Current-amplitude discrimination with calibrated uncertainty

Prospective decision and scope

The five-child specification gives a phase-free current statistic for an exact model. This gate asks whether one fixed input and one fixed observation separate that amplitude from a doubled amplitude after admitting nonzero uncertainty. It does not ask which law nature follows. No physical record, apparatus or new acquisition is evaluated.

Item	Fixed declaration
Models	Full C9-by-C5 laws (3) with mu equal to 1 or 2; 45 forms and phases, 90 unit edges, held unit capacities, full-circle sine, neutral uniform-child inputs. The C5 carrier is an exact electrical reference; discarded fine state remains in the full model
Preparation	Independently supplied observation-start class: zero global form mean, abs(x_i)<=X=.01, theta_cj=psi_j+Phi_c+z_cj, zero child means of z, abs(z_cj)<=zeta=1e-6, principal mean gaps at most Aphase=7/5, winding in {-1,0,1}. Common phase is retained. This is not a formation certificate for the alternative or for changed calibration coefficients
Input	Same nominal cubic word for both laws: fifteen unit-duration ramps, each preceded by a plateau h=1e-5; epsilon .1, marked increments .1,-.2,.1; duration T=15.00015. Nominal coefficients and sensor conversion are fixed before any response
Calibration and clock	Independent positive ratios C,G,A,D defined below, each in [1-1e-8,1+1e-8]. Exact C5 component shapes and a constant command-clock ratio; arbitrary parasitics or time-dependent clock drift are outside this first domain
Tracking	Neutral child-uniform discrepancy d in (3), with total integral(norm_inf(d))<=F=1e-4; on each nominal plateau its integral is at most P=1e-15. No derivative or pointwise bound is presumed
Observation	Same 150 nominal-voltage-to-form scalars in (5,3,2,5) order for both laws; arbitrary correlated additive error eta_raw<=1e-14. Each reading is a sample or positive normalized average within its own inward endpoint aperture of radius delta<=1e-13 in nominal clock units
Statistic	Five unnormalized current norms (13), computed with nominal coefficients; no measured or inferred phase and no amplitude-dependent redefinition of the observed statistic
Decision	Require every whole-history identity/collar margin to be positive for both laws and the upper mu=1 current envelope to be strictly below the lower mu=2 envelope. Failed sufficient bounds mean unavailable separation, not impossibility or physical rejection
Evidence	Commit declaration, helper and driver before one analytical assessment of the stated joint box using outward dyadic128 intervals. No trajectory, scan, source replay, fitting or reserved experimental response

The error allowances are prospective mathematical premises, not measured instrument capabilities. A common initial-state class need not specify identical hidden states under the two hypotheses: separation must hold uniformly for either admissible initial state. The existing acquired source is included at mu=1 and nominal coefficients through its earlier proof. That proof does not establish acquisition under mu=2 or perturbed laws.

1. One nominal laboratory input and observation

Choose independently calibrated nominal c0,g0,a00>0 and keep

\[ \gamma=1/(1023\pi),\quad b_0=4g_0/c_0,\quad \kappa_0=a_{00}\gamma g_0/9,\quad v_0=a_{00}\gamma g_0/c_0, \quad j_0=a_{00}\gamma^2g_0^2/c_0. \tag{1} \]

Here a00 is the nominal voltage-phase constant, not a derived TNFR constant. Let C=c/c0, G=g/g0, A=a0/a00. A positive constant D relates the command coordinate to actual laboratory time by s=b0*D*t_lab. Thus D is instrument clock calibration, separate from the matching relation b=4g/c.

The physically specified current is I_cmd=9*b0*kappa0*f(s), with the same f for every calibration member and both laws. Delivery discrepancy is retained explicitly. Readout always uses the nominal voltage coordinate U=L5+ V/v0 and nominal inferred charge q_nom=c0*L5+ V=9*kappa0*U. Actual charge is q=C*q_nom. A common voltage gauge cancels; different unknown offsets require the declared sensor error. For per-island voltage error eta_V the nominal form error is at most 4*eta_V/(5*v0). Calibration is not silently refitted in this conversion.

For a circuit with j_actual=mu*a0*gamma^2*g^2/c, transforming both rows to these fixed coordinates gives the coefficients

\[ \lambda=G/(CD),\quad \rho=1/(CD),\quad \eta=1/(AD),\quad \beta=AG^2/(C^2D). \tag{2} \]

The full fine comparison explicitly supplies the corresponding laws

\[ x'=(-\lambda L_{45}x+\mu\gamma\beta S(\theta))/4 +R(\rho f+d),\qquad \theta'=\gamma\eta L_{45}x/4. \tag{3} \]

Prime denotes s. R repeats a child coordinate across its nine slots. On the helical carrier these equations give exactly the C5 circuit rows. Off it, internal cancellation leaves the actual mean external currents, not an assumed closed sine law. The complete fine extension is a model premise; a passive physical fine-node circuit has not been constructed. In particular the full fine source is not replaced by an ideal carrier.

Physical current discrepancy maps to d=delta_I/(9*b0*kappa0*C*D). The total integral F must be admitted from independent delivery evidence, including switch timing and leakage. The separate bound P is essential: a small total input error can otherwise concentrate in a reading window and mimic its signal. Neutrality preserves the zero form mean; nonneutral errors require another law and observation.

2. Whole-history bound for each law

Reuse only the explicit waveform, not the old response bound. Let z(s) be its known form offset and H(s) its phase command:

\[ f=z'+L_5z/4,\qquad H'=\gamma L_5z/4, \quad \|H\|_\infty\le4\epsilon/5, \quad J_z:=\int_0^T\|L_5z\|_\infty ds=32/(5\gamma). \tag{4} \]

The last equality follows from L5*z=4*a*k_c*s_cubic'/gamma, norm_inf(k_c)=4/5, nonnegative cubic derivative and total absolute increment two. Each offset vanishes at its ramp endpoints and throughout plateaus. H is continuous, starts and ends at zero, and each ring edge has nominal excursion at most epsilon.

Write y=x-rho*R*z. Its actual residual input is w=rho*(1-lambda)*R*L5*z/4+R*d. Heat contraction on the unchanged support and norm_inf(S)<=4 give, for each fixed calibration member,

\[ J=\rho|1-\lambda|J_z/4+F,\qquad M_\mu=X+\mu\gamma\beta T+J,\qquad \|y\|_\infty\le M_\mu. \tag{5} \]

The phase drift after subtracting the actual command etarhoH satisfies

\[ e_b'=\gamma\eta L_{45}y/4 =-\frac{\gamma\eta}{\lambda}y' +\frac{\mu\gamma^2\eta\beta}{4\lambda}S +\frac{\gamma\eta}{\lambda}w. \tag{6} \]

Integration across the whole word, including every switch, bounds the departure from the nominal initial shifted state by

\[ \|\theta(s)-\theta(0)-RH(s)\|_\infty\le E_\mu=2\gamma\eta M_\mu/\lambda +|\eta\rho-1|\,4\epsilon/5. \tag{7} \]

No stationary background or hidden-phase input is used. The sufficient identity margins are, separately for mu=1 and mu=2,

\[ 2\pi/9-2\zeta-2E_\mu,\quad \pi/2-2\pi/9-2\zeta-2E_\mu,\quad \pi/2-A_{phase}-\epsilon-2\zeta-2E_\mu,\quad \pi/2-2\zeta-4E_\mu. \tag{8} \]

Positive margins retain all five internal and nine outer identities and the circular collar. They neither select a winding nor prove formation. At final release the command and offset vanish; residual state is retained.

For boxes, compute positive upper bounds using shared outward gamma and coefficient intervals, lambda's positive lower endpoint where divided, and the absolute maxima of 1-lambda and eta*rho-1. Discarding box correlations enlarges the bounds; it cannot silently improve a verdict.

3. Readings with aperture and arbitrary integrable tracking

On a nominal plateau, actual child means obey

\[ U'=-\lambda L_5U/4+\mu\gamma\beta\bar S/4+d, \qquad \|U\|_\infty\le M_\mu,\quad \|\bar S\|_\infty\le2. \tag{9} \]

An inward aperture stays entirely within that plateau. Relative to its endpoint the form error of a sample, or positive normalized average, is bounded by

\[ n_\mu=\eta_{raw}+(\lambda M_\mu+\mu\gamma\beta/2)\delta+P. \tag{10} \]

The P term needs no pointwise tracking bound. Signed observation kernels need their own total-variation bound and are outside this declaration. Two-sided jitter crossing a ramp cannot use (10). The first sample looks forward into its plateau and the last backward; windows may overlap but must remain inside it. We require 0<=delta<=h.

Keep the same nominal corrected increment at receiver (c+1) mod 5: D=Q_r-P_r+h*[L5(P+Q)]_r/8. Its reference current is the initial fine current shifted by nominal H. Define

\[ B_{2,\mu}=4\lambda^2 M_\mu+2\lambda\mu\gamma\beta +4\mu\gamma^2\beta\eta M_\mu. \tag{11} \]

The corrected-increment error is bounded by

\[ b_\mu=\mu\gamma\beta hE_\mu +|1-\lambda|hM_\mu+B_{2,\mu}h^3/12 +P(1+h/2+\lambda h^2/4)+(2+h/2)n_\mu. \tag{12} \]

The first term bounds phase/current transport; the second is the unmatched diffusion coefficient in the nominal correction. The last is the exact absolute sum of the six raw endpoint coefficients. Arbitrary correlations between raw errors are permitted.

The tracking/trapezoid terms do not differentiate d. To see this, set v(s)=integral_plateau_start^s R*d and w=x-v on that plateau. The smooth drift differentiated along the absolutely continuous state has norm at most B2 apart from -lambda*L45*R*d/4. The trapezoid Peano kernel has maximum h^2/8; norm_inf(L45)<=8 therefore contributes at most lambda*h^2*P/4. The integral of d in v has trapezoid kernel s-h/2, bounded by h/2, contributing h*P/2. Direct input in the mean increment contributes P. This proves (12) for integrable tracking, without assuming finite derivative, exact switches or an unforced actual plateau.

4. A common phase-free amplitude observation

For every marked block form O=D_+-D_-, W=D_++D_--2*D_0 from its same three corrected readings. Define the physical nominal-current statistic with coefficients fixed before comparison:

\[ A_c=\frac{2j_0 O_c}{\gamma h\sin\epsilon},\qquad B_c=\frac{2j_0 W_c}{\gamma h(1-\cos\epsilon)},\qquad m_c=\sqrt{A_c^2+B_c^2}. \tag{13} \]

Neither candidate changes the raw readings, the input or this conversion. There is no division by its tested amplitude and no reconstructed phase. Equivalently compute the quadratures directly from nominal charge using the nominal plateau duration h/b0. Gamma cancels from the resulting charge conversion because 2*j0/(9*kappa0*gamma)=b0/2; it also cancels from the physical nominal waveform after substituting kappa0 and the cubic f. Thus choosing the command and observation does not fit the tested amplitude. Since q_nom=q/C and nominal instrument seconds equal D*t_lab, the reference amplitude in this statistic is j_actual/(C*D)=mu*beta*j0. It is not the uncorrected physical current amplitude when C or D differs from one.

Let Cc+iSc be the initial fine interface phasor. The initial residual prior gives cos(2*zeta)<=abs(Cc+iSc)<=1. For each model,

\[ |A_c/j_0-\mu\beta C_c|\le a_\mu=4b_\mu/(\gamma h\sin\epsilon),\qquad |B_c/j_0-\mu\beta S_c|\le b_\mu^*=8b_\mu/[\gamma h(1-\cos\epsilon)]. \tag{14} \]

The normalized bounds below are only a dimensionless way of displaying the same measured current, not an amplitude fit. With r_mu=sqrt(a_mu^2+(b_mu*)^2) and coefficient endpoints beta_minus/plus,

\[ m_c/j_0\in I_\mu= [\max(0,\mu\beta_-\cos(2\zeta)-r_\mu), \mu\beta_++r_\mu]. \tag{15} \]

Use outward sine, cosine and square-root enclosures. If the lower endpoint of I2 exceeds the upper endpoint of I1, the complete 150-reading record sets are disjoint: even one marked norm cannot belong to both intervals. This is uniform over the admitted unknown phases, fine deviations and correlated errors. It is a prospective discriminator, not an identification of either law by an unevaluated experiment. Real records outside both envelopes can invalidate premises or exclude both hypotheses; records inside one envelope still need complete physical admission.

5. Storage and boundaries of physical interpretation

With fixed positive coefficients each model has its own storage

\[ H_\mu=\tfrac12x^TL_{45}x+ \frac{\mu\beta}{\eta}\sum_e(1-\cos\delta_e),\qquad H_\mu'=-\lambda\|L_{45}x\|_2^2/4 +(L_{45}x)^TR(\rho f+d). \tag{16} \]

This follows by cancellation using both rows. It is not the earlier mu=1 storage when the coefficients change. Explicit finite bounds follow without a pointwise tracking assumption. Set Zpeak=24*epsilon/(5*gamma_lower), Gpeak=48*epsilon/(5*gamma_lower), Xpeak=rho*Zpeak+M_mu, Gfull=rho*Gpeak+8*M_mu and F1=56/(5*gamma_lower), an upper bound on the nominal input integral. Then

\[ \sup H_\mu\le180X_{peak}^2+180\mu\beta/\eta,\quad Loss\le45\lambda T G_{full}^2/4,\quad W_{abs}\le45G_{full}(\rho F_1+F). \tag{17} \]

The last bound also covers the sum of positive powers over all fine input channels integrated over time; no slot cancellation hides cost. Use positive interval upper endpoints throughout. The earlier numerical work ceilings are not inherited, and an L1 tracking bound does not supply an actuator peak or bandwidth limit. Preparation costs under the alternative and a physical fine-state energy map remain separate. All inputs and their timing are supplied; no autonomous force or event occurrence is inferred.

Constant scalar component ratios preserve the reference's exact shapes. Parasitics, nonuniform junction variation, nonlinear capacitance, phase noise and a time-dependent clock require additional terms and their own admission. This first comparison changes one scalar current amplitude; it is not a classification of possible laws or a proof that the sine profile is uniquely selected. The ontology continues to separate mathematical organization from physical identity.

6. Execution and evidence boundary

The pure bounds and raw statistic admit primitive intervals, nonnegative errors and ordered readings using shared owners. They rebuild all consumed fields; no cached margin or source report certifies the present law. The fixed corner is reserved to the exclusive analytical driver. The instrument tests and independent theory controls use other synthetic inputs and algebraic counterexamples, without assessing that corner or executing previous frozen producers.

The execution plan owns the result-dependent next gate. No physical observation or claim of practical sensor capability follows from evaluating this sufficient bound.

7. Frozen joint-box decision

Declaration, consuming helper and driver were committed at 98ccd8686592eea15ad22151802d731d683d70e2 before one assessment. The preserved attempt and result record verdict conditional_current_discrimination_admitted. Both models retain all four positive identity/collar margins, their current envelopes are strictly separated, and no unresolved condition remains within this declared sufficient-bound decision. The read-only evidence controls reconstruct its receipt, coefficient intervals, both laws' errors and identity margins, and the strict separation from archived primitives. They do not import or rerun the bound, driver or assessor.

The following summaries round outwards. Current quantities are divided by the same fixed nominal j0 only for display; the observable itself is (13).

Uniform quantity	Fixed amplitude, mu=1	Doubled amplitude, mu=2
Mean form magnitude at the nominal plateau endpoints	<0.014922	<0.019589
Phase departure from the nominal command	<9.290e-6 radians	<1.220e-5 radians
Total sample error including tracking and aperture	<1.251e-14	<1.300e-14
Corrected-increment error	<5.941e-14	<1.088e-13
Radial current-error allowance / j0	<0.030580	<0.055983
Necessary observed current norm / j0	[0.969420,1.030580]	[1.944017,2.055983]
Minimum outer acute-gap margin	>0.070775 radians	>0.070769 radians

The exact archived gap is greater than 0.913437*j0. Hence no complete valid observation record can be generated under both declared hypotheses with their permitted calibration and error boxes. The conclusion is uniform over unknown initial phases and all admitted fine deviations; the inverse has not supplied hidden phase information to test itself.

This is a finite analytical decision about prospective record sets. It generated no trajectory, physical raw record or choice of the true law. The preparation class remains a supplied premise under the alternative and under calibration changes. The absolute tolerances, inward apertures, linear component shapes and delivery bounds still need independent physical evidence. Positive mathematical tolerances do not establish that an apparatus can meet them, nor select a universal current law.