Geometry, sensor gain and a declared uncertain clock¶
Question and conditional result¶
The two-input inverse constrains an original source angle and held sensor gain under a known complete law and structural clock. Its reserved evaluation retains that premise. Here the conversion from the observation clock to structural time is an additional positive, unknown constant. The result is a necessary outer enclosure of geometry, an effective response scale, and separate gain/clock marginals. It neither proves that these marginals are jointly realizable nor determines an absolute physical clock.
Two distinctions matter. An unknown multiplier of every evolution row can be exactly confounded with clock conversion. An unknown sensor gain can instead compensate the clock in the leading short-time response, without preserving the full nonlinear response. An explicit curvature witness below proves that these are different statements.
This is a mathematical extension and implementation contract, not a new reserved experiment. It changes none of the earlier frozen protocols, source archives, responses or verdicts. The ontology permits collective measurement maps while retaining their independent operational premises; the parameter foundation requires a clock change to transform the complete law.
F1-F2: full source, observation clock and complete event law¶
Retain the eighteen-node support and full source family of the
two-input owner: two
unit C9 rings on 0,...,8 and 9,...,17, joined at (0,9) and (1,10).
Let \(M\) be the degree matrix, \(L\) its graph Laplacian,
\(A=M^{-1}L\), and \(q=e_4-e_5\). The source is
where \(P_Mu=u\), \(P_Mv=v\), \(\|u\|_M\le X\), \(\|v\|_M\le Y\). The original priors obey
The affine phase family, its single weighted centering, continuous lifts and original acute guard are unchanged. All thirty-six coordinates, including arbitrary common means and receiver geometry, remain in the model. An uncertain clock does not authorize a different preparation, recentered intermediate state or hidden clock-dependent residual.
In fast structural time \(\tau\), retain the complete supplied law
Support, unit capacity and both coefficients are held premises of this normalized model. They are not inferred from the readings.
Let \(s\) be the declared observation clock, with common origin zero, and
The same constant \(\rho\) applies to both windows. Its units are structural-time units per observation-time unit. In fixed chosen units the equations consumed by an observation-clock trajectory are therefore
Multiplying only the form row, changing \(\gamma\) alone, or treating \(\rho\) as a measured phase frequency gives a different model. The general clock owner also explains why state-dependent and varying conversions require further laws. They are outside this constant-conversion admission.
Supply cumulative amplitudes \(0<a_1\le a_2\le1\). Apply the phase jump \(a_1q\) at \(s=0\), evolve (4) until \(s=H>0\), then apply \((a_2-a_1)q\) and evolve until \(s=2H\). Both jumps leave form unchanged. There is no other forcing or support event. Their structural times are \(0,\rho H\); all three observation times are \(0,\rho H,2\rho H\). The full first endpoint starts the second window. Both common means remain conserved under these declared rows and events, irrespective of \(\rho\).
The three recorded scalars use one held sensor:
Here \(x(jH)\) means the solution in the observation clock. Reading intervals \([r_j^-,r_j^+]\) enclose those recorded values; their exact midpoints \(c_j\) and half-widths \(\epsilon_j\) differ from the sensor allowance \(\delta\). One offset \(O\) and one middle reading are retained. The units of \(G\) are readout units per form unit; \(J=G\rho\) has the additional clock-conversion factor and is not itself a calibrated sensor gain. No laboratory identification of \(s\), the input amplitudes or this affine sensor is supplied.
The gate admits any positive compact clock prior with
The range \([1/2,2]\), for example, is an admissible supplied uncertainty when \(H\le1/4\); it is not a measured clock range or a universal bound.
F3: one uniform envelope for the entire uncertain history¶
For a structural window length \(0<h\le h_*\), write the earlier complete-flow bounds as functions of their duration:
The denominator is positive on (6). The numerator is nonnegative and nondecreasing, while the denominator is positive and nonincreasing. Thus \(Q(h)\), and consequently
are nondecreasing. This is a bound on the complete original-source trajectory, including the first-window form and phase evolution carried into the second. It assumes neither a reset nor later acuteness.
For the actual \(h=\rho H\), put \(\xi=\rho/\rho_+\in(0,1]\). Monotonicity gives the useful normalized error bound
The actual form and dynamical phase displacement on both windows obey \(\|P_Mx\|_M\le Q(h_*)\) and \(\|d\|_M\le4gh_*Q(h_*)\). In particular, the horizon used to bound all hidden evolution is the maximum structural horizon, not \(2H\) unless the chosen units make \(\rho_+=1\).
Let \(y_j=q^T[x(jH)-x((j-1)H)]\). The inherited finite response identity and (9) imply
Consequently, with the auxiliary radial scale
the true recorded increments satisfy
These are precisely the necessary algebraic inequalities consumed by the existing two-input enclosure, with duration \(h_*\) and radial prior (11). This justifies sharing that calculation. It does not assert that the actual readings were produced by another trajectory running at \(\rho_+\), by sensor gain \(\ell\), or by a newly prepared source. The auxiliary report is a constraint envelope with that explicit meaning.
For either exact row \(w=(w_1,w_2)\) of the inherited inverse matrix, the radius remains
The common offset cancels in exact midpoint differences first. The middle reading/error and the factor-based rank checks are unchanged. The original source acute guard is required to interpret its geometry; the sufficient whole-window guard at \(h_*\) stays optional.
F4: necessary geometry, effective scale and quotient marginals¶
Apply the auxiliary two-input enclosure only after admitting the original primitives and (3), (6). It returns necessary bounds on \(b\), the original actual long-arc mean \(B=b-(v_1-v_0)/8\), and \(\ell\). The actual-angle expansion remains \(Y/8\), using the original phase radius, independently of the clock uncertainty.
If \(\ell\in[\ell_-,\ell_+]\) survives the auxiliary projection, then define an outward effective-scale enclosure
Every compatible sensor gain and clock belongs to the necessary positive-product constraint
Its separate coordinate projections are
The implementation retains outward bounds and the relation \(J=G\rho\). It does not fill the rectangle between these projections with compatible trajectories. Even \(\mathcal P\) discards correlations with geometry, the complete hidden state and the three errors. Nonempty marginal intervals prove no joint existence or exact point identification.
A narrow effective scale may coexist with wide gain and clock bounds. For example, even the exact product \(J=3/2\), combined only with \(G\in[1,2]\) and \(\rho\in[1/2,2]\), permits the entire positive-product curve with \(G\in[1,2]\) and \(\rho\in[3/4,3/2]\). This is an algebraic limitation of this necessary envelope; it is not an exact full-response collision between all those pairs. A singleton clock prior recovers the known-clock calculation at \(h=\rho H\), up to outward projection arithmetic.
Equal amplitudes or unresolved positive matrix factors retain
unavailable, distinct from exclusion. A strict necessary exclusion
gives incompatible; bounded_candidate means only the outer constraints
survive. No clock grid, selected best fit, regenerated response or
cached incoming report supplies the result.
A finite resolution regime with a broad clock prior¶
A conditional arithmetic example shows that uncertain clock conversion need not erase all geometric information. Choose
Then \(h_*=2^{-21}\) and the auxiliary radial prior is \([1/4,2]\). The earlier conditioning calculation bounds the Euclidean norm of the two ideal coordinate radii by
The upper auxiliary gain remains two, so that radius calculation is unchanged. Its lower gain is now \(1/4\). For compatible readings the rectangle contains a true vector of norm at least \(1/4\), has diameter at most \(2R\), and therefore stays at norm at least \(1/4-2R>0\). It also stays in the first quadrant: the true first coordinate exceeds \(1/64\) and the second exceeds \(9/40\), both larger than \(2R\). The norm and argument gradient bounds consequently give
Prior intersections cannot enlarge those marginals. These are conditional exact-arithmetic budgets for compatible observations, not evaluated readings. A numerical report must separately admit outward coefficient and projection arithmetic. No comparable narrow separate \(G\) or \(\rho\) guarantee follows, as the product example above demonstrates.
Exact equivalence with a common multiplier of every row¶
Consider a separately specified extension of (2): a positive constant \(\lambda\) multiplies its entire vector field. If \(z=(x,\theta)\) and \(F\) denotes (2), the observation-clock model is
Retain the same full source, support, held sensor \(G,O\), input amplitudes, and events scheduled at the same observation-clock times. Any two admissible parameter pairs with \(\rho_1\lambda_1=\rho_2\lambda_2\) have exactly the same right-hand side and initial state. Uniqueness gives identical full states up to the first event; the same jump map gives identical post-event states. Repeating this argument gives identical full histories and all noiseless readings. Coupling the same allowed additive errors preserves the recorded readings.
Equivalently, \((\rho,\lambda)\mapsto(k\rho,\lambda/k)\), for any positive \(k\) admitted by the respective priors, is an exact equivalence. Where a prior fiber contains multiple pairs, no observation of these identical state histories can distinguish its individual factors. The product \(\rho\lambda\) is an invariant of this equivalence; this proof does not establish that the selected scalar readings identify even that product. A singleton prior or an additional independent clock reference can alter the identification problem.
This argument requires the complete-row multiplier, common preparation and observation-clock event convention. Holding events at fixed structural times instead generally changes their observed schedule. A changed relative loss/exchange coefficient, capacity law, source, forcing or sensor is not covered by (17). The current inference API retains \(\lambda=1\); it must not interpret its fixed \(\gamma\) as a free global multiplier.
Equal leading gain-clock products need not preserve the full response¶
Here is an exact counterexample inside the fixed-law source domain, with no numerical trajectory evaluation. Choose
The nominal donor short angle is \(4\pi-8b=1\); receiver long angles are \((2\pi-3/4)/8\) and the port origin difference is \(1/8\). Together with the stated long donor and short receiver angles these all lie strictly inside the original acute chart and priors. Common form and phase means can be arbitrary and equal in both compared preparations. Any later declared second event is irrelevant to this right-hand local calculation before \(H\).
Immediately after the first phase jump, write \(f=f(\theta^-+a_1q)\). The four local rows, all of degree two, satisfy
Other rows, including both ports and the receiver, retain their actual values; they are not set to equilibrium. Locality of \(A\) gives
Since the initial form is constant, \(Ax(0)=0\), and the complete structural rows imply \(x_\tau(0^+)=\gamma f\), \(\theta_\tau(0^+)=0\), and \(x_{\tau\tau}(0^+)=-\gamma Af\). For the noiseless held readout \(r(s)=Gq^Tx(s)+O\), therefore
These are derivatives on the actual complete flow after the event, not derivatives of a reduced fitted response or of the discrete sensor errors.
Compare \((G,\rho)=(3/2,1)\) with \((1,3/2)\), using exactly the same source, first jump and offset. Both pairs fit the priors \(G\in[1,2]\), \(\rho\in[1/2,2]\), have the same initial reading and slope, and satisfy \(G\rho=3/2\). The second pair's curvature exceeds the first by \(9\gamma F_0/8>0\). Its readout minus the first consequently has leading term \((9\gamma F_0/16)s^2\) as \(s\downarrow0\), so it is nonzero for every sufficiently small positive time before the second event. The product-preserving gain/clock change is not an exact response symmetry.
For this exact preparation with exact continuous derivative access, \(-2r''(0^+)/[3r'(0^+)]=\rho\). That observation is conditional on the known geometry and vanishing form residual; it does not provide a uniform clock estimator for uncertain sources or three noisy samples. Unknown initial form, phase residuals and later phase feedback enter the general curvature. The witness rules out an overly strong impossibility claim; it does not prove complete finite-record identifiability.
Implementation, evidence and continuation boundary¶
The clock inverse admits the observation duration, clock/gain priors and the original source/reading primitives, then constructs the auxiliary constraint envelope through the existing two-input owner. Its report distinguishes the observed clock, maximum structural horizon, normalized radial scale, effective response scale, sensor-gain marginal and clock marginal. Exact rational endpoint arithmetic precedes interval materialization; an extremely small positive clock is not replaced by a zero denominator. The contract owns the executable domains and availability rules, and the tests check independent full-state responses and analytic controls rather than reevaluating any reserved source.
This owner extends one fixed nonlinear source/measurement problem. The earlier bridge clock/law discriminator uses different preparations and observations, and the mode/pole identification uses its own modal closure and measurement premises. Their methods motivate careful separation of rate and readout, but their coefficients or verdicts do not transfer here.
The new enclosure can lose resolution while remaining sound. It does not claim that gain/clock ambiguity is fundamental to every observation, that the exact common-rate equivalence selects a physical law, or that event funding, formation, recovery or maintenance follows from an inferred geometric interval. A physical clock and gain need their own measurement bridge. The execution plan alone owns admission of further derivative bounds or reserved responses.