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Shadow Theory

Section 18 4 October 2026

Model, stock, and scope

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23131081

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18 Model, stock, and scope

The active configuration is (S,r,Z)∈R×R+×R(S,r,Z)\in\mathbb R\times\mathbb R_+\times \mathbb R, with a two-component spinor. The relative radius rr is the exact ss-wave reduction of the frozen two-body scalar model. The free centre coordinate and constant spherical harmonic factor from the active Hamiltonian. The clock SS and pointer ZZ are effective scalar coordinates; no three-dimensional lifting error for those two coordinates is asserted.

Write Π0,Π1\Pi_0,\Pi_1 for the conserved qubit projectors. Retain the exact radial preparation baseline bounded-phase preparation, smooth activation, finite radial potentials, and all clock self terms. Denote that operator by HradH_{\mathrm{rad}}. Its physical clock mass is M=10 kgM=10\,\mathrm{kg}, its carrier speed is v=1 m/sv=1\,\mathrm{m/s}, and its reduced radial mass is

mr=252ℏT0/ℓ02,T0=0.03 s,ℓ0=10−4 m. m_r=2^{52}\hbar T_0/\ell_0^2,\qquad T_0=0.03\,\mathrm{s},\quad \ell_0=10^{-4}\,\mathrm m.

The scalar preparation operator is defined in Section 17. Its fixed cutoffs and admitted laws are part of the theorem. The appendices derive the specialized coefficient and current estimates used below.

The enlarged autonomous Hamiltonian is

H=Hrad+ℏT0[−∂Z2+Z2−12μZ+Π1{−F(s)Z+Cb(s)}], H=H_{\mathrm{rad}}+\frac{\hbar}{T_0} \left[\frac{-\partial_Z^2+Z^2-1}{2\mu_Z} +\Pi_1\{-F(s)Z+C_b(s)\}\right], (7)

where

μZ=0.01,s=S−SonvT0,Son=0.14145 m,w=0.003,b(s)=24B9(s/w),F=b/μZ+μZb′′,Cb=b2/(2μZ)+μZbb′′/2.\begin{gathered}\mu_Z=0.01,\quad s=\frac{S-S_{\rm on}}{vT_0},\quad S_{\rm on}=0.14145\,\mathrm m,\quad w=0.003,\\ b(s)=24 B_9(s/w),\qquad F=b/\mu_Z+\mu_Z b'',\qquad C_b=b^2/(2\mu_Z)+\mu_Z b b''/2. \end{gathered}

Here B9(x)=126x5−420x6+540x7−315x8+70x9B_9(x)=126x^5-420x^6+540x^7-315x^8+70x^9 on [0,1][0,1], extended by 00 and 11. Primes denote ss derivatives. Thus the pulse width is 90 μs90\,\mu\mathrm s in the characteristic clock, and its last spatial point is 0.14154 m0.14154\,\mathrm m. The physical pointer mass is 10−19 kg10^{-19}\,\mathrm{kg}; its length unit is ℏμZT0/10−19kg≃0.562469 nm\sqrt{\hbar\mu_Z T_0/10^{-19}\mathrm{kg}}\simeq0.562469\,\mathrm{nm}. Every displayed compensation term remains in (7).

At laboratory time t0=−0.106 st_0=-0.106\,\mathrm s, the state is precisely the original full small-Gaussian radial and smooth compact clock stock, with its Galilean carrier and arbitrary normalized qubit, tensored with

g0(Z)=π−1/4exp⁡(−Z2/2). g_0(Z)=\pi^{-1/4}\exp(-Z^2/2).

There is no wave truncation or later preparation reset. The original radial width is 10−4 m10^{-4}\,\mathrm m; the clock is the positive 10−10 m10^{-10}\,\mathrm m mollification of its normalized cos⁡4\cos^4 packet of half-width 10−3 m10^{-3}\,\mathrm m. Its initial centre is S=−0.002 mS=-0.002\,\mathrm m. The analytical handoff is th=−0.002 st_h=-0.002\,\mathrm s, with clock centre 0.102 m0.102\,\mathrm m.

The narrowed nominal reference-law class of Section 17.1 is retained exactly. The pointer is included in the same normalized, input-independent joint auxiliary conditional rule gnew≤2wnewg_{\rm new}\leq2w_{\rm new}. This is one rule for the whole auxiliary configuration, allowing the declared correlations; it is not a separate cap or independent Born assignment for each added coordinate. Its disintegration gives

νmicro,0≤C∣Ψ0∣2 dq,C=27000067499. \nu_{\rm micro,0}\leq C|\Psi_0|^2\dd q, \qquad C=\frac{270000}{67499}. (8)

The retained periodic reference marginal, its allowed neighbourhood, and its target-calibration allowance remain part of the hypotheses. The cap alone is not substituted for that complete reference-coupling contract.