Operational identification of the collective electrical model

Question and decision scope

The electrical construction and actual-source comparison establish a conditional post-contact description. This owner asks which independently measurable relation remains after calibrating that description, and whether the existing preparation is already an admissible electrical experiment.

There are two results. Independently calibrated capacitance and conductance leave a prospective, unfitted sine-current amplitude for the fixed law. However, its dimensionless coefficient is supplied, and the literal instantaneous phase events in the existing source history have no finite-work realization in the unchanged electrical reference. This is a scoped preparation obstruction, not an obstruction to all physical preparation or to the valid finite post-contact theorem.

The mathematical results below are symbolic. No physical data, new source response, parameter search, numerical campaign or laboratory apparatus is admitted. Numerical controls use synthetic records and independent algebra; the earlier frozen source evidence remains unchanged.

1. Complete reference, observation and calibration boundary

Retain the six marked electrical islands, floating neutral charge leaf, fixed same-slot wheel support and collective circular observation of the electrical owner. Use its notation L for the wheel Laplacian, B for an oriented wheel incidence, P=I-11^T/6, and the supplied Josephson constant a_0=hbar/(2e)>0. Physical quantities use coulombs, volts, seconds, farads, siemens and amperes; structural quantities retain their original units.

The reference rows are

\[ q=CV,\qquad \dot q=-GV-B\operatorname{diag}(j_e)\sin(B^T\Phi), \qquad \dot\Phi=V/a_0. \tag{1} \]

The declared dictionary remains q=9*kappa*(X-r*1), tau=tau_0+b*t_lab, and unchanged circular Phi, with positive common scales kappa,b and retained neutral reference r. Arbitrary sensor mixing, changed phase coordinates, hidden dynamic circuitry and a different background-charge leaf are outside this particular comparison. Voltage differences require one consistent floating gauge; a common unmeasured phase is not set to a new physical value for every record.

An operational comparison separates the following information:

Information	Independent premise and use
Node labels, component connections and reference	Fix before measuring or comparing records; preserve all fifteen capacitor/resistor pairs and ten wheel junctions
C and G	Calibrate the full effective linear component matrices from separate records, with the junction contribution, loading, frequency/temperature regime and uncertainty accounted for. A single terminal trace does not determine these matrices
Charge and voltage readings	Supply calibrated transfer, offsets, aperture and noise. Computing q from independently calibrated C V is a model-based charge inference; it cannot independently validate that same capacitor law
Relative phase readings	Supply an independent phase-sensitive transfer model, or retain known initial phase and integrate V/a_0. The latter consumes the Josephson law and cannot test it; it also does not recover unknown initial relative phases
Reserved junction observation	Measure the current-phase profile with the independent phase readout. Its sine amplitude is reserved when testing (4); a maximum current alone does not establish a sinusoidal profile
Preparation, input and laboratory time	Record actual initial collective state, physical actuation, support changes, laboratory clock and all work; preserve their uncertainty when predicting later records

Reconstructing phase from arcsin(I/j) while testing the sine-current law would be circular. Integrating measured voltage requires the initial phase and does not resolve that missing information by itself.

The constitutive reference is supported by equations (1)-(2) of Babenko et al. (2020). That paper describes separate impedance and instrument calibration, and reports model discrepancies attributed to nonsinusoidal behavior and junction variation. Its apparatus and measured parameters are not evidence for this TNFR realization or its tolerances. Direct phase-sensitive measurements of nonsinusoidal profiles, including their loop/sensor model, are reported by English et al. (2016), arXiv v2. These primary sources justify keeping profile and sensor assumptions explicit; neither supplies a TNFR calibration or reserved dataset. Sources reviewed 2026-10-10.

2. An unfitted relation after independent linear calibration

Conditional characterization. On an open neutral charge and relative-phase domain, the fixed dictionary realizes the complete quarter sine law if and only if, for positive scalars c, g, j,

\[ C=cL^+,\qquad G=gP,\qquad j_e=j\text{ on all ten edges},\qquad \frac{cj}{a_0g^2}=\gamma^2, \quad \gamma=\frac1{1023\pi}. \tag{2} \]

The statement is within the declared floating six-island reference and observation class, not a classification of every possible electrical map. Its necessity follows from the phase and form rows in the electrical owner:

\[ c=\frac{36\kappa}{a_0b\gamma},\qquad g=\frac{9\kappa}{a_0\gamma},\qquad j=\frac{9b\kappa\gamma}{4}. \tag{3} \]

Equality of the nodal sine fields on an open relative-phase domain forces each edge coefficient. Differentiate a node's current with respect to one neighbor's phase: that single edge coefficient is isolated. Cycle-current ambiguity in a single record does not create a freedom in the complete nonlinear field. Eliminating the two scales in (3) yields

\[ \boxed{b=4g/c,\qquad \kappa=a_0\gamma g/9, \qquad j_{\rm pred}=a_0\gamma^2g^2/c.} \tag{4} \]

Conversely, inserting the positive scales (4) and relation (2) into (3) recovers both full rows and their storage/work relation. This proves sufficiency. After exact independent C/G calibration, both scales are unique; changing either to fit a reserved j would contradict that calibration. The scalar c has units farads, g siemens, j amperes; cj/(a_0*g^2) is dimensionless. It is independent of the arbitrary charge and clock scales.

The complete component shape must be checked before using (4). From the proved wheel inverse, the mutual-capacitance coefficients are 5/396, 41/396, 11/396 times c for adjacent rim, nonadjacent rim and rim-hub pairs, respectively. Every resistor branch has conductance g/6. Equivalently, symmetric floating matrices satisfy C L=cP and G=gP. Although c=tr(CL)/5, g=tr(G)/5, traces alone cannot establish those matrix identities. Changing two branch values can preserve a trace and still violate the required shape.

The same restriction must hold across independently prepared states of the same fixed assembly. Preparation cannot retune c, g, j, gamma or sensor gains. A conventional circuit with calibrated C and G but all amplitudes 2*j_pred is a positive passive reference outside this fixed-law dictionary; it gives a concrete distinguishable alternative on an open phase domain. A conventional circuit satisfying (2) is exactly equivalent on the ideal carrier. No response in that common model can distinguish the two descriptions.

3. Why the number is not a derived universal constant

The fixed coefficient comes from the earlier specified weights e=1023/1024, w=1/1024, giving gamma=w/(e*pi) in the complete-law owner. It is not selected by the nodal identity, neutrality, positive dissipation or the passive circuit construction. The constitutive comparison class explicitly retains positive coefficient and current-profile freedom.

If gamma is allowed to be fitted after measuring a positive triple c,g,j, then

\[ \gamma_{\rm fit}=\sqrt{\frac{cj}{a_0g^2}},\qquad b=4g/c,\qquad \kappa=a_0\gamma_{\rm fit}g/9 \tag{5} \]

matches every such triple with the required component shapes. The scalar restriction has disappeared. Thus (2) tests the previously fixed complete law, not TNFR with unrestricted constitutive coefficients. Replacing the sine profile by another admitted current law also needs its own comparison; its formation, finite source bounds and sensor inference do not transfer merely because equilibria or tangents match.

There is a limited composition statement. If a different organization of m nodes independently admits the same exact collective law, phase map and passive coarse inverse, replacing 9 by m in the aggregate charge map gives

\[ c=\frac{4m\kappa}{a_0b\gamma},\quad g=\frac{m\kappa}{a_0\gamma},\quad j=\frac{mb\kappa\gamma}{4}. \]

The ratio (2) is unchanged. This is conditional cancellation of an admitted replication factor; it does not prove arbitrary-scale formation, physical scale invariance or the same law at every scale.

4. Uncertainty, actual source and a possible reserved balance

Let c in [c_-,c_+], g in [g_-,g_+], and a_0 in [a_-,a_+] be independently admitted positive calibration boxes. The exact monotone envelope for the fixed mathematical gamma is

\[ j_{\rm pred}\in \left[\frac{a_-\gamma^2g_-^2}{c_+}, \frac{a_+\gamma^2g_+^2}{c_-}\right]. \tag{6} \]

The implementation uses the shared outward enclosure of gamma. Disjoint reserved amplitude bounds exclude the fixed-map relation under the supplied premises. Overlap is necessary compatibility only: it proves neither the physical premises nor a joint noisy trajectory. Correlations discarded by these boxes can weaken exclusion. If capacitance or conductance shape intervals have empty intersection, calibration already excludes this dictionary; averaging them into a best fit would erase that finding.

The actual-source theorem starts its ideal comparator at the actual collective coordinates. Its very small intrinsic error is not an experimental initialization error budget. With initial Euclidean coordinate uncertainty e_X,e_Phi, its same coupled argument gives

\[ E_X\le\frac{e_X+c_We_\Phi+h\rho}{1-c_W^2},\qquad E_\Phi\le\frac{e_\Phi+c_We_X+c_Wh\rho}{1-c_W^2}, \quad c_W=3\gamma_+h/2. \tag{7} \]

Physical sensor, coefficient and clock errors require their own additional terms; (7) holds for the otherwise unchanged model. On that same source, ||R||_1<=rho and sum R=0. For any union A of complete children, one possible reserved observable is the integrated balance

\[ \Delta q_A+\int (GV)_A\,dt+ j_{\rm pred}\int[B\sin(B^T\Phi)]_A\,dt =9b\kappa\int R_A\,dt. \tag{8} \]

Across the original structural horizon h its intrinsic right-hand bound is 9*kappa*h*rho/2. Independent measurements, integration aperture, inputs, initial phases and uncertainty must be declared before (8) becomes a physical test. It avoids differentiating noisy charge but does not eliminate the phase observation requirement. The bound is an allowed model discrepancy, not an observable lower bound or a TNFR-only signal. Higher precision on that intrinsic remainder cannot replace the missing physical information.

5. A concrete preparation obstruction: instantaneous phase events

The original fifteen observation events occur on the five-child host before donor attachment. The six-island contact capacitance cannot simply be used for those earlier events. Its ideal host carrier does, however, have a separate exact passive realization: repeat the electrical derivation for C5. The centered inverse of the C5 Laplacian has diagonal 2/5, adjacent entries zero and nonadjacent entries -1/5, so

\[ C_5=cL_{C5}^+,\qquad G_5=gP_5. \tag{9} \]

Five positive mutual capacitors c/5 join the nonadjacent pairs, all ten resistor pairs have conductance g/5, and five ring junctions have amplitude j. These matrices have the same scalar formulas (3) if the same dictionary is stipulated. This admits the ideal fixed-support host circuit, not the uncertain acquisition, observation actuator or later support switch.

The source's event changes phase by Delta Phi=a*k_c, where k_c=e_c-1/5, and leaves form unchanged. Its three amplitudes in each block are a=epsilon,-2*epsilon,epsilon, epsilon=1/10. Each is a nonzero relative circular change, not a common gauge or an integer full turn.

Finite-work obstruction. On a fixed floating circuit with positive capacitance and conductance on the neutral subspace, the Josephson row dot Phi=V/a_0, imposed in its integral solution sense, makes relative phase absolutely continuous when voltage has finite squared integral. Finite dissipated work guarantees that property. It therefore excludes a nonzero relative phase jump in zero time.

More quantitatively, any pulse of laboratory duration d>0 realizing a declared continuous lifted increment Delta Phi satisfies

\[ \begin{split} \int_0^d V^TGV\,dt&\ge \frac{a_0^2}{d}\Delta\Phi^TG\Delta\Phi,\\ \int_0^d E_C(t)\,dt&\ge \frac{a_0^2}{2d}\Delta\Phi^TC\Delta\Phi,\\ \mathop{\rm ess\,sup}_{[0,d]} E_C(t)&\ge \frac{a_0^2}{2d^2}\Delta\Phi^TC\Delta\Phi, \qquad E_C=\tfrac12V^TCV. \end{split} \tag{10} \]

Each follows from the positive quadratic-form Cauchy--Schwarz inequality and integral V=a_0*Delta Phi. The integrated capacitor energy is an energy-time quantity, not work; Joule heat in the first row is dissipated energy. A finite initial energy reservoir cannot make the first row bounded as d tends to zero. The result concerns the fixed reference and declared lift; other lifts or supports need their own endpoint and work accounting.

For the host event, k_c^T C_5 k_c=2c/5 and k_c^T G_5 k_c=4g/5, so these lower bounds become respectively

\[ \frac{4a_0^2a^2g}{5d},\qquad \frac{a_0^2a^2c}{5d},\qquad \frac{a_0^2a^2c}{5d^2}. \tag{11} \]

This proves that the literal instantaneous phase-only word has no finite-work implementation in this reference. It does not rule out a finite-duration current pulse or another preparation reaching an admissible endpoint. No endpoint form restoration, fine-source inclusion, winding retention or finite-pulse budget follows from the lower bounds alone. The original structural event-work budget cannot be converted into a payment for (11).

The separate finite-input construction now supplies those source, identity and resource obligations for one approximate intervention, retaining a nonzero endpoint form error. It does not admit the original complete observation word or a physical actuator.

There is also a conditional clock restriction. If a future model extends one constant calibrated b over the entire old preparation and contact, its isolated recovery/contact duration ratio remains 8e10/1e-3=8e13. Changing b cannot alter this ratio. For an independently admitted minimum resolvable contact time d_min and maximum recovery time D_max, that direct history requires D_max/d_min>=8e13. These physical limits have not been supplied. The current theorem only maps post-contact evolution, so neither this clock extension nor physical preparation energy is already certified.

6. Detached instrument and gate closure

The component inference owner accepts only primitive positive component values or shared outward intervals. Capacitances and conductances use all fifteen lexicographically ordered island pairs. It intersects the individual scalar-shape constraints before predicting j; the comparison consumes the ten independently supplied wheel sine amplitudes in the same pair ordering restricted to wheel edges. Reserved amplitudes never enter calibration. A comparison rebuilds calibration from primitives rather than consuming a cached report or fitted scale.

Malformed scalars, unsupported positive interval domains and wrong coordinate counts reject explicitly. Empty shape or amplitude intersections give scoped exclusions. A surviving interval relation is only necessary compatibility; the instrument admits no apparatus, phase sensor, source history or current profile. In particular a supplied sine amplitude is not proof that a physical current-phase curve is sinusoidal. The synthetic inference controls and exact pulse-bound controls exercise independent algebra, invalid domains and counterexamples without executing an acquired-source or physical response.

The operational gate closes with an unfitted conditional relation (2)-(4) and a specific obstruction (10)-(11) to directly implementing the existing instantaneous preparation word. No physical evaluation is admitted. The admitted finite observation still requires a complete physical observation, clock and preparation bridge before the circuit comparison can support an experimental protocol. Those requirements do not activate a physical campaign; the execution plan alone selects further work. The supplied gamma and Josephson constant remain independent constitutive premises throughout.