The positive particle-sourced response fails spectral transparency when it has no independent bias. A natural extension is to add one common bias per depth. This note determines exactly what that buys and where it stops.

Orthogonality with a common bias

Let A>0A>0 be the response stiffness, SiS_i the quadratic source of the selected mode ii, and bb a depth-wide bias. Eliminating the response gives the source energy

Ueff,i=−12⟨Si+b,A−1(Si+b)⟩.(1) U_{\mathrm{eff},i} =-\frac12\langle S_i+b,A^{-1}(S_i+b)\rangle. \tag{1}

Under a field-amplitude variation, SiS_i scales as ∣a∣2|a|^2. Vanishing of the resulting frequency shift requires

⟨Si,A−1b⟩=−⟨Si,A−1Si⟩.(2) \boxed{\langle S_i,A^{-1}b\rangle =-\langle S_i,A^{-1}S_i\rangle.} \tag{2}

Writing ui=A−1/2Siu_i=A^{-1/2}S_i and q=A−1/2bq=A^{-1/2}b, these are linear conditions

⟨ui,q⟩=−∥ui∥2.(3) \langle u_i,q\rangle=-\|u_i\|^2. \tag{3}

For a finite linearly independent set of uiu_i, a solution exists. The minimum-norm solution is obtained from the exact Gram system

Gc=−r,Gij=⟨ui,uj⟩,ri=∥ui∥2,q=∑jcjuj.(4) Gc=-r,\qquad G_{ij}=\langle u_i,u_j\rangle, \qquad r_i=\|u_i\|^2, \qquad q=\sum_jc_ju_j. \tag{4}

This is a common bias for the depth, not an independently fitted bias for each selected mode. It can therefore be a legitimate inverse construction if a single action derives both the source vectors and the bias.

The finite-set boundary

The construction does not automatically support the next mode. Applying the same qq to an additional source u∗u_* requires

⟨u∗,q⟩+∥u∗∥2=0.(5) \langle u_*,q\rangle+\|u_*\|^2=0. \tag{5}

Using the maintained source controls

(ui)ℓ=i+ℓ+di+2ℓ+d+1,⟨ui,uj⟩=∑ℓ∈H(ui)ℓ(uj)ℓLd(ℓ),(6) (u_i)_\ell=\frac{i+\ell+d}{i+2\ell+d+1}, \qquad \langle u_i,u_j\rangle =\sum_{\ell\in H}\frac{(u_i)_\ell(u_j)_\ell} {\mathcal L_d(\ell)}, \tag{6}

the first calculation used the finite harmonic control

H=Hd,I={2,4,…,2max⁡I}.(7) H=H_{d,I}=\{2,4,\ldots,2\max I\}. \tag{7}

Its Gram matrices are nonsingular for every maintained depth. The exact finite-dimensional solution of (4) satisfies (3) for all selected indices at depths 2,3,4,5,6,102,3,4,5,6,10, but (5) is nonzero for the next index at every depth. This establishes a finite-control obstruction. It does not by itself establish the same statement for the infinite even-harmonic space.

The minimum bias norm ∥q∥2\|q\|^2 from (4) quantifies the cost of this finite construction. For the algebraic source controls in (6) and cutoff (7), it is:

Depth Selected indices Minimum ∥q∥2\|q\|^2
2 76, 81, 95 5.25
3 1, 4, 16, 17 2.21×1032.21\times10^3
4 3, 20, 72 56.7
5 10, 15, 22 65.1
6 13, 35 13.5
10 23 1.19

These values are controls of the stated source model, not physical support energies. They show why the bias cannot be treated as an innocuous constant: its norm is a derived part of the action and must be included in the total energy and stress.

Infinite even-harmonic completion

The projected Jacobi inverse has the exact factorization

1Ld(2k)=d(2k−1)(2k+d).(8) \frac{1}{\mathcal L_d(2k)} =\frac{d}{(2k-1)(2k+d)}. \tag{8}

Consequently the infinite inner product in (6), now with H={2,4,6,…}H=\{2,4,6,\ldots\}, converges because its summand is O(k−2)O(k^{-2}). Direct high-precision summation at 50 and 80 decimal digits gives the following cutoff-independent controls:

Depth Minimum ∥q∥2\|q\|^2 Gram condition estimate Next-mode residual
2 4.97618 1.19×1091.19\times10^9 1.11×10−61.11\times10^{-6}
3 79.7667 4.21×1084.21\times10^8 2.43×10−52.43\times10^{-5}
4 43.4310 1.87×1041.87\times10^4 4.46×10−44.46\times10^{-4}
5 35.0233 2.97×1062.97\times10^6 1.21×10−41.21\times10^{-4}
6 13.3677 7.50×1037.50\times10^3 −7.02×10−4-7.02\times10^{-4}
10 1.22498 1 3.61×10−33.61\times10^{-3}

The next-mode residual is the left side of (5). It remains nonzero at every maintained depth, so the finite-set boundary survives completion of the stated harmonic space. The large depth-two and depth-three condition numbers are a separate limitation: their solved biases are highly sensitive to small changes in the assumed source law. The calculation therefore establishes compatibility for these specified algebraic controls, not a robust physical selection mechanism.

Stationarity is not amplitude stability

The orthogonality condition cancels the first amplitude derivative, but the eliminated positive response has a fixed second derivative. Write s=∣a∣2s=|a|^2, so the selected source is suis u_i, and subtract the constant energy of the bias alone. The response contribution is

ΔUi(s)=−s⟨q,ui⟩−12s2∥ui∥2.(9) \Delta U_i(s) =-s\langle q,u_i\rangle-\frac12s^2\|u_i\|^2. \tag{9}

Using (3) gives

dΔUids∣s=1=0,d2ΔUids2∣s=1=−∥ui∥2<0.(10) \left.\frac{d\Delta U_i}{ds}\right|_{s=1}=0, \qquad \left.\frac{d^2\Delta U_i}{ds^2}\right|_{s=1} =-\|u_i\|^2<0. \tag{10}

For a real amplitude varied about a=1a=1, the corresponding curvature is

d2ΔUi(∣a∣2)da2∣a=1=−4∥ui∥2.(11) \left.\frac{d^2\Delta U_i(|a|^2)}{da^2}\right|_{a=1} =-4\|u_i\|^2. \tag{11}

Thus the bias makes the normalized amplitude stationary but does not stabilize it. Within the response sector alone, the stationary point is a strict local maximum in the amplitude direction. A viable joint action needs a bare-field or constrained contribution whose positive curvature exceeds 4∥ui∥24\|u_i\|^2 for every selected mode in the common normalization. This is the amplitude block of the full Schur-complement condition, not a complete instability proof for a model that includes such an additional term.

This is a boundary of the common-bias transparency calculation. A physical action could select occupations through a nonlinear law, a finite band, a conserved charge sector, or a boundary condition. The failed cancellation does not establish that the next mode is absent: it could have a shifted frequency, remain unoccupied, be unstable, or describe an exotic branch. Continue that branch under the same action and test its stability and observable response. The bias and any occupation rule require a same-field derivation; choosing the bias after listing observed modes fits the spectrum.

There is also a simple incompatibility when two sources are proportional. If uj=αuiu_j=\alpha u_i, equations (3) demand both ⟨ui,q⟩=−∥ui∥2\langle u_i,q\rangle=-\|u_i\|^2 and α⟨ui,q⟩=−α2∥ui∥2\alpha\langle u_i,q\rangle=-\alpha^2\|u_i\|^2. For a nonzero source, these agree only when α=1\alpha=1. A common bias cannot transparently support arbitrary rescalings of one source direction.

What this adds to the support problem

The result leaves three conditional routes:

  1. an independently present background with a derived bias and a finite mode-selection law;
  2. a constrained or indefinite response for which the positive overlap argument no longer applies; or
  3. a nonlinear same-field support whose stationary equations generate the orthogonality conditions without mode-by-mode tuning.

The first route is algebraically consistent for a finite selected set, but it introduces a new physical obligation: derive the bias and the selection mechanism together. It does not establish the mass operator, a complete particle, or a neutrino interaction rate.

Verification

biased_source_compatibility_checks.py solves the exact finite Gram systems and tests the next unselected mode at every maintained depth. The normal and optimized runs each pass 40 conditions. The selected exact report contains the source vectors, solved bias coefficients, and unselected-mode residuals.

biased_source_infinite_checks.py performs the infinite even-harmonic sums independently at 50 and 80 decimal digits. Its normal and optimized runs each pass 48 convergence, compatibility and amplitude-curvature conditions. The selected infinite-space report contains the converged norms, coefficients, condition estimates and next-mode residuals. The source components are algebraic controls; they are not measured particle densities.