bρ),
Γ(ρ) = κxy,
psrc(ρ) = 1 −e−ηxy,
η = κ∆t.
A state-dependent total intensity or susceptibility must be treated as a further source/controller
variable; it cannot be silently absorbed into a state-independent detector. Point evaluation on an
unrestricted continuum is not a bounded effect, so the finite bin is also an explicit addition.
Proposition 4.1 (Single-copy obstruction and two-copy effect). The product Tr(Aρ) Tr(Bρ) is affine
on the full density-matrix state space if and only if at least one of A, B is scalar. In general it is not a
single-copy click effect. Nevertheless, for 0 ≤η ≤1 the two-copy effect
Ec = ηA ⊗B,
E0 = I −Ec
is positive and complete, and on independently prepared copies it gives pc = ηxy. Its difference from
the source frame rule obeys
0 ≤pc −psrc ≤1
2η2x2y2 ≤1
2η2.
Proof. Write ρ = I/d+εX for arbitrary traceless Hermitian X and sufficiently small ε. Affinity requires
Tr(AX) Tr(BX) = 0 for every X. Two nonzero linear functionals cannot have their kernels cover the
whole vector space: choose a vector off the first kernel and, if necessary, add a multiple of a vector
off the second. Thus one functional vanishes on all traceless matrices, equivalently its effect is scalar.
The converse is immediate. For an explicit obstruction even to an ensemble-independent extension of
pure-state rates, take A = B = |0⟩⟨0|. The equally weighted {|0⟩, |1⟩} and {|+⟩, |−⟩} ensembles both
prepare I/2, but their averaged pure-state product rates are 1/2 and 1/4. Positivity and completeness
of the two-copy effects follow from 0 ≤A ⊗B ≤I. The bound uses 0 ≤z −(1 −e−z) ≤z2/2 for
z ≥0.
An explicit retained-reference instrument is Ic(ω) = √Ec ω√Ec and I0(ω) = √E0 ω√E0, with the
identity on an arbitrary reference, and orthogonal recorded flags. Its probabilities on a correlated
two-register input are Tr(Ecω12); they factor as ηxy only under the declared product preparation.
This needs two preparations, not a machine cloning an unknown input.
The standard isometry
V = |c⟩√Ec + |0⟩√E0 provides a finite apparatus dilation, but a material coupling remains to be
supplied. The square roots specify a state update not determined by the scalar rate alone.
Proposition 4.2 (Exact exponential needs an unbounded stock or more input). For η > 0, no
fixed finite number m of identically prepared copies and a fixed quantum effect implements 1 −e−ηp2
for every unknown qubit pure state with p = |⟨0|ψ⟩|2. An exact protocol using a random number of
independent two-copy tests exists: sample K ∼Poisson(η) independently, perform the effect A ⊗B
on each independently prepared pair, and report a click if any test clicks, performing all K tests.
Then pc = 1 −e−ηxy and the expected number of consumed copies is 2η (early stopping can reduce
consumption but is not used in this count). If the protocol instead has at most m pairs and retains a
separate stock-loss flag when K > m, its classical report law is within Pr{K > m} in total variation of
the exact law (padded with a zero-probability loss flag). For m + 1 > η,
Pr{K > m} ≤e−η

eη
m + 1
m+1
.
14


## Source page 15

Proof. Apply a putative m-copy effect to |ψp,φ⟩= √p |0⟩+ eiφ√1 −p |1⟩. Since the desired result is
independent of φ, average the effect over the common phase. On these symmetric inputs its expectation
becomes Pm
k=0
m
k

pk(1 −p)m−kek, with fixed 0 ≤ek ≤1. This is a polynomial in p, whereas 1 −e−ηp2
is not. Extra state-independent ancillas or randomization do not change the polynomial obstruction.
For the random-stock construction the no-click probability is E(1 −xy)K = exp(−ηxy). Couple the
exact and truncated protocols using the same K and tests; their classical reports differ only on K > m.
The tail bound is the usual Poisson exponential bound, optimized at et = (m + 1)/η.
For η = 0.8, xy = 0.24 and at most six tests (twelve copies), the exact click probabil-
ity is 0.1746931315; the truncated value is 0.1746753459, with retained stock-loss probability
2.0746838 × 10−5.
The analytic tail certificate is 1.2547853 × 10−4.
Standard-library code
supplement/operational/10_verify_product_reader.py reproduces these values. This is an exact
preparation-cost interface for one interpretation of the source, not a claim that a classical driven field
needs duplicated unknown quantum states. It supplies neither fresh independence from a reset nor a
primitive Born-event selection. The comparison bound concerns the declared classical reports; it does
not identify quantum residual outputs of two incompatible apparatuses. All loss flags, controller draws,
input stock, clock and retained references remain part of the complete model.
5
Symmetric-family calibration with explicit confidence and drift
5.1
Source, trusted assays and all nulls
Use HS = span{|vac⟩, |0⟩, |1⟩}. The qubit marginal block is wρ, with
ρ = (I + tX + vY + uZ)/2,
u2 + t2 + v2 ≤1,
p = (1 + u)/2.
For hypothesis h = −, apply the stipulated qubit Z phase flip to the h = + preparation, reversing
the transverse components. The source/controller relation is an assumption. Its hidden true label is
not available to the classifier. Sequential use must also obey its declared conditional-state conditions;
calibration does not establish them. Vacuum/qubit coherence may exist, since dark-free label effects
annihilate vacuum and hence do not see that coherence.
Predetermine NZ, NX calibration trials using independently prepared copies of one stationary
source marginal. A common trusted calibrator has the same state-independent efficiency ηc > 0 in
both settings and known contrast 0 < γ ≤1:
Z :
E0 = ηc|0⟩⟨0|,
E1 = ηc|1⟩⟨1|,
(36)
X :
E± = ηc(Pq ± γX)/2,
E∅= I −ηcPq.
(37)
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.
15


## Source page 16

5.2
One global finite confidence event
For binomial count k in n trials, let [Lβ(k, n), Uβ(k, n)] be the two-sided Clopper–Pearson interval of
failure at most β. In the interior,
Lβ = B−1(β/2; k, n −k + 1),
Uβ = B−1(1 −β/2; k + 1, n −k).
Use L(0, n) = 0, U(n, n) = 1, and [0, 1] if n = 0. Define the corruption envelope
Cβ(f
K, n, kf) = [Lβ((f
K −kf)+, n), Uβ(min(n, f
K + kf), n)].
(38)
Allocate source failure αS, set β = αS/3, and take
Ip = Cβ(f
KZ, nZ, kZ),
Iq = Cβ(f
KX, nX, kX),
It = [(2IL
q −1)/γ, (2IU
q −1)/γ] ∩[−1, 1].
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],
with upper endpoint one when ηL = 0. Then
GS = {p0 ∈Ip, t0 ∈It, w0 ∈Iw},
Pr(GS) ≥1 −αS.
(39)
A statistical efficiency calibration requires its own allocated failure.
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.
5.3
Exact source-class minimax certificate
For every future actual conditional input, given the preceding record, assume
|p −p0| ≤dp,
t ≥t0 −dt,
w ≥w0 −dw.
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
I = [2(I′
p)L −1, 2(I′
p)U −1] ∩[−R, R],
R =
p
1 −l2.
(40)
If l < 0 this positive-direction theorem is not used; an empty set means model/confidence inconsistency,
not a vacuous certificate. Use the real symmetric reader
E± = 1
2
 
x
±z
±z
y
!
,
E∅= I −diag(x, y),
0 ≤x, y ≤1,
0 ≤z ≤√xy.
(41)
The reader acts as null on vacuum. Its transverse axis can differ from the calibration axis by any
|θ| ≤∆≤π/2. Only differential phase matters; global phase convention is irrelevant. The actual
coefficients must be known or calibrated. Set d = (x + y)/2, e = (x −y)/2, with A(u) = d + eu > 0 on
I.
16


## Source page 17

C2: sharp error and success.
For a = l cos ∆, b = sin ∆,
ε∗= 1
2
"
1 −z min
u∈I
a −b
√
R2 −u2
d + eu
#
,
(42)
Q∗= wL min
u∈I (d + eu).
(43)
These are the exact worst conditional error and guaranteed unconditional conclusive probability over
the stated physical source/drift/axis class. The conditional-error statement applies only to members
with positive conclusive probability. In particular, Q∗= 0 supplies no positive-rate guarantee and
assigns no value to a 0/0 conditional error. They hold for every inert-reference extension, as statements
about marginal labels, not full output-state equality.
Proof.
At fixed u, Bloch positivity gives t2 + v2 ≤1 −u2. The smallest reader component is
Xmin(u) = l cos ∆−
p
1 −u2 −l2 sin ∆.
Choose the adverse sign of v, saturate its allowed magnitude, and take t = l, |θ| = ∆: the expression
increases in t ≥0 and decreases in |θ| ≤π/2. A pure state attains it. For either hypothesis the exact
error is [1−zXreader/A(u)]/2; minimizing over u proves the first claim. Conclusive probability is wA(u),
minimized by wL and an endpoint. References do not change these marginals.
□
The finite minimization evaluates endpoints and the admissible interior roots of
b(du + eR2) = ea
p
R2 −u2.
(44)
Squaring gives
(b2d2 + e2a2)u2 + 2b2deR2u + b2e2R4 −e2a2R2 = 0.
(45)
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,
√
0.27): the worst point is u = 0.3685962371, error
0.3671399216.
C3: sharper coherence interface.
Let u∗be the point of I closest to zero, rmax =
p
1 −u2∗. For
rmax > 0, define q = l/rmax. The exact source-only combined visibility/phase lower bound is
mcal = q cos ∆−
q
1 −q2 sin ∆.
(46)
Divide Xmin(u) by
√
1 −u2. The result increases with l/
√
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.
17


## Source page 18

5.4
Actual-reader probes and a measured phase-axis bound
A held uncertain loss rate may alter a tuned amplitude ratio, so a worst scalar transfer factor is not an
actual-effect certificate. Instead freeze the source-data-selected pulse, then calibrate
E± = 1
2
 
x
±c
±c∗
y
!
,
|c|2 ≤xy.
(47)
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:
x = Pcon|0,
y = Pcon|1,
Re c = P+| + −P−|+,
Im c = P−| + i −P+| + i.
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
zL =
q
(IL
r )2 + dist(0, Ii)2,
∆D = arctan max(|IL
i |, |IU
i |)
IL
r
.
Future hardware allowances widen x, y, lower zL, and add to ∆D. They must bound actual conditional
instruments, not merely averaged dynamics. The global event is
G = GS ∩GD,
Pr(G) ≥1 −α,
α = αS + αD.
(48)
Device coverage is conditional on the preceding source-based choice. Accepting or rejecting the frozen
pulse afterward is allowed. Retuning requires fresh calibration or simultaneous catalogue coverage;
source-confidence uniformity does not calibrate every hardware setting.
When all admitted source reader components are nonnegative, use xU, yU, zL in Eq. (43) for a valid
device-box error certificate. It is sharp for that outer Cartesian coefficient class when the corner is
physical. Use xL, yL for a safe success lower bound, which need not be attained: positivity or raw-data
correlations may exclude that lower corner. No joint optimum over the original multinomial confidence
region is claimed. If the source component can be negative, this monotone device-box rule is not used.
Effects certify nominated labels on arbitrary inert-reference inputs; they do not identify the retained
field, recoil, controller or bath. The complete operational construction still supplies those maps. Their
agreement cannot be inferred from count calibration.
6
Continuum validation of a phase-grid design
Proposition 6.1 (Continuum validation of a phase-grid reader). Let ρ±(θ) be the pure states of
Uz(θ)(a|0⟩± b|1⟩), where a, b ≥0, a2 + b2 = 1, s = a2 −b2, and Uz is a rotation by angle θ about the
Bloch z axis. Let a fixed reader and bounded nonnegative decision loss give operators 0 ≤C± ≤CmaxI
and risk
R(M, θ) = 1
2 Tr[C+(M)ρ+(θ)] + 1
2 Tr[C−(M)ρ−(θ)].
18


## Source page 19

For phase differences at most π,
|R(M, θ) −R(M, θ′)| ≤Cmax
p
1 −s2 sin(|θ −θ′|/2).
If a finite grid G has covering radius h ≤π in a compact phase interval Θ, and the admissible reader
class is the same for both optimizations, then
VG ≤VΘ ≤VG + Cmax
p
1 −s2 sin(h/2),
VA := inf
M sup
θ∈A
R(M, θ).
Proof. The pure-state trace distance is exactly
D(ρ±(θ), ρ±(θ′)) =
p
1 −s2 | sin((θ −θ′)/2)|.
For trace-zero Hermitian X and 0 ≤C ≤CmaxI, | Tr(CX)| ≤Cmax∥X∥1/2 by the positive and negative
parts of X. Apply this to each state difference, average, and choose a grid point within h of each
phase. Take infima over the identical admissible reader class. If the grid infimum is not attained, use
an arbitrarily close reader and pass to its limiting risk bound.
This is a standard trace-distance metric-net argument newly specialized to this phase reader, not
a new general minimax theorem. If constraints themselves are imposed only on grid points, they
need their own modulus margin. In particular, equality of an inconclusive rate at grid points is not
proof of that equality between grid points. For an n-copy joint reader one may use the safe modulus
Cmaxn
√
1 −s2 sin(h/2), capped by Cmax, from telescoping trace distance; the one-copy formula must
not be reused unchanged.
7
One calibration event and a finite adaptive record
The calibration certificate can be used over a finite sequence without charging its failure probability at
every use. This is an application of established martingale concentration, not a new probability law or
an independence theorem. It makes the batch-calibration and robust-reader constructions compatible
at the level of the complete declared classical record.
Let F0 contain all original calibration settings, counts, nulls, chosen reader settings and admitted
calibration metadata. For the fixed true calibration parameters, let G ∈F0 be the simultaneous
confidence event with Pr(G) ≥1 −α. The drift, access and preparation conditions of the calibration
section are physical assumptions: on G, every future actual input conditional on its preceding declared
history must remain in the certified set. Neither an unchanged wave, a reset flag nor a null outcome
proves this condition for a reused source.
For use t, let Ct indicate a conclusive flag and Wt a wrong conclusive flag; hence 0 ≤Wt ≤Ct ≤1.
The source-label truth needed to score Wt must be defined in the test model and recorded in a validation
experiment; an unknown production label does not become observable from this theorem. For t ≥1,
take Ft to contain the completed-trial outcomes, scoring truths and original calibration record. The
physical controller is adapted only to its permitted accessible-record subfiltration. The conditional
contracts below are assumed on this augmented mathematical past; they do not follow merely by
enlarging an observed-record filtration. Scoring access in the proof grants the reader no access to
an unread preparation label. Let εt ∈[0, 1] and qt ∈[0, 1] be predictable, calibrated bounds for the
selected reader. On G suppose
E[Wt | Ft−1] ≤εtE[Ct | Ft−1],
E[Ct | Ft−1] ≥qt.
(49)
The constructed symmetric robust reader supplies these inequalities label by label under the calibrated
conditional-state contract. Consequently its certificate permits arbitrary, history-dependent label
19


## Source page 20

priors when both label-conditioned source marginals satisfy that contract. This does not transfer an
optimality claim from a different prior-weighted task. Arbitrary inert quantum references may be
retained. Their additional measurement or feedback is outside the contract unless the augmented
conditional input bounds have also been established.
Theorem 7.1 (Finite-use certificate with a single calibration fee). For any predetermined horizon N
and 0 < β < 1, put bN =
p
(N/2) log(2/β). With probability at least 1 −α −β, simultaneously
N
X
t=1
Wt ≤
N
X
t=1
εtCt + bN,
(50)
N
X
t=1
Ct ≥
N
X
t=1
qt −bN.
(51)
No independence of the future trials is required. With fixed εt = ε and qt = q, if Nq > bN this implies
P
t Wt
P
t Ct
≤ε +
bN
Nq −bN
.
Every null and loss event remains in the record and in the horizon cost.
Proof. On each calibration outcome in G, set Zt = Wt −εtCt. It has nonpositive conditional mean
and its conditional range is contained in [−εt, 1 −εt], of length one. The conditional Hoeffding lemma
therefore gives E[eθZt | Ft−1] ≤eθ2/8 for θ > 0. Induction and exponential Markov give
Pr
 X
t
Zt ≥b | F0
!
≤e−2b2/N.
Apply the same argument to qt −Ct, whose conditional mean is nonpositive and whose range also has
length one. Taking b = bN bounds each failure by β/2. The union bound supplies β on G; the original
calibration failure adds α once. This proof permits adaptation to all admitted preceding classical
outcomes, not retroactive selection of a better calibration confidence event.
The range-dependent lemma can be checked directly: among bounded variables with fixed mean,
convexity maximizes the moment generating function on the two endpoints. The centered endpoint
Bernoulli log-mgf has second derivative at most 1/4; integrating twice from zero bounds it by θ2/8.
This proves the concentration constant used above.
An anytime version is also available. For any fixed θ > 0, with failure at most α + β,
n
X
t=1
(Wt −εtCt) < θn
8 + log(1/β)
θ
for every n ≥1.
Indeed exp[θ P
t≤n Zt −nθ2/8] is a nonnegative supermartingale starting at one; stopping it at its first
crossing and using the expectation bound proves the displayed probability inequality. This supports an
admitted stopping policy without falsely applying a fixed-horizon bound to a data-selected horizon.
8
General qutrit effect monitoring under a conditional drift contract
8.1
Operational contract and units
The input space is H = span{|v⟩, |0⟩, |1⟩}. It may include physical vacuum. The nominated terminal
labels are {+, −, ∅}. At clock slot t, conditional on the preceding declared classical history, there is a
common linear measurement
E+(t) ≥0,
E−(t) ≥0,
E∅(t) = I −E+(t) −E−(t) ≥0.
20


## Source page 21

It is independent of the current probe’s classical preparation tag; the probe influences it through its input
quantum state only. All conditional source/controller, ready-stock, preparation and independence of the
joint input and inert reference from the ready stock conditions needed for such an effect representation
are admitted. Correlation of an unknown input with an uncontrolled device stock cannot be cured by
tomography.
The device may otherwise depend on the entire past. A known pathwise bound is required for each
nominated conclusive effect:
∥Ey(t) −Ey(s)∥∞≤ν|t −s|,
y = +, −.
(52)
Here effects and ν|t −s| are dimensionless probabilities; ν has units inverse clock slot. If an external
clock measures seconds, replace ν by the corresponding rate per second and use actual elapsed times.
Calibration uses, production uses, delays, controller changes and finite reference depletion all count
toward elapsed time. A bound for an averaged master equation does not prove Eq. (52) for actual
conditional instruments. A known deterministic drift envelope d(t, s) can replace ν|t −s| in the
confidence construction.
Every +, −, ∅, time, probe setting and deployment decision is retained. Neither missed clicks nor
accepted labels are selected away. This section certifies the declared labels on inert-reference extensions
and the explicit pointer/reference distance stated below. It does not identify the postmeasurement
source, coherent field, bath or complete quantum instrument. Those require the operational completion
and separate retained-output proofs.
8.2
Nine probes and a general confidence body
Use the three basis projectors and, for each i < j, the two projectors onto (|i⟩+|j⟩)/
√
2 and (|i⟩+i|j⟩)/
√
2.
Number these nine trusted states σj, j = 0, . . . , 8. In every trial their actual conditional preparation
has trace distance at most δprep from its nominated state, and the conditional joint input/reference is
independent of the device’s ready stock. This trusted preparation condition is not inferred from the
same unknown detector.
When |v⟩is physical vacuum and the other modes contain one particle, the four vacuum/occupied
superpositions require a phase reference and compatible preparation access. The finite-reference
resource model applies when appropriate. If number superselection and block-diagonal source/effects
are separately admitted, five probes suffice for the five-dimensional real algebra C ⊕M2; unrestricted
vacuum coherences cannot be certified with that reduced access.
Let Xy,j,k be the indicator of label y on the kth use of probe j. Its true conditional mean py,j,k
is predictable and may change with every preceding record. Define the retained empirical frequency
bpy,j,n = n−1 Pn
k=1 Xy,j,k. Suppose at most K monitoring epochs will be opened. Within each epoch
use any predictable probe schedule, with a finite total budget; our example uses round-robin cycles. Set
δ =
α
2JK ,
J = 9,
u(n) =
s
log(π2n2/(3δ))
2n
(n ≥1).
(53)
A probe with no observations gives no constraint and cannot trigger a complete certificate.
M1: finite-budget, all-prefix coverage.
With probability at least 1 −α, simultaneously for every
opened epoch, both conclusive labels, every probe, and every observed prefix n,
bpy,j,n −1
n
n
X
k=1
py,j,k
 ≤u(n).
21


## Source page 22

Consequently, at any data-selected calibration completion time T, for probe times sj,k < T, the actual
next-slot effect lies in the physical confidence body
PT = {(E+, E−) : E± ≥0, E+ + E−≤I,
(54)
| tr(Eyσj) −bpy,j,nj| ≤bj(T),
y = ±, j = 0, . . . , 8},
(55)
bj(T) = u(nj) + δprep + ν
nj
X
k
(T −sj,k).
(56)
If the body is empty, decline deployment; do not call it a vacuous success. Future time T + a may use
the same formula with every radius enlarged by νa, or the operator-ball construction below.
Proof.
For a chosen stream, X −p is a martingale difference of conditional range one. The conditional
Hoeffding lemma gives the exponential supermartingale with compensator nθ2/8, also when sampling
times are predictable and chosen from the preceding record. At a fixed observed sample index n,
optional stopping, or extension of a stopped stream, gives
Pr



X
k≤n
(Xk −pk)

> nb

≤2e−2nb2.
At b = u(n) the right side is 6δ/(π2n2), summing to δ over all n ≥1. Union over 2JK streams proves the
claim, conditional on every history at which an epoch is opened and then averaged over those histories.
No stationarity or fresh statistical independence has been used. Conditional probe-preparation error
changes an event probability by at most δprep; pathwise drift changes it by at most ν(T −sj,k). Average
these inequalities to obtain the body. The same simultaneous event permits arbitrary inspection and
stopping within the stated epoch budget.
□
The construction remains conservative if only a predetermined finite set of prefixes is inspected. If
just one fixed terminal sample size per epoch is used, the simpler width
ufixed(n) =
q
log(4JK/α)/(2n)
is sufficient; it must not be substituted after selecting a favorable sample size. Tighter established
confidence sequences may replace Eq. (53); the displayed one was chosen for a self-contained proof and
transparent implementation.
8.3
An exact finite reconstruction constant
For any Hermitian H, write its nine probe expectations as v. The inverse map R is
Hii = vi,
(57)
Re Hij = vij,R −(vi + vj)/2,
(58)
Im Hij = (vi + vj)/2 −vij,I.
(59)
This fixes the sign convention for the +i probe. The exact rectangular uncertainty radius is the finite
computation
C(b) =
max
s∈{−1,+1}9 ∥R(s0b0, . . . , s8b8)∥∞.
(60)
Convexity gives equality: the norm of every point of a box is bounded by its largest vertex value, and
the vertices are included. Thus ∥Ey(T) −R(bpy)∥∞≤C(b) on M1. This is sharp for an unconstrained
Hermitian box; enforcing the physical joint confidence body can make it smaller.
22


## Source page 23

M2: exact constant for a uniform box.
For bj = b,
C(b) = (1 + 2
√
5)b.
(61)
A short reproducible algebraic proof accompanies the code.
For every one of 512 sign vertices,
form H = R(s). Its entries are Gaussian integers. All principal minors of (1 + 2
√
5)I ± H are
nonnegative, checked in exact integer arithmetic in Z[
√
5]. For clarity, with diagonal entries di, t = tr H,
c2 = P
i<j(didj −|Hij|2), and d = det H, the determinant is
(61 −21t + c2 −d) + (46 −4t + 2c2)
√
5.
A 2 × 2 minor for i, j is
21 −di −dj + didj −|Hij|2 + (4 −2di −2dj)
√
5.
One-dimensional minors are immediate. Their signs are decided exactly by integer squares for A+B
√
5;
no floating-point tolerance is used in these 7168 checks. The vertex s = (1, 1, 1, −1, −1, −1, 1, −1, −1)
gives
H =



1
−2 + 2i
−2
−2 −2i
1
−2 + 2i
−2
−2 −2i
1


,
whose characteristic polynomial (λ −1)[(λ −1)2 −20] attains the claimed radius. The finite sign checks
plus this witness prove equality. They do not claim that every sign vertex is itself a physical effect.
Set bE± = R(bp±) and bE∅= I −bE+ −bE−. If these are not physical, define
λ = max(0, −min
y
λmin( bEy)),
Fy =
bEy + λI
1 + 3λ .
Then F is a physical center. On M1, for y = ±,
∥Ey(T + a) −Fy∥∞≤ry(T) + νa,
ry(T) = C(b) + ∥Fy −bEy∥∞.
(62)
This explicit repair need not be an optimal fit; it costs the computed shift. Using r = maxy ry + νa
gives a physical operator-ball outer class, denoted B(F, r). It is not the same class as the full body PT ,
so sharpness for that outer class is not a sharp multinomial-data claim.
8.4
Sharp conditional error for general asymmetric effects
Fix known normalized source marginals ρ+, ρ−and conditional label priors π+, π−, summing to one.
Let
 ̄ρ = π+ρ+ + π−ρ−,
WF = π+ tr(F−ρ+) + π−tr(F+ρ−),
QF = tr[(F+ + F−) ̄ρ].
Wrong outcomes remain a subset of conclusive outcomes, including dark-count labels if the input is
vacuum. Put
A+(ε) = π−ρ−−ε ̄ρ,
A−(ε) = π+ρ+ −ε ̄ρ.
M3: operational support certificate and its sharp regime.
For every E ∈B(F, r),
WE −εQE ≤B(ε) := WF −εQF + r(∥A+(ε)∥1 + ∥A−(ε)∥1),
QE ≥QF −2r.
(63)
If QF > 2r, any ε ∈[0, 1] with B(ε) ≤0 certifies conditional error at most ε. No source/excitation
postselection is used. These statements hold on every inert-reference extension having the nominated
marginals.
23


## Source page 24

Both support and success bounds are sharp over the physical operator-ball class whenever
r ≤rint := min{λmin(F+), λmin(F−), 1
2λmin(F∅)}.
(64)
If the radius exceeds this sufficient condition, the bounds remain valid; their physical attainment is no
longer asserted.
Proof.
Write E± = F± +H±. Then WE −εQE = WF −εQF + P
y tr(HyAy). The duality of operator
and trace norms bounds each term by r∥Ay∥1, with equality at Hy = r sign(Ay). Condition (64) makes
these effects positive and keeps F∅−H+ −H−≥0, so they are a real physical POVM. For success use
H+ = H−= −rI, which is also physical in that regime. The denominator is positive when QF > 2r;
the first inequality then implies the conditional-error statement. References do not alter nominated
label probabilities.
□
There is also a complete pointer/reference guarantee. For the channel ME(ρ) = P
y tr(Eyρ)|y⟩⟨y|,
1
2∥ME −MF ∥⋄≤2r.
For every joint input, each partial-trace block for a Hermitian effect perturbation H has trace norm
at most ∥H∥∞, by trace-norm duality. The two conclusive blocks and their negative sum give the
bound. In the interior regime it is sharp: choose H+ = H−= rI, with null perturbation −2rI. All
three blocks are scalar multiples of the reference marginal, giving half trace distance exactly 2r on
every input. This is a bound for the declared pointer plus an arbitrary inert reference; it makes no
statement about retained output source or bath.
For unequal known radii, replace r∥Ay∥1 by ry∥Ay∥1 and 2r by r+ + r−; the analogous margin
condition applies. The proof supplies a finite implementation V = P
y
pEy ⊗|y⟩, retaining the output
source and arbitrary inert reference. It demonstrates attainability within ordinary quantum instruments,
not a derivation of Born statistics or identity with the physically nominated detector’s full bath.
Balanced pure hypotheses: a quadratic.
Let π± = 1/2, ρ± = |ψ±⟩⟨ψ±|, and s = |⟨ψ+|ψ−⟩| ∈
[0, 1]. In their two-dimensional span,
∥A+(ε)∥1 + ∥A−(ε)∥1 =
q
1 −4ε(1 −ε)s2.
This follows from tr Ay = (1 −2ε)/2 and det Ay = −ε(1 −ε)(1 −s2)/4. A zero eigenvalue in unused
qutrit directions causes no difficulty. Thus the sharp error in the interior regime is the root in [0, 1] of
WF −εQF + r
q
1 −4ε(1 −ε)s2 = 0.
(65)
Equivalently,
(Q2
F −4r2s2)ε2 + (4r2s2 −2WF QF )ε + W 2
F −r2 = 0,
retaining the unsquared sign εQF −WF ≥0. Since QE > 0 for every member, the physical support
function is strictly decreasing in ε, and the root is the exact worst ratio. Orthogonal hypotheses give
ε∗= (WF + r)/QF ; identical hypotheses have no informative balanced-label advantage. Outside the
interior regime use the support inequality as a conservative certificate, not a minimax claim.
Source uncertainty and arbitrary priors.
The example fixes source marginals. If each actual
conditional ρ′
h has trace distance at most δS from ρh, the valid repair is B(ε) + δS ≤0 and QE ≥
QF −2r −δS. Indeed Ewrong −ε(E+ + E−) lies between −εI and (1 −ε)I; its spectral oscillation is
at most one. This term bounds source error without mistaking a measured detector for a measured
source. The combined radius/source certificate need not be sharp.
24


## Source page 25

For priors that may vary arbitrarily with history, certify each label separately. If wh = tr(F−hρh),
qh = tr[(F+ + F−)ρh] > 2r, then
εh ≤min{1, (wh + r)/qh},
Qh ≥qh −2r.
(66)
The support is wh −εqh +r, attained by Hwrong = rI, Hcorrect = −rI in the same interior regime. Using
maxh εh, minh Qh covers arbitrary label priors. The sharper balanced expression requires balanced
conditional priors; neither calibration nor a reset establishes them.
8.5
Finite-budget deployment and certificate expiry
A complete protocol is now explicit. Predetermine α, K, a maximum number of probe trials per epoch,
trusted preparations, the drift contract, target error  ̄ε, and target conclusive probability  ̄q > 0. Open
an epoch, retain all outcomes, and inspect M1 at any prefixes. Construct F, r; deploy only if B( ̄ε) ≤0
and QF −2r ≥ ̄q (or their source-uncertain/labelwise versions). Continue only while the aged radius in
Eq. (62) passes the same gate. Otherwise take more probes, open the next budgeted epoch, or stop. A
different physical pulse requires its own calibration or a separately established transformation bound;
an online choice among uncalibrated pulse settings is not licensed.
For balanced pure inputs the largest admitted error radius at a chosen threshold is
rerr =
 ̄εQF −WF
p
1 −4 ̄ε(1 − ̄ε)s2 ,
(67)
when numerator and denominator are positive. Also impose r ≤(QF − ̄q)/2. The allowed monitoring
gap follows by subtracting r(T) and dividing by ν > 0; if sharpness is desired rather than just a valid
certificate, additionally impose r ≤rint. This is an operational expiry time, not a detector change-point
localization theorem.
There is a real statistical/drift tradeoff even before deployment. For n complete round-robin cycles
with probe slots j + 9k, j = 0, . . . , 8, and next-slot anchor T = 9n, the largest mean age is (9n + 9)/2.
A transparent sufficient radius is
runif(n) = (1 + 2
√
5)[u(n) + δprep + ν(9n + 9)/2].
(68)
For a fixed terminal sample size set u(n) = A/√n, A =
p
log(4JK/α)/2. The continuous optimum of
this displayed design bound is
nopt = (A/(9ν))2/3
(ν > 0).
Check neighboring integers and the declared finite budget. Thus blindly collecting older data eventually
worsens a current detector certificate. This is an exact optimum of the stated bound, not of every
tomography schedule or confidence method. For the anytime width the optimum is an elementary
one-dimensional minimization, provided in the code.
8.6
Online calibration events and adaptive record guarantees
The simultaneous M1 event spans future calibration records; it need not be measurable at the initial
time. Conditioning on that future event and reusing the symmetric-family initial-event proof would be
invalid. A stopped-process repair gives the correct sequential conclusion.
For analysis let τ be the first completed calibration prefix at which the true current effects leave
the certified body. It is a stopping time for the full mathematical filtration, even though an operator is
unknown to the experimenter. M1 gives Pr(τ < ∞) ≤α. At each nominated production opportunity
25


## Source page 26

let Dt be the predictable deploy gate, Ct, Wt its conclusive and wrong indicators, and qt, εt its reported
bounds. A skipped opportunity has all three indicators zero. Prior to τ, drift plus the gate gives
E[Wt −εtCt | Ft−1] ≤0,
E[qtDt −Ct | Ft−1] ≤0.
Use zero increments after the first failed prefix in the analysis. The stopped increments have nonpositive
conditional mean and range length at most one. For any fixed θ > 0, a nonnegative exponential
supermartingale and its first-crossing inequality yield, with failure at most α + β,
X
t≤n
(Wt −εtCt) < θn
8 + log(1/β)
θ
for every n ≥1.
(69)
The identical construction applies to P
t≤n(qtDt −Ct). For simultaneous two-sided objectives split β
between them. No conditional law on the future success event was used; on {τ = ∞} the stopped and
actual production records agree. The fee α is paid once for the entire finite monitoring budget, not
once per production use. This is standard stopped-supermartingale reasoning newly supplied for the
online calibration interface.
Here the proof filtration includes completed-trial scoring truths and outcomes, as in Section 7; the
controller sees only the permitted accessible-record subfiltration. The admitted conditional source,
ready-stock and drift promises must hold on that augmented past. Such a promise is not implied by
calibration on a smaller observable history. An inert retained quantum reference does not change the
label bounds; measuring that reference or a bath and feeding it back changes the conditional contract
unless separately included. Physical resets and fresh source independence are not inferred.
9
Two-template monitoring and task-relative conditional-error certi-
fication
9.1
The same conditional-device contract, a smaller task
Let E+(t), E−(t), E0(t) be any conditional three-outcome POVM on a finite-dimensional input space,
summing to identity. Qutrit vacuum and occupied modes are an example. The POVM is independent
of the current probe’s classical preparation tag; the probe affects it through its quantum state only. The
actual conditional input and inert reference are independent of the device’s ready stock. Uncontrolled
input/stock correlation is outside the linear effect representation.
Nominate normalized density matrices ρ+, ρ−and trusted probe preparations within trace distance
δprep of them in every calibration trial. Future actual label-conditioned marginals must lie within
δS,h of ρh. These are independently admitted preparation conditions, not conclusions of the unknown
detector. Effects may depend on all preceding accessible history. Admit pathwise operator-norm drift
envelopes
∥Ey(t) −Ey(s)∥∞≤dy(t, s),
y = +, −.
A common rate model is dy(t, s) = ν|t −s|, with ν in inverse clock slots (or use actual seconds and
rates per second). Every calibration, production and waiting slot consumes that clock. The envelopes
must constrain actual conditional effects, not only an averaged master equation. Neither probes nor
resets establish the envelopes.
In a trial of template h, record all three outcomes. Let W indicate a wrong conclusive label, V
a correct conclusive label, and C = W + V any conclusive label. Nulls have W = V = C = 0. For
threshold e ∈[0, 1],
Le = W −eC = (1 −e)W −eV ∈[−e, 1 −e].
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## Source page 27

Thus it has range length one, even if clicks are rare. No clicked-only sample selection or Bernoulli
assumption on thinned clicks is used. All raw outcomes, settings, times and calibration/deploy decisions
are part of the physical retained record. The numerical supplement supplies aggregated synthetic count
tables, not an ordered experimental archive.
9.2
One event certifies all thresholds and all inspected prefixes
Permit at most K epochs, opened adaptively. Each epoch may use any predictable template schedule
and a declared finite budget. For n ≥1 uses of one template, let hats denote empirical means and bars
the averages of their true conditional means. Set
u(n) =
s
log(2π2n2K/α)
2n
.
(70)
With probability at least 1 −α, simultaneously for all epochs, both templates, all observed prefixes,
and X ∈{W, V, C},
| bXn −Xn| ≤u(n).
Indeed the centered indicators are martingale differences of range one. Conditional Hoeffding and
exponential Markov give each two-sided failure at most 2e−2nu(n)2 = α/(π2n2K). Summing over n
gives α/(6K) for each stream; union over 6K streams proves the claim. Predictable sampling can be
handled by stopping each stream at its observed sample index, or appending zero centered increments
after its finite budget ends. No iid outcome law is required.
Crucially, the same event gives
|bLe −Le| ≤(1 −e)|c
W −W| + e| bV −V | ≤u(n)
for every e ∈[0, 1].
(71)
There is no threshold-grid union cost and an error target can be chosen after observing the retained
data. The third counter C supplies a separate range-one conclusive-rate bound rather than the looser
sum of two counter errors.
At a future deployment time t after template-h probes at times sh,k, define
Dh,y(t) = 1
nh
X
k
dy(t, sh,k),
bh = u(nh) + δprep + δS,h.
Write Dw = Dh,−h and Dc = Dh,h.
Theorem 9.1 (Direct general-reader deployment certificate). On the simultaneous event, every actual
label-conditioned future input obeys
wh(t) −eqh(t) ≤c
Wh −e bCh + bh + (1 −e)Dw + eDc,
(72)
qh(t) ≥bCh −bh −Dw −Dc.
(73)
If the latter lower bound is positive, define
ecert
h
=
c
Wh + bh + Dw
bCh + Dw −Dc
.
(74)
When ecert
h
≤1, the conditional error wh/qh is at most ecert
h
. Otherwise the trivial bound one applies. The
maximum of the two label bounds and minimum of the two rate bounds cover arbitrary history-dependent
label priors. No balanced prior is required.
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## Source page 28

Proof. The score effect is Ah,e = (1 −e)E−h −eEh, whose spectral range lies within [−e, 1 −e]. Each
probe-preparation error changes its expectation by at most δprep, and future source deviation changes
it by at most δS,h. Its drift is at most (1 −e)d−h + edh by operator-norm duality. Combine with
Eq. (71) and average probe ages to prove Eq. (72). The conclusive effect E+ + E−lies between zero
and identity, so its preparation/source deviations cost the same trace-distance margins, and its drift
costs Dw + Dc. The C counter proves Eq. (73).
Positivity of that rate lower bound ensures bCh + Dw −Dc > 0. Setting the right side of Eq. (72)
to zero then gives exactly Eq. (74). Its threshold can be selected from the data because the coverage
event is uniform over e. Every statement is about the nominated label marginal and holds for any
inert-reference extension of that input marginal. It is not equality of reference assemblages or of
complete quantum instruments.
For the common rate model, with mean probe age Ah(t),
ecert
h
=
c
Wh + u(nh) + δprep + δS,h + νAh(t)
bCh
,
qh ≥bCh −u(nh) −δprep −δS,h −2νAh(t).
The absence of a subtracted confidence width in the ratio’s denominator is justified by the uniform
score argument; it is not a ratio of two independent one-sided confidence intervals. The coefficients ν
and 2ν are sharp for their respective local drift tasks: at an interior POVM change the wrong effect
by +dI and the correct effect by −dI to increase every score by d, or decrease both effects by dI
to decrease conclusive probability by 2d. These different witnesses establish individual bounds, not
simultaneous physical attainment of all statistical and drift corners.
9.3
Budget, expiry and a quantified comparison
Alternate the two templates for n cycles and anchor deployment after slot T = 2n. Their mean ages are
n + 1 and n, hence at further age a the worst age is n + 1 + a. The common-rate numerator budget is
B(n, a) = u(n) + δprep + δS + ν(n + 1 + a).
For fixed data and target  ̄e,  ̄q, deploy only while c
Wh+B ≤ ̄e bCh and bCh−u−δprep−δS −2ν(n+1+a) ≥ ̄q
for both labels. The smaller of the two resulting gap bounds is an explicit expiry time. A negative gap
means stop or take new probes, not extrapolate stability.
At a predetermined terminal sample size the fixed-time width is u(n) = A/√n, A =
p
log(12K/α)/2.
The displayed bound A/√n + νn has continuous optimum n∗= (A/(2ν))2/3. This is an exact optimum
of this specified bound and schedule; check neighboring integers and the finite budget. For the all-prefix
width use one-dimensional minimization. Neither claim optimizes over every possible calibration
protocol.
For a separate specified stationary asymmetric/dark-count qutrit detector, use the known occupied
inputs
√
0.8|0⟩±
√
0.2|1⟩. Keep α = 0.01, K = 3, ν = 10−11 per clock slot, δprep = 5 × 10−5, δS = 0,
and deployment age 108 slots. Both procedures pay actual calibration time in the same drift clock.
The nine-probe example used 36 million trials and gave the arbitrary-prior certificate e ≤0.169748239,
q ≥0.268980153.
The new predetermined two-template count-table diagnostics give:
Total probe trials
arbitrary-prior error bound
conclusive lower bound
200,000
0.1641071006
0.2820958913
1,000,000
0.1440878798
0.2913177448
2,000,000
0.1387493968
0.2921346968
36,000,000
0.1292163991
0.2948488929
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## Source page 29

These are synthetic count tables, not experimental data or a proof of sample optimality. Their protocols
have different output tasks: the nine-probe method additionally estimates a general qutrit POVM and
a pointer/reference channel bound, which two templates do not identify.
There is also a non-simulation sufficiency statement for this stationary benchmark law with exactly
nominated calibration preparations. Write its true template means as wh, qh. The same simultaneous
event guarantees that the deployment gate succeeds if
2u(n) + δprep + δS + ν(n + 1 + a) ≤ ̄eqh −wh,
qh −2u(n) −δprep −δS −2ν(n + 1 + a) ≥ ̄q.
This follows by bounding the empirical score and conclusive frequency with their opposite sides on
that same absolute-confidence event. For the full-POVM arbitrary-prior targets above, n = 500, 000
per template satisfies both inequalities: one million probes suffice to guarantee deployment and the
stated future conditional performance with probability at least 99 percent for this specified stationary
benchmark. This is 36 times fewer trials than that nine-probe count-table demonstration, not a
universal 36-fold improvement over all tomography methods or a guarantee for every drift/preparation
law in the larger admitted class.
9.4
What two probes can and cannot identify
For arbitrary unknown effects, exact response on a set of target states can be determined from probe
expectations if and only if the target states lie in the real linear span of the probe density matrices.
Sufficiency follows by linearity. If a target lies outside that span, there is a Hermitian H orthogonal to
every probe and not to that target. Starting with a strictly interior POVM, perturb one effect by εH
and the null effect by −εH. For sufficiently small nonzero ε this remains physical, leaves every measured
probe response unchanged and changes the target response. This is an exact indistinguishability
counterexample, not a failed numerical search. Two distinct template density matrices are independent,
so two probes are minimal for full label-response identification in this unrestricted effect model. This is
not a minimum for every loose one-sided claim.
Our occupied templates require only the already nominated source’s relative-mode preparations.
No vacuum/occupied superposition probe is needed. Their unmeasured vacuum response remains
unidentified: adding d|v⟩⟨v| to a conclusive effect and subtracting it from the null is invisible to both
templates. It can change another input’s response and the full qutrit measurement-channel distance.
For a mixed target, an invisible traceless effect direction can likewise change a purifying reference’s
conditional assemblage while preserving its label probabilities. No full-POVM or arbitrary-input
diamond guarantee is claimed here. Future source deviations are charged through δS,h rather than
assumed harmless.
9.5
Adaptive monitoring and retained records
Open at most K epochs; retain complete outcomes and apply the gate at any covered prefix. Expire it
as the drift clock advances. Retuning the physical reader requires new calibration or an independently
justified transformation rule. Neither template counts nor old calibration identify an unknown bath
law or path-blind Hamiltonian.
Let Dt ∈{0, 1} be the predictable production/deploy gate at global clock slot t, with Wt = Ct = 0
outside production. Let et and qt be its predictable certified error upper bound and conclusive lower
bound; values outside production are immaterial. Write Nn = P
t≤n Dt for the number of deployed
production trials. The global confidence event includes future calibration and must not be conditioned
on as initial information. For analysis stop at the first completed probe prefix whose true counter
means miss their intervals; its probability is at most α. Before that stopping time, every predictable
production gate has the certified conditional error and rate. Stop the production scores there. They
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## Source page 30

have nonpositive conditional mean and conditional range length at most Dt, so the exponential
supermartingale with compensator Nnθ2/8 gives, for any fixed θ > 0, a uniform record bound
X
t≤n
(Wt −etCt) < θNn
8
+ log(1/β)
θ
for every n
with failure at most α+β. The corresponding conclusive lower bound uses qtDt −Ct; split β if both are
required. On no confidence failure the stopped and actual production records agree. Thus calibration
risk is paid once across the finite budget, without conditioning on future success or inferring fresh
independence. The proof filtration includes scoring truths for completed trials, while controller choices
use only the permitted accessible-record subfiltration. Conditional device, source and drift promises
are assumed on this augmented past, not inferred by enlarging an observed-history filtration. Known
truth labels are required to score wrong outcomes in validation; an unknown production label is not
made observable by this theorem.
The code independently checks the continuum score identity, predictable non-iid exponential
moments, the blind-direction example, exact drift coefficients and all benchmark budgets. Numerical
tables corroborate the proofs and synthetic protocol; they do not certify physical drift or trusted state
preparation. All source/controller, clock, null/loss, reference and correction contracts remain. No
Born-selection, biological channel or consciousness mechanism is established.
10
Preservation and scientific status
The companion’s proof interfaces differ in observable task and access permissions. Their numerical
examples are synthetic; tests corroborate the proofs rather than constitute empirical evidence. Original
source/controller, reference, clock, preparation, null/loss and retained-output conditions survive every
reuse. The general qutrit method estimates a full effect family; the two-template method certifies a
smaller task and does not identify the same quantum output. Their sample counts are therefore not a
universal tomography comparison. No physical horizon-to-biological channel has been supplied; no
actual-event affinity is inferred from an averaged density equation. The source–readout completion
framework remains compatible with unrestricted operational surrogates, so a constrained detector
distinction does not establish unique ontology.
Research assistance.
AI assistance was used in reconstruction, checking and exposition. Supporting
code and verification instructions accompany the draft. These are checked drafts with declared scope,
not externally peer-reviewed empirical papers.
References
[1] Mehdi Ahmadi, David Jennings, and Terry Rudolph. The wigner–araki–yanase theorem and
the quantum resource theory of asymmetry. New Journal of Physics, 15:013057, 2013. doi:
10.1088/1367-2630/15/1/013057. URL https://arxiv.org/abs/1209.0921.
[2] Vahid Ansari, Georg Harder, Markus Allgaier, Benjamin Brecht, and Christine Silberhorn.
Temporal-mode measurement tomography of a quantum pulse gate. Physical Review A, 96:
063817, 2017. doi: 10.1103/PhysRevA.96.063817. URL https://arxiv.org/abs/1702.03336.
[3] Todd A. Brun. Measuring polynomial functions of states. Quantum Information and Computation,
4(5):401–408, 2004. URL https://arxiv.org/pdf/quant-ph/0401067.
30


## Source page 31

[4] Merlin Cooper, Michał Karpiński, and Brian J. Smith. Local mapping of detector response for
reliable quantum state estimation. Nature Communications, 5:4332, 2014. doi: 10.1038/ncomms
5332. URL https://www.nature.com/articles/ncomms5332.
[5] Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon. Time-uniform, nonpara-
metric, nonasymptotic confidence sequences. The Annals of Statistics, 49(2):1055–1080, 2021. doi:
10.1214/20-AOS1991. URL https://arxiv.org/pdf/1810.08240. Author preprint version 9.
[6] Dennis Kretschmann, Dirk Schlingemann, and Reinhard F. Werner. A continuity theorem for
stinespring’s dilation. Journal of Functional Analysis, 255:1889–1904, 2008. doi: 10.1016/j.jfa.20
08.07.023. URL https://arxiv.org/abs/0710.2495.
[7] Jason D. Padgett. Formal operator-level equivalence between photoelectric detection and a qih-
inspired microtubule readout model, . Supplied unpublished manuscript, 35 pages. Relevant reader
construction pp.18–21; reduced equivalence pp.10–15.
[8] Jason D. Padgett. Exact reduced-operator equivalence between the photoelectric effect, a QIH
horizon readout, and a microtubule quantum readout, . Supplied unpublished manuscript, 74
pages, filename Photoelectric Origin Microtubules.pdf. Matched-reader construction, Section 32,
pp.42–43.
[9] Jeremy Rodgers. Source–readout non-equivalence: Descent and equivariant reconstruction ob-
structions, 2026. URL https://doi.org/10.5281/zenodo.21369967.
[10] Laura Serino, Jano Gil-Lopez, Michael Stefszky, Raimund Ricken, Christof Eigner, Benjamin
Brecht, and Christine Silberhorn. Realization of a multi-output quantum pulse gate for decoding
high-dimensional temporal modes of single-photon states. PRX Quantum, 4:020306, 2023. doi:
10.1103/PRXQuantum.4.020306. URL https://arxiv.org/abs/2211.05693.
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