Physical feasibility of the organized current discriminator¶
Decision and scope¶
Physical status: not admitted. Experimental evaluation: not tested. The reviewed primary sources do not supply a laboratory realization of the frozen current discriminator. The failure has three distinct parts: its voltage observation has an extreme clock-independent precision/aperture requirement; ordinary junction capacitance changes an exact matrix premise; and independent preparation of the full organized state has not been established. This is a scoped admission result, not a physical rejection of either candidate law or TNFR.
The two conditional current envelopes and their frozen evidence remain unchanged. This audit derives laboratory-unit consequences and reviews methods/specifications; it acquires no experimental response, fits no coefficient and reruns no frozen producer. A published method is not a calibration certificate for a proposed assembly. The research program alone owns the next task.
1. Complete object and physical reference¶
The subject remains the 45-node C9-by-C5 organization, its two complete evolution rows, fixed support and held capacity. The comparison supplies mu equal to 1 or 2, a common observation-start state class and one finite input word. It does not prove alternative-law acquisition. Its exact five-coordinate electrical carrier and bounded fine departures are different objects: a prepared ideal five-island circuit cannot stand in for the formation of five constituent C9 organizations.
Use the fixed-support dictionary and the robust owner's fixed nominal coordinates. Write L for the C5 Laplacian, P for its neutral projection and L+ for its centered inverse. The supplied physical reference has capacitance matrix c L+, conductance matrix g P, five sine-current branches and voltage-phase constant a0. Nominal calibration fixes
The ratios C=c/c0, G=g/g0, A=a0/a00 and D are constant and independently bounded by 1 +/- 1e-8. Laboratory time maps to s=b0 D t. Readout always uses U=L+ V/v0 and nominal charge q_nom=c0 L+ V; actual charge is C q_nom. Changing nominal conversion after viewing the current would change the test.
For numerical illustrations only, set a00=h_P/(4 pi e), where h_P denotes Planck's constant, not the plateau length. The SI definitions, NIST SP 330 Section 2 give h_P=6.62607015e-34 J s and e=1.602176634e-19 C exactly. Thus a00 is approximately 3.291059785e-16 V s. Those supplied physical constants and the Josephson reference law are not derived from TNFR. An exact SI constant does not eliminate uncertainty in its realization or prove that a device obeys the reference law.
2. Voltage precision and time cannot be chosen separately¶
The frozen word has h=1e-5, fifteen unit-duration ramps, duration T=15.00015, raw nominal-form error eta_raw=1e-14 and inward endpoint aperture delta=1e-13. For every constant D,
All 150 scalars require their declared marking and windows. Clock accuracy does not substitute for sensor bandwidth or an inward averaging kernel.
Because the rows of L+ contain 2/5, zero, -1/5, -1/5, zero, its infinity norm is 4/5. Arbitrarily correlated voltage errors of size e_V therefore give form error at most 4 e_V/(5 v0). The bound is sharp for the admitted error class: signs aligned with one row attain it, whereas a common offset cancels. Allocating the entire raw allowance to voltage requires
Any other raw conversion error consumes part of this allowance. Known sensor correlations may permit a different bound, but cannot be assumed from independent channel labels.
Suppose an independently specified sensor mode needs an averaging support of duration at least t_min and offers a uniform absolute error guarantee e_V for that mode. Fitting its support inside each frozen aperture and using (3) requires
For one apparatus guarantee over the entire D box, use D_max. This is a necessary compatibility condition for the specified error-box admission, not a universal uncertainty principle or a lower bound on every possible sensor. It does not apply to an unmodeled charge, flux or current readout. Positive averages must stay inside the window; a manufacturer's response time alone does not establish their support or positivity.
The following unit conversions have D=1. They are illustrative scales, not independently realizable component choices or new assessed protocols.
| b0 (s^-1) | Maximum aperture (s) | Maximum voltage error (V) | Word duration (s) |
|---|---|---|---|
| 1e-12 | 1e-1 | 3.2001e-46 | 1.500015e13 |
| 1 | 1e-13 | 3.2001e-34 | 15.00015 |
| 1e12 | 1e-25 | 3.2001e-22 | 1.500015e-11 |
Merely slowing the experiment lengthens the aperture while lowering the allowed voltage error by the same factor. Increasing precision in the existing interval calculation does not repair (4).
The nominal waveform has peak voltage (6/25) a00 b0. In the actual robust decomposition its commanded part is multiplied by rho=1/(C D), with background bounded by a00 b0 gamma M_mu. A single range covering the nominal ramps and resolving the plateaus therefore encounters the nominal commanded-component/error ratio
This is not an ADC bit-depth theorem. Separate ranges or gated protection change the range problem and need their own settling/loading evidence; they do not alone remove (4).
3. Delivery, leakage and work in the same units¶
The robust law consumes u=rho f+d. Its physical current discrepancy obeys d=delta_I/(9 b0 kappa0 C D). Consequently, for any part J of the word,
The total F=1e-4 and each plateau P_leak=1e-15 thus require respective integrated absolute current errors at most a00 gamma g0 C F and a00 gamma g0 C P_leak, in coulombs. A bound uniform over C uses C_min. These are integrals of absolute error, not the absolute value of net delivered charge; cancellation cannot certify them.
On a plateau the permitted mean absolute leakage is
This is not a pointwise leakage bound. For the prescribed cubic r(ramp)=3 ramp^2-2 ramp^3, k_c=e_c-1/5 and v=L+ k_c,
The largest increment has absolute amplitude 1/5. Substitution of the five coordinates gives the peak at a ramp endpoint. Dividing (8) by the allowed plateau average gives 48 h/(25 gamma C D P_leak), approximately 6.17059e13/(C D). This is an on/off integrated-delivery demand; it does not require that many DAC levels. An L1 discrepancy budget supplies neither a peak bound nor a bandwidth. Every switch, transient, sensor loading current and nonneutral leakage needs its own admission.
For resources let K0=kappa0 v0 and use the robust coefficients lambda=G/(C D), beta=A G^2/(C^2 D), eta=1/(A D). Direct power pairing gives
Indeed I_actual=9 b0 kappa0 C D u and (L45 x)^T R u=9 (L U)^T u. The positive part of total supplied power scales by the same positive factor. Resistive heat equals C K0 times fine loss minus hidden loss, each with coefficient lambda. The existing finite resource majorants translate only on this same model and time interval. K0 times full fine storage is not the energy of a constructed fine electrical substrate. The dictionary starts at observation and does not convert earlier constituent recovery/formation into lab time or joules without an extended map.
4. Component shape is an independent obstruction¶
The exact capacitance matrix c L+ has zero adjacent off-diagonal entries. Suppose the five ring junctions also have a positive mutual capacitance c_J each, as permitted by the supplied RCSJ reference. Then
At each adjacent off-diagonal the left side is -c_J and the right side is zero. This remains a disagreement on the neutral leaf: both matrices annihilate the common mode, so equality there would imply full equality. It cannot be covered by widening only the scalar capacitance interval. Unequal branches or ground capacitance introduce additional changes; sensor loading can also change the dynamical law.
Equation (10) excludes exact shape matching for this explicit passive realization with nonzero ring capacitance. It neither sets a universal parasitic lower bound nor excludes a controlled approximate reduction. A small shape departure needs a matrix perturbation theorem carrying both rows, preparation, memory, observation and discrimination error. It cannot be assigned the intrinsic fine-state remainder by terminology.
On the neutral collective leaf, V=v0 L U gives
q=c*v0*(I+(c_J/c)*L^2)*U. The
matrix-response comparison
owns the resulting changed rows, storage and charge-coordinate contract.
This collective reference supplies no automatic transfer of the fine-source
acquisition or frozen discrimination certificates.
5. Independent source review and preparation gap¶
The methods receipt records the reviewed sources and retrieval availability. Retrieved-byte hashes, where available, are provenance associations, not authentication. The following are source facts; their admission consequences are our inferences. No reviewed figure or response becomes reserved evidence.
| Primary source | Supported facts | Remaining admission |
|---|---|---|
| Babenko et al. (2020), pp. 937-939 | RCSJ and voltage-phase equations; a 1500-junction array at 4 K with 5 GHz excitation and calibrated microwave measurements. The authors discuss junction variation and nonsinusoidal response. | No C5 matrix, 45-state preparation or error guarantee for this word. Junction capacitance is explicit in the reference equation. |
| English et al., arXiv v2, pp. 1-2 | Phase-sensitive observation through a superconducting loop, flux transformer and DC SQUID; 10 mK base temperature and a nonhysteretic operating condition. | An independent phase method is possible, but its loop state, loading and calibration must be retained. It does not establish our initial phase collar. |
| AH2700A brochure, pp. 1-2 | Published 1 kHz capacitance accuracy of +/-5 ppm. | This specification alone is 500 times wider than the frozen 0.01 ppm scalar allowance, and is not a matrix calibration at the proposed operating conditions. |
| NIST impedance metrology, Major Accomplishments | A specialized resistor/capacitor comparison reports combined standard uncertainty of 5 ppb near 1592 Hz. | The scalar budget is not universally unattainable; this is not a complete cryogenic matrix certificate or a deterministic error bound. |
| Keithley 2182A manufacturer page, Features | Two channels, 1 nV sensitivity, typical 15 nV peak-to-peak noise at 1 s response and 40-50 nV at 60 ms. | No five-coordinate synchronized transfer, positive inward kernel or simultaneous absolute guarantee meeting (3)-(4). Response time, sensitivity and typical noise must not be substituted for those distinct premises. |
| JCGM 100:2008, Sections 2.3.5, 6.2, 7.2.5 | Expanded uncertainty carries a stated coverage interpretation; multiple outputs require their uncertainty/covariance information. | RMS, typical noise and marginal coverage intervals do not supply the frozen simultaneous deterministic error box. |
For magnitude context only, the hypothetical pair e_V=1 nV and t_min=1 ms has product 1e-12 V s, exceeding (4) by about 3.1e34 at D=1. This pair is not asserted to be a manufacturer guarantee or a fundamental noise floor. The reviewed voltmeter specifications cannot certify the required pair, and do not supply the observation kernel in the first place. Neither a finite typical peak-to-peak sample nor calibration uncertainty may be promoted to a hard bound on every future reading. A statistical protocol would instead need a joint confidence event over all readings, preparation and calibration; the deterministic theorem could apply conditionally on that event if all its other premises were met.
Preparation remains separate from precision. The model needs independently justified bounds on the complete initial signed forms, hidden constituent deviations and admitted relative phase set. Zero voltage does not establish relative phases or internal deviations. Even the ideal carrier allows identical U=0 with Phi=0 or Phi=(d,-d,0,0,0), with small positive d: both have winding zero and zero voltage, but different sine-current vectors. Winding alone is insufficient. The prior need not identify an exact phase, yet must be independently justified as a set under both candidate laws. None of the reviewed sources supplies that preparation or extends the physical map through the TNFR acquisition.
6. Consequence for research¶
There is no admitted physical candidate in this review. Equation (4) identifies a limitation of the frozen endpoint-voltage method, (10) an exact shape mismatch for a specified junction-capacitance realization, and the preparation audit missing independent evidence. These are different conclusions and must not be combined into a universal no-go.
A successor must explicitly change the observation/protocol or prove an appropriate approximation; repeating the same arithmetic at higher precision supplies neither. Finite-window observables, direct current or flux sensing, a prepared collective reference and a generatively acquired organization have different obligations. Any successor must keep the source distinction, declare loading and matrix deviations, retain its own law and work, and freeze its unfitted prediction before response selection. The conditional current result is useful as an ideal comparison, not as evidence that the required physical apparatus already exists.
The independent algebra controls check inverse-matrix error propagation, scale cancellation, current/work conversion and a positive-capacitance counterexample. They establish no instrument performance. No runtime capability or public API is added by this documentary and mathematical admission audit.