The first-order parent fixes a conserved norm and its spatial flux. That flux is different from the endpoint bilinear used in the quartic control. The distinction can be calculated exactly. The nearly equal endpoint coefficients therefore do not, on their own, derive physical coupling strengths from the kinetic equation.

This comparison preserves the preceding endpoint-current calculation as a declared interaction model. An interaction need not be built from a conserved current, but choosing another bilinear requires its own origin in the master action. Neither current is identified here with electric charge or an observed force.

The current fixed by canonical evolution

Let S exchange the x and y halves of the chain:

S=(0IdId0).(1) S=\begin{pmatrix}0&I_d\\I_d&0\end{pmatrix}. \tag{1}

All derivative links in the Hodge parent have the same principal symbol c gamma_a. Consequently its complete spatial norm flux is

ja=cΨ†(S⊗γa)Ψ=2cRe⁡∑r=1dyr†γaxr−1.(2) \boxed{j_a=c\Psi^\dagger(S\otimes\gamma_a)\Psi =2c\operatorname{Re}\sum_{r=1}^d y_r^\dagger\gamma_a x_{r-1}.} \tag{2}

The canonical density includes every chain field. The local continuity equation is partial_t rho+div j=0. Algebraic curvature shifts and gap terms cancel in this identity because they are Hermitian.

By comparison, the declared endpoint current is

jaend=x0†γax0+yd†γayd.(3) j_a^{\rm end}=x_0^\dagger\gamma_a x_0+ y_d^\dagger\gamma_a y_d. \tag{3}

Its coefficient near one is a legitimate matrix element of (3), but (3) is not the flux of the canonical norm. A single stationary harmonic can have zero divergence for either vector; interference between two different frequencies makes the distinction explicit.

An exact divided-difference identity

Write b=beta/R for an eigenvalue of the actual spatial Hodge operator B. On its normalized form eigenfield, the chain matrix is H(b)=c b S+H_0, where H_0 contains the shifts and the common gap. Let u_a,u_b be two unit positive-frequency chain eigenvectors. Subtracting their two eigenvalue equations gives

c ua†Sub=ωb−ωabb−baua†ub.(4) \boxed{c\,u_a^\dagger S u_b =\frac{\omega_b-\omega_a}{b_b-b_a} u_a^\dagger u_b.} \tag{4}

For equal b the diagonal limit is the Hellmann–Feynman identity c u^dagger S u=d omega/db. If dimensionless beta is used instead, the right side of (4) carries an additional R.

The complete pointwise cross-current is the coefficient on the left of (4), multiplying chi_a^dagger gamma chi_b. The Hodge equation implies div(chi_a^dagger gamma chi_b)=i(b_b-b_a)chi_a^dagger chi_b. Equation (4) therefore verifies the full interference continuity law

div⁡jab=i(ωb−ωa)ρab.(5) \operatorname{div}j_{ab} =i(\omega_b-\omega_a)\rho_{ab}. \tag{5}

This is a constraint from the actual evolution, independent of how an interaction is subsequently chosen.

In units R=c=1, at G=10,000, the diagonal kinetic-current coefficients for depth-six degrees 14 and 36 are respectively 5.2381149445e-14 and 4.4309346846e-12. Their endpoint coefficients are 0.9999975350 and 0.9999858750. The mixed kinetic coefficient is 1.2501008459e-12, while the mixed endpoint coefficient is 0.9999917050. These are values in the specified inverse parent, not empirical interaction strengths.

At large gap, the kinetic coefficient behaves as

dωdb≃Rω∑j=0d−11β+j−(d−1)/2.(6) \frac{d\omega}{db} \simeq R\omega\sum_{j=0}^{d-1} \frac{1}{\beta+j-(d-1)/2}. \tag{6}

It therefore retains strong mode dependence. The norm density still has its own canonical unit integral; neither that conserved norm nor its current is automatically ordinary electric charge.

Exact parity of the endpoint overlap

The signed Hodge branches have equal positive low frequencies. For the positive beta branch choose real endpoint coefficients

eℓ+=(Aℓ,(−1)d−1Bℓ),Aℓ,Bℓ>0.(7) e_\ell^+=(A_\ell,(-1)^{d-1}B_\ell),\qquad A_\ell,B_\ell>0. \tag{7}

Reversal of the full bidiagonal chain gives, on negative beta at the same positive parent frequency,

eℓ−=(Bℓ,−Aℓ).(8) e_\ell^-=(B_\ell,-A_\ell). \tag{8}

This follows by reversing the x and y equations and their alternating signs; it is also checked with the complete chain eigenvectors. Hence

(eℓ+)†eℓ−={0,d odd,2AℓBℓ,d even.(9) (e_\ell^+)^\dagger e_\ell^-= \begin{cases}0,&d\text{ odd},\\2A_\ell B_\ell,&d\text{ even}. \end{cases} \tag{9}

For an operator acting equally on both endpoints, the odd-depth off-diagonal matrix element vanishes throughout that fixed-degree polarization pair. This is an operator selection within the specified subspace, rather than merely a vanishing expectation for one vector. It does not remove the diagonal channels.

At unequal degrees, the odd-depth overlap instead obeys

(ea+)†eb−=(d−1)(a−b)2η2+O(η−4),η=GR/c.(10) (e_a^+)^\dagger e_b^-= \frac{(d-1)(a-b)}{2\eta^2}+O(\eta^{-4}), \qquad \eta=GR/c. \tag{10}

The expansion uses A_ell=2^(-1/2)[1-(ell-1)^2/(2 eta^2)+...] and B_ell=2^(-1/2)[1-(ell+m)^2/(2 eta^2)+...]. The effect holds at other odd depths as well as five. An endpoint operator with opposite weights would have a different rule. Equal endpoint weighting remains part of this trial action, not a derived universal force law.

The kinetic current provides a useful control: equation (4) makes its matrix element between opposite b at the same omega vanish at every depth. Thus the even/odd rule in (9) belongs to the selected endpoint interaction and cannot be attributed to every current of the geometry.

Resolve the small components before computing flux

The earlier scaled-inverse singular value calculation accurately resolves the low frequency, canonical norm and endpoint weights. A standard floating-point singular vector can nevertheless lose relative accuracy in intermediate entries many orders smaller than its norm. The kinetic flux in (2) depends on those small entries.

The scaled inverse has separable row and column signs. Taking its entrywise absolute value gives a positive matrix P. Alternating the normalized products u=P^T v and v=P u restores relative precision in the small components without cancellation; the signs are then restored. Eight iterations suffice in the tested large-gap cases. This is a numerical method for those cases, not a uniform iteration bound for arbitrary gap.

Independent 100-digit singular decomposition checks every component. The old unrefined vectors would give kinetic-flux errors of about 17% and 11% in the depth-ten degree-two and degree-24 controls. Those were not previously adopted flux results. The newly resolved vectors agree with the high-precision calculation and the derivative identity. The earlier frequencies, endpoint interaction and norm results are unchanged at their stated accuracy.

curvature-noether-current-20260914T003117_040924Z.json passes 75 checks, including complete small components, high-precision flux, exact divided differences, endpoint reversal and parity, and direct pointwise continuity in the actual surface coordinates. This comparison narrows what the endpoint calculation establishes; it does not derive a master interaction or complete particle inertia.