The exact holonomy Hessian and its two-block constraint

Separate the small-angle expansion from weak hopping

There are two different expansions in the singlet-loop model. Weak hopping means |t|/Δ is small and gives a series for the entire holonomy-dependent energy. Small curvature means W lies near identity, W=e^iX with a small Hermitian X, while t and Δ are held fixed. The exact Hessian concerns the second expansion and can be calculated at arbitrarily strong finite hopping.

Define the normalized energy response ΔE_m(X)=E_m(e^iX)−E_m(I). The coefficient convention is

For a direction X=sT with TrT²=1, κ̂_m is the second derivative d²ΔE/ds² at zero. It has energy units. If a doubled cost C=2ΔE is separately defined, C=κ̂_mTrX²+⋯. The explorer and energy formulas use ΔE directly, with the factor 1/2 displayed. Theorem 2.3, equation (8), p.10.

This distinction prevents an apparent factor-of-two discrepancy: the weak-hopping loop term is κ_m^loop(m−ReTrW), and the trace deficit itself is TrX²/2+O(‖X‖⁴).

Write the exact coefficient in terms of the identity root

Let z₀>2|t| be the physical identity root of z₀³−Δz₀²−4t²z₀+2Δt²=0, with z₀=Δ when t=0. The exact stiffness is

The numerator and denominator are positive for nonzero t because z₀>2|t|. Thus the identity is locally stiff in every nonzero Hermitian direction. At t=0 the coefficient vanishes, as the Hamiltonian then has no holonomy dependence. Simultaneously scaling t and Δ by a positive c scales z₀ and κ̂ by c, confirming that κ̂ has energy units. Theorem 2.3, pp.10–11.

For ε=t/Δ, the expansion is κ̂_m=(Δ/m)[2ε⁴−12ε⁶+O(ε⁸)]. The leading coefficient is κ_m^loop=2t⁴/(mΔ³); higher orders change the static stiffness. At large |t|/Δ, z₀=2|t|+Δ/4+O(Δ²/|t|) and κ̂_m∼|t|/(8m). Extrapolating the quartic weak-hopping law into that regime would be incorrect.

Derive the Hessian from the resolvent

For eigenvalues x_a of X, the exact secular denominator contains 2t⁴(1−cosx_a). Set N(z)=z(z²−2t²) and D₀(z)=z²(z²−4t²). Expanding at small X gives

The identity derivative is R′_I(z)=−(z⁴−2z²t²+8t⁴)/[z²(z²−4t²)²], strictly negative above the free spectrum. Write z=z₀+δz in ΔR=1. The linear root shift is δz=t⁴N(z₀)TrX²/[mD₀(z₀)²R′_I(z₀)]+O(‖X‖⁴). Since ΔE=−δz, simplifying gives the stated κ̂/2 formula. Theorem 2.3 and proof, pp.10–11.

The simple root makes this implicit expansion legitimate. The remainder is local in X and its constants may depend on the fixed Δ and t. It is not the same estimate as the uniform weak-hopping remainder over all W. The two results control different limits and should be used for different questions.

Why an independent trace-square term is absent

Conjugation invariance alone permits both TrX² and (TrX)² as quadratic functions on u(m). The exact singlet model has an additional symmetry: its secular equation is unchanged when any individual eigenangle x_a is replaced by −x_a. The analytic physical root inherits that symmetry.

A mixed quadratic monomial x_ax_b, a≠b, changes sign under one such reversal and must therefore have zero coefficient. Permutation symmetry makes the remaining x_a² coefficients equal. The quadratic term is consequently proportional to Σx_a²=TrX², with no independent (TrX)² contribution for m≥2. For m=1 these two expressions coincide, so their distinction is unavailable. Theorem 2.3, pp.10–11.

This argument does not exclude higher-order multitraces. At fourth order in X, both Σx_a⁴ and products such as (Σx_a²)² respect individual sign reversal. The exact finite-angle potential therefore need not be a single Wilson trace, despite its single-trace Hessian. “All hopping orders” means exact parameter dependence of the quadratic term, not an all-curvature form of the action.

Derive the central response of the 3+2 pair

For independent blocks define V=ΔE₃+ΔE₂. Near identity, impose the determinant-linked decomposition X₃=X₃⁰−2fI₃ and X₂=X₂⁰+3fI₂, with both X_m⁰ traceless. Then

The cross terms vanish by tracelessness. Adding the two exact Hessians yields

The central defining charges −2 and +3 are the primitive integral choice compatible with the product determinant. The factors 12 and 18 are rank times charge squared. This is an exact static identity for the specified independent auxiliaries, with no weak-hopping assumption. Changing central normalization rescales f and the reported central coefficient, so any comparison must state the same convention. Corollary 2.4, p.11.

Equal parameters give an exact ratio; unequal ones need the full formula

Let F(t,Δ)=mκ̂_m(t,Δ). The exact expression shows that F is independent of rank. If t₂=t₃≠0 and Δ₂=Δ₃, both blocks have the same z₀ and κ̂₂/κ̂₃=3/2 exactly. Equality of those microscopic parameters is an assumption.

Only in the weak-hopping limit does this reduce to (3/2)(t₂/t₃)⁴(Δ₃/Δ₂)³, with relative corrections of order (t₂/Δ₂)²+(t₃/Δ₃)². §2.5, equation (23), p.18.

These values are calculated from the exact identity root. The leading expression is shown to illustrate its departure outside weak hopping, without extending the perturbative error theorem beyond |t|/Δ<1/4.

Gauge symmetry permits an independent central stiffness

Define Q=2V_quadratic to keep the energy’s factor 1/2 explicit. A general invariant quadratic form on the declared Lie algebra can be written

The simple-sector coefficients must be positive for a positive form, and the total central coefficient must also remain positive. The last term does not need χ>0 by itself if the total form stays positive. Since TrX₃=−6f, the residual C=κ₁−12κ₃−18κ₂ is 36χ. In this convention the extra term in the energy V is χ(TrX₃)²/2. Adding χ(TrX₃)² directly to V instead would shift the reported energy Hessian residual by 72χ.

The exact independent defining-carrier singlet model has χ=0. Additional auxiliary representations, collective determinant probes, or other gauge-invariant interactions may generate χ≠0. Thus a nonzero calibrated residual rejects the restricted static model; it does not reject gauge invariance or all SMM/DTT realizations. Proposition 17.6, pp.54–55; §2.5, p.18.

Give a residual test an explicit error budget

Suppose reported coefficients have absolute calibration errors ε₁,ε₃,ε₂. The difference between the measured residual and its true value is bounded by ε₁+12ε₃+18ε₂. A claimed failure of the model relation must exceed this budget and any independently assessed modeling corrections. Equal-parameter assumptions must also be tested before using the separate ratio 3/2.

The source gives a sharper operator-level statement for invariant quadratic forms. With the direct-sum Hilbert–Schmidt norm, let the operator representing Q differ from a primitive form by norm at most η. The primitive central generator diag(−2I₃,3I₂) has squared norm 30. Hence |δκ₁|≤30η, while each normalized simple-sector coefficient changes by at most η. The triangle inequality gives |C|≤60η, and the constant is sharp within the stated class when positivity permits the extremal shifts. Proposition 17.7, p.55.

These are deterministic bounds with a stated normalization. They do not estimate the physical size of η or substitute for measurement. Corrections that only change independent block rates or penalties preserve the two-trace relation, as do analytic additions with zero quadratic term. A general gauge-invariant correction need not preserve it.

Static stiffness is not yet a measured gauge coupling

The same central weights appear in several constructions, but their coefficients have different meanings. A primitive rejection-cost κ is chosen through a protocol and penalty. κ^loop is derived at fourth order in weak hopping. κ̂ is the exact frozen-link energy Hessian. A short-time completion probability has another dimensionless coefficient involving t⁴(τ/ℏ)⁴. Equating any of them requires a matching calculation. §2.4, p.17; Theorem 20.4, pp.61–62.

Electric coefficients remain independent inputs. Once electric motion is included, dressing and operator-order effects can modify effective response; hard representation compression can introduce boundary terms already at lower orders. Quantum renormalization is not constrained to preserve this static two-parameter class. The relation is not a general Ward identity and does not predict observed weak mixing or unification couplings by itself. §§2.3–2.5, pp.12–18.

The exact Hessian remains a useful diagnostic because it is stronger than a leading series and has clear failure modes. A proposed microscopic realization should reproduce the singlet Hamiltonian’s assumptions, calibrate its energy units and central generator, and then compare the central and traceless response. Physical continuum identification is the next independent step.

Continue reading

• The exact resolvent behind the Hessian
• Fourth-order virtual loops
• Physical normalization and matching
• Corrections and effective dynamics