% En01-A.tex
% Self-contained input for pdfLaTeX.
% Compile twice:
%   pdflatex En01-A.tex
%   pdflatex En01-A.tex
%
% Compilation also writes the embedded regression program:
%   En01-A-check.py
%
% Optional numerical checks:
%   python En01-A-check.py
%
% No Python execution or pdfLaTeX execution is claimed in this source.
% The mathematical proofs do not depend on the numerical appendix.

\documentclass[11pt]{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[margin=1.05in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{microtype}
\usepackage{booktabs,array,tabularx}
\usepackage{listings}
\usepackage{tikz}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}

\hypersetup{
  pdftitle={Accessible Interference in Recorded Quantum Circuits},
  pdfauthor={Ngo Cheung},
  pdfsubject={En01-A: exact interfaces, record architecture, and collapse-model discrepancies}
}

\setlength{\emergencystretch}{3em}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\Ree}{Re}
\newcommand{\HH}{\mathcal H}
\newcommand{\KK}{\mathcal K}
\newcommand{\Alg}{\mathcal A}
\newcommand{\BB}{\mathcal B}
\newcommand{\id}{\mathbb 1}
\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\bra}[1]{\langle #1|}
\newcommand{\braket}[2]{\langle #1|#2\rangle}
\newcommand{\ketbra}[2]{|#1\rangle\langle #2|}
\newcommand{\norm}[1]{\lVert #1\rVert}
\newcommand{\Sbar}{\overline S}
\newcommand{\Nset}{\mathcal N}
\newcommand{\Pcoh}{P^{\mathrm{coh}}}
\newcommand{\Pcoll}{P^{\mathrm{coll}}}
\newcommand{\Pseq}{P^{\mathrm{seq}}}
\newcommand{\Cprod}{\mathsf C_{\mathrm{prod}}}
\newcommand{\tnorm}[1]{\left\lVert #1\right\rVert_1}
\newcommand{\dnorm}[1]{\left\lVert #1\right\rVert_\diamond}

\lstset{
  language=Python,
  basicstyle=\small\ttfamily,
  columns=fullflexible,
  keepspaces=true,
  breaklines=true,
  showstringspaces=false,
  frame=single,
  framesep=4pt,
  tabsize=4
}

% The starred environment suppresses generated header comments.
% The overwrite option prevents an earlier artifact from being silently reused.
\begin{filecontents*}[overwrite]{En01-A-check.py}
"""En01-A-check.py

Deterministic regression checks for the examples and finite-dimensional
identities in En01-A.tex.

Dependencies: Python 3 and NumPy.
This program is supplied as a reproducible check, not as a formal proof.
Its execution is not claimed by the manuscript source.

On successful execution it writes En01-A-run.json, recording:
  * SHA-256 of this exact generated Python file;
  * Python and NumPy versions;
  * UTC execution time;
  * the fixed seed and tolerances;
  * completion of the assertions in this local invocation.
"""

from itertools import combinations, product
from pathlib import Path
import datetime
import hashlib
import json
import platform
import numpy as np

SEED = 20261006
TOL = 1e-10
GOLDEN_TOL = 1e-12
rng = np.random.default_rng(SEED)


def subsets(items):
    items = tuple(items)
    for size in range(len(items) + 1):
        yield from combinations(items, size)


def ket(bits):
    bits = tuple(bits)
    value = 0
    for bit in bits:
        value = (value << 1) | int(bit)
    out = np.zeros(2 ** len(bits), dtype=complex)
    out[value] = 1.0
    return out


def density(psi):
    return np.outer(psi, psi.conj())


def nqubits_matrix(Y):
    n = int(round(np.log2(Y.shape[0])))
    assert Y.shape == (2 ** n, 2 ** n)
    return n


def ptrace(Y, keep):
    n = nqubits_matrix(Y)
    keep = list(keep)
    assert len(set(keep)) == len(keep)
    rest = [q for q in range(n) if q not in keep]
    perm = keep + rest + [q + n for q in keep] + [q + n for q in rest]
    tensor = np.transpose(Y.reshape([2] * (2 * n)), perm)
    dkeep, drest = 2 ** len(keep), 2 ** len(rest)
    return np.trace(
        tensor.reshape(dkeep, drest, dkeep, drest),
        axis1=1,
        axis2=3,
    )


def pinch(Y, sites):
    n = nqubits_matrix(Y)
    labels = np.zeros(2 ** n, dtype=np.int64)
    for q in sites:
        labels = 2 * labels + (
            (np.arange(2 ** n) >> (n - 1 - q)) & 1
        )
    return Y * (labels[:, None] == labels[None, :])


def trace_norm(Y):
    return float(np.linalg.svd(Y, compute_uv=False).sum())


def delta(Y, sites, keep=None):
    X = Y - pinch(Y, sites)
    if keep is not None:
        X = ptrace(X, keep)
    return 0.5 * trace_norm(X)


def mobius(Y, J):
    out = np.zeros_like(Y)
    for K in subsets(J):
        out += (-1) ** (len(J) - len(K)) * pinch(Y, K)
    return out


def psqrt(Y):
    Y = (Y + Y.conj().T) / 2
    values, vectors = np.linalg.eigh(Y)
    values = np.where(values > 1e-13, values, 0.0)
    return (vectors * np.sqrt(values)) @ vectors.conj().T


def fidelity(Y, Z):
    return trace_norm(psqrt(Y) @ psqrt(Z))


def visibility(phi, phip, keep):
    return trace_norm(ptrace(np.outer(phi, phip.conj()), keep))


def effect_bias(Y, sites, M):
    return float(np.trace(M @ (Y - pinch(Y, sites))).real)


def ghz(c):
    return (ket([0] * c) + ket([1] * c)) / np.sqrt(2)


def ghz_effect(c):
    return density(ghz(c))


def tensor_all(items):
    out = np.ones((1, 1), dtype=complex)
    for item in items:
        out = np.kron(out, item)
    return out


def apply_unitary(psi, U, targets):
    n = int(round(np.log2(psi.size)))
    targets = list(targets)
    k = len(targets)
    tensor = np.tensordot(
        U.reshape([2] * (2 * k)),
        psi.reshape([2] * n),
        axes=(list(range(k, 2 * k)), targets),
    )
    return np.moveaxis(tensor, range(k), targets).reshape(-1)


def full_unitary(U, targets, n):
    eye = np.eye(2 ** n, dtype=complex)
    return np.column_stack([
        apply_unitary(eye[:, j], U, targets)
        for j in range(2 ** n)
    ])


H = np.array([[1, 1], [1, -1]], dtype=complex) / np.sqrt(2)
XPAULI = np.array([[0, 1], [1, 0]], dtype=complex)
CNOT = np.array([
    [1, 0, 0, 0],
    [0, 1, 0, 0],
    [0, 0, 0, 1],
    [0, 0, 1, 0],
], dtype=complex)


def fr_state(p=(1 / 3, 1 / 3, 1 / 3), phases=(0.0, 0.0)):
    a, b, c = np.sqrt(np.asarray(p, dtype=float))
    return np.array([
        a,
        0,
        b * np.exp(1j * phases[0]),
        c * np.exp(1j * phases[1]),
    ], dtype=complex)


def fr_effect():
    ok = np.array([1, -1], dtype=complex) / np.sqrt(2)
    return density(np.kron(ok, ok))


def cancellation_state(theta):
    psi = np.zeros(32, dtype=complex)
    for a, b in product((0, 1), repeat=2):
        psi += 0.5 * np.exp(1j * theta * a * b) * ket([a, a, b, b, b])
    return psi


def noisy_state(c, gamma):
    e0 = np.array([1, 0], dtype=complex)
    e1 = np.array([gamma, np.sqrt(1 - gamma ** 2)], dtype=complex)
    left = ket([0])
    right = ket([1])
    for _ in range(c - 1):
        left = np.kron(left, e0)
        right = np.kron(right, e1)
    return (left + right) / np.sqrt(2)


def random_channel(din, dout=2, denv=None):
    if denv is None:
        denv = din
    A = (
        rng.normal(size=(dout * denv, din))
        + 1j * rng.normal(size=(dout * denv, din))
    )
    V = np.linalg.qr(A, mode="reduced")[0]

    def channel(Y):
        tensor = (V @ Y @ V.conj().T).reshape(
            dout, denv, dout, denv
        )
        return np.trace(tensor, axis1=1, axis2=3)

    return channel


def fmt(values):
    return "[" + ", ".join(
        f"{0.0 if abs(v) < TOL else float(v):.6f}"
        for v in values
    ) + "]"


def run_checks():
    # Perfect read-chain: four members, three copying events.
    Yghz = density(ghz(4))
    ghz_values = [delta(Yghz, [0], list(range(m))) for m in range(1, 5)]
    assert np.allclose(
        ghz_values, [0, 0, 0, 0.5], atol=GOLDEN_TOL, rtol=0
    )

    # Independent noisy fragments, always retaining the perfect root.
    Ynoisy = density(noisy_state(4, 0.5))
    noisy_values = [
        delta(Ynoisy, [0], list(range(m))) for m in range(1, 5)
    ]
    assert np.allclose(
        noisy_values,
        [0.0625, 0.125, 0.25, 0.5],
        atol=GOLDEN_TOL,
        rtol=0,
    )

    # FR profile and operator/question weights.
    Yfr, Mfr = density(fr_state()), fr_effect()
    record_sets = [(), (0,), (1,), (0, 1)]
    fr_profile = [
        float(np.trace(Mfr @ pinch(Yfr, J)).real)
        for J in record_sets
    ]
    Zfr = [mobius(Yfr, J) for J in record_sets[1:]]
    fr_state_weights = [0.5 * trace_norm(Z) for Z in Zfr]
    fr_question_weights = [
        abs(float(np.trace(Mfr @ Z).real)) for Z in Zfr
    ]
    assert np.allclose(
        fr_profile, [1 / 12, 1 / 12, 1 / 12, 1 / 4],
        atol=GOLDEN_TOL, rtol=0,
    )
    assert np.allclose(
        fr_state_weights, [np.sqrt(2) / 3, np.sqrt(2) / 3, 1 / 3],
        atol=GOLDEN_TOL, rtol=0,
    )
    assert np.allclose(
        fr_question_weights, [0, 0, 1 / 6],
        atol=GOLDEN_TOL, rtol=0,
    )

    eps = 1e-3
    Ypert = density(fr_state((1 / 3, 1 / 3 + eps, 1 / 3 - eps)))
    first_order_bias = -effect_bias(Ypert, [0], Mfr)
    assert abs(first_order_bias - eps / 2) < 2e-9

    # One extra perfect copy of each FR record.
    psifr_copy = np.zeros(16, dtype=complex)
    wencoded = np.zeros(16, dtype=complex)
    for a, b in product((0, 1), repeat=2):
        if (a, b) != (0, 1):
            psifr_copy += ket([a, b, a, b]) / np.sqrt(3)
        wencoded += ((-1) ** (a + b)) * ket([a, b, a, b]) / 2
    Yfr_copy = density(psifr_copy)
    assert abs(
        np.trace(Mfr @ ptrace(Yfr_copy, [0, 1])).real - 1 / 4
    ) < GOLDEN_TOL
    assert abs(
        np.trace(density(wencoded) @ Yfr_copy).real - 1 / 12
    ) < GOLDEN_TOL

    # Same class structure, probabilities, and scalar visibilities;
    # different relative cross-operator phases.
    Ycancel = density(cancellation_state(np.pi))
    Ynocancel = density(cancellation_state(0.0))
    cancellation_values = [
        delta(Ynocancel, [0, 1], [0, 1]),
        delta(Ycancel, [0, 1], [0, 1]),
        delta(Ycancel, [0, 1], [0, 1, 4]),
    ]
    assert np.allclose(
        cancellation_values, [0.5, 0, 0.5],
        atol=GOLDEN_TOL, rtol=0,
    )
    assert delta(Ycancel, [0, 1], [4]) < GOLDEN_TOL
    for J in [(0, 1), (2, 3), (0, 1, 2, 3)]:
        assert trace_norm(ptrace(mobius(Ycancel, J), [0, 1])) < TOL
    for theta in (0.0, np.pi):
        for b in (0, 1):
            phi0 = ket([0, 0, b, b, b])
            phi1 = np.exp(1j * theta * b) * ket([1, 1, b, b, b])
            assert abs(visibility(phi0, phi1, [0, 1]) - 1) < GOLDEN_TOL

    # Correlated GHZ-phase environment: a retained perfect root.
    E0 = ghz(3)
    E1 = (ket([0, 0, 0]) - ket([1, 1, 1])) / np.sqrt(2)
    phi0 = np.kron(ket([0]), E0)
    phi1 = np.kron(ket([1]), E1)
    secret_values = [
        visibility(phi0, phi1, list(range(4 - lost)))
        for lost in range(4)
    ]
    assert np.allclose(
        secret_values, [1, 1, 1, 0],
        atol=GOLDEN_TOL, rtol=0,
    )

    # Product conjunction versus locally measured parity.
    plus_effect = density(np.array([1, 1], dtype=complex) / np.sqrt(2))
    product_values, parity_values = [], []
    for c in range(1, 5):
        Y = density(ghz(c))
        product_effect = tensor_all([plus_effect] * c)
        parity_effect = (
            np.eye(2 ** c) + tensor_all([XPAULI] * c)
        ) / 2
        product_values.append(abs(effect_bias(Y, [0], product_effect)))
        parity_values.append(abs(effect_bias(Y, [0], parity_effect)))
    assert np.allclose(
        product_values, [0.5, 0.25, 0.125, 0.0625],
        atol=GOLDEN_TOL, rtol=0,
    )
    assert np.allclose(
        parity_values, [0.5] * 4,
        atol=GOLDEN_TOL, rtol=0,
    )

    # Sequential pinching versus final pinching.
    c = 4
    initial = density(ket([0] * c))
    UH = full_unitary(H, [0], c)
    coherent = UH @ initial @ UH.conj().T
    sequential = pinch(coherent, [0])
    for source in range(c - 1):
        UC = full_unitary(CNOT, [source, source + 1], c)
        coherent = UC @ coherent @ UC.conj().T
        sequential = UC @ sequential @ UC.conj().T
    assert trace_norm(sequential - pinch(coherent, range(c))) < TOL

    # A concrete gate-error check.
    # Each noisy gate is (1-eta) G + eta R, with R replacing by I/d.
    # Its diamond distance from G is at most 2*eta.
    eta = 1e-4
    noisy = initial.copy()
    ideal = initial.copy()
    gates = [UH] + [
        full_unitary(CNOT, [q, q + 1], c) for q in range(c - 1)
    ]
    for U in gates:
        ideal = U @ ideal @ U.conj().T
        noisy = (
            (1 - eta) * (U @ noisy @ U.conj().T)
            + eta * np.eye(2 ** c) / (2 ** c)
        )
    error_budget = 2 * eta * len(gates)
    assert trace_norm(noisy - ideal) <= error_budget + TOL
    assert abs(
        delta(noisy, range(c)) - delta(ideal, range(c))
    ) <= error_budget + TOL

    # Every prescribed nonempty-record support family is realizable
    # by a binary measurement channel on this full-support state.
    Yfull = density(np.ones(4, dtype=complex) / 2)
    Js = [(0,), (1,), (0, 1)]
    Zs = [mobius(Yfull, J) for J in Js]
    Gram = np.array([
        [np.trace(A @ B).real for B in Zs] for A in Zs
    ])
    assert np.linalg.eigvalsh(Gram).min() > 0
    for flags in product((0, 1), repeat=3):
        target = np.array(flags, dtype=float)
        coefficients = np.linalg.solve(Gram, target)
        HH = sum(
            (a * Z for a, Z in zip(coefficients, Zs)),
            np.zeros_like(Yfull),
        )
        scale = 1 / max(1.0, np.max(np.abs(np.linalg.eigvalsh(HH))))
        M = (np.eye(4) + scale * HH) / 2
        values = np.array([np.trace(M @ Z).real for Z in Zs])
        assert np.allclose(values, scale * target / 2, atol=TOL, rtol=0)
        assert np.linalg.eigvalsh(M).min() >= -TOL
        assert np.linalg.eigvalsh(np.eye(4) - M).min() >= -TOL

    # A nonsymmetric point on the complex-amplitude order-two stratum.
    alpha, beta = 0.1, 0.2
    A, B, C = np.cos(alpha), np.cos(beta), np.cos(alpha - beta)
    L = np.sqrt(A * A + B * B + C * C)
    probs = ((C / L) ** 2, (B / L) ** 2, (A / L) ** 2)
    Ycomplex = density(fr_state(probs, (alpha, beta)))
    etas = [np.trace(Mfr @ mobius(Ycomplex, J)).real for J in Js]
    assert abs(etas[0]) < TOL and abs(etas[1]) < TOL
    assert etas[2] > 0

    # Random finite-dimensional checks with a fixed seed.
    for _ in range(12):
        psi = rng.normal(size=8) + 1j * rng.normal(size=8)
        psi /= np.linalg.norm(psi)
        Y = density(psi)
        for J in subsets((0, 1)):
            reconstructed = sum(
                (mobius(Y, K) for K in subsets(J)),
                np.zeros_like(Y),
            )
            assert trace_norm(reconstructed - pinch(Y, J)) < TOL

        channel = random_channel(8)
        XJ = Y - pinch(Y, [0])
        out = channel(XJ)
        out = (out + out.conj().T) / 2
        eigenvalues, eigenvectors = np.linalg.eigh(out)
        Pplus = density(eigenvectors[:, -1]) if eigenvalues[-1] > 0 else np.zeros_like(out)
        assert abs(
            np.trace(Pplus @ out).real - 0.5 * trace_norm(out)
        ) < TOL
        further = pinch(out, [0])
        assert trace_norm(further) <= trace_norm(out) + TOL

        # Normalized branches for roots 0 and 1, residual qubit 2.
        branches = []
        for r in range(4):
            branch = np.zeros(8, dtype=complex)
            branch[2 * r:2 * r + 2] = psi[2 * r:2 * r + 2]
            branch /= np.linalg.norm(branch)
            branches.append(branch)

        for keep in subsets((0, 1, 2)):
            rest = [q for q in range(3) if q not in keep]
            for r in range(4):
                for rp in range(r + 1, 4):
                    v = visibility(branches[r], branches[rp], keep)
                    sig = ptrace(density(branches[r]), rest)
                    sigp = ptrace(density(branches[rp]), rest)
                    assert abs(v - fidelity(sig, sigp)) < TOL

        # Subsystem Mobius support is down-closed.
        for keep in subsets((0, 1, 2)):
            support = {
                J for J in Js
                if trace_norm(ptrace(mobius(Y, J), keep)) > TOL
            }
            for J in support:
                for K in subsets(J):
                    if K:
                        assert K in support

    chernoff = np.log(9 / 5)
    stein = np.log(3) - (2 / 3) * np.log(2)
    assert -0.5 * np.log(1 - 2 / 9) <= chernoff
    assert chernoff <= -np.log(1 - np.sqrt(2) / 3)

    print("GHZ:", fmt(ghz_values))
    print("NOISY:", fmt(noisy_values))
    print("FR_PROFILE:", fmt(fr_profile))
    print("FR_STATE_WEIGHTS:", fmt(fr_state_weights))
    print("FR_QUESTION_WEIGHTS:", fmt(fr_question_weights))
    print("CANCELLATION:", fmt(cancellation_values))
    print("SECRET:", fmt(secret_values))
    print("PRODUCT_EFFECT:", fmt(product_values))
    print("LOCAL_PARITY:", fmt(parity_values))
    print(f"FR_CHERNOFF: {chernoff:.6f}")
    print(f"FR_STEIN: {stein:.6f}")
    print("STRUCTURAL_CHECKS: PASS")
    print("ALL CHECKS PASSED")


if __name__ == "__main__":
    run_checks()
    source = Path(__file__).read_bytes()
    digest = hashlib.sha256(source).hexdigest()
    manifest = {
        "artifact": Path(__file__).name,
        "sha256": digest,
        "execution_status": "completed by this local invocation",
        "utc": datetime.datetime.now(datetime.timezone.utc).isoformat(),
        "python": platform.python_version(),
        "numpy": np.__version__,
        "seed": SEED,
        "random_check_tolerance": TOL,
        "golden_value_tolerance": GOLDEN_TOL,
    }
    manifest_path = Path("En01-A-run.json")
    manifest_path.write_text(
        json.dumps(manifest, indent=2, sort_keys=True) + "\n",
        encoding="utf-8",
    )
    print("SOURCE_SHA256:", digest)
    print("MANIFEST:", manifest_path.name)
\end{filecontents*}

\title{Accessible Interference in Recorded Quantum Circuits:\\
Exact Interfaces, Record Architecture, and Collapse-Model Hierarchies}
\author{
Ngo Cheung, MBBS(HK), FHKAM(Psychiatry)\\
Independent Researcher\\
Hong Kong SAR, China\\
\texttt{info@cheungngomedical.com}
}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We give a self-contained finite-dimensional theory of access to
interference in nested-measurement circuits with undisturbed records.
Internal recording is coherent copying, and established records are
subsequently used only as copy sources or computational-basis controls.
For a record set \(J\), the discrepancy between a coherent state
\(\rho\) and its record-dephased model is the traceless Hermitian operator
\(X_J=\rho-\Delta_J\rho\).
For every quantum channel \(\Lambda\), agreement on all available effects
is equivalent to \(\Lambda(X_J)=0\), and the maximal probability bias is
\(\frac12\|\Lambda(X_J)\|_1\).
Subsystem and algebraic interfaces follow as special cases.

A branch cross-operator calculus separates perfect-copy screening,
residual distinguishability, operator-level cancellation, and
question-specific agreement.
Pairwise visibility is the fidelity of the inaccessible branch states.
This gives exact product laws for independent noisy fragments and
all-or-nothing behavior for correlated phase records.
Operator-valued M\"obius elements distinguish state, interface, and
question orders.
State and subsystem supports are down-closed; arbitrary channel supports
need not be, and every prescribed support family is realizable by a
binary measurement channel on a full-support tower.
For the symmetric Frauchiger--Renner state, order two belongs to Wigner's
question, not to the state.

We prove sequential agreement on the record-invariant algebra,
diamond-norm perturbation bounds, a same-data counterexample to scalar
visibility characterizations, and the codimension-two geometry of the
complex-amplitude Frauchiger--Renner order-two stratum.
The read-chain bias \(2^{-c}\) is established for single product effects,
while local parity measurements attain the unrestricted bias \(1/2\).
Standard hypothesis-testing theorems give a trace-distance Chernoff
sandwich and a Pinsker lower bound for the Stein exponent.
An embedded, hash-identifiable regression program is supplied without
claiming an execution record.
\end{abstract}


\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Coherent record copy};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Wigner's measurement channel};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical no-go};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Order two at one question is not a physical no-go.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A question-level order is not a physical no-go. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:scope-En01-A}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Coherent record copy};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Wigner's measurement channel};
\node[draw,dashed,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical no-go};
\draw[->] (a) -- (b);
\draw[->,dashed] (b) -- (c);
\end{tikzpicture}
\caption{A question-level order is not a physical no-go. Reading downward is not a transfer. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-En01-A}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Coherent record copy};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Wigner's measurement channel};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical no-go};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\end{tikzpicture}
\caption{Reading downward is not a transfer. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-En01-A}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.4cm,font=\small,inner sep=3pt] (a) at (0,0) {Coherent record copy};
\node[draw,rounded corners,align=center,text width=3.4cm,font=\small,inner sep=3pt] (b) at (6.2,0) {Physical no-go};
\draw[dashed] (a) -- (b);
\end{tikzpicture}
\caption{Coherent record copy is not Physical no-go. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-En01-A}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Coherent record copy};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Wigner's measurement channel};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical no-go};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Order two at one question is not a physical no-go.};
\end{tikzpicture}
\caption{Three statuses stay separate. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-En01-A}
\end{figure}

Figure~\ref{fig:scope-En01-A}, Figure~\ref{fig:split-En01-A}, Figure~\ref{fig:chain-En01-A}, Figure~\ref{fig:pair-En01-A}, and Figure~\ref{fig:gate-En01-A} name the stop. Not a proof, and not an admission of a quantum coefficient.

\section{Introduction and scope}\label{sec:intro}

Nested-observer experiments distinguish two statistical descriptions of
the same recorded quantum process: a coherent description of the entire
laboratory and a model in which selected records have been dephased.
The Frauchiger--Renner experiment is a prominent example
\cite{FrauchigerRenner2018}.
The present question is operational:
\emph{which accessible measurements can distinguish these descriptions?}

This article consolidates the subsystem and channel developments of two
earlier drafts \cite{DraftSubsystem,DraftChannel}.
All definitions and proofs needed for the finite-dimensional results are
included here. There is no dependence on an unproved companion theorem.

The organizing chain is
\begin{equation}\label{eq:chain}
X_J
\ \xrightarrow{\ \Lambda\ }\
\Lambda(X_J)
\ \xrightarrow{\ \mathsf C\ }\
\sup_{M\in\mathsf C}|\tr(M\Lambda(X_J))|.
\end{equation}
Its three stages describe interference content, accessible interference,
and implementable probability bias, respectively.

The channel-interface theorem is the standard operational
characterization of trace distance, applied to a structured family of
state differences. We do not claim a new proof of the Born rule or a new
state-discrimination principle. The circuit-specific results concern
record anchoring, copy screening, cross-operator cancellation,
M\"obius localization, and the consequences of restricting the interface.

Several distinctions are essential.
\begin{enumerate}
\item Internal recording is a coherent isometry, not destructive
readout. A nonselective dephasing channel is obtained by discarding a
fresh coherent copy.
\item Agreement of one effect is weaker than agreement of every effect
on a subsystem or after a channel.
\item The copy budget is necessary, not sufficient, for detection.
Relative cross-operator phases are not determined by scalar visibilities.
\item Classicality order is explicitly indexed by the object being
tested: state, interface, or question.
\item Exact vanishing is not generally stable under arbitrary
perturbations. Quantitative smallness and thresholded support are stable
in the senses proved below.
\end{enumerate}

\paragraph{Physical and mathematical boundary.}
The Born rule and the completely positive trace-preserving
(CPTP) channel formalism are assumed.
The dephased model is a specified statistical comparator, not an
additional physical collapse mechanism.
The probability calculations for the Frauchiger--Renner state do not
reproduce, resolve, or replace the original theorem concerning agents'
reasoning assumptions.
No claim is made about destructive-primitive uniqueness, quantum field
theory, gravitational coefficient admission, or a higher-order protocol
calculus.

The new finite-dimensional statements are proved below.
The asymptotic quantum Chernoff and Stein theorems are standard external
results, invoked with their assumptions and references in
Section~\ref{sec:testing}.

\section{Recorded circuits and the two statistical models}
\label{sec:towers}

All Hilbert spaces are finite-dimensional.
A state is a positive operator of trace one.
An effect is an operator \(0\le M\le\id\).
We use
\[
\|A\|_1=\tr\sqrt{A^\dagger A},
\qquad
T(\rho,\sigma)=\tfrac12\|\rho-\sigma\|_1.
\]
A finite qubit set \(Q\) has Hilbert space
\(\HH_Q=\bigotimes_{q\in Q}\mathbb C^2\).
For \(S\subseteq Q\), write \(\Sbar=Q\setminus S\).

\subsection{Three different recording interfaces}

For a measured factor \(S\), an untouched factor \(B\), and a
computational basis \(\{\ket z_S\}\), coherent recording is the isometry
\begin{equation}\label{eq:coherentcopy}
V\bigl(\ket z_S\otimes\ket b_B\bigr)
=
\ket z_S\otimes\ket z_A\otimes\ket b_B.
\end{equation}
It retains \(S\) and a quantum carrier \(A\).
It does not make \(A\) a classical register in the sense of a joint
block-diagonal output state.

By contrast, destructive readout is the channel
\begin{equation}\label{eq:destructive}
\mathcal M_S(Y)
=
\sum_z\ketbra zz_R\otimes
(\bra z_S\otimes\id_B)Y(\ket z_S\otimes\id_B),
\end{equation}
whose output does not contain \(S\).

A represented nondestructive projective instrument has a third output:
\begin{equation}\label{eq:nondestructive}
\mathcal I_S^{\mathrm{nd}}(Y)
=
\sum_z\ketbra zz_R\otimes P_zYP_z,
\qquad
P_z=\ketbra zz_S\otimes\id_B.
\end{equation}
It retains \(S\), but its output record \(R\) is classical.
These three maps have different declared outputs and are not identified.

The circuits studied here use \eqref{eq:coherentcopy}.
The nonselective comparator arises by tracing its additional carrier,
not by renaming \eqref{eq:coherentcopy} as \eqref{eq:destructive}.

\subsection{Compiled towers}

\begin{definition}[Compiled recorded circuit]\label{def:tower}
A \emph{compiled tower} is a finite event sequence on \(Q\), initialized
in a pure lab state and fresh zero-state ancillas, with evolving pairwise
disjoint \emph{copy classes}
\(K_1,\ldots,K_N\), each having a designated root.
The admissible events are:
\begin{enumerate}
\item[(T1)] \emph{Designation:} a qubit outside all classes becomes the
root of a new singleton class. This is bookkeeping, not a gate.
\item[(T2)] \emph{Perfect copy:} a CNOT from a class member to a fresh
ancilla in \(\ket0\); the target is added to the source's class.
\item[(T3)] \emph{Record-controlled processing:} a unitary on qubits
outside all classes, controlled in the computational basis by class
members.
\item[(T4)] \emph{Residual processing:} a unitary acting only on qubits
outside all classes.
\end{enumerate}
After designation or copy creation, a class member may appear only as a
(T2) source or a (T3) control.

Write
\[
\Nset=\{1,\ldots,N\},
\qquad c_j=|K_j|,
\qquad
E=Q\setminus\bigcup_{j=1}^N K_j.
\]
The final pure state is \(\ket\Psi\), with
\(\rho=\ketbra\Psi\Psi\).
\end{definition}

The discipline means that an established record is not edited.
It is a restriction on internal gates, not on an external detecting
effect.
An external reversal circuit is allowed only through an interface
containing all factors on which that circuit acts.

A partition into observer levels may be supplied, but the results depend
on the final classes, not on the level names.
A record designated at one level and copied once at each of \(d-1\)
later levels has \(c=d\) members.
Throughout this paper a \(c\)-member read-chain contains one root and
\(c-1\) copying events.

\begin{theorem}[Branch normal form]\label{thm:bnf}
Every compiled tower has the form
\begin{equation}\label{eq:bnf}
\ket\Psi
=
\sum_{r\in\{0,1\}^N}
\left(\bigotimes_{j=1}^N\ket{r_j}^{\otimes K_j}\right)
\otimes\ket{\xi_r},
\qquad
p_r=\|\xi_r\|^2,
\qquad
\sum_rp_r=1.
\end{equation}
For supported branches \(p_r>0\), define
\[
\ket{\hat\xi_r}=\ket{\xi_r}/\sqrt{p_r},
\qquad
\ket{\Phi_r}
=
\left(\bigotimes_j\ket{r_j}^{\otimes K_j}\right)
\otimes\ket{\hat\xi_r}.
\]
Then
\[
\ket\Psi=\sum_r\sqrt{p_r}\ket{\Phi_r},
\qquad
\braket{\Phi_r}{\Phi_{r'}}=\delta_{rr'}.
\]
Phases are included in \(\hat\xi_r\).
All subsequent branch sums range over supported branches.
\end{theorem}

\begin{proof}
Induct on the event sequence.
With no designated classes the assertion is immediate.
For (T1), expand the residual vector in the computational basis of the
new root, creating an additional branch bit.
For (T2), a source in class \(K_j\) is \(\ket{r_j}\) in branch \(r\);
the fresh target becomes another factor \(\ket{r_j}\).
A (T3) gate acts on the residual vector by the unitary selected by \(r\).
A (T4) gate acts on each residual vector without touching the record
factors.
These exhaust the allowed events.
Distinct branch labels differ on a class carrying orthogonal
computational factors, proving orthonormality.
\end{proof}

\subsection{Pinching, anchoring, and dilation}

For a qubit set \(A\), let
\[
\Delta_A(Y)=\sum_aP_aYP_a
\]
be computational-basis pinching on \(A\), with identity factors elsewhere.
It is CPTP, idempotent, and Hilbert--Schmidt self-adjoint.
For \(J\subseteq\Nset\), define the fixed channel
\begin{equation}\label{eq:fixedpinch}
\Delta_J
=
\Delta_{\bigcup_{j\in J}K_j}.
\end{equation}
Thus \(\Delta_J\) is defined on all operators, not just tower states.

\begin{lemma}[Class anchoring]\label{lem:anchor}
Let \(A\subseteq\bigcup_jK_j\), with
\(A\cap K_j\ne\emptyset\) exactly when \(j\in J\).
On a tower state,
\begin{equation}\label{eq:anchoring}
\Delta_A(\rho)=\Delta_J(\rho)
=
\sum_{r,r':\,r_J=r'_J}
\sqrt{p_rp_{r'}}\ketbra{\Phi_r}{\Phi_{r'}}.
\end{equation}
In particular, pinching one member of each selected class is sufficient.
The equality need not hold on an arbitrary operator outside the
tower-supported subspace.
\end{lemma}

\begin{proof}
On a member \(q\in K_j\), a branch cross term carries
\(\ketbra{r_j}{r'_j}\).
Pinching that member preserves the term precisely when \(r_j=r'_j\).
The number and choice of members pinched within the class are therefore
irrelevant on the branch expansion.
The condition \(A\subseteq\bigcup_jK_j\) excludes additional pinching of
residual degrees of freedom.
\end{proof}

\begin{definition}[Coherent and dephased models]\label{def:laws}
For an effect \(M\) and \(J\subseteq\Nset\), define
\[
\Pcoh(M)=\tr(M\rho),
\qquad
\Pcoll_J(M)=\tr(M\Delta_J\rho).
\]
The difference operator is
\begin{equation}\label{eq:X}
X_J=\rho-\Delta_J\rho
=
\sum_{\substack{r\ne r'\\r_J\ne r'_J}}
\sqrt{p_rp_{r'}}\ketbra{\Phi_r}{\Phi_{r'}}.
\end{equation}
It is Hermitian and traceless, and the probability bias of \(M\) is
\(\tr(MX_J)\).
\end{definition}

\begin{proposition}[Discarded-copy dilation]\label{prop:deeper}
Choose one anchor in each class \(j\in J\).
Let \(C\) copy those anchors into a fresh register \(R'\).
Then
\[
\Delta_J\rho=\tr_{R'}(C\rho C^\dagger).
\]
Consequently, the dephased model is the coherent Born model of an extended
tower when its last copies are inaccessible.
\end{proposition}

\begin{proof}
For one anchor,
\[
\ketbra{b}{b'}\otimes\ketbra00
\longmapsto
\ketbra{b}{b'}\otimes\ketbra{b}{b'}.
\]
Tracing the copy yields the factor \(\delta_{bb'}\), hence pinching.
Repeat for each anchor and apply Lemma~\ref{lem:anchor}.
\end{proof}

\begin{remark}[What the dilation does and does not show]
The proposition identifies two specified statistical descriptions within
the Born formalism. It does not prove that exactly two probability maps
are possible under all interpretations or assumptions.
Likewise,
\[
\tr(MV\rho V^\dagger)=\tr(V^\dagger MV\rho)
\]
shows that external unitary processing can be absorbed into the effect;
it does not constitute a uniqueness theorem for measurement laws.
\end{remark}

\section{Exact channel, subsystem, and algebraic interfaces}
\label{sec:interface}

\begin{lemma}[Helstrom optimization]\label{lem:helstrom}
For traceless Hermitian \(Y\),
\[
\max_{0\le M\le\id}|\tr(MY)|=\tfrac12\|Y\|_1.
\]
The positive-spectral projector \(P_+(Y)\) attains the positive maximum.
\end{lemma}

\begin{proof}
Write \(Y=Y_+-Y_-\), with positive parts of orthogonal supports.
Since \(\tr Y=0\),
\(\tr Y_+=\tr Y_-=\frac12\|Y\|_1\).
For an effect \(M\),
\[
-\tr Y_-\le\tr(MY)\le\tr Y_+.
\]
Taking \(M=P_+(Y)\) attains the upper endpoint.
\end{proof}

This is the standard two-state discrimination optimization
\cite{Helstrom1976,Watrous2018}.

\begin{lemma}[Hermitian trace-norm contractivity]\label{lem:contract}
For a CPTP map \(\mathcal E\) and Hermitian \(Y\),
\(\|\mathcal E(Y)\|_1\le\|Y\|_1\).
\end{lemma}

\begin{proof}
Positivity and trace preservation give
\[
\|\mathcal E(Y)\|_1
\le\|\mathcal E(Y_+)\|_1+\|\mathcal E(Y_-)\|_1
=\tr Y_++\tr Y_-.
\]
\end{proof}

\begin{definition}[Interface channel]\label{def:channel}
An interface channel is a CPTP map
\(\Lambda:\BB(\HH_Q)\to\BB(\KK)\).
An input effect is available after \(\Lambda\) if it equals
\(\Lambda^*(M)\) for an output effect \(M\).
Define
\[
\delta_J(\Lambda)=\tfrac12\|\Lambda(X_J)\|_1.
\]
\end{definition}

\begin{theorem}[Channel-interface theorem]\label{thm:channel}
For a fixed tower, record set \(J\), and channel \(\Lambda\), the following
are equivalent:
\begin{enumerate}
\item The coherent and \(J\)-dephased models agree on every effect
available after \(\Lambda\).
\item \(\delta_J(\Lambda)=0\).
\item \(\Lambda(X_J)=0\).
\end{enumerate}
Moreover,
\begin{equation}\label{eq:channelbias}
\sup_{0\le M\le\id_\KK}
\left|
\tr(M\Lambda(\rho))-\tr(M\Lambda(\Delta_J\rho))
\right|
=
\delta_J(\Lambda).
\end{equation}
The supremum is attained by \(P_+(\Lambda(X_J))\).
\end{theorem}

\begin{proof}
The operator \(\Lambda(X_J)\) is Hermitian and traceless.
For every output effect,
\[
\tr(M\Lambda(X_J))=\tr(\Lambda^*(M)X_J).
\]
Apply Lemma~\ref{lem:helstrom} to \(\Lambda(X_J)\).
The three conditions are equivalent because the resulting maximum
vanishes exactly when that operator vanishes.
\end{proof}

\begin{corollary}[Data processing]\label{cor:dp}
For every further CPTP map \(\Gamma\),
\[
\delta_J(\Gamma\circ\Lambda)\le\delta_J(\Lambda).
\]
Exactly agreeing channels are closed under post-processing.
For subsystem interfaces the agreement family
\[
\mathcal D_J=\{S\subseteq Q:\delta_J(S)=0\}
\]
is a down-set.
\end{corollary}

\begin{proof}
Use Lemma~\ref{lem:contract}.
For \(S'\subseteq S\), restriction to \(S'\) is a further partial trace
after restriction to \(S\).
\end{proof}

\begin{corollary}[Exact subsystem interface theorem]\label{thm:interface}
Take \(\Lambda_S=\tr_{\Sbar}\).
Then
\[
\delta_J(S)=\tfrac12\|\tr_{\Sbar}X_J\|_1
=
\sup_{0\le M_S\le\id_S}
|\tr((M_S\otimes\id_{\Sbar})X_J)|.
\]
All effects on \(S\) agree if and only if
\(\tr_{\Sbar}X_J=0\).
A Helstrom projector on \(S\) attains the maximum.
\end{corollary}

\begin{proof}
The available effects after \(\tr_{\Sbar}\) are exactly
\(M_S\otimes\id_{\Sbar}\).
Apply Theorem~\ref{thm:channel}.
\end{proof}

\begin{lemma}[Finite-dimensional conditional expectation]
\label{lem:expectation}
For a unital \( * \)-subalgebra
\(\Alg\subseteq\BB(\HH_Q)\), the Hilbert--Schmidt orthogonal projection
\(E_\Alg\) onto \(\Alg\) is CPTP, unital, and self-adjoint.
Hermitian elements of \(\Alg\) have spectral projectors in \(\Alg\).
\end{lemma}

\begin{proof}
After a unitary change of coordinates, write
\[
\HH_Q=\bigoplus_\ell
\mathbb C^{n_\ell}\otimes\mathbb C^{m_\ell},
\qquad
\Alg=\bigoplus_\ell
\BB(\mathbb C^{n_\ell})\otimes\id_{m_\ell}.
\]
If \(P_\ell\) are the block projectors, then
\[
E_\Alg(Y)
=
\bigoplus_\ell
\tr_{m_\ell}(P_\ell YP_\ell)\otimes
\frac{\id_{m_\ell}}{m_\ell}.
\]
Block pinching, partial trace, and replacement by a maximally mixed
factor are completely positive.
The formula preserves trace and the identity and is the
Hilbert--Schmidt orthogonal projection.
Spectral projectors belong to \(\Alg\) by finite-dimensional functional
calculus.
\end{proof}

\begin{corollary}[Exact algebraic interface theorem]\label{cor:algebra}
For effects restricted to \(\Alg\),
\[
\sup_{\substack{0\le M\le\id\\M\in\Alg}}|\tr(MX_J)|
=
\tfrac12\|E_\Alg(X_J)\|_1.
\]
All such effects agree if and only if \(E_\Alg(X_J)=0\).
The corresponding Helstrom projector belongs to \(\Alg\).
\end{corollary}

\begin{proof}
For \(M\in\Alg\), self-adjointness gives
\(\tr(MX_J)=\tr(ME_\Alg(X_J))\).
The spectral projector is admissible by Lemma~\ref{lem:expectation}.
Apply Lemma~\ref{lem:helstrom}.
\end{proof}

For \(\Alg=\BB(\HH_S)\otimes\id_{\Sbar}\),
\[
E_\Alg(Y)=\tr_{\Sbar}(Y)\otimes
\frac{\id_{\Sbar}}{\dim\HH_{\Sbar}},
\]
so the algebraic formula recovers the subsystem formula.
Agreeing algebras and subsystems form ordered families; no general
closure under joins or unions is asserted.

\begin{corollary}[Measurement channels]\label{cor:measurement}
For a POVM \(\mathsf M=\{M_o\}\), let
\[
\Lambda_{\mathsf M}(Y)=\sum_o\tr(M_oY)\ketbra oo.
\]
Then
\[
\delta_J(\Lambda_{\mathsf M})
=
\tfrac12\sum_o|\tr(M_oX_J)|.
\]
For the binary POVM \(\{M,\id-M\}\), denoted by \(\Lambda_M\),
\[
\delta_J(\Lambda_M)=|\tr(MX_J)|.
\]
Thus a fixed question is itself a particular interface channel.
\end{corollary}

\begin{proof}
The output difference is diagonal.
For the binary case its two entries are
\(\tr(MX_J)\) and its negative.
\end{proof}

\begin{remark}[Noise in the interface]
Detector noise and classical post-processing can be included by
composition of channels.
They cannot increase \(\delta_J\).
This is a contraction statement about the total Hermitian difference;
it does not imply that every general branch cross operator is multiplied
by one universal scalar damping factor.
\end{remark}

\begin{corollary}[Record-commuting questions]\label{cor:commutant}
If \(M\) commutes with every computational record projector defining
\(\Delta_J\), then \(\tr(MX_J)=0\).
\end{corollary}

\begin{proof}
Such an \(M\) satisfies \(\Delta_J(M)=M\).
Hilbert--Schmidt self-adjointness of pinching gives
\(\tr(M\rho)=\tr(M\Delta_J\rho)\).
\end{proof}

For a fixed subsystem, individually agreeing effects form the closed
convex slice
\[
\{0\le M_S\le\id_S:
  \tr(M_S\,\tr_{\Sbar}X_J)=0\}.
\]
When the reduced difference is nonzero, this is a hyperplane slice of
the effect body, not the whole body.

\section{Cross operators, fidelity, screening, and noisy architecture}
\label{sec:visibility}

For distinct branch labels define the difference set
\[
D(r,r')=\{j:r_j\ne r'_j\}.
\]
A pair is \(J\)-split if \(D(r,r')\cap J\ne\emptyset\).
Pairs below are unordered unless an ordering is explicitly displayed.

\begin{definition}[Cross operators and visibility]\label{def:visibility}
For supported \(r\ne r'\), set
\[
\Gamma^S_{rr'}=\tr_{\Sbar}\ketbra{\Phi_r}{\Phi_{r'}},
\qquad
v_{rr'}(S)=\|\Gamma^S_{rr'}\|_1.
\]
Then
\begin{equation}\label{eq:cross-expansion}
\tr_{\Sbar}X_J
=
\sum_{\substack{r<r'\\J\text{-split}}}
\sqrt{p_rp_{r'}}
\left(\Gamma^S_{rr'}+\Gamma^{S\dagger}_{rr'}\right).
\end{equation}
\end{definition}

\begin{lemma}[Variational visibility]\label{lem:variational}
For every pair,
\[
v_{rr'}(S)
=
\max_{U_S\ {\rm unitary}}
\left|
\bra{\Phi_{r'}}(U_S\otimes\id_{\Sbar})\ket{\Phi_r}
\right|
\le1.
\]
Visibility is nondecreasing when \(S\) is enlarged.
\end{lemma}

\begin{proof}
For a square matrix \(A\), polar decomposition and trace duality give
\[
\|A\|_1=\max_{U\ {\rm unitary}}|\tr(UA)|.
\]
Apply this identity to \(\Gamma^S_{rr'}\) and use the defining property of
partial trace.
The overlap of two unit vectors has modulus at most one.
A unitary on a smaller interface extends by the identity to a larger
one, proving monotonicity.
\end{proof}

We use the square-root fidelity convention
\[
F(\sigma,\tau)=\|\sqrt\sigma\sqrt\tau\|_1.
\]

\begin{theorem}[Visibility is inaccessible-side fidelity]
\label{thm:fidelity}
For
\[
\sigma^{\Sbar}_r=\tr_S\ketbra{\Phi_r}{\Phi_r},
\]
one has
\begin{equation}\label{eq:visibility-fidelity}
v_{rr'}(S)=F(\sigma^{\Sbar}_r,\sigma^{\Sbar}_{r'}).
\end{equation}
\end{theorem}

\begin{proof}
This is Uhlmann's variational identity
\cite{Uhlmann1976}.
For completeness, write Schmidt decompositions
\[
\ket{\Phi_r}=\sum_k\sqrt{\lambda_k}\ket{u_k}\ket{x_k},
\qquad
\ket{\Phi_{r'}}=\sum_l\sqrt{\mu_l}\ket{w_l}\ket{y_l}.
\]
The overlap optimized in Lemma~\ref{lem:variational} is
\(\tr(UB)\), where
\[
B=\sum_{k,l}
\sqrt{\lambda_k\mu_l}\braket{y_l}{x_k}\ketbra{u_k}{w_l}.
\]
Its maximum is \(\|B\|_1\).
In the respective orthonormal Schmidt bases, the nonzero singular values
of \(B\) are those of the entrywise conjugate of the matrix of
\(\sqrt{\sigma^{\Sbar}_r}\sqrt{\sigma^{\Sbar}_{r'}}\).
Complex conjugation preserves singular values.
Thus the two trace norms coincide.
\end{proof}

\begin{proposition}[Perfect-copy screening and doubling]
\label{prop:screening}
For a perfect-record tower:
\begin{enumerate}
\item If some \(j\in D(r,r')\) has a member outside \(S\), then
\(\Gamma^S_{rr'}=0\).
\item If every member of every differing class lies in \(S\), then the
left and right supports of \(\Gamma^S_{rr'}\) are orthogonal, and
\[
\|\Gamma^S_{rr'}+\Gamma^{S\dagger}_{rr'}\|_1
=
2v_{rr'}(S).
\]
\end{enumerate}
\end{proposition}

\begin{proof}
An inaccessible differing member contributes
\(\tr\ketbra{r_j}{r'_j}=0\), proving the first assertion.
Otherwise, the reduced cross operator retains the differing perfect
record factors on \(S\).
They place its left and right supports in orthogonal computational
subspaces.
For an operator \(P\Gamma Q\) with \(PQ=0\), the Hermitian sum has block
form
\[
\begin{pmatrix}
0&B\\ B^\dagger&0
\end{pmatrix}.
\]
Its singular values are those of \(B\), each occurring twice.
\end{proof}

\begin{theorem}[Visibility envelope and two-branch identity]
\label{thm:visibility}
For every \(J,S\),
\begin{equation}\label{eq:visibility-bound}
\delta_J(S)
\le
\sum_{\substack{r<r'\\J\text{-split}}}
\sqrt{p_rp_{r'}}\,v_{rr'}(S).
\end{equation}
If exactly two branches \(r,r'\) are supported, then
\begin{equation}\label{eq:two-branch}
\delta_J(S)
=
\begin{cases}
\sqrt{p_rp_{r'}}\,v_{rr'}(S),
  &D(r,r')\cap J\ne\emptyset,\\
0,&D(r,r')\cap J=\emptyset.
\end{cases}
\end{equation}
The equality in the first line uses perfect-record structure.
\end{theorem}

\begin{proof}
Apply the triangle inequality to
\eqref{eq:cross-expansion} and use
\(\|\Gamma+\Gamma^\dagger\|_1\le2\|\Gamma\|_1\).
For two branches, an unsplit pair contributes nothing to \(X_J\).
A split pair is either screened, making both sides zero, or has
orthogonal supports as in Proposition~\ref{prop:screening}, giving
equality.
\end{proof}

\begin{corollary}[Combinatorial screening and copy budget]
\label{cor:budget}
If every supported \(J\)-split pair has an inaccessible perfect member of
a differing class, then \(\delta_J(S)=0\).
Conversely, detection requires a supported \(J\)-split pair satisfying
\[
S\supseteq\bigcup_{j\in D(r,r')}K_j.
\]
In particular,
\[
\delta_J(S)>0
\quad\Longrightarrow\quad
|S|\ge
\min_{\substack{r<r'\\J\text{-split}}}
\sum_{j\in D(r,r')}c_j.
\]
The minimum of an empty family is interpreted as \(+\infty\).
\end{corollary}

\begin{proof}
Screening kills every summand of \eqref{eq:cross-expansion}.
If its sum is nonzero, at least one cross operator is nonzero.
Proposition~\ref{prop:screening} then forces the stated containment.
\end{proof}

The budget is a necessary threshold.
Even after it is met, inaccessible residuals may lower visibility or
different cross operators may cancel.
Section~\ref{sec:cancellation} gives a same-data counterexample to
sufficiency.

\begin{example}[Perfect read-chain]\label{ex:ghz}
Start with a designated \(\ket+\) root and make \(c-1\) perfect copies.
The state is
\[
\ket{\mathrm{GHZ}_c}
=
\frac{\ket{0^c}+\ket{1^c}}{\sqrt2}.
\]
For an interface containing \(m<c\) class members, every effect has zero
bias.
At \(m=c\),
\[
\delta_J(S)=\tfrac12.
\]
The projector onto \(\ket{\mathrm{GHZ}_c}\) has coherent probability one
and dephased probability \(1/2\).
Thus the perfect-copy transition is a hard step at the full budget.
This is an example at every chain length, not a universal statement
that every budget-complete interface has maximal discrepancy.
\end{example}

\subsection{Independent noisy fragments}

Nonorthogonal pointer states are not interchangeable with perfect
computational records.

\begin{definition}[Independent pointer fragment]\label{def:fragment}
A fresh residual fragment \(a\) may be prepared under control of a
perfect record root \(q\) by
\[
\ket b_q\ket0_a\longmapsto\ket b_q\ket{e_a^{(b)}},
\qquad
\gamma_a=|\braket{e_a^{(0)}}{e_a^{(1)}}|.
\]
The fragment is not added to a perfect copy class unless it is actually
a computational-basis copy.
A product-law assertion additionally assumes that the branch states
retain the specified fragment tensor factors.
\end{definition}

The map is an isometry because the source labels are orthogonal.
However, a nonorthogonal fragment cannot generally be used as a perfect
source that copies its underlying branch bit.
For example, an isometry satisfying
\[
\ket{e^{(b)}}\ket0
\longmapsto
\ket{e^{(b)}}\ket b
\]
would preserve an input overlap that is nonzero while producing zero
output overlap.
Thus a noisy-fragment ``read-chain'' must specify how each fragment is
prepared; the independent construction below uses a retained perfect
root.

\begin{proposition}[Exact fragment product law]\label{prop:product-law}
Suppose the inaccessible branch states factorize as
\[
\sigma_r^{\Sbar}
=
\bigotimes_a\ketbra{e_a^{(r)}}{e_a^{(r)}}\otimes\tau_r,
\qquad
\sigma_{r'}^{\Sbar}
=
\bigotimes_a\ketbra{e_a^{(r')}}{e_a^{(r')}}\otimes\tau_{r'}.
\]
Then
\[
v_{rr'}(S)
=
\prod_a|\braket{e_a^{(r')}}{e_a^{(r)}}|
\;F(\tau_r,\tau_{r'}).
\]
In particular, the overlap product alone is an upper bound, and is an
equality when the residual fidelity is one.
\end{proposition}

\begin{proof}
Fidelity is multiplicative under tensor products:
both the square root and the trace norm factorize.
For pure factors,
\(F(\ketbra ee,\ketbra{e'}{e'})=|\braket{e'}e|\).
Apply Theorem~\ref{thm:fidelity}.
\end{proof}

\begin{corollary}[Exponential suppression with an accessible root]
\label{cor:noisy}
Consider
\[
\ket\Psi
=
\sqrt{p_0}\ket0_R\bigotimes_{a=1}^{c-1}\ket{e_a^{(0)}}
+
\sqrt{p_1}\ket1_R\bigotimes_{a=1}^{c-1}\ket{e_a^{(1)}}.
\]
Dephase the perfect root.
If \(S\) contains that root and \(m-1\) fragments, then
\[
\delta_J(S)=
\sqrt{p_0p_1}
\prod_{a\notin S}\gamma_a.
\]
For uniform \(\gamma_a=\gamma\),
\[
\delta_J(S)=\sqrt{p_0p_1}\,\gamma^{c-m}.
\]
Complex overlap phases do not change this equality.
\end{corollary}

\begin{proof}
Tracing the inaccessible fragments yields
\[
\Gamma^S=z\ketbra uv,
\qquad
|z|=\prod_{a\notin S}\gamma_a.
\]
The accessible vectors \(u,v\) are orthogonal because \(S\) retains the
perfect root.
The Hermitian cross sum therefore has trace norm \(2|z|\).
\end{proof}

If the root is inaccessible, exact screening remains.
Consequently the expression \(\gamma^{c-m}\) is not an architecture-free
formula for arbitrary choices of \(m\) accessible qubits.

\begin{example}[Correlated phase records]\label{ex:secret}
Let a retained perfect root control the two residual environment states
\[
\ket{E_b}
=
\frac{\ket{0^c}+(-1)^b\ket{1^c}}{\sqrt2},
\qquad b=0,1.
\]
Prepare the environment with residual gates, then apply a
record-controlled \(Z\) to one fragment.
If \(k<c\) environment fragments are inaccessible, their two reduced
states coincide, so \(v=1\).
If all \(c\) fragments are inaccessible, the two environment states are
orthogonal, so \(v=0\).
For \(k=0\), the inaccessible environment is the one-dimensional trivial
factor.

This differs sharply from independent redundant fragments.
The relationship to environment-based classicalization
\cite{Zurek2009} is architectural: fidelity measures information in the
inaccessible remainder, not merely the number of fragments.
\end{example}

\section{Operator-valued M\"obius hierarchy and three orders}
\label{sec:mobius}

\begin{definition}[M\"obius elements]\label{def:mobius}
For \(J\subseteq\Nset\), define
\[
Z_J=\sum_{K\subseteq J}(-1)^{|J|-|K|}\Delta_K\rho,
\qquad
\omega_J(\Lambda)=\tfrac12\|\Lambda(Z_J)\|_1,
\qquad
\eta_M(J)=\tr(MZ_J).
\]
One has \(Z_\emptyset=\rho\), and \(Z_J\) is traceless for nonempty \(J\).
\end{definition}

\begin{theorem}[Inversion and operator localization]
\label{thm:localization}
For every \(J\),
\[
\Delta_J\rho=\sum_{K\subseteq J}Z_K.
\]
For nonempty \(J\),
\begin{equation}\label{eq:localization}
Z_J
=
(-1)^{|J|}
\sum_{\substack{r\ne r'\ {\rm ordered}\\J\subseteq D(r,r')}}
\sqrt{p_rp_{r'}}\ketbra{\Phi_r}{\Phi_{r'}}.
\end{equation}
Equivalently,
\[
Z_J=\prod_{j\in J}(\Delta_{\{j\}}-\mathrm{id})(\rho).
\]
\end{theorem}

\begin{proof}
For inversion, interchange the subset sums. The coefficient of
\(\Delta_L\rho\) is
\[
\sum_{K:\,L\subseteq K\subseteq J}(-1)^{|K|-|L|},
\]
which is one when \(L=J\) and zero otherwise.

For localization, the diagonal terms cancel when \(J\ne\emptyset\).
A cross term with difference set \(D\) receives the coefficient
\[
\sum_{K\subseteq J\setminus D}(-1)^{|J|-|K|}
=
(-1)^{|J|}[J\subseteq D].
\]
The product formula follows by expanding the commuting factors.
\end{proof}

\begin{corollary}[Discrepancy decomposition]\label{cor:decomposition}
For every \(J\),
\[
X_J=-\sum_{\emptyset\ne K\subseteq J}Z_K,
\qquad
\delta_J(\Lambda)\le
\sum_{\emptyset\ne K\subseteq J}\omega_K(\Lambda).
\]
For a singleton,
\[
X_{\{j\}}=-Z_{\{j\}},
\qquad
\delta_{\{j\}}(\Lambda)=\omega_{\{j\}}(\Lambda).
\]
\end{corollary}

\begin{proof}
Subtract the inversion formula from \(Z_\emptyset=\rho\), then apply the
triangle inequality.
\end{proof}

\begin{proposition}[State interference complex]\label{prop:statecomplex}
For nonempty \(J\),
\[
Z_J\ne0
\quad\Longleftrightarrow\quad
J\subseteq D(r,r')
\text{ for some supported pair}.
\]
Moreover,
\[
\|Z_J\|_{\mathrm{HS}}^2
=
\sum_{\substack{r\ne r'\ {\rm ordered}\\J\subseteq D(r,r')}}
p_rp_{r'}.
\]
Thus the state support, after adjoining the empty face, is the
simplicial complex generated by supported branch difference sets.
\end{proposition}

\begin{proof}
The branch matrix units are Hilbert--Schmidt orthonormal:
\[
\tr\!\left[
(\ketbra{\Phi_r}{\Phi_{r'}})^\dagger
\ketbra{\Phi_s}{\Phi_{s'}}
\right]
=
\delta_{rs}\delta_{r's'}.
\]
Hence no distinct terms in \eqref{eq:localization} cancel globally.
The norm formula and support characterization follow.
Subset closure is immediate.
\end{proof}

\begin{definition}[State, interface, and question orders]
\label{def:orders}
Set
\[
\ord(\Lambda)
=
\min\{|J|:J\ne\emptyset,\ \Lambda(Z_J)\ne0\},
\]
and
\[
\ord(M)
=
\min\{|J|:J\ne\emptyset,\ \eta_M(J)\ne0\},
\]
with \(\min\emptyset=\infty\).
The state order is \(\ord(\mathrm{id})\).
An interface is \emph{fully classical for this hierarchy} if its order
is infinite.
\end{definition}

\begin{proposition}[Orders and adequacy]\label{prop:orders}
For every channel \(\Lambda\) and output effect \(M\),
\[
\ord(\mathrm{id})
\le
\ord(\Lambda)
\le
\ord(\Lambda^*(M)).
\]
Post-processing can only increase order and decrease each weight.
The state order is one whenever two supported branches coexist, and is
infinite otherwise.

The question order is at least \(m\) if and only if all dephasings of
fewer than \(m\) records leave its probability unchanged.
The corresponding statement holds for interface order with agreement
on every output effect.
\end{proposition}

\begin{proof}
If \(Z_J=0\), then \(\Lambda(Z_J)=0\).
If \(\Lambda(Z_J)=0\), then
\(\tr(M\Lambda(Z_J))=0\).
Contractivity proves the post-processing statement.

Any two distinct labels differ on some \(j\), and
Proposition~\ref{prop:statecomplex} then gives \(Z_{\{j\}}\ne0\).
With one supported branch all nonempty elements vanish.

Finally,
\[
\tr(M\Delta_J\rho)-\tr(M\rho)
=
\sum_{\emptyset\ne K\subseteq J}\eta_M(K).
\]
Vanishing below size \(m\) implies adequacy below size \(m\).
Conversely, induction on \(|J|\) recovers the vanishing of the
coefficients from adequacy.
The operator argument is identical after applying \(\Lambda\).
\end{proof}

\begin{corollary}[Questions as hierarchy interfaces]
\label{cor:question-order}
For a measurement channel,
\[
\omega_J(\Lambda_{\mathsf M})
=
\tfrac12\sum_o|\eta_{M_o}(J)|.
\]
For a binary question and nonempty \(J\),
\[
\omega_J(\Lambda_M)=|\eta_M(J)|,
\qquad
\ord(\Lambda_M)=\ord(M).
\]
\end{corollary}

\begin{proof}
Apply the measurement channel to \(Z_J\).
In the binary case its two diagonal entries are
\(\eta_M(J)\) and \(-\eta_M(J)\).
\end{proof}

\subsection{Subsystem support is down-closed}

The channel hierarchy and the subsystem hierarchy have different
combinatorial restrictions.

\begin{theorem}[Subsystem M\"obius support]\label{thm:subsystemsupport}
For a perfect-record tower and fixed subsystem \(S\), the family
\[
\mathcal I_S
=
\{J\ne\emptyset:\tr_{\Sbar}Z_J\ne0\}
\]
is down-closed under nonempty subsets.
In particular,
\[
\ord(\tr_{\Sbar})\in\{1,\infty\}.
\]
\end{theorem}

\begin{proof}
For each class meeting \(S\), choose an accessible member and let
\(Q_j\) be pinching that member on \(\HH_S\).
If \(K_j\cap S=\emptyset\), put \(Q_j=\mathrm{id}\).
The \(Q_j\) commute.

Inspect a branch cross term.
If a class has an inaccessible differing member, that term is already
zero after partial trace.
Every surviving term therefore has the same survival rule under global
class pinching and under its selected accessible anchor.
A class wholly outside \(S\) cannot distinguish surviving reduced terms.
Thus, for all \(K\subseteq\Nset\),
\[
\tr_{\Sbar}(\Delta_K\rho)
=
\prod_{j\in K}Q_j(\rho_S).
\]
Consequently,
\[
\tr_{\Sbar}Z_J
=
\prod_{j\in J}(Q_j-\mathrm{id})(\rho_S).
\]
For \(L\subseteq J\),
\[
\tr_{\Sbar}Z_J
=
\prod_{j\in J\setminus L}(Q_j-\mathrm{id})
\bigl(\tr_{\Sbar}Z_L\bigr).
\]
If the rightmost argument were zero, the result would be zero.
Hence a nonzero \(J\)-element forces every nonempty subset element to be
nonzero.
A nonempty support therefore contains a singleton.
\end{proof}

\subsection{Arbitrary channels can realize arbitrary support families}

General channel supports need not be simplicial complexes.
The order-two measurement channel in
Section~\ref{sec:fr} already supplies a counterexample.
In fact, there is no down-closure restriction for unrestricted binary
measurement channels.

\begin{theorem}[Realization of every channel support family]
\label{thm:anysupport}
Fix \(N\ge1\).
There is a full-support compiled tower such that, for every family
\[
\mathcal F\subseteq
2^{\Nset}\setminus\{\emptyset\},
\]
a binary measurement channel \(\Lambda_M\) satisfies
\[
\{J\ne\emptyset:\Lambda_M(Z_J)\ne0\}=\mathcal F.
\]
Thus every order in \(\{1,\ldots,N,\infty\}\) is realizable at the
question and arbitrary-channel levels.
\end{theorem}

\begin{proof}
Prepare \(\ket+^{\otimes N}\) and designate all \(N\) qubits.
The classes are singleton, every branch is supported, and
\(p_r=2^{-N}\).

For each nonempty \(D\subseteq\Nset\), define
\[
T_D
=
2^{-N}\sum_r
\ketbra{\Phi_r}{\Phi_{r\oplus 1_D}},
\]
where \(1_D\) flips precisely the bits in \(D\).
These are nonzero Hermitian operators with mutually disjoint matrix-unit
supports, hence are Hilbert--Schmidt orthogonal.
Localization gives
\[
Z_J=(-1)^{|J|}\sum_{D\supseteq J}T_D.
\]
The Boolean-lattice transformation on nonempty subsets is triangular
with nonzero diagonal, so the \(Z_J\), \(J\ne\emptyset\), are linearly
independent over the reals.

Their real Gram matrix
\(G_{JK}=\tr(Z_JZ_K)\) is therefore positive definite.
Set \(a_J=1\) for \(J\in\mathcal F\) and \(a_J=0\) otherwise.
Choose a Hermitian
\[
H=\sum_K h_KZ_K,
\qquad
Gh=a,
\]
so that \(\tr(HZ_J)=a_J\).
Take \(t>0\) with \(t\|H\|_\infty\le1\), and put
\[
M=\tfrac12(\id+tH).
\]
This is an effect.
Since \(Z_J\) is traceless for nonempty \(J\),
\[
\eta_M(J)=\tfrac t2 a_J.
\]
Corollary~\ref{cor:question-order} gives the prescribed support.
For the empty family, \(M=\id/2\) suffices.
\end{proof}

\begin{remark}[Resolved and unresolved combinatorics]
Down-closure for arbitrary channel support is false, not an open
question. Unrestricted support-family realizability is settled by
Theorem~\ref{thm:anysupport}.
Realizability with a prescribed subsystem, branch support, locality
restriction, or implementability class is a different problem and is not
settled by that theorem.
\end{remark}

\section{Exact operator cancellation with unchanged scalar data}
\label{sec:cancellation}

\begin{example}[Five-qubit phase family]\label{ex:cancellation}
Use qubits \(s_1,a_1,s_2,a_2,A\), initially zero.
Apply:
\[
H_{s_1},\quad
\mathrm{CNOT}_{s_1\to a_1},\quad
H_{s_2},\quad
\mathrm{CNOT}_{s_2\to a_2},
\]
then \(X\) on \(A\) controlled by \(s_2=1\), and
\(\operatorname{diag}(1,e^{i\theta})\) on \(A\) controlled by
\(s_1=s_2=1\).
These are legal compiled events.
The classes are
\[
K_1=\{s_1,a_1\},
\qquad
K_2=\{s_2,a_2\},
\]
and the final state is
\begin{equation}\label{eq:phasefamily}
\ket{\Psi_\theta}
=
\tfrac12\sum_{a,b\in\{0,1\}}
e^{i\theta ab}\ket{aa}_{K_1}\ket{bb}_{K_2}\ket b_A.
\end{equation}
Let \(J=\{1\}\), \(S=K_1\), and \(S'=K_1\cup\{A\}\).
\end{example}

\begin{proposition}[Same visibilities, different discrepancy]
\label{prop:cancellation}
For the family \eqref{eq:phasefamily},
\[
\delta_{\{1\}}(S)
=
\frac{|1+e^{-i\theta}|}{4}
=
\tfrac12|\cos(\theta/2)|,
\qquad
\delta_{\{1\}}(S')=\tfrac12.
\]
The copy classes, branch probabilities, and all pairwise visibilities on
\(S\) are independent of \(\theta\).
At \(\theta=\pi\), every nonempty M\"obius element vanishes on \(S\), so
that subsystem is fully classical for the hierarchy.
\end{proposition}

\begin{proof}
Use the normalized branch vectors
\[
\ket{\Phi_{ab}}=
e^{i\theta ab}\ket{aa}\ket{bb}\ket b,
\qquad p_{ab}=\tfrac14.
\]
Pairs with different \(b\)-values are screened by inaccessible \(K_2\).
The two surviving split pairs are
\((0b,1b)\), and
\[
\Gamma^S_{0b,1b}
=
e^{-i\theta b}\ketbra{00}{11}.
\]
Each has trace norm one.
Therefore
\[
\tr_{\Sbar}X_{\{1\}}
=
\frac{1+e^{-i\theta}}4\ketbra{00}{11}
+
\frac{1+e^{i\theta}}4\ketbra{11}{00}.
\]
Its eigenvalues are
\(\pm|1+e^{-i\theta}|/4\).

On \(S'\), the two contributions occupy orthogonal \(A\)-blocks:
\[
\tr_{\overline{S'}}X_{\{1\}}
=
\tfrac14\sum_b
\left(
e^{-i\theta b}\ketbra{00}{11}
+
e^{i\theta b}\ketbra{11}{00}
\right)\otimes\ketbra bb_A.
\]
Each block has eigenvalues \(\pm1/4\), giving total discrepancy \(1/2\).

At \(\theta=\pi\), the singleton \(Z_{\{1\}}\) vanishes on \(S\).
Every element involving record \(2\) consists of pairs differing on
inaccessible \(K_2\), hence is screened.
All nonempty reduced elements therefore vanish.
\end{proof}

\begin{corollary}[Scalar data do not determine agreement]
\label{cor:noscalar}
No function of the copy-class structure, branch probabilities, and
pairwise scalar visibilities alone decides whether \(\delta_J(S)=0\).
The states at \(\theta=0\) and \(\theta=\pi\) have exactly the same such
data on the same interface, but discrepancies \(1/2\) and zero.
\end{corollary}

\begin{proof}
The data equality and discrepancy inequality are
Proposition~\ref{prop:cancellation}.
\end{proof}

At \(\theta=\pi\), the qubit \(A\) carries the relative sign of the two
surviving cross terms. A Helstrom effect on \(S'\) projects onto
\[
\operatorname{span}\left\{
\frac{\ket{00,0}+\ket{11,0}}{\sqrt2},
\frac{\ket{00,1}-\ket{11,1}}{\sqrt2}
\right\}.
\]

\begin{corollary}[Agreement is not union-closed]\label{cor:union}
At \(\theta=\pi\),
\[
\delta_{\{1\}}(K_1)=0,
\qquad
\delta_{\{1\}}(\{A\})=0,
\qquad
\delta_{\{1\}}(K_1\cup\{A\})=\tfrac12.
\]
Thus \(\mathcal D_{\{1\}}\) is a down-set but not closed under unions.
\end{corollary}

\begin{proof}
The first and third values were proved above.
On \(\{A\}\), inaccessible \(K_1\) screens every split pair.
\end{proof}

\paragraph{Layers of agreement.}
Perfect-copy screening is a termwise sufficient certificate.
Residual distinguishability changes the norms of surviving terms.
Operator cancellation concerns their weighted Hermitian sum.
Question-specific agreement concerns the final linear functional
\(Y\mapsto\tr(MY)\), and includes commuting questions as well as
cancellation between nonzero scalar contributions.
Higher-order M\"obius behavior concerns the dependence on record sets.
These are useful distinctions, not an exclusive five-way partition:
attenuation need not yield exact agreement, and several mechanisms may
coexist.

\section{Frauchiger--Renner probabilities and parameter geometry}
\label{sec:fr}

\subsection{The real-positive family}

Designate two lab qubits \(a,b\).
A preparation followed by a record-controlled rotation produces
\begin{equation}\label{eq:fr}
\ket\Psi
=
\sqrt{p_{00}}\ket{00}
+
\sqrt{p_{10}}\ket{10}
+
\sqrt{p_{11}}\ket{11},
\qquad
p_{00}+p_{10}+p_{11}=1.
\end{equation}
The record classes may be singleton.
Let
\[
\ket{\mathrm{ok}}=\frac{\ket0-\ket1}{\sqrt2},
\qquad
\ket w=\ket{\mathrm{ok}}\otimes\ket{\mathrm{ok}},
\qquad
M=\ketbra ww.
\]
For a pair \(\pi=\{r,r'\}\), define its scalar cross contribution
\[
C_M(\pi)
=
2\sqrt{p_rp_{r'}}\Ree\bra{\Phi_{r'}}M\ket{\Phi_r}.
\]
Then
\[
\tr(M\Delta_J\rho)
=
D_M+\sum_{\pi:\,D(\pi)\cap J=\emptyset}C_M(\pi),
\qquad
D_M=\sum_rp_r\bra{\Phi_r}M\ket{\Phi_r}.
\]

\begin{proposition}[FR profile and interface discrepancy]
\label{prop:frprofile}
At \(p_{00}=p_{10}=p_{11}=1/3\),
\[
D_M=\tfrac14,
\quad
C_M(00,10)=-\tfrac16,
\quad
C_M(10,11)=-\tfrac16,
\quad
C_M(00,11)=\tfrac16.
\]
Consequently,
\[
\begin{array}{c|cccc}
J&\emptyset&\{1\}&\{2\}&\{1,2\}\\ \hline
\tr(M\Delta_J\rho)&
1/12&1/12&1/12&1/4
\end{array}
\]
but, on the full two-qubit interface,
\[
\delta_{\{1\}}=\delta_{\{2\}}=\frac{\sqrt2}{3}.
\]
\end{proposition}

\begin{proof}
The overlaps with \(00,10,11\) are \(1/2,-1/2,1/2\).
Substitution gives the cross contributions and the four probabilities.

For record \(1\),
\[
X_{\{1\}}
=
\ketbra{\alpha}{\beta}+\ketbra{\beta}{\alpha},
\quad
\ket\alpha=\frac{\ket{00}}{\sqrt3},
\quad
\ket\beta=\frac{\ket{10}+\ket{11}}{\sqrt3}.
\]
The vectors are orthogonal, with norm product \(\sqrt2/3\).
The nonzero eigenvalues are therefore \(\pm\sqrt2/3\).
The record-\(2\) calculation is the corresponding exchanged
decomposition.
\end{proof}

\begin{theorem}[FR order two is question-level]\label{thm:frorder}
For positive real amplitudes,
\begin{align}
\eta_M(\{1\})
&=
\tfrac12\sqrt{p_{00}}
\left(\sqrt{p_{10}}-\sqrt{p_{11}}\right),\label{eq:freta1}\\
\eta_M(\{2\})
&=
\tfrac12\sqrt{p_{11}}
\left(\sqrt{p_{10}}-\sqrt{p_{00}}\right),\label{eq:freta2}\\
\eta_M(\{1,2\})
&=
\tfrac12\sqrt{p_{00}p_{11}}.\label{eq:freta12}
\end{align}
At the symmetric point,
\[
\ord(\mathrm{id})=1,
\qquad
\ord(M)=\ord(\Lambda_M)=2,
\]
with weights
\[
\begin{array}{c|ccc}
&\{1\}&\{2\}&\{1,2\}\\ \hline
\omega_J(\mathrm{id})&
\sqrt2/3&\sqrt2/3&1/3\\
\omega_J(\Lambda_M)&0&0&1/6
\end{array}
\]
The order-two assertion is about Wigner's effect or its binary measurement
channel, not about the state.
\end{theorem}

\begin{proof}
Localization gives
\[
\eta_M(\{1\})=-C_M(00,10)-C_M(00,11),
\]
\[
\eta_M(\{2\})=-C_M(00,11)-C_M(10,11),
\qquad
\eta_M(\{1,2\})=C_M(00,11).
\]
Substitute the overlaps.
The state singleton weights equal the discrepancies of
Proposition~\ref{prop:frprofile}.
The two-record operator \(Z_{\{1,2\}}\) contains only the pair
\((00,11)\), so its half trace norm is \(\sqrt{p_{00}p_{11}}\).
The question weights follow from
Corollary~\ref{cor:question-order}.
\end{proof}

\begin{proposition}[Non-robust single-record adequacy]
\label{prop:frrobustness}
On the positive real simplex, friend \(1\)'s dephasing is adequate for
\(M\) exactly when \(p_{10}=p_{11}\), and friend \(2\)'s exactly when
\(p_{00}=p_{10}\).
Each condition is codimension one.
Their simultaneous satisfaction is the symmetric point, of codimension
two in the two-dimensional simplex.

For
\[
p_{00}=\tfrac13,\quad
p_{10}=\tfrac13+\varepsilon,\quad
p_{11}=\tfrac13-\varepsilon,
\]
\[
\tr(M\Delta_{\{1\}}\rho)-\tr(M\rho)
=
\tfrac{\varepsilon}{2}+O(\varepsilon^3).
\]
The two-record coefficient remains nonzero for sufficiently small
\(\varepsilon\), although the question order generally changes to one.
\end{proposition}

\begin{proof}
Use \eqref{eq:freta1}--\eqref{eq:freta12} and positivity.
The square-root difference satisfies
\[
\sqrt{\tfrac13+\varepsilon}
-
\sqrt{\tfrac13-\varepsilon}
=
\sqrt3\,\varepsilon+O(\varepsilon^3).
\]
Multiply by \(1/(2\sqrt3)\).
Continuity preserves the positive two-record coefficient locally.
\end{proof}

Thus ``no single scapegoat'' denotes equality of these unconditional
probabilities for this question at the symmetric point.
It is not the absence of a detectable single-record difference on the
same laboratory interface.

\begin{proposition}[Additional copies screen the laboratory question]
\label{prop:frcopies}
Add one perfect copy of each record.
On the original laboratory interface \(\{a,b\}\), all branch cross
terms are screened, and Wigner's original effect has coherent
probability \(1/4\).
On all four qubits, the effect projecting onto
\[
\frac{\ket{00}-\ket{11}}{\sqrt2}\bigg|_{K_1}
\otimes
\frac{\ket{00}-\ket{11}}{\sqrt2}\bigg|_{K_2}
\]
has coherent probability \(1/12\) at the symmetric point.
\end{proposition}

\begin{proof}
Every differing class has a copy outside the laboratory interface.
The reduced state is therefore its record-diagonal mixture.
Each supported basis branch has overlap squared \(1/4\) with \(M\).

The four-qubit effect has overlaps \(1/2,-1/2,1/2\) with the three
encoded branches, reproducing the coherent amplitude \(1/(2\sqrt3)\).
\end{proof}

\subsection{Complex amplitudes and the order-two stratum}

Write
\[
\ket\Psi
=
x\ket{00}
+
y e^{i\alpha}\ket{10}
+
z e^{i\beta}\ket{11},
\qquad
x,y,z>0,
\qquad
x^2+y^2+z^2=1.
\]

\begin{proposition}[Closed forms]\label{prop:frcomplex}
For the same effect \(M\),
\begin{align}
\eta_M(\{1\})
&=\tfrac12\left(xy\cos\alpha-xz\cos\beta\right),\\
\eta_M(\{2\})
&=\tfrac12\left(yz\cos(\alpha-\beta)-xz\cos\beta\right),\\
\eta_M(\{1,2\})
&=\tfrac12xz\cos\beta.
\end{align}
\end{proposition}

\begin{proof}
The three cross contributions are
\[
-\tfrac12xy\cos\alpha,\qquad
-\tfrac12yz\cos(\alpha-\beta),\qquad
+\tfrac12xz\cos\beta.
\]
Apply the scalar localization identities used above.
\end{proof}

\begin{theorem}[Exact FR order stratification]\label{thm:frgeometry}
On the positive-amplitude simplex with two phase parameters:
\begin{enumerate}
\item The order-one region is open and dense.
\item The order-two region is a smooth codimension-two locus.
It is real-algebraic with nonvanishing inequalities when expressed in
real and imaginary amplitude coordinates.
Its intersection with \(\alpha=\beta=0\) is exactly
\(x=y=z=1/\sqrt3\).
\item There is no infinite-order point: the three nonempty question
coefficients cannot all vanish.
\end{enumerate}
\end{theorem}

\begin{proof}
Set
\[
A=\cos\alpha,\qquad
B=\cos\beta,\qquad
C=\cos(\alpha-\beta).
\]
Order two requires
\[
yA=zB,\qquad yC=xB,\qquad B\ne0.
\]
Since \(x,y,z>0\), these equations require \(A,B,C\) to be nonzero and
have the same sign.
Conversely, on that open phase domain the unique positive normalized
solution is
\begin{equation}\label{eq:frgraph}
(x,y,z)
=
\frac{(|C|,|B|,|A|)}{\sqrt{A^2+B^2+C^2}}.
\end{equation}
Thus the order-two set is the smooth graph of amplitudes over an open
two-dimensional phase domain, inside the four-dimensional parameter
space. Its codimension is two.
At zero phases, \eqref{eq:frgraph} gives the symmetric point.

For the algebraic description, write the amplitudes as
\(x\), \(u+iv\), and \(w+it\), with \(x>0\).
The two equations become
\[
u=w,\qquad uw+vt=xw,
\]
with normalization
\(x^2+u^2+v^2+w^2+t^2=1\) and \(w\ne0\).
These are polynomial equations and inequalities.
In angular coordinates the same locus is real-analytic, rather than
polynomial in the angles themselves.

If all three coefficients vanished, \(\cos\beta=0\) would follow.
Then \(\cos\alpha=0\).
Both phases would be odd multiples of \(\pi/2\), so
\(|\cos(\alpha-\beta)|=1\).
The second coefficient would be nonzero, a contradiction.

Finally, the order-two locus has empty interior, while the nonvanishing
of at least one singleton coefficient defines an open set.
The explicit graph description and the absence of an infinite-order
point establish density of that open order-one set.
\end{proof}

No Monte-Carlo observation is needed for the codimension or
nonexistence assertions.

\section{Sequential agreement and accessible adaptive sectors}
\label{sec:sequential}

\begin{definition}[Sequential dephased execution]\label{def:sequential}
The sequential execution inserts a nonselective computational pinching
immediately after each (T1) designation.
All gates are otherwise unchanged.
Its final state is \(\rho_{\mathrm{seq}}\).
\end{definition}

\begin{lemma}[Record covariance]\label{lem:covariance}
After designation of a root \(s\), every later compiled gate commutes,
as a conjugation channel, with pinching on \(s\).
\end{lemma}

\begin{proof}
A gate touching \(s\) can only use it as a computational control or a
CNOT source, so it is block-diagonal in the basis of \(s\).
Every other gate acts away from \(s\).
In either case, conjugation commutes with the pinching.
\end{proof}

\begin{definition}[Record-invariant algebra]\label{def:invariant}
Let
\[
\Alg_{\mathrm{inv}}
=
\{M:\Delta_{\Nset}(M)=M\}.
\]
Equivalently, its operators commute with every computational projector
on the established record members.
\end{definition}

\begin{theorem}[Exact sequential agreement]\label{thm:sequential}
For every compiled tower,
\[
\rho_{\mathrm{seq}}=\Delta_{\Nset}\rho.
\]
For every \(M\in\Alg_{\mathrm{inv}}\),
\[
\Pseq(M)=\Pcoll_{\Nset}(M)=\Pcoh(M).
\]
The assertion holds at every circuit depth.
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:covariance}, each root pinching can be moved past all
later gates.
Pinchings on different roots commute.
Their final composition is pinching on all roots, which equals
\(\Delta_{\Nset}\rho\) by anchoring.

For invariant \(M\), self-adjointness gives
\[
\tr(M\Delta_{\Nset}\rho)
=
\tr(\Delta_{\Nset}(M)\rho)
=
\tr(M\rho).
\]
\end{proof}

Computational readouts of arbitrary register subsets commute with record
projectors, as do their classical coarse-grainings.
Record-block-diagonal adaptive effects are also covered.
This is a statement for each tower, not a claim that adding levels can
never change the circuit's ordinary outcome probabilities.

\begin{corollary}[Record distributions]\label{cor:recorddist}
The coherent and sequential executions have the same joint distribution
\(\{p_r\}\) of established record values.
Conditional probabilities among those record variables agree whenever
the conditioning event has positive probability.
\end{corollary}

\begin{proof}
Record-value projectors belong to \(\Alg_{\mathrm{inv}}\).
Their probabilities are \(p_r\) by the normal form.
Classical conditioning then uses the same joint distribution.
\end{proof}

This corollary concerns record variables.
It does not identify every conditional prediction for a later
non-record-diagonal interference measurement with a classical
conditional.

\begin{proposition}[Flagged adaptive access]\label{prop:adaptive}
Let \(\{P_s\}\) be orthogonal sector projectors commuting with
\(\Delta_J\), and define
\[
w_s=\tr(P_s\rho),\qquad
\rho_s=\frac{P_s\rho P_s}{w_s}
\]
for \(w_s>0\).
Suppose the interface supplies an accessible orthogonal sector flag:
\[
\Lambda_{\mathrm{flag}}(Y)
=
\bigoplus_s\Lambda_s(P_sYP_s),
\]
where \(\Lambda_s\) is CPTP on the \(s\)-sector.
Then
\[
\delta_J(\Lambda_{\mathrm{flag}})
=
\sum_sw_s\,
\tfrac12\|\Lambda_s(\rho_s-\Delta_J\rho_s)\|_1.
\]
Sector-wise Helstrom effects attain this value.
After forgetting the flag, only the upper bound by this sum is
guaranteed, and signed sector contributions may cancel.
\end{proposition}

\begin{proof}
Commutation gives
\[
P_sX_JP_s=w_s(\rho_s-\Delta_J\rho_s).
\]
Off-diagonal sectors are removed by the flagged interface.
Trace norm is additive on orthogonal direct sums, and the resulting
blocks are traceless.
Assemble the positive-spectral projectors blockwise.
Forgetting the flag is CPTP post-processing, so
Corollary~\ref{cor:dp} gives the upper bound.
Without the flag, the block effects need not be separately
implementable.
\end{proof}

\section{Robustness and the instability of exact order}
\label{sec:robustness}

For a linear map \(\Phi\), let \(\|\Phi\|_\diamond\) denote the completely
bounded trace norm.
The same fixed channels \eqref{eq:fixedpinch} are used for ideal and
perturbed states.
Anchoring is not assumed off the ideal tower state.

\begin{lemma}[Telescoping state error]\label{lem:telescope}
Let
\[
\rho=\mathcal G_L\cdots\mathcal G_1(\rho_0),
\qquad
\tilde\rho=\widetilde{\mathcal G}_L\cdots
\widetilde{\mathcal G}_1(\rho_0),
\]
where all maps are CPTP and
\[
\|\widetilde{\mathcal G}_\ell-\mathcal G_\ell\|_\diamond
\le\varepsilon_\ell.
\]
Then
\[
\|\tilde\rho-\rho\|_1\le
\epsilon,
\qquad
\epsilon=\sum_{\ell=1}^L\varepsilon_\ell.
\]
\end{lemma}

\begin{proof}
Expand the difference of the compositions as a telescoping sum, using
ideal maps before the differing factor and perturbed maps after it.
The differing factor contributes at most \(\varepsilon_\ell\) in trace
norm. Subsequent CPTP maps contract the Hermitian difference.
Sum the bounds.
\end{proof}

\begin{theorem}[Discrepancy stability]\label{thm:robustness}
For a fixed interface \(\Lambda\),
\[
\left|
\tfrac12\|\Lambda(\tilde\rho-\Delta_J\tilde\rho)\|_1
-
\tfrac12\|\Lambda(\rho-\Delta_J\rho)\|_1
\right|
\le
\|\tilde\rho-\rho\|_1
\le\epsilon.
\]
Thus ideal exact agreement implies perturbed discrepancy at most
\(\epsilon\), and an ideal discrepancy larger than \(\epsilon\)
remains nonzero.
\end{theorem}

\begin{proof}
Put \(Y=\tilde\rho-\rho\).
The reverse triangle inequality and contractivity give
\[
|\tilde\delta_J-\delta_J|
\le
\tfrac12\|\Lambda(Y-\Delta_JY)\|_1
\le
\tfrac12\bigl(\|Y\|_1+\|\Delta_JY\|_1\bigr)
\le\|Y\|_1.
\]
Apply Lemma~\ref{lem:telescope}.
\end{proof}

\begin{corollary}[M\"obius-weight stability]\label{cor:weightstability}
For nonempty \(J\),
\[
|\tilde\omega_J(\Lambda)-\omega_J(\Lambda)|
\le
2^{|J|-1}\epsilon.
\]
A nonzero weight larger than this bound remains nonzero.
An ideal zero weight becomes at most this bound.
\end{corollary}

\begin{proof}
The change in \(Z_J\) is a signed sum of \(2^{|J|}\) pinched copies of
\(Y\).
Each has trace norm at most \(\|Y\|_1\).
Apply the triangle inequality, \(\Lambda\), and the prefactor \(1/2\).
\end{proof}

\begin{remark}[Exact order is not generally stable]
The weight bound does not preserve exact classicality order.
Arbitrarily small perturbations can create nonzero singleton weights
where exact cancellation previously gave zero.
The FR perturbation in Proposition~\ref{prop:frrobustness} is an explicit
example.

For thresholds \(\tau_J>0\), membership in
\[
\{J:\omega_J(\Lambda)>\tau_J\}
\]
is preserved whenever the ideal distance from each threshold exceeds
\(2^{|J|-1}\epsilon\).
Exact order can instead be preserved under additional restrictions that
enforce the lower-order zeros exactly.
\end{remark}

\begin{proposition}[Additional interface error]\label{prop:interfaceerror}
If
\(\|\tilde\Lambda-\Lambda\|_\diamond\le\eta\), then
\[
|\delta_J(\tilde\Lambda)-\delta_J(\Lambda)|
\le\tfrac\eta2\|X_J\|_1\le\eta.
\]
Combining state and interface perturbations yields
\[
|\tilde\delta_J(\tilde\Lambda)-\delta_J(\Lambda)|
\le\epsilon+\eta.
\]
\end{proposition}

\begin{proof}
Use the reverse triangle inequality and the diamond-norm bound on
\(X_J\).
Since \(X_J\) is a difference of states, \(\|X_J\|_1\le2\).
For the combined statement, first change the state at fixed
\(\tilde\Lambda\), then change the interface at fixed ideal state.
\end{proof}

\section{Implementability constraints and accessibility hypergraphs}
\label{sec:resources}

\begin{definition}[Restricted implementable bias]\label{def:cost}
For a class \(\mathsf C\) of output effects,
\[
\delta_J^{(\mathsf C)}(\Lambda)
=
\sup_{M\in\mathsf C}|\tr(M\Lambda(X_J))|
\le\delta_J(\Lambda).
\]
The quantity is monotone under enlargement of \(\mathsf C\).
A data-processing comparison with restricted effects requires a
specified pullback relation between the two implementability classes.
\end{definition}

If
\[
\mathsf C=\{M:0\preceq M\preceq\id,\ \mathcal F(M)\preceq C\}
\]
for an affine Hermitian-matrix-valued map \(\mathcal F\), the restricted
bias is the larger optimum of two semidefinite programs with objectives
\(\pm\tr(M\Lambda(X_J))\).
Convexity alone does not imply that an arbitrary effect class has an
exact semidefinite representation.

\begin{theorem}[Single-product-effect read-chain bias]\label{thm:productgap}
For the \(c\)-member GHZ read-chain at full access, let
\[
\Cprod=
\left\{
\bigotimes_{q=1}^cM_q:0\le M_q\le\id
\right\}.
\]
Then
\[
\delta_J^{(\Cprod)}=2^{-c},
\qquad
\delta_J(S)=\tfrac12.
\]
\end{theorem}

\begin{proof}
Here
\[
X_J=\tfrac12(\ketbra{0^c}{1^c}+\ketbra{1^c}{0^c}),
\]
so a product effect has bias
\[
\Ree\prod_{q=1}^c\bra1M_q\ket0.
\]
For a two-dimensional effect, the eigenvalue gap is at most one.
Writing it as
\(\left(\begin{smallmatrix}a&z\\\bar z&b\end{smallmatrix}\right)\),
this gap is \(\sqrt{(a-b)^2+4|z|^2}\), hence \(|z|\le1/2\).
The bias is therefore at most \(2^{-c}\).
Taking every \(M_q=\ketbra++\) attains it.
The unrestricted value follows from the two eigenvalues
\(\pm1/2\) of \(X_J\).
\end{proof}

\begin{proposition}[Local parity attains the unrestricted value]
\label{prop:parity}
The effect
\[
M_{\mathrm{parity}}=\tfrac12(\id+X^{\otimes c})
\]
is implementable by measuring each qubit in the \(X\)-basis and accepting
the even-parity outcomes.
It has bias \(1/2\).
Thus the exponential bound in Theorem~\ref{thm:productgap} is not a
limitation on general product POVMs, separable effects, or local
measurements with classical post-processing.
\end{proposition}

\begin{proof}
The even-parity effect is the sum of the corresponding tensor-product
\(X\)-basis projectors, so it has the stated local implementation.
The coherent GHZ state has \(X^{\otimes c}\)-expectation one.
Its record-dephased mixture has expectation zero.
Their acceptance probabilities are therefore one and \(1/2\).
\end{proof}

\subsection{The perfect-copy hypergraph}

\begin{definition}[Accessibility hypergraph]\label{def:hypergraph}
For each supported \(J\)-split pair \(\pi=\{r,r'\}\), define
\[
E_\pi=\bigcup_{j\in D(r,r')}K_j.
\]
The hypergraph has vertex set \(Q\) and these labelled hyperedges.
Visibility is evaluated at an interface \(S\); it is not a fixed
interface-independent edge weight.
\end{definition}

\begin{proposition}[Covering and a conditional tight budget]
\label{prop:covering}
Detection requires \(E_\pi\subseteq S\) for some split pair.
If \(\pi^*\) minimizes edge size, has
\(v_{\pi^*}(E_{\pi^*})>0\), and is the unique split pair whose edge is
contained in \(E_{\pi^*}\), then
\[
\delta_J(E_{\pi^*})
=
\sqrt{p_rp_{r'}}\,v_{\pi^*}(E_{\pi^*})>0.
\]
Under these hypotheses the minimal edge-size lower bound is attained.
\end{proposition}

\begin{proof}
The necessity is Corollary~\ref{cor:budget}.
At \(S=E_{\pi^*}\), all other split pairs are screened.
The surviving pair has orthogonal record supports, so
Proposition~\ref{prop:screening} gives the displayed equality.
\end{proof}

The uniqueness hypothesis cannot be omitted: the two surviving pairs in
Example~\ref{ex:cancellation} share their minimal edge and cancel.

\begin{proposition}[Copy-certified screening is a hitting-set condition]
\label{prop:hitting}
For a deleted set \(C\subseteq Q\), the perfect-copy certificate screens
every split pair exactly when
\[
C\cap E_\pi\ne\emptyset
\quad\text{for every split pair }\pi.
\]
Thus minimum copy-certified screening sets are minimum hitting sets of
the hypergraph.
This equivalence concerns the copy certificate, not every possible
reason for a reduced cross operator to vanish.
\end{proposition}

\begin{proof}
The intersection condition says precisely that each pair has a deleted
perfect member of a differing class.
Apply Proposition~\ref{prop:screening}.
If the intersection condition fails, that particular copy certificate
does not screen the unhit pair, although residual distinguishability may
still do so.
\end{proof}

The computable visibility envelope is
\[
\delta_J(S)
\le
\sum_{\substack{\pi\ J\text{-split}\\E_\pi\subseteq S}}
\sqrt{p_rp_{r'}}\,v_\pi(S).
\]
It is not an exact formula when several terms survive.
No claim of fixed-parameter tractability in edge size alone is needed
or made.

\section{Statistical detection through a fixed interface}
\label{sec:testing}

Fix a tower, record set, and channel, and write
\[
\varrho=\Lambda(\rho),
\qquad
\varsigma=\Lambda(\Delta_J\rho),
\qquad
\delta=T(\varrho,\varsigma).
\]
All logarithms in this section are natural.

\begin{lemma}[Pinching domination]\label{lem:domination}
Let \(k\) be the number of record-\(J\) sectors with nonzero weight in
\(\ket\Psi\).
Then \(1\le k\le2^{|J|}\) and
\[
\rho\preceq k\Delta_J\rho,
\qquad
\varrho\preceq k\varsigma.
\]
Consequently,
\[
\delta\le1-\frac1k,
\qquad
\supp\varrho\subseteq\supp\varsigma,
\qquad
D(\varrho\Vert\varsigma)\le\log k.
\]
\end{lemma}

\begin{proof}
Write \(\ket\Psi=\sum_{z=1}^k\ket{v_z}\) in orthogonal record sectors.
For every vector \(u\), Cauchy--Schwarz gives
\[
|\braket u\Psi|^2
\le
k\sum_z|\braket u{v_z}|^2,
\]
which is the first operator inequality.
A positive channel preserves it.

For \(k>1\),
\[
\varsigma=\frac1k\varrho+
\left(1-\frac1k\right)\tau
\]
for a state \(\tau\), so
\(\delta\le1-1/k\).
For \(k=1\), the states coincide.
The support inclusion also follows from domination.

Operator monotonicity of the logarithm, with a positive regularization
and then a limit on the support of \(\varrho\), gives
\[
\tr(\varrho\log\varsigma)
\ge
\tr(\varrho\log\varrho)-\log k.
\]
This proves the relative-entropy bound.
The underlying pinching inequality is standard
\cite{OgawaHayashi2002}.
\end{proof}

\begin{theorem}[Single-copy and asymptotic detection]\label{thm:testing}
For independent identically prepared outputs of the fixed interface:
\begin{enumerate}
\item With equal priors, the optimal single-copy success probability is
\(\frac12(1+\delta)\).
\item The optimal symmetric error exponent is
\[
\xi_C
=
-\log\min_{0\le s\le1}
\tr(\varrho^s\varsigma^{1-s}),
\]
with endpoint powers interpreted as support projections.
It satisfies
\begin{equation}\label{eq:chernoff}
-\tfrac12\log(1-\delta^2)
\le\xi_C\le-\log(1-\delta).
\end{equation}
\item At fixed type-I error in \((0,1)\), the Stein exponent, with
\(\varrho\) as the null state, is \(D(\varrho\Vert\varsigma)\), and
\begin{equation}\label{eq:stein}
2\delta^2
\le
D(\varrho\Vert\varsigma)
\le
\log k
\le
|J|\log2.
\end{equation}
\end{enumerate}
Thus the two models are distinguishable at a positive exponential rate
through this interface if and only if \(\delta>0\).
The Chernoff exponent has a two-sided bound using \(\delta\) alone;
the Stein upper bound requires additional information, here the
pinching-sector count.
\end{theorem}

\begin{proof}
The first assertion is Helstrom optimization.

The quantum Chernoff theorem identifies the symmetric exponent
\cite{Audenaert2007,NussbaumSzkola2009}.
Put
\(q_*=\min_s\tr(\varrho^s\varsigma^{1-s})\).
The standard trace inequality used in its proof gives
\(q_*\ge1-\delta\).
Also,
\[
q_*
\le
\tr(\sqrt\varrho\sqrt\varsigma)
\le
F(\varrho,\varsigma)
\le
\sqrt{1-\delta^2},
\]
where the last inequality is Fuchs--van de Graaf
\cite{FvdG1999}.
Taking negative logarithms proves \eqref{eq:chernoff}.

The quantum Stein theorem gives the asymmetric exponent
\cite{HiaiPetz1991,OgawaNagaoka2000}.
Quantum Pinsker gives
\(D\ge\frac12\|\varrho-\varsigma\|_1^2=2\delta^2\)
\cite{Watrous2018}.
Lemma~\ref{lem:domination} gives the upper bound and ensures the required
support inclusion.
If \(\delta=0\), the output states are identical at every sample size.
If \(\delta>0\), the lower bounds are positive.
\end{proof}

\begin{remark}[Why no universal Stein upper bound follows from
trace distance alone]\label{rem:stein-upper}
Consider pinching-related output states
\[
\varrho_k
=
(1-t)\ketbra00\oplus t\ketbra{+_k}{+_k},
\qquad
\varsigma_k
=
(1-t)\ketbra00\oplus t\,\frac{\id_k}{k},
\]
where \(\ket{+_k}=k^{-1/2}\sum_{r=1}^k\ket r\).
They can be obtained from a pure recorded preparation by making the
sector flag classical at the interface.
Their values are
\[
T(\varrho_k,\varsigma_k)=t(1-1/k),
\qquad
D(\varrho_k\Vert\varsigma_k)=t\log k.
\]
For fixed \(0<\delta_0<1\), choose
\(t=\delta_0/(1-1/k)\) for sufficiently large \(k\).
The trace distance is exactly \(\delta_0\), while relative entropy grows
without bound.
One may take \(k\) to be powers of two to realize the example using
qubit registers.
\end{remark}

\begin{example}[Exact FR exponents]\label{ex:fr-exponents}
At the symmetric point, dephasing record \(1\) gives two orthogonal pure
sectors of weights \(1/3\) and \(2/3\).
The coherent state is pure.
Therefore
\[
\min_s\tr(\rho^s(\Delta_{\{1\}}\rho)^{1-s})
=
(1/3)^2+(2/3)^2
=
5/9,
\]
and
\[
\xi_C=\log(9/5)\approx0.587787,
\]
\[
D(\rho\Vert\Delta_{\{1\}}\rho)
=
-\tfrac13\log\tfrac13-\tfrac23\log\tfrac23
\approx0.636514.
\]
The Chernoff minimum occurs at the \(s=0\) support-projection endpoint.
\end{example}

\begin{corollary}[Robust detection certificate]\label{cor:robusttest}
If the ideal discrepancy is \(\delta\) and the combined state/interface
error bound is \(e\), then \(\delta>e\) certifies
\[
\tilde\xi_C
\ge
-\tfrac12\log\!\left(1-(\delta-e)^2\right).
\]
An exactly agreeing actual interface transmits no statistical
distinction at any sample size.
Ideal agreement alone, followed by implementation error, instead gives
only an upper bound on the actual discrepancy; small leakage can still
be detected with enough samples.
\end{corollary}

\begin{proof}
The actual discrepancy is at least \(\delta-e\) by
Theorem~\ref{thm:robustness} and
Proposition~\ref{prop:interfaceerror}.
Apply the increasing lower-bound function in \eqref{eq:chernoff}.
Exact actual agreement means equality of the output states.
\end{proof}

\section{Discussion and dependency boundary}\label{sec:discussion}

The operator \(X_J\) is the exact difference between two specified
statistical models.
A channel transmits that difference as \(\Lambda(X_J)\), and an
implementability class selects which linear functionals can test it.
This separation prevents several otherwise tempting identifications.

The perfect-copy budget is a syntactic lower bound, not a complete
accessibility criterion.
Fidelity measures residual branch distinguishability.
Scalar visibility does not retain the relative phases needed to decide
operator cancellation.
Agreement on one question can conceal a large discrepancy on the same
subsystem.

The M\"obius hierarchy makes the location of higher-order behavior
precise.
The coherent state is order one whenever more than one record branch
is supported.
Perfect-record subsystem interfaces likewise have order one or
infinity.
Higher finite orders arise through more restrictive channels, including
binary measurement channels.
For the FR example, the value \(1/6\) is the two-record weight of
Wigner's question at the symmetric point, not a state-level
classification and not a new physical no-go theorem.

The distinction between a single product conjunction and a product
measurement with classical processing is equally important.
An exponentially small conjunction probability bias does not imply an
exponential obstacle to a local parity witness.
Additional experimental restrictions must therefore be stated as an
actual effect or protocol class, not inferred from the word
``unentangled.''

\paragraph{Relation to supplied landmark terminology.}
The comparison record accompanying the earlier drafts refers to
Landmark I, VII-E, and Em01-A.
None is a formal premise here.
The typed record equation described for Landmark I concerns
uniqueness of a destructive primitive in its specified strategy class;
it is not the copying-and-discarding identity
of Proposition~\ref{prop:deeper}, and is not used as a physical
Frauchiger--Renner no-go.
The three recording interfaces distinguished in the supplied VII-E
description are kept separate in
\eqref{eq:coherentcopy}--\eqref{eq:nondestructive}.
The supplied Em01-A witness-to-graph non-implication concerns a different
distributional construction, not the finite-dimensional cancellation
of Section~\ref{sec:cancellation}.

Likewise, the down-set in Corollary~\ref{cor:union} is a family of
subsystems for fixed \(\rho,J\).
It is not a family of authorized characteristic extensions, a declared
gravitational path, or an admission condition in another manuscript.
No theorem transports the results of this paper to those objects.
The supplied landmark summaries are comparison material, not a
verification of their full source manuscripts.

\paragraph{Further problems.}
The unrestricted channel-support question is not left open:
Theorem~\ref{thm:anysupport} realizes every support family.
Remaining restrictions worth studying include subsystem realizability
with prescribed branch support, geometric locality, and resource
constraints.
Continuous-variable and thermodynamic records require an
infinite-dimensional domain and convergence theory absent here.
Higher-order protocols require an explicit higher-order type and
composition system, rather than being obtained by renaming a CPTP
access channel.
No result in the supplied distributional or gravitational comparison
material supplies these missing structures.

\appendix

\section{Reproducible numerical checks and execution status}
\label{app:reproducibility}

The proofs above are analytic.
The program embedded in this source checks the listed examples and
several finite-dimensional identities.
It is a regression artifact, not an exhaustive proof and not a formal
machine verification.

\subsection{Artifact, hashes, and status}

This single \texttt{.tex} input writes
\texttt{En01-A-check.py} during pdfLaTeX compilation, without shell
escape.
The exact generated script is printed in
Section~\ref{app:script}.

The artifact's build-computed MD5 identifier is:
\begin{center}
\texttt{\pdfmdfivesum file{En01-A-check.py}}
\end{center}
This identifier fingerprints the generated bytes in the compiled
document; MD5 is not used as a security guarantee.

A successful Python execution computes the script's SHA-256 and writes
\texttt{En01-A-run.json}.
The manifest records the digest, environment versions, UTC execution
time, fixed seed, tolerances, and completion of that local invocation.
The SHA-256 line is intentionally not replaced by a fabricated digest
or an unrecorded execution claim.

\paragraph{Execution status of this source.}
No execution of the regression program and no pdfLaTeX compilation is
claimed by the source as supplied.
The output below is a specified expected output, derived from the
proved closed forms.
A submission accompanied by an execution record should include the
generated manifest and captured standard output from the actual run.

\paragraph{Commands.}
\begin{lstlisting}[language={}]
pdflatex En01-A.tex
pdflatex En01-A.tex
python En01-A-check.py
\end{lstlisting}
Python requires NumPy.
The numerical appendix uses tolerance \(10^{-12}\) for deterministic
golden values and \(10^{-10}\) for the fixed-seed random matrix checks.

\subsection{Golden values}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}lX@{}}
\toprule
Check & Analytic value\\
\midrule
Four-member perfect read-chain &
Discrepancies \(0,0,0,1/2\) at accessible sizes \(1,2,3,4\).\\
Independent fragments, \(\gamma=1/2\), root retained &
Discrepancies \(1/16,1/8,1/4,1/2\).\\
FR profile &
\(1/12,1/12,1/12,1/4\) for
\(\emptyset,\{1\},\{2\},\{1,2\}\).\\
FR state M\"obius weights &
\(\sqrt2/3,\sqrt2/3,1/3\).\\
FR binary-question weights &
\(0,0,1/6\).\\
Same-data phase family &
\(\delta_{\{1\}}(S)=1/2\) at \(\theta=0\),
zero at \(\theta=\pi\), and
\(\delta_{\{1\}}(S')=1/2\).\\
Three-fragment GHZ-phase environment &
Visibility \(1,1,1,0\) with \(0,1,2,3\) inaccessible fragments.\\
Single product effect &
Bias \(1/2,1/4,1/8,1/16\) for chain sizes \(1,2,3,4\).\\
Local parity effect &
Bias \(1/2\) for each of those sizes.\\
FR symmetric exponents &
Chernoff \(\log(9/5)\);
Stein \(\log3-\frac23\log2\).\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Specified expected output}

The checksum and environment-dependent manifest lines follow these
fixed lines:
\begin{lstlisting}[language={}]
GHZ: [0.000000, 0.000000, 0.000000, 0.500000]
NOISY: [0.062500, 0.125000, 0.250000, 0.500000]
FR_PROFILE: [0.083333, 0.083333, 0.083333, 0.250000]
FR_STATE_WEIGHTS: [0.471405, 0.471405, 0.333333]
FR_QUESTION_WEIGHTS: [0.000000, 0.000000, 0.166667]
CANCELLATION: [0.500000, 0.000000, 0.500000]
SECRET: [1.000000, 1.000000, 1.000000, 0.000000]
PRODUCT_EFFECT: [0.500000, 0.250000, 0.125000, 0.062500]
LOCAL_PARITY: [0.500000, 0.500000, 0.500000, 0.500000]
FR_CHERNOFF: 0.587787
FR_STEIN: 0.636514
STRUCTURAL_CHECKS: PASS
ALL CHECKS PASSED
\end{lstlisting}

The structural assertions include inversion, Helstrom saturation for
fixed-seed random channels, data processing, the visibility--fidelity
identity, subsystem support down-closure, all eight binary-channel
support families on two records, a nonsymmetric complex-amplitude
order-two FR point, sequential agreement, and one explicit
gate-error bound.
These are finite tests of the written theorems, not evidence that
every possible circuit has been enumerated.

\subsection{Embedded regression program}\label{app:script}

\lstinputlisting{En01-A-check.py}

\section{Assumption and correction checklist}\label{app:checklist}

For clarity, the following boundaries are part of the statements in
this consolidated version.
\begin{enumerate}
\item Class anchoring pinches only record members and is a statement
about tower-supported states, not arbitrary off-model states.
\item The two-branch discrepancy identity contains the \(J\)-split
condition.
\item Nonorthogonal pointer fragments are not treated as perfect
classical controls or as perfect copy sources.
\item Depth conventions count one root and \(c-1\) subsequent copies in
a \(c\)-member chain.
\item The cancellation comparison uses the same circuit architecture,
probabilities, interface, and pairwise visibilities at two phases.
\item FR order two is a question-level or binary-channel statement.
\item Individual positive-real FR adequacy conditions have codimension
one; simultaneous adequacy has codimension two.
The complex order-two locus is proved through an explicit graph.
\item Arbitrary channel M\"obius supports need not be down-closed.
Perfect-record subsystem supports are down-closed.
\item Exact zeros and exact order are not inferred to survive generic
small errors.
\item The product-effect bound is not a bound on local measurements
with classical parity processing.
\item The hitting-set equivalence refers to perfect-copy certificates,
not all sources of termwise or operator-level agreement.
\item The Stein exponent is not assigned a universal upper bound from
trace distance alone.
\item Ideal screening with implementation error is approximate
screening, not statistical impossibility at all sample sizes.
\item No numerical execution, hash manifest, or compiler run is claimed
without an actual record.
\end{enumerate}

\begin{thebibliography}{99}

\bibitem{DraftSubsystem}
N.~Cheung,
\emph{Observer-Relative Interference in Reflection Towers:
An Exact Accessibility Theory via Reduced Difference Operators},
unpublished author-supplied draft,
\texttt{085WvGp1\_2 (1).tex}.
Consolidated and revised in the present article.

\bibitem{DraftChannel}
N.~Cheung,
\emph{Accessible Interference in Nested Quantum Measurements:
Channel Interfaces, Operator-Valued M\"obius Hierarchies,
and Robust Classicalization},
unpublished author-supplied draft,
\texttt{085WvGp3v2 (1).tex}.
Consolidated and revised in the present article.

\bibitem{FrauchigerRenner2018}
D.~Frauchiger and R.~Renner,
\emph{Quantum theory cannot consistently describe the use of itself},
Nature Communications \textbf{9}, 3711 (2018).
doi:10.1038/s41467-018-05739-8.

\bibitem{Helstrom1976}
C.~W.~Helstrom,
\emph{Quantum Detection and Estimation Theory},
Academic Press, New York (1976).

\bibitem{Watrous2018}
J.~Watrous,
\emph{The Theory of Quantum Information},
Cambridge University Press, Cambridge (2018).

\bibitem{Uhlmann1976}
A.~Uhlmann,
\emph{The ``transition probability'' in the state space of a
\( * \)-algebra},
Reports on Mathematical Physics \textbf{9}, 273--279 (1976).
doi:10.1016/0034-4877(76)90060-4.

\bibitem{FvdG1999}
C.~A.~Fuchs and J.~van de Graaf,
\emph{Cryptographic distinguishability measures for quantum-mechanical
states},
IEEE Transactions on Information Theory \textbf{45}, 1216--1227 (1999).
doi:10.1109/18.761271.

\bibitem{Zurek2009}
W.~H.~Zurek,
\emph{Quantum Darwinism},
Nature Physics \textbf{5}, 181--188 (2009).
doi:10.1038/nphys1202.

\bibitem{Audenaert2007}
K.~M.~R.~Audenaert, J.~Calsamiglia, R.~Mu\~noz-Tapia, E.~Bagan,
Ll.~Masanes, A.~Ac\'in, and F.~Verstraete,
\emph{Discriminating states: the quantum Chernoff bound},
Physical Review Letters \textbf{98}, 160501 (2007).
doi:10.1103/PhysRevLett.98.160501.

\bibitem{NussbaumSzkola2009}
M.~Nussbaum and A.~Szko{\l}a,
\emph{The Chernoff lower bound for symmetric quantum hypothesis testing},
Annals of Statistics \textbf{37}, 1040--1057 (2009).
doi:10.1214/08-AOS593.

\bibitem{HiaiPetz1991}
F.~Hiai and D.~Petz,
\emph{The proper formula for relative entropy and its asymptotics in
quantum probability},
Communications in Mathematical Physics \textbf{143}, 99--114 (1991).

\bibitem{OgawaNagaoka2000}
T.~Ogawa and H.~Nagaoka,
\emph{Strong converse and Stein's lemma in quantum hypothesis testing},
IEEE Transactions on Information Theory \textbf{46}, 2428--2433 (2000).

\bibitem{OgawaHayashi2002}
T.~Ogawa and M.~Hayashi,
\emph{On error exponents in quantum hypothesis testing},
IEEE Transactions on Information Theory \textbf{50}, 1368--1372 (2004).
The pinching-based direct proof is also given in the authors'
preprint \emph{A New Proof of the Direct Part of Stein's Lemma in
Quantum Hypothesis Testing}, arXiv:quant-ph/0110125 (2001).

\end{thebibliography}

\end{document}