Five-child electrical observation protocol

Question and result scope

The finite-input observation identifies collective winding on one continuing acquired organization. This owner translates that complete history to the fixed five-island electrical reference, including its actual fine deviations, observation uncertainty, initialization and supplied work. No donor or support change is required.

The result is a conditional specification and symbolic source-error theorem. It also gives a necessary current-amplitude consequence without requiring measured hidden phases. It does not admit a physical preparation, sensor, actuator or dataset. All earlier frozen declarations, producers and responses remain unchanged; no new reserved response is evaluated here.

The exact sine-current and voltage-phase laws are supplied constitutive premises. Equations (1)-(2) of Babenko et al. (2020) state that reference. Its measured apparatus is not this five-island assembly. English et al., arXiv v2 illustrates why independently measuring a current-phase profile requires a declared phase-sensitive observation model. Neither source supplies TNFR calibration or evidence. The identification owner retains the physical-law and constant boundary.

1. Complete state, support, source and clock

Keep the acquired five-child source immediately after outer formation, before the first plateau. The complete fine state has 45 signed forms and 45 circular phases on the unit-weight C9-by-C5 support, held unit capacities, and both rows

\[ x'=(-L_{45}x+\gamma S(\theta))/4+Rf,\qquad \theta'=\gamma L_{45}x/4,\qquad \gamma=1/(1023\pi). \tag{1} \]

R repeats five child coordinates over nine slots. Prime denotes tau=(1023/1024)*t, not laboratory seconds. Use the entire fixed word: 15 cubic ramps of duration one, 15 preceding plateaus of length h=1e-5, total T=15.00015, marked increments epsilon,-2*epsilon,epsilon, epsilon=1/10, and all 150 endpoint mean-form readings. Its input, phase command, release state, identity and work proof remain with the finite-word owner. The waveform consumes no phase measurement or future response.

On the admitted circular collar, use continuous lifts and the zero-mean template psi_j=2*pi*(j-4)/9 to write

\[ x=RU+u,\qquad \theta=\psi+R\Phi+z,\qquad R^Tu=R^Tz=0. \tag{2} \]

The shared circular observer defines these coordinates without averaging arbitrary principal representatives. A common phase is retained. The actual acquired hosts have zero total form and lifted phase; in the larger inference class retain r=sum(U)/5 and the conserved common phase. Neutral inputs do not change either. A physical circuit observes relative voltage and charge; it does not measure the discarded constant r or absolute common phase.

Let L be the five-cycle Laplacian, P=I-11^T/5, and B an oriented ring incidence with S5(Phi)=-B sin(B^T Phi). Mean cancellation gives

\[ U'=(-LU+\gamma S_5(\Phi))/4+f+\mathcal R,\qquad \Phi'=\gamma LU/4,\qquad \mathbf1^T\mathcal R=0, \tag{3} \]

\[ \mathcal R_c=\frac\gamma4\sum_{d\sim c}\left\{ \frac19\sum_j\sin(\Phi_d-\Phi_c+z_{d,j}-z_{c,j}) -\sin(\Phi_d-\Phi_c)\right\}. \tag{4} \]

This remainder belongs to the retained fine state. It is not an external current invented from a measured response. On the exact carrier u=z=0 it vanishes; actual acquisition only bounds it.

2. Passive C5 dictionary and independent calibration

On the neutral floating leaf define

\[ q=9\kappa(U-r\mathbf1),\quad \tau=b t_{lab},\quad V=v_* LU,\quad v_*=a_0b\gamma/4, \qquad a_0,\kappa,b>0. \tag{5} \]

Laboratory time zero is the start of observation, not substrate acquisition. Both rows transform. The exact identities are

\[ C=cL^+,\quad G=gP,\quad c=\frac{36\kappa}{a_0b\gamma},\quad g=\frac{9\kappa}{a_0\gamma},\quad j=\frac{9b\kappa\gamma}{4}, \tag{6} \]

\[ q=CV,\quad \dot q=-GV-jB\sin(B^T\Phi)+I_{ext}+I_{res},\quad \dot\Phi=V/a_0, \qquad I_{ext}=9b\kappa f(bt_{lab}),\ I_{res}=9b\kappa\mathcal R. \tag{7} \]

The centered inverse L+ has diagonal 2/5, adjacent entries zero and nonadjacent entries -1/5. Thus five positive capacitors c/5 join the pentagram pairs, ten resistor branches have conductance g/5, and five ring junctions have amplitude j. Multiplying L+ by L gives P; the pentagram and complete resistor graph are connected, so both matrices are positive on the neutral subspace. The absent adjacent/ground capacitances are exact reference premises; parasitics are not automatically part of this model.

Independent calibration must admit the full shapes, not just their traces. For positive branch boxes intersect all five capacitance multiples and all ten conductance multiples before computing

\[ b=4g/c,\quad \kappa=a_0\gamma g/9,\quad v_*=a_0\gamma g/c,\quad j_{pred}=a_0\gamma^2g^2/c,\quad K=\frac{a_0b\kappa\gamma}{4} =\frac{a_0^2\gamma^2g^2}{9c}. \tag{8} \]

Empty shape intersections exclude this dictionary. Nonempty intervals establish only necessary compatibility, not component calibration or a joint physical realization. The later current response must not fit c, g, gamma, sensor gain or clock. Fitting gamma removes the scalar restriction c*j/(a0*g^2)=gamma^2, as in the identification theorem. A conventional circuit satisfying these rows is an equivalent description, not an empirically distinguishable rival to the same mathematical model.

3. Actual-source transport through every finite input

Let ||u(0)||2<=R0, ||z(0)||2<=Z0, and gamma<=gbar. The within-child projection Q commutes with L45 and annihilates the complete input Rf. On the carrier Q S is zero. The full-circle sine Hessian and ||L45||2<=8 give exactly the shared coupled transport majorant

\[ s=2\bar g T<1,\qquad A=\frac{R_0+sZ_0}{1-s^2},\quad Z=\frac{Z_0+sR_0}{1-s^2}. \tag{9} \]

It holds throughout all ramps and plateaus, including their switches; large uniform commanded form does not directly drive hidden deviations. The strict observation collar follows when 2Z<pi/2.

The linear term in each edge's sine departure averages to zero. Since ||L||2<=4, Taylor's theorem yields

\[ \sum_{\{c,d\}}|e_{cd}|\le\frac1{18} z^T(L\otimes I_9)z\le2Z^2/9,\qquad \|\mathcal R\|_1\le\bar g Z^2/9=:\rho. \tag{10} \]

Initialize the ideal driven circuit at the actual U and Phi, use the same f and compare on the same clock. This initialization defines a mathematical comparator; it does not assert knowledge of the hidden phases. Heat contraction and the five-cycle sine Lipschitz constant four imply

\[ k=\bar g T<1,\quad E_U=\frac{T\rho}{1-k^2},\quad E_\Phi=kE_U. \tag{11} \]

These are uniform Euclidean trajectory differences. With independently bounded initial errors e_U,e_Phi, replace them by

\[ E_U=\frac{e_U+k e_\Phi+T\rho}{1-k^2},\qquad E_\Phi=\frac{e_\Phi+k e_U+kT\rho}{1-k^2}. \tag{12} \]

Physical component, timing and tracking errors are absent from (11)-(12). They cannot be hidden in the very small intrinsic source remainder.

For the already proved source with q0=1e-8, outer acquisition supplies Z0=6*q0*(1+gamma_upper) and R0=3*q0+24*gamma_upper*Z0. The elementary bounds gamma_upper<1/3000, T<16 imply R0<3.1e-8, Z0<6.01e-8, then A<3.2e-8, Z<6.1e-8 and E_U<2.3e-18. These loose symbolic inequalities follow from (9)-(11) and the existing source inclusion; they are not a replay of its assessor or a new reserved numerical observation. They do not set a laboratory noise floor. The frozen 1e-14 total reading allowance must include any comparator mismatch when transferring its readings to the actual source.

4. Accessible readings, uncertainty and initialization

With the independent linear calibration, each five-voltage endpoint gives

\[ U=r\mathbf1+\alpha L^+V,\quad \alpha=1/v_* =c/(9\kappa),\qquad (L^+V)_c=(2V_c-V_{c+2}-V_{c-2})/5. \tag{13} \]

One common voltage gauge cancels. Differing unknown sensor offsets do not. All five simultaneous coordinates, in the declared marking and endpoint order, are required. This is a model-based inference of charge from CV; it cannot independently test that capacitor law. It needs no phase reading.

For independently bounded alpha in [alpha_-,alpha_+], take midpoint alpha_m and radius alpha_r, primitive readings Vhat, and baseline rhat with error eta_r. Admit a uniform voltage error eta_V, including its sensor transfer and offsets. With w=L+ Vhat, use the exact rational center Uhat=rhat+alpha_m*w. Since ||L+||infinity=4/5, a uniform bound over all 150 inferred scalars is

\[ \eta_U=\eta_r+\frac45\alpha_+\eta_V +\alpha_r\max_{samples,c}|w_c|+\eta_{additional}. \tag{14} \]

The additional allowance must independently cover any source-comparator, aperture, timing or other admitted discrepancy. A nonnegative supplied number is not proof of that bound. No input or clock error is automatically admitted by (14). The adapter retains the frozen requirement eta_U<=1e-14 before using the existing finite-word inverse. Over-budget records leave that transfer unavailable; they do not justify changing the frozen noise.

The actual source's baseline is r=0. A constant unknown r cancels from the diffusion-corrected increments, because L annihilates constants, but voltage data cannot thereby establish the absolute initial-form prior. Retain a preparation bound on r and |x_i(0)|<=.01.

No exact relative-phase initialization follows from a winding label. For example U=0, Phi=0 and U=0, Phi=(delta,-delta,0,0,0) have the same zero winding and initial voltage for sufficiently small nonzero delta, but different immediate sine currents and form derivatives. A valid prediction therefore uses the complete preparation set: the initial mean-form bounds, fine-deviation bounds, all principal mean gaps in [-7/5,7/5], correlations and conserved origins. It may retain a set of possible initial phases instead of supplying unobserved values. The winding inverse is uniform over this set; an individual voltage trajectory is not determined by its returned label.

Laboratory durations are 1/b, h/b and T/b; the input is (7). An uncertain clock changes the applied waveform as well as timestamps. For an aperture confined to an unforced plateau, the exact mean row bounds |U'|infinity<=M+gamma/2, where M=.01+gamma_upper*T, so a structural timing radius delta_tau contributes at most (M+gamma_upper/2)*delta_tau. A nominal endpoint is also an input switch: two-sided jitter may enter a ramp, invalidating that smaller bound. One-sided plateau apertures or a separately proved whole-word bound are necessary. No finite-bandwidth or exact-switch implementation is certified.

5. Full resource account and physical units

Let H5=U^T L U/2+sum_ring(1-cos(delta)) and H_int=45*(1-cos(2*pi/9)). The C5 coefficients give

\[ H_{45}=H_{int}+9H_5+R_H,\quad -4Z^2\le R_H\le4(A^2+Z^2),\quad \int_0^T\|L_{45}u\|^2/4\le16TA^2. \tag{15} \]

The remainder is signed. It is not a free energy reservoir. If |U|<=M_* over the complete driven history, including its commanded form offset,

\[ \left|\int_0^T(LU)^T\mathcal R\,d\tau\right| \le4M_*T\rho,\quad H_5'=-\|LU\|^2/4+(LU)^T(f+\mathcal R), \quad R_H'=-\|L_{45}u\|^2/4-9(LU)^T\mathcal R. \tag{16} \]

The collective electrical energy is E_co=9K H5. Its lab-time balance contains resistive heat, supplied input power and the signed endogenous correction 9K*b*(LU)^T R. The input pairing is exact even off carrier:

\[ (L_{45}x)^TRf=9(LU)^Tf,\qquad \int V^TI_{ext}\,dt_{lab}=K\int(L_{45}x)^TRf\,d\tau. \tag{17} \]

The same factor converts the positive part of total supplied power integrated over time, as well as signed net work and absolute work. The full finite-word bounds therefore remain part of the mapped actual-source specification. Resistive heat corresponds to K*(Loss_fine-Loss_hidden), not all fine loss. Its lower bound must subtract the upper hidden loss in (15), with zero as a floor. Likewise collective energy is K*(H45-H_int-R_H), not an assertion that K H45 is the energy of a realizable passive fine-node circuit. No earlier acquisition or support-event work is mapped to joules by extending (5) without proof.

Those equalities concern actual collective coordinates with I_res retained. For the ideal comparator with I_res absent, its input-work difference is at most 36*K*E_U*integral(||f||1) over structural time; use ||L*(U_ideal-U)||infinity<=4*E_U in the power pairing. The same bound controls the difference of positive-work integrals because the positive-part map is 1-Lipschitz. The frozen input peak F_m and 15 active units give integral(||f||1)<=75*F_m. Thus ideal comparator work needs this additional allowance; equal initial winding alone cannot transfer the actual-source work certificate.

For exact calibration and only voltage readout error, (14) permits eta_V<=5*a0*b*gamma*eta_U/16. The largest commanded voltage peak is (6/25)*a0*b; the actual background changes it by at most a0*b*gamma*M. The commanded-peak to allowed-noise ratio is consequently (96/125)/(gamma*eta_U), independent of b and kappa. Changing units or clock scale cannot remove this ratio. It constrains a single sensor range covering both ramps and plateau resolution; gated sampling or separate ranges have different requirements. It is not a proof of instrumental impossibility or a physical noise estimate.

6. An unfitted consequence and the remaining admission

Use the same endpoint combinations as the finite-word inverse, now on charges inferred using independently calibrated C: D_q=9*kappa*D_U. The constant baseline cancels. If d_h=h/b, define the two unnormalized current quadratures

\[ A_c=\frac{D_{q,+}-D_{q,-}}{2d_h\sin\epsilon},\qquad B_c=\frac{D_{q,+}+D_{q,-}-2D_{q,0}} {2d_h(1-\cos\epsilon)}. \tag{18} \]

In the supplied model these enclose j_pred*C_c and j_pred*S_c, where C+iS is the initial fine interface phasor. Its modulus lies between cos(2*zeta) and one, with zeta=1e-6. Thus a radial current statistic can test an amplitude interval without first inserting a reconstructed phase into the very law being tested. If beta_D is the full corrected-form error from the finite-word theorem, then

\[ |A_c-j_{pred}C_c|\le e_A=\frac{9b\kappa\,\beta_D}{h\sin\epsilon},\qquad |B_c-j_{pred}S_c|\le e_B=\frac{18b\kappa\,\beta_D} {h(1-\cos\epsilon)}. \tag{19} \]

Set e_J=sqrt(e_A^2+e_B^2) and m_c=sqrt(A_c^2+B_c^2). The reverse triangle inequality gives the necessary prediction

\[ \max(0,j_{pred}\cos(2\zeta)-e_J)\le m_c\le j_{pred}+e_J. \tag{20} \]

No division by the predicted amplitude enters the measured statistic. With calibration/reading intervals, enclose its norm by [m_-,m_+] and retain their correlations where known. A necessary current-amplitude interval is [max(0,m_--e_J),(m_++e_J)/cos(2*zeta)], using outward bounds. Nonintersection with the independently predicted amplitude excludes the supplied law and map under their observation premises. Intersection is only compatibility, not a complete trajectory or physical identification. The model-derived beta_D cannot be transferred to a different current law.

The coefficients, error budget and complete finite input remain model assumptions; this is not a standalone experimental verification of the sinusoidal profile. In particular, an inverse normalized by j_pred and returned as a phase is not independent phase evidence for validating j_pred.

The conditional specification is now complete for exact input, support, clock and admitted endpoint errors. Before a physical test it still needs independent preparation information, component shape/calibration, a voltage transfer model, clock and finite-aperture evidence, and an actuator tracking bound valid at every switch. The same complete uncertainty must be carried into any competing constitutive model. A supplied physical coefficient cannot be derived by recognizing the equivalent circuit.

The execution plan alone selects the next gate. This owner adds no hardware campaign or physical identification, and selects no universal TNFR law.

7. Detached implementation and controls

The C5 component and departure helper reuses shared scalar/interval admission and hidden-deviation transport. It accepts primitive bounds and components, not cached source verdicts. Its separate amplitude comparison consumes five independently supplied ring amplitudes after rebuilding calibration; it does not measure those amplitudes or their current-phase profile.

The voltage adapter admits all 150 raw scalars before returning even an unavailable component result. It reconstructs (13)-(14) and delegates to the unchanged finite inverse. Component exclusions, over-budget uncertainty and incompatible records remain distinct. Derived scales use fixed outward arithmetic; unsupported positive scales raise explicitly rather than silently becoming zero or changing precision. No helper installs an electrical evolution law.

The instrument controls and independent theorem controls exercise algebra, uncertainty, counterexamples and rejected domains using synthetic inputs. An admitted numerical record remains conditional on its preparation, full law, clock, input and observation premises. No source assessment or physical experiment is invoked by these controls.