Comparing current positive-response routes on common controls

The current response proposals should be compared before adding another support term. This calculation puts their immediate algebraic tests in one script, with the same mode assignments and source vectors at physical depths two, three, four, five, six and ten. It distinguishes a diagnostic identity from a complete physical solution.

Shared calculation

For each maintained mode index $i$, the algebraic source control and positive inverse stiffness are

$$s_{i,\ell}=\frac{i+\ell+d}{i+2\ell+d+1},\qquad w_{d,\ell}=\frac{d}{(\ell-1)(\ell+d)},\qquad K_{ij}=\sum_{\ell=2,4,\ldots}s_{i,\ell}s_{j,\ell}w_{d,\ell}. \tag{1}$$

These are the existing source controls, not derived particle densities. The cumulative simplex count is $\mathcal S(i,d)=\binom{i+d-1}{d}$. The single script computes the following tests from (1):

1. The fixed-norm unbiased response changes the standing frequency by $-g^2Q_iK_{ii}$. At equal $Q_i$, a common rescaling of a bare simplex spectrum requires $K_{ii}/\mathcal S(i,d)$ to be constant within each depth.
2. A common minimum-norm bias for the selected modes solves $Kc=-\operatorname{diag}K$. Its selected residuals, the next-mode residual, condition estimate and real-amplitude curvature $-4K_{ii}$ are all computed from the same matrix.
3. For the one-scale energy $E=Q c\mu/L-g^2Q^2K_{ii}/(2L^{d-2})$, the script records the stationary size curvature and the stationary frequency factor $(d-4)/(d-2)$ when a finite stationary dilation exists.
4. For a spectrally transparent nonzero background, the joint Hessian requires $G>F^*A^{-1}F+(M+vN)^*P^{-1}(M+vN)$. If its metric-force source is normalized to the same $s_i$, with $v=1$ and the same inverse $w$, then $F^*A^{-1}F=K_{ii}$. The script reports $\max_i K_{ii}$ as a necessary common geometric stiffness at each depth in that unit-control comparison. The actual metric force, field-mixing term and geometric stiffness must come from a same-field action before it can be called a physical pass.

Results and decision boundary

The infinite harmonic sums were calculated at 50 and 80 decimal digits. The table gives the largest-to-smallest spread of $K_{ii}/\mathcal S$, the signed next-mode bias residual, and the unit-control lower bound on a shared geometric stiffness. The depth-ten sector has one maintained mode and cannot test within-depth proportionality.

The unbiased positive response does not preserve the proposed ratios at equal norm under these source controls. Constraining the norm retains its frequency shift. The fitted common bias cancels selected first derivatives but fails the next mode and has negative response amplitude curvature. The scale-covariant response has no stable finite size by itself at any maintained depth. Thus these variants do not yet give the requested joint mass, size and support mechanism.

The nonzero background route remains algebraically possible in the angular sector because it can cancel a metric force without shifting the reference field equation. Its reported stiffness numbers are necessary lower bounds only in the stated unit-control normalization. The subsequent common-background radius calculation specifies one finite-carrier inverse action and derives its radius force, field--radius mixing and radial curvature. It leaves the angular second variation of that carrier, master-field origin and reciprocal pair response unresolved. The separate whole-depth current calculation derives its conformal Hessian from its own action and tests a wider angular basis.

The combined check initially passed 108 identities and convergence comparisons. That initial report contains every retained depth and mode; the current script also includes the later radial and conformal-current actions as separate candidates. A passing diagnostic is evidence for the stated equations, not for the physical realization of a candidate.

One quick equation comparison for the next decision

The later joint candidate script recalculates the equations needed to compare the current routes in one run. For every maintained mode at both common strength controls it places the positive-response Gram norm, linked-carrier radius pressure, whole-depth current frequency, conformal metric force, geometric Hessian and correctly normalized background threshold in one row. The depth-two current row also checks nine form-carrier zonal directions; the higher-depth rows record the pointwise coefficients that establish positive bare spinor curvature for all smooth conformal directions. Its joint report contains 32 rows and common within-depth summaries.

The two carrier actions remain separate. The linked radial threshold is reported per unit wave norm. The current model's tested conformal threshold is reported per unit phase charge and is only a lower bound for untested metric directions. The report keeps finite-strength frequency-unit spreads visible; neither carrier has an exact simplex ratio at these finite strengths. The script is a compact decision calculation, while the larger verification script retains the independent precision and quadrature tests.

The finite-chain current analysis extends that same quick calculation. It proves that both fixed positive hopping chains have a strictly decreasing frequency per simplex count as the harmonic index rises. It then checks an exact finite-band depth-two current law and a dimension-aware current correction on all maintained modes. A further inverse-depth batch tests finite branches and both shared-gap interpretations of the six-to-ten frequency unit. A component-Hessian batch retains every signed component direction under the fractional correction. Those inverse laws change the reference action, so the saturated current's conformal Hessian and support threshold do not transfer to them.