% En01-B.tex
% Standalone pdfLaTeX source.
%
% Compile twice:
%   pdflatex En01-B.tex
%   pdflatex En01-B.tex
%
% Compilation writes the embedded regression artifact:
%   En01-B-check.py
%
% Optional numerical checks:
%   python En01-B-check.py
%
% Optional exact-rational quadrature enclosure:
%   python En01-B-check.py --validated
%
% Python dependencies: NumPy.
% Matplotlib is optional and generates diagnostic figures.
%
% Verification status:
% No pdfLaTeX or Python execution is claimed by this source.
% Analytical certificates do not depend on numerical execution.
% No apparatus-specific sensitivity or experimental exclusion is claimed.

\documentclass[11pt]{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[margin=1.02in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{microtype}
\usepackage{booktabs,array,tabularx}
\usepackage{listings}
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\usepackage{pgfplots}
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\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}

\hypersetup{
  pdftitle={Calibrated Obstructions and Nuisance-Cancelling Controls
  for Gravity-Related Decoherence},
  pdfauthor={Ngo Cheung},
  pdfsubject={En01-B: superconducting microsphere interferometry,
  calibrated force noise, regularization, and detector statistics}
}

\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{assumption}[theorem]{Assumption}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

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\begin{filecontents*}[overwrite]{En01-B-check.py}
"""En01-B-check.py

Analytical regressions and explicitly synthetic diagnostics.

NOT a reproduction of the Pino apparatus.
NOT an experimental sensitivity or exclusion calculation.

Default checks use Python exact rational arithmetic for the golden
contrast and analytic small-smearing certificate, and NumPy for
finite-matrix and floating-point diagnostics.

--validated additionally constructs an exact-rational spectral
quadrature enclosure at alpha=1/2. This uses rational interval
Taylor evaluations, a proved Simpson error, and a proved tail bound.
It is slower than the default checks.

Dependencies: NumPy.
Matplotlib is optional.

Successful execution writes En01-B-run.json. No execution is claimed
by the manuscript source.
"""

from fractions import Fraction as F
from pathlib import Path
import argparse
import datetime
import hashlib
import json
import math
import platform
import numpy as np

TOL = 2e-10
WEIGHTS = np.array([5.0, -8.0, 3.0])
SEPARATIONS = np.array([0.25, 0.5, 0.75])


def require(condition, message):
    if not bool(condition):
        raise RuntimeError(message)


def sphere_H_exact(u):
    return u*u/2 - 3*u**3/16 + u**5/160


def sphere_H(u):
    u = np.asarray(u, dtype=float)
    return np.where(
        u <= 2,
        u*u/2 - 3*u**3/16 + u**5/160,
        6/5 - 1/np.maximum(u, 1e-300),
    )


def sphere_form(u):
    u = np.asarray(u, dtype=float)
    out = np.empty_like(u)
    mask = np.abs(u) < 1e-3
    z = u[mask]
    out[mask] = 1 - z*z/10 + z**4/280 - z**6/15120
    z = u[~mask]
    out[~mask] = 3*(np.sin(z) - z*np.cos(z))/z**3
    return out


def contrast_kernel(u):
    u = np.asarray(u, dtype=float)
    out = np.empty_like(u)
    mask = np.abs(u) < 0.05
    z = u[mask]
    # Constant and quadratic terms cancel exactly.
    out[mask] = -z**4/256 + z**6/12288
    z = u[~mask]
    out[~mask] = -sum(
        w*np.sinc(v*z/np.pi)
        for w, v in zip(WEIGHTS, SEPARATIONS)
    )
    return out


def simpson(values, step):
    require((len(values)-1) % 2 == 0, "even Simpson interval count")
    return step/3 * (
        values[0] + values[-1]
        + 4*values[1:-1:2].sum()
        + 2*values[2:-1:2].sum()
    )


def floating_contrast(alpha, intervals=32768):
    """Diagnostic central value only; no floating-point certification."""
    upper = min(512.0, 10.0/alpha)
    u = np.linspace(0, upper, intervals+1)
    y = (
        sphere_form(u)**2
        * np.exp(-(alpha*u)**2)
        * contrast_kernel(u)
    )
    return float(2/np.pi * simpson(y, upper/intervals))


def floating_Ashape(alpha, intervals=32768):
    """Returns A_rc / [G M^2/(hbar R^3)], diagnostically."""
    upper = min(512.0, 10.0/alpha)
    u = np.linspace(0, upper, intervals+1)
    y = u*u*sphere_form(u)**2*np.exp(-(alpha*u)**2)
    return float(simpson(y, upper/intervals)/(3*np.pi))


def tv(p, q):
    return float(np.abs(np.asarray(p)-np.asarray(q)).sum()/2)


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


def unitary(H, dt):
    values, vectors = np.linalg.eigh((H+H.conj().T)/2)
    return (vectors*np.exp(-1j*dt*values)) @ vectors.conj().T


def diffusion_multiplier(x, h, b, strength, A):
    difference = x[:, None] - x[None, :]
    if h == 0:
        coefficient = b
    else:
        # Independently assembled alternative.
        coefficient = b + strength*A
    return -coefficient*difference**2


def staged_surrogate(h, b, strength, A):
    """Seven-label synthetic matrix protocol.

    Potentials, packet, units, instruments and detector are synthetic.
    Both hypotheses are separately assembled. All outcomes and
    preparation rejections remain in the attempted-run law.
    """
    n = 20
    x = np.linspace(-3, 3, n)
    dx = x[1]-x[0]
    kinetic = (
        2*np.eye(n)-np.eye(n, k=1)-np.eye(n, k=-1)
    )/(2*dx*dx)
    potential = np.diag(x*x/2)
    Hs = [
        kinetic+potential, kinetic-potential, kinetic, kinetic,
        kinetic+potential, kinetic-potential, kinetic,
    ]
    durations = np.array([25, 17, 483, 12.7, 2.5, 61, 20])
    durations = durations/durations.sum()
    multiplier = diffusion_multiplier(x, h, b, strength, A)

    def evolve(Y, stage):
        out = Y.copy()
        dt = durations[stage]/4
        U = unitary(Hs[stage], dt)
        for _ in range(4):
            out = out*np.exp(dt*multiplier)
            out = U @ out @ U.conj().T
        return out

    psi = np.exp(-x*x/(4*0.35**2)).astype(complex)
    psi /= np.linalg.norm(psi)
    Y = density(psi)
    for stage in range(4):
        Y = evolve(Y, stage)

    zvalues = np.array([-0.2, 0.7, 1.8])
    likelihood = np.array([
        np.exp(-0.7*(x*x-z)**2) for z in zvalues
    ])
    likelihood /= likelihood.sum(axis=0)
    probs = []
    efficiency = 0.8
    edges = [0, 0.6, 1.2, 2.0, 3.01]

    for zindex, row in enumerate(likelihood):
        K = np.diag(np.sqrt(row))
        block = K @ Y @ K
        if zindex == 0:
            probs.append(float(np.trace(block).real))
            continue
        for stage in range(4, 7):
            block = evolve(block, stage)
        require(
            np.linalg.eigvalsh((block+block.conj().T)/2).min() > -TOL,
            "surrogate block positivity",
        )
        diagonal = np.diag(block).real
        probs.append(float((1-efficiency)*diagonal.sum()))
        for left, right in zip(edges[:-1], edges[1:]):
            mask = (np.abs(x) >= left) & (np.abs(x) < right)
            probs.append(float(efficiency*diagonal[mask].sum()))

    p = np.asarray(probs)
    require(abs(p.sum()-1) < TOL, "attempted-run normalization")
    require(p.min() > -TOL, "nonnegative attempted law")
    return p


def phase_attempt_law(V, success=0.8, efficiency=0.8, damping=0.9):
    """Synthetic detector: separate rejection and no-detection flags."""
    require(0 <= V <= 1, "visibility range")
    vv = damping*V
    cells = np.array([
        0.25+vv/(2*np.pi), 0.25-vv/(2*np.pi),
        0.25-vv/(2*np.pi), 0.25+vv/(2*np.pi),
    ])
    return np.concatenate((
        [1-success, success*(1-efficiency)],
        success*efficiency*cells,
    ))


def visibility_interval(qlo, qhi, alo, ahi, klo, khi, error, Vmin):
    require(0 < klo <= khi, "positive detector sensitivity")
    lower = max(Vmin, (qlo-ahi-error)/khi)
    upper = min(1.0, (qhi-alo+error)/klo)
    return lower, upper


def log_contrast_interval(Vbounds):
    Ybounds = [(-math.log(hi), -math.log(lo))
               for lo, hi in Vbounds]
    lower = upper = 0.0
    for w, (lo, hi) in zip(WEIGHTS, Ybounds):
        if w >= 0:
            lower += w*lo
            upper += w*hi
        else:
            lower += w*hi
            upper += w*lo
    return lower, upper


# Exact rational interval operations used by --validated.
def interval(a, b=None):
    a = F(a)
    return (a, a if b is None else F(b))


def iadd(A, B):
    return (A[0]+B[0], A[1]+B[1])


def ineg(A):
    return (-A[1], -A[0])


def imul(A, B):
    values = [A[i]*B[j] for i in (0, 1) for j in (0, 1)]
    return (min(values), max(values))


def iscale(A, c):
    return imul(A, interval(c))


def isquare(A):
    if A[0] <= 0 <= A[1]:
        return (F(0), max(A[0]**2, A[1]**2))
    return (min(A[0]**2, A[1]**2), max(A[0]**2, A[1]**2))


def sin_cos_interval(x, sine, terms=40):
    x = F(x)
    degree0 = 1 if sine else 0
    term = x if sine else F(1)
    total = term
    for j in range(1, terms):
        degree = degree0+2*j
        term *= -x*x/F((degree-1)*degree)
        total += term
    # Taylor's theorem, with all relevant derivatives bounded by one.
    degree = degree0+2*(terms-1)
    error = abs(x)**(degree+1)/math.factorial(degree+1)
    return (total-error, total+error)


def exp_negative_interval(x, terms=100):
    """Exact enclosure for exp(-x), x>=0, by reciprocal positive series."""
    x = F(x)
    require(x >= 0 and x < terms+2, "exponential tail ratio")
    term = total = F(1)
    for j in range(1, terms+1):
        term *= x/j
        total += term
    next_term = term*x/(terms+1)
    remainder = next_term/(1-x/F(terms+2))
    return (1/(total+remainder), 1/total)


def atan_interval(x, terms=32):
    x = F(x)
    total = sum(
        ((-1)**j)*x**(2*j+1)/F(2*j+1)
        for j in range(terms)
    )
    next_term = ((-1)**terms)*x**(2*terms+1)/F(2*terms+1)
    return (min(total, total+next_term), max(total, total+next_term))


def pi_interval():
    # Machin identity and alternating-series remainders.
    return iadd(
        iscale(atan_interval(F(1, 5)), 16),
        iscale(atan_interval(F(1, 239)), -4),
    )


def exact_integrand_interval(u, alpha):
    u, alpha = F(u), F(alpha)
    if u == 0:
        return interval(0)
    sin_u = sin_cos_interval(u, True)
    cos_u = sin_cos_interval(u, False)
    form = iscale(iadd(sin_u, ineg(iscale(cos_u, u))), 3/u**3)
    S = interval(0)
    for w, v in zip((5, -8, 3), (F(1, 4), F(1, 2), F(3, 4))):
        sinc_iv = iscale(sin_cos_interval(v*u, True), 1/(v*u))
        S = iadd(S, iscale(sinc_iv, -w))
    exponential = exp_negative_interval(alpha*alpha*u*u)
    return imul(imul(isquare(form), exponential), S)


def gaussian_derivative_bounds(alpha, upper, order=4):
    # E^(j)=P_j E, P_{j+1}=P_j'-2 alpha^2 u P_j.
    polynomial = {0: F(1)}
    bounds = []
    for j in range(order+1):
        bounds.append(sum(abs(c)*upper**d for d, c in polynomial.items()))
        derivative = {}
        for d, c in polynomial.items():
            if d:
                derivative[d-1] = derivative.get(d-1, F(0))+d*c
            derivative[d+1] = (
                derivative.get(d+1, F(0))-2*alpha*alpha*c
            )
        polynomial = derivative
    return bounds


def exact_simpson_enclosure(alpha=F(1, 2), upper=F(8), n=1024):
    require(n % 2 == 0 and alpha > 0 and upper >= 1,
            "validated quadrature inputs")
    step = upper/n
    total = interval(0)
    for j in range(n+1):
        weight = 1 if j in (0, n) else (4 if j % 2 else 2)
        total = iadd(
            total, iscale(exact_integrand_interval(j*step, alpha), weight)
        )
    integral = iscale(total, step/3)
    pi = pi_interval()
    normalization = (2/pi[1], 2/pi[0])
    central = imul(normalization, integral)

    E = gaussian_derivative_bounds(alpha, upper)
    v = [F(1, 4), F(1, 2), F(3, 4)]
    weights = [5, 8, 3]
    S = [
        sum(F(w)*x**j/F(j+1) for w, x in zip(weights, v))
        for j in range(5)
    ]
    M4 = F(0)
    for a in range(5):
        for b in range(5-a):
            c = 4-a-b
            multinomial = (
                math.factorial(4)
                // (math.factorial(a)*math.factorial(b)*math.factorial(c))
            )
            M4 += multinomial*E[a]*(2**b)*S[c]

    quadrature_error = 2/pi[0]*upper**5*M4/F(180*n**4)
    exp_upper = exp_negative_interval(alpha*alpha*upper*upper)[1]
    gaussian_tail = 16*exp_upper/(pi[0]*alpha*alpha*upper)
    sphere_tail = 384/(pi[0]*upper**3)
    error = quadrature_error+min(gaussian_tail, sphere_tail)
    return central[0]-error, central[1]+error


def run_checks():
    v = [F(1, 4), F(1, 2), F(3, 4)]
    w = [5, -8, 3]
    require(sum(w) == 0, "constant cancellation")
    require(sum(wi*vi**2 for wi, vi in zip(w, v)) == 0,
            "quadratic cancellation")
    exact = sum(wi*sphere_H_exact(vi) for wi, vi in zip(w, v))
    require(exact == -F(5041, 81920), "exact sphere contrast")
    lower = F(5041, 81920)-48*F(1, 100)**2
    require(lower > F(1, 20), "uniform small-smearing certificate")

    bminus, bplus = F(1, 5), F(4, 5)
    width = bplus-bminus
    for shift in [F(0), F(1, 10), width, width+F(1, 100)]:
        overlap = max(bminus, bminus+shift) <= min(bplus, bplus+shift)
        require(overlap == (shift <= width), "calibration width boundary")

    b1, strength, A = 0.25, 0.5, 0.4
    b0 = b1+strength*A
    p0 = staged_surrogate(0, b0, strength, A)
    p1 = staged_surrogate(1, b1, strength, A)
    require(tv(p0, p1) < TOL, "independent staged quadratic absorption")

    # The continuum heating identity is analytic in the paper.
    require(abs(2*b0-2*(b1+strength*A)) < TOL,
            "matched total heating coefficient")
    x = np.linspace(-2, 2, 9)
    L0 = diffusion_multiplier(x, 0, b0, strength, A)
    L1 = diffusion_multiplier(x, 1, b1, strength, A)
    require(np.max(np.abs(L0-L1)) < TOL, "matched generators")
    bad = diffusion_multiplier(x, 0, b0+0.02, strength, A)
    require(np.max(np.abs(bad-L1)) > 0.01, "mismatch negative control")

    u = SEPARATIONS
    gamma = sphere_H(u)
    strength = 0.7
    Y0 = 0.15+0.3*u*u
    Y1 = 0.15+0.3*u*u+strength*gamma
    require(abs(WEIGHTS @ Y0) < TOL, "nuisance-only contrast")
    require(abs(WEIGHTS @ (Y1-Y0)-strength*float(exact)) < TOL,
            "nonquadratic contrast")
    require(abs(WEIGHTS @ (u*u/2)) < TOL,
            "quadratic-gravity negative control")

    # Free preparation offsets absorb each separation response.
    free_offsets = 0.15+strength*gamma
    matched_null = free_offsets+0.3*u*u
    require(np.max(np.abs(matched_null-Y1)) < TOL,
            "free preparation offset negative control")

    # Separation uncertainty: derivative bound on the specified interval.
    error_d = 1e-4
    derivative_bound = float(
        sum(abs(wi)*(vi+F(1, 1000)) for wi, vi in zip(w, v))
    )
    perturbed = u+np.array([error_d, -error_d, error_d])
    actual_error = abs(WEIGHTS @ (sphere_H(perturbed)-gamma))
    require(actual_error <= derivative_bound*error_d, "separation bound")

    # Event calibration and logarithmic interval propagation.
    a, kappa = 0.32, 0.64*0.9/np.pi
    V = np.exp(-Y1)
    bounds = []
    for value in V:
        q = a+kappa*value
        iv = visibility_interval(
            q-1e-4, q+1e-4, a, a, kappa, kappa, 0, 0.5
        )
        require(iv[0] <= value <= iv[1], "visibility interval coverage")
        bounds.append(iv)
    Cinterval = log_contrast_interval(bounds)
    require(Cinterval[0] <= WEIGHTS @ Y1 <= Cinterval[1],
            "log-contrast interval coverage")

    # A bounded event on complete synthetic histograms.
    # This is a finite-family diagnostic, not a full apparatus certificate.
    families = [[], []]
    for h in (0, 1):
        for beta0 in (0.10, 0.12):
            for beta2 in (0.20, 0.22):
                Ys = beta0+beta2*u*u+h*gamma
                laws = [phase_attempt_law(float(np.exp(-y))) for y in Ys]
                families[h].append(np.concatenate([p/3 for p in laws]))
    statistic = np.zeros(18)
    for setting in range(3):
        statistic[6*setting+3] = 1
        statistic[6*setting+4] = 1
    q0 = [statistic @ p for p in families[0]]
    q1 = [statistic @ p for p in families[1]]
    histogram_gap = min(q1)-max(q0)
    require(histogram_gap > 0, "complete-histogram common event gap")
    flat0 = phase_attempt_law(0.8, damping=0)
    flat1 = phase_attempt_law(0.2, damping=0)
    require(tv(flat0, flat1) < TOL, "flat-detector negative control")

    print("EXACT_SPHERE_CONTRAST:", str(exact))
    print("SMALL_SMEARING_LOWER_BOUND:", str(lower))
    print("CALIBRATION_WIDTH: PASS")
    print("INDEPENDENT_STAGED_ABSORPTION: PASS")
    print("HEATING_MATCH: PASS")
    print("LOG_INTERVALS: PASS")
    print("NEGATIVE_CONTROLS: PASS")
    print("SYNTHETIC_HISTOGRAM_GAP:", f"{histogram_gap:.9f}")
    print("ALL CHECKS PASSED")

    return {
        "exact_sphere_contrast": str(exact),
        "uniform_small_smearing_lower_bound": str(lower),
        "staged_absorption_tv": tv(p0, p1),
        "synthetic_histogram_gap": float(histogram_gap),
    }


def make_figures(folder):
    try:
        import matplotlib.pyplot as plt
    except ImportError:
        print("FIGURES: skipped; Matplotlib not installed")
        return []
    folder.mkdir(exist_ok=True)
    files = []

    def save(fig, name):
        path = folder/name
        fig.tight_layout()
        fig.savefig(path)
        plt.close(fig)
        files.append(str(path))

    alphas = np.logspace(-3, 0.3, 70)
    contrast = np.array([floating_contrast(a) for a in alphas])
    fig, ax = plt.subplots()
    ax.semilogx(alphas, -contrast, label="floating central diagnostic")
    ax.axhline(1/20, color="black", linestyle=":")
    ax.axvspan(1e-3, 1e-2, color="green", alpha=0.15,
               label="analytic bound: -b(alpha)>1/20")
    ax.set_xlabel("alpha = rc/R")
    ax.set_ylabel("-B / [G M^2/(hbar R)]")
    ax.set_title("Homogeneous-sphere control response; not apparatus output")
    ax.legend()
    save(fig, "01-regularized-control.pdf")

    aa = np.logspace(-2, 0.3, 45)
    strength = np.linspace(0, 2, 55)
    Avalues = np.array([floating_Ashape(a) for a in aa])
    width = np.outer(strength, Avalues)
    fig, ax = plt.subplots()
    im = ax.pcolormesh(aa, strength, width, shading="auto")
    ax.set_xscale("log")
    fig.colorbar(im, ax=ax, label="critical calibrated width Delta b")
    ax.set_xlabel("alpha = rc/R")
    ax.set_ylabel("lambda")
    ax.set_title("Quadratic-model overlap surface in units G M^2/(hbar R^3)")
    save(fig, "02-calibration-obstruction-surface.pdf")

    Ys0 = 0.15+0.3*SEPARATIONS**2
    Ys1 = Ys0+sphere_H(SEPARATIONS)
    fig, ax = plt.subplots()
    for i in range(3):
        p0 = phase_attempt_law(float(np.exp(-Ys0[i])))
        p1 = phase_attempt_law(float(np.exp(-Ys1[i])))
        ax.plot(np.arange(6)+7*i, p0, "o-", label=f"H0 setting {i+1}")
        ax.plot(np.arange(6)+7*i, p1, "x--", label=f"H1 setting {i+1}")
    ax.set_xlabel("complete synthetic outcome index")
    ax.set_ylabel("attempted-run probability within setting")
    ax.set_title("Synthetic records: rejection, no detection, four bins")
    ax.legend(fontsize=7)
    save(fig, "03-synthetic-histograms.pdf")

    widths = np.linspace(0, 2, 100)
    signal = 5041/81920
    kappa = 0.64*0.9/np.pi
    Vmin = np.exp(-0.15-(0.3+widths)*0.75**2-sphere_H(0.75))
    epsilon_q = kappa*Vmin*signal/(4*16)
    per_setting = np.ceil(np.log(6/0.05)/(2*epsilon_q**2))
    fig, ax = plt.subplots()
    ax.semilogy(widths, 3*per_setting)
    ax.set_xlabel("hypothetical quadratic-nuisance interval width")
    ax.set_ylabel("sufficient attempted runs, all three settings")
    ax.set_title("Log-contrast budget: calibration width lowers visibility")
    save(fig, "04-diagnostic-log-budget.pdf")
    return files


if __name__ == "__main__":
    parser = argparse.ArgumentParser()
    parser.add_argument("--validated", action="store_true")
    args = parser.parse_args()
    results = run_checks()
    if args.validated:
        lower, upper = exact_simpson_enclosure()
        require(upper < 0, "validated negative contrast at alpha=1/2")
        results["validated_alpha"] = "1/2"
        results["validated_contrast_lower"] = str(lower)
        results["validated_contrast_upper"] = str(upper)
        print("VALIDATED_ALPHA=1/2:", float(lower), float(upper))
    figures = make_figures(Path("En01-B-diagnostics"))
    source = Path(__file__).read_bytes()
    manifest = {
        "artifact": Path(__file__).name,
        "sha256": hashlib.sha256(source).hexdigest(),
        "utc": datetime.datetime.now(datetime.timezone.utc).isoformat(),
        "python": platform.python_version(),
        "numpy": np.__version__,
        "execution_status": "completed by this local invocation",
        "scope": "analytical regressions and synthetic diagnostics",
        "reference_apparatus_reproduced": False,
        "experimental_exclusion": False,
        "results": results,
        "figures": figures,
    }
    Path("En01-B-run.json").write_text(
        json.dumps(manifest, indent=2, sort_keys=True)+"\n",
        encoding="utf-8",
    )
    print("SOURCE_SHA256:", manifest["sha256"])
    print("MANIFEST: En01-B-run.json")
\end{filecontents*}

\title{
Calibrated Obstructions and Nuisance-Cancelling Controls\\
for Gravity-Related Decoherence\\[0.4em]
\large A Superconducting-Microsphere Interferometer:
Regularized Dynamics, Preparation Records, and Detector Certificates
}

\author{
Ngo Cheung, MBBS(HK), FHKAM(Psychiatry)\\
Independent Researcher\\
Hong Kong SAR, China\\
\texttt{info@cheungngomedical.com}
}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We develop a paired negative and constructive identifiability analysis
for the superconducting-microsphere interferometer proposed by Pino
and collaborators.
For a shared residual force-diffusion amplitude independently restricted
to an interval, proportional quadratic environmental and gravitational
responses produce an exact complete-record match whenever the
gravitational coefficient shift does not exceed the calibration width.
This establishes family overlap, not containment of the entire
alternative family.
We distinguish calibration of the environmental source from calibration
of total centre-of-mass heating, and derive a regularized-model
near-obstruction bound with explicit domain and implementation errors.

A complementary three-separation contrast cancels a common preparation
offset and quadratic environmental attenuation.
For a continuous homogeneous sphere at separations
\(R/4,R/2,3R/4\), its unsmeared gravitational coefficient is
\(-5041/81920\), in units \(GM^2/(\hbar R)\).
We prove that the contrast remains strictly negative for every finite
Gaussian regularization length, obtain a uniform quantitative lower
bound for \(0<r_{\mathrm c}\le R/100\), and give validated spectral
quadrature and large-smearing bounds.
A robust signed contrast margin survives only when setting-dependent
preparation, environmental-shape, trajectory, and sensing errors are
sufficiently small.

Log-visibility inference is connected to complete attempted-run records
through confidence intervals; it is not treated as a bounded per-run
statistic.
A separate bounded-statistic route provides uniform sample certificates.
The analytical results and embedded regressions do not claim reproduced
apparatus performance or an experimental exclusion.
\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) {Quadratic calibration overlap};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Nonquadratic sphere contrast};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Experimental exclusion};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {A signed contrast is not an experimental exclusion.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A signed contrast is not an experimental exclusion. 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-B}
\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) {Quadratic calibration overlap};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Nonquadratic sphere contrast};
\node[draw,dashed,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Experimental exclusion};
\draw[->] (a) -- (b);
\draw[->,dashed] (b) -- (c);
\end{tikzpicture}
\caption{A signed contrast is not an experimental exclusion. Reading downward is not a transfer. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-En01-B}
\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) {Quadratic calibration overlap};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Nonquadratic sphere contrast};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Experimental exclusion};
\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-B}
\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) {Quadratic calibration overlap};
\node[draw,rounded corners,align=center,text width=3.4cm,font=\small,inner sep=3pt] (b) at (6.2,0) {Experimental exclusion};
\draw[dashed] (a) -- (b);
\end{tikzpicture}
\caption{Quadratic calibration overlap is not Experimental exclusion. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-En01-B}
\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) {Quadratic calibration overlap};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Nonquadratic sphere contrast};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Experimental exclusion};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {A signed contrast is not an experimental exclusion.};
\end{tikzpicture}
\caption{Three statuses stay separate. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-En01-B}
\end{figure}

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

\section{Question, contribution, and scope}
\label{sec:intro}

A reduced fringe contrast does not identify the source of decoherence.
The relevant question is whether a calibrated environmental family can
reproduce the same recorded outcomes as a specified additional
gravity-related dynamical model.

We use the staged superconducting-microsphere proposal
of Pino and collaborators \cite{Pino2018} as the experimental reference.
Its long-wavelength localization model motivates a residual
force-diffusion nuisance, while its preparation and sensing stages
require explicit accounting of squared-position outcomes and failures.
The reference is a proposal, not a completed gravitational-decoherence
experiment.

The contribution has two linked parts:
\begin{enumerate}
\item \textbf{Calibrated obstruction.}
A shared uncertain residual amplitude can leave an exact
non-identifiable region even after an environmental calibration
interval is supplied.
The matching condition is imposed across all stages and settings.

\item \textbf{Nuisance-cancelling control.}
A specified separation contrast eliminates selected constant and
quadratic attenuation terms, but retains a nonquadratic gravitational
response with a proved regularization-dependent margin.
\end{enumerate}

These conclusions concern different models and must not be conflated.
The obstruction is exact in the declared quadratic dynamical model.
The positive contrast is a controlled attenuation diagnostic whose
application to the full apparatus requires a residual ledger.
Coefficient separation alone is not detector separation.

All analytical results are proved here.
External inputs are the reference architecture, the motivation for
regularized mass-density models, and the radiation constraints cited
in Section~\ref{sec:radiation}.
Neither a manuscript registry nor an unverified companion result is a
mathematical premise.

En01-A \cite{En01A} provides conceptual background on accessible state
differences.
Here the difference is between dynamically constructed outputs,
not a state and its record pinching.
Its copy-screening and M\"obius results are not imported.

\section{Reference protocol and observation boundary}
\label{sec:protocol}

\begin{table}[ht]
\centering
\small
\begin{tabularx}{\linewidth}{@{}lXX@{}}
\toprule
Input & Reference benchmark & Application status\\
\midrule
Sphere & Niobium, \(R=1\,\mu\mathrm m\) &
Geometry and material must be measured\\
Mass & Approximately \(2\times10^{13}\) amu &
Independent mass characterization\\
Nominal separation & \(d=500\) nm &
Preparation-outcome dependent\\
Cooling & \(25\) ms & Initial preparation retained\\
Boost & \(17\) ms & Stage retained\\
Free expansion & \(483\) ms & Stage retained\\
Split sensing & \(12.7\) ms & Finite squared-position instrument\\
Rotation & \(2.5\) ms & Stage retained\\
Inflation & \(61\) ms & Stage retained\\
Final sensing & \(20\) ms & Finite squared-position readout\\
Post-cooling duration & Approximately \(596\) ms &
Not an attempt repetition period\\
\bottomrule
\end{tabularx}
\caption{Proposed reference values from \cite[Sec.~V and Appendix]{Pino2018}.
The listed rounded post-cooling durations sum to \(596.2\) ms.
They are not demonstrated capabilities.}
\label{tab:protocol}
\end{table}

The order is
\[
\text{cool}\to\text{boost}\to\text{free}\to
\text{split}\to\text{rotate}\to\text{inflate}\to\text{sense}.
\]
The three control separations introduced below are proposed settings;
their feasibility is not established by the single reference separation.

An attempted run includes its preparation record, its accept/reject
flag, sensing failures, and detector bins.
A quantum instrument is a countably additive family of completely
positive maps whose total map is trace preserving.
For a continuous outcome, instrument integrals, rather than individual
density elements, are trace nonincreasing.

Let \(\Phi_{j,h}\) be propagation maps and
\(\II_{4,h}(\dd z),\II_{7,h}(\dd y)\) the finite sensing instruments.
The successful record law is schematically
\begin{equation}\label{eq:record}
\pi_h(\dd z,\dd y)=
\tr\!\left[
\II_{7,h}(\dd y)\Phi_{6,h}\Phi_{5,h}
\II_{4,h}(\dd z)\Phi_{3,h}\Phi_{2,h}\Phi_{1,h}
(\rho_{\mathrm{init}})
\right].
\end{equation}
Rejection and failure outcomes complete its mass to one.
Later maps may depend on earlier accessible outcomes.

The \(h\)-dependence of a sensing instrument includes dynamics during
its finite interaction time.
It is not a choice of different electronics for the two hypotheses.
A common terminal detector channel is applied only after completing
the hypothesis-dependent sphere--sensor evolution.

For an impulsive diagnostic, a normalized squared-position instrument
has Kraus density
\[
M_z=\left(\frac2\pi\right)^{1/4}
e^{-(\chi(\widehat x/\sigma)^2-z)^2}e^{i\phi(\widehat x)},
\qquad
\int M_z^\dagger M_z\dd z=\id.
\]
In a resolved positive-outcome regime its packet-centre separation is
approximately \(2\sigma\sqrt{z/\chi}\).
Replacing finite split sensing by this impulsive map requires a
separate approximation bound.

\section{Declared regularized model and its dynamics}
\label{sec:model}

\subsection{Convention and rigid-body specification}

Let \(\mu\) be a positive finite intrinsic mass measure on \(\R^3\),
with total mass \(M\), and let
\[
g_{\rc}(r)=
(2\pi\rc^2)^{-3/2}e^{-|r|^2/(2\rc^2)},
\qquad
f_{\rc}=g_{\rc}*\mu,
\qquad \rc>0.
\]
The width \(\rc\) is the standard deviation in each Cartesian direction.
A fixed-orientation rigid body moves along a unit vector \(e\), with
centre-of-mass position operator \(\widehat x\) on \(L^2(\R)\):
\[
\widehat\mu_{\rc}(r)=f_{\rc}(r-\widehat x e).
\]

We declare the generator
\begin{equation}\label{eq:DP}
\GG_{\rc}(\rho)=
-\frac{G}{2\hbar}\int\!\!\int
\frac{[\widehat\mu_{\rc}(r),
[\widehat\mu_{\rc}(r'),\rho]]}{|r-r'|}
\dd r\,\dd r'.
\end{equation}
The alternative adds \(\lambda\GG_{\rc}\), with
\(\lambda\ge0\).
Unit strength means this normalization, not every convention called
D\'iosi--Penrose in the literature.
Regularization and normalization are physical model specifications
\cite{Bahrami2014,TilloyDiosi2017}.

Our Fourier convention is
\[
\widehat f(k)=\int e^{-ik\cdot r}f(r)\dd r,
\qquad
f(r)=(2\pi)^{-3}\int e^{ik\cdot r}\widehat f(k)\dd k.
\]
Thus
\(\widehat f_{\rc}(k)=e^{-\rc^2|k|^2/2}\widehat\mu(k)\).

\begin{theorem}[Bounded regularized kick generator]
\label{thm:kick}
Put \(k_e=k\cdot e\), \(U_k=e^{ik_e\widehat x}\), and
\[
Q_{\rc}(\dd k)=
\frac{G}{2\pi^2\hbar}
\frac{|\widehat f_{\rc}(k)|^2}{|k|^2}\dd^3k.
\]
Then \(Q_{\rc}\) is a finite positive even measure and
\begin{equation}\label{eq:kick}
\GG_{\rc}(\rho)=
\int(U_k\rho U_k^\dagger-\rho)\,Q_{\rc}(\dd k).
\end{equation}
Writing \(q_{\rc}=Q_{\rc}(\R^3)\),
\[
q_{\rc}\le\frac{GM^2}{\sqrt\pi\hbar\rc},
\qquad
\|\GG_{\rc}\|_{1\to1}\le2q_{\rc}.
\]
Its exponential is a CPTP semigroup on trace-class operators.
The position coherence rate is
\begin{equation}\label{eq:rate}
\Gamma_{\rc}(d)=
\int[1-\cos(k_ed)]\,Q_{\rc}(\dd k).
\end{equation}
\end{theorem}

\begin{proof}
Since \(|\widehat\mu(k)|\le M\),
\[
q_{\rc}\le
\frac{2GM^2}{\pi\hbar}
\int_0^\infty e^{-\rc^2k^2}\dd k
=\frac{GM^2}{\sqrt\pi\hbar\rc}.
\]
Fourier transforming \(1/|r|\) gives \(4\pi/|k|^2\).
On smooth finite-rank position kernels, substitution in
\eqref{eq:DP} gives multiplication by
\(-\Gamma_{\rc}(x-x')\).
Evenness removes the sine part in \eqref{eq:kick}.
The bounded integral extends the identity to trace-class operators.

For \(q_{\rc}>0\), define
\[
\mathcal J(\rho)=q_{\rc}^{-1}
\int U_k\rho U_k^\dagger Q_{\rc}(\dd k).
\]
Then
\[
e^{t\GG_{\rc}}=
e^{-tq_{\rc}}\sum_{n=0}^{\infty}
\frac{(tq_{\rc})^n}{n!}\mathcal J^n
\]
is a norm-convergent CPTP mixture.
The zero-rate case is the identity.
The norm bound follows from the norm-one unitary conjugations.
\end{proof}

\begin{assumption}[Admitted baseline and sensing dynamics]
\label{ass:baseline}
For every tested setting and nuisance value, the common Hamiltonian,
environmental evolution, and apparatus couplings define a strongly
continuous CPTP propagator with the composition property.
Its instruments are well defined.
Unbounded diffusion terms, when used, have admitted realizations and
the same operation domains in matched comparisons.
\end{assumption}

\begin{proposition}[Bounded addition to an admitted baseline]
\label{prop:addition}
Under Assumption~\ref{ass:baseline}, adding
\(\lambda\GG_{\rc}\) gives a unique strongly continuous CPTP
bounded-perturbation propagator.
\end{proposition}

\begin{proof}
Interleave the baseline propagator with Poisson-distributed jumps of
rate \(\lambda q_{\rc}\), applying \(\mathcal J\) at each jump.
The sum over ordered jump times converges in trace norm, with factorial
bounds, and is CPTP.
It solves the bounded variation-of-constants equation.
Picard iteration with bound \(2\lambda q_{\rc}\) proves uniqueness.
\end{proof}

The gravitational contribution is included during cooling,
pre-split evolution, and finite sensing.
Boundedness of \(\GG_{\rc}\) does not make the kinetic Hamiltonian or
the position operator bounded.

\subsection{Different mass and packet prescriptions}

For two displaced intrinsic profiles,
\[
E_G(d)=\frac G2\int\!\!\int
\frac{\delta f(r)\delta f(r')}{|r-r'|}\dd r\,\dd r',
\qquad
\delta f(r)=f_{\rc}(r)-f_{\rc}(r-de),
\]
and Fourier substitution gives \(E_G(d)=\hbar\Gamma_{\rc}(d)\ge0\).

A continuous homogeneous sphere has
\[
\widehat\mu(k)=M F(|k|R),
\qquad
F(u)=3\frac{\sin u-u\cos u}{u^3},
\qquad F(0)=1,
\]
so
\begin{equation}\label{eq:spectral}
\Gamma_{\rc}(d)=
\frac{2GM^2}{\pi\hbar}
\int_0^\infty
e^{-\rc^2k^2}F(kR)^2[1-\sinc(kd)]\dd k.
\end{equation}

A microscopic constituent specification instead has
\[
\mu=\sum_jm_j\delta_{a_j},
\qquad
\widehat f_{\rc}(k)=
e^{-\rc^2|k|^2/2}\sum_jm_je^{-ik\cdot a_j}.
\]
Its cross terms cannot be discarded without an approximation argument.
Writing
\[
W_{\rc}(r)=\frac{\erf(r/(2\rc))}{r},
\qquad W_{\rc}(0)=\frac1{\sqrt\pi\rc},
\]
its rate is
\[
\hbar\Gamma_{\rc}(d)=
G\sum_{j,l}m_jm_l
\left[
W_{\rc}(|a_j-a_l|)
-W_{\rc}(|a_j-a_l+de|)
\right].
\]
This follows by convolving two Gaussian mass profiles and using their
Newtonian potential.

Finite-width centre-of-mass packets are states of \(\widehat x\),
not intrinsic mass profiles.
Their coherence kernel experiences
\(\Gamma_{\rc}(x-x')\) at every position pair.
Packet width is not interchangeable with \(\rc\).

\begin{proposition}[Quadratic coefficient and remainder]
\label{prop:quadratic}
Define
\[
A_{\rc}=\frac12\int k_e^2Q_{\rc}(\dd k),
\qquad
D_{\rc}^{(4)}=\frac1{24}\int k_e^4Q_{\rc}(\dd k).
\]
For \(\rc>0\), these are finite and
\[
0\le A_{\rc}d^2-\Gamma_{\rc}(d)
\le D_{\rc}^{(4)}d^4.
\]
The quadratic generator is
\[
\GG_{\rc}^{(2)}=-A_{\rc}[\widehat x,[\widehat x,\cdot]].
\]
For isotropic profiles,
\[
A_{\rc}=
\frac{G}{3\pi\hbar}
\int_0^\infty k^2|\widehat f_{\rc}(k)|^2\dd k,
\qquad
D_{\rc}^{(4)}=
\frac{G}{60\pi\hbar}
\int_0^\infty k^4|\widehat f_{\rc}(k)|^2\dd k.
\]
\end{proposition}

\begin{proof}
Integrate
\(0\le z^2/2-(1-\cos z)\le z^4/24\).
Angular means of \(k_e^2\) and \(k_e^4\) are \(k^2/3\) and \(k^4/5\).
Gaussian decay controls all moments.
\end{proof}

\section{Residual force noise under independent calibration}
\label{sec:calibration-obstruction}

\subsection{One shared amplitude, not a free amplitude per setting}

Let \(\mathcal K_x(\rho)=[\widehat x,[\widehat x,\rho]]\).
The narrow residual model is
\begin{align}
\LL_{0,s}(t;b)&=
\LL_{\mathrm{known},s}(t)-b f_s(t)\mathcal K_x,
\label{eq:scalar0}\\
\LL_{1,s}^{(2)}(t;b)&=
\LL_{\mathrm{known},s}(t)
-\left[b f_s(t)+\lambda A_{\rc}g_s(t)\right]\mathcal K_x.
\label{eq:scalar1}
\end{align}
Here \(f_s,g_s\ge0\) are declared functions of stage, time, and
accessible history.
The same amplitude \(b\) is used across settings.
Independent environmental characterization supplies
\[
b\in I=[b_-,b_+],\qquad 0\le b_-\le b_+,
\qquad \Delta b=b_+-b_-.
\]

The reference's vibrational localization has a potential-dependent
transfer function and can vanish during free motion
\cite[Sec.~II]{Pino2018}.
Thus proportionality to a persistent gravitational contribution is
not inferred from the words ``force noise''.
It is an explicit model condition.

\begin{theorem}[Calibrated quadratic obstruction]
\label{thm:width}
Suppose the two hypotheses use the same interval \(I\), all known
dynamics and apparatus specifications match, and
\[
g_s(t)=c f_s(t)
\]
at every relevant stage, setting, and history, with one constant
\(c\ge0\).
Put \(a_G=\lambda A_{\rc}c\).

The coefficient intervals \(I\) and \(I+a_G\) overlap exactly when
\begin{equation}\label{eq:obstruction}
a_G\le\Delta b.
\end{equation}
Whenever \eqref{eq:obstruction} holds, at least one null--alternative
pair has identical complete attempted-run laws, including preparation,
heralding, finite sensing, and failures.
For that pair, every test satisfies
\[
\alpha+\beta=1.
\]
Consequently no test has both errors uniformly below \(1/2\) over the
entire two families.

This is family overlap, not containment of every alternative member
in the null.
When \(a_G>\Delta b\), the coefficient intervals are disjoint, but
observable separation does not follow without a detector argument.
\end{theorem}

\begin{proof}
Matching requires \(b_0=b_1+a_G\).
Such values in \(I\) exist precisely when
\([b_-,b_+]\cap[b_-+a_G,b_++a_G]\ne\emptyset\), which for \(a_G\ge0\)
is \eqref{eq:obstruction}.
One explicit pair is \(b_1=b_-\), \(b_0=b_-+a_G\).

For this pair the total generators in
\eqref{eq:scalar0}--\eqref{eq:scalar1} agree.
Uniqueness gives identical propagation.
The same apparatus coupling and instrument give identical
unnormalized outcome states.
Induct over stages and accessible histories to obtain identical
complete laws.

For a randomized test \(\varphi\in[0,1]\) declaring alternative,
\[
\alpha+\beta=\mathbb E_0\varphi+\mathbb E_1(1-\varphi)=1.
\]
The remaining qualifications follow from interval overlap alone.
\end{proof}

If \(c=0\), or all \(f_s\) vanish on the effective observation domain,
there is no gravitational shift in this quadratic comparison.
Otherwise nonproportional \(f_s,g_s\) cannot be matched by this scalar
shift at the generator level.
That failure is not, by itself, an identifiability theorem.

\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{axis}[
width=.72\linewidth,height=5.5cm,
xmin=0,xmax=1,ymin=0,ymax=1,
xlabel={Calibration width \(\Delta b/D_*\)},
ylabel={Shift \(\lambda A_{\rc}c/D_*\)},
legend style={at={(.03,.97)},anchor=north west,font=\small}
]
\addplot[draw=none,fill=blue!15] coordinates
{(0,0) (1,0) (1,1) (0,0)} \closedcycle;
\addlegendentry{Coefficient overlap}
\addplot[black,thick,domain=0:1]{x};
\addlegendentry{Exact boundary}
\end{axis}
\end{tikzpicture}
\caption{Quadratic-model obstruction for any chosen positive unit \(D_*\).
Regularization enters through \(A_{\rc}\).
Above the boundary only coefficient overlap is excluded; detector
separation still needs proof.
The optional artifact plots the corresponding homogeneous-sphere
surface as a function of \((\lambda,\rc/R,\Delta b)\).}
\label{fig:width}
\end{figure}

\subsection{Calibration records versus a calibrated interval}

Let \(C\) denote calibration data and \(Z\) the interferometer record.

\begin{proposition}[Joint-calibration distinction]
\label{prop:joint}
Write the joint laws as
\[
P_h(\dd C,\dd Z)=\kappa_h(\dd C)\,\pi_h(\dd Z\mid C).
\]
They are equal if and only if their calibration marginals are equal
and their conditional interferometer laws agree almost everywhere
under the common marginal.

A match of interferometer laws after fixing an interval does not imply
a match of the joint laws when an external calibration distinguishes
the corresponding environmental amplitudes.
\end{proposition}

\begin{proof}
Equality of joint laws implies equality of marginals.
Uniqueness of conditional probabilities up to marginal null sets gives
the conditional equality.
Conversely, integrate the common conditional law against the common
marginal.
\end{proof}

Theorem~\ref{thm:width} is therefore an obstruction for inference using
the declared interval and interferometer records.
If raw external calibration data are also used, their laws belong in
the comparison.
Coverage of an interval alone does not establish equality of those
laws.

\begin{proposition}[Heating is not source identification]
\label{prop:heating}
For a pure force-diffusion term \(-D(t)\mathcal K_x\), its contribution
to momentum heating is
\[
\left.\frac{\dd}{\dd t}\langle \widehat p^2\rangle\right|_{\mathrm{diff}}
=2\hbar^2D(t).
\]
If two models have identical centre-of-mass generators throughout a
heating calibration and the interferometer, all admitted
centre-of-mass calibration instruments have identical laws.
Total heating alone does not identify the environmental part of
\(D_{\mathrm{total}}\).
\end{proposition}

\begin{proof}
On a common moment domain,
\[
[\widehat x,[\widehat x,\widehat p^2]]=-2\hbar^2\id.
\]
Move the double commutator to the observable.
This is the diffusion contribution; Hamiltonian or damping terms may
also change \(\langle p^2\rangle\).

Identical generators give identical states before each heating
instrument and hence identical recorded distributions, as in
Theorem~\ref{thm:width}.
\end{proof}

An external force sensor, gas characterization, or bath-resolved
measurement can be an independent source calibration.
A measurement of total centre-of-mass diffusion is not automatically
such a calibration.

\section{Near-obstruction in the full regularized model}
\label{sec:near}

\begin{proposition}[Bounded-aperture generator remainder]
\label{prop:aperture}
In a model with bounded self-adjoint position operator
\(\|\widehat x\|\le L\),
\[
\|\GG_{\rc}-\GG_{\rc}^{(2)}\|_{1\to1}
\le D_{\rc}^{(4)}(2L)^4.
\]
\end{proposition}

\begin{proof}
Evenness writes the generator as
\(\int[\cos(k_e\ad_x)-\mathrm{id}]Q_{\rc}(\dd k)\).
The cosine family is the average of two unitary-conjugation
trace-norm isometries.
Taylor's integral remainder after the quadratic term is bounded by
\(|k_e|^4\|\ad_x\|^4/24\).
Since \(\|\ad_x\|_{1\to1}\le2L\), integration proves the result.
\end{proof}

\begin{theorem}[Full-model near-obstruction certificate]
\label{thm:near}
Choose \(b_0=b_1+\lambda A_{\rc}c\) in the calibration interval.
Replace the gravitational quadratic term by
\(\lambda g_s(t)\GG_{\rc}\).
Suppose bounded-aperture comparison models have
\(\|\widehat x\|\le L_s(t)\), and the actual complete laws are within
total variation \(e_h\) of those comparison laws.

With uniform history-wise bounds when feedback is used,
\begin{equation}\label{eq:zeta}
\TV(\pi_0,\pi_1)\le\zeta,
\end{equation}
where
\[
\begin{split}
\zeta=e_0+e_1
&+\frac{\lambda}{2}\sum_s\int
g_s(t)D_{\rc}^{(4)}(2L_s(t))^4\dd t\\
&+2\lambda A_{\rc}\sum_s\int
L_s(t)^2|g_s(t)-cf_s(t)|\dd t .
\end{split}
\]
The sums denote the stages of an attempted run, not an unweighted sum
over alternative experimental designs.
A bound exceeding one is replaced by the trivial bound one.

For \(N\) independent attempted runs,
\[
\alpha_N+\beta_N\ge1-N\zeta.
\]
Thus uniformly small errors \(\alpha_N,\beta_N\le\alpha<1/2\)
require \(N\zeta\ge1-2\alpha\).
\end{theorem}

\begin{proof}
After matching the scalar shift, the generator difference is
\[
\lambda g_s(\GG_{\rc}-\GG_{\rc}^{(2)})
+\lambda(g_s-cf_s)\GG_{\rc}^{(2)}.
\]
The first norm is bounded by Proposition~\ref{prop:aperture}.
The second is at most
\(4\lambda A_{\rc}L_s^2|g_s-cf_s|\).
Variation of constants and CPTP contractivity bound the state trace
distance by half the integrated generator norms.
Complete instruments and terminal processing contract the same norm.
Add the two law-approximation errors.

For product laws,
\(\TV(p^{\otimes N},q^{\otimes N})\le N\TV(p,q)\) by telescoping.
Every test has error sum at least one minus product-law total variation.
\end{proof}

A scalar rate remainder is not a full-protocol trace-distance estimate.
The \(e_h\) terms must cover aperture replacement, boundary dynamics,
preparation, sensing, and any other approximation.
Spatial tail probabilities alone need not control the change of the
Hamiltonian at a truncation boundary.

\begin{proposition}[State-specific residual certificate]
\label{prop:residual}
Let \(\rho(t)\) be an admitted exact evolution and
\(\widehat\rho(t)\) a Hermitian normalized absolutely continuous
approximation on a common variation-of-constants domain.
If
\[
r(t)=\dot{\widehat\rho}(t)-\LL(t)\widehat\rho(t)
\]
is integrable in trace norm, then
\[
T(\rho(t),\widehat\rho(t))
\le T(\rho(0),\widehat\rho(0))
+\frac12\int_0^t\|r(u)\|_1\dd u.
\]
The statement also applies to direct sums of unnormalized instrument
blocks.
\end{proposition}

\begin{proof}
Propagate the initial difference and the residual integral.
The admitted propagator contracts Hermitian trace norm.
\end{proof}

\section{Beyond quadratic localization: an explicit control}
\label{sec:contrast}

\subsection{Diagnostic attenuation model}

For a setting with fixed effective separation \(d_i\), suppose
\begin{equation}\label{eq:Y}
Y_{h,i}=-\log V_{h,i}
=\beta_{0,h}+\beta_{2,h}d_i^2
+h\lambda t\Gamma_{\rc}(d_i)+r_{h,i},
\qquad V_{h,i}>0.
\end{equation}
The common offset and quadratic coefficient can differ between fitted
hypotheses, but each is shared across the settings of its hypothesis.

The constant and quadratic forms are motivated by saturated
short-wavelength and long-wavelength localization, respectively
\cite[Sec.~II]{Pino2018}.
Actual gas-scattering localization and setting-dependent preparation
are not assumed to obey them exactly; their departures enter \(r_{h,i}\).

Equation~\eqref{eq:Y} is a controlled diagnostic, not the complete
seven-stage prediction.

Choose
\[
(d_1,d_2,d_3)=R(1/4,1/2,3/4),
\qquad
w=(5,-8,3).
\]
Then
\[
\sum_iw_i=0,
\qquad
\sum_iw_id_i^2=0.
\]
Define
\[
C_h=\sum_iw_iY_{h,i},
\qquad
\mathcal B_{\rc}=\sum_iw_i\Gamma_{\rc}(d_i).
\]

\begin{theorem}[Nuisance cancellation and robust margin]
\label{thm:contrast}
Under \eqref{eq:Y},
\[
C_0=\sum_iw_ir_{0,i},
\qquad
C_1=\lambda t\mathcal B_{\rc}+\sum_iw_ir_{1,i}.
\]
If \(|r_{h,i}|\le\epsilon_{h,i}\), put
\(E_h=\sum_i|w_i|\epsilon_{h,i}\).
The contrast intervals are
\[
C_0\in[-E_0,E_0],
\qquad
C_1\in[\lambda t\mathcal B_{\rc}-E_1,
       \lambda t\mathcal B_{\rc}+E_1].
\]
For a fixed alternative they are disjoint exactly when
\[
|\lambda t\mathcal B_{\rc}|>E_0+E_1,
\]
with remaining gap
\[
m_C=|\lambda t\mathcal B_{\rc}|-E_0-E_1.
\]

For an alternative family, a common oriented certificate instead
requires one sign \(\sigma\in\{-1,1\}\) and
\[
\inf_{\Theta_1}\sigma\lambda t\mathcal B_{\rc}>E_0+E_1.
\]
An infimum of absolute values does not establish a common oriented
threshold when the sign varies.
\end{theorem}

\begin{proof}
Multiply \eqref{eq:Y} by the weights and sum.
The two nuisance terms vanish.
The residual magnitude is bounded by the weighted sum of its component
bounds.
Two centred closed intervals with radii \(E_0,E_1\) and centre distance
\(|\lambda t\mathcal B_{\rc}|\) are disjoint precisely under the
displayed strict inequality.
The family statement applies the same orientation to every alternative.
\end{proof}

\subsection{Exact homogeneous-sphere calculation}

Put
\[
\Kscale=\frac{GM^2}{\hbar R}.
\]

\begin{proposition}[Unsmeared continuous-sphere rate]
\label{prop:sphere}
For a continuous homogeneous sphere,
\[
\Gamma_0(d)=\Kscale H(d/R),
\]
where
\[
H(u)=
\begin{cases}
u^2/2-3u^3/16+u^5/160,&0\le u\le2,\\
6/5-1/u,&u\ge2.
\end{cases}
\]
In particular \(A_0=GM^2/(2\hbar R^3)\).
\end{proposition}

\begin{proof}
The intrinsic density autocorrelation is
\[
w_0(r)=
\left(\frac{3M}{4\pi R^3}\right)^2
\frac{\pi}{12}(4R+r)(2R-r)^2
\mathbf1_{\{r\le2R\}}.
\]
It is the overlap volume of two spheres times the squared density.
Its Newtonian pairing at displacement \(d\) is
\[
J(d)=4\pi\int_0^{2R}
\frac{r^2w_0(r)}{\max(r,d)}\dd r.
\]
Integrating the polynomial separately below and above \(r=d\) gives
\[
J(d)=\frac{M^2}{R}
\left(\frac65-\frac{u^2}{2}
+\frac{3u^3}{16}-\frac{u^5}{160}\right)
\]
for \(u\le2\), and \(J(d)=M^2/d\) otherwise.
The rate is \(G[J(0)-J(d)]/\hbar\).
Its quadratic coefficient follows at \(d=0\).
\end{proof}

\begin{corollary}[Exact three-separation signature]
\label{cor:exactcontrast}
With \(\hstar=5041/81920\),
\begin{equation}\label{eq:exactcontrast}
\mathcal B_0=-\Kscale\hstar,
\qquad
C_G=-\lambda t\Kscale\hstar.
\end{equation}
\end{corollary}

\begin{proof}
The quadratic contribution cancels.
The remaining moments are
\[
\sum_iw_i(d_i/R)^3=\frac{15}{32},
\qquad
\sum_iw_i(d_i/R)^5=\frac{435}{1024}.
\]
Therefore
\[
-\frac3{16}\frac{15}{32}
+\frac1{160}\frac{435}{1024}
=-\frac{5041}{81920}.
\]
\end{proof}

Replacing gravity by a purely quadratic kernel makes the contrast zero.
Allowing an independently free preparation offset at each setting also
destroys the cancellation premise.

\section{Gaussian regularization: sign, margin, and validated integration}
\label{sec:regularization}

Set
\[
\alpha=\rc/R,\quad
v=(1/4,1/2,3/4),\quad
b(\alpha)=\mathcal B_{\rc}/\Kscale.
\]
Equation~\eqref{eq:spectral} gives
\begin{equation}\label{eq:b}
b(\alpha)=\frac2\pi\int_0^\infty
e^{-\alpha^2u^2}F(u)^2S(u)\dd u,
\qquad
S(u)=-\sum_iw_i\sinc(v_i u).
\end{equation}
The constant term vanishes, and \(S(u)=O(u^4)\) at the origin.
Its pointwise oscillations do not determine the sign of the complete
integral.

\subsection{A sign theorem for every finite Gaussian width}

\begin{lemma}[Radial autocorrelation after Gaussian smearing]
\label{lem:radial}
For a homogeneous sphere and \(\rc>0\), its smeared autocorrelation
\[
w_{\rc}=w_0*g_{\sqrt2\rc}
\]
is positive, smooth, radial, and strictly decreasing at every radius
\(r>0\).
\end{lemma}

\begin{proof}
The unsmeared \(w_0\) is nonnegative, radial, and decreasing on
\(0<r<2R\), and vanishes outside that ball.
Its layer-cake representation is a positive mixture of indicators of
balls of nonzero radii.

For a ball of radius \(a>0\), let
\[
L_a(r)=\int_{|z|\le a}g_{\sqrt2\rc}(re-z)\dd z.
\]
By the divergence theorem,
\[
L_a'(r)=
-\int_{|z|=a}(e\cdot n_z)g_{\sqrt2\rc}(re-z)\dd S(z).
\]
Pair a point with \(e\cdot n_z>0\) with its reflection across the plane
orthogonal to \(e\).
For \(r>0\), the Gaussian value at the nearer point is strictly larger.
The paired integral before the minus sign is positive.
Thus \(L_a'(r)<0\).
Integrate this strict inequality against the positive layer-cake
mixture.
Smoothness and positivity follow from Gaussian convolution.
\end{proof}

\begin{theorem}[Strictly negative regularized control response]
\label{thm:sign}
For every finite \(\rc>0\), the function
\[
z\longmapsto\Gamma_{\rc}(\sqrt z)
\]
is strictly concave on \(z>0\).
Consequently
\[
\mathcal B_{\rc}<0
\]
for the three separations used here.
On every compact interval of finite positive regularization lengths,
its negative magnitude has a strictly positive minimum.
\end{theorem}

\begin{proof}
Let
\(I(d)=\int_0^d r^2w_{\rc}(r)\dd r\).
The shell theorem gives
\[
\Gamma_{\rc}'(d)=\frac{4\pi G}{\hbar d^2}I(d).
\]
Writing \(\psi(z)=\Gamma_{\rc}(\sqrt z)\), differentiation gives
\[
\psi''(z)=
\frac{\pi G}{\hbar d^5}
\left[d^3w_{\rc}(d)-3I(d)\right],
\qquad d=\sqrt z.
\]
Strict decrease of \(w_{\rc}\) implies
\(3I(d)>d^3w_{\rc}(d)\), so \(\psi''<0\).

For \(z_i=d_i^2\), strict concavity gives
\[
\frac{\psi(z_3)-\psi(z_2)}{z_3-z_2}
<
\frac{\psi(z_2)-\psi(z_1)}{z_2-z_1}.
\]
Since \(z_2-z_1=3R^2/16\) and \(z_3-z_2=5R^2/16\),
their slope difference equals \(16\mathcal B_{\rc}/(15R^2)\).
This proves the sign.

Continuity in \(\rc\) follows by dominated convergence in the spectral
integral on compact positive intervals.
A continuous strictly positive \(-\mathcal B_{\rc}\) has a positive
minimum on such an interval.
\end{proof}

The last assertion is an existence result.
The next theorem supplies an explicit usable lower bound.

\begin{theorem}[Uniform small-smearing certificate]
\label{thm:small}
For all \(\alpha\ge0\),
\[
|b(\alpha)-b(0)|\le48\alpha^2.
\]
Hence, on \(0<\rc\le R/100\),
\begin{equation}\label{eq:small}
-\mathcal B_{\rc}\ge
\Kscale\left(\frac{5041}{81920}-\frac{48}{10000}\right)
>
\frac{\Kscale}{20}.
\end{equation}
More generally,
\[
|b(\alpha)-b(\beta)|\le48|\alpha^2-\beta^2|.
\]
\end{theorem}

\begin{proof}
Because \(\sum_iw_i=0\),
\(|S(u)|\le\sum_i|w_i|=16\).
Parseval's identity for the uniform unit-ball density gives
\[
\int_0^\infty u^2F(u)^2\dd u=\frac{3\pi}{2}.
\]
Indeed, \(\int\mu^2=3M^2/(4\pi R^3)\); applying the stated Fourier
normalization proves the identity.

Use
\[
|e^{-\alpha^2u^2}-e^{-\beta^2u^2}|
\le|\alpha^2-\beta^2|u^2
\]
in \eqref{eq:b}.
The bound is
\((2/\pi)16(3\pi/2)|\alpha^2-\beta^2|\).
Set \(\beta=0\) and apply Corollary~\ref{cor:exactcontrast}.
The final rational inequality is direct.
\end{proof}

\begin{corollary}[A uniform constructive margin]
\label{cor:uniformmargin}
For alternatives with \(\lambda\ge\lambda_{\min}>0\),
\(t\ge t_{\min}>0\), and \(0<\rc\le R/100\), the oriented contrast
\(-C\) has a common deterministic gap at least
\[
m_{\mathrm{uniform}}
=
\lambda_{\min}t_{\min}\frac{\Kscale}{20}-E_0-E_1.
\]
A positive value certifies separation within the declared attenuation
model and residual bounds.
\end{corollary}

\begin{proof}
Use the fixed negative sign and \eqref{eq:small} in
Theorem~\ref{thm:contrast}.
\end{proof}

\subsection{Large smearing reduces the available margin}

\begin{proposition}[Large-smearing upper bound]
\label{prop:large}
For the homogeneous sphere,
\[
|\mathcal B_{\rc}|
\le
\Kscale\frac{47}{5120\sqrt\pi\,\alpha^5}.
\]
Thus its constructive margin tends to zero as \(\alpha\to\infty\),
despite remaining strictly signed for every finite \(\alpha\).
\end{proposition}

\begin{proof}
The quadratic terms cancel and
\[
|\mathcal B_{\rc}|
\le D_{\rc}^{(4)}\sum_i|w_i|d_i^4.
\]
Using \(|\widehat\mu(k)|\le M\),
\[
D_{\rc}^{(4)}
\le\frac{GM^2}{160\sqrt\pi\hbar\rc^5}.
\]
The weighted fourth moment is \(47R^4/32\).
Substitute and express the result in units \(\Kscale\).
\end{proof}

\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{axis}[
width=.82\linewidth,height=5.7cm,
xmin=0,xmax=.04,ymin=0,ymax=.065,
xlabel={Gaussian width \(\alpha=r_{\mathrm c}/R\)},
ylabel={Certified lower bound for \(-\mathcal B_{\rc}/K_G\)},
legend style={at={(.98,.98)},anchor=north east,font=\small},
samples=80
]
\addplot[blue,thick,domain=0:.0358]
{max(0,5041/81920-48*x*x)};
\addlegendentry{Analytical certificate}
\addplot[black,dashed,domain=0:.01]{.05};
\addlegendentry{Uniform \(1/20\) bound on \(0<\alpha\le.01\)}
\end{axis}
\end{tikzpicture}
\caption{A proved lower certificate, not an approximation to the full
response curve.
The certificate becoming zero does not mean the response changes sign;
Theorem~\ref{thm:sign} proves otherwise.}
\label{fig:small}
\end{figure}

\subsection{Validated spectral quadrature}

\begin{proposition}[Explicit spectral integration enclosure]
\label{prop:quadrature}
Let \(U\ge1\), \(\alpha>0\), and use composite Simpson quadrature with
an even number \(n\) of equal intervals on \([0,U]\).
For \(j=0,\ldots,4\), define
\[
s_j=\sum_i\frac{|w_i|v_i^j}{j+1}.
\]
Let \(P_0=1\) and
\[
P_{j+1}(u)=P_j'(u)-2\alpha^2uP_j(u).
\]
If \(P_j(u)=\sum_\ell p_{j,\ell}u^\ell\), put
\[
e_j=\sum_\ell|p_{j,\ell}|U^\ell,
\qquad
M_4=\sum_{a+b+c=4}
\frac{4!}{a!b!c!}e_a2^bs_c.
\]
Then the finite-interval quadrature error for \(b(\alpha)\) is at most
\[
\epsilon_{\mathrm{quad}}
=
\frac2\pi\frac{U^5M_4}{180n^4}.
\]
The omitted tail is bounded by
\[
\epsilon_{\mathrm{tail}}
\le
\min\left\{
\frac{16e^{-\alpha^2U^2}}{\pi\alpha^2U},
\frac{384}{\pi U^3}
\right\}.
\]
Validated nodal intervals and an interval for \(2/\pi\), combined with
these errors, give a rigorous enclosure for \(b(\alpha)\).

An enclosure at \(\alpha_0\) extends to a whole regularization cell by
adding \(48\sup_{\alpha\text{ in cell}}|\alpha^2-\alpha_0^2|\).
\end{proposition}

\begin{proof}
The normalized sphere form factor is an average of
\(e^{-iuZ}\) with \(|Z|\le1\), so \(|F^{(j)}|\le1\).
Leibniz's rule gives \(|(F^2)^{(j)}|\le2^j\).
Since \(\sinc(vu)=\int_0^1\cos(vtu)\dd t\),
\(|S^{(j)}|\le s_j\).
The Gaussian derivative is \(P_je^{-\alpha^2u^2}\), bounded by \(e_j\).
The product derivative bound is therefore \(M_4\).
The standard Simpson remainder, obtained by integrating the
fourth-order interpolation remainder panel by panel, gives the stated
finite-interval error.

For the first tail bound, use \(|F|\le1\), \(|S|\le16\), and
\[
\int_U^\infty e^{-\alpha^2u^2}\dd u
\le\frac{e^{-\alpha^2U^2}}{2\alpha^2U}.
\]
For \(u\ge1\), \(|F(u)|\le6/u^2\).
Integrating \(16\cdot36/u^4\) gives the second bound after multiplying by
\(2/\pi\).
The cell extension is Theorem~\ref{thm:small}.
\end{proof}

The embedded optional validated calculation uses exact rational interval
Taylor evaluations, including a Machin-identity enclosure for \(\pi\).
It does not assume floating-point trigonometric evaluations are
rigorous.
Ordinary floating central values are separately labelled diagnostic.

\section{Preparation and detector conditions for the control}
\label{sec:implementation}

\subsection{Residual ledger}

\begin{table}[ht]
\centering
\small
\begin{tabularx}{\linewidth}{@{}p{.32\linewidth}X@{}}
\toprule
Departure from \eqref{eq:Y} & Required bound or modification\\
\midrule
Setting-dependent preparation visibility &
Bound its log-visibility departure from one shared offset\\
Finite packet width and shape change &
State-specific packet or full-protocol error certificate\\
Unsaturated gas localization &
Bound departure from the constant template\\
Nonquadratic environmental attenuation &
Bound it or include an additional nuisance template\\
Setting-dependent sensing response &
Joint response calibration, including efficiencies and failures\\
Separation uncertainty &
Derivative or interval propagation with calibrated separation windows\\
Moving branch separation &
Trajectory-integrated templates, validated before use\\
Different pre-split dynamics &
Full protocol comparison, not a post-split attenuation insertion\\
Outcome-dependent preparation &
Retain records; bound windowing and heralding effects\\
\bottomrule
\end{tabularx}
\caption{The constructive theorem requires every relevant departure to
enter its residual budget.
An independently free offset at each setting invalidates the
cancellation rather than merely increasing precision requirements.}
\label{tab:residual}
\end{table}

\begin{proposition}[Separation-error propagation]
\label{prop:separation}
If \(|d_i-\bar d_i|\le\delta_i\) and
\(|\Gamma_{\rc}'(d)|\le L_i\) on the corresponding windows, then
\[
\left|
\sum_iw_i\Gamma_{\rc}(d_i)
-\sum_iw_i\Gamma_{\rc}(\bar d_i)
\right|
\le\sum_i|w_i|L_i\delta_i.
\]
For the homogeneous sphere, Gaussian smearing gives \(A_{\rc}\le A_0\),
and
\[
|\Gamma_{\rc}'(d)|\le2A_{\rc}|d|\le2A_0|d|.
\]
For nominal weights cancelling \(\bar d_i^2\), the quadratic nuisance
leakage is at most
\[
|\beta_2|\sum_i|w_i|(2|\bar d_i|\delta_i+\delta_i^2).
\]
\end{proposition}

\begin{proof}
Apply the mean-value theorem termwise.
Differentiate \eqref{eq:rate} and use
\(|\sin(k_ed)|\le|k_ed|\).
Smearing multiplies the positive moment integral by a factor at most
one.
The last bound expands \(d_i^2-\bar d_i^2\).
\end{proof}

A separation error is therefore not harmless when the cancelled
nuisance coefficient is arbitrarily large.

\begin{proposition}[Finite-width packet diagnostic]
\label{prop:packet}
Let an isometry \(W:\mathbb C^2\to L^2(\R)\) embed two orthonormal
packets with disjoint supports, each within distance \(\ell\) of its
centre \(x_j\).
For a static dephasing interval \(t\), comparison with the two-centre
kick model gives trace-distance error at most
\[
\lambda t\int\min(2,|k_e|\ell)\,Q_{\rc}(\dd k).
\]
\end{proposition}

\begin{proof}
Disjoint supports give
\[
\|U_kW-W\diag(e^{ik_ex_0},e^{ik_ex_1})\|
\le\min(2,|k_e|\ell).
\]
The conjugated state difference is bounded by twice this norm.
Integrate the generator residual, propagate it contractively, and
divide by two.
\end{proof}

The disjoint-support condition is essential to this stated operator
norm estimate.
Gaussian packets require a tail/truncation certificate.
Moving and deforming packets require a propagation certificate, not
repeated informal use of the static result.

\subsection{Moving trajectories and additional nuisance templates}

Suppose a validated branch description gives
\[
Y_{h,i}=\beta_{0,h}+\beta_{2,h}F_i
+h\lambda B_i+r_{h,i},
\]
where
\[
F_i=\int d_i(u)^2\dd u,
\qquad
B_i=\int\Gamma_{\rc}(d_i(u))\dd u.
\]
Choose weights with
\[
w^\top\mathbf1=0,\qquad w^\top F=0.
\]
Then Theorem~\ref{thm:contrast} applies with
\(\lambda t\mathcal B_{\rc}\) replaced by \(\lambda w^\top B\).

More generally, for a matrix \(N\) of declared nuisance templates,
a nuisance-cancelling linear contrast exists precisely when
\[
B\notin\range N.
\]
One choice is \(w=(I-P_N)B\), giving
\(w^\top B=\|(I-P_N)B\|^2>0\).
This is linear algebra on a validated attenuation model, not proof that
a branch-trajectory model describes the entire interferometer.

\section{From detector records to log-visibility confidence}
\label{sec:routeA}

\subsection{The common terminal detector}

Let \(\Sigma_h\) be normalized completed sphere--sensor outputs and
\(\DD\) their common finite-outcome detector channel.
For a static squared-position approximation,
\[
p_h(b)=
\int_{B_b}\dd y\int_{\R}\dd x\,
K(y|x)\langle x|\rho_h^{\mathrm{out}}|x\rangle,
\]
where \(K\ge0\) is sub-Markov and principally depends on \(x^2\).
Add the no-detection effect to normalize the measurement.

\begin{proposition}[Accessible distinguishability and heralding]
\label{prop:detector}
For \(X=\Sigma_0-\Sigma_1\),
\[
\TV(p_0,p_1)=\tfrac12\|\DD(X)\|_1
\le\tfrac12\|X\|_1.
\]
Every fixed event bias is at most this detector total variation.
For specified distributions, the positive-difference bins attain it.

If an attempted law is
\[
p_h=(1-s_h)\delta_{\mathrm{reject}}+s_hr_h,
\]
with \(r_h\) the normalized successful detector law, then equal success
\(s_0=s_1=s\) gives
\[
\TV(p_0,p_1)=s\,\TV(r_0,r_1).
\]
For unequal success, \(\TV(p_0,p_1)\ge|s_0-s_1|\).
\end{proposition}

\begin{proof}
The measurement output is diagonal, giving the first identity.
Positivity and trace preservation contract Hermitian trace norm.
The positive signed entries sum to half the absolute sum.
For heralding, compare the explicit rejection and success entries.
\end{proof}

This is the trace-distance measurement principle
\cite{Helstrom1976}, not a license to compare unbounded generators in
diamond norm.

Ideal \(x^2\) readout preserves position total variation for even
densities: folding sends \(f(x)\) to \(2f(|x|)\).
An odd density difference is lost.
Finite response, binning, and jitter can lose more information.

\subsection{Route A: invert calibrated event laws}

The log contrast \(C=\sum_iw_i(-\log V_i)\) is not a bounded
single-run statistic.
It must not be inserted directly into a per-run Hoeffding bound.

\begin{assumption}[Calibrated visibility response]
\label{ass:visibility}
At setting \(i\), a fixed attempted-run histogram event satisfies
\[
q_i=a_i+\kappa_iV_i+\delta_i,
\qquad
|\delta_i|\le\epsilon_{\mathrm{det},i},
\]
with independently or jointly validated ranges
\[
a_i\in[a_i^-,a_i^+],\qquad
0<\kappa_i^-\le\kappa_i\le\kappa_i^+,
\qquad
V_i\in[V_{\min},1],\quad V_{\min}>0.
\]
Preparation acceptance and detection efficiencies are included in the
response ranges or estimated jointly from retained flags.
\end{assumption}

\begin{theorem}[Visibility and log-contrast confidence intervals]
\label{thm:logCI}
At \(k\) settings, use \(N_i\) independent attempted runs and empirical
event frequencies \(\widehat q_i\).
Let
\[
u_i=\sqrt{\frac{1}{2N_i}\log\frac{2k}{\alpha}},
\quad
q_i^-=\max(0,\widehat q_i-u_i),
\quad
q_i^+=\min(1,\widehat q_i+u_i).
\]
With probability at least \(1-\alpha\), all event probabilities lie in
these intervals.

On the calibration coverage event, valid visibility enclosures are
\[
V_i^-=
\max\left\{V_{\min},
\frac{q_i^--a_i^+-\epsilon_{\mathrm{det},i}}{\kappa_i^+}\right\},
\]
\[
V_i^+=
\min\left\{1,
\frac{q_i^+-a_i^-+\epsilon_{\mathrm{det},i}}{\kappa_i^-}\right\}.
\]
If an enclosure is empty, the response model and data are inconsistent
at the declared confidence level.
Otherwise
\[
Y_i\in[-\log V_i^+,-\log V_i^-].
\]
The interval for \(C=\sum_iw_iY_i\) is obtained by selecting lower or
upper endpoints according to the sign of each weight.
It has simultaneous coverage at least \(1-\alpha-\beta\) if the
calibration event has probability at least \(1-\beta\).
\end{theorem}

\begin{proof}
Apply two-sided Hoeffding to each event and a union bound.
For positive \(V_i\), the response equation implies
\[
q_i\le a_i^++\kappa_i^+V_i+\epsilon_{\mathrm{det},i},
\quad
q_i\ge a_i^-+\kappa_i^-V_i-\epsilon_{\mathrm{det},i}.
\]
Rearrange using the observed probability bounds.
The logarithm is monotone, and a signed linear combination of intervals
has the stated endpoint rule.
Combine the two coverage events by a union bound.
\end{proof}

A lower visibility bound is an application condition, not a numerical
convenience.
Without it, logarithmic errors can become arbitrarily large.

\begin{corollary}[A transparent sufficient log-contrast budget]
\label{cor:logbudget}
Suppose \(a_i,\kappa_i\) are known, with
\(\kappa_i\ge\kappa_{\min}>0\), and all systematic errors have already
been included in the deterministic contrast gap \(m_C>0\).
Estimate \(V_i\) by inverting the event law and clipping to
\([V_{\min},1]\).
With equal \(N\) per setting, the empirical contrast error is at most
\[
\frac{\sum_i|w_i|}{\kappa_{\min}V_{\min}}
\sqrt{\frac{1}{2N}\log\frac{2k}{\alpha}}
\]
with probability at least \(1-\alpha\).
For the three-setting weights, \(\sum|w_i|=16\).
Thus
\[
N\ge
\frac{2(16)^2}{\kappa_{\min}^2V_{\min}^2m_C^2}
\log\frac{6}{\alpha}
\]
is sufficient to keep this error at most \(m_C/2\).
The total attempted-run count is \(3N\).
\end{corollary}

\begin{proof}
Clipping is nonexpansive when the true visibility belongs to the clipped
interval.
The inverse response has Lipschitz constant \(1/\kappa_{\min}\).
On \([V_{\min},1]\), the log has Lipschitz constant \(1/V_{\min}\).
Sum the three errors and solve the resulting inequality.
\end{proof}

The midpoint decision orientation must use the corresponding
deterministic contrast intervals.
Detector calibration uncertainty should use
Theorem~\ref{thm:logCI}, rather than silently treating fitted response
parameters as exact.

\section{Bounded complete-record certificates}
\label{sec:routeB}

For randomized settings with known probabilities \(\omega_s>0\), define
\[
p_h(s,z;\theta_h)=\omega_sp_{h,s}(z;\theta_h),
\qquad
\PP_h=\{p_h(\theta_h):\theta_h\in\Theta_h\}.
\]
The outcomes include setting labels, preparation records, rejections,
and sensing failures.
Nuisance parameters are jointly constrained across settings.

For a predeclared \(f(s,z)\in[0,1]\), set
\(q_h(\theta_h)=\mathbb E_hf\).

\begin{theorem}[Route B: robust bounded-statistic certificate]
\label{thm:bounded}
Suppose
\[
g=
\inf_{\Theta_1}q_1-\sup_{\Theta_0}q_0>0.
\]
If actual laws are within total variation \(e_h\) of the corresponding
specified family members, put
\[
g_*=g-e_0-e_1.
\]
For \(g_*>0\), the midpoint threshold between the two enlarged
expectation ranges, applied to \(N\) independent attempted runs, has
each error at most
\[
e^{-Ng_*^2/2}.
\]
Consequently
\[
N\ge\frac{2}{g_*^2}\log\frac1\alpha
\]
is sufficient for each error to be at most \(\alpha\).
The reversed orientation is equally valid.
\end{theorem}

\begin{proof}
For \(0\le f\le1\), total variation bounds the expectation difference.
This follows by integrating the events \(\{f>u\}\), \(u\in[0,1]\).
The enlarged ranges are separated by \(g_*\).
The threshold is at least \(g_*/2\) from either range.
Hoeffding's one-sided bound gives
\(e^{-2N(g_*/2)^2}\) \cite{Hoeffding1963}.
\end{proof}

\begin{proposition}[Common-statistic feasibility]
\label{prop:LP}
For finite recorded alphabets and compact closed convex hulls
\(C_h=\overline{\conv}\PP_h\), a positive common bounded-statistic gap
exists if and only if \(C_0\cap C_1=\emptyset\).

For finite lists of family vertices, its optimal gap is the linear
program
\[
\begin{split}
\max_{f,u,v}\quad&v-u\\
\text{subject to}\quad&
0\le f_z\le1,\\
&f\cdot p_{0,j}\le u,\qquad
f\cdot p_{1,l}\ge v
\quad\text{for all listed vertices}.
\end{split}
\]
A finite nuisance grid represents a continuous family only after a
uniform interpolation or optimization error is supplied.
\end{proposition}

\begin{proof}
A positive gap survives convexification and excludes an intersection.
Disjoint compact convex sets admit a strict separating linear
functional.
Adding a constant to every coordinate and rescaling puts its
coefficients in \([0,1]\), since all law vectors sum to one.
The displayed program is exactly the finite common-gap optimization.
\end{proof}

A positive log-contrast margin does not automatically imply a positive
single-copy linear-program gap.
The logarithm is nonlinear; convexification can introduce intersections
not present in the original law families.
Route A remains a valid separate inferential construction under its
assumptions.
Route B is the preferred apparatus-facing certificate when its positive
gap has actually been established.

For specified laws, the full-record likelihood ratio is
\[
L=\sum_zn_z\log\frac{p_1(z)}{p_0(z)},
\]
and their classical Chernoff information is
\[
-\log\min_{0\le u\le1}\sum_zp_0(z)^up_1(z)^{1-u}.
\]
Endpoint terms use support conventions.
This is an asymptotic simple-hypothesis exponent
\cite{Chernoff1952}, not a uniform composite-family certificate.

Local nuisance-rank diagnostics can guide settings.
With all probability coordinates retained, define
\(v_G=\partial_\lambda p\), \(J_\nu=\partial_\nu p\), and
\(W=\diag(1/p_z)\) for positive \(p_z\).
The nuisance-orthogonal information is
\[
I_{\mathrm{eff}}=
\min_a(v_G-J_\nu a)^\top W(v_G-J_\nu a).
\]
It vanishes precisely when \(v_G\in\range J_\nu\).
The normalization constraints are retained in the derivative columns;
one must not drop a probability coordinate and then use an unmodified
diagonal Fisher weight.
This tangent-space condition is not a global exclusion theorem.

\section{Calibration--control trade-offs}
\label{sec:tradeoff}

\begin{table}[ht]
\centering
\small
\begin{tabularx}{\linewidth}{@{}p{.30\linewidth}XX@{}}
\toprule
Regime & Established result & Required next certificate\\
\midrule
Proportional quadratic responses,
\(\lambda A_{\rc}c\le\Delta b\) &
Exact interferometer-family overlap &
Joint external calibration may remove the match\\
Full response close to matched quadratic dynamics &
Near-obstruction with bound \(\zeta\) &
Domain, aperture, sensing and residual errors\\
Nonquadratic response survives nuisance cancellation &
Signed contrast margin \(m_C\) &
Visibility-response confidence or complete-histogram gap\\
Broad residual spectral nuisance &
Exact absorption if its spectrum admits the gravity addition &
Independent constraints excluding that spectrum\\
\bottomrule
\end{tabularx}
\caption{The negative and constructive results have compatible but
different hypotheses.
No row is an apparatus-specific experimental verdict.}
\label{tab:tradeoff}
\end{table}

The calibration width can be large while an ideal log contrast still
cancels the quadratic nuisance.
Nevertheless, larger attenuation lowers \(V_{\min}\) and increases the
required sample count.
Cancellation of a nuisance coefficient is not cancellation of its
cost to visibility.

Additional settings help only when their validated response is
transverse to the admitted nuisance templates.
A time-only scan of
\[
Y_s=a t_s+\lambda b_G t_s
\]
has proportional derivatives and cannot distinguish the two
coefficients.
A separation scan likewise fails in a purely quadratic model.
The three-setting construction exploits the exact sphere's
nonquadratic response, not the number of settings alone.

If each setting admits a free offset \(\beta_{0,i}\), choose
\(\beta_{0,i}^{(0)}=
\beta_{0,i}^{(1)}+\lambda t\Gamma_{\rc}(d_i)\).
This matches all diagnostic visibilities and eliminates the proposed
positive identification.
Likewise a flat response \(\kappa_i=0\) eliminates visibility information
at the detector even when the underlying contrast is nonzero.

\section{Existing radiation constraints and interpretation}
\label{sec:radiation}

Radiation limits apply to a universal microscopic collapse model with
the corresponding charge dynamics, noise spectrum, mass smearing, and
normalization.
They are not automatically limits on a continuous macroscopic
rigid-body prescription.

Donadi and collaborators excluded the natural prescription considered
in their underground radiation study \cite{Donadi2021}.
Their displayed coefficient \(4\pi G/\hbar\) is \(8\pi\) times
\eqref{eq:DP}'s coefficient.
The XENONnT analysis reports a source-native \(95\%\) lower bound
\(R_0>4.5\times10^{-10}\) m for its Markovian DP model and includes
charge-cancellation effects \cite{XENON2026}.
These limits cannot be transferred by replacing a macroscopic sphere
radius or packet width with the microscopic cutoff.

A source template proportional to
\(\lambda_{\mathrm{ext}}\rc^{-3}\) permits a cubic-strength rescaling
only when that template remains valid throughout the scan.
When the spectral shape changes, the full external likelihood must be
used.
A prospective study should show separately the applicable external
exclusion set, the certified interferometer sensitivity set, and their
difference.
This article produces no physical numerical exclusion contour.

\section{Conclusion and submission boundary}
\label{sec:conclusion}

The paired contribution is precise.
A calibrated common force-noise amplitude leaves an exact quadratic
family-overlap region under a declared proportionality condition.
Centre-of-mass heating of matched total generators cannot identify the
source.
Beyond the quadratic approximation, a full-protocol near-obstruction
requires operator-level and implementation error bounds.

Conversely, three proposed separations cancel a shared constant and
quadratic attenuation while retaining a nonquadratic continuous-sphere
response.
Its coefficient is evaluated exactly.
Gaussian regularization preserves its sign at every finite width, and
an explicit uniform magnitude certificate is available for
\(0<\rc\le R/100\).
The positive conclusion survives only below the stated residual margin.

The constructive attenuation theorem is not yet a validation of the
reference apparatus's three-setting operation.
Actual sensitivity requires calibrated preparation laws, transfer
functions, detector response, and uniform propagation errors.
The paper supplies the mathematical certificates and a reproducible
test artifact; it does not invent those empirical inputs.

No result is imported from characteristic-product, free-shell,
retarded-graph, or field-theory admission programmes.
A positive detector certificate would not resolve their separate
analytic obligations.
No Born-rule uniqueness result, ultraviolet completion, or experimental
gravity measurement is claimed.

\appendix

\section{Regression artifact, evidence, and execution status}
\label{app:artifact}

The embedded program checks:
\begin{enumerate}
\item exact interval-overlap boundaries;
\item independently assembled matched quadratic generators in a
synthetic seven-label protocol;
\item the matched total heating coefficient;
\item constant and quadratic nuisance cancellation;
\item the rational \(-5041/81920\) sphere contrast;
\item the exact-rational uniform small-smearing lower certificate;
\item separation-error and visibility-to-log interval propagation;
\item a bounded common event gap for complete synthetic histograms;
\item negative controls: quadratic gravity, free preparation offsets,
and a flat detector.
\end{enumerate}

The default spectral integrations are floating-point diagnostics.
They are not labelled validated continuum computations.
The optional \texttt{--validated} calculation encloses
\(b(1/2)\) using exact rational nodal intervals and
Proposition~\ref{prop:quadrature}.
Its printed decimal endpoints are presentations of the enclosing
rational numbers recorded in the manifest.

The staged matrix model is synthetic.
Its Hamiltonians, units, packet, instruments, and efficiencies are not
the reference apparatus.
The survival of exact matching in this regression illustrates the
written theorem but is not its proof.

Optional Matplotlib output includes:
\begin{enumerate}
\item regularized contrast diagnostics with the analytic certified
small-width region;
\item the quadratic overlap surface versus strength and regularization;
\item complete synthetic histograms;
\item a hypothetical log-contrast sample budget versus calibration width.
\end{enumerate}

A successful local invocation writes the SHA-256, environment,
execution time, and results in \texttt{En01-B-run.json}.
No invocation or compilation is claimed here.

\begin{lstlisting}[language={}]
pdflatex En01-B.tex
pdflatex En01-B.tex
python En01-B-check.py
python En01-B-check.py --validated
\end{lstlisting}

The build-computed generated-script MD5 identifier is:
\begin{center}
\texttt{\pdfmdfivesum file{En01-B-check.py}}
\end{center}
This fingerprints generated bytes; it is not a proof or security
certificate.

\subsection{Embedded program}

\lstinputlisting{En01-B-check.py}

\section{Corrections and assumption audit}
\label{app:audit}

\begin{enumerate}
\item Undefined state-label notation is replaced by explicit
\(\mathrm{state}\) or named quantities.
\item Continuous instruments are defined through their integrated maps;
an outcome density is not incorrectly required to be trace
nonincreasing pointwise.
\item The scalar calibrated obstruction establishes overlap, not
whole-family containment.
\item Matching is required across stages, controls, and histories;
proportionality is not inferred from a generic force-noise label.
\item Raw external calibration and interval-constrained inference are
different experiments.
\item Heating denotes the diffusion contribution unless other
dynamical contributions have been subtracted.
\item The homogeneous-sphere nonquadratic coefficient is exactly
\(-5041/81920\).
\item Regularized sign is proved for the complete integral, not inferred
from selected spectral nodes.
\item Numerical sign certificates retain quadrature, tail, nodal, and
parameter-cell errors.
\item Packet operator-norm estimates explicitly require disjoint
supports and an isometric embedding.
\item Nominal separation cancellation does not remove uncertainty in
the actual squared-separation template.
\item Positive log contrast is not automatically a positive
single-copy bounded-statistic gap.
\item Log-confidence propagation requires a positive visibility floor
and positive calibrated detector sensitivity.
\item Fisher weighting uses complete constrained probability
coordinates, or an appropriately transformed reduced metric.
\item All statistical sample counts refer to attempted runs;
calibration coverage, success probabilities, reset time, and
correlations require separate accounting.
\item No numerical or compiler execution record is fabricated.
\end{enumerate}

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\end{document}