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Shadow Theory

Section 4 4 October 2026

A finite coherent source and resource module

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23131069

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4 A finite coherent source and resource module

Here is a nontrivial receptor that is physically in the same inventory. On a finite factor DD use orthogonal states

∣r⟩,∣pa⟩,∣ca⟩,∣la⟩(a=0,1). \ket r,\quad\ket{p_a},\quad\ket{c_a},\quad\ket{l_a}\quad(a=0,1).

They include the following actual material degrees in their internal wave description:

StateProductionFuelSiteExcitationMemoryRemnant
rr11ready0blankvacuum
pap_a01readymode aablankvacuum
cac_a00spent0aacapture aa
lal_a01ready0blankloss aa

Assign energy E>0E>0 to a production cofactor, fuel unit, excitation and loss remnant, and 2E2E to a capture remnant; the displayed labels are degenerate. Every row has total resource energy 2E2E. Hence the conversion gates conserve this resource energy exactly while spending readiness and retaining energy in products. Their full tensor-factor implementation is defined to be zero outside the indicated equal-energy active subspace. Other exhausted sectors stay present.

For source projectors PaP_a, set

Gw=i∑aPa⊗(∣pa⟩⟨r∣−∣r⟩⟨pa∣),∣ba⟩=η∣ca⟩+1−η∣la⟩,0<η<1,Gr=i∑a(∣ba⟩⟨pa∣−∣pa⟩⟨ba∣).\begin{align} G_w&=i\sum_aP_a\otimes(\ket{p_a}\bra r-\ket r\bra{p_a}),\tag{15}\\ \ket{b_a}&=\sqrt\eta\ket{c_a}+\sqrt{1-\eta}\ket{l_a},\qquad 0<\eta<1,\notag\\ G_r&=i\sum_a(\ket{b_a}\bra{p_a}-\ket{p_a}\bra{b_a}). \notag\end{align}

These Hermitian generators have norm one on their active subspaces. Nonoverlapping pulses ℏgw(t)Gw\hbar g_w(t)G_w and ℏgr(t)Gr\hbar g_r(t)G_r with areas θ,φ\theta,\varphi give exactly

ΨD=cos⁡θ ψ∣r⟩+sin⁡θ∑aPaψ(cos⁡φ∣pa⟩+sin⁡φη∣ca⟩+sin⁡φ1−η∣la⟩). \begin{aligned} \Psi_D={}&\cos\theta\,\psi\ket r+\sin\theta\sum_aP_a\psi \left(\cos\varphi\ket{p_a}+\sin\varphi\sqrt\eta\ket{c_a} +\sin\varphi\sqrt{1-\eta}\ket{l_a}\right). \end{aligned} (16)

Indeed each generator is a two-dimensional σy\sigma_y rotation, and the active aa sectors are orthogonal. There was no sampled reaction time in this calculation. Reduced excitation populations are not actual level trajectories in the adopted ontology. The actual event is a subsequent spatial registration governed by Sections 2–3.

The finite response has four exact orthogonal status weights (and ideal resolved-pointer probabilities):

(Pr,Pp,Pc,Pl)=(cos⁡2θ, sin⁡2θcos⁡2φ, ηsin⁡2θsin⁡2φ, (1−η)sin⁡2θsin⁡2φ). (P_r,P_p,P_c,P_l)= (\cos^2\theta,\ \sin^2\theta\cos^2\varphi,\ \eta\sin^2\theta\sin^2\varphi,\ (1-\eta)\sin^2\theta\sin^2\varphi). (17)

The pending excitation remains a vector in the actual model at a finite cutoff. Continuing GrG_r processes it coherently; a finite closed receptor can recur. A zero response window does not erase it. Neither an absorbing boundary nor a restart clock is imposed when a coefficient begins to populate pap_a.

4.1 Physical null, capture and loss

Use three spatial readout centres −L,0,L-L,0,L for captured a=0a=0, null, and captured a=1a=1. The same forced oscillator construction applies to each orthogonal control projector, using signed trajectories. Noncaptured r,p,lr,p,l components share the null packet. Nearest-centre cells have worst tail at most 2δ2\delta, with δ=F‾(L/(2σ))\delta=\Ntail(L/(2\sigma)). The ideal orthogonal-label comparator has capture coefficient

q Paψ∣ca⟩,q=ηsin⁡2θsin⁡2φ, \sqrt q\,P_a\psi\ket{c_a},\qquad q=\eta\sin^2\theta\sin^2\varphi,

and complete null vector

ΨN=cos⁡θ ψ∣r⟩+sin⁡θ∑aPaψ(cos⁡φ∣pa⟩+sin⁡φ1−η∣la⟩). \Psi_N=\cos\theta\,\psi\ket r+ \sin\theta\sum_aP_a\psi\left(\cos\varphi\ket{p_a} +\sin\varphi\sqrt{1-\eta}\ket{l_a}\right). (18)

Only for a declared reduced comparison, tracing DD gives

N(ρ)=cos⁡2θ ρ+sin⁡2θ(cos⁡2φ+(1−η)sin⁡2φ)∑aPaρPa. \mathcal N(\rho)=\cos^2\theta\,\rho+ \sin^2\theta\bigl(\cos^2\varphi+(1-\eta)\sin^2\varphi\bigr) \sum_aP_a\rho P_a. (19)

Equation (18), along with pointer and all receiving systems, is the retained continuation. Equation (19) is not a global collapse rule. Finite spatial classifiers approximate these ideal labels; Section 8 bounds the complete output error.

4.2 Finite stock and exhaustion

For a promised mm-epoch experiment allocate mm ready cells, their blank archives, and receivers. Gate the active conversion only on sectors containing the required cofactor, fuel, site and blank capacity. Extend the unitary by identity on explicitly exhausted sectors, which can be spatially flagged by the same writer. A failed or null attempt does not receive a new ∣r⟩\ket r for free. The consumed-ready-cell count is bounded by the allocated finite schedule; no infinite Poisson bath is hidden in this implementation.