An exact projector with variable curvature and a fixed spectrum

Start from matrices whose spectrum is exact

Let x,y and β be real, and let δ>0. Define Z=diag(x+iβy,x−iβy) and s=1+x²+β²y². Since Z†Z=(s−1)I₂, the normalized graph frame and its projector are particularly simple:

Semicolons separate block rows. Direct multiplication gives u†u=I₂ and P²=P=P†. Every column of u has energy zero. The complementary two-dimensional space has energy δ, so spec H={0,0,δ,δ} for every embedding input. Equivalently H²=δH and Tr H=2δ.

This family specializes the source’s graph construction. The diagonal choice makes the full curvature calculable at every point, extending its origin-level witness. It supplies a controlled mathematical example; the coordinates have not been identified with spacetime or derived from the seven-state source.

Source: Sevenfold Gauge, PDF pp. 25–27.

Differentiate the normalized frame

For one graph line write z=x+iβy. Differentiating (1,z)/√s produces both a derivative of z and a real normalization term. The real part of z̄ dz equals ds/2, so it cancels that normalization term in u†du. Only the imaginary part remains. The conjugate line carries the opposite sign.

The result is anti-Hermitian and traceless. Its two components commute, but they vary with x and y. This already distinguishes nonzero curvature from noncommuting transport: the example can be curved while remaining inside a fixed Abelian subgroup of SU(2).

At y=0 the frame and projector are independent of β, whereas their y derivatives are not. Therefore knowing P at one point, even exactly, cannot determine its connection in nearby directions. Geometric information resides in the family of retained subspaces and their derivatives, not in a single spectral snapshot.

Source: Sevenfold Gauge, PDF p. 24. The displayed diagonal formula is derived here from the source’s projected connection.

Calculate curvature and its trace norm

Because the components commute, A∧A=0 in this frame. Differentiating βx/s and −βy/s gives

Hence F_xy=diag(2iβ/s²,−2iβ/s²). The ambient version is uF_xyu†=P[∂_xP,∂_yP]P. Either expression can be checked without choosing a physical gauge action.

At the origin, F_xy=diag(2iβ,−2iβ), matching the source’s rule F_xy(0)=2B for B=diag(iβ,−iβ). Reversing β reverses the signed curvature while preserving its squared norm and every eigenvalue of H. Setting β=0 makes the family flat. Varying the positive number δ changes neither P nor A nor F.

Source: Sevenfold Gauge, PDF pp. 25–26.

Compare identical spectra and identical point projectors

At x=1,y=0, the projector is P=(1/2)[I₂,I₂;I₂,I₂] for every β. The Hamiltonian is correspondingly δ/2 times [I₂,−I₂;−I₂,I₂]. Nevertheless the surrounding two-dimensional geometry depends on β.

The β=0 and β=1 cases agree on the complete instantaneous matrix H at this point. They differ in how that matrix extends away from the x-axis. This is a stronger diagnostic than comparing eigenvalues alone: no function of the single matrix H(1,0) can recover the missing transverse derivative data.

The source likewise separates bare clock-frame variation, which can be flat, from an additional embedding-control class. The table makes that separation explicit rather than inferring curvature from rank or spectral protection.

Source: Sevenfold Gauge, PDF pp. 26–27.

A change of frame cannot create new holonomy directions

Replace u by u′=ug⁻¹, where g(x,y) is any smooth unitary two-by-two matrix. The component connection becomes A′=gAg⁻¹−(dg)g⁻¹, while F′=gFg⁻¹. A non-diagonal, position-dependent g can give A′ complicated off-diagonal entries. Those entries do not establish a new physical non-Abelian effect.

Every loop based at the same point transforms by the same conjugation. In the original frame every loop is diagonal, so W₁W₂=W₂W₁. Consequently the transformed loops still commute. The simultaneous-conjugation invariant commutator is a direct diagnostic that survives all frame choices.

The band admits an SU(2) frame description because its connection is traceless, but this particular connection preserves two conjugate line subbundles. Its holonomy therefore lies in a fixed U(1) subgroup, up to conjugation. A rank-two protected band, a traceless connection and a nonzero curvature matrix are each insufficient to establish full SU(2) holonomy.

Source: Sevenfold Gauge, PDF pp. 24–25. The commuting-loop test is worked out here for the diagonal model.

Integrate an exact counterclockwise loop

Choose a coordinate circle x=r cosφ,y=r sinφ with φ increasing from zero to 2π. Along it, x dy−y dx=r²dφ. The remaining integral is elementary: ∫₀²π dφ/(a+b sin²φ)=2π/√[a(a+b)] for positive denominators.

The minus sign on the first eigenphase follows from forward transport P exp(−∮A). The connection is diagonal, so path ordering adds no complication here. Reversing the path inverts W. At fixed r, β→0 gives Φ→0 continuously.

For β=1,r=1, Φ=π and W=−I₂. This central loop is fully compatible with determinant one. The phase can be reported unwrapped, but the matrix depends only on its value modulo 2π. The explorer takes r=√(x²+y²); its circle is a supplied comparison path through that radius. This closed formula is a calculation for the present model, not a quoted source theorem.

Source: Sevenfold Gauge, PDF pp. 25, 36–37. Source setting: forward holonomy and explicit finite comparison protocols.

Recover the local sign and control the interpretation

Near r=0, the circle flux is Φ=2πβr²[1−(1+β²)r²/2+O(r⁴)]. Its leading term is the origin curvature coefficient times the oriented disk area. Therefore W=I−F_xy(0)πr²+O(r⁴), consistent with the section’s transport convention.

For a small square centered at a general point, target-first polar overlaps give the same local limit. Form M_yx=u(y)†u(x), take its unitary polar part when nonsingular, and multiply the earliest link on the right. Independent matrix differentiation and small-square products can check the sign without relying on the closed circle integral.

This example samples one diagonal Lie-algebra direction. The source’s twelve-direction accessibility result instead allows a broader family of independently chosen block generators and an explicit determinant-compatible control prescription. Neither that pointwise accessibility theorem nor this exact circle calculation establishes arbitrary global field configurations or a quantum measure.

Source: Sevenfold Gauge, PDF pp. 25–27.

Flat curvature can retain geometric projection loss

Let B_μ=(I−P)∂_μu. The normal frame derivatives contain information discarded by the connection. In this model they satisfy

The antisymmetric cross term gives F, while the diagonal terms govern leading projection loss. A normalized state projected from x to x+dx survives with probability 1−dx²/s²+O(dx³). Along y the corresponding coefficient is β²/s². These probabilities are per normalized state, without an extra factor from tracing over both carrier directions.

At β=0 the curvature vanishes, but the x-dependent projector still moves. From x=0 to x=h at y=0, the exact survival is 1/(1+h²), below one for h≠0. Flat connection geometry therefore does not imply a constant retained subspace or zero geometric leakage. The source’s curvature–loss bound concerns these normal derivatives under a declared projection protocol.

Source: Sevenfold Gauge, PDF pp. 35–36.

What the invariant measures and what it does not

The quantity −Tr(F_xy²) is invariant under internal frame changes at the same coordinate point. It is not a coordinate scalar by itself. Under x′=λx and y′=μy, the curvature component becomes F_x′y′=F_xy/(λμ), so its displayed square rescales by 1/(λ²μ²).

A physical action would require a metric, volume element, coefficient normalization and a relation between the parameters and physical coordinates. It would also need dynamics for the connection or underlying embedding. The protected spectral gap is a property of the supplied four-state Hamiltonian; it is not the excitation mass of a gauge field.

The useful conclusion is falsifiable and complete at its intended level: these matrices have fixed spectrum, their projected connection has the stated curvature, and the chosen circle has the stated holonomy. The curvature explorer exposes the matrices and comparisons. Identifying them with a physical weak interaction is a separate programme whose missing inputs remain visible.

Source: Sevenfold Gauge, PDF pp. 26–27.

Source: Interactions Unification, PDF pp. 24–27.

Continue reading

• Explore the exact projector matrices
• Separate Kato transport from physical evolution
• Choose a genuine spectral projection
• Impose global determinant compatibility