Regional phase/amplitude measurement contract

Status: 2026-09-26; specification complete, physical admission not_admitted, physical prediction not_tested; exact-model data search parked after the bounded review below. This is the observation annex for the derived phase/amplitude candidate. The execution plan owns scheduling; the existing source inventory owns general dataset descriptions. This document owns the candidate-specific measurement map, resolution criterion and source admission decision. It does not add a simulator, a physical law or a second task queue.

1. What would be measured

The candidate has six scalar states: two equally oriented directed triangles, with reciprocal links between corresponding vertices. All twelve directed edges have unit relative weight. In the engine's row convention, i -> j means node i reads node j; this must not be confused with a physical material-flow arrow. Each row reads two neighbors. Pressure is only p_i=(x_next+x_cross)/2-x_i, capacity is fixed at a>0 or b>0 within each triangle, and dx_i/dt=nu_i*p_i. There are no operator events, Gamma terms, other pressure channels or independently evolving primitive phases.

An admissible observation would supply six simultaneous scalar channels, fixed channel-to-node labels and their directed connection map. Independently calibrated sensor offsets and gains must recover a common scalar chart; they may not be selected to make the predicted phase appear. One conditional electrical chart is

\[ \widehat V_i=(y_i-\beta_i)/g_i,\qquad \widehat x_i=(\widehat V_i-V_{\rm ref})/V_*,\qquad t=(s-s_0)/\tau_*. \]

Here y is the recorded sensor output; beta_i,g_i are independently calibrated instrument offset/gain with g_i!=0, V_ref is a common reference, V_*>0 is a common scale, and s is physical time in seconds. tau_*>0 is a fixed number of seconds per declared structural time unit. A measured temperature or concentration could use an analogous chart, but would need its own constitutive and support admission. This voltage map is a hypothesis for a suitable apparatus, not evidence that such data exist.

For each ordered triangle, apply the already derived orthonormal contrasts:

\[ \mu=(x_0+x_1+x_2)/3,\quad u=(x_0-x_1)/\sqrt2,\quad v=(x_0+x_1-2x_2)/\sqrt6,\quad z=u+iv. \]

Retain both means and both Cartesian contrasts. Where amplitudes are resolved, report r_0=abs(z_0), r_1=abs(z_1), q=r_1/r_0, and delta=Arg(z_1*conj(z_0)). This is a spatial form angle, not a Hilbert phase of one temporal signal, primitive theta, electrical superconducting phase, or a tetrad replacement. A common calibrated offset cancels from each contrast; unequal calibration errors do not. Fixed node ordering determines orientation and may not be permuted after evaluation to improve agreement. No future-dependent filtering, phase unwrapping or numerical differentiation is needed for this instantaneous observation.

2. Constitutive and clock evidence

The nodal product alone does not identify its factors. An apparatus must independently justify the stated pressure row and absence or bounded effect of additional state, leakage, forcing and delays over the declared interval. One conditional linear realization would have a separately verified law

\[ S_i\frac{dV_i}{ds} =G(V_{\rm next}+V_{\rm cross}-2V_i),\qquad \nu_i=2\tau_*G/S_i. \]

S_i is physical storage (capacitance in an electrical realization), not TNFR capacity itself. G is the equal effective directed transfer coefficient; G/S_i has units of inverse seconds. Both the equality of links and the two regional rate classes require independent evidence and error bounds. Measured components or separate calibration preparations may determine these rates; fitting them to the reserved phase response is prohibited. A calibrated effective rate alone does not identify G and S_i separately. Likewise responses in seconds determine a/tau_* and b/tau_*, not an independent absolute structural clock: fix tau_* before calibration instead of fitting redundant clock and capacity scales.

This equation is a prospective apparatus requirement, not a claim that reciprocal resistors implement it. The reciprocity obstruction proves that a passive reciprocal first-order scalar RC/thermal network with positive storage cannot realize this generator even through an invertible linear state chart. Its spectrum is real, while the directed model's contrast spectrum is complex. Nonreciprocal circuitry or directed transport would need its own justified reduction and recorded external operating conditions. An active apparatus may supply energy while its effective state equation is homogeneous; that is not autonomous substrate creation or a derived TNFR energy source. No apparatus construction is scheduled for the computer-only research route.

The graph and its generator must be specified from wiring/transport evidence, not inferred solely from phase correlation. Removing unobserved nodes or selecting six channels from a larger graph requires an exact closed reduction or an independently bounded memory/error term. The existing memory owner supplies that distinction; it does not certify closure for a new dataset.

3. Sensor, amplitude and phase uncertainty

Use the joint-error conventions already specified by P2MeasurementBounds: calibrated coordinate error includes gain/offset error, digitization and any bounded channel skew. Two timestamp errors contribute to an elapsed interval; timestamp accuracy alone does not make different channels simultaneous. Its executable two-node enclosure is not a six-node predictor and is not reused outside that domain.

For one triangle suppose x_hat=x+e, with simultaneous deterministic bounds abs(e_i)<=epsilon_i. Writing the contrast map as z=U^T*x in real two-vector form gives U^T*U=I and U*U^T=I-11^T/3. Consequently

\[ |\widehat z-z|\le\rho,\qquad \rho=\max_{\sigma_i\in\{-1,1\}} \left\|U^T(\sigma_i\epsilon_i)_i\right\|_2 \le\sqrt{\sum_i\epsilon_i^2}. \]

The finite maximum is valid because a convex norm on a box attains a maximum at a vertex. For a common node bound epsilon, the exact box radius is rho=sqrt(8/3)*epsilon: a mixed-sign corner has squared contrast 8*epsilon^2/3, whereas a common-offset corner has zero contrast. This uses simultaneous channel bounds, not an assumption of independent Gaussian sensor noise.

If r_hat=abs(z_hat)>rho, the disk around the observation excludes zero, and

\[ r\in[\widehat r-\rho,\widehat r+\rho],\qquad d_{S^1}(\widehat\psi,\psi)\le \arcsin(\rho/\widehat r). \]

The angular bound follows from the tangent from the origin to that disk. Relative-angle error is at most the sum of the two regional bounds, capped at pi. The ratio lies in [(r_hat_1-rho_1)/(r_hat_0+rho_0), (r_hat_1+rho_1)/(r_hat_0-rho_0)] when both lower amplitudes are positive. If either disk contains zero, the corresponding phase and the relative-angle comparison are unavailable. A ratio bound still exists when the denominator disk excludes zero: use max(0,r_hat_1-rho_1) for its lower numerator. If the denominator disk contains zero, no uniformly defined finite ratio bound follows. Retain Cartesian observations and report the separate availability of angle and ratio. Neither zeroing an unavailable angle nor dropping low-amplitude endpoints after evaluation is permitted.

A design budget must use a justified lower bound on the future observed amplitudes, including initial-state, parameter, clock and observation errors; nominal predicted amplitudes alone do not guarantee resolution. Capacity and clock uncertainty must be propagated jointly through the declared fine generator. Independent solver agreement is not a rigorous enclosure. Any statistical alternative must predeclare joint coverage over the complete comparison; per-sample standard deviations do not imply this deterministic bound. No sensor accuracy has yet been admitted for this candidate.

4. Frozen finite prediction and decision

Use the full six-node initial-state observation and the independently fixed rate/clock information to forecast both models before reading later responses. The comparator freezes q(t)=q_0, where q_0 is the same initial amplitude ratio, with its uncertainty, used by the full model. Its initial ratio interval must have a strictly positive lower bound and a finite upper bound, since the angular comparator uses both q_0 and 1/q_0:

\[ \dot\delta_{\rm fr} =\frac{\sqrt3}{4}(a-b)-\frac12(b/q_0+a q_0)\sin\delta_{\rm fr}. \]

The nominal equal-amplitude, quadrature preparation gives the previously derived acceleration difference, but a real initial record must not be forced to q_0=1,delta_0=pi/2. No subsequent measured amplitude is fed into either forecast. The main observable is the finite circular phase separation, with amplitudes and complete nodal response retained as checks of the full model. Estimating a second derivative from the reserved signal is unnecessary.

Let the two nominal phase forecasts have independently justified error radii B_full and B_fr, including initialization, parameters, clock, admitted model-reduction error and numerical error. Let B_obs bound the relative-angle observation error. A sufficient robust resolution condition is

\[ d_{S^1}(\delta_{\rm full},\delta_{\rm fr}) >B_{\rm full}+B_{\rm fr}+2B_{\rm obs}. \]

This makes the two possible observation tubes disjoint. It is a sufficient design criterion, not evidence that either model is true. Predeclare finite times, run identities, initial-state admission and joint uncertainty before opening reserved responses. At scoring, use circular distances to the two tubes: full-compatible and comparator-incompatible favors the full reduction over that approximation; both compatible is inconclusive; full-incompatible rejects the tested realization or one of its declared premises. Also check the full predicted scalar traces; a coincidental phase match alone does not admit the nodal model. Pointwise compatibility with outer error tubes need not imply that one joint parameter/preparation realization explains the whole record; it means only that this necessary check did not reject it. An unresolved amplitude yields an unavailable angular test, not support for either model. Reserve independent preparations and retain all declared failures and missing observations.

Scale of the already retained mathematical example. At (a,b)=(1,2), z(0)=(1,i)/16, and t=0.5, the existing finite control predicts a phase gap of about 0.05947 radians (3.407 degrees), with full-model amplitudes about 0.036327 and 0.024267 in its chosen EPI chart. This gives a prospective resolution target; it is not 0.5 physical seconds without a clock bridge. For illustration, if future observed amplitudes are independently bounded below by 0.036 and 0.024, a joint node error epsilon<=0.0001 gives B_obs<=0.011341 radians and leaves about 0.03678 radians for the sum of the two forecast error radii. Those lower amplitudes and errors are hypothetical requirements, not instrument specifications or measured performance. Decay eventually destroys angular resolution; waiting indefinitely for the limiting lag is not the proposed test. This distinguishes a full model from a lossy approximation, not TNFR from mathematically equivalent external physics.

5. Existing-source admission decision

The bounded review on 2026-09-26 rechecked source descriptions and apparatus equations, not reserved arrays. The earlier inventory retains licenses, archive sizes and repeat identities. No signal archive was downloaded, no new response scored and no calibration changed in this review. Published qualitative summaries are prior exposure, not blind evidence.

Existing source	Finding for this particular candidate	Decision
TCLab apparatus/model	Two measured temperatures; heater states, supplied inputs and ambient exchange enter its model. It does not provide the six measured scalar states or declared directed generator.	Not admitted for this prediction. Preserve the separate frozen exploratory comparison; its scored files are not new holdouts.
NED3-008 thermal record and README	Six thermocouple channels serve a steady-state conductivity apparatus. Channel count does not establish six closed storage nodes, the required directed support or capacities; fresh-conversion timing remains unresolved.	Not admitted. A reciprocal scalar conduction realization would additionally face the proved spectral obstruction. This is not a finding about uninspected transient arrays.
Rössler record, primary circuit equations	The measured second coordinate has hidden-state and local circuit terms besides diffusive coupling. One observed coordinate per three-state oscillator does not close the specified scalar law. Nominal acquisition/filter specs do not supply its state or clock bridge.	Not admitted. Neither temporal phase extraction nor selecting six of the 28 channels removes those missing terms or exterior couplings.
Analog learning network record and README	Published 4x4 free/clamped node snapshots accompany changing gate voltages and learning steps; the stated continuous scope record contains time, input voltage and output voltage. The documented schema does not establish simultaneous six-node free evolution under the fixed directed law.	Not admitted. Gate storage is not an independently identified nodal capacity. Admitting another driven/adaptive model would be a separate decision.

The conclusion is restricted: none of the existing inventory admits this particular phase/amplitude test. It does not show that no suitable public dataset exists, that these sources are scientifically useless, or that TNFR is physically refuted. The missing bridge is both constitutive (known closed nonreciprocal scalar dynamics) and observational (joint state, rate, clock and uncertainty evidence). Six sensor traces or oscillatory-looking signals alone would not close it. The protocol is ready to assess a matching source; the physical objective remains open.

The subsequent bounded directed-source review examined six new studies and admitted none for this exact test. That closes the scheduled search and parks this physical experiment; it does not discard the conditional theorem or its observation/error machinery. MOSTR is retained as a deferred three-tank transport lead under the user's collective-emergence priority. Its directed open cascade has different boundaries and spectrum and cannot inherit the present phase-locking prediction. The source inventory owns that candidate's metadata and missing input/measurement dependencies; the plan owns the next action.

Boundary for a future primitive-phase information test

This annex's six-state spatial form angle does not identify primitive phase in a different model. Only its calibration, provenance, uncertainty and reserved-evaluation conventions can be reused for that purpose. The first-moment controls are exact algebraic designs, not instruments or noisy measured phasors; their static source equality does not imply equal future trajectories. A finite test must retain the actual support, companion laws, hidden state and phase/rate association. Imposing unit norm or equal moments on noisy readings would not supply an independent preparation.

A preliminary metadata review on 2026-10-06 located Nijholt et al., Emergent hypernetworks in weakly coupled oscillators. Its four-electrode ring uses nonlinear delayed feedback. The authors' data repository describes time in seconds, four electrode potentials and four extracted phases in coupled and uncoupled files. Only published descriptions were inspected; no signal file was downloaded, no reserved prediction frozen and no response scored. The phase/closure bridge, uncertainty and controlled matched preparations are unadmitted. Published descriptions are prior exposure, not blind evidence.

The information theorem also supplies a design restriction: two incident phasors with nonzero sum are determined up to permutation by that sum. At zero sum, the selected sine/cubic currents both cancel by their half-turn symmetry. Thus a degree-two instantaneous equal-moment witness cannot separate those two kernels. This algebraic restriction is not a rejection of the authors' different delayed-feedback model or of all odd currents.

The following metadata remain a supporting source inventory for a future admitted comparison. The execution plan alone selects whether their missing bridges become active work. They supplement the phase-information option; they do not admit this annex's original six-state law, the two-C6 discriminator or a particle model. No raw signal archive was downloaded or decoded in this review. Published methods and qualitative results are prior exposure.

Supporting source	Metadata inspected on 2026-10-06	Remaining admission boundary
28 Rössler circuits	CC BY 4.0; documented supports and three repetition labels; only the second circuit coordinate is recorded	Direct-source admission below closes application of the existing star test. Additional state, a proved invariant preparation or controlled approximation would require separate admission.
Two resonant accelerometers	CC0; about 48.7 MB, figure-organized recordings including phase and mechanical perturbations	Independent acquisition identities, phase extraction and feedback/input laws are not yet admitted. Published curves cannot be recast as unseen predictions.
Two coupled MEMS resonators	CC BY 4.0; 125 kHz sensor records, coupling/feedback settings, about 35 GB total with smaller regime archives	External driving and feedback remain supplied. No verified independent-repeat allocation or TNFR measurement bridge.
Three Rössler circuits with higher-order interactions	CC BY 4.0; x/y channels for three oscillators, 30,000 samples per record; one block about 23.5 GB	Hidden z coordinates, programmed nonlinear coupling and unverified repeated acquisitions. Interaction order is not a phase-harmonic order.

The four-electrode repository above was inspected at revision fe79902503e59f8fba1c635b5e01518d13b904dc. Its coupled and uncoupled records are not repeated independent acquisitions; cropping a record does not create one. Delayed feedback and the published phase/filter procedure need causal admission, including possible future-sample access. No explicit data license was found in that repository during this inspection.

All candidates retain physical_status=not_admitted and evaluation=not_tested here. An oscillator phase experiment could test a specified effective law; it would not derive that apparatus's material constituents. The collective-property comparison has the main generative scope. The execution plan owns selection; no archive size, available repetition or attractive waveform starts a campaign.

Selected source: 28 electronic oscillators

Decision: direct source route closed; physical_status=not_admitted, evaluation=not_tested. The 2026-10-06 review selected this existing lead because it documents connections and repetition identities, unlike a phase trace with unknown support. This is an admission of methods and observation scope, not an evaluation of either TNFR law against measured responses.

The dataset and primary methods, Eqs. (1)-(3) describe three-state circuits, recording only voltage v2. Acquisition uses 30 kS/s, a 16-bit range of +/-5 V and a third-order 1,500 Hz Butterworth filter. Coupling is increased in steps after an initial zero-coupling settling interval. This is not a documented preparation of our two matched full states. The inspected sources do not supply per-channel timing, calibrated uncertainty, filter realization/initialization or independently reset hidden states. Repetition labels alone do not establish those properties. Nominal component values are not measured error bounds; ADC bit depth is not sensor accuracy.

The retained admission record and source archive preserve the primary article XML, dataset metadata and small Structure.zip with their hashes and licenses. No signal archive was downloaded, no response was fitted and no measured trajectory became a reserved evaluation. Interpreting each support file as undirected endpoint pairs, all twenty graphs have 28 vertices, 42 edges and are connected. At nonzero coupling, selecting seven channels therefore discards exterior contacts rather than producing the isolated seven-node star of the finite test. Column/label correspondence has not been independently checked against signal files.

A precise observation obstruction

This deduction uses the published nominal circuit law, not the measured arrays. Write u=v1, y=v2, w=v3, with fixed support and coupling. The first two circuit rows have the vector form

\[ u'=-d u-e y-f w,\qquad y'=-a u+M y, \qquad M=bI-cL,\qquad a,d,e,f>0, \]

where L is the unnormalized support Laplacian. In physical seconds, the nominal second row is

\[ y_i'=-10000u_i+\frac{44200}{29}y_i +2000\kappa\sum_j A_{ij}(y_j-y_i). \]

Two circuit states can have identical y and different u, hence different y'. The existing closure criterion therefore excludes an autonomous instantaneous law for y alone on an open source-state family. This is not pressure reconstructed from the response: the different derivatives follow from the independently supplied source law.

The same rows give a stronger information result. The unfiltered observation and its first two derivatives determine all three coordinate blocks:

\[ u=(My-y')/a,\qquad w=(y''-My'-ad u-ae y)/(af). \]

Thus the observation map Psi=(y,y',y'') has rank 3N: in state order (y,u,w), its Jacobian is block lower triangular with diagonal I,-aI,afI. The third evolution row can be nonlinear; it does not enter this identity. Suppose a sufficiently smooth autonomous target of dimension m, with smooth initialization J and readout H, reproduced these voltage histories for every source state in an open set and a fixed clock. Its first two output derivatives would factor through J, so rank(DPsi)<=m. Consequently m>=3N. A same-node-count, fixed-capacity form/phase target has only 2N coordinates: 56 instead of 84 for this source. It cannot be an exact open-family realization of the full unfiltered voltage output.

This does not determine the dimension of a recorded attractor, exclude a separately proved invariant preparation family, or rule out approximations, additional nodes/state or causal memory. It also does not identify the filtered recordings with y: filter dynamics and noisy differentiation need their own causal observation model. An extracted temporal phase is not an automatic escape from the state requirement. The source-matching condition must hold for both mapped rows, with an independently admitted clock. The shared observation kernel controls check the block-rank mechanism with exact rational instances; the rank argument above supplies its general scope, not a numerical reconstruction of these data.

Consequence for the finite information test

The star theorem retains exact reference support, capacity, clock and form/phase laws. Its initial-state and endpoint-readout errors do not cover missing continuous source terms, sensor memory or discarded exterior interactions. The existing proof does not cover those effects merely by relabeling them as preparation error or renaming a voltage as form. A collective observation remains allowed, but must supply its own state, law and uncertainty justification.

A potential approximate bridge would instead expose the two mapped-law defects r_x and r_theta, bound them independently throughout the prediction window and propagate their effect into the reference prediction intervals. These are errors relative to a declared law, not new fundamental TNFR variables or parameters fitted to an evaluated response. No such bounds are established for this source. The execution plan retains this as an auxiliary, deferred route. The direct circuit comparison stays closed unless new preparation, reduction and measurement evidence resolves it; this review does not start another data or approximation campaign.