We study the precision geometry generated by three self-adjoint observables in a faithful quantum state. The centered GNS Gramian Γ = C + iΣ, its inverse Q, normalized precision coefficients B, and conditional levels pi = ΓiiQii define a weighted magnetic triangle. The paper separates exact structure, reconstruction, and open prediction. First, a reduction identity gives pi = (1 − b 2 i )/ det(I − B). For the logarithmic embedding this implies an exact global rigidity theorem: if all three induced edge weights are positive, then the logarithmic circulant deviation obeys D > J∗ = 0.394202716596 . . . , with sharp infimum J ∗ approached only in the degenerate equal-b limit. The former numerical global-floor claim is therefore replaced by an analytic theorem, independently cross-checked by an interval branch-and-bound certificate. Second, the Pauli triple has Q = (I −i[n]×)/(1−|n| 2 ) and can realize only Hol = ±1. Real precision geometry gives instead Hol = ±i; the charged- lepton phase lies outside both quantized sectors. Third, the Koide–Foot square-root mass family is exactly the spectrum of an offset uniform magnetic Schr¨odinger operator on C3. Current 2026 PDG masses show that r = √ 2 and δ = 2/9 are each individually compatible after refitting the other parameter, but their simultaneous exactness is not: it fixes mµ/me about 451.7 quoted standard deviations from the measured ratio. Fourth, for the explicit modelling prescription µi = pi , a Krawczyk interval certificate proves an exact realization of the measured square-root mass ratios at the idealized Brannen flux Φ0 = π+2/3 on a fixed transversal slice; further Krawczyk certificates prove realizations at the exact decimal fluxes Φ = 1.105 and Φ = 1.4, and a continuation scan locates the feasibility boundary at Φ∗ ≈ 1.1044604 (record level). Finally, an implicit-function theorem turns a former caveat into a structural result: near the certified witness, the exact mass-ratio solutions form a three-dimensional manifold when (b1, Φ, α) are allowed to vary in the family µ = p α. Hence the two mass ratios cannot locally select the flux, the measure exponent, or the witness coordinate. The framework therefore provides exact rigidity, exclusions, and realizability, while identifying a parameter-free upstream selector as the precise missing ingredient required for prediction.
We study the precision geometry generated by three self-adjoint observables in a faithful quantum state. The centered GNS Gramian Γ = C + iΣ, its inverse Q, normalized precision coefficients B, and conditional levels pi = ΓiiQii define a weighted magnetic triangle. The paper separates exact structure, reconstruction, and open prediction. First, a reduction identity gives pi = (1 − b 2 i )/ det(I − B). For the logarithmic embedding this implies an exact global rigidity theorem: if all three induced edge weights are positive, then the logarithmic circulant deviation obeys D > J∗ = 0.394202716596 . . . , with sharp infimum J ∗ approached only in the degenerate equal-b limit. The former numerical global-floor claim is therefore replaced by an analytic theorem, independently cross-checked by an interval branch-and-bound certificate. Second, the Pauli triple has Q = (I −i[n]×)/(1−|n| 2 ) and can realize only Hol = ±1. Real precision geometry gives instead Hol = ±i; the charged- lepton phase lies outside both quantized sectors. Third, the Koide–Foot square-root mass family is exactly the spectrum of an offset uniform magnetic Schr¨odinger operator on C3. Current 2026 PDG masses show that r = √ 2 and δ = 2/9 are each individually compatible after refitting the other parameter, but their simultaneous exactness is not: it fixes mµ/me about 451.7 quoted standard deviations from the measured ratio. Fourth, for the explicit modelling prescription µi = pi , a Krawczyk interval certificate proves an exact realization of the measured square-root mass ratios at the idealized Brannen flux Φ0 = π+2/3 on a fixed transversal slice; further Krawczyk certificates prove realizations at the exact decimal fluxes Φ = 1.105 and Φ = 1.4, and a continuation scan locates the feasibility boundary at Φ∗ ≈ 1.1044604 (record level). Finally, an implicit-function theorem turns a former caveat into a structural result: near the certified witness, the exact mass-ratio solutions form a three-dimensional manifold when (b1, Φ, α) are allowed to vary in the family µ = p α. Hence the two mass ratios cannot locally select the flux, the measure exponent, or the witness coordinate. The framework therefore provides exact rigidity, exclusions, and realizability, while identifying a parameter-free upstream selector as the precise missing ingredient required for prediction.
We study the precision geometry generated by three self-adjoint observables in a faithful quantum state. The centered GNS Gramian Γ = C + iΣ, its inverse Q, normalized precision coefficients B, and conditional levels pi = ΓiiQii define a weighted magnetic triangle. The paper separates exact structure, reconstruction, and open prediction. First, a reduction identity gives pi = (1 − b 2 i )/ det(I − B). For the logarithmic embedding this implies an exact global rigidity theorem: if all three induced edge weights are positive, then the logarithmic circulant deviation obeys D > J∗ = 0.394202716596 . . . , with sharp infimum J ∗ approached only in the degenerate equal-b limit. The former numerical global-floor claim is therefore replaced by an analytic theorem, independently cross-checked by an interval branch-and-bound certificate. Second, the Pauli triple has Q = (I −i[n]×)/(1−|n| 2 ) and can realize only Hol = ±1. Real precision geometry gives instead Hol = ±i; the charged- lepton phase lies outside both quantized sectors. Third, the Koide–Foot square-root mass family is exactly the spectrum of an offset uniform magnetic Schr¨odinger operator on C3. Current 2026 PDG masses show that r = √ 2 and δ = 2/9 are each individually compatible after refitting the other parameter, but their simultaneous exactness is not: it fixes mµ/me about 451.7 quoted standard deviations from the measured ratio. Fourth, for the explicit modelling prescription µi = pi , a Krawczyk interval certificate proves an exact realization of the measured square-root mass ratios at the idealized Brannen flux Φ0 = π+2/3 on a fixed transversal slice; further Krawczyk certificates prove realizations at the exact decimal fluxes Φ = 1.105 and Φ = 1.4, and a continuation scan locates the feasibility boundary at Φ∗ ≈ 1.1044604 (record level). Finally, an implicit-function theorem turns a former caveat into a structural result: near the certified witness, the exact mass-ratio solutions form a three-dimensional manifold when (b1, Φ, α) are allowed to vary in the family µ = p α. Hence the two mass ratios cannot locally select the flux, the measure exponent, or the witness coordinate. The framework therefore provides exact rigidity, exclusions, and realizability, while identifying a parameter-free upstream selector as the precise missing ingredient required for prediction.
Let X1, . . . , XN be a fixed finite family of centered self-adjoint observables in a faithful state ω, and let Γij = ω(XiXj ) be their full Hermitian GNS Gram matrix. We study the normalized inverse Gramian Bij = −Qij/ p QiiQjj , Q = Γ−1 , and show that it is the negative Gram matrix of a canonical normalized dual frame formed by conditional residual directions in the GNS Hilbert space. Consequently, cyclic products of the entries of B are Bargmann invariants of this conditional-residual frame. Combining B with the scale-free conditional-embedding levels pi = ΓiiQii yields a Hermitian commutator generator F = i[B, Hg], where Hg = diag(g(pi)) and g is strictly increasing. The phases of the nonzero entries of F define a magnetic connection on the precision graph. We prove four principal results. First, the induced cycle holonomy factorizes into a discrete ordering skeleton and the continuous Bargmann phase of the inverse GNS Gramian. Second, at a real Gramian C, the commutator-visible antisymmetric perturbation Γ(ε) = C + iεΣ maps with full local rank onto the entire cycle-flux space of the precision graph; hence the flat locus has local codimension equal to the first Betti number. Third, the lowest magnetic-Laplacian eigenvalue opens quadratically from a flat point, with coefficient equal to the weighted Hodge norm of the non-exact edge phase divided by total vertex measure. A complete perturbative proof is supplied. Fourth, every positive Hermitian Gramian admits an explicit faithful finite-dimensional realization by centered self-adjoint observables, so the construction is not merely formal. The relation to covariance selection and complex partial spectral coherence is stated explicitly: Hermitian precision matrices and complex partial-coherence phases are estab- lished objects. The proposed contribution is the GNS provenance, the conditional-residual Bargmann interpretation, the state-to-magnetic-connection map, the full-cycle-rank theo- rem, and the resulting spectral-gap transduction law. A deterministic four-cycle witness and a bitwise reproducible verification package are included. No identification of the resulting gap with a particle mass is claimed.
The companion theory paper Quantum Precision Holonomy constructs a magnetic graph Laplacian LU from inverse GNS-Gramian data and proves a quadratic opening law for its lowest eigenvalue. The present paper addresses a narrower physical question: what does the dimensionless operator mean, and how is it assigned physical units in a system where the same operator class is an actual Hamiltonian? We use the programmable nano-optomechanical network of Slim, del Pino, and Verhagen. In the opposed-flux diamond at the special ratio Jc = √ 2J, one antisymmetric apex mode decouples and the remaining three modes are exactly those of an equal-coupling Aharonov– Bohm loop. For the unit-weight magnetic triangle, LU = 2I − AU , and the experimentally realized coupled-mode Hamiltonian satisfies Hphys h = νrefI + jeff(2I − LU ). Thus spectral splittings are linear in differences of eigenvalues of LU ; no particle-mass map is introduced. The independently reported perimeter coupling is J/(2π) = 5.000 kHz, implying jeff = 7.071 067 812 kHz. A one-parameter calibration on ten digitized gaps gives jeff = 6.877 941 342 kHz and hence J/(2π) = 4.863 438 964 kHz, a difference of −2.731%. On nine frozen held-out gaps, MAE = 0.344 229 kHz and RMSE = 0.423 010 kHz. The raster scale is approximately 0.303 kHz per vertical pixel; reported mechanical linewidths are approximately 1 kHz to 7 kHz. The small-flux QPH law becomes ∆νQPH = jeff 9 Φ 2 + O(Φ4 ), with independently calibrated coefficient 0.785 674 201 kHz rad−2 . This is a figure-level held- out consistency test of the downstream map (G, w, U) → LU → physical spectrum. It does not test the upstream derivation of (w, U) from a measured complex GNS Gramian. Keywords: magnetic graph Laplacian; optomechanics; Aharonov–Bohm phase; spectral calibra- tion; held-out test; Quantum Precision Holonomy. Scope. This paper is a self-contained empirical companion to the mathematical QPH paper. It defines every object needed for the calibration, derives the C3 spectral map, freezes one fit parameter and a prospective train/test split, and reports physical units and held-out errors. It does not repeat the full inverse-GNS-Gramian proofs and does not claim that the
Quantization Isn’t Magic: Holonomy + Rigidity in the Phase–Defect Dual Channel View AbstractWe present a structural, topological-field-theory motivated account of why quantization is an unavoidable feature of matter, written in a dual-channel architecture that makes the wave–particle (phase–defect) duality explicit. The key claim is that the discrete labels of quantum states are not “postulates of magic,” nor artifacts of semiclassical orbit pictures, but emerge from two coupled structural requirements: (i) global phase coherence on a compact configuration space (holonomy) and (ii) energetic enforcement by stiffness (rigidity) of the underlying phase medium. This provides a unified interpretation of “quantized orbits” as globally consistent stationary phase patterns (eigenmodes), while simultaneously explaining how discrete “particle-like” events arise as topological defects or singular phase-slip processes required to change topological sector.Channel I (Phase/Wave): Holonomy and stationary eigenmodes. We start from the polar (Madelung) decomposition of a complex amplitude Ψ = √ρ e^{iθ}, highlighting that the kinetic term contains a phase-gradient penalty proportional to ρ(∇θ)^2. In a rigidity-first vacuum, this term is read as a stiffness functional that suppresses incoherent phase gradients and stabilizes coherent global patterns. Single-valuedness requires that when traversing any closed loop C fully contained in a region with ρ > 0, the phase must close modulo 2π, enforcing an integer winding number ∮_C ∇θ·dℓ = 2πn, n ∈ ℤ. This integer is topological: it cannot change continuously without leaving the smooth phase manifold. Bound-state quantization then appears as the selection of globally admissible eigenmodes under regularity and normalizability boundary conditions, providing the standard anchor for discrete atomic spectra and angular multiplets.A critical precision: fluxoid (canonical circulation) vs kinematic circulation. In the presence of a gauge field A, the quantized object must be identified carefully. The topological integer remains the phase-winding n, but the physically measured kinematic circulation is shifted continuously by the magnetic flux Φ = ∮ A·dℓ. Defining the kinematic velocity v = (ħ/m)(∇θ − (q/ħ)A), one finds ∮ m v·dℓ = n h − qΦ, while the canonical/fluxoid circulation is the quantized object ∮ (m v + qA)·dℓ = n h. This distinction is essential for gauge invariance: Bohr–Sommerfeld quantization applies to the canonical momentum, and in flux-threaded geometries the integer label remains discrete while energies shift by continuous holonomy, as in the standard ring-with-flux spectrum E_n ∝ (n − Φ/Φ_0)^2.Angular momentum refinement: component vs total. For central potentials, phase closure under φ → φ + 2π directly quantizes the azimuthal component m of angular momentum (the L_z eigenvalue), while the total L^2 eigenvalues follow from the full SO(3) algebra. In other words, phase closure gives the discrete “component label,” while representation theory supplies the complete multiplet structure L^2 = ħ^2 ℓ(ℓ + 1). This emphasizes that quantization is simultaneously a global closure rule and an algebraic symmetry statement.Channel II (Defect/Particle): integer topological charges and phase slips. The same compactness that forces integer holonomy also forces defects to carry integer-valued charges. In three spatial dimensions, the fundamental topological defects of a compact phase are string-like vortex lines; their vorticity is distributional and quantized by the same integer winding. Crucially, changes of the winding sector require singular events where ρ → 0 at a defect core, allowing phase slips. This supplies a physically explicit mechanism for “quantum jumps”: discrete transitions are mediated not by arbitrary discontinuities but by topological defect processes that temporarily lift the phase coherence constraint.Duality and topological field theory completion. We formalize the equivalence of the two channels by decomposing the phase into smooth and singular parts, θ = θ_smooth + θ_sing, and noting that the holonomy integer in Channel I is exactly the defect charge sourced in Channel II. In 3+1 dimensions a compact scalar (Goldstone-like phase) is dual to a Kalb–Ramond two-form gauge field B_{μν}, with field strength H = dB, and defects couple through a conserved worldsheet current Σ^{μν} via ∫ B_{μν} Σ^{μν}. This places quantized “orbits” and quantized “matter” within a single modern TFT-compatible framework: smooth phase coherence corresponds to propagation of the dual gauge sector, while defect matter corresponds to quantized sources.Structural conclusion. Quantization emerges as a dual invariant of a compact, stiff vacuum: integers are enforced by global phase closure (holonomy) and stabilized by stiffness, while changes in those integers require defect-mediated singular events. The same logic unifies canonical gauge-invariant quantization (fluxoids vs kinematic circulation), angular momentum multiplets (component quantization vs full algebra), and the physical origin of discrete transitions (phase slips). This two-channel architecture is intended as a rigorous structural explanation that complements “shut up and calculate” quantum mechanics with a transparent topological and energetic rationale for why the universe admits only discrete stationary patterns and integer-labeled matter sectors. ## Abstract (Version 2 Addendum: N-Body Configuration Space & Entanglement) We address the standard “configuration space catastrophe” objection faced by hydrodynamic or medium-like reformulations of quantum mechanics: while a single-particle wavefunction admits a 3D polar (Madelung) representation, the fundamental state of an N-body system lives in the tensor-product Hilbert space and generically supports entanglement and Bell-inequality violations. In the R⁸ framework, the phase–closure (holonomy/winding) quantization mechanism developed for the 3D leaf fields is therefore interpreted as a *local* constraint acting on one-body marginals or conditional amplitudes derived from the full state, rather than as an ontological replacement of the N-body wavefunction by a single classical fluid on 3D space. We formulate this distinction explicitly via reduced density matrices and conditional wavefunctions, and show how local quantization indices can be enforced on these leaf-level representatives while nonlocal correlations arise from the globally maintained (projector-selected) entangled state supported by the Bank. This removes the local-hidden-variable ambiguity, preserves no-signalling, and clarifies how the two-channel (phase/defect) quantization picture coexists with genuine quantum information structure. We also state the remaining open microphysical task: an explicit model for how the Bank dynamically stabilizes the relevant projector subspaces while maintaining luminality constraints. # Minimal Patch Insertions ## A) Abstract (replace the existing one with this 3‑paragraph version) **Abstract** Quantization is often presented as an axiomatic rule (“allowed energies are discrete”) layered on top of classical mechanics. Here we give a structural account in which discreteness follows from two coupled ingredients: **holonomy** (global phase closure on compact configuration space) and **rigidity** (an energetic penalty for incoherent phase gradients). The outcome is that “quantized orbits” are not literal trajectories but globally admissible stationary phase patterns (eigenmodes). We organize the explanation into two dual channels. **Channel I (Phase/Wave):** single‑valuedness forces integer winding,∮C∇θ⋅dℓ=2πn,n∈Z,while boundary and regularity conditions select a discrete set of globally consistent modes (atomic spectra, angular multiplets). A crucial precision is the distinction between **kinematic circulation** and **canonical/fluxoid circulation** in the presence of gauge holonomy: the integer label remains discrete while measurable kinematics shifts continuously with flux. **Channel II (Defect/Particle):** compactness of the phase implies that topological defects carry integer charges and that changes of sector require singular events where the phase description fails (typically where ρ→0), enabling **phase slips**. We formalize the equivalence of the two channels via the standard compact‑scalar ↔ two‑form duality, placing quantized “orbits” and defect‑mediated “jumps” within a single TFT‑compatible framework. --- ## B) “Why is the phase compact?” (insert near the first appearance of Ψ=ρeiθ) **Add immediately after** Ψ=ρeiθ: > **Why θ is compact.** The overall phase of a quantum state is a gauge redundancy: physical states are equivalence classes under> >Ψ∼eiαΨ.>> Therefore the phase variable is defined only modulo 2π, i.e.> >θ≡θ+2π,>> which is the structural origin of integer winding and topological sectors. --- ## C) Spinors: explicit SU(2) double cover + Berry phase (insert in the spinor subsection) **Replace / augment the current spinor paragraph with:** A spin‑21 state transforms under SU(2), the double cover of SO(3). A rotation by angle φ about axis n^ is represented byU(φ)=exp(−2iφn^⋅σ),so that\[U(2\pi)=-\mathbb{1},\qquadU(4\pi)=+\mathbb{1}.\]Thus the configuration-space loop for a spinor closes only after 4π, giving the geometric origin of half‑integer sectors. Equivalently, for an adiabatic closed path of a spin‑21 on the Bloc
Preprint. Third paper of the contextuality-obstructions program, alongside Paper A (the canonical class of the MASA context stack), Paper B (the obstruction spectrum of Weyl context families), Paper D (the observer-context category) and a synthesis note. The slogan "contextual holonomy encodes quantum contextuality" is attractive and, as usually stated, false. This paper gives it a sharp form. On the context stack of maximal abelian subalgebras the canonical degree-2 class has order exactly n, and its cocycle holonomy along closed context loops detects EXACTLY the all-versus-nothing sector of contextuality for closed-triple Weyl families at even local dimension. Both containments are strict, each by an explicit certificate: at d=3 the class is nontrivial of order 3 while the achievable state-independent AvN value is 0, and KCBS exhibits state-dependent contextuality where the class shadow vanishes. The Wigner-friend loop generated by H and S splits by level: its projective (Pancharatnam) holonomy is +1 while its determinant-section holonomy is exactly -i, so any interpretational claim about "the" holonomy of that loop must state which one it means. ATTRIBUTION. The classification of state-independent values for qudit Weyl families, the even/odd dichotomy and the fixed value d/2, on which the state-independent side of the equivalence rests, is DUE TO Abramsky, Cercelescu and Constantin (FSCD 2024). Attainment at general even d is the Raussendorf-Okay-Zurel-Feldmann construction. This paper uses both and rediscovers neither. A RETRACTION IS CARRIED IN THIS PAPER, NOT BURIED. An earlier draft presented the tier-two facets of the d=4 census on a (1/(2*sqrt2))Z grid with a single coefficient magnitude 3/(2*sqrt2), and read the resulting unit-coefficient bounds 4*sqrt2 and 14*sqrt2/3 as placing Tsirelson's sqrt2 inside the tier-two geometry. The whole presentation was a normalization artifact and is withdrawn (Retraction 1): the stored lift has coefficients in {0,+-1} with integer bounds 6 and 7, every facet functional in the census is rational, and Q(sqrt2) is not among the eighteen canonical arithmetic fields. Certificate: d4_arithmetic_profile.py. WHERE THE EVIDENCE IS THIN, STATED IN THE PAPER. Two sweeps quoted here drew no positive instances at all: the d=4 half of the 2,100-family equivalence test, and the 6,500-trial false-certificate sweep, which draws no contextual family at any of d=2,4,8 and therefore establishes the absence of false certificates only vacuously. Both are disclosed where they appear. Theorem 2 is labelled machine-validated rather than proved: the closed form is established end to end for the deletion families treated, while the general closed-triple d=2 statement in compressed coordinates remains open and is recorded as open in the verification ledger. HARDWARE CAVEAT, STATED UP FRONT. The quoted significances, about 56 sigma for the ray-level Wigner-friend holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented witness model and conditional on compatibility and nondisturbance assumptions. They are not systematics-aware and neither measurement is loophole-free; a roughly 3.5 sigma preparation offset on one strand is itself evidence of unmodelled systematics. We also record that the pinned value S = 4.6125 cannot be recomputed from the stored counts (a naive recomputation gives 4.5662, a gap of about 2.7 standard errors); the archived value 4.558 +- 0.018 is the one an independent reader can reproduce from the deposited data. Read these as consistency checks on real hardware, not as precision measurements or certifications. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. It was held back from the first publication batch and released only after three adversarial review passes; what each pass found, and how each item was closed, is recorded in verification/INDEX.md rather than quietly fixed. One defect found there is NOT closed and belongs to the companion Paper B, which cites a false-certificate artifact that is not in fact in the repository; it is recorded rather than left for a reader to discover. COMPANIONS. Paper A: https://doi.org/10.5281/zenodo.21630431 . Paper B: https://doi.org/10.5281/zenodo.21630630 . Paper D: https://doi.org/10.5281/zenodo.21630697 . Note: https://doi.org/10.5281/zenodo.21630739 . Code archive: https://doi.org/10.5281/zenodo.21498498 AI-ASSISTANCE DISCLOSURE. Large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance by the scripts named in the text, and the author takes full responsibility for the results, including any errors. VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
Parts VIII–XI of this series reduced the program to a single missing object: a quantum measure μ(C) over causal histories. This Part XII specifies that object, realizes it in prototype, and tests whether it selects the observed effective spacetime dimension. The fundamental object is a Hermitian, positive, bi-additive Sorkin decoherence functional D(A,B) with physical measure μ(A)=D(A,A); classical measures obey the two-event sum rule I2=0, while quantum measures have I2≠0 but obey the Sorkin triple sum rule I3=0. An explicit such measure is constructed on the growth histories of small causal sets and verified numerically to satisfy I3=0 to machine precision while I2≠0. The fourth gate is sharpened beyond the standard Lorentzian-continuum requirement: the measure must concentrate on causal orders carrying coherent local proper time (a stable interval volume–proper-time law), the distinctive addition of this program. A systematic test on controlled ensembles then asks whether any non-circular intrinsic ingredient selects the 3+1 sector, with a uniformly negative result. The free measure is dominated by the entropic Kleitman–Rothschild phase (d_MM ≈ 2.37); proper-time coherence with propagation escapes that phase but settles on the effective 2+1 sector; the Benincasa–Dowker quantum phase produces only seed-unstable windows that vanish at the physically natural phase scale; and the Pauli–Jordan operator and the Sorkin–Johnston state, though intrinsic and phase-based, are monotone in dimension. A circularity audit shows that every selector whose optimum lies at four dimensions either inserts the dimension (local gravitational degrees of freedom, bound-matter stability, a (d−4)² term) or was tuned to it. The honest conclusion is that proper-time coherence explains the exit from the entropic phase but not the final 3+1 selection: the bare measure with coherence predicts an effective 2+1 phase, and the open problem is localized as the emergence of the third spatial direction (2+1 → 3+1). This yields a falsifiable prediction and identifies the missing object as a covariant, Bell-causal growth law whose continuum limit would fix the gravitational phase normalization. Supplementary Python scripts reproducing all numerical tests are included.