Microscopic Closure, Functional Nuisance, and Predictive Rank in the Compact-Phase Framework

On its regular branch the minimal CHC scalar sector is locally Einstein gravity plus one canonical scalar. A nontrivial extension must therefore enter through global target topology or through couplings that survive the canonical field redefinition. Let

\chi:M\longrightarrow S^1_L,
 \qquad \chi\sim\chi+L,

be the canonical compact field. A general sector description may contain periodic constitutive functions

c_A(\chi),\qquad A=1,\ldots,p,

in matter masses, kinetic coefficients, probe metrics, boundary susceptibilities, and detector couplings. Periodicity alone leaves infinitely many Fourier coefficients. Consequently compactness is not a parameter-selection principle.

The closure question is sharper. It asks whether the sector observables are fixed by one predeclared finite parameter vector, whether the same microscopic interaction determines both phase loading and detector response, whether the reduced event dynamics is probability complete, and whether at least one observable direction remains unavailable to nuisance retuning.

definition: Microscopically closed observation window. A declared observation window is microscopically closed when it is supplied with

- a compact-phase action and boundary data; - a finite parameter vector fixed before the observable basket is evaluated; - a specified system--detector interaction and detector initial state; - a controlled reduction with an error bound; - a completely positive trace-preserving event description; and - at least one nonzero observable covector annihilating the fitted tangent space after nuisance profiling.

Let $X$TeXX be a Banach space of admissible constitutive functions and let $\Theta\subset\mathbb R^q$TeX\Theta\subset\mathbb R^q be a finite-dimensional parameter domain. A finite observable basket is represented by a continuously differentiable map

F:X\times\Theta\longrightarrow\mathbb R^m.

theorem: Functional-nuisance saturation. Suppose that $D_uF(u_0,\theta_0):X\to\mathbb R^m$TeXD_uF(u_0,\theta_0):X\to\mathbb R^m is surjective. Then the image of $F$TeXF contains an open neighborhood of $F(u_0,\theta_0)$TeXF(u_0,\theta_0). Hence no nonzero equality germ $C(y)=0$TeXC(y)=0 whose zero set contains that attainable neighborhood follows from the model near the reference point. Any equality satisfied throughout the local image vanishes identically on an open set.

proof. Choose $x_1,\ldots,x_m\in X$TeXx_1,\ldots,x_m\in X such that $D_uF(u_0,\theta_0)x_j=e_j$TeXD_uF(u_0,\theta_0)x_j=e_j, where $e_j$TeXe_j is the standard basis of $\mathbb R^m$TeX\mathbb R^m. The map $R:\mathbb R^m\to X$TeXR:\mathbb R^m\to X defined by $Ra=\sum_ja_jx_j$TeXRa=\sum_ja_jx_j is a bounded right inverse of $D_uF$TeXD_uF. Define

\phi(a)=F(u_0+Ra,\theta_0).

Then $D\phi(0)=I_m$TeXD\phi(0)=I_m. The finite-dimensional inverse-function theorem implies that $\phi$TeX\phi, and therefore $F$TeXF, maps a neighborhood onto an open neighborhood of $F(u_0,\theta_0)$TeXF(u_0,\theta_0). A continuously differentiable equality holding on that image holds on an open set and supplies no nontrivial local restriction.

corollary: Necessity of finite constitutive closure. If an augmented response map remains surjective after further observables are added, those observables create no local equality prediction. Predictive content requires a symmetry, microscopic construction, or independently fixed restriction that makes the augmented functional derivative fail to be onto.

proof. The first assertion is reference. Its contrapositive gives the second: a nonzero local equality requires a proper image tangent space.

Write $f=L/(2\pi)$TeXf=L/(2\pi) and let $z$TeXz be a complex spurion. Under a compact shift by $\alpha f$TeX\alpha f, impose

\chi\mapsto\chi+\alpha f,
 \qquad z\mapsto e^{-i\alpha}z.

The product $z e^{i\chi/f}$TeXz e^{i\chi/f} is invariant. This is an additional symmetry hypothesis; it is not a consequence of target compactness.

theorem: Single-spurion first-harmonic normal form. Let $c_A(\chi,z,\bar z)$TeXc_A(\chi,z,\bar z) be real, $L$TeXL-periodic in $\chi$TeX\chi, analytic in $(z,\bar z)$TeX(z,\bar z) near the origin, and invariant under reference. Then

c_A(\chi,z,\bar z)
 =c_A^{(0)}+2\operatorname{Re}\!\left(\beta_Az e^{i\chi/f}\right)
 +O(|z|^2),

where $c_A^{(0)}\in\mathbb R$TeXc_A^{(0)}\in\mathbb R and $\beta_A\in\mathbb C$TeX\beta_A\in\mathbb C. If the microscopic representation fixes the coefficients $\beta_A$TeX\beta_A, the first-order constitutive freedom has at most two real directions, $\operatorname{Re}z$TeX\operatorname{Re}z and $\operatorname{Im}z$TeX\operatorname{Im}z. If the spurion phase is fixed independently, it has at most one.

proof. Expand $c_A$TeXc_A in Fourier modes in $\chi/f$TeX\chi/f and in a convergent power series in $z$TeXz and $\bar z$TeX\bar z. A monomial $z^p\bar z^q e^{in\chi/f}$TeXz^p\bar z^q e^{in\chi/f} transforms by $e^{i(n-p+q)\alpha}$TeXe^{i(n-p+q)\alpha} and is invariant only when $n-p+q=0$TeXn-p+q=0. At order zero this requires $n=0$TeXn=0. At first order the allowed monomials are $z e^{i\chi/f}$TeXz e^{i\chi/f} and its complex conjugate. Reality pairs their coefficients, proving reference. Once the $\beta_A$TeX\beta_A are fixed, differentiation with respect to the two real components of $z$TeXz gives a response matrix of rank at most two; fixing its phase leaves one real component.

remark. Allowing each $\beta_A$TeX\beta_A to be fitted independently restores a separate parameter direction for each coupling and can saturate the observable space. The normal form becomes predictive only when a specified microscopic representation fixes these coefficients or their ratios.

Consider a finite-dimensional detector Hilbert space $\cH_D$TeX\cH_D, an initial density operator $\rho_D$TeX\rho_D, and a bounded continuously differentiable Hamiltonian $H_D(t;\chi(t))$TeXH_D(t;\chi(t)). Let $U_\chi(t,s)$TeXU_\chi(t,s) be its unitary propagator. For two phase histories define [citation]

\cF[\chi_+,\chi_-]
 =\Tr\!\left(U_{\chi_+}(T,0)\rho_DU_{\chi_-}(T,0)^\dagger\right),
 \qquad S_{\rm IF}=-i\log\cF,

on a neighborhood where the logarithm is defined. Units with $\hbar=1$TeX\hbar=1 are used in this section.

theorem: Influence-functional phase-loading identity. At coincident histories $\chi_+=\chi_-=\chi$TeX\chi_+=\chi_-=\chi, one has $\cF=1$TeX\cF=1 and

\left.\frac{\delta S_{\rm IF}}{\delta\chi_+(t)}\right|_{+=-}
 =-\Tr\!\left(\rho_D(t)\,\partial_\chi H_D(t;\chi(t))\right),
 \qquad
 \rho_D(t)=U_\chi(t,0)\rho_DU_\chi(t,0)^\dagger.

For a linear coupling $H_D=H_{D0}-\chi B$TeXH_D=H_{D0}-\chi B, differentiation of the right-hand side with respect to an earlier phase perturbation gives the retarded response [citation]

G_R(t,t')=i\Theta(t-t')\Tr\!\left(\rho_D[B(t),B(t')]\right).

Thus the phase-loading expectation and the causal detector response are fixed by the same microscopic operator $B$TeXB.

proof. Duhamel differentiation gives

\frac{\delta U_{\chi_+}(T,0)}{\delta\chi_+(t)}
 =-iU_{\chi_+}(T,t)\partial_\chi H_D(t;\chi_+(t))U_{\chi_+}(t,0).

At coincident histories, unitarity and cyclicity of the trace reduce the derivative of $\cF$TeX\cF to $-i\Tr(\rho_D(t)\partial_\chi H_D)$TeX-i\Tr(\rho_D(t)\partial_\chi H_D). Since $S_{\rm IF}=-i\log\cF$TeXS_{\rm IF}=-i\log\cF and $\cF=1$TeX\cF=1, reference follows. A second Duhamel differentiation of the expectation value, followed by time ordering, gives the commutator in reference. Causality supplies the factor $\Theta(t-t')$TeX\Theta(t-t').

corollary: No independent retuning of loading and response. Within a declared microscopic model, a phenomenological PLE loading and a detector susceptibility cannot be varied independently if both are claimed to descend from $H_D(t;\chi)$TeXH_D(t;\chi). A fit requiring independent changes rejects that microscopic closure on the declared window.

proof. Both quantities are first and second derivatives of the same influence action by reference. Independent variation would change one derivative without the corresponding change in the common primitive.

Let a real finite-dimensional memory state satisfy

\dot x=Ax+Bu,
 \qquad y=Cx+Du,
 \qquad G(s)=D+C(sI-A)^{-1}B.

The eigenvalues of $A$TeXA may include complex-conjugate pairs and therefore damped oscillatory modes.

theorem: Certified passive finite-memory realization. Suppose $A$TeXA is Hurwitz and there exists a symmetric matrix $P>0$TeXP>0 such that

A^{\mathsf T}P+PA  PB-C^{\mathsf T}

 B^{\mathsf T}P-C  -(D+D^{\mathsf T})
 \preceq0.

Then the storage function $V(x)=x^{\mathsf T}Px/2$TeXV(x)=x^{\mathsf T}Px/2 satisfies

\dot V\le u^{\mathsf T}y.

If $(A,B)$TeX(A,B) is controllable and $(C,A)$TeX(C,A) is observable, the realization is minimal: every finite-dimensional realization of the same impulse-response operator has state dimension at least $\dim x$TeX\dim x. Equivalently, the input--output Hankel operator has rank $\dim x$TeX\dim x.

proof. Direct differentiation yields

\dot V-u^{\mathsf T}y
 =\frac12
 x
u^{\mathsf T}
 
 A^{\mathsf T}P+PA  PB-C^{\mathsf T}

 B^{\mathsf T}P-C  -(D+D^{\mathsf T})
 
 x
u,

which is nonpositive by reference. For minimality, controllability makes the reachable subspace the entire state space and observability makes distinct reachable states distinguishable by future outputs. The past-input--to--future-output Hankel map therefore factors through an $n=\dim x$TeXn=\dim x dimensional space with no smaller quotient; its rank is $n$TeXn. Any realization must factor the same Hankel map through its state space and hence has dimension at least $n$TeXn.

remark. The theorem strictly extends the completely monotone positive-real-pole subclass: complex poles are allowed whenever the real realization and the certificate reference hold [citation]. A transfer function with a branch cut generally requires an infinite-dimensional realization or a declared finite-band approximation with an error bound [citation]. No finite Hankel-rank claim is made for an exact branch-cut kernel.

Let the propagation system and $r$TeXr explicit memory modes have joint state $\varrho$TeX\varrho and generator

\mathcal L_\chi(\varrho)
 =-i[H_\chi,\varrho]
 +\sum_{a=1}^{s}\left(
 J_a\varrho J_a^\dagger-\frac12\{J_a^\dagger J_a,\varrho\}
 \right).

theorem: Enlarged-instrument reduction. For bounded $H_\chi=H_\chi^\dagger$TeXH_\chi=H_\chi^\dagger and $J_a$TeXJ_a, reference generates a completely positive trace-preserving semigroup on the enlarged system [citation]. For a fixed memory initial state $\sigma_M$TeX\sigma_M, the reduced map

\Phi_t(\rho)=\Tr_M\!\left[e^{t\mathcal L_\chi}(\rho\otimes\sigma_M)\right]

is completely positive and trace preserving at every fixed time, although it need not be a semigroup. Conditional removal of the recycling terms gives the no-event evolution

\dot\varrho_0=-i(H_{\rm eff}\varrho_0-\varrho_0H_{\rm eff}^\dagger),
 \qquad
 H_{\rm eff}=H_\chi-\frac{i}{2}K,
 \qquad K=\sum_aJ_a^\dagger J_a,

and event rate $\Tr(K\varrho_0)$TeX\Tr(K\varrho_0). Hence the DBC sink and the PCD event channels are the no-event and resolved-event components of one enlarged instrument.

proof. The GKSL theorem gives complete positivity and trace preservation of the enlarged semigroup. Tensoring with a fixed memory state and taking a partial trace are completely positive trace-preserving operations, so their composition with that semigroup has the same property at each time. Expanding reference and deleting $\sum_aJ_a\varrho J_a^\dagger$TeX\sum_aJ_a\varrho J_a^\dagger leaves reference. Taking the trace gives $\dd\Tr\varrho_0/\dd t=-\Tr(K\varrho_0)$TeX\dd\Tr\varrho_0/\dd t=-\Tr(K\varrho_0), while restoration of the recycling terms assigns the removed trace to the resolved outcomes.

corollary: Finite-mode memory and multitime consistency. If the enlarged memory modes have the passive linear response reference, their minimal number is fixed by reference. The reduced non-Markovian evolution possesses a completely positive multitime process because it is obtained from a fixed enlarged CPTP evolution with interventions inserted before the final partial trace.

proof. The first assertion is the minimality part of reference. For the second, compose each intervention with the enlarged CPTP propagators and trace out the memory only after the final step. Complete positivity is preserved under composition and partial trace, producing a positive process-tensor Choi operator [citation].

Let $\theta\in\Theta\subset\mathbb R^q$TeX\theta\in\Theta\subset\mathbb R^q collect the parameters remaining after the microscopic representation, spurion rule, and detector architecture are fixed. Let $\eta\in\mathbb R^k$TeX\eta\in\mathbb R^k be a declared nuisance vector. Form a dimensionless standardized observable $y_A$TeXy_A as $\log(O_A/O_{A0})$TeX\log(O_A/O_{A0}) for a positive observable, or as $(O_A-O_{A0})/s_A$TeX(O_A-O_{A0})/s_A with a fixed nonzero scale $s_A$TeXs_A otherwise, and define

F(\theta,\eta)=(y_1,\ldots,y_m)

the observable map. At a reference point let $J_\theta$TeXJ_\theta and $J_\eta$TeXJ_\eta be the whitened Jacobians and let $P_\eta$TeXP_\eta project onto $\operatorname{im}J_\eta$TeX\operatorname{im}J_\eta.

theorem: Profiled predictive codimension. Put $S=(I-P_\eta)J_\theta$TeXS=(I-P_\eta)J_\theta and $r=\rank S$TeXr=\rank S. The linearized model leaves exactly

m-\rank[\,J_\eta\;J_\theta\,]

observable directions unavailable to either nuisance or theory-parameter retuning. Every vector

w\in\ker[\,J_\eta\;J_\theta\,]^{\mathsf T}

gives the first-order dimensionless restriction

w^{\mathsf T}\delta y=0.

No local equality prediction survives if the combined Jacobian has full row rank.

proof. The fitted tangent space is the column space of $[J_\eta\;J_\theta]$TeX[J_\eta\;J_\theta]. Its orthogonal complement is the kernel in reference. Rank--nullity gives its dimension as reference, and orthogonality gives reference. If the combined Jacobian has full row rank, the orthogonal complement is zero.

corollary: Single-spurion test. If the microscopic coefficients in reference are fixed and only the amplitude of a fixed-phase spurion is varied, then before nuisance profiling the first-order theory response has rank at most one. An $m$TeXm-observable basket therefore supplies at least $m-1$TeXm-1 left-null restrictions. After nuisance profiling, the empirical restrictions are precisely the covectors in $\ker J_\theta^{\mathsf T}\cap\ker J_\eta^{\mathsf T}$TeX\ker J_\theta^{\mathsf T}\cap\ker J_\eta^{\mathsf T}.

proof. The fixed-phase conclusion of reference leaves one real theory direction. Apply reference before and after adjoining the nuisance tangent space.

theorem: Compact-phase microscopic closure. Fix a declared finite observation window. Assume:

- the canonical phase is compact as in reference; - every sector coupling is generated by a finite microscopic parameter vector and, where continuous shift breaking is present, obeys the spurion hypothesis of reference; - the detector coupling is specified by a bounded Hamiltonian of the class in reference; - the admitted finite-memory approximation has a certified realization satisfying reference and is controllable and observable; - the event dynamics has an enlarged GKSL realization reference; and - all approximation errors are bounded uniformly on the window.

Then the specified common microscopic data determine the phase-loading expectation, the causal response kernel, the passive memory order, the DBC no-event sink, the PCD resolved event channels, and the finite-dimensional tangent space of every declared observable basket. A nonzero vector in reference is therefore a dimensionless, non-calibration prediction up to the declared reduction error. Failure of the loading--response derivative identity, KYP certificate, CPTP completion, or profiled null restriction rejects the proposed closure on that window.

proof. The compact and spurion assumptions restrict the constitutive family by reference. The first and second derivatives of the influence action give the loading and causal response by reference. The KYP certificate and controllable--observable minimality give passivity and memory order by reference. The enlarged dynamics gives the CPTP instrument and its no-event branch by reference. All observable derivatives therefore factor through the same finite microscopic parameter vector and the declared nuisance vector. reference supplies the surviving null restrictions. Uniform reduction bounds transfer each exact reduced statement to the observation window with at most the declared error. Violation of any necessary conclusion rejects at least one corresponding hypothesis.

A fixed numerical construction checks the algebra used in each component of the closure theorem without replacing its proof. The functional-nuisance example has a four-dimensional identity derivative; a four-observable single-spurion response has ranks two and one before and after fixing the spurion phase; a two-state oscillatory realization satisfies the KYP inequality and both minimality ranks; and an amplitude-damping channel has an exactly complete Kraus representation with a positive-semidefinite Choi matrix. A discrete unitary insertion additionally checks the influence-loading derivative, while a four-observable Jacobian checks the predicted one-dimensional left null space. No fitted coefficient or random draw enters these calculations.

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

The existence conditions above are realized explicitly in Paper CHC-MHB by an externally controlled compact phase on a symmetric three-mode loop. Two spectral nuisance parameters, the common frequency and hopping scale, are eliminated globally, leaving the exact dimensionless identity

\frac{3\sqrt6\prod_j(\Omega_j-\bar\Omega)}
 {\bigl[\sum_j(\Omega_j-\bar\Omega)^2\bigr]^{3/2}}=\cos\theta.

The same finite model supplies the Hamiltonian phase load, a passive three-port response, and a probability-complete amplitude-loss instrument. Its 1001-point numerical witness reaches normalized residuals below $3.3\times10^{-10}$TeX3.3\times10^{-10}, and a 39-point retrospective extraction from a published superconducting-circuit spectrum has invariant RMSE $0.02464$TeX0.02464. Because the published figure has no raw covariance and the phase-cycle convention is retrospective, the latter result is classified as external figure-level concordance rather than a likelihood confirmation. A new raw-data test is fixed separately in CHC-MHB.

The closure theorem is conditional on an explicit microscopic action. It does not derive that action from compactness, select the canonical circumference, prove a defect solution, determine Standard-Model representations, or establish ultraviolet completion. It also does not turn a generic one-pole response into a distinctive CHC prediction. Distinctive content appears only when the compact-phase representation fixes coefficients shared by more observables than there are fitted directions.

The theorem changes the admissible order of construction. Sector-wise response functions may no longer be introduced and fitted first and described as unified afterward. One must specify the microscopic coefficients, derive the reduced maps with error control, compute the combined nuisance rank, and then evaluate the surviving null directions on information not used in calibration.

Unrestricted constitutive freedom is locally nonpredictive whenever it spans the observable space. A compact target supplies periodicity but does not remove that freedom. A single charged spurion, a fixed microscopic representation, and a controlled detector dilation reduce the freedom to a finite parameter map. The resulting influence action binds phase loading to causal response; a passive state-space certificate fixes admissible memory and its minimal order; an enlarged GKSL model completes the event probabilities; and the profiled observable Jacobian identifies the exact directions that remain predictions. This chain is the closure standard for subsequent compact-phase applications.

No observational or experimental data are introduced. The proofs use only the stated finite-dimensional operator, Banach-space, state-space, and differential assumptions. The accompanying files scripts/validate_microscopic_closure_v2.py and reports/microscopic_closure_validation_v2.0.json reproduce the deterministic checks in reference.