Sine environmental state and causal memory

Complete sine mediator geometry, cancellation, conserved hidden inventory, exact causal elimination and conditional state/capacity inference with its sampling boundaries.

Part of Native pattern reduction and memory. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

Common mediator geometry does not select its transmitted pressure

The complete-law comparison permits a direct test of whether native argument pressure is an effective version of primitive pairwise sine exchange. Retain one supplied intermediary m, attached to k>=2 visible ports, and any reciprocal unit edges among visible nodes. Use the sine law with held capacities, nu_m=mu>0 and e>=0,w,beta>0. Visible degrees retain their original mediator incidence. No edge is created by the following elimination.

Set Z=sum_p exp(i*theta_p)=R*exp(i*Psi) with R>0. The conditional storage minimum is again x_m=bar(x),theta_m=Psi, now without requiring every incident gap to be acute. It is the unique minimum modulo a turn, not the only stationary state: theta_m=Psi+pi is a maximum of the hidden phase cost and a saddle of the hidden dynamics when e>0. At R=0,x_m=bar(x) every hidden phase is stationary and no direction is selected.

At the minimum the visible gradients are

\[ \widetilde q_p=q_{{\rm internal},p}+x_p-\bar x,\qquad \widetilde S_p=S_{{\rm internal},p} +\frac{\sum_r\sin(\theta_r-\theta_p)}R , \]

with internal gradients unchanged at nonports. The visible law is

\[ \dot x_i=\nu_i(-e\widetilde q_i/d_i+ w\widetilde S_i/(\pi d_i)),\qquad \dot\theta_i=w\nu_i\widetilde q_i/(\beta\pi d_i). \]

This is the full sine field evaluated at the conditional minimum. In particular, the transmitted phasor contributes sin(Psi-theta_p), not Arg(Z) as a force. Where the complete reconstructed native state is also admitted, both primitive candidates have the same conditional geometry and inherited storage,

\[ E_{\rm eff}=E_{\rm internal} +\frac1{2k}\sum_{p<r}(x_p-x_r)^2+\beta(k-R). \]

The hidden gradients vanish, so the envelope chain rule and visible exchange cancellation give Edot_eff=-e*sum_visible nu_i*q_tilde_i^2/d_i. This is a balance of the conditional reduced equation. It is not equality to a general finite-capacity full trajectory's storage.

Collective response survives primitive additivity. The previous nonpairwise storage proof applies to k=3 unchanged. The actual sine port response is also collective. In the same three-C5 preparation, hold A fixed, rotate B and C by s,t, keep all forms zero and visible capacities one. With Psi=Arg(1+exp(i*s)+exp(i*t)),

\[ \dot x_A=\frac w{3\pi}\sin\Psi,\qquad \left.\partial_s\partial_t\dot x_A\right|_{s=t} =-\frac{2w\sin t(1-\cos t)(2+\cos t)} {3\pi(5+4\cos t)^{5/2}}<0\quad(0<t<\pi/4). \]

Independent pair responses would have zero mixed derivative. The native mixed derivative in the preceding subsection is positive. Thus a primitive additive law can yield effective collective interaction, but this mediator does not make the two effective laws equal. The sign comparison uses a frozen family of initial states and instantaneous fields, without fitting or running a new response. It is not a physical identification.

Finite capacity retains a tracking defect. Let u=x_m-bar(x),v=theta_m-Psi on a local lift, r=R/k and alpha_p=theta_p-Psi. The exact moving-boundary equations are

\[ \begin{pmatrix}\dot u\\\dot v\end{pmatrix} =\mu\begin{pmatrix}-eu-(w/\pi)r\sin v\\wu/(\beta\pi)\end{pmatrix} -\begin{pmatrix}\dot{\bar x}\\\dot\Psi\end{pmatrix},\qquad \dot\Psi=\frac1R\sum_p\cos\alpha_p\,\dot\theta_p . \]

On u=v=0 the defect is the negative velocity of the moving conditional minimum, generally nonzero. For example, on a two-port path take endpoint (x,nu)=(1,1),(0,2), all phases zero and x_m=1/2. The hidden rows vanish, but (udot,vdot)=(-e/4,w/(4*beta*pi)). A freshly minimized environment therefore does not remain minimized merely because its instantaneous hidden rates vanish.

A damped-pendulum equation follows from these two rows. Freeze the visible boundary, for example by zero visible capacity, and use fast time tau=mu*t. Eliminating u gives exactly

\[ \frac{d^2v}{d\tau^2} +e\frac{dv}{d\tau} +\frac{w^2R}{\beta\pi^2 k}\sin v=0,\qquad \frac{d}{d\tau}\left[\frac k2u^2+\beta R(1-\cos v)\right] =-ek u^2 . \]

This damped-pendulum form is derived from the supplied nodal comparison; no oscillator is added to it. Its form alone does not establish oscillatory decay. A moving environment supplies the explicit boundary terms above. At the minimum the fast exponents solve

\[ \lambda^2+e\lambda+\frac{w^2r}{\beta\pi^2}=0 \quad\hbox{(sine)},\qquad \lambda^2+e\lambda+\frac{w^2}{\beta\pi^2r}=0 \quad\hbox{(native)}. \]

The native comparison retains admission of every visible resultant; a positive hidden resultant alone does not supply that premise. For R>0,e>0 the minimum attracts a sufficiently small hidden perturbation; oscillatory versus real decay depends on the discriminant. Decreasing resultant strength softens the sine restoring term, whereas the native tangent term grows. At cancellation the sine full field is still smooth, but its conditional direction and uniform attraction are lost. A singular reduced description can therefore coexist with regular fine dynamics; that fact does not derive native Arg pressure or its singular response.

A controlled fast limit can be proved with the existing method. At e=w=1/2,beta=1 require |alpha_p|<=1/4,|v|<=1/4 in a compact smooth tube about the new reduced visible solution. Then r*sinc(v)>=2945/3072. Relative to the existing A_0 and P in the fast-mediator proof, the sine remainder obeys ||R_s(z)||<=127*||z||/18432. This follows from 1-r*sinc(v)<=127/3072 and 1/(2*pi)<1/6. Since ||P||<14, 2*||P||*127/18432<1/2, which reproves the same sufficient quadratic decay estimate for this different law.

With W=sqrt(z^T*P*z), newly justified bounds B,L for its moving boundary and visible field, and the same first-exit hypotheses, it follows that

\[ W(t)\le e^{-\mu t/56}W(0) +\frac{784B}{\mu}(1-e^{-\mu t/56}),\qquad \sup_{[0,T]}\|y-y_*\| \le\frac{Le^{LT}}{\mu}[56W(0)+784BT]. \]

Require W(0)<rho<1/4, 784B/mu<rho and the visible bound strictly below the chosen tube margin. These conditions prevent a first exit as in the earlier proof. Its native numerical constants B,L, trajectory, clock samples and records do not transfer. The estimate is finite-horizon, includes the initial layer and is not uniform near R=0.

The detached sine mediation owner encloses the conditional visible field, storage, work and tracking defects using captured represented inputs. It uses Z/R directly instead of rounding Psi into a new graph phase. A certified positive lower bound on R is mandatory; a zero-containing enclosure is unresolved and rejects this reduction even though the fine sine law remains defined. The controls verify the independent field, inherited work, mixed response, hidden oscillator and noninvariance. The report advances no trajectory, certifies no fast-limit tube and selects no primitive pressure law.

Finite environmental state, cancellation and causal-response reuse

The stationary-minimum reduction and the memory/chart owners support a causal description with retained environmental state. Keep the sine comparison, one supplied mediator of capacity mu>0, fixed unit incidence and the original signed form and circular phase. The following are conditional results of that complete law.

A faithful state that remains regular at cancellation

The existing cylinder encoding retains signed form and unit phase separately. Put X=mean(x_ports), Z=A+i*B=sum_p exp(i*theta_p), u=x_m-X and (c,s)=(cos(theta_m),sin(theta_m)). Direct substitution into the sine rows gives

\[ \dot u=\mu[-eu+\tfrac{w}{\pi k}(Bc-As)]-\dot X,\qquad \omega=\tfrac{\mu w}{\beta\pi}u,\qquad \dot c=-\omega s,\quad \dot s=\omega c . \]

The circle constraint c^2+s^2=1 is preserved. This is a faithful encoding of the two existing hidden coordinates, not three independent coordinates or a new law. It never divides by |Z| or chooses Arg(Z) and remains regular at cancellation. In a coupled network X,A,B and their derivatives come from the visible nodal rows; they are not automatically externally prescribed inputs. With a supplied differentiable visible path, the same identity instead describes an explicitly driven hidden subsystem.

An absolute local lift of the hidden phase obeys

\[ \ddot\theta_m+\mu e\dot\theta_m =\frac{\mu^2w^2}{\beta\pi^2k}(B\cos\theta_m-A\sin\theta_m) -\frac{\mu w}{\beta\pi}\dot X . \]

There is no derivative of an undefined mean angle here. Define H=k*u^2/2+beta*[k-(A*c+B*s)]. Its exact work identity is

\[ \dot H=-\mu e k u^2-ku\dot X-\beta(\dot A c+\dot B s). \]

The actual incident-edge storage additionally contains sum_p(x_p-X)^2/2. Its derivative is equivalently

\[ \dot E_{\rm star}=-\mu ek u^2 +\sum_p(x_p-x_m)\dot x_p +\beta\sum_p\sin(\theta_p-\theta_m)\dot\theta_p . \]

Thus boundary work is explicit; a moving environment need not decrease the hidden storage by itself. Full-network exchange retains its existing global balance.

For a frozen visible boundary with Z=0,e>0 there is an exact response:

\[ u(t)=u_0e^{-\mu et},\qquad \theta_m(t)=\theta_{m0} +\frac{wu_0}{\beta\pi e}(1-e^{-\mu et}). \]

The hidden form relaxes, but the limiting phase retains the initial hidden orientation and form impulse. Positive capacity changes the relaxation time, not this limiting phase. At mu=0 the hidden state is frozen instead; at e=0 division by damping is unavailable and that formula is not used. The infinite-time frozen-boundary conclusion requires the ports really to remain fixed, for example through zero visible capacities.

A separate instantaneous visible-closure counterexample permits active ports: take ideal port phases (0,pi), all forms zero, and hidden phase +pi/2 or -pi/2. Both hidden rates vanish and both preparations have the same visible state and storage. Yet the visible form-rate pair is (+nu_0*w/pi,-nu_1*w/pi) or its negative. Their subsequent ports need not stay frozen. This distinguishes stationary hidden state from dispensable hidden state. The exact mathematical phases in this proof are not assertions that represented float(pi) has an exact zero sine.

Static agreement hides different memory clocks

Let rho=R/k>0, a=w/pi, b=w/(beta*pi). Linearize at one frozen conditional minimum, retaining its geometry. If h denotes absolute hidden perturbations and eta=(delta X,delta Psi) the visible displacement of that minimum, the two hidden tangents are

\[ \dot h=D(h-\eta),\qquad D_s=\mu\begin{pmatrix}-e&-a\rho\\b&0\end{pmatrix},\qquad D_n=\mu\begin{pmatrix}-e&-a\\b/\rho&0\end{pmatrix}. \]

This is a linearization, not a global nonlinear closure. Full native comparison still requires every native resultant to be admitted. With zero initial hidden perturbation the Laplace response is H(z)=(zI-D)^-1*(-D). For the sine model,

\[ H_s(z)= \frac{\begin{pmatrix} \mu ez+\mu^2ab\rho&\mu a\rho z\\ -\mu bz&\mu^2ab\rho \end{pmatrix}}{z^2+\mu ez+\mu^2ab\rho}. \]

Both models have H(0)=I. Their stationary reconstruction therefore cannot discriminate them; their poles and finite responses can. Nonzero hidden initial state adds exp(Dt)*h(0) and must be retained. In an autonomous network the visible equations close the feedback around these rows; treating eta as a freely set input would be a different preparation.

For e>0 and 0<4ab*rho<=e^2 the sine slow decay rate satisfies

\[ \gamma=\frac{2\mu ab\rho}{e+\sqrt{e^2-4ab\rho}},\qquad \frac{\mu ab\rho}{e}\le\gamma\le\frac{2\mu ab\rho}{e}. \]

Hence large mu alone does not justify an instantaneous approximation uniformly near cancellation; the weak-resultant relaxation scale involves mu*rho. At the default e=w=1/2,beta=1, e^2-4ab*rho=1/4-rho/pi^2>0 for every 0<rho<=1. The local hidden sine modes at the frozen conditional minimum are therefore distinct negative real modes, not small-amplitude oscillations. This does not classify a whole network's spectrum or finite nonlinear trajectory. Native hidden modes instead become nonreal for rho<4/pi^2.

The pole ratio det(D)/trace(D)^2 cancels positive hidden capacity and a common affine clock scale: it is ab*rho/e^2 for sine and ab/(rho*e^2) for native. This is a prospective tangent discriminator at declared geometry, not a physical measurement or a derived clock.

Reuse the signed memory owner, not diffusion-only shortcuts

The full sine field has a particularly simple derivative. With B now denoting the support Laplacian and K(theta) the cosine-weighted Laplacian, its form/phase Jacobian is

\[ J_s=\begin{pmatrix} -eND^{-1}B&-(w/\pi)ND^{-1}K(\theta)\\ (w/(\beta\pi))ND^{-1}B&0 \end{pmatrix}. \]

At a fixed equilibrium this supplies the joint tangent for the existing linear-memory algebra. The native phase mobility must not be substituted into this different law. At a nonequilibrium state the displayed derivative is still correct, but freezing it does not give the exact evolving tangent along a nonlinear trajectory.

For a fixed joint tangent partitioned as y'=A_v*y+B_v*h, h'=C_v*y+D_v*h, the existing coordinate-memory owner retains B_v*exp(D_v*t)*h(0) and kernel K(t)=B_v*exp(D_v*t)*C_v exactly at the symbolic level. It evaluates rational blocks rather than a certified exponential. The same owner's output-Krylov construction already supplies a finite test for identically zero kernel: with m hidden coordinates,

\[ K\equiv0\ \Longleftrightarrow\ B_vD_v^jC_v=0\quad(j=0,\ldots,m-1). \]

Taylor coefficients prove necessity and Cayley-Hamilton proves sufficiency. Replace C_v by a particular h(0) to test its initial-source contribution; the two tests are distinct. A zero K(0) alone is insufficient. No new exponential or stability assumption is needed for this algebraic test. The static response controls reuse these owners on explicitly declared rational tangent families; their exact coefficients are not promoted to exact transcendental TNFR coefficients.

The reversible positivity/Gram criterion in EPI memory, the P5 finite-history formula and the REMESH echo law have narrower, different hypotheses. They cannot select or truncate this signed joint kernel automatically. Likewise, the Jacobi result supplies a conditional mobility restriction only when Poisson structure is an additional premise. Work cancellation alone does not select that premise. These reuse boundaries replace speculative new closures with existing state, algebra and independently checkable responses.

Conserved inventory constrains both memory and its initial source

For the preceding fixed linear block law, suppose the full quantity I=l_v^T*y+l_h^T*h is conserved for every state. There is no input, and all blocks and covectors are held. Equivalently,

\[ \ell_v^{\mathsf T}A_v=-\ell_h^{\mathsf T}C_v, \qquad \ell_v^{\mathsf T}B_v=-\ell_h^{\mathsf T}D_v. \]

The exact hidden-state reconstruction therefore retains the inventory as

\[ I(t)=\ell_v^{\mathsf T}y(t) +\ell_h^{\mathsf T}e^{D_vt}h(0) +\int_0^t\ell_h^{\mathsf T}e^{D_v(t-s)}C_vy(s)\,ds=I(0). \]

In particular the visible charge alone need not be constant. Its memory kernel and initial source satisfy

\[ \ell_v^{\mathsf T}K(t) =-\frac d{dt}\left[\ell_h^{\mathsf T}e^{D_vt}C_v\right],\qquad \ell_v^{\mathsf T}B_ve^{D_vt}h(0) =-\frac d{dt}\left[\ell_h^{\mathsf T}e^{D_vt}h(0)\right]. \]

These identities need neither stability nor an evaluated exponential. At each derivative order j>=0 they reduce to the exact matrix checks l_v^T B_v D_v^j C_v=-l_h^T D_v^(j+1) C_v and the same expression with C_v replaced by h(0). Dropping a nonzero hidden initial source changes the visible dynamics. Its contribution to the chosen inventory requires a separate projection: a charge-neutral source can still drive visible motion. For unrestricted hidden states, the full inventory is a function of the visible coordinates alone only if l_h=0; a constrained preparation can supply a separate relation, which must remain in its contract.

Even an instantaneous stationary substitution can change the accounting. If D_v is invertible, put H=-D_v^-1 C_v and A_s=A_v-B_v D_v^-1 C_v. Substituting h=Hy gives y'=A_s y and l_v^T A_s=0. But the reconstructed full inventory has covector l_eff^T=l_v^T+l_h^T H, and l_eff^T A_s need not vanish. The stationary hidden row is zero, whereas its reconstruction moves at H A_s y. This graph is invariant for every visible state only if C_v A_s=0.

For an explicit same-law tangent control, take unit P4 with unit capacities at sine consensus, e,a=w/pi,b=w/(beta*pi)>0, and eliminate node 1's form and phase. Retain nodes (0,2,3), with all forms before all phases. Its stationary-substitution generator and form covectors are

\[ T=\begin{pmatrix}-1/2&1/2&0\\1/4&-3/4&1/2\\0&1&-1\end{pmatrix}, \qquad A_s=\begin{pmatrix}eT&aT\\-bT&0\end{pmatrix}, \]

\[ \ell_v^{\mathsf T}=(1,2,1,0,0,0),\qquad \ell_{\rm eff}^{\mathsf T}=(2,3,1,0,0,0),\qquad \ell_{\rm eff}^{\mathsf T}A_s= (-e/4,-e/4,e/2,-a/4,-a/4,a/2)\ne0. \]

The extra weights account for the eliminated form x_1=(x_0+x_2)/2. Thus apparent conservation in the substituted visible system need not preserve the original inventory, even when the initial hidden state satisfies that stationary relation. The static response controls use the shared block owner and independently declared exact rational coefficients to check this counterexample and the hidden initial source. This is a held-coefficient tangent statement, not a nonlinear elimination theorem or a contradiction of the controlled fast-mediator approximation, which retains its finite-capacity error and initialization obligations.

An autonomous three-node crossing closes the moving-environment obstruction

The preceding frozen-boundary result does not require a new simulation to test whether autonomous neighbors can invalidate instantaneous minimization. Reuse the reflection/uniqueness method on the smaller path 1--h--2, with unit edges, equal endpoint capacities nu>0 and any fixed finite hidden capacity mu>0. Use the complete sine comparison with held e,w,beta>0 and no input or support event.

On a local phase lift, remove the common form and phase offsets by defining

\[ m=(x_1+x_2)/2-x_h,\quad b=(\theta_1+\theta_2)/2-\theta_h,\quad p=(x_1-x_2)/2,\quad a=(\theta_1-\theta_2)/2 . \]

Direct projection of the full nodal rows gives the exact four-coordinate quotient

\[ \begin{aligned} \dot m&=-(\nu+\mu)[em+(w/\pi)\cos a\sin b],& \dot b&=(\nu+\mu)wm/(\beta\pi),\\ \dot p&=\nu[-ep-(w/\pi)\cos b\sin a],& \dot a&=\nu wp/(\beta\pi). \end{aligned} \]

This is an exact symmetry quotient for this support and equal endpoint capacities, not a closure of arbitrary regional averages. The half-angle coordinates retain their chosen lift; adding an endpoint turn changes both a and b consistently. Its storage and loss are

\[ E=m^2+p^2+2\beta(1-\cos a\cos b),\qquad \dot E=-2e[\nu p^2+(\nu+\mu)m^2]. \]

The subspace m=b=0 is invariant. If x_h=theta_h=0 initially, both hidden rows remain exactly zero. The endpoint motion then obeys pdot=nu*(-e*p-w*sin(a)/pi), adot=nu*w*p/(beta*pi) independently of hidden capacity. The hidden relative resultant is z_h=2*cos(a).

One rational preparation and an explicit finite crossing bound

Fix e=w=1/2,beta=nu=1 and the exact rational preparation

\[ (x_1,x_h,x_2)=(1,0,-1),\qquad (\theta_1,\theta_h,\theta_2)=(3/2,0,-3/2). \]

All initial edge gaps are acute. Until a=pi/2, and while p>=1/2, adot=p/(2*pi)>0 and

\[ \frac{dp}{da}=-\pi-\frac{\sin a}{p}\ge-\pi-2>-\frac{36}{7}. \]

Using 3<pi<22/7 gives pi/2-3/2<1/14 and hence

\[ p>1-\frac{36}{7}\frac1{14}=\frac{31}{49}>\frac12. \]

Thus p cannot reach 1/2 first. The globally smooth full sine law and adot>31/308 force a crossing at

\[ 0<T<\frac{22}{31},\qquad z_h(T)=0,\qquad \dot z_h(T)=-p(T)/\pi<-\frac{31}{154}. \]

This is a rigorous finite-time sign change from a fully specified autonomous preparation, not an inferred crossing from a numerical step. Mathematical pi/2 identifies the event surface; no represented graph phase is rounded to that value. Full-state rates stay finite through it.

Before the crossing the hidden state is its exact conditional minimum and both tracking defects vanish identically. Immediately after it, that same hidden phase is the conditional maximum: the minimum has moved from 0 to pi while the actual hidden phase remains 0. Its phase storage exceeds the minimized value by 4*beta*|cos(a)|. This happens for every fixed finite mu>0, however large. An imposed switch to the new minimum would add an undeclared phase jump. With visible state held at the event, the chosen +pi switch changes the conserved lifted weighted phase sum by 2*pi/mu. Any odd-pi representative changes that lifted sum by a nonzero odd multiple of 2*pi/mu; this weighted lift is not a phase sum with a generally defined torus modulus.

This proves that zero tracking defect and high capacity are insufficient without a uniform attraction/resultant margin. Both conserved weighted means remain zero on the actual reflected solution, so conservation does not remove the obstruction.

The same initial state is a native boundary-access control, within that law's admitted interval only. Its reflected rows are

\[ \dot p=-p/2-a/(2\pi),\qquad \dot a=\frac{pa}{2\pi\sin a},\qquad \frac{dp}{da}=-\pi\operatorname{sinc}a-\frac{\sin a}{p}. \]

The same lower bound on p and adot>=p/(2*pi) give a native endpoint before 22/31 as well. Its full law is undefined at the hidden zero resultant. This does not identify the two trajectories or their crossing times, and gives no native continuation convention.

Transient cancellation, not persistent formation

For the sine preparation, E(0)=3-2*cos(3/2)<3 and Edot=-p^2. The reflected trajectory cannot reach a=+-2*pi/3, where phase storage is three. Its sublevel component is compact in (p,a) on that lift. The only invariant subset of zero loss is p=0,sin(a)=0, hence p=a=0. LaSalle therefore gives eventual consensus on this reflected branch. The cancellation crossing is a transient failure of instantaneous elimination, not evidence of a newly maintained pattern.

Along the reflected orbit the transverse variational block is

\[ (\nu+\mu)\begin{pmatrix} -e&-(w/\pi)\cos a(t)\\ w/(\beta\pi)&0 \end{pmatrix}. \]

When cos(a)<0 its frozen restoring sign is reversed. The moving reference does not permit an all-future instability conclusion from those instantaneous eigenvalues alone. For each fixed finite mu, continuity of the smooth flow and transversality preserve a nearby cancellation crossing under small initial perturbations: the full path has z_h=2*exp(i*b)*cos(a) even outside reflection. No perturbation radius uniform in unbounded capacity or nonlinear amplification bound is claimed.

The static path controls compare the exact quotient and work identities against the full sine field, and check the analytic rational bounds. The comparison report's opt-in resultant_kinematics() reuses the existing chain-rule owner: every ideal sine phase rate is a_i/pi with exact rational a_i=(w/beta)*nu_i*q_i/d_i, so one common pi division follows the rational directional calculation. It retains ideal-rate provenance and admits kinematics at cancellation without changing the native runtime. These instantaneous bounds are not the finite-time crossing proof or a solver.

Exact causal environmental pressure with retained internal state

The cancellation obstruction rules out unqualified instantaneous-minimum replacement. It does not obstruct controlled reductions under their stated uniform margins or an exact causal representation. Keep the declared normalized-sine law on fixed simple unit support, held capacities, no source or support event, and e>=0,w,beta>0. Let one supplied hidden node h have k>=2 visible neighbors P and capacity mu>=0. All other nodes remain visible, including their internal edges. No equal-capacity, reflection, nonzero-resultant or stationary-minimum premise is needed.

Exact interface and a sufficient hidden state

Write y=x_h, theta=theta_h, X=sum_P x_p/k, a=w/pi, b=w/(beta*pi) and T=sum_P sin(theta_p-theta). The already derived hidden rows are

\[ \dot y=-\mu e(y-X)+\frac{\mu a}{k}T,\qquad \dot\theta=\mu b(y-X). \]

They retain two coordinates, with circular phase or a continuous chosen lift. In particular, T uses the actual hidden phase, not the argument of the neighbor resultant. For each port p the environmental pressure and phase-rate contribution are

\[ P^h_p=-\frac{e}{d_p}(x_p-y) +\frac{a}{d_p}\sin(\theta-\theta_p),\qquad V^h_p=\frac{\nu_p b}{d_p}(x_p-y). \]

The original degree d_p includes the hidden edge. Let q^V_p=sum_(j visible neighbor of p)(x_p-x_j) and S^V_p=sum_(j visible neighbor of p)sin(theta_j-theta_p). The internal contributions are P^V_p=(-e*q^V_p+a*S^V_p)/d_p and V^V_p=nu_p*b*q^V_p/d_p. Therefore

\[ \dot x_p=\nu_p(P^V_p+P^h_p),\qquad \dot\theta_p=V^V_p+V^h_p. \]

Nonports keep their original rows. This decomposition exactly reproduces the complete fine field, including ports that also share visible edges. Removing the hidden edge from degree normalization would change the law. The hidden node is retained as state; no live support mutation, independent pair approximation or new pressure postulate occurs.

For mu>0 the change to hidden phase and its velocity v=dot(theta) is reversible: y=X+v/(mu*b). The equivalent second-order row is

\[ \ddot\theta+\mu e\dot\theta =\frac{\mu^2ab}{k}T-\mu b\dot X, \qquad v(0)=\mu b[y(0)-X(0)]. \]

This is a same-information representation, not removal of a degree of freedom. At mu=0 that inversion is unavailable: both hidden coordinates freeze and their supplied values still affect the ports.

Derivative-free nonlinear memory

Set lambda=mu*e and

\[ F_\lambda(t)=\int_0^t e^{-\lambda s}\,ds =\begin{cases}(1-e^{-\lambda t})/\lambda,&\lambda>0,\\ t,&\lambda=0.\end{cases} \]

Here theta(t) denotes the continuous lift starting at supplied theta_0. Variation of constants in the form row, followed by integration of the phase row, gives

\[ \begin{aligned} y(t)&=e^{-\lambda t}y_0+ \int_0^t e^{-\lambda(t-s)} \left[\lambda X(s)+\frac{\mu a}{k}T(s)\right]ds,\\ \theta(t)&=\theta_0+\mu bF_\lambda(t)y_0 -\mu b\int_0^t e^{-\lambda(t-s)}X(s)\,ds +\frac{\mu^2ab}{k}\int_0^t F_\lambda(t-s)T(s)\,ds . \end{aligned} \]

The second equation is a causal nonlinear Volterra equation: T(s) depends on the unknown hidden phase at that same earlier time and on the visible phase history. It is not a convolution with a fixed linear memory kernel or a formula using future response. Together with the visible rows and the supplied initial hidden state it is equivalent to the original autonomous network. If the visible path is independently prescribed instead, it represents an explicitly driven subsystem. These two uses must not be interchanged.

The identities follow by integrating the form row and swapping finite continuous integrals; conversely differentiation with the given initial values recovers both hidden rows. The globally Lipschitz complete sine field supplies uniqueness. Thus the representation inherits its well-posedness, including Z=sum_P exp(i*theta_p)=0, without division by Z or a resultant-argument branch convention. It also covers e=0 through F_0 and mu=0 through its zero coefficients. No fitted memory timescale has been introduced: lambda is fixed by the supplied capacity and dissipation. A common constant form shift cancels between y_0 and X in the phase formula because F_lambda(t)=integral_0^t exp(-lambda*(t-s)) ds.

Linearizing this exact realization at a frozen conditional minimum recovers the preceding D_s block and its hidden initial source. The signed tangent memory owner remains reusable in that stated limit; replacing T by a fitted linear kernel would require a separate error argument.

Work must be accounted for across the interface

Let the incident-edge storage be

\[ E_h=\sum_{p\in P}\frac{(x_p-y)^2}{2} +\beta\sum_{p\in P}[1-\cos(\theta_p-\theta)]. \]

Differentiating the actual edges gives the same boundary-work identity as the cylinder calculation:

\[ \dot E_h=-\mu e k(y-X)^2+\mathcal W_P,\qquad \mathcal W_P=\sum_P(x_p-y)\dot x_p +\beta\sum_P\sin(\theta_p-\theta)\dot\theta_p . \]

The boundary rates are the full port rates, including visible internal neighbors. E_h need not decrease independently. In the full-network identity the visible loss is e*nu_p*(q^V_p+x_p-y)^2/d_p, not the sum of separately squared internal and environmental gradients. Discarding that cross term or the boundary work creates a false passivity claim for the interface.

A non-reflected initial-state discriminator

Take a path with ports (x_1,theta_1,nu_1)=(1,0,1) and (x_2,theta_2,nu_2)=(0,1/2,2), hidden form y=1/4 and fixed mu>0. Compare hidden phase 0 with 1/2 under the same coefficients. These are rational, non-reflected preparations with identical visible states. Switching the hidden phase from 0 to 1/2 changes the visible form rates by

\[ (\Delta\dot x_1,\Delta\dot x_2) =a\sin(1/2)(1,2), \]

both strictly positive. It changes the hidden form rate by -mu*a*sin(1/2). Every phase rate is unchanged; the hidden phase rate is -mu*b/4. This follows directly from the pressure law before evaluating any trajectory. Hence an instantaneous visible-only pressure cannot represent both preparations. Retaining phase is necessary here, while the exact causal representation predicts the distinction. This is not physical identification or a theorem that one primitive law is unique.

Shared implementation and evidence scope

comparison.mediated_pressure(mediator=...) derives a SineMediatedPressure report through the existing sine mediation owner. It retains the full captured comparison, actual hidden state, original port incidence, separated pressure/phase contributions and memory coefficients. It computes reconstruction residuals, hidden acceleration and incident storage/work from the existing field and interval kernels. The report does not replace the hidden state by a minimum or claim to evaluate the nonlinear memory integrals.

The independent controls check the non-reflected response, internal-edge normalization, full-field reconstruction, cross-term work, degenerate capacities, phase cancellation and exact export. This integrates the reusable instantaneous realization; the memory/equivalence theorem supplies its continuous interpretation. No numerical trajectory, native execution switch or primitive connection creation is implied.

Prior visible observations and hidden-state observability

The causal representation requires initial hidden form and phase. Their presence in a full simulator state does not make them observable. This section fixes the same sine law, one hidden node and its known incident ports, visible internal support, coefficients, visible capacities and clock. The observations below precede any reserved response. The ideal theorem treats visible state and rates as exact; the implementation separately propagates supplied rate intervals.

Two observed rows remove the diffusive contribution

Let V_p=dot(x_p) and Omega_p=dot(theta_p) be visible port rates at the declared observation time. For an active observed port nu_p>0, the known phase row gives

\[ y_p=x_p+q^V_p-\frac{d_p\Omega_p}{\nu_p b},\qquad b=w/(\beta\pi). \]

All ports used for this reconstruction must give the same hidden form y. Substituting into the form row gives sin(theta_h-theta_p)=r_p, with the cancellation-improved expression

\[ r_p=\frac{d_p}{\nu_p a} \left[V_p+\frac e b\Omega_p\right]-S^V_p,\qquad a=w/\pi. \]

Thus the combination of observed form and phase rates removes the diffusive term without subtracting large form coordinates or inserting the reconstructed form into every phase equation. This is an identity of the supplied law, not a pressure fitted to the evaluated future. Internal gradients and currents keep the original degree d_p=degree_visible(p)+1. A different hidden incidence changes the inverse problem.

Set h=(cos(theta_h),sin(theta_h)). The remaining exact conditions are

\[ A_ph=r_p,\qquad A_p=(-\sin\theta_p,\cos\theta_p),\qquad h^\mathsf Th=1. \]

Two ports p,q with sin(theta_q-theta_p)!=0 determine h uniquely, provided all other rows and the unit-circle equality agree. A useful reference-relative form puts delta=theta_q-theta_p:

\[ s_{\rm rel}=r_p,\qquad c_{\rm rel}=\frac{r_p\cos\delta-r_q}{\sin\delta}. \]

These are sin(theta_h-theta_p) and cos(theta_h-theta_p). No inverse-trigonometric branch or normalization is needed. One active phase-rate observation plus two active form-rate observations at nonparallel phases is generically sufficient; supplying both rates at every observed port permits the simplified formula and cross-checks.

Rank, consistency and an exceptional unique branch

At rank one, the observed phase rows are equal up to sign. After checking the corresponding signs of r_p, choose one unit row A_0 and a perpendicular unit vector B_0. The circular possibilities are

\[ h=r_0 A_0\ \mathord{\pm}\ \sqrt{1-r_0^2}\,B_0. \]

For |r_0|<1 there are two global possibilities; a separately supplied branch prior may distinguish them, but an arbitrary arcsine convention may not. At |r_0|=1 there is one exceptional solution. It is not a regular inverse: perturbing the datum toward the interior splits the solutions with square-root sensitivity. For |r_0|>1 there is no circular solution. Rank deficiency alone therefore does not prove two distinct states in every case. The bounded implementation abstains from this branch inversion rather than silently selecting one.

With no active observed ports there is no information from these rate equations. A zero-capacity port must have both rates exactly zero; otherwise the data contradict this unforced law. Its phase can still affect the hidden dynamics, but its zero output supplies no state constraint here. Missing measurements and unknown capacities are not replaced by zero.

Observation geometry differs from the phase resultant

Let m count active observed phase rows, and define the derived second-harmonic resultant Z_2=sum exp(2i*theta_p) over those rows. Then

\[ A^\mathsf TA=\frac12 \begin{pmatrix} m-\operatorname{Re}Z_2&-\operatorname{Im}Z_2\\ -\operatorname{Im}Z_2&m+\operatorname{Re}Z_2 \end{pmatrix}, \qquad \lambda_\pm=\frac{m\pm|Z_2|}{2}. \]

Equivalently, by the two-column Gram determinant identity,

\[ \det(A^\mathsf TA) =\sum_{p<q}\sin^2(\theta_q-\theta_p) =\frac{m^2-|Z_2|^2}{4}. \]

For m>=2 this determinant is positive exactly when the observed rows have rank two. The ideal least-squares phasor error amplification is 1/sqrt(lambda_-); it is not controlled by the ordinary neighbor resultant |Z|=|sum exp(i*theta_p)|. Three active ports at phases 0,2*pi/3,4*pi/3 have Z=Z_2=0 and A^T A=(3/2)I: the conditional-minimum direction is undefined but hidden phase remains observable from their responses. Aligned ports have maximal |Z| yet rank one. For two ports only, exact resultant cancellation implies antipodal directions and rank one.

This is a derived observation-conditioning diagnostic. It neither adds a primitive phase coordinate nor drives a nodal row, selects an operator or supplies a new physical constant. Exact ideal-angle examples are distinct from represented numerical probes near those configurations.

What instantaneous observation still cannot identify

Hidden capacity mu does not appear in any instantaneous visible port-rate equation above. For the same hidden form/phase and visible state, every admitted hidden capacity gives the same visible first rates. Therefore even exact form/phase recovery does not identify mu or generally determine a unique future unless its value or law is independently supplied. A complete equilibrium can remain identical for every capacity. The hidden rows, and hence later visible rates, can differ. This is nonidentifiability under this information budget, not absence of a capacity effect.

These observations also do not identify the primitive pressure law among competing models. Derivative evidence and a clock/preparation bridge must be independently justified; inferred pressure from the reserved future cannot be relabeled prediction.

Sound bounded admission and shared implementation

The sine observation owner accepts a visible-only graph and declared incident ports. The hypothetical hidden node and its coordinates are not input. Shared relational admission retains signed scalar form, circular phase values, nonnegative capacity, unit support and absent Gamma. Visible components may be disconnected, but each must touch a supplied port so that the declared hidden incidence would complete a connected support. No fictitious hidden state is staged.

Paired rate intervals retain all uncertainty supplied by the caller. The observer intersects per-port form bounds, evaluates the cancellation- improved phase projections, and uses a pair whose determinant interval is separated from zero. It intersects proven component ranges with [-1,1], checks the other phase rows and tests unit-norm compatibility. It never rescales an estimate onto the unit circle.

An empty form intersection, impossible projection, or residual/norm interval excluding its required value proves inconsistency with the declared premises. The reverse does not hold: interval overlap ignores some shared-data correlations and does not prove that one joint state exists. A surviving bounded_candidate is a conditional enclosure for every consistent state, not an exact identification or existence certificate. inconsistent and unavailable remain distinct. An unresolved sine interval is not a rank-one proof; exact equal captured phases or a single active row can establish that rank. The Gram determinant report uses only active observed rows.

Source identifier, clock identifier, observation time, evidence window and forecast start are explicit. The evidence window must contain the observation time and end strictly before the forecast starts. The existing three-sample rate observer estimates the rate at its first sample, not at its window endpoint. An inferred state cannot be silently moved to the end of that window or to a later forecast start. Declared provenance is not authentication; the supplied error budget must include the derivative-estimation error.

The independent observation controls withhold hidden coordinates from inference, supply independently bounded prior port rates and check recovery, rank loss, impossible data, source isolation and uncertainty. This establishes conditional software-state observability. It does not constitute a physical measurement bridge, hidden-capacity identification or an evaluated temporal prediction.

Prior acceleration identifies capacity only through an active hidden response

Keep the same fixed support, held capacities, coefficients and clock as the causal sine model. The preceding visible-only inverse bounds hidden form and phase but its instantaneous port equations do not consume hidden capacity. Differentiating those already supplied equations supplies the missing capacity dependence without postulating a new acceleration law.

A shared tangent chain rule

Use a=w/pi, b=w/(beta*pi), y=x_h, X=mean_P x_p, and define the hidden response per unit capacity

\[ f=-e(y-X)+\frac ak\sum_{p\in P}\sin(\theta_p-\theta_h),\qquad g=b(y-X). \]

Then dot(y)=mu*f and dot(theta_h)=mu*g. The sum includes all structural ports, even ports with zero capacity and zero output rates. For visible state v and hidden state h=(y,theta_h), the full rows have the form dot(v)=F_v(v,h), dot(h)=mu*F_h(v,h), hence

\[ \ddot v=D_vF_v\,F_v+\mu D_hF_v\,F_h. \]

The sensitivity to capacity is precisely the hidden-to-visible tangent block applied to the hidden response, reusing the same geometry as the existing memory and observation calculations. This identity holds at the current state; it is not a linearized replacement for its trajectory.

Let V_p=dot(x_p), Omega_p=dot(theta_p) and c_p=cos(theta_h-theta_p). Define quantities that exclude the unknown hidden rates:

\[ D_p=d_pV_p-\sum_{\substack{j\sim p\\j\ {\rm visible}}}V_j, \qquad C_p=\sum_{\substack{j\sim p\\j\ {\rm visible}}} \cos(\theta_j-\theta_p)(\Omega_j-\Omega_p)-c_p\Omega_p . \]

The original incidence degree includes the hidden neighbor. Differentiating the form gradient and sine current gives dot(q_p)=D_p-mu*f and dot(S_p)=C_p+mu*c_p*g, so

\[ \begin{aligned} \ddot\theta_p&=\frac{\nu_pb}{d_p}(D_p-\mu f),\\ \ddot x_p&=\frac{\nu_p}{d_p} [-eD_p+aC_p+\mu(e f+a c_p g)]. \end{aligned} \]

Every acceleration is affine in mu. Visible nonports have no hidden neighbor: their first rates can be calculated from visible state and the already supplied law. Port first rates remain prior observations. Using a nonport's model-derived rate is a declared use of the law, not an additional measurement or a hidden-state read.

A second cancellation improves the inference

When both acceleration channels are observed, their combination eliminates the form-gradient rate and the hidden form-response contribution:

\[ \ddot x_p+\frac e b\ddot\theta_p =\frac{\nu_pa}{d_p}(C_p+\mu c_p g). \]

This is the differentiated version of the paired-rate cancellation in the state inverse. Computing it directly avoids duplicating uncertain terms that cancel algebraically. It supplies a complementary phase-exchange channel; it is not statistically independent of its two input measurements. Intersecting all channel enclosures is sound, but cannot recover correlations that the input intervals did not provide.

For any measured channel write A=B+mu*S. Exact known S!=0 gives mu=(A-B)/S. In particular, an active port's phase acceleration identifies capacity whenever f!=0. If f=0,g!=0, a form acceleration with c_p!=0 supplies the information instead. When both channels are observed on rank-two active port geometry, all their sensitivities vanish if and only if f=g=0: phase rows first force f=0, and the remaining form rows cannot all have c_p=0 at rank two unless g=0. This is sufficient, not a necessary rank condition for every individual preparation or measurement subset.

There is no division by mu. Informative data can therefore identify mu=0, and e=0 does not by itself prevent identification. This determines a capacity under the declared clock and law; it does not derive its value universally, establish a capacity evolution law or identify laboratory units.

Capacity information can survive phase ambiguity

A unique hidden phase is sufficient but not always necessary for capacity inference. Each already bounded projection r_p=sin(theta_h-theta_p) contributes

\[ f=-e(y-X)-\frac ak\sum_P r_p . \]

If form and these projections are known, a phase acceleration can identify capacity even when the complete phase has two branches. For example, aligned ports with r_p=0 and y-X!=0,e>0 have hidden phase alternatives separated by a half-turn, but the same nonzero f=-e(y-X). Their phase acceleration still reveals mu. This does not resolve the phase needed for a general future response.

In bounded admission, an unobserved projection or cosine can retain its proved range [-1,1]. This permits a conservative capacity bound if a sensitivity remains separated from zero. It does not fabricate a unit phase, discard inactive structural ports or certify that all independently enclosed projections share one possible phase.

Informative and blind preparations

One rational non-reflected example has hidden state (y,theta_h)=(1,0) and ports (x,theta,nu)=(0,0,1),(0,1/2,2). Here f=-e+(a/2)*sin(1/2) and g=b. Changing hidden capacity by Delta mu preserves every visible first rate but changes the phase accelerations by -Delta mu*b*f*(1,2). At the default coefficients f is nonzero; it is also nonzero when e=0. These facts can be checked before any trajectory is evaluated.

The earlier autonomous path quotient supplies a stronger blind control. Take hidden form and phase zero, endpoints (x,theta)=(1,1/2),(-1,-1/2) and equal positive endpoint capacities. The endpoint phase rows have rank two, so form and phase are identifiable, yet reflection keeps f=g=0 and the hidden state fixed for every finite held mu>=0. The entire visible temporal history is independent of that capacity, even though the endpoints move. Higher derivatives or a longer observation of this same preparation cannot identify it. Unequal endpoint capacities need not preserve that invariant branch; instantaneous blindness alone is not an all-future theorem.

Bounded prior admission and its limits

state_inference.infer_capacity(...) in the existing sine observation owner retains the prior state report and accepts supplied form and/or phase acceleration intervals on a subset of its ports. Missing channels remain unobserved. The clock and observation time must match the state evidence; both windows must contain that time and end strictly before the inherited forecast start. The report derives required nonport first rates from the visible snapshot and shared sine kernel, with explicit provenance.

The observer bounds f,g,D_p,C_p, preserves actual port incidence and constructs the form, phase and available combined exchange channels. It inverts only sensitivities certified away from zero, intersects their capacity enclosures with mu>=0, and checks all supplied channels against the surviving interval. A zero-containing sensitivity is unresolved, not proof of exact stationarity. A separated impossible residual or negative-only capacity excludes the declared premises. Inconsistent source evidence cannot become a successful capacity estimate.

A surviving bounded_candidate encloses every jointly consistent capacity; it does not prove existence, exact uniqueness under uncertain data, independence of channels or authenticated provenance. Form information may support capacity bounds while phase stays unavailable, so capacity status alone does not admit a complete predictive state. All bounds belong to the original observation time, not automatically to the forecast start.

The independent controls compute full fine-field accelerations by differentiating edge gradients and currents, withholding hidden capacity from inference. They distinguish informative, zero-capacity, no-damping, phase-ambiguous and blind cases, including visible internal edges and inactive structural ports. These are software-state observations under a supplied law, not a physical bridge or an evaluated reserved forecast.

A finite forecast from jointly admitted prior environmental evidence

The state and capacity observers supply necessary outer bounds. Their next use is a finite forecast under the same supplied normalized sine law, with the response source's hidden coordinates and capacity withheld from the predictor. Neither an overlap of inferred intervals nor a successful forward computation establishes that the original observations have a joint realization. This admission obligation precedes the reserved response.

Joint admission and a regular circular coordinate

Let Y, (C,S) and M be the inferred hidden form, relative unit-phase rectangle and nonnegative capacity interval at the actual observation time. For this preparation require C.lo>0. The existing rational argument enclosure then gives

\[ \Theta=\theta_{\rm anchor}+\operatorname{atan}(S/C). \]

Every admissible circular phase represented by (C,S) has a lift in Theta; replacing its lift by an integer turn leaves the sine-law visible response unchanged. This is a chart enclosure, not normalization of a possibly nonunit midpoint. The rectangle can contain points off the unit circle, and the resulting angle interval can enlarge the feasible set. That enlargement is acceptable for an outer forecast enclosure.

Construct one rational candidate (y_*,theta_*,mu_*) from the prior inference alone, with a declared rounding rule. Require membership in Y x Theta x M and certify that its forward form/phase first rates and all supplied acceleration channels lie wholly inside the original prior evidence intervals. The forward checks use the same full sine equations, including the original degrees and held-capacity chain rule. They do not read the source's hidden truth or fit any reserved response. If these inclusions hold, this one circular state witnesses a nonempty joint realization of the supplied evidence. If they fail, admission fails; interval overlap alone cannot repair the missing witness. This verifies mathematical compatibility, not the authenticity of the declared source.

The witness does not replace the predictive state by a point. Retain the product outer enclosure Y x Theta x M with the captured visible coordinates. It contains every jointly compatible hidden state, although it discards correlations and may be conservative. For the three-node path below, propagate the seven coordinates

\[ (x_L,x_R,y,\theta_L,\theta_R,\theta_h,\mu),\qquad \dot\mu=0. \]

An interval/jet field with this held-capacity row lets the shared validated Taylor owner propagate initial state and capacity uncertainty together. Every phase is an ordinary continuous lift of a circular state; no stationary hidden minimum or native Arg pressure enters the field. Propagation must begin at the observation time and include the complete gap to the reserved observation, including any declared forecast-start boundary. Prior error, propagated initial uncertainty and integration remainder remain separate.

A prepared source and an explicit counterfactual

Take a unit path L--h--R, with no direct visible edge, and set

\[ e=w=\tfrac12,\quad \beta=1,\quad (x_L,\theta_L,\nu_L)=(0,0,1),\quad (x_R,\theta_R,\nu_R)=(0,\tfrac12,2),\quad (y,\theta_h,\mu)=(1,0,1). \]

These exact source coordinates specify the software preparation, not predictor inputs. The predictor receives visible state and separately bounded prior first/second rates, then uses the observers and joint admission above. The reserved observable is the left-port phase at elapsed time H=1/16 in the declared structural clock.

The control retains the same initial visible and hidden coordinates but sets mu=0, freezing the mediator's form and phase. Its initial visible first rates coincide with those of the source; its accelerations generally do not. It is therefore an explicit counterfactual capacity intervention, not an equally fitting alternative to the complete prior evidence. The practical control inherits the same inferred initial-coordinate enclosure and changes only this stated capacity premise. It cannot claim to have independently fitted the source's acceleration record.

An analytic separation before any response is evaluated

Write a=b=1/(2*pi) and consider both exact arms, mu=1 and mu=0. Use the whole-time candidate tube

\[ |x_L|,|x_R|\le\tfrac18,\qquad |y-1|\le\tfrac18,\qquad |\theta_i-\theta_i(0)|\le\tfrac18. \]

Each incident form difference has magnitude at most 5/4. Since 3<pi<22/7, the full degree-normalized rows obey

\[ |\dot x_i|\le\frac{19\nu_i}{24},\qquad |\dot\theta_i|\le\frac{5\nu_i}{24},\qquad (\nu_L,\nu_R,\nu_h)=(1,2,\mu). \]

For H=1/16, these bounds permit at most 19/192 form displacement and 5/192 phase displacement, both strictly below 1/8. The usual first-exit argument therefore certifies the tube for both arms throughout this window; no evaluated trajectory is needed to establish it.

Differentiating the rows within that tube, and using |cos|<=1, gives

\[ \begin{aligned} |\ddot x_L| &\le \tfrac12\left(\tfrac{19}{24}+\tfrac{19}{24}\right) +\tfrac16\left(\tfrac5{24}+\tfrac5{24}\right) =\tfrac{31}{36},\\ |\ddot y| &\le \tfrac12\left(\tfrac{19}{24}+\tfrac{19+38}{48}\right) +\tfrac16\left(\tfrac5{24}+\tfrac{5+10}{48}\right) =\tfrac{155}{144}. \end{aligned} \]

The hidden row is exactly zero in the control, so these same upper bounds remain valid there. The left-port phase row is dot(theta_L)=b(x_L-y), hence each arm satisfies

\[ |\theta_L^{(3)}| \le \tfrac16\left(\tfrac{31}{36}+\tfrac{155}{144}\right) =\tfrac{31}{96}. \]

At the exact initial preparation the hidden response per unit capacity is

\[ f=-\tfrac12+\frac{\sin(1/2)}{4\pi}<-\tfrac{11}{24}. \]

Both arms have identical initial left phase and phase rate. Their initial phase-acceleration contrast is -b*f. Since b>7/44, this contrast is strictly greater than 7/96. Taylor's theorem, with a separate third-order remainder for each arm, therefore yields

\[ \begin{aligned} \theta_L^{(\mu=1)}(H)-\theta_L^{(\mu=0)}(H) &>\frac7{192}H^2-\frac{31}{288}H^3\\ &=\frac{137}{1179648}>\frac1{10000},\qquad H=\tfrac1{16}. \end{aligned} \]

This is an exact conditional separation theorem for the stated preparation. It supplies a prospective direction and scale; it does not automatically transfer the margin to uncertain reconstructed states. Their forecast and counterfactual enclosures must retain the prior uncertainty and certified numerical errors, and meet the separately frozen width and separation budgets before comparison with the reserved response. No response result is asserted by this derivation.

Retained prospective software response

The fixed v1 protocol was prepared, predicted and evaluated in separate invocations on 2026-10-03. The public prior contains outward first/second derivative bounds padded by 2^-30, obtained independently from the exact source's forward edge equations at t=0. These are synthetic instantaneous derivatives, with zero finite-difference truncation by construction; they are not observations acquired from time samples. The predictor receives no source hidden state or capacity.

The prior-derived grid witness passed joint forward containment. The full inferred form/phase/capacity box was retained, including hidden phase width approximately 4.85e-8 and capacity width approximately 3.56e-8. Both prediction and counterfactual were issued before the source response was evaluated. Each used eight fixed 1/128 steps, order six and 128-bit outward rational arithmetic from t=0 through H=1/16, including the declared preforecast gap to 1/32.

Frozen check	Retained outcome
Joint prior witness and full-horizon inclusion	Admitted; all three chains have eight strict Picard/Taylor steps
Maximum endpoint width at most 1e-6	Prediction <4.89e-8; control <4.85e-8; source response <2.00e-18
Issued left-phase prediction	Outward displayed interval [-0.009654176181, -0.009654176112]
Reserved left-phase response	Approximately -0.00965417614677594; its complete certified interval lies inside the issued prediction
Issued frozen-control phase	Outward displayed interval [-0.009793204164, -0.009793204102]
Response minus control greater than 1e-5	Lower bound >0.000139027955; positive and separated before response evaluation

The exact rational intervals, whole-time tubes, initial-radius propagation and remainder bounds remain in artifacts/research/relational_sine_prior_forecast/response-v1.prediction.json and response-v1.json. Sibling .protocol.json, .source-state.json and .source.zip retain the public declaration, separate source preparation and source archive. The protocol SHA-256 is 0c5bcf6497f16ec0c4e2438c3af87901b17f7d80d1809fd822563acbc7d3a882; the issued prediction file SHA-256 is 24fef37f090607b6e58d057cdeba2f76cd57a301d0ce313dc186ba35b35facda. These bind retained bytes, not authenticated chronology. Missing local artifacts remain unavailable; tests do not regenerate the producer.

This result closes one finite prior-to-future software gate under the supplied sine law. It establishes neither a physical measurement bridge nor unknown-law discrimination: the control is a changed-capacity intervention, and source and predictor share the declared dynamics and validated numerical owner. Finite sampled evidence is a separate admission obligation; the execution plan owns its next bounded step.

Finite samples, derivative errors and the remaining observation boundary

The retained forecast uses synthetic instantaneous derivatives. The following contract instead bounds first and second derivatives from the same three earlier samples. It is a conditional observation result, not a new response evaluation or a claim that suitable samples have already been acquired.

One time, one stencil and separate error sources

Let a real coordinate f have three recorded values z_j at nominal times t_0+jh, j=0,1,2, with h>0. Its actual acquisition time may differ by at most tau_j, and its value error at that actual time is at most epsilon_j. Independently assume |f'|<=B_1 on the enlarged time window containing both actual and nominal times, and |f'''|<=M_3 throughout the nominal stencil window. The mean-value bound transfers the timing error to the nominal sample:

\[ |z_j-f(t_0+jh)|\le E_j:=\epsilon_j+B_1\tau_j. \]

This transfer does not identify the actual timestamps or synchronize a physical clock. All derivative units refer to the declared common clock. Define the two forward estimates

\[ \widehat v=\frac{-3z_0+4z_1-z_2}{2h},\qquad \widehat a=\frac{z_0-2z_1+z_2}{h^2}. \]

Both refer to the initial time t_0, not the middle or final sample. Their simultaneous conditional enclosures are

\[ \begin{aligned} |\widehat v-f'(t_0)| &\le \frac{3E_0+4E_1+E_2}{2h}+\frac{M_3h^2}{3},\\ |\widehat a-f''(t_0)| &\le \frac{E_0+2E_1+E_2}{h^2}+M_3h. \end{aligned} \]

For the first formula, Taylor's integral remainder gives a Peano kernel, in relative time s, equal to (3*s^2-4*h*s)/(4*h) on [0,h] and -(2*h-s)^2/(4*h) on [h,2*h]. It is nonpositive and its absolute integral is h^2/3. For the second formula, the noiseless second difference is the average of f''(t_0+s) with triangular density s/h^2 on [0,h] and (2*h-s)/h^2 on [h,2*h]. That density has mass one and mean h; the M_3 Lipschitz bound on f'' therefore supplies the stated remainder. Applying the absolute stencil weights to each sample error proves the remaining terms.

Measurement error, clock error and differentiation remainder stay separate in the report even though their sum provides the final interval. The same sample errors affect both estimates. A rectangular pair of intervals encloses the joint possibilities conservatively; it does not declare independent errors or assert that every point of the rectangle is realizable. The observed state at t_0 has its own interval [z_0-E_0,z_0+E_0], which must not be discarded.

The existing uniform-error rate and coefficient observers are special cases with tau_j=0 and epsilon_j=epsilon. Their constants 4*epsilon/h+M_3*h^2/3 and 4*epsilon/h^2+M_3*h are preserved by the shared joint stencil. For positive epsilon,M_3 these bounds are minimized, respectively, at h^3=6*epsilon/M_3 and h^3=8*epsilon/M_3. Their minima are (6*epsilon)^(2/3)*M_3^(1/3) and 3*epsilon^(1/3)*M_3^(2/3). Reducing sample spacing indefinitely amplifies fixed observation error; it does not guarantee better inference. These optima describe the bounds, not a prescription to retune an evaluated record.

Smoothness from an independently declared nodal class

The complete normalized-sine law supplies a trajectory-free smoothness bound when its whole network, including any hidden node, satisfies independent preparation and capacity limits. Retain fixed simple unit support without isolates, held capacities 0<=nu_i<=N, fixed e>=0,w>0,beta>0, no forcing and no events. Put a=w/pi and b=w/(beta*pi). Suppose the initial form diameter max_i x_i-min_i x_i is at most D_0.

At a maximum-form node the diffusive term is nonpositive and the normalized sine current is at most one. At a minimum it gives the opposite bound. The upper Dini derivative of the diameter is therefore at most 2*N*a, even when a maximizing node changes. On 0<=t<=T,

\[ D(t)\le D:=D_0+2NaT. \]

This depends on relative form rather than an unnecessary absolute form origin. The normalized edge differences then yield the uniform bounds

\[ \begin{aligned} |\dot x_i|&\le V:=N(eD+a),& |\dot\theta_i|&\le\Omega:=NbD,\\ |\ddot x_i|&\le A:=N(2eV+2a\Omega),& |\ddot\theta_i|&\le B:=2NbV,\\ |x_i^{(3)}|&\le M_x:=N[2eA+a(4\Omega^2+2B)],& |\theta_i^{(3)}|&\le M_\theta:=2NbA. \end{aligned} \]

To obtain the second row, differentiate each form difference and each sin(theta_j-theta_i) in the supplied nodal law. Their normalized sums are bounded by 2*V and 2*Omega. Differentiating once more gives 2*A for the form sum and 4*Omega^2+2*B for the sine sum: the latter contains both the squared phase-rate difference and the phase-acceleration difference. No stationary, acute-phase or nonzero resultant premise is used. Zero capacity is admitted and the corresponding row remains zero.

These are supplied-class bounds, not reconstructed hidden-state estimates. The ceiling N includes hidden capacity; it cannot be inferred using derivative errors whose justification already presumes that ceiling. Similarly, the form diameter limit must include the hidden node. A validated whole-window tube can supply sharper bounds under its own premises, but three recorded values alone do not authenticate smoothness: A_0*t*(t-h)*(t-2*h) vanishes at all three sample times while its initial rate 2*A_0*h^2 and acceleration -6*A_0*h are unbounded as A_0 varies. This last example is an observation-level obstruction; it is not asserted to solve the sine law.

A fixed prospective derivative budget

One declared candidate uses the default e=w=1/2,beta=1 with D_0<=2,N<=2, t_0=0, h=1/4096, sample errors (0,2^-44,2^-44) and timing errors (0,2^-50,2^-50). The first visible state is independently prepared exactly; its zero error is an additional premise, not a conclusion from measurement precision. Since 2^-50<h, all acquisitions lie in the nonnegative window [0,T], T=2*h+2^-50.

Using only pi>3 in the preceding bounds gives rational upper estimates a,b<1/6, M_x<11.853637 and M_theta<3.407890. The separate source terms above imply the following outward displayed total errors:

Coordinate	First derivative at zero	Second derivative at zero
Form	<2.362e-7	<0.002897
Lifted phase	<6.830e-8	<0.0008350

The shared exact-arithmetic owner may use a sharper certified pi enclosure. This table is a prospective error budget computed without samples or a trajectory. It does not reuse the earlier 2^-30 derivative padding, inherit the earlier forecast-width gate or certify that an instrument meets these errors. Conditional composition with the inverse requires actual prior records, the stated exact visible preparation and all its other support, law and capacity premises.

Boundaries that prevent a false observation claim

The current hidden-state inverse stores visible form and phase as exact point coordinates. General noisy first samples cannot be inserted as those points. A simple same-law ambiguity is the common form shift x_i(t)->x_i(t)+c at every node: it preserves all form differences, phase dynamics and derivatives, while changing every absolute form. Finite form errors may admit both histories. Choosing the recorded center as an exact state discards such uncertainty; a downstream outer-state forecast is then no longer justified for the complete measurement class. The exact prepared anchor above avoids this issue only within its declared scope.

The common phase shift theta_i(t)->theta_i(t)+c is another exact symmetry of this law. Uncertain absolute phase therefore persists even when all phase differences and derivative evidence agree. A forecast of an absolute lifted port phase must retain its initial reference uncertainty; a relative-phase observable has a different, explicitly stated contract.

Circular samples also need a declared consistent lift or separately proved unwrapping rule. At nominal spacing h, phases differing by 2*pi*k*t/h have identical circle samples but different rates. This is another observation-level aliasing obstruction, not a claim that both functions solve a fixed supplied nodal law. Whole-window phase-rate limits can constrain lifting; treating a wrapping jump as acceleration cannot.

In particular, if circular observation errors are bounded by epsilon_j and the phase speed by Omega, the strict margins

\[ m_j=\pi-\Omega(h+\tau_j+\tau_{j+1}) -\epsilon_j-\epsilon_{j+1}>0,\qquad j=0,1, \]

are sufficient for unique nearest-increment lifting. The actual phase change between the two acquisition times has magnitude at most Omega*(h+tau_j+tau_(j+1)). Adding the two observation errors still leaves the corresponding measured lifted difference strictly between -pi and pi, so its principal circular difference selects that unique increment. An initial reference selects the common lift; its error is not removed. A certified lower bound for pi yields conservative margins, and a nonpositive margin means this sufficient condition is unresolved, not that every data record is ambiguous. This is a conditional lifting criterion; the budget observer does not unwrap or authenticate actual circular samples.

Finally, instantaneous derivative intervals are necessary consequences of the raw sample constraints. A witness accepted by admit_sine_prior proves compatibility with those relaxed derivative intervals, not that its entire trajectory passes through every sample/time/error box. Conservative inverse and forecast boxes can still enclose every genuinely compatible state, but raw-record existence remains a separate obligation. No sample source, acquisition, future response or physical bridge has been established here. The bounded result is the joint derivative contract and its explicit preparation requirements; general noisy-anchor inference and any new sample-based forecast retain their own admission.