Geometric phase and finite projection protocols

A two-state system with a moving retained line

Consider the normalized complex vector u(θ,φ)=(cos(θ/2),e^iφsin(θ/2))^T, and the projector P=uu†. It selects one line in C². The same projector can be written P=(I+n·σ)/2 using the Pauli matrices and the unit vector n=(sinθ cosφ,sinθ sinφ,cosθ). This is a two-state geometric construction, not yet a weak-isospin field.

The Hamiltonian H=δ(I−P), with δ>0, has an isolated zero-energy line and a complementary state at δ. Changing θ or φ rotates the eigenspace without changing its gap. An adiabatic realization would slowly move these parameters; a projection realization instead prepares the line and tests successively against nearby lines.

These are distinct protocols sharing the same projector geometry. The exact survival formula below belongs to the ideal projection protocol. An arbitrary noisy preparation, dissipative completion channel or driven two-level system needs its own probability calculation.

Source: Sevenfold Gauge, §§11, 16.1, pp. 25–26, 36.

The connection and forward geometric phase

Put b=sin²(θ/2). Direct differentiation gives the anti-Hermitian connection A=u†du=ib dφ and curvature F=(i/2)sinθ dθ∧dφ. Along a fixed latitude, b is constant. Forward parallel transport solves dψ+Aψ=0, so one positive turn in φ has holonomy exp(−2πib).

The geometric phase is therefore −π(1−cosθ), equal to minus one half of the enclosed oriented solid angle. Choosing another local frame changes the connection by a gauge term; the loop phase modulo 2π remains the same. Reversing the loop reverses its phase.

The source writes its finite phase for the overlap u_x†u_y, an endpoint-frame comparison. The guide uses forward transport from x to y, whose overlap is u_y†u_x. These are conjugates for a line, so the phases differ by a sign. Making the direction explicit prevents apparent disagreements between equivalent calculations.

Source: Sevenfold Gauge, §16.1, pp. 36–37; forward-transport convention.

Exact finite-record phase and survival

Sample one turn at N equally spaced azimuths, with h=2π/N. Each backward comparison overlap is the same number m_N=1−b+be^ih. A forward polar link is its conjugate divided by its modulus. The product of N such links gives the forward loop phase on a continuous unwrapped branch.

For the ideal projection process, start in u₀ and perform N successive rank-one projections, including the closing projection onto the initial line. Each successful step has probability |m_N|². Conditional normalization after success then gives

Polar transport removes the overlap modulus to retain a norm-preserving comparison; sequential projection retains that modulus as a success probability. The two observables answer different questions. A large loop phase and a survival probability near one are compatible, because the first accumulates signed phases while the second records losses at each short step.

Source: Sevenfold Gauge, Proposition 16.1, pp. 36–37.

Worked example: a latitude at sixty degrees

Choose θ=π/3, giving b=1/4. The limiting forward phase is −π/2. The following values are obtained directly from the finite formulas, not from a fit to measurements.

Doubling N from 32 to 64 reduces the phase error by almost four, while the survival loss approaches its limit more slowly. Seven records are a possible sampling choice in this example; the formula applies to any permitted N and does not identify N with the separate seven-state source recurrence.

The projection explorer lets readers change latitude and record count. It should report the exact probability, a continuous phase branch where available, and singular-link warnings. The displayed output is a calculation in the declared carrier protocol.

Source: Sevenfold Gauge, §16.1 and Figure 1, pp. 36–37; direct numerical evaluation.

Different convergence laws and exceptional cases

Expanding the logarithm of the overlap at large N gives

The geometric phase error is generically second order in inverse record count, while the cumulative projection loss is first order. These exponents refer to a fixed latitude and this equal-step protocol. They do not automatically apply to a different path, sampling rule or noisy measurement model.

At b=1/2 the leading phase-error coefficient vanishes. For N>2 the unwrapped forward phase is exactly −π. At b=0 or b=1 the line is constant and survival is one; the unwrapped phase at the latter pole can differ by a full turn, which is physically equivalent. At b=1/2 and N=2, adjacent lines are orthogonal: m_N=0, survival is zero and no polar link phase exists. A plotting tool must not assign a continuous phase through that singularity.

Source: Sevenfold Gauge, Proposition 16.1 and discussion, pp. 36–37.

Curvature and leakage probe the same embedding

For a general orthonormal band frame u, define B_μ=(I−P)∂_μu. This is the component of a frame derivative that leaves the retained space. The nonnegative quantity q_μ=||B_μ||_HS² measures infinitesimal projection leakage in that direction.

The identity follows by inserting the complementary projection between frame derivatives. The norm bound is a Hilbert–Schmidt product estimate and the triangle inequality. It links two operationally distinct measurements: a loop curvature signal and losses under short projections must be compatible with the same underlying band motion.

This is a geometric consistency test, not a unique DTT signature. It also does not bound curvature solely by the spectral gap: the graph construction keeps its gap fixed while changing B_μ. For a determinant-corrected pair, central redistribution can change individual block bounds, while the corresponding direct-sum bound survives the orthogonal total-trace projection.

Source: Sevenfold Gauge, Theorem 15.1, pp. 35–36.

An exact determinant-linked central family

A line bundle L with its connection can generate the central sector of the rank-three/rank-two construction. Set E₃=(L⁻²)^⊕3 and E₂=(L³)^⊕2, using dual connections for negative powers. Their product determinant is L⁻⁶⊗L⁶, with a canonical parallel pairing. This controls every loop, including flat periods.

If the line's forward holonomy is e^iΦ, the block holonomies are e^−2iΦI₃ and e^3iΦI₂. Define a Wilson defect D_m=m−Re Tr W_m. Then

The ratio is defined only when D₃ is nonzero. Its small-loop limit is 3/2; finite loops have the explicitly calculable correction. The charge weights −2 and 3 express the product-volume constraint. This ratio compares trace defects in a specified carrier family, not weak and strong gauge couplings or particle masses.

Source: Sevenfold Gauge, §16.2, pp. 37–38.

What this example can establish

The latitude carrier demonstrates nontrivial holonomy, exact finite sampling effects and a clean separation between unitary comparison and projection probability. It can test a proposed experimental implementation of that geometry. The two-dimensional ambient space and Pauli-matrix description do not make the retained line an SU(2) gauge multiplet; its Berry transport is Abelian.

A rank-two retained band can carry non-Abelian Wilczek–Zee transport, but a background connection still needs further structure to become an autonomous gauge field. A physical theory must supply independently variable configurations, an action or Hamiltonian, couplings to matter and a state space with the desired constraints. External control parameters are not already propagating gauge degrees of freedom.

A useful interpretation therefore proceeds in stages: specify the carrier and protocol, check its exact geometric predictions, derive a completion dynamics that selects them, and establish the physical field limit. Success at one stage is meaningful without being presented as evidence for all the others.

Source: Sevenfold Gauge, §§16.4, 19, pp. 38, 44–45.

Continue reading

• Projection explorer
• Protected spectral bundles
• Winding, charge and topology
• Holonomy and loop observables