Microscopic Compact-Holonomy Benchmark and Spectral-Invariant Test

The microscopic-closure criterion of CHC-MCL requires more than a periodic constitutive function. It requires one finite Hamiltonian whose derivatives determine loading and response, one probability-complete event process, and at least one observable restriction that survives nuisance removal. The present construction meets those requirements on a controlled finite system.

The construction has two distinct evidential roles. First, it is an exact realization theorem and a reproducible numerical witness. Second, it permits a retrospective parameter-free shape comparison with an independently published quantum-circuit experiment that implemented a three-qubit loop with a controlled synthetic flux [citation]. The resulting concordance is consistent with the finite compact-holonomy relation in that emulator class. It neither establishes that the CHC cosmological scalar is realized in the device nor selects the complete CHC framework over standard synthetic-gauge theory.

Let $\theta\sim\theta+2\pi$TeX\theta\sim\theta+2\pi be an externally prescribed compact control coordinate. Let $\cH_{01}$TeX\cH_{01} be the four-dimensional vacuum--one-excitation space with basis

|0\rangle,\quad |1\rangle,\quad |2\rangle,\quad |3\rangle,

where $|j\rangle$TeX|j\rangle denotes one excitation at site $j$TeXj. Put

H_1(\theta)=\hbar\bigl[\omega_0P_1+K(\theta)\bigr],
 \qquad P_1=\sum_{j=1}^3|j\rangle\langle j|,

where $g>0$TeXg>0 and

K(\theta)=g
 
 01e^{-i\theta}

 101

 e^{i\theta}10

on the ordered basis $(|1\rangle,|2\rangle,|3\rangle)$TeX(|1\rangle,|2\rangle,|3\rangle) and is extended by zero on the vacuum. Thus $H_1(\theta)$TeXH_1(\theta) is a bounded self-adjoint operator for every control phase. An autonomous compact rotor may instead be introduced on $L^2(S^1)$TeXL^2(S^1) by

H_{\rm aut}=\frac{p_\theta^2}{2I_\theta}+V(\theta)+H_1(\theta),
 \qquad I_\theta>0,

with periodic $V$TeXV. The momentum $p_\theta$TeXp_\theta is unbounded, and none of the frozen-phase spectral statements below is asserted for the full operator reference. They concern the finite controlled Hamiltonian reference; the autonomous extension is recorded only to distinguish control-phase loading from total rotor force.

definition: Microscopic benchmark window. The benchmark window consists of the vacuum--one-excitation sector, a symmetric on-site frequency $\omega_0$TeX\omega_0, a common positive hopping magnitude $g$TeXg, one loop phase $\theta$TeX\theta, and uniform amplitude loss. Unequal site frequencies, unequal hopping magnitudes, leakage to higher excitation sectors, and nonuniform loss are deviations to be tested rather than nuisance functions to be fitted separately at every phase.

Consider a connected hopping graph with terms $g_{jk}e^{i\alpha_{jk}}|j\rangle\langle k|+\mathrm{h.c.}$TeXg_{jk}e^{i\alpha_{jk}}|j\rangle\langle k|+\mathrm{h.c.}, where $\alpha_{kj}=-\alpha_{jk}$TeX\alpha_{kj}=-\alpha_{jk}. A local basis change $|j\rangle\mapsto e^{i\beta_j}|j\rangle$TeX|j\rangle\mapsto e^{i\beta_j}|j\rangle changes

\alpha_{jk}\mapsto\alpha_{jk}-\beta_j+\beta_k.

theorem: Tree-phase no-go and cycle completeness. Every edge phase on a connected tree can be removed by a local basis change. On a connected graph, two phase assignments are locally unitarily equivalent if and only if their oriented phase sums agree modulo $2\pi$TeX2\pi on a cycle basis. In particular, a triangle has exactly one independent phase, its loop holonomy

\theta=\alpha_{12}+\alpha_{23}+\alpha_{31}\pmod{2\pi}.

No spectrum of a one-edge or open-chain model can depend on its apparent hopping phase.

proof. Choose a root and set its $\beta$TeX\beta to zero. On a tree, every other vertex is reached by a unique path. Define $\beta_k$TeX\beta_k recursively along that path so that reference vanishes on each traversed edge. Because there is no cycle, the recursion is consistent and removes every phase. On a general connected graph, apply the same construction to a spanning tree. Only the chords remain phased. Each chord closes one fundamental cycle, and its residual phase is precisely the oriented phase sum on that cycle. Such sums are invariant under reference because the vertex phases telescope. Equality of all fundamental-cycle sums therefore gives the same residual chord phases after tree gauge fixing, proving unitary equivalence. A triangle has one chord relative to a spanning tree and hence one invariant. Spectral phase dependence on a tree would contradict the constructed unitary equivalence.

This theorem separates physical compact holonomy from a phase convention. It also supplies a compulsory experimental control: the same phase sweep on an open three-site chain must leave its spectrum unchanged.

Let $\Omega_1,\Omega_2,\Omega_3$TeX\Omega_1,\Omega_2,\Omega_3 be the three one-excitation angular frequencies of $H_1(\theta)/\hbar$TeXH_1(\theta)/\hbar at fixed $\theta$TeX\theta, and put $\bar\Omega=\tfrac13\sum_j\Omega_j$TeX\bar\Omega=\tfrac13\sum_j\Omega_j.

theorem: Trimer characteristic polynomial. The centered frequencies $\lambda_j=\Omega_j-\omega_0$TeX\lambda_j=\Omega_j-\omega_0 are the three roots of

\lambda^3-3g^2\lambda-2g^3\cos\theta=0,

and may be written

\Omega_k(\theta)=\omega_0+2g\cos\!\left(\frac{\theta+2\pi k}{3}\right),
 \qquad k=0,1,2.

Consequently

\sum_j\lambda_j=0,
 \qquad \sum_j\lambda_j^2=6g^2,
 \qquad \prod_j\lambda_j=2g^3\cos\theta.

proof. The matrix $K$TeXK has trace zero, $\Tr K^2=6g^2$TeX\Tr K^2=6g^2, and determinant $2g^3\cos\theta$TeX2g^3\cos\theta. The characteristic polynomial of a traceless $3\times3$TeX3\times3 matrix is $\lambda^3-\tfrac12\Tr(K^2)\lambda-\det K$TeX\lambda^3-\tfrac12\Tr(K^2)\lambda-\det K, which gives reference. Substitution of reference into reference uses $4\cos^3u-3\cos u=\cos3u$TeX4\cos^3u-3\cos u=\cos3u and gives $2g^3[\cos(\theta+2\pi k)-\cos\theta]=0$TeX2g^3[\cos(\theta+2\pi k)-\cos\theta]=0. The three values exhaust the cubic roots. Vieta's relations and $\Tr K^2$TeX\Tr K^2 give reference.

theorem: Scale-free compact-holonomy invariant. For any nonzero spectral spread define

\cI(\bm\Omega)=
 \frac{3\sqrt6\prod_{j=1}^3(\Omega_j-\bar\Omega)}
 {\left[\sum_{j=1}^3(\Omega_j-\bar\Omega)^2\right]^{3/2}}.

For the microscopic ring,

\boxed{\cI(\bm\Omega)=\cos\theta.}

The identity is unchanged by $\Omega_j\mapsto a+b\Omega_j$TeX\Omega_j\mapsto a+b\Omega_j for every $a\in\mathbb R$TeXa\in\mathbb R and $b>0$TeXb>0, and it is unchanged by a permutation of the modes. Thus the common frequency origin, positive frequency calibration, hopping magnitude, and branch labeling are absent from the prediction.

proof. By reference, $\bar\Omega=\omega_0$TeX\bar\Omega=\omega_0, the numerator before its prefactor is $2g^3\cos\theta$TeX2g^3\cos\theta, and the bracket in the denominator is $6g^2$TeX6g^2. Therefore the numerical prefactor reduces the ratio to $\cos\theta$TeX\cos\theta. Centering removes $a$TeXa, while a positive rescaling multiplies both numerator and denominator by $b^3$TeXb^3. Both the product and sum of squares are symmetric under branch permutation.

corollary: Exact predictive codimension. At fixed independently known $\theta$TeX\theta, the spectral map $(\omega_0,g)\mapsto(\Omega_1,\Omega_2,\Omega_3)$TeX(\omega_0,g)\mapsto(\Omega_1,\Omega_2,\Omega_3) has rank two for $g>0$TeXg>0 and hence codimension one. Equation reference is a global, not merely linearized, representative of the surviving restriction.

proof. The derivative with respect to $\omega_0$TeX\omega_0 is $(1,1,1)$TeX(1,1,1). The derivative with respect to $g$TeXg is the nonzero centered vector with components $2\cos[(\theta+2\pi k)/3]$TeX2\cos[(\theta+2\pi k)/3]; it is orthogonal to $(1,1,1)$TeX(1,1,1) by reference. The two derivative columns are therefore independent. The invariant theorem eliminates both parameters exactly.

Define the phase-loading operator and dimensionless loop current by

\cL_\theta=-\partial_\theta H_1,
 \qquad J_\theta=\cL_\theta/\hbar.

theorem: Spectral load--current identity. For a differentiable normalized eigenstate $|k;\theta\rangle$TeX|k;\theta\rangle away from a degeneracy,

\langle \cL_\theta\rangle_k
 =-\hbar\frac{\partial\Omega_k}{\partial\theta}
 =\frac{2\hbar g}{3}\sin\!\left(\frac{\theta+2\pi k}{3}\right).

Moreover, the linear response of this same load to a small phase perturbation is determined by the retarded commutator of $\partial_\theta H_1$TeX\partial_\theta H_1 [citation]. The static loading, persistent current, and causal response therefore cannot be retuned independently within this microscopic model. In the optional autonomous extension, the total generalized force is $-\partial_\theta H_{\rm aut}=-V'(\theta)+\cL_\theta$TeX-\partial_\theta H_{\rm aut}=-V'(\theta)+\cL_\theta and therefore includes the separate rotor-potential term.

proof. The Hellmann--Feynman identity gives $\partial_\theta(\hbar\Omega_k)=\langle k;\theta|\partial_\theta H_1|k;\theta\rangle$TeX\partial_\theta(\hbar\Omega_k)=\langle k;\theta|\partial_\theta H_1|k;\theta\rangle. Combining it with reference and differentiating reference proves reference. Duhamel differentiation of the propagator with respect to the phase history gives the Kubo retarded commutator. Both the first derivative and its response variation contain the same operator $\partial_\theta H_1$TeX\partial_\theta H_1 and the same propagator, which excludes independent adjustment.

At a degeneracy the individual derivative depends on the diagonalization of $\partial_\theta H$TeX\partial_\theta H in the degenerate subspace. The characteristic polynomial and the invariant remain valid without choosing that basis.

Let $\eta=e^{-\kappa\Delta t}$TeX\eta=e^{-\kappa\Delta t} for $\kappa\ge0$TeX\kappa\ge0, and define the unitary step

U_\theta=\exp[-iH_1(\theta)\Delta t/\hbar]

and loss operators

M_0=|0\rangle\langle0|+\sqrt\eta P_1,
 \qquad M_j=\sqrt{1-\eta}\,|0\rangle\langle j|,\qquad j=1,2,3.

theorem: Exact event-instrument completion. The operators $K_\mu=M_\mu U_\theta$TeXK_\mu=M_\mu U_\theta, $\mu=0,1,2,3$TeX\mu=0,1,2,3, define a completely positive trace-preserving instrument. Outcome $0$TeX0 is the no-loss branch and outcomes $j=1,2,3$TeXj=1,2,3 are resolved loss events. In the limit $\Delta t\to0$TeX\Delta t\to0 the repeated process has generator

\dot\rho=-\frac{i}{\hbar}[H_1(\theta),\rho]
 +\sum_{j=1}^3\left(L_j\rho L_j^\dagger-\frac{1}{2}\{L_j^\dagger L_j,\rho\}\right),
 \quad L_j=\sqrt\kappa\,|0\rangle\langle j|,

where $H_1$TeXH_1 is defined in reference. The non-Hermitian no-event Hamiltonian is $H_{\rm eff}=H_1-i\hbar\kappa P_1/2$TeXH_{\rm eff}=H_1-i\hbar\kappa P_1/2, so every spectral pole has the common decay displacement $-\kappa/2$TeX-\kappa/2 [citation].

proof. Equation reference gives

M_0^\dagger M_0+\sum_{j=1}^3M_j^\dagger M_j
 =|0\rangle\langle0|+\eta P_1+(1-\eta)P_1=I.

Unitary right multiplication preserves this completeness relation. The corresponding outcome maps are completely positive and their sum is trace preserving. Expanding $\eta=1-\kappa\Delta t+O(\Delta t^2)$TeX\eta=1-\kappa\Delta t+O(\Delta t^2) and $U_\theta=I-iH_1\Delta t/\hbar+O(\Delta t^2)$TeXU_\theta=I-iH_1\Delta t/\hbar+O(\Delta t^2) gives the no-event and jump terms in reference. Since $\sum_jL_j^\dagger L_j=\kappa P_1$TeX\sum_jL_j^\dagger L_j=\kappa P_1, the stated $H_{\rm eff}$TeXH_{\rm eff} follows. Uniform loss is scalar on the one-excitation sector, so it shifts every pole equally.

The same ring also has an exact passive input--output realization. Let $\bm a$TeX\bm a be the three complex mode amplitudes and let $\bm u,\bm y$TeX\bm u,\bm y denote incident and outgoing waves at three matched ports:

\dot{\bm a}=\left[-i(\omega_0I+K)-\frac\kappa2I\right]\bm a-\sqrt\kappa\,\bm u,
 \qquad \bm y=\bm u+\sqrt\kappa\,\bm a.

proposition: Exact passivity balance. For reference,

\frac{d}{dt}\|\bm a\|^2=\|\bm u\|^2-\|\bm y\|^2.

Consequently the scattering realization is passive, all poles lie at $-\kappa/2-i\Omega_j$TeX-\kappa/2-i\Omega_j, and three-port access makes the state controllable and observable for $\kappa>0$TeX\kappa>0.

proof. Differentiate $\bm a^\dagger\bm a$TeX\bm a^\dagger\bm a. The Hermitian frequency matrix cancels, leaving

-\kappa\|\bm a\|^2-\sqrt\kappa(\bm a^\dagger\bm u+\bm u^\dagger\bm a),

which equals $\|\bm u\|^2-\|\bm u+\sqrt\kappa\bm a\|^2$TeX\|\bm u\|^2-\|\bm u+\sqrt\kappa\bm a\|^2. The pole statement follows by diagonalizing the Hermitian frequency matrix. The input and output matrices are nonsingular scalar multiples of the identity, so every state direction is directly reachable and directly visible.

The supplied program evaluates 1001 equally spaced phases in $[-\pi,\pi]$TeX[-\pi,\pi] at $\omega_0=5.3$TeX\omega_0=5.3 and $g=0.17$TeXg=0.17. It diagonalizes reference, evaluates the cubic and invariant, compares with reference, verifies the Hellmann--Feynman derivative by a centered finite difference, and repeats the spectral sweep on an open chain. It also checks affine and permutation invariance, the analytic gradient, unitary propagation, Kraus completeness, and the passivity balance. No random draw or fitted coefficient enters the computation.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

These numbers are floating-point checks of the preceding identities, not independent empirical evidence.

Roushan et al. implemented a triangular loop of three superconducting qubits with parametrically modulated complex hopping. The loop phase is gauge invariant, whereas the phase of a two-site link can be removed by a local transformation. Their Supplementary Figure S4(c,d) reports energy expectations obtained after full density-matrix measurement and displays three reordered one-excitation energy branches as the synthetic flux is swept [citation].

The extraction uses the immutable arXiv PDF with SHA-256 digest center 4e9833dc600b49ecb5c594095e7a60209a27867d108ad974c52a1149295721ba. center Page 12 is rendered at 180 dpi. The printed abscissa $x\in[-1,1]$TeXx\in[-1,1] is sampled at the 39 fixed points $-0.95,-0.90,\ldots,0.95$TeX-0.95,-0.90,\ldots,0.95. Median centers of the red, green, and blue branch pixels are converted through the printed axis ticks. Because the displayed spectrum completes one period when $x$TeXx advances by one, the retrospective phase convention is $\theta=2\pi x$TeX\theta=2\pi x. Each branch center is displaced over five equally spaced levels spanning a four-pixel vertical half-width, and all $5^3$TeX5^3 combinations are evaluated. The resulting grid range is a digitization-sensitivity diagnostic, not a continuous extremal bound or an experimental confidence interval.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The comparison has no spectral offset, scale, or branch-assignment fit because those quantities cancel in $\cI$TeX\cI. It is nevertheless retrospective: the displayed-period convention and extraction procedure were fixed after the source existed, and no raw experimental covariance is available. It establishes quantitative concordance of the exact holonomy invariant with the published emulator figure. It does not constitute a preregistered discovery test.

For measured resonance vector $\bm\Omega$TeX\bm\Omega, define $P=I-\one\one^{\mathsf T}/3$TeXP=I-\one\one^{\mathsf T}/3, $\bm z=P\bm\Omega$TeX\bm z=P\bm\Omega, $q=\bm z^{\mathsf T}\bm z$TeXq=\bm z^{\mathsf T}\bm z, and $p=z_1z_2z_3$TeXp=z_1z_2z_3. The derivative required for covariance propagation is

\frac{\partial\cI}{\partial z_i}
 =3\sqrt6\left(
 \frac{z_jz_k}{q^{3/2}}-\frac{3pz_i}{q^{5/2}}
 \right),
 \qquad \nabla_{\bm\Omega}\cI=P\nabla_{\bm z}\cI,

where $(i,j,k)$TeX(i,j,k) is a permutation of $(1,2,3)$TeX(1,2,3). This formula makes offset covariance cancel automatically. For $r_m=\cI(\bm\Omega_m)-\cos\theta_m$TeXr_m=\cI(\bm\Omega_m)-\cos\theta_m, the phase-reference derivative is

\frac{\partial r_m}{\partial\theta_n}=\delta_{mn}\sin\theta_m.

The joint Jacobian formed from reference and reference propagates frequency--phase cross-covariances as well as covariance across distinct phase points.

The following protocol is fixed for a new raw-data realization.

- Set twelve control phases $\theta_m=2\pi m/12$TeX\theta_m=2\pi m/12, $m=0,\ldots,11$TeXm=0,\ldots,11, using a phase reference calibrated independently of the resonance spectrum. - At every phase fit all three resonances jointly and retain the full cross-mode and cross-phase covariance. A phase point is resolved only if every adjacent resonance separation is at least three times its propagated standard uncertainty; unresolved points are reported and not replaced. - Form $r_m=\cI(\bm\Omega_m)-\cos\theta_m$TeXr_m=\cI(\bm\Omega_m)-\cos\theta_m. Propagate the full joint frequency and phase-reference covariance with reference and reference; no $\omega_0$TeX\omega_0, $g$TeXg, scale, offset, or branch label is fitted in this statistic. - With the full residual covariance $\Sigma_r$TeX\Sigma_r, compute equation T= r^ T\Sigma_r^+ r, equation using the Moore--Penrose inverse and $\rank\Sigma_r$TeX\rank\Sigma_r degrees of freedom. The chi-square reference distribution is used only after calibrated simulations of the frozen resonance likelihood validate the whitened Gaussian approximation. Otherwise the null distribution is obtained from $10^6$TeX10^6 parametric-bootstrap draws from that frozen likelihood. In either case reject the symmetric one-holonomy trimer closure if the upper-tail probability is below $0.0027$TeX0.0027. - Repeat the identical phase sweep on the open three-site chain. Any phase-dependent chain spectrum rejects the assumed phase control or graph model before the ring result is interpreted.

The resolution rule is fixed before acquisition and prevents unresolved degeneracies from being silently converted into fitted branches. Equation reference tests the whole covariance-weighted residual vector rather than accepting a visually close curve.

The benchmark provides the following concrete closure within one finite observation class:

- a compact variable whose observable content is a genuine loop holonomy; - a tree no-go theorem that distinguishes physical phase from basis convention; - an exact microscopic Hamiltonian and analytic spectrum; - a global dimensionless restriction after eliminating both spectral nuisance parameters; - a common Hamiltonian derivative for phase loading, current, and response; - a probability-complete loss instrument and a passive three-port realization; - a deterministic numerical witness, a reproducible external figure comparison, and a frozen raw-data rejection protocol.

It does not derive a control-circle circumference, an autonomous rotor potential, or the trimer coupling from the gravitational root action. It does not show that a laboratory drive phase is the cosmological field $\chi/f$TeX\chi/f. The external comparison concerns a standard synthetic-gauge implementation and is consistent with the finite trimer relation, not the complete CHC ontology. A cross-sector fundamental claim would additionally require one independently justified map from the root compact scalar to multiple physical systems and a successful held-out comparison against their standard models.

The minimum observable compact phase is not an arbitrary periodic coefficient but a loop holonomy that survives every local rephasing. On the symmetric trimer that holonomy produces an exact cubic spectrum and a scale-free invariant equal to the cosine of the independently controlled phase. The identity has one exact predictive codimension after the frequency origin and hopping scale are removed, while an open chain supplies a strict null control. The same finite model fixes phase loading, current, causal response, passive poles, norm loss, and resolved events. Its analytic claims are verified numerically and its invariant agrees quantitatively with a published quantum-circuit spectrum at figure level. The prospective protocol states the additional raw-data and covariance conditions required to turn that concordance into a formal rejection test.

The accompanying repository provides the digitized table, its JSON provenance, the structured validation result, the vector figure, and two executable scripts:

- data/chc-mhb/roushan_fig_s4d_digitized.csv; - data/chc-mhb/roushan_fig_s4d_provenance.json; - reports/trimer_holonomy_validation_v2.0.json; - scripts/digitize_trimer_s4d_v2.py; - scripts/validate_trimer_holonomy_v2.py.

The digitization script downloads and verifies arXiv:1606.00077, renders only the cited page, and reproduces the table. The source PDF and rasterized source figure are not redistributed.