Resonance: reciprocal dynamics, measurable response and limits

This owner distinguishes reciprocal exchange, response to a probe, autonomous motion and pattern formation under declared complete laws. Its main results concern the normalized-sine relational law with supplied finite support and held capacity. Each result states whether form loss is positive or zero; those are different constitutive premises.

Read by claim

Claim	Proof owner	Required distinction
Reciprocal work and a measurable resonance response	Complete law, equilibrium derivative, collocated response, modal examples and clock	A finite-frequency gain maximum requires a declared stable probe/readout, not necessarily complex poles
Response through a retained intermediary	Remote response	Its noncollocated proof and autonomous transient do not establish a periodic orbit
Persistent motion versus dissipation	Positive-loss obstruction and zero-loss P2 pulse	Preparation, loss and the complete law determine which result applies
Regional exchange without total loss	Finite conservative memory	Eliminated state retains memory and initial-state forcing; it does not become exact irreversible damping
Finite-horizon effective loss	Common setup, channel-covariance obstruction	Equal form/phase observation and preparation impose a nonzero error floor for positive form-only loss; this does not exclude every approximate or asymmetric reduction
Structural distinction between form and phase	Bridge response and general initial-loss obstruction	Geometry changes the observed exchange and memory; it does not select instantaneous damping or make form reconstructible from instantaneous phase
Pattern protection under admitted storage freedom	Storage-family robustness	The same critical geometry remains locally protected while its observed tangent response changes; neither the law nor formation is selected
Distinguishing laws with a finite observation	Finite bridge response	Opposite finite form preparations give disjoint response intervals with explicit preparation and reading errors; the structural clock is declared, and no physical law is selected without data
Separating a constitutive change from clock uncertainty	Finite two-channel comparison	A common constant clock factor in an independently declared interval permits disjoint response-ratio intervals; finite-time errors still depend on that interval, and the signed phase denominator must be admitted
Complete memory of the admitted bridge	Exact shell reduction and causal kernel	Hidden modes retain preparation and oscillatory exchange; the static and low-frequency coefficients describe reactive response, not a derived local loss law
Nonlinear full-state return	Invariant-volume recurrence	Almost-everywhere return gives no selected-state guarantee, exact period or deadline
Formation rather than prepared maintenance	Conservative capture boundary	A two-sided invariant family cannot be entered from outside; observation and finite-time organization remain separate questions
Geometry and compatible phase rhythms	Acquired-state handoff obstruction, invariant phase-offset families	Regional storage alone does not control environmental rates. Matched external counts and phase moments with zero internal current preserve a prepared collective family; neither formation nor perturbation stability follows. Relative phase motion is not capacity or a proved resonance response
Moving-pattern robustness and actual collective feedback	Full-state retention window, mean-contact balance	Finite maintenance retains environmental errors. Away from the exact compatible family, contact deviations change the collective pulse; its signed remainder is not a positive reservoir, a resonance measurement or a formation certificate
Reusing a spectral rhythm calculation	Generator and observation audit	Auxiliary graph waves and the actual joint nodal tangent have different modal frequencies; interference requires an excited and observable pair of modes

The scale owner owns collective-state inheritance, the prepared doubled-C5 pulse and its conditional transverse instability, and relative-geometry protection. In particular, its alternative-mobility result preserves the shared storage barrier without establishing the invariant measure needed to transfer the recurrence theorem below. A protected geometry need not preserve one waveform or prove autonomous formation.

At phase consensus, the native Arg law shares the sine law's equilibrium derivative. The full-state consensus result also admits a nonuniform-form native source and bounds the associated complete linear comparison. It does not transfer that bound to the nonlinear native trajectory. The foundation audit connects this distinction to loss selection and dynamical scale inheritance; those premises remain explicit when interpreting a resonance result.

This owner defines what a resonance claim must establish. The parameter foundation owns the state and closure premises; the pattern-memory owner owns the complete sine state, storage and recovery results. The execution plan remains the sole research queue.

1. Distinguish the mechanism from the response

The following statements are different mathematical claims.

Claim	Required evidence
Reciprocal form--phase exchange	A complete pair of evolution rows, their actual coupling and compatible storage balance
Oscillatory local mode	A nonreal temporal pole of the correct equilibrium derivative
Causal oscillatory response	An excited oscillatory mode visible in the declared output; a nonzero transfer alone need not oscillate
Frequency-selective response	A declared stable input/output transfer whose gain is greatest at a nonzero frequency
Autonomous periodic pulse	A nonstationary state returning exactly after a fixed positive period under the specified autonomous law
Nonlinear recurrence	Arbitrarily close full-state returns along unbounded times, under the stated pointwise or measure-theoretic hypotheses; no exact period follows
Parametric pulse instability	A transverse unstable return multiplier of the full time-periodic variational law; a response gain peak alone does not prove it
Resonance operator RA	The registered event's live admission, configured update and execution evidence

Here operational resonance means the fourth statement: a positive finite-frequency gain maximum relative to a specified probe and readout. It need not be sharp, large or accompanied by an oscillatory free transient. Calling an interaction reciprocal does not by itself establish that response, and a named operator does not establish any modal theorem.

Every return claim must specify the retained state and observation. In the exact doubled-C5 pulse, labeled fine nodes return after T, the complete unordered-pair state returns after T/2, and the collective means stay constant. These are different observations of one evolution. The larger description retains the internal states; removing interchangeable member labels does not erase constituents or continuous state coordinates. The scale and symmetry proofs own these distinctions, including the closed unordered state.

RA is a configured event with EPI mixing, circular phase adjustment and a separately triggered capacity update. Its operator contract remains separate from continuous dynamics. None of the results below automatically invokes RA or derives when that event must occur.

2. The declared law retains the full exchange

Fix a finite connected simple unit-support graph with at least two nodes. Let L be its combinatorial Laplacian, d_i its degrees, and let all held capacities nu_i be strictly positive. Write

\[ K=\operatorname{diag}(\nu_i/d_i),\qquad q=Lx,\qquad S_i(\theta)=\sum_{j\sim i}\sin(\theta_j-\theta_i). \]

Form is real and signed, phase is circular, and the common clock is declared. For e>0,w>0,beta>0, set

\[ a=\frac w\pi,\qquad b=\frac{w}{\beta\pi},\qquad a=\beta b. \]

The complete unforced sine law is

\[ \dot x=-eKq+aKS(\theta),\qquad \dot\theta=bKq. \]

It is distinct from native neighbor-argument pressure and its phase mobility. Holding capacity and support fixed is a premise, not an inferred universal law.

Define structural storage

\[ E(x,\theta)=\frac12x^\mathsf TLx+ \beta V(\theta),\qquad V(\theta)=\sum_{\{i,j\}}\bigl(1-\cos(\theta_j-\theta_i)\bigr). \]

The existing exact decomposition uses grad E=(q,-beta S) and gives

\[ \binom{\dot x}{\dot\theta} = \left[ \begin{pmatrix}0&-bK\\bK&0\end{pmatrix} - \begin{pmatrix}eK&0\\0&0\end{pmatrix} \right]\nabla E, \qquad \dot E=-e\,q^\mathsf TKq. \]

Both consumed rows appear in this identity. The reversible terms exchange storage between form and phase; they cancel in the total balance because the first matrix is skew. They have not been deleted from the dynamics. The constant skew matrix supplies the same-law Poisson structure discussed in that owner, rather than an auxiliary Hamiltonian assigned to an unrelated graph.

This balance justifies reciprocal exchange within the supplied model. It does not derive the sine kernel, its coefficient or the dissipation strength uniquely from xdot=nu*p. Nor does the structural storage become physical energy without an independent measurement bridge.

The exact nonlinear representations retain this same balance when form is represented by phase velocity or by initial-state forcing and nonlinear memory. Their quotient equation reduces to the pencil below at equilibrium; the nonlinear equivalence itself neither requires an equilibrium nor discards the initial form. The controlled phase comparison supplies a finite-horizon approximation with that initial form retained as a transient and shifted initialization; it is distinct from the local resonance linearization below.

3. The full equilibrium derivative

Take an exact critical pattern x_*=c*1,S(theta_*)=0 and define its phase Hessian

\[ H=\nabla^2 V(\theta_*) =L_{\cos(\theta_{*,j}-\theta_{*,i})}. \]

Assume H is positive definite after removal of common phase rotation. Strictly acute critical edge gaps are a sufficient condition on connected support. This hypothesis is essential; not every phase configuration is a stable pattern.

Let

\[ h=K^{-1/2}\mathbf1,\qquad \mathcal Q=h^\perp,\qquad B=K^{1/2}LK^{1/2},\qquad C=K^{1/2}HK^{1/2}. \]

Both matrices annihilate h and are positive definite on mathcal Q. Every inverse and square root below is restricted to this quotient, rather than inverting a singular full Laplacian. In transformed perturbations xi=K^(-1/2) delta x and eta=K^(-1/2) delta theta, with their common-origin components removed, the exact tangent is

\[ \dot\xi=-eB\xi-aC\eta,\qquad \dot\eta=bB\xi. \]

Consequently

\[ \ddot\eta+eB\dot\eta+abBC\eta=0, \qquad B^{-1}\ddot\eta+e\dot\eta+abC\eta=0. \]

This is derived from the joint nodal rows. It does not add a primitive oscillator, identify capacity with an angular frequency or reconstruct pressure from a response.

The quotient tangent is asymptotically stable. For example, testing the second quadratic pencil on a possibly complex eigenvector v gives

\[ m s^2+e n s+ab c=0,\qquad m=v^*B^{-1}v>0,\quad n=v^*v>0,\quad c=v^*Cv>0. \]

A real root is negative. For a nonreal root, its imaginary part gives Re(s)=-e*n/(2m)<0. This establishes stability, but does not classify every root as real or nonreal.

In general B and C do not commute. An isolated-ring discriminant cannot classify a different full graph without checking that reduction. If H=rho*L, then C=rho*B even with heterogeneous positive capacity: capacity changes modal rates and eigenvectors, but not the resulting common pole-class condition. Otherwise use the whole pencil and actual observations. The native collective-mode result and the sine derivative have different coefficients and cannot be interchanged merely because both contain phase feedback.

Reusing spectral rhythms requires the actual joint generator

At the conservative boundary e=0, phase consensus, unit held capacity and w=beta=1, the same rows in clock tau=t/pi give

\[ \xi'=-L_{\rm rw}\eta,\qquad \eta'=L_{\rm rw}\xi, \qquad \xi''=-L_{\rm rw}^{\,2}\xi. \]

Thus a positive eigenvalue lambda of the normalized Laplacian has angular frequency lambda in tau, or lambda/pi in original structural time. This is a consensus tangent statement, not a general finite-amplitude wave equation. For other capacities, phase geometries or laws, use the full joint derivative and declared clock; the graph spectrum alone is insufficient. The exact path-memory construction and independent full-field controls already retain this generator and its initialization.

The archived emergent_rhythm.py at e6d5e731 instead supplies xi''=-L_sym xi, assigns frequencies sqrt(lambda), and constructs its self-similar support in advance. Its reusable question is whether interference of excited modes appears in a specified readout. A quadratic detector can contain sum and difference frequencies, but that calculation selects neither an autonomous preparation nor a fundamental rhythm. A small pressure diagnostic does not establish full-state equilibrium; a prebuilt repeated graph does not establish dynamical fractality. The auxiliary wave correspondence retains the wave model's scope. Same-law claims instead use the pulse, recurrence and scale-inheritance proofs with their separate premises.

4. A finite-frequency maximum for every active collocated port

Declare the probe and the measured response

Introduce, solely for this response question, a supplied form-rate input f*u(t):

\[ \dot x=-eKq+aKS+f\,u(t),\qquad \dot\theta=bKq. \]

The fixed real vector f and scalar waveform u specify the probe. This does not derive a sinusoidal drive from an autonomous environment or install an input in the unforced engine. The exact balance now includes

\[ \dot E=-e\,q^\mathsf TKq+u\,y,\qquad y=f^\mathsf Tq=f^\mathsf TLx. \]

Thus y is the output conjugate to this input in structural storage work. It is not necessarily a laboratory power measurement. Changing the output to a single phase or a different region changes the response claim.

Let P_Q project onto mathcal Q and put

\[ F=P_{\mathcal Q}K^{-1/2}f,\qquad G=B^{1/2}F,\qquad \zeta=B^{1/2}\xi,\qquad A=abB^{1/2}CB^{1/2}. \]

The common-origin input component is tracked separately in the weighted form mean and contributes no work to y. An active port means F!=0, equivalently G!=0. A nonzero spatially uniform form-rate input has F=0 and is an explicit excluded case: it moves only the common form origin.

At the tangent level the transfer from u to y, with zero initial perturbation, is

\[ \boxed{ \mathcal H(s)=sG^\mathsf T (s^2I+esB+A)^{-1}G.} \]

Indeed, zeta'= -eB zeta-aB^(1/2)C eta+G*u and eta'=bB^(1/2)zeta; eliminating eta yields the displayed transfer, and y=G^T zeta. This is a tangent frequency response around the exact pattern. A finite nonlinear experiment needs its own amplitude, error and preparation admission.

Conditional theorem and proof

For every active port under the stated hypotheses, the scalar gain |mathcal H(i*omega)| has a positive global maximum at at least one finite omega>0.

First, A and B are positive definite. For omega>0, the matrix

\[ M(\omega)=A-\omega^2I+ie\omega B \]

is nonsingular: if M v=0, the imaginary part of v^*M v would be e*omega*v^*Bv=0, forcing v=0. There is no imaginary-axis pole, in agreement with the stability argument above.

For z=M(omega)^(-1)G, direct multiplication gives

\[ \operatorname{Re}\mathcal H(i\omega) =e\omega^2 z^*Bz>0. \]

The response is therefore nonzero at every positive frequency. It is continuous there, and its endpoint asymptotics are

\[ \mathcal H(i\omega) =i\omega\,G^\mathsf TA^{-1}G+O(\omega^2) \quad(\omega\to0), \]

\[ \mathcal H(i\omega) =\frac{G^\mathsf TG}{i\omega}+O(\omega^{-2}) \quad(\omega\to\infty). \]

The gain tends to zero at both ends. Choose any positive interior value; outside a sufficiently large compact interval the gain is smaller than that value. Continuity on the interval supplies a positive finite-frequency global maximum.

The gain is also bounded in terms of the admitted input and loss:

\[ |\mathcal H(i\omega)|^2 \le\omega^2(G^\mathsf TB^{-1}G)(z^*Bz) =\frac{\|F\|^2}{e}\operatorname{Re}\mathcal H(i\omega), \]

and hence

\[ \boxed{|\mathcal H(i\omega)|\le\frac{\|F\|^2}{e}.} \]

This is frequency selectivity of a specified structural-work port through reciprocal form--phase dynamics. It holds for noncommuting B,C and does not require complex temporal poles. It does not prove a narrow peak, large gain, a particular maximum frequency, selective response at every remote node or a maintained autonomous pulse. The normalization of f fixes the numerical gain.

Multiple ports: orientation readout is not linear Hall response

Keep the exact critical background, fixed support and positive held capacities above. For this algebra also allow the conservative boundary e=0, excluding resolvent poles. Let the columns of a fixed real matrix F_in specify multiple form-rate inputs. Their conjugate storage-work outputs are y=F_in^T L delta_x; inputs of another type or arbitrary readouts are not covered. Set G=B^(1/2) P_Q K^(-1/2) F_in on the same quotient. With zero initial perturbation the scalar calculation extends to

\[ \mathcal H(s)=sG^{\mathsf T} [s^2I+esB+abB^{1/2}CB^{1/2}]^{-1}G, \qquad \boxed{\mathcal H(s)^{\mathsf T}=\mathcal H(s)}. \]

The pencil is symmetric under transpose, and so is its inverse wherever it exists. No commutation of B and C is required; complex symmetry is not Hermitian symmetry. Consequently the antisymmetric transfer between these matched same-type work ports is zero. Replacing a critical theta_* by -theta_* also leaves its cosine Hessian and entire tangent generator unchanged. With the same linear input/output maps, opposite winding therefore cannot supply a winding-odd linear response in this experiment.

This reuses the exact nonlinear phase representation: B^-1 vartheta_ddot + e vartheta_dot + ab grad(Phi)=0. Its mass and damping are symmetric; no skew velocity force is present. The constant same-law Poisson tensor pairs form and phase, which does not by itself produce a gyroscopic force between physical spatial coordinates. Around a static background, exact elimination of hidden linear coordinates also preserves the symmetric pencil through its Schur complement. Retain the transformed ports and hidden initial-state contribution; treating that free response as part of an input transfer can give a false asymmetry.

Port-type boundary. Collocation alone is insufficient. The ordinary canonical oscillator p_dot=-q+u_p, q_dot=p+u_q, with storage (p^2+q^2)/2 and conjugate output (p,q), has

\[ \mathcal H_{\rm mixed}(s)=\frac1{s^2+1} \begin{pmatrix}s&-1\\1&s\end{pmatrix}. \]

Its antisymmetric mixed entries distinguish phase-space channels, not spatial Hall transport. A TNFR comparison must identify the actual port types and their physical reversal operations before interpreting an off-diagonal sign.

The two-contact orientation control is compatible with this restriction. Its seven-node preparation is not a full equilibrium: the probes immediately have nonzero form pressure. It measures an autonomous signed response, not a linear susceptibility about equilibrium. The full field also contains orientation-sensitive nonlinear information:

\[ D^2S_i(\theta_*)[v,v] =-\sum_{j\sim i}\sin(\theta_{*,j}-\theta_{*,i})(v_j-v_i)^2. \]

This coefficient reverses under theta_* -> -theta_*, even though the first derivative does not. Moving internal states likewise give a time-dependent tangent. Neither finite nonlinear response nor response about a moving pattern is excluded by the static transfer theorem. Their input, retained motion, readout and finite error bounds require their own derivation. The moving doubled-C5 pulse provides such a finite-time infinitesimal response: time ordering of its varying symmetric stiffnesses produces a nonzero matched form/work contrast. Its distributed probes, retained pulse phase and finite horizon are essential premises. This is compatible with the stationary theorem. A separate full nonlinear kick bound now supplies a sufficient finite amplitude radius; the complete-history reversal identity prevents interpreting that contrast as failure of microscopic reversibility. No magnetic identification follows.

The full-support regression controls exercise noncommuting form/phase geometry, winding reversal and the mixed-port boundary using the shared tangent. The physical comparison records why the present orientation signal is not yet an admitted Hall test.

5. Modal examples make the output distinction explicit

If B,C commute, choose a common normalized mode psi with B*psi=lambda*psi and C*psi=mu*psi, where lambda,mu>0. With F=r_in*psi and modal coordinates X=psi^T xi,Theta=psi^T eta,

\[ \dot X=-e\lambda X-a\mu\Theta+r_{\rm in}u,\qquad \dot\Theta=b\lambda X. \]

Write d=e*lambda and k=ab*lambda*mu. The three distinct outputs have transfers

\[ \frac{X(s)}{u(s)}=\frac{r_{\rm in}s}{s^2+ds+k},\qquad \frac{\Theta(s)}{u(s)}=\frac{b\lambda r_{\rm in}}{s^2+ds+k},\qquad \frac{y(s)}{u(s)}=\frac{r_{\rm in}^2\lambda s}{s^2+ds+k}. \]

The last identity uses y=r_in*lambda*X. An unweighted form coordinate is not the structural-work output unless the required factor is retained.

For a nonzero modal input:

• Free perturbations oscillate precisely when d^2<4k, or e^2*lambda<4ab*mu.
• Form and conjugate-work gain peak at omega^2=k for every d>0, including real-pole cases. Their peak gains are respectively |r_in|/d and r_in^2/e.
• Phase gain has a nonzero-frequency maximum only when d^2<2k. Then omega_peak^2=k-d^2/2; otherwise its maximum is at zero frequency.

These statements follow by differentiating the squared gain with respect to omega^2. They demonstrate why an oscillatory pole, a form-response maximum and a phase-response maximum are not interchangeable resonance tests.

At phase consensus, C=B. For a uniform cycle twist with acute edge increment alpha, C=cos(alpha)*B. Writing rho=mu/lambda, the modal poles satisfy

\[ s^2+e\lambda s+ab\rho\lambda^2=0. \]

At the default e=w=1/2,beta=1, every such mode with 0<rho<=1 has distinct negative real poles because e^2-4ab*rho=1/4-rho/pi^2>0. Its phase-output gain has no nonzero-frequency maximum, while its active form and conjugate-work outputs still do. Thus real poles do not negate the theorem in Section 4, and that theorem does not imply underdamped free oscillation.

The same scalar reduction is unavailable for a general attached network whose B,C do not commute. Neither an isolated cycle nor a frozen intermediary classifies all its modes.

Unit form-input specialization on a cycle

For C_n with n>=3, common positive capacity nu, degree two, and an exact acute twist of integer winding j, write

\[ \rho=\cos\frac{2\pi j}{n}>0,\qquad 4|j|<n. \]

Choose a real Euclidean-unit nonconstant Laplacian eigenmode psi, indexed by an integer 1<=m<n, with

\[ L\psi=\ell\psi,\qquad \ell=2-2\cos\frac{2\pi m}{n}>0,\qquad r=\frac{\nu\ell}{2}. \]

This r is the spatial rate; the supplied input is normalized in the original form coordinates as f=psi. Writing delta x=X*psi,delta theta=Theta*psi gives

\[ \dot X=-erX-a\rho r\Theta+u,\qquad \dot\Theta=brX,\qquad y=\ell X. \]

Consequently, for D(s)=s^2+er*s+ab*rho*r^2,

\[ \mathcal H_X(s)=\frac{s}{D(s)},\qquad \mathcal H_\Theta(s)=\frac{br}{D(s)},\qquad \mathcal H_y(s)=\frac{\ell s}{D(s)}. \]

The form and structural-work peaks occur at omega^2=ab*rho*r^2, with gains

\[ \max_\omega|\mathcal H_X(i\omega)|=\frac1{er},\qquad \max_\omega|\mathcal H_y(i\omega)| =\frac{\ell}{er}=\frac2{e\nu}. \]

Here ||F||^2=2/nu, so this mode attains the general collocated bound. The phase-output peak and pole classification still have their distinct conditions e^2<2ab*rho and e^2<4ab*rho respectively. The constant mode has ell=0 and is a common-origin input, not an active-port member of this theorem.

6. Constitutive changes and clock changes are different

The coefficient audit already derives the dimensionless combination chi=beta*(e/w)^2. For the commuting mode above,

\[ \zeta^2=\frac{e^2\lambda}{4ab\mu} =\frac{\pi^2\chi}{4\rho}. \]

The squared damping ratio therefore depends on both the supplied coefficient combination and the phase geometry. Consensus is underdamped when chi<4/pi^2; this is a model-dependent condition for the declared law and consensus geometry, not a universal physical constant. For example, changing default beta=1 to beta=1/4 changes the consensus pole class while preserving the declared positive-storage family.

For the sine rows, the precise unit transformation y=s*x,tau=c*t with s,c>0 gives

\[ \beta' =s^2\beta,\qquad e'=\frac ec,\qquad w'=\frac{sw}{c},\qquad K'=K. \]

If the engine convention e+w=1 is retained instead, put kappa=e+s*w and use

\[ e_{\mathrm{new}}=\frac e\kappa,\quad w_{\mathrm{new}}=\frac{sw}{\kappa},\quad K_{\mathrm{new}}=\frac{\kappa}{c}K,\quad \beta_{\mathrm{new}}=s^2\beta. \]

Transform the input, output and their units consistently as well. Both representations preserve chi. A common positive affine clock change rescales every pole and response frequency together; it cannot turn real poles into nonreal ones or create a gain peak for the same transformed scalar input/output observation. An invertible state chart likewise preserves the transfer when the input and observation are transformed with it. Replacing a phase output with a form output is a different observation, not such a coordinate repair.

At an equilibrium a smooth positive clock change scales the Jacobian by its local common factor, preserving pole class. Along a nonstationary trajectory, a nonconstant clock need not preserve a time-invariant frequency-response problem. Changing chi, support, capacity ratios or phase geometry requires a new declared constitutive/preparation comparison.

7. Remote response through a retained intermediary

Preparation, ports and the complete tangent

Take two disjoint oriented five-cycles D=(d_0,...,d_4) and R=(r_0,...,r_4), one intermediary h, and exactly the additional edges {d_0,h} and {r_0,h}. All edges have unit conductance. These eleven nodes and twelve edges are the complete supplied support. No direct donor--receiver edge is added or inferred.

Use uniform form and the exact critical phases

\[ \theta_{d_j,*}=\theta_{r_j,*}=\frac{2\pi j}{5},\qquad \theta_{h,*}=0 \]

up to common origins and circular lifts. Both bridge gaps are zero; each cycle has winding +1. The phase Hessian has ring edge weight rho=cos(2*pi/5)>0 and bridge weight one. Consequently it is positive on the common-phase quotient. Every capacity is held and strictly positive; equality of capacities is unnecessary for the transmission result. Set k_i=nu_i/d_i, so the two ports have degree three and the intermediary has degree two.

Declare the donor pattern and receiver observation

\[ p_D=e_{d_0}-\frac14\sum_{j=1}^{4}e_{d_j},\qquad r_R=e_{r_0}-\frac14\sum_{j=1}^{4}e_{r_j},\qquad f=Kp_D. \]

The hypothetical input adds f*u(t) to the form row only. The output is the receiver phase contrast y=r_R^T delta theta, taken in continuous perturbation lifts about the declared target. It is not the work-conjugate output of Section 4. The capacity factors belong to the declared spatial input; removing them would change the probe.

In the original coordinates the full tangent and its port vectors are

\[ J=\begin{pmatrix}-eKL&-aKH\\ bKL&0\end{pmatrix},\qquad v_D=\binom{f}{0},\qquad C_R=(0,r_R^\mathsf T). \]

Because 1^T p_D=0, the input preserves the weighted form mean: 1^T K^(-1) f=0. Both unforced weighted means are constant, and the receiver contrast discards common phase origin. Thus

\[ \mathcal T_R(s)=C_R(sI-J)^{-1}v_D \]

means the transfer on the stable quotient from Section 3. The singular full-state resolvent at s=0 is not inverted. For a held capacity family this theorem applies to each member and its own capacity-weighted input; it does not identify one capacity-independent probe or a uniform peak frequency. The specialized reader currently requires an exact held capacity at every node and does not consume an uncertain-capacity family.

Zero static contrast and a nonzero two-hop response

The constant-input tangent equilibrium on the quotient has delta x=0 and solves

\[ H\,\delta\theta=\frac{p_D}{a}. \]

Sum these equations over the donor nodes. Internal edge terms cancel, and the donor source has total zero, so the only outgoing bridge current vanishes. Thus delta theta_h=delta theta_d0. The intermediary equation then gives delta theta_r0=delta theta_h. The source-free receiver has constant phase perturbation, since its positively weighted connected Laplacian has only the constant null direction. Its measured contrast is therefore zero:

\[ \boxed{\mathcal T_R(0)=0.} \]

This cancellation follows from the source, observation and bridge geometry. It is not a statement that phase is always a high-pass or band-pass observation; the local phase output in Section 5 can instead have its maximum at zero frequency.

There is nonetheless a nonzero remote response. Direct differentiation of the full rows gives

\[ C_Rv_D=C_RJv_D=0,\qquad \boxed{C_RJ^2v_D=-be\,k_{d_0}k_hk_{r_0}<0.} \]

Indeed, the lower-left block of J^2 is -be*KLKL. The only length-two path from the donor input to a receiver node is d_0--h--r_0. Its two off-diagonal Laplacian factors multiply to one. Other donor coordinates and other receiver coordinates cannot contribute at this order. The initial form weight at the donor port is f_d0=k_d0. With common capacity nu, the coefficient is -be*nu^3/18; positivity of e,b makes it nonzero without any condition on e^2-ab.

Although the displayed preparation uses two +1 twists, the proof also applies to any independently admitted acute critical target on this complete support whose two bridge gaps are zero. The DC argument requires only positive edge Hessian weights, and the two-hop coefficient uses only the bridge weights, which remain one. This is the broader target class consumed by assess_sine_mediated_response; the captured observation is never substituted for its exact recovery target. In fact criticality and zero bridge current force all oriented sine currents on each ring to agree. Strict acuity makes sine injective on the principal edge gaps, so each ring has one common increment. Circular closure then leaves precisely winding -1, 0 or +1. Opposite ring windings and consensus members therefore satisfy the same argument without a different supplied law. The complete local-stability classification shows that these nine zero-bridge acute geometries exhaust the locally attracting relative equilibria on this support under the positive-loss sine law. Which one a preparation reaches remains a separate basin question. At fixed support and capacity, reversing either ring winding from +1 to -1 leaves every edge cosine, the full tangent J and all linear transfers unchanged. These responses therefore cannot by themselves identify the winding's handedness, even though the exact target phase pattern is different.

For comparison, the same receiver form contrast has second Markov coefficient (e^2-ab)*k_d0*k_h*k_r0. That coefficient can vanish within the admitted law family. The phase route avoids this cancellation; it does not establish that every remote observation is responsive.

The intermediary also explains the response as exact memory. Partition the tangent into visible donor/receiver coordinates and the two hidden intermediary coordinates. With

\[ M=\begin{pmatrix}e&a\\-b&0\end{pmatrix},\qquad D_h=2k_h\begin{pmatrix}-e&-a\\ b&0\end{pmatrix}, \]

the receiver--donor part of the hidden-state kernel is

\[ \mathcal K_{RD}(t) =k_{r_0}M\exp(D_ht)k_hM, \]

with the displayed rows and columns referring to the two contact-node form/phase coordinates. In particular,

\[ \mathcal K_{RD}(0) =k_{r_0}k_h \begin{pmatrix} e^2-ab&ea\\-be&-ab \end{pmatrix}. \]

This is the exact block-elimination identity owned by derive_coordinate_memory. Its lower-left entry supplies the nonvanishing phase route. The hidden initial-state contribution and the remaining visible dynamics are retained; this kernel is not a new instantaneous edge. The shared implementation accepts rational/represented generators; rounding pi or the target cosine for that interface proves identities for the supplied matrix, not for the ideal transcendental generator. The ideal statements here follow from the displayed exact blocks.

A remote frequency maximum and an autonomous rebound

Stability on the quotient makes mathcal T_R(i*omega) continuous for all real omega. The static calculation gives its zero-frequency limit, while the exact Markov coefficient gives

\[ \mathcal T_R(s) =-\frac{be\,k_{d_0}k_hk_{r_0}}{s^3}+O(s^{-4}) \quad(|s|\to\infty). \]

It is therefore not identically zero and its gain tends to zero at both frequency endpoints. A nonzero rational transfer cannot vanish on the entire positive imaginary axis. Choose a positive interior gain; outside a sufficiently large compact frequency interval the gain is smaller. Continuity then proves a positive global gain maximum at some finite omega>0.

This remote, noncollocated theorem requires neither a positive-real transfer nor complex temporal poles. It imports neither the collocated gain ceiling nor nonvanishing response at every positive frequency. Destructive interference, zeros and an output dependent peak location remain possible. No frequency or gain is selected by this existence proof.

Now remove the input and prepare a donor-only initial disturbance,

\[ \delta x(0)=\epsilon f,\qquad \delta\theta(0)=0, \qquad \epsilon>0. \]

At the tangent level its autonomous output is

\[ y_{\epsilon}(t)=\epsilon g_R(t),\qquad g_R(t)=C_R\exp(Jt)v_D. \]

This uses an initial condition, not a persistent or secretly supplied periodic drive. Stability makes g_R exponentially decaying and integrable, and the preceding identities imply

\[ g_R(t)=-\frac{be\,k_{d_0}k_hk_{r_0}}2t^2+O(t^3),\qquad \int_0^\infty g_R(t)\,dt=\mathcal T_R(0)=0. \]

It is strictly negative for sufficiently small positive time. Its zero integral then forces it to be strictly positive at some later time: otherwise its integral would be negative. The receiver phase contrast must consequently change sign at least once before decaying. This is a necessary transient rebound, not a proof of a complex pole, repeated oscillations or a maintained autonomous pulse.

The full nonlinear sine law has the same initial second derivative for this exact critical-phase preparation, since its initial sine currents vanish. The zero-area identity is a tangent identity. Smooth dependence on the initial state additionally implies a qualitative finite-time extension: after choosing one negative and one positive tangent-response time, sufficiently small positive epsilon preserves those two signs in the nonlinear response. This gives no computed amplitude allowance, response time or validated nonlinear horizon. Such an admission still requires its own bounds; the report does not silently promote the tangent result to that quantitative prediction.

Silent controls and what has not been created

The donor-ring reflection fixes d_0 and swaps d_1,d_4 and d_2,d_3. If capacities are equal on each swapped pair, its odd form/phase perturbations, supported only on the donor, are an invariant four-dimensional tangent subspace. Their port coordinate is zero; neighbor contributions at the port cancel in both L and H. They produce no intermediary or receiver response. Similarly, reflection-paired receiver capacities make every receiver-odd observation unexcited by a donor-only disturbance in the tangent law. The declared even receiver contrast is chosen separately from those silent observations. These symmetry controls require the paired-capacity hypotheses; the nonzero phase-route proof does not.

Thus a particular donor pattern can causally affect a separate receiver and have a frequency-selective response through the retained environment, while other patterns are silent. The support and critical patterns remain supplied. This proves neither their formation nor a new physical interaction, autonomous forcing, or the origin of an edge. The existing full nonlinear storage loss excludes nonstationary recurrent full-state motion under the stated unforced premises.

The shared reader and its detached-report boundary are documented in the mediated-response contract. It retains the full target, capacities, declared ports and low-order derivatives; it does not evaluate a trajectory or locate the maximum or the sign reversal.

8. Permanent motion, irreversible loss and the reversible boundary

A finite closed dissipative system has no nonstationary recurrence

Retain finite connected unit support, fixed nonnegative capacities, w,beta>0 and no input, support event, clipping or added phase velocity. Unlike the stable-response theorems above, this subsection allows zero capacities. Let A={i:nu_i>0} and Z be its complement. The sine law remains globally smooth on real form and circular phase, and its storage balance is

\[ \dot E=-e\sum_{i\in A}k_iq_i^2,\qquad k_i=\nu_i/d_i. \]

For e>0, a full-state return modulo common form and phase origins returns the storage. Continuity and nonnegativity therefore force q_i=0 at every active node throughout the return interval. Every phase row then vanishes; inactive form rows also vanish.

There is no loophole through a zero-capacity anchor. If both A and Z are nonempty, differentiate the active constraints:

\[ 0=\dot q_A=L_{AA}\dot x_A. \]

The grounded Laplacian L_AA is positive definite. Its quadratic form is the sum of squared differences on active--active edges and squared active coordinates on edges to the frozen set; connectivity makes its null space trivial. Thus xdot_A=0. If all capacities are positive, q=0 instead gives x=c(t)*1. The sine law's weighted form-mean identity,

\[ \mathbf1^\mathsf TK^{-1}\dot x =-e\,\mathbf1^\mathsf Tq+a\,\mathbf1^\mathsf TS=0, \]

forces c'=0. If all capacities are zero, both rows are frozen from the outset. In each case the recurrent trajectory is stationary. Zero capacity can preserve nonzero pressure or form contrast at an equilibrium, but cannot create a loss-free moving sector of this law.

The argument also excludes recurrence along a sequence of times returning to the initial state modulo origins: monotone storage converging back to its initial value must have been constant at every earlier time. It does not confuse decreasing-amplitude transients with recurrent full-state motion.

For the native argument-pressure completion, the same recurrence conclusion holds while the full solution remains in its admitted regular domain with positive phase metric. Its loss is the same -e*q^T Kq, and its selected phase row freezes at zero capacity. With some inactive nodes the grounded-Laplacian proof is unchanged. With all nodes active, write the native positive phase metric as M=diag(m_i) and its source as g. At q=0 the form row is xdot_i=w*nu_i*g_i=c', so

\[ c'\sum_i\frac{m_i}{\nu_i} =w\sum_i m_i g_i=0. \]

The final equality follows from Mg=-grad V and phase-origin invariance. This extends the existing native recurrence proof to held zero capacities without substituting the sine phase law for the native one. It does not continue native trajectories through their undefined resultant or branch boundary.

The sine obstruction also controls nonperiodic long-time motion

For the globally smooth sine law with e>0, every such finite-system trajectory approaches the equilibrium set. This is stronger than excluding exact periodic motion, but does not select one limiting phase pattern or a convergence rate.

Here is a compactness proof retaining the full state. Storage bounds all form differences on connected support. If every capacity is positive, the conserved weighted form mean also bounds the common form origin. If some capacity is zero, the frozen form of any such node anchors that origin instead. Circular phases lie on a compact torus. Hence the trajectory stays in a compact set of real form and circular phase and has global continuation.

Its nonempty compact omega-limit set is invariant and has constant storage E_infinity. Every complete trajectory within that set therefore satisfies q_A=0 and thetadot=0. Its sine currents are constant, and its form velocity is the constant vector aKS. A nonzero constant velocity would leave the compact limit set, so that velocity must vanish. Every state of the omega-limit set is an equilibrium; equivalently the distance of the original state to the equilibrium set tends to zero.

This proof permits heterogeneous and zero held capacities. It does not establish convergence to one equilibrium, a uniform relaxation time or a theorem for another runtime. In particular, the compact smooth sine argument cannot be imported into native argument dynamics without separately controlling that model's regular-domain boundary.

For the specific two-C5/live-intermediary support with strictly positive held capacities, the finite critical-geometry classification strengthens this result to convergence to one relative equilibrium. An explicit conserved-lift reconstruction also fixes its limiting common phase origin. That support-specific theorem does not select which equilibrium a preparation reaches or supply a convergence deadline; it does not replace the broader equilibrium-set statement above.

A finite passive environment is not an unlimited source

Retaining additional environmental nodes does not supply energy without accounting for it. More generally, if a closed full system has storage bounded below and obeys

\[ \dot E_{\rm total}=-D_{\rm total},\qquad D_{\rm total}\ge0, \]

then

\[ \int_0^T D_{\rm total}(t)\,dt \le E_{\rm total}(0)-E_{\min},\qquad \frac1T\int_0^T D_{\rm total}(t)\,dt\longrightarrow0. \]

A subsystem loss included in D_total cannot remain positive on average forever by receiving work from a finite passive environment. It may receive a finite transient supply or exchange storage reversibly. Neither possibility derives a perpetual drive for a dissipative periodic response. For example, at an active sine node,

\[ e k_i q_i^2=\frac{e}{b^2k_i}\dot\theta_i^2. \]

A repeated nonconstant phase cycle with a fixed period consumes a strictly positive loss per cycle. An initially finite closed storage cannot pay that cost indefinitely.

This obstruction concerns the stated total balance. It does not assume that an observed subsystem is closed, or that irreversible loss is fundamental rather than an effective description after discarding degrees of freedom. The origin and admissibility of that loss are separate research questions; naming an environment does not answer them.

Exact nonlinear periodic exchange on the sine P2 boundary

Now change the declared constitutive premise to e=0, retaining w,beta>0, fixed support and held capacity. This boundary is already admitted by RelationalExchangeModel; under its weight-normalization convention the effective phase weight is then one. No primitive oscillator or external pump is appended.

On a single unit edge P2, write

\[ u=x_0-x_1,\qquad \delta=\theta_1-\theta_0,\qquad \sigma=\nu_0+\nu_1. \]

If sigma>0, including one frozen endpoint, the complete nonlinear relative rows reduce exactly to

\[ \dot u=a\sigma\sin\delta,\qquad \dot\delta=-b\sigma u,\qquad \ddot\delta+ab\sigma^2\sin\delta=0, \]

\[ E=\frac12u^2+\beta(1-\cos\delta),\qquad \dot E=0. \]

Thus the pendulum equation is a consequence of the two consumed nodal rows in this preparation, not an assumed equation for a new state variable. The full node coordinates are recovered without dividing by either capacity:

\[ \begin{aligned} x_0(t)&=x_0(0)+\frac{\nu_0}{\sigma}[u(t)-u(0)],\\ x_1(t)&=x_1(0)-\frac{\nu_1}{\sigma}[u(t)-u(0)],\\ \theta_0(t)&=\theta_0(0)-\frac{\nu_0}{\sigma}[\delta(t)-\delta(0)],\\ \theta_1(t)&=\theta_1(0)+\frac{\nu_1}{\sigma}[\delta(t)-\delta(0)]. \end{aligned} \]

For every level 0<E<2*beta, choose the relative phase lift centered on its energy well 2*pi*n. Its deviation from that center librates between -A and A, where

\[ A=\arccos(1-E/\beta)<\pi,\qquad |u|_{\max}=\sqrt{2E},\qquad m=\frac{E}{2\beta}\in(0,1). \]

The level curve is compact, regular and contains no equilibrium, so it is traversed periodically. Both relative coordinates return, and the reconstruction proves full-state periodicity even when precisely one capacity is zero. Define the complete elliptic integral using its parameter m, not its square-root modulus:

\[ \mathbf K(m)=\int_0^{\pi/2} \frac{d\psi}{\sqrt{1-m\sin^2\psi}}. \]

Separation of the conserved energy, followed by sin((delta-2*pi*n)/2)=sqrt(m)*sin(psi), gives the exact period

\[ \boxed{ T(E)=\frac{4}{\sigma\sqrt{ab}}\,\mathbf K(m) =\frac{4\pi\sqrt{\beta}}{w\sigma}\,\mathbf K(m).} \]

This is the model-clock period. With

\[ T_0=\frac{2\pi}{\sigma\sqrt{ab}} =\frac{2\pi^2\sqrt{\beta}}{w\sigma}, \]

the integrand bounds give

\[ \boxed{T_0<T(E)\le\frac{T_0}{\sqrt{1-m}}} \qquad(0<m<1). \]

Also T(E)->T0 as E->0 and T(E)->infinity as E->2*beta from below. For common capacity nu, T0=pi^2*sqrt(beta)/(w*nu). These analytic bounds permit a detached period enclosure from independently captured initial storage; they require no sampled oscillation or fitted frequency.

At E=0 the state is stationary. At E=2*beta the antipodal equilibrium and nonperiodic separatrix need separate admission. Above that energy the relative phase rotates; full circular state return must additionally account for individual phase windings. Neither the libration period formula nor a native phase-branch claim is silently extended to those cases. If sigma=0, both nodes are frozen regardless of initial storage.

Reuse the native result without replacing its clock

The existing native P2 reversible result already proves periodic exchange on 0<E<beta inside its acute domain. Here delta denotes the centered acute gap. Its two relative rates at e=0 are the sine rates multiplied by 1/sinc(delta):

\[ \dot u_{\rm native}=a\sigma\delta,\qquad \dot\delta_{\rm native} =-\frac{b\sigma u}{\operatorname{sinc}\delta}. \]

Thus the two laws share the relative energy curves there, but

\[ dt_{\rm native}=\operatorname{sinc}(\delta)\,dt_{\rm sine}. \]

Their nonzero-amplitude periods differ. The common small-amplitude limit does not make them the same constitutive law or the same clock. The sine law's smooth phase-circle continuation also does not remove the native law's separate boundary obligations.

What a permanent-pulse claim has and has not established

These results establish a sharp conditional distinction: finite closed sine systems with positive form dissipation approach their equilibrium set, whereas the admitted reversible P2 law has exact nonlinear periodic states prepared at nonzero storage. The replica pulse proof extends periodic-state existence to a supplied ten-node preparation with static collective means and moving internal constituents.

Energy conservation prevents one such periodic orbit from attracting an open neighborhood containing other energy levels. Its amplitude, phase of motion and period are determined by preparation; no amplitude-selection mechanism or asymptotically attracting limit cycle has been derived. An exactly prepared equilibrium remains an equilibrium by uniqueness. Periodic-state existence is therefore not spontaneous creation from an equilibrium or from an absent substrate. It also does not prove robustness: the complete replica variation and its small-amplitude splitting establish transverse instability of sufficiently small nonzero members of that prepared family. Neither its exact orbit nor an instantaneous oscillator spectrum certifies stability under general fine perturbations.

Nor does the existence proof select e=0 as the fundamental law. Choosing that boundary solely to obtain the desired pulse would leave the selection question unanswered. Whether an irreversible loss term is independently justified at the fundamental level, derived through a declared reduction with retained memory and preparation, or restricted to an effective sector remains a separate closure question. A finite Euler trajectory, an added drive, a hidden energy reserve or an auxiliary graph spectrum cannot substitute for that derivation.

9. Finite conservative reduction gives memory, not irreversible loss

Begin with the same nonlinear law and retained environment

Use the already admitted e=0 sine law on the complete unit path left--h--right. The intermediary is a live TNFR node, not an external waveform. Hold nonnegative capacities nu_left,mu,nu_right, the positive storage scale beta and effective phase weight w>0 fixed. Declare the constant clock conversion

\[ \tau=\frac w\pi t. \]

In this clock the exact nonlinear equations are dx/dtau=KS(theta) and dtheta/dtau=KLx/beta. No coefficient is dropped: the factor w/pi is carried by the declared conversion back to the structural clock t.

Define oriented endpoint deviations from the intermediary,

\[ r=x_{\rm left}-x_h,\quad s=x_{\rm right}-x_h,\qquad \alpha=\theta_h-\theta_{\rm left},\quad \gamma=\theta_h-\theta_{\rm right}, \]

and their actual nonlinear edge storages

\[ E_{\rm left}=\frac12r^2+\beta(1-\cos\alpha),\qquad E_{\rm right}=\frac12s^2+\beta(1-\cos\gamma). \]

Substitution of all three nodal rows gives

\[ \boxed{ \frac{dE_{\rm left}}{d\tau} =\frac\mu2\bigl(r\sin\gamma-s\sin\alpha\bigr) =-\frac{dE_{\rm right}}{d\tau}.} \]

The endpoint capacities cancel from this exchange identity; the intermediary capacity remains. In the structural clock multiply its right-hand side by w/pi. Thus a decreasing regional storage can be exact reversible transfer to the retained neighboring region. It is not by itself evidence of dissipative microscopic loss. Reversing both phase contrasts preserves the two edge storages and reverses the transfer. The local storage values alone therefore do not determine even its sign.

The visible endpoints alone do not close the full nonlinear law for arbitrary hidden preparation. For example, keep both visible forms and phases zero and the hidden form zero. Hidden phases eta and -eta, with 0<eta<pi, give the same visible state and the same edge storages, but opposite visible form rates plus/minus nu_i*sin(eta) in clock tau. When at least one endpoint capacity is positive, these rates differ. A single autonomous function of that visible state cannot equal both rates. Supplying the hidden preparation, retaining its state, or proving a separately restricted invariant family is essential.

Exact tangent memory with a rational clock representation

For an independently specified reduction question, linearize at uniform form and phase consensus, and now take common capacity nu>0. The consensus target is not the currently observed state unless that equality is independently known. In full node order (left,h,right) put

\[ L_{\rm rw}= \begin{pmatrix} 1&-1&0\\ -1/2&1&-1/2\\ 0&-1&1 \end{pmatrix},\qquad J_\tau= \begin{pmatrix} 0&-\nu L_{\rm rw}\\ \nu L_{\rm rw}/\beta&0 \end{pmatrix}. \]

This is the exact derivative of the same nonlinear law, ordered as (delta x_all,delta theta_all). If beta,nu are rational, every generator entry is rational. The constant clock change thus permits exact use of derive_coordinate_memory without replacing mathematical pi by a rounded coefficient. It does not establish finite-amplitude tangent accuracy.

For the following displayed blocks, group each node's form and phase together. Let y contain both endpoint pairs and let z=(delta x_h,delta theta_h) be the hidden pair. Set

\[ M=\begin{pmatrix}0&1\\-1/\beta&0\end{pmatrix},\qquad D=-\nu M,\qquad \omega=\frac{\nu}{\sqrt\beta}. \]

Then

\[ y'=Ay+Bz,\qquad z'=Cy+Dz, \]

\[ A=\operatorname{diag}(D,D),\qquad B=\binom{\nu M}{\nu M},\qquad C=\begin{pmatrix}\nu M/2&\nu M/2\end{pmatrix}, \]

where primes in this subsection denote d/dtau. Exact elimination retains both terms generated by the hidden system:

\[ \boxed{ y'(\tau)=Ay(\tau)+B e^{D\tau}z(0) +\int_0^\tau B e^{D(\tau-s)}C\,y(s)\,ds.} \]

In particular D^2=-omega^2 I, so

\[ e^{D\tau}=I\cos(\omega\tau)+\frac D\omega\sin(\omega\tau). \]

Every endpoint-to-endpoint 2 by 2 block of the kernel is

\[ \boxed{ \mathcal K_{ij}(\tau) =-\frac{\omega^2}{2}e^{D\tau}, \qquad i,j\in\{\mathrm{left},\mathrm{right}\}.} \]

It is a periodic signed matrix, not a decaying friction kernel. The independent hidden source B exp(D*tau) z(0) is also periodic and cannot be set to zero merely because the hidden coordinates are unobserved. The exact source and kernel recurrence is f''=-omega^2 f and K''=-omega^2 K, with initial data f(0)=Bz(0),f'(0)=BDz(0),K(0)=BC,K'(0)=BDC. These identities determine the response without a simulated trajectory.

The endpoints' retained form and phase coordinates also admit a simple all-state linear closure check. Their first derivatives reveal both hidden coordinates:

\[ \delta\theta_h =\delta\theta_{\rm left} +\frac1\nu\,\delta x_{\rm left}',\qquad \delta x_h =\delta x_{\rm left} -\frac\beta\nu\,\delta\theta_{\rm left}'. \]

The four visible coordinate rows and their first output-Krylov extension consequently span all six coordinates. The shared exact linear-observation owner can recover this closure. It does not supply a new four-state memoryless law; additional state or history carries the information that projection discarded.

A prepared disturbance leaves, returns and loses no total storage

The normalized path Laplacian has exact eigenpairs

\[ \lambda_0=0,\ v_0=(1,1,1),\qquad \lambda_1=1,\ v_1=(1,0,-1),\qquad \lambda_2=2,\ v_2=(1,-1,1). \]

Thus the nonconstant tangent frequencies are omega and 2*omega in clock tau. The full tangent, including its constant-origin modes, returns after

\[ T_\tau=\frac{2\pi\sqrt\beta}{\nu},\qquad T_t=\frac{2\pi^2\sqrt\beta}{w\nu}. \]

For example, prepare only the left form perturbation: delta x(0)=(epsilon,0,0),delta theta(0)=0. With s=omega*tau the exact tangent solution is

\[ \delta x =\frac\epsilon4v_0+\frac\epsilon2\cos(s)v_1 +\frac\epsilon4\cos(2s)v_2, \]

\[ \sqrt\beta\,\delta\theta =\frac\epsilon2\sin(s)v_1 +\frac\epsilon4\sin(2s)v_2. \]

Use the quadratic tangent edge storages, replacing the phase cost on each edge by beta*(delta theta_gap)^2/2. Direct substitution gives

\[ E_{\rm left}^{\rm lin} =\frac{\epsilon^2}{4}(1+\cos s),\qquad E_{\rm right}^{\rm lin} =\frac{\epsilon^2}{4}(1-\cos s),\qquad E_{\rm total}^{\rm lin}=\frac{\epsilon^2}{2}. \]

At s=pi the full form perturbation is localized at the right endpoint and all phase perturbations vanish; at s=2*pi the initial state returns. Reading only the left regional storage during its first decrease would mistake reversible transfer for irreversible loss. This is an exact witness for the tangent model, not an asserted finite-amplitude nonlinear trajectory. The preceding nonlinear exchange balance remains exact without that tangent interpretation.

Why exact finite conservative reduction cannot manufacture damping

The obstruction is broader than this path. Consider an e=0 critical target on any finite connected support, with positive held capacity and positive phase Hessian on the mean quotient. In the transformed coordinates of Section 3, but clock tau=w*t/pi,

\[ J_q= \begin{pmatrix}0&-C\\B/\beta&0\end{pmatrix},\qquad W=\operatorname{diag}(B,\beta C)>0. \]

Here B,C retain their quotient Laplacian/Hessian meaning, not the visible--hidden blocks used above. Their exact balance is

\[ J_q^\mathsf TW+WJ_q=0. \]

Consequently W^(1/2) J_q W^(-1/2) is a real skew-symmetric matrix. The quotient generator is diagonalizable with purely imaginary eigenvalues. If common origins are appended, they add constant modes rather than decaying ones. Every linear observation is a finite sum of those harmonics and constants.

Suppose an all-state exact linear Markov closure z=Oz_full exists with independent observation rows and OJ=G O. The kernel of O is invariant, so G is the induced quotient of the same diagonalizable generator. Its active poles are inherited imaginary or zero poles, not new strictly negative-real-part poles. The linear-observation owner tests this precise all-state intertwining identity, rather than inferring a closure from one fitted decay.

Likewise, a nonzero observed response of the finite conservative tangent cannot tend to zero at infinite time. After combining equal frequencies, the long-time average of its squared norm is the sum of nonnegative constant and harmonic contributions. That average is strictly positive unless the response vanishes identically, whereas a bounded response converging to zero has average squared norm zero. Thus even a selected nonzero preparation cannot equal a strictly decaying Markov response for all future times.

This does not forbid apparent attenuation over a finite observation window, or a reduced approximate loss law with a declared limiting procedure, preparation, error and horizon. It proves that finite conservative elimination alone has not justified permanent positive memoryless damping. An exact kernel cannot silently become such a law by dropping its hidden source, deleting its recurrence or calling the removed degrees of freedom an environment. The general argument is a tangent statement at the specified critical targets; the nonlinear endpoint counterexample and nonlinear edge balance above have their separately stated scope.

Common coordinates for a controlled effective-loss comparison

This setup fixes the comparison problem; it is not a derived damping law. Use the finite connected unit support and positive held capacities of Section 2, at phase consensus, with fine loss e_0=0, phase weight w_0>0 and held storage scale beta>0. Phase perturbations are real tangent coordinates at that target, not global unwrapped replacements for circular phase.

On the mean quotient of Section 3, consensus gives C=B. Put

\[ u=B^{1/2}\xi,\qquad v=\sqrt\beta B^{1/2}\eta, \qquad z=(u,v),\qquad \tau=\frac{w_0}{\pi}t. \]

The constant means have been removed, not inverted. Their values must be retained separately if a later observation or contact uses absolute origins. Substitution in both tangent rows gives

\[ E_2=\tfrac12(\|u\|^2+\|v\|^2),\qquad z'=Jz,\qquad J=\frac1{\sqrt\beta} \begin{pmatrix}0&-B\\B&0\end{pmatrix},\qquad J^\mathsf T=-J. \]

Primes denote d/dtau; the norm is the Euclidean norm in this energy chart. This symbolic change is valid for heterogeneous positive held capacities. It supplies neither a nonlinear error bound nor a rational representation of an arbitrary matrix square root.

For an explicitly restricted first observation class, take an orthonormal column map P from m visible spatial coordinates into the r=n-1 quotient coordinates, with P^T P=I_m. Apply the same map to both channels:

\[ \mathcal O=\operatorname{diag}(P^\mathsf T,P^\mathsf T),\qquad \mathcal E=\mathcal O^\mathsf T,\qquad z_0=\mathcal E y_0+h_0,\qquad \mathcal O h_0=0. \]

Thus O E=I and ||z_0||^2=||y_0||^2+||h_0||^2. The actual observation is

\[ y(\tau)=\mathcal O e^{J\tau}\mathcal E y_0 +\mathcal O e^{J\tau}h_0. \]

Choosing h_0=0 is a preparation premise. More generally, specify the allowed hidden states and their norm bound H_0; absence of observation never sets them to zero. A genuine hidden quotient requires m<r, hence at least three nodes for one visible form/phase pair. An invariant spatial subspace (I-PP^T)BP=0 instead supplies a no-exchange control, not a reservoir.

The candidate form-only-loss generator must use these same coordinates and this same clock. Write it as

\[ G=\begin{pmatrix}-\Gamma&-\Omega\\\Omega&0\end{pmatrix}. \]

Require Gamma=Gamma^T positive semidefinite and nonzero, and Omega=Omega^T; the first single-pair target has scalar Gamma>0,Omega>0. This makes the Euclidean storage derivative -y_form^T Gamma y_form, with no phase loss row.

If it represents the consensus sine law on a declared coarse support with the same beta, positive coarse spatial matrix B_v on its mean quotient and positive effective weights e_v,w_v, then

\[ \Gamma=\frac{\pi e_v}{w_0}B_v,\qquad \Omega=\frac{w_v}{w_0\sqrt\beta}B_v. \]

These relations convert both target rows from the common structural clock. The model constructor normalizes its two weights together: a normally constructed conservative model has w_0=1; introducing positive EPI weight does not leave its normalized phase weight unchanged. Read the authoritative effective coefficients, not constructor inputs, and do not normalize again. An independently specified G, different storage scale or different coarse chart needs its own declaration and transformation. Neither matrix above has been derived by eliminating the fine state.

A uniform error claim must specify the preparation family and, for example, bound the full response quantity

\[ \mathcal E_T= \sup_{\substack{0\leq\tau\leq T,\ \|y_0\|\leq1\\ \mathcal O h_0=0,\ \|h_0\|\leq H_0}} \left\|\mathcal O e^{J\tau}(\mathcal E y_0+h_0) -e^{G\tau}y_0\right\|. \]

Here T is in clock tau; the unit ball fixes the tangent amplitude normalization, not a claim of finite-amplitude nonlinear validity. A smaller hidden family must be stated instead of silently changing this supremum. An amplitude envelope or one initial form excitation cannot establish this two-channel estimate. A proposed tolerance and any discarded initial layer need independent admission. The lower bounds below address this admitted class; they do not establish a successful effective law.

Exact three-node coordinate control

There is a small rational control without numerical square roots. Use the existing conservative P3 generator with beta=nu=w_0=1, ordered nodes (left,h,right), and tau=t/pi. Define

\[ r=(\delta x_{\rm left}-\delta x_h,\delta x_{\rm right}-\delta x_h), \qquad s=(\delta\theta_{\rm left}-\delta\theta_h, \delta\theta_{\rm right}-\delta\theta_h). \]

These four edge contrasts remove precisely the two common origins and give E_2=(||r||^2+||s||^2)/2. Direct subtraction of the full nodal rows yields

\[ J_{\rm edge}=\begin{pmatrix}0&-B_{\rm edge}\\B_{\rm edge}&0\end{pmatrix}, \qquad B_{\rm edge}=\begin{pmatrix}3/2&1/2\\1/2&3/2\end{pmatrix}. \]

Taking (r_left,s_left) visible and (r_right,s_right) hidden, the exact coordinate-memory owner gives blocks A=D=3R/2, B_h=C_h=R/2, with R=((0,-1),(1,0)). Consequently K(0)=-I/4; a hidden preparation contributes B_h h_0 already to the initial visible rate. These are retained edge coordinates, not newly created physical nodes. The shared P3 controls check the full-generator intertwining, storage and memory blocks exactly.

For a general observation, the coordinate-memory owner requires an explicit coordinate chart before partitioning; its indices are not arbitrary measurement rows. Its all-state Krylov companion tests exact closure, not a preparation-dependent approximation. Replacing an ideal square root or pi by an admitted binary64 value certifies only that represented matrix. The P3 control avoids this replacement; it does not select an effective coefficient, prove the error estimate or alter engine dynamics.

Equal channel observations impose a finite-horizon loss error floor

Retain the preceding state, clock, Euclidean norm and transpose preparation. Write R_k=((0,-I_k),(I_k,0)) for the quarter rotation between form and phase. The conservative consensus generator satisfies JR_r=R_rJ. The equal-channel observation and preparation obey

\[ \mathcal O R_r=R_m\mathcal O,\qquad R_r\mathcal E=\mathcal E R_m. \]

Therefore the complete visible propagator T(tau)=O exp(J tau) E commutes with R_m for every time. These are algebraic identities, not a near-consensus assertion about a nonlinear trajectory. For any proposed generator G, let F(tau)=exp(G tau) and use the spectral operator norm. Since R_m is orthogonal,

\[ \boxed{\|T(\tau)-F(\tau)\|_2 \geq\tfrac12\|[F(\tau),R_m]\|_2.} \]

Indeed [T,R_m]=0, so the norm of [F,R_m] is at most twice the response error. For the form-only-loss target above,

\[ [G,R_m]=\begin{pmatrix}0&\Gamma\\\Gamma&0\end{pmatrix}. \]

Nonzero loss thus prevents exact equality on any interval starting at zero: the propagator commutator has this nonzero first derivative there. A larger fine support or a different modal mixture respecting these same intertwiners cannot remove that obstruction. This is stronger than the earlier all-time finite-system argument within this narrower observation class.

A rational bound for a declared single-pair target

For m=1, fix gamma>0,omega>0 and G=((-gamma,-omega),(omega,0)) in the admitted clock. Its symmetric part is diag(-gamma,0), so ||exp(G u)||_2<=1 for u>=0. With M=gamma+omega, ||G||_2<=M and ||exp(G u)-I||_2<=M u. Put D=[G,R_1], so ||D||_2=gamma. The exact commutator identity and its remainder give

\[ [e^{G\tau},R_1]=\int_0^\tau e^{G(\tau-s)}D e^{Gs}\,ds, \qquad \|[e^{G\tau},R_1]-\tau D\|_2\leq\gamma M\tau^2. \]

To obtain the remainder, subtract D under the integral and split the difference as (exp(G(tau-s))-I)D exp(Gs)+D(exp(Gs)-I). Contractivity bounds the integrand norm by gamma M tau. Consequently, for any horizon T>=0,

\[ s_* = \min\{T,1/(2M)\},\qquad \boxed{\mathcal E_T\geq \frac{\gamma s_*(1-Ms_*)}{2}.} \]

This positive lower bound for gamma,T>0 uses an observation time within the requested horizon; it is not an assertion of the error at its endpoint. Zero loss or zero horizon gives zero and supplies no exclusion. The bound also applies when an independently bounded hidden family includes h_0=0, since enlarging a supremum cannot reduce it. It does not silently discard the nonzero hidden-state source. An approximation fails a declared uniform budget only when a justified lower bound is strictly greater than that budget.

The shared observation kernel implements this rational bound as bound_form_loss_observation_error. It checks skewness, commutation, the output intertwiner and isometry exactly on the supplied matrices, then uses their transpose preparation. Its scope can include other supplied generators obeying those identities; it does not authenticate that they came from a TNFR graph or an exact energy chart.

For an independently declared algebraic discriminator, fix the exact P3 edge observation above, h_0=0, the unit visible ball, gamma=omega=1, T=1/4 in clock tau, and error tolerance 1/32. Before any response evaluation the formula gives s_*=1/4 and lower bound 1/16>1/32. Thus this comparison fails its tolerance. These target coefficients and budget are control inputs, not inferred effective constants or a physical calibration. No trajectory, fit or floating transcendental coefficient is needed for the verdict. The independent controls also check underdamped, critical and overdamped target exponentials.

Sharp pointwise boundary and retained hidden source

For comparison with the conservative rational bound, set a=gamma/2, q=a^2-omega^2, and

\[ s_q(\tau)= \begin{cases} \sin(\sqrt{-q}\,\tau)/\sqrt{-q},&q<0,\\ \tau,&q=0,\\ \sinh(\sqrt q\,\tau)/\sqrt q,&q>0. \end{cases} \]

Because (G+aI)^2=qI, its exponential gives the sharper boundary

\[ \|T(\tau)-e^{G\tau}\|_2 \geq a e^{-a\tau}|s_q(\tau)|. \]

It is exactly the distance of exp(G tau) from the space of real 2 by 2 matrices commuting with R_1: remove its term -a exp(-a tau) s_q(tau) diag(1,-1) to attain the bound in that matrix space. Such a projection need not itself be realizable by a given fine TNFR system. Isolated zeros in the underdamped expression cannot establish agreement over an interval. The shared kernel uses the rational estimate and does not claim to evaluate or maximize this sharper transcendental expression.

For the exact P3 preparation, direct diagonalization of its two spatial modes gives, including the hidden pair,

\[ y(\tau)=\cos(\tau/2)e^{3\tau R_1/2}y_0 +\sin(\tau/2)e^{3\tau R_1/2}R_1h_0. \]

The homogeneous response attenuates both channels together before returning. It is not form-only damping. For an independent hidden ball of radius H_0, the fixed-time worst error equals ||T(tau)-exp(G tau)||_2+H_0*|sin(tau/2)|: the hidden map is a scalar times an orthogonal map, so it can align with the worst homogeneous error. Ignoring that source can only understate this full-family discrepancy.

What changing the observation or preparation does and does not show

The following exact P3 controls retain the same fine generator and clock. They delimit the theorem rather than select a new law.

• Observe (r_left,(3*s_left+4*s_right)/5) and use the observation's transpose as preparation. The rows remain orthonormal, but no longer intertwine the channel rotation. Here T'(0)=13R_1/10 and T''(0)=diag(-5/2,-197/50), which breaks the covariance. Nevertheless any orthogonal observation with transpose preparation has skew initial derivative O J O^T. Merely changing the readout cannot give a nonzero symmetric loss term at that initial instant.
• Observe the original left pair but prepare hidden (r_right,s_right) as (0,r_left). The resulting lift still gives the requested initial visible state, but supplies correlated hidden storage. It yields T'(0)=((-1/2,-3/2),(3/2,0))=G. However T''(0)=((-5/2,0),(-3/2,-5/2)) differs from G^2=((-2,3/4),(-3/4,-9/4)). Matching the initial damping derivative through chosen preparation does not establish the complete reduced law.

At a nonconsensus critical geometry, the positive-energy chart instead has cross-block D=B^(1/2)C^(1/2)/sqrt(beta) and its transpose. The canonical rotation still commutes when B,C commute, even if B!=C. Noncommuting Hessians can remove this particular symmetry; the energy generator remains skew and the transpose-preparation initial-derivative restriction remains. Neither nonconsensus nor asymmetric observation is by itself a loss mechanism.

The result excludes exact form-only damping for the admitted symmetric reduction and bounds its finite-horizon approximation error. It does not exclude loose budgets, tiny loss, different justified preparations, a declared initial layer, nonlinear effective laws or separately admitted limits. An independent structural reason for any changed premise is still required.

A structural channel distinction, with its constitutive boundary

The complete sine law already distinguishes the coordinates: signed form has quadratic edge storage, while circular phase has periodic edge storage. Their second variations are L and H=L_cos(theta_j*-theta_i*). This is a consequence of the supplied storage and reciprocal law, not a unique derivation from xdot=nu*p. The distinction must survive an actual declared observation before it explains a measured or reduced response.

At consensus H=L. At a uniform acute winding on an isolated cycle, H=cL with the same positive cosine on every edge. Even heterogeneous held capacities then give C=cB, so these examples retain the energy-channel rotation. Nonconsensus alone is insufficient. Joining a winding component through an aligned bridge introduces edges with different phase stiffness: the bridge has cosine one and winding edges have cosine less than one. The existing bridge-tree criticality proof admits this without a new law.

An admitted nodal tangent port without numerical matrix square roots

Keep zero loss, positive held capacities, strictly acute exact critical phase, the full supplied unit support and clock tau=w*t/pi. Let Z be the oriented node-by-edge incidence matrix, Q=diag(c_e) with exact positive edge cosines, and A=Z^T K Z. Edge energy coordinates obey

\[ u=Z^\mathsf T\delta x,\quad v=\sqrt\beta\,Q^{1/2}Z^\mathsf T\delta\theta,\quad E_2=\tfrac12(\|u\|^2+\|v\|^2),\qquad J=\begin{pmatrix}0&-D\\D^\mathsf T&0\end{pmatrix},\quad D=\frac{A Q^{1/2}}{\sqrt\beta}. \]

This is an exact energy embedding with constraints u in im(Z^T) and v in Q^(1/2)im(Z^T); arbitrary independent edge states are generally not nodal states. These subspaces are invariant under J. Now select an aligned bridge b. Its unit edge vector e_b is a cut vector: opposite constant nodal offsets on the two components produce only that edge difference. Since c_b=1, it is admissible in both channel subspaces. Means can be centered independently without changing the port. Thus observation (u_b,v_b) and its transpose preparation are admissible nodal orthonormal tangent maps. The two retained raw contrasts are the form gap and sqrt(beta) times the phase gap; no desired loss coefficient selects them.

For T(tau)=O exp(J tau) O^T, direct multiplication gives

\[ T'(0)=\frac{A_{bb}}{\sqrt\beta} \begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad T''(0)=-\frac1\beta \operatorname{diag}\left(\sum_e c_e A_{be}^2,\sum_e A_{be}^2\right). \]

All square roots cancel from the second derivative. Therefore

\[ \boxed{T_{uu}''(0)-T_{vv}''(0) =\frac1\beta\sum_e(1-c_e)A_{be}^2\geq0.} \]

With positive capacities on simple support this is strictly positive exactly when a nonaligned target edge meets an endpoint of the selected bridge. Remote winding alone can leave this second-order difference zero; that does not prove equality of the complete responses at later orders. Reversing the bridge orientation reverses both observed coordinates and preserves the jets.

For a fixed exact witness, take two C6 cycles at winding one, joined by an aligned bridge, unit capacities and beta=w=1. Declare both bridge nodes at turn zero and each cycle in turns j/6. Internal cosines are exactly 1/2, the bridge cosine is one, and symbolic sine currents cancel. Each bridge endpoint has full degree three; hence A_bb=2/3 and the four incident cycle edges have |A_be|=1/3. In clock tau=t/pi,

\[ T'(0)=\tfrac23R_1,\qquad T''(0)=\operatorname{diag}(-2/3,-8/9),\qquad T_{uu}''(0)-T_{vv}''(0)=2/9. \]

The consensus control on the same support has both second derivatives -8/9. In the winding witness, the phase response has the larger initial curvature magnitude; this is not a universal preference for form loss. The bridge reader reuses complete source and exact-target admission, retains held capacities and bounds these jets on an actual declared bridge. Its independent controls differentiate the shared full nodal law. The report concerns the supplied critical target, not a claim that the captured source has reached it.

Exact hidden elimination interprets these derivatives as retained exchange: the two zero-lag memory coefficients after removing the visible pair are -sum(c_e A_be^2)/beta+omega^2 and -sum(A_be^2)/beta+omega^2, where omega=A_bb/sqrt(beta). The complete kernel and initial hidden-state source still have to be retained. A second derivative is neither a Markov loss coefficient nor a kernel decay certificate. The field tetrad does not replace these state and law data.

Geometry does not remove the initial conservative obstruction

For any real skew energy generator J, orthonormal observation O and transpose preparation, T'(0)=O J O^T is skew. This remains true when the previous channel-rotation covariance fails. A form-only-loss target G=((-gamma,-omega),(omega,0)), with gamma>0,omega>0, has nonzero symmetric part and cannot agree on an interval starting at zero.

There is a quantitative statement independent of that rotation. Let N>=||J||_2, M=gamma+omega>=||G||_2. Fine and target propagators are contractions. Their integral Taylor remainders give

\[ \|T(\tau)-e^{G\tau}\|_2 \geq \gamma\tau-\tfrac12(N^2+M^2)\tau^2. \]

Indeed the symmetric part of T'(0)-G has norm gamma, while the two second-order remainders have norms at most N^2 tau^2/2 and M^2 tau^2/2. Evaluate at s=min(T,gamma/(N^2+M^2)) for a strictly positive uniform lower bound when T,gamma>0. The shared bound_orthogonal_form_loss_observation_error admits the skew and isometry hypotheses and uses the exact maximum absolute row sum as N. It preserves the more specific covariance bound as a separate method rather than silently weakening that reader's contract.

For the admitted acute sine tangent, 0<C<=B and ||B||_2<=2*max(nu_i) give N=2*max(nu_i)/sqrt(beta). The latter follows also from the Gershgorin bound for the similar matrix KL. Thus increasing finite support while retaining bounded capacities, fixed positive beta, the same target coefficients, clock and positive comparison horizon does not remove this initial-time floor. A controlled discarded initial layer, different justified preparation, or a limit with unbounded rates would be a changed problem, not a consequence of the geometric distinction.

There is also no disappearance of conservative modal structure. On the positive-energy quotient, the singular value decomposition D=U Sigma V^T transforms the two channels by different orthogonal maps to J'=((0,-Sigma),(Sigma,0)). Equivalently the original law commutes with R_D=((0,-UV^T),(VU^T,0)). This transformation can change the spatial readout; it does not erase the observed bridge distinction. It shows why that distinction is not irreversible loss. The two squared blocks DD^T,D^TD have identical spectra and traces despite their different directions.

The supported structural result is therefore unequal observed exchange and memory under a specified phase geometry. Selection of a memoryless loss law remains open. Likewise, interpreting form as phase-motion information requires the retained velocity and mean described by the form reconstruction result; it does not derive EPI from the other instantaneous variables.

A protected geometry can survive a changed phase-storage law

The existing alternative-storage family retains form balance and the consensus tangent without selecting cosine storage uniquely. Its first-moment distinction separates the information consumed by its primitive source. The present question is different: which pattern and response properties survive this already admitted constitutive freedom?

Keep exactly the preceding two C6 cycles, their supplied aligned bridge and unit capacities. Set e=0,beta=w=1, with no inputs or events, and retain tau=t/pi. For each independently specified finite epsilon>=0, use

\[ U_\epsilon(\delta)=1-\cos\delta+ \epsilon\left(\frac23-\cos\delta+\frac{\cos^3\delta}{3}\right),\quad j_\epsilon(\delta)=U_\epsilon'(\delta)=\sin\delta+\epsilon\sin^3\delta. \]

Writing S_epsilon,i=sum_j j_epsilon(theta_j-theta_i), the complete rows in this normalized clock and their storage are

\[ x'=K S_\epsilon(\theta),\qquad \theta'=KLx,\qquad K=\operatorname{diag}(1/d_i),\qquad E_\epsilon=\tfrac12x^{\mathsf T}Lx+V_\epsilon(\theta),\quad V_\epsilon=\sum_e U_\epsilon(\delta_e),\qquad E_\epsilon'=0. \]

These are different supplied complete laws with one common clock and support. The phase row is unchanged; pressure and phase storage change together. No coefficient is fitted to the bridge response.

Criticality and nonlinear protection remain

Give both cycles winding one with uniform oriented gap pi/3, align their bridge endpoints, and take constant form. The two cycle currents at each node cancel for every epsilon; the bridge current is zero. Thus the same geometry is an exact critical target throughout the family. Its storage is

\[ E_\epsilon^*=12U_\epsilon(\pi/3)=6+\tfrac52\epsilon. \]

The phase stiffness is

\[ U_\epsilon''(\delta)=\cos\delta\,[1+3\epsilon\sin^2\delta],\qquad c_\epsilon=U_\epsilon''(\pi/3)=\tfrac12+\tfrac98\epsilon,\qquad U_\epsilon''(0)=1. \]

Independently reversing either cycle's winding preserves cancellation, storage and curvature, by oddness of the current and evenness of the potential. The same proof therefore covers each pair of winding signs; (1,1) fixes the displayed example.

Hence H_epsilon=c_epsilon*L_cycles+L_bridge is positive definite modulo common rotation. The form Hessian remains L, positive modulo constant form. This gives a strict local storage minimum on the relative-state quotient for every member, rather than an assertion that two equal consensus tangents have the same nonlinear stability.

A common geometric neighborhood gives an explicit protection statement. Choose continuous phase perturbation lifts zeta=theta-theta_* and centered form xi=x-x_*. Fix their degree-weighted means; conservation of both means follows from sum S_epsilon=0 and sum Lx=0. The phase mean refers to this local lift, not to a global real-valued observable on the torus. With node-by-edge incidence Z, set

\[ r=\pi/12,\qquad m=\cos(5\pi/12)>0,\qquad \|Z^{\mathsf T}\zeta\|_\infty\le r. \]

Every lifted edge gap along the segment from the target lies strictly inside (-pi/2,pi/2) and has cosine at least m. Since epsilon>=0, the full phase Hessian on this neighborhood is at least mL. Criticality cancels the summed first-order phase term, although individual winding edges have nonzero first derivatives. Taylor's integral identity therefore gives the local bound

\[ \Delta E_\epsilon:=E_\epsilon-E_\epsilon^* \geq\tfrac12\|Z^{\mathsf T}\xi\|^2 +\tfrac m2\|Z^{\mathsf T}\zeta\|^2. \]

If the initial phase is in the strict neighborhood and its own full excess storage satisfies

\[ \boxed{\Delta E_\epsilon(0)<\frac{m\pi^2}{288},} \]

then conservation forbids a first boundary crossing: any boundary point has one phase-edge perturbation of magnitude r and excess storage at least mr^2/2. Throughout the solution,

\[ \|Z^{\mathsf T}\xi\|\le\sqrt{2\Delta E_\epsilon(0)},\qquad \|Z^{\mathsf T}\zeta\|\le \sqrt{2\Delta E_\epsilon(0)/m}<r. \]

Connectedness and the retained means bound the nodal relative state. The smooth complete field has global continuation, and the trajectory stays in this neighborhood with both winding identities (1,1) preserved. For each fixed epsilon this is nonlinear Lyapunov protection modulo the two common origins. The numerical radius is a declared sufficient proof neighborhood, not a universal TNFR scale. The threshold uses each law's own excess storage; it is not a common initial-amplitude ball for unbounded epsilon. Outside this neighborhood E_epsilon-E_epsilon* need not be nonnegative.

The same nodal bridge preparation has different responses

Observe the raw bridge form and phase gaps, with the same orientation in every model. Let p be the nodal preparation vector equal to -1/2 on the left cycle and 1/2 on the right. Then Z^T p=e_bridge after choosing that bridge orientation, so

\[ Lp=H_\epsilon p=z_{\rm bridge},\qquad p^{\mathsf T}Lp=p^{\mathsf T}H_\epsilon p=1. \]

The two nodal preparation columns (p,0) and (0,p) are therefore the same energy-orthonormal transpose preparation for every epsilon. There is no hidden change of experiment from using a different internal energy chart. This agreement is tangent normalization: equal finite phase offsets can have different nonlinear storage beyond quadratic order.

For T_epsilon(tau)=O exp(J_epsilon*tau)P, using those fixed nodal maps and the full tangent J_epsilon=((0,-K H_epsilon),(KL,0)), direct differentiation gives

\[ T_\epsilon'(0)=\tfrac23R_1,\qquad T_\epsilon''(0)= \operatorname{diag}(-2/3-\epsilon/2,-8/9),\qquad \boxed{T_{uu,\epsilon}''(0)-T_{vv,\epsilon}''(0) =\tfrac29-\tfrac\epsilon2.} \]

The bridge has unit stiffness and the four incident cycle edges have stiffness c_epsilon; their incidence-mobility entries retain magnitudes 2/3 and 1/3. Thus no angle fitting or trajectory is needed. At epsilon=4/9, H_epsilon=L and the complete tangent regains the same energy-channel rotation as consensus, despite its nonzero winding. The sign of the bridge distinction reverses across this parameter value.

This is not a loss of pattern stability or a phase transition: both Hessians remain positive on the quotient. Nor does it restore an identical nonlinear law. For example, at that value U_epsilon'''(pi/3)=-2*sqrt(3)/3, whereas the consensus edge expansion has zero third derivative. Coincident Hessians do not remove higher-order geometric response.

The protected critical identity is therefore robust within this supplied storage family, while the observed channel distinction is constitutively dependent. Protection assumes a nearby preparation; it neither forms the pattern from outside its sector nor gives attraction under zero loss. The previous cosine-specific capture, memory and rate bounds cannot be transferred unchanged. In particular, H<=L fails when epsilon>4/9. No physical constituent, preferred storage law or autonomous bridge event has been selected.

The shared assess_bridge_storage_family rebuilds the full nodal tangent and energy for the declared alternative, retaining the fixed bridge observation and preparation. Its outward local threshold and geometric radius state a conditional bound; they do not certify that the reference comparison's captured state satisfies this preparation. Admission controls and independent dynamics controls keep that distinction separate from the exact target and response identities.

Finite bridge observations distinguish the declared storage laws

The preceding jets distinguish the two complete laws mathematically. A finite observation also distinguishes them with an explicit nonzero error budget, retaining all twenty-four form and phase coordinates. Keep the two C6 cycles, aligned bridge, unit capacities, zero loss and absence of inputs/events. Compare eta=0 and eta=1/100 in j_eta(delta)=sin(delta)+eta*sin(delta)^3. Each law consumes its own reciprocal phase storage. The clock remains declared as tau=t/pi; this result does not cancel or identify an unknown physical clock scale.

Let theta_* be the common exact phase pattern with winding one on both cycles and zero bridge phase gap. Either cycle may independently reverse orientation: odd current cancellation, the internal cosine and the argument below are unchanged. Its angles are specified as rational turns times 2*pi, rather than treating rounded angles as exact. Let p equal -1/2 on the left cycle and +1/2 on the right. Prepare two nominal full states

\[ z_+(0)=(ap,\theta_*),\qquad z_-(0)=(-ap,\theta_*),\qquad a>0. \]

The support and all other coordinates are identical. Observe the same oriented bridge form gap u=x_right-x_left after positive scaled time h>0 in both preparations, and form the finite statistic

\[ \boxed{C_{a,h}=\frac{2a-u_+(h)+u_-(h)}{ah^2}.} \]

Both a and h are declared before reading the response. No pressure is reconstructed from this statistic. The two observations concern forward evolutions of opposite preparations, not a physical negative-time experiment or a changed law.

The finite-amplitude quadratic coefficient is exact

Form reversal R(x,theta)=(-x,theta) reverses the complete flow and the form readout. Since z_-(0)=Rz_+(0), it follows that u_-(h)=-u_+(-h). The statistic is therefore the centered-time curvature of one mathematical solution. This identity needs no ring-swap symmetry or nonlinear reduction to an observed shell.

At the common target the full current sum vanishes. Write b for the oriented bridge incidence column. Then Lp=b, and differentiating the complete field at the finite preparation gives

\[ u_+(0)=a,\quad u_+'(0)=0,\quad u_+''(0)=-a\,b^{\mathsf T}KH_\eta KLp =-a\left(\frac23+\frac\eta2\right). \]

Indeed Kb is -1/3,+1/3 at the bridge ports and zero elsewhere. Its bridge contrast contributes 4/9 to the Hessian quadratic; the four incident cycle edges contribute 4c_eta/9, with c_eta=1/2+9eta/8. This computation uses the full nodal field. The coefficient is exact for every finite a, because the initial phases are exactly the target and the phase row is linear in form. There is no additional limit-in-amplitude truncation. Amplitude-dependent higher time derivatives still enter the remainder below.

A full-state nonlinear remainder

The following bounds hold uniformly for 0<=eta<=1/100. With c=cos(delta),

\[ j_\eta'(\delta)=c(1+3\eta-3\eta c^2),\qquad |j_\eta'|\le1. \]

For 0<=c<=1, the right-hand polynomial is increasing because its derivative is at least 1-6eta>0, and its maximum is one; oddness in c gives the absolute bound. The harmonic identity j_eta=(1+3eta/4)sin(delta)-(eta/4)sin(3delta) further gives

\[ |j_\eta''|\le J_2:=103/100,\qquad |j_\eta'''|\le J_3:=43/40. \]

In the full maximum norm, the complete field has global Lipschitz bound two. Its equilibrium is (0,theta_*), so both ideal trajectories, in both time directions through |tau|<=h, obey

\[ \|x\|_\infty,\ \|\theta-\theta_*\|_\infty \le R_h:=\frac a2e^{2h}. \]

Let g(theta)=KS_eta(theta) and A=KL. Then ||A||_infinity<=2, ||Dg||<=2, ||D^2g||<=4J_2 and ||D^3g||<=8J_3, for the corresponding induced multilinear norms. Along the whole solution, theta'=Ax, theta''=Ag and theta'''=A Dg theta' have bounds 2R_h,4R_h,8R_h respectively. Thus the identity

\[ x^{(4)}=D^3g[\theta',\theta',\theta']+ 3D^2g[\theta',\theta'']+Dg\theta''' \]

and the two-coordinate bridge readout give

\[ |u^{(4)}|\le U_4:=128J_3R_h^3+192J_2R_h^2+32R_h. \]

The symmetric Taylor formula cancels the odd powers and proves the finite prediction

\[ \left|C_{a,h}-\left(\frac23+\frac\eta2\right)\right| \le \frac{U_4h^2}{12a}. \]

No sampled trajectory, truncated linear shell or assumed future symmetry of perturbed sources enters this inequality.

Independent preparation and observation errors

Allow errors of size at most rho in each of the twenty-four initial form and lifted-phase coordinates of each preparation. These errors may break every preparation symmetry. The full-field Lipschitz bound compares each actual source with its ideal counterpart, giving bridge-readout error at most 2rho exp(2h) per run. Let each observed bridge gap additionally have absolute reading error at most sigma. Then

\[ \boxed{\left|\widehat C_{a,h}-\left(\frac23+\frac\eta2\right)\right| \le E:=\frac{U_4h^2}{12a} +\frac{4\rho e^{2h}}{ah^2} +\frac{2\sigma}{ah^2}.} \]

Here sigma bounds a complete bridge-gap reading; an instrument returning two individual node readings must first combine their error bounds. Numerical phase representation error belongs in rho. Capacities, support, clock and the declared amplitude in the statistic remain fixed premises.

For both ideal and perturbed trajectories the full-window phase-edge displacement from the target is at most 2(R_h+rho exp(2h)). Requiring it to be less than pi/12 places both complete flows inside the preceding protected local chamber, with all environmental and transverse errors retained. This is an additional verified domain statement, not a loss of hidden coordinates in the error calculation.

One analytically admitted finite budget

Choose, without evaluating a reserved trajectory,

\[ a=\frac1{10},\qquad h=\frac1{50},\qquad \rho=10^{-10},\qquad\sigma=10^{-8}. \]

The elementary bound exp(2h)<=1/(1-2h)=25/24 gives

\[ R_h\le\frac5{96},\quad U_4\le\frac{122899}{55296},\quad 2(R_h+\rho e^{2h})\le\frac{166666667}{1600000000}<\frac\pi{12}. \]

The separately retained error budgets are

\[ E_{\rm time}\le\frac{122899}{165888000},\qquad E_{\rm preparation}\le\frac1{96000},\qquad E_{\rm reading}\le\frac1{2000}, \]

so E<=207571/165888000, approximately 0.001251272. The two response intervals have centers 2/3 and 403/600; their strict separation is at least

\[ \boxed{\frac1{200}-2E\ge\frac{207149}{82944000}>0.00249745.} \]

This closes finite-amplitude, finite-time discrimination at the declared uncertainty scale by an analytic prediction. It does not require selecting smaller amplitudes after inspecting a response. An observed value outside both intervals would contradict the admitted two-model comparison or its preparation/error premises; it would not justify selecting the nearer law. No actual physical measurement is supplied here. Without an independent measurement and clock bridge, the result selects neither law as physical, and adds no formation or indefinite-lifetime claim.

The detached reader assess_bridge_finite_law_discrimination returns BridgeFiniteLawDiscrimination. It admits the common support, symbolic target turns and the declared finite budget, reuses both storage-family owners, and exposes the two full preparations and separate error terms. Its source comparison anchors the law and support; its captured state is not silently used as the target. This reader evaluates the analytic inequalities rather than forecasting a trajectory.

A finite two-channel observation separates law from bounded clock uncertainty

The previous form-only coefficient cannot by itself distinguish a changed law from an unknown time scale. Reuse its support, full-state preparations, derivative bounds and error meanings, and add the existing phase-preparation column. Let s be the sampling clock and assume

\[ \tau=ks,\qquad 0<k_{\rm lo}\le k\le k_{\rm hi}, \]

where k is constant and common to all three experiments. This interval must be declared independently of their responses. Neither an unbounded unknown scale nor a preparation-dependent clock factor is covered. The argument does not require knowing the particular k within the interval.

Alongside the opposite form preparations (ap,theta_*) and (-ap,theta_*), prepare the full phase experiment

\[ z_P(0)=(0,\theta_*+ap). \]

Use the signed bridge phase relative to its target lift, denoted v. If the target bridge phases differ by 2*pi*m, subtract that declared integer turn. The admitted phase chamber makes this the same signed principal gap near zero, and v(0)=a. An unsigned phase difference or an undeclared choice of branch cannot replace this readout.

At the same positive sampling time s, form

\[ \widehat C_F^{(s)}= \frac{2a-\widehat u_+(s)+\widehat u_-(s)}{as^2},\qquad \widehat C_P^{(s)}=\frac{2(a-\widehat v(s))}{as^2},\qquad \widehat Q=\frac{\widehat C_F^{(s)}}{\widehat C_P^{(s)}}. \]

Each column is normalized by its declared amplitude in its own model coordinate. These are three positive-time preparations and three readings; no derivative or zero-time response is measured. The phase denominator must be certified positive before taking the ratio.

The exact finite phase coefficient and its remainder

The phase experiment has zero initial form, so its initial state is fixed by form reversal. Reversibility makes v(tau) exactly even. Initially only the bridge contributes a nonzero net current: the unchanged cycle gaps still cancel. In the preceding incidence notation,

\[ v''(0)=-j_\eta(a)\,b^{\mathsf T}KLKb =-\frac89j_\eta(a). \]

Thus its finite-amplitude curvature center is

\[ P_\eta(a)=\frac89\frac{j_\eta(a)}a, \qquad F_\eta=\frac23+\frac\eta2 \]

for the phase and form channels respectively. The finite sin(a)/a factor remains in P_eta; replacing it by one would introduce an unaccounted amplitude approximation.

Let h=ks, h_lo=k_lo*s, h_hi=k_hi*s, and choose an outward G>=exp(2h_hi). Set R=aG/2. Differentiating the full phase row gives

\[ \theta^{(4)}=A\left(D^2g[\theta',\theta']+Dg\theta''\right), \qquad |v^{(4)}|\le M_P:=64J_2R^2+32R. \]

The form-channel bound remains M_F=128J_3R^3+192J_2R^2+32R. Both use the complete twenty-four-state field throughout the common window, not a closed two-channel approximation.

For each channel, normalized by the actual structural h^2, the error is bounded by

\[ E_c(h)=\frac{M_c}{12a}h^2+ \frac{4\rho G+2\sigma_c}{ah^2},\qquad c\in\{F,P\}. \]

Here sigma_F bounds each of the two bridge-form readings and sigma_P the bridge-phase reading. The latter receives a factor two in the statistic; independent full-coordinate preparation errors give the stated 4rho G bound in both channels. The actual preparations need not preserve the nominal reversal symmetries.

Importantly, the same h appears in both error terms. As a function of z=h^2, Az+B/z is convex on the positive axis. A uniform bound is therefore

\[ \boxed{E_c^*=\max\{E_c(h_{\rm lo}),E_c(h_{\rm hi})\}.} \]

Separately maximizing the time term at h_hi and the uncertainty term at h_lo is valid but unnecessarily loses their shared-time constraint.

Algebraically, C_c^(s)=k^2 C_c^(h) for each finite observed statistic, so their common normalization factor cancels in the ratio. This does not make the finite response independent of the clock: the response and its remainders still depend on h, bounded above by the admitted interval. If P_eta^-<=P_eta<=P_eta^+, and both F_eta-E_F^*>0 and P_eta^--E_P^*>0, interval division gives

\[ \boxed{\widehat Q\in \left[\frac{F_\eta-E_F^*}{P_\eta^++E_P^*}, \frac{F_\eta+E_F^*}{P_\eta^--E_P^*}\right].} \]

This admits the actual noisy denominator. Nonpositive or unadmitted denominators provide no law-discrimination verdict.

The same finite budget tolerates ten-percent clock uncertainty

Keep the preceding amplitude, sample duration and error sizes,

\[ a=\frac1{10},\quad s=\frac1{50},\quad \rho=10^{-10},\quad\sigma_F=\sigma_P=10^{-8},\quad k\in\left[\frac9{10},\frac{11}{10}\right]. \]

The new premise is the independently supplied clock interval. No reserved response is evaluated to select it or to change the earlier amplitude or precision budget. Now h_lo=9/500, h_hi=11/500. One may take the elementary rational bounds

\[ G=\frac{250}{239},\quad R=\frac{25}{478},\quad M_F\le\frac{30502250}{13651919},\quad M_P\le\frac{105900}{57121}. \]

The full perturbed phase-edge displacement is at most 2(R+rho G)<=500000001/4780000000<pi/12, preserving the common signed phase chart throughout all three experiments. The two complete endpoint error maxima are bounded by

\[ E_F^*\le\frac{2622585209}{1982258638800},\qquad E_P^*\le\frac{8084073}{6911641000}. \]

Using only a-a^3/6<=sin(a)<=a-a^3/6+a^5/120 to enclose the exact phase centers, both denominator lower bounds exceed 0.8862. Conservatively rounded ratio predictions are

Declared law	Finite ratio interval, uniform in the clock factor
eta=0	[0.74877366184, 0.75373642785]
eta=1/100	[0.75432554878, 0.75930247738]

The unrounded exact separation exceeds 1/2000. A shared change of clock within the declared interval cannot make these admitted finite observations indistinguishable. This is a three-preparation analytic comparison with explicit noise and denominator bounds, not a new trajectory or a fit of clock or pressure to data. It still does not select either law as physical, cover arbitrary clock uncertainty, identify the complete current function, or establish formation from these already patterned preparations.

The detached assess_bridge_clock_law_discrimination returns BridgeClockLawDiscrimination. It reuses the same full-support and symbolic-target admission, retains all three preparations and the signed target bridge turn, and evaluates separate endpoint error budgets before admitting the ratio. It does not estimate the clock from the response, reconstruct pressure, or execute a new trajectory.

Exact bridge memory and the boundary of local loss

Return to the normalized-sine member epsilon=0, with the same two-C6 winding-one target, aligned bridge, unit capacities, e=0,beta=w=1 and tau=t/pi. This section concerns that member's exact linear tangent. Reflection of each cycle and interchange of the two cycles preserve both tangent Laplacians. No corresponding finite-amplitude nonlinear reduction is asserted.

An eight-coordinate nodal reduction

For either form or lifted phase perturbations, let d_k be the right-cycle mean minus the left-cycle mean over vertices at cycle distance k=0,1,2,3 from the bridge endpoint. The shell sizes on each cycle are 1,2,2,1. Define r=(d_0,d_1-d_0,d_2-d_1,d_3-d_2) for form and s by the same map for phase. Then the full nodal tangent induces exactly

\[ \binom{r'}{s'}=J_8\binom r s,\qquad J_8=\begin{pmatrix}0&-A_4Q_4\\A_4&0\end{pmatrix},\quad Q_4=\operatorname{diag}(1,1/2,1/2,1/2), \]

\[ A_4=\begin{pmatrix} 2/3&-2/3&0&0\\ -2/3&7/6&-1/2&0\\ 0&-1/2&1&-1/2\\ 0&0&-1/2&3/2 \end{pmatrix}. \]

This is an exact intertwining, not independent admission of arbitrary edge coordinates. A nodal section is obtained by cumulatively reconstructing d_k and setting the left and right shell values to -d_k/2 and d_k/2. The represented sector is odd under cycle interchange and even under cycle reflection. Its edge-energy coordinates are the bridge vector and three orthonormal cut vectors, each with coefficients of magnitude 1/2 on the four edges of one shell. The other symmetry sectors have identically zero bridge observation and remain invariant under the tangent. Thus arbitrary fine initial perturbations have the same bridge response as their shell projection; the section does not reconstruct those discarded states or means.

The represented sector has storage E_8=(r^T r+s^T Q_4 s)/2, with metric diag(I_4,Q_4), and E_8'=0. Invisible fine modes can retain additional storage. With v=Q_4^(1/2)s, the eight-coordinate generator is skew and its bridge pair y=(r_0,s_0) is an orthonormal energy observation. Setting the other six coordinates to zero is precisely the preceding transpose preparation: opposite constant offsets on the two cycles. A general hidden preparation is retained below instead of silently setting it to zero.

Exact kernel and hidden initial-state source

Order the hidden coordinates as h=(r_1,r_2,r_3,s_1,s_2,s_3) and put

\[ F=\begin{pmatrix}7/6&-1/2&0\\-1/2&1&-1/2\\0&-1/2&3/2\end{pmatrix}, \quad b=\begin{pmatrix}-2/3\\0\\0\end{pmatrix},\quad A_0=\tfrac23R_1,\quad R_1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}. \]

\[ y'=A_0y+Bh,\quad h'=Cy+D_hh,\qquad B=\begin{pmatrix}0&-b^{\mathsf T}/2\\b^{\mathsf T}&0\end{pmatrix},\quad C=\begin{pmatrix}0&-b\\b&0\end{pmatrix},\quad D_h=\begin{pmatrix}0&-F/2\\F&0\end{pmatrix}. \]

These are rational matrices in a declared nonidentity hidden metric diag(I_3,I_3/2). Variation of constants gives the complete causal identity

\[ \boxed{y'(\tau)=A_0y(\tau)+B e^{D_h\tau}h_0 +\int_0^\tau \mathcal K(\tau-\sigma)y(\sigma)\,d\sigma, \qquad \mathcal K(\ell)=B e^{D_h\ell}C.} \]

Every lag ell below uses the normalized clock tau. Let C_F(ell)=cos(F*ell/sqrt(2)), S_F(ell)=sin(F*ell/sqrt(2)) and c(ell)=e_1^T C_F(ell)e_1, s(ell)=e_1^T S_F(ell)e_1, where e_1=(1,0,0). Then

\[ \boxed{\mathcal K(\ell)=\frac49 \begin{pmatrix}-c(\ell)/2&s(\ell)/\sqrt2\\-s(\ell)/\sqrt2&-c(\ell)\end{pmatrix}.} \]

For independently declared h_0=(r_h,s_h), the retained source is

\[ B e^{D_h\ell}h_0= \binom{-b^{\mathsf T}S_F(\ell)r_h/\sqrt2-b^{\mathsf T}C_F(\ell)s_h/2} {b^{\mathsf T}C_F(\ell)r_h-b^{\mathsf T}S_F(\ell)s_h/\sqrt2}. \]

In particular K(0)=diag(-2/9,-4/9) and K'(0)=((0,7/27),(-7/27,0)). The homogeneous response satisfies T'(0)=A_0 and T''(0)=A_0^2+K(0), recovering the bridge jets above. The two-channel kernel cannot be replaced by its diagonal at zero lag. Its off-diagonal entries and hidden source carry additional information.

These eight coordinates are minimal for the all-state linear bridge response. Successive output derivative levels recover the first, second and third hidden shell pairs because both neighbor entries of the tridiagonal F and the bridge coupling are nonzero. The cumulative observation ranks are 2,4,6,8; no other fine symmetry sector enters the bridge output.

Actual hidden frequencies and nondecay

The eigenvalues lambda_j of F are the three simple roots of

\[ P(\lambda)=12\lambda^3-44\lambda^2+47\lambda-13, \]

one each in the rational intervals (2/5,1/2), (6/5,4/3) and (7/4,2). Sign changes prove existence in these disjoint intervals and the cubic degree exhausts the roots. The exact spectral weights are

\[ \mu_j=\frac{3(4\lambda_j^2-10\lambda_j+5)}{P'(\lambda_j)},\quad \mu_j>0,\quad \sum_j\mu_j=1,\qquad \omega_j=\lambda_j/\sqrt2, \]

\[ c(\ell)=\sum_j\mu_j\cos(\omega_j\ell),\qquad s(\ell)=\sum_j\mu_j\sin(\omega_j\ell). \]

The weights follow by residues of e_1^T(zI-F)^(-1)e_1=3(4z^2-10z+5)/P(z). Positivity follows independently from the symmetric spectral theorem: the nonzero off-diagonal entries of F prevent an eigenvector from vanishing at its first coordinate. The slowest hidden frequency satisfies omega_1<1/(2*sqrt(2))<2/3, and its weight exceeds 75/439>1/6, by bounding the displayed residue on (2/5,1/2). An appreciably weighted hidden mode is therefore slower than the visible instantaneous exchange; this is not an admitted fast reservoir.

Distinct positive frequencies also give

\[ \lim_{T\to\infty}\frac1T\int_0^T c(\ell)^2\,d\ell =\lim_{T\to\infty}\frac1T\int_0^T s(\ell)^2\,d\ell =\tfrac12\sum_j\mu_j^2\geq\tfrac16. \]

Consequently the kernel does not decay. Its ordinary integral over the positive half-line does not converge. The fixed geometry supplies no limit parameter making the hidden memory short; rescaling a common clock changes all frequencies together. These statements do not exclude an explicitly bounded loose approximation on a separately admitted finite window.

Static response and the first frequency correction are conservative

For Re(z)>0, let Khat(z)=B(zI-D_h)^(-1)C. Direct inversion gives

\[ F^{-1}=\frac1{13} \begin{pmatrix}15&9&3\\9&21&7\\3&7&11\end{pmatrix},\qquad \widehat{\mathcal K}(0)=-\frac{20}{39}R_1,\qquad A_0+\widehat{\mathcal K}(0)=\frac2{13}R_1. \]

Here Khat(0) denotes the analytic zero-frequency continuation, equivalently the Abel limit with exp(-epsilon*ell) as epsilon tends to zero from above. It is not a convergent memory integral. This static Schur complement changes the reactive exchange and contributes no damping.

The next frequency coefficient retains a positive rate matrix:

\[ zI-A_0-\widehat{\mathcal K}(z) =zM_{\rm eff}-\frac2{13}R_1+O(z^2),\qquad \boxed{M_{\rm eff}=\operatorname{diag}(309/169,449/169).} \]

Indeed M_eff=I+B D_h^(-2)C and e_1^T F^(-2)e_1=315/169. The same matrix is the storage metric of the stationary hidden section r_h=-F^(-1)b*r_0, s_h=-2F^(-1)b*s_0. That section need not be dynamically invariant. The formal local equation M_eff*y'=(2/13)R_1*y preserves y^T M_eff y/2; the expansion therefore supplies a reactive frequency correction, not a time-domain approximation theorem or a selected loss law.

The coefficient expansion has a quantitative bound on the real Laplace axis. Put S=F/sqrt(2). The leading principal minors of S^2-I/16 are 107/144,167/768,1543/36864, so its smallest frequency exceeds 1/4. In hidden energy coordinates D_h is skew and ||(zI-D_h)^(-1)||_2<=1/sqrt(z^2+omega_min^2) for real z>=0. The resolvent identity (zI-D_h)^(-1)=-D_h^(-1)-z*D_h^(-2)+z^2*D_h^(-2)*(zI-D_h)^(-1) and energy coupling norm squared 4/9 therefore give

\[ \left\|zI-A_0-\widehat{\mathcal K}(z) -\left(zM_{\rm eff}-\tfrac2{13}R_1\right)\right\|_2 \leq \frac{256}{9}z^2\qquad(z\geq0). \]

This bounds the coefficient pencil, not its inverse transfer, imaginary-axis response or a time-domain error. The formal collective frequency 26/sqrt(309*449) is below the hidden frequencies; comparing them only with the direct rate 2/3 does not exclude a separately prepared slow sector. Its admissible preparation and approximation error remain separate questions.

The shared bridge-memory reader admits the source support, capacities, law and declared critical target, checks the nodal intertwining, and reuses the shared exact observation and coordinate-memory owners. It supplies matrices, not a trajectory, fitted coefficient or observed initial perturbation. The comparison's captured state is neither automatically the target nor the independent tangent initial state. Nodal admission controls and kernel controls check those distinct contracts.

The bounded question closes with an exact memory and its preparation source. No fast/decaying-memory premise or controlled discarded initial layer has been supplied for positive local loss. The earlier initial-response error floor remains applicable, while arbitrary finite-window tolerances are not declared impossible. Memory can remain part of a collective model; deriving damping is one scoped route, not a prerequisite for every persistent pattern.

Periodic, constant-frequency and universal are different requirements

The P3 tangent's common period is a property of its two commensurate modal rates. It does not make a universal TNFR pulse. Even with the same reversible law and common capacity, a consensus C5 tangent has the two nonzero normalized Laplacian eigenvalues

\[ \frac{5-\sqrt5}{4},\qquad \frac{5+\sqrt5}{4}, \]

whose ratio is (3+sqrt(5))/2. A preparation exciting both frequencies has no nonzero common tangent period. This is a finite conservative superposition, not dissipative decay; it does not establish the corresponding finite-amplitude nonlinear motion.

The exact nonlinear P2 result separately has a constant return period along each supplied libration, but that period depends on its energy, capacities and storage scale. Its instantaneous relative phase speed varies, vanishing at the turning points. Neither result selects one universal frequency, an absolute clock, or spontaneous motion from an equilibrium.

Nor does repeated exchange or a collection of modes establish fractality. A recursive scale claim needs its own state map, composition law and preservation or controlled reduction of dynamics. The existing scale-bridge owner separates exact intertwining from discarded-state memory for its declared models; pure-EPI closure cannot simply be transferred to coupled form/phase dynamics. Those are the appropriate obligations for a later scale question, not consequences silently added to the present conservative-memory result.

The conservative-path contract owns assess_sine_path_memory and its shared exact-matrix integration. That reader requires the common-positive-capacity P3 preparation for its combined report. The wider nonlinear edge-flux identity above does not extend its accepted API domain to heterogeneous capacities.

10. Almost-everywhere recurrence of the full nonlinear reversible law

State, law and the measure being used

Fix a finite connected simple unit-support graph with n>=2, strictly positive held capacities, w,beta>0 and declared e=0. There are no inputs, support changes, resets, clipping or other operator events. Use the complete sine rows in the structural clock,

\[ \dot x=aKS(\theta),\qquad \dot\theta=bKLx,\qquad a=w/\pi,\quad b=w/(\beta\pi), \]

on the state manifold

\[ \mathcal M=\mathbb R^n\times(\mathbb R/2\pi\mathbb Z)^n. \]

The form coordinates are real and the phases are circles. Recurrence below means nearness in this full product state space, rather than return of separately chosen real phase lifts. Use Lebesgue measure in form and Haar measure on the phase torus, normalized to total torus measure one. A different constant normalization has the same null sets.

This theorem uses the displayed constant degree/capacity mobility in both rows. Conservation of the same storage under another reciprocal mobility does not establish conservation of this measure or the bounded mean slab. The alternative-mobility audit owns those separate obligations.

The two conserved quantities needed here are

\[ E=\frac12x^\mathsf TLx+\beta V(\theta),\qquad m(x)=\frac{\sum_i\rho_i x_i}{\sum_i\rho_i},\qquad \rho_i=\frac{d_i}{\nu_i}>0. \]

The storage identity gives Edot=0, and pairwise cancellation of the sine currents gives

\[ \dot m =\frac{a\sum_i S_i}{\sum_i\rho_i}=0. \]

No weighted mean of unwrapped phases is needed as a global torus coordinate. In particular, arbitrary real capacity ratios do not make such a lifted mean a single-valued observable on the torus.

Compact invariant slabs and two-sided continuation

Choose an energy ceiling E_max>0 and an interval m_lo<m_hi of positive width. Define the family

\[ \mathcal S= \{(x,\theta):E(x,\theta)\le E_{\max}, \quad m_{\rm lo}\le m(x)\le m_{\rm hi}\}. \]

Let D_G be the unweighted graph diameter. A shortest path and Cauchy--Schwarz give

\[ |x_i-x_j|^2 \le {\rm dist}_G(i,j)\sum_{\text{path}}(x_u-x_v)^2 \le 2D_G E_{\max}. \]

Since m is a positive weighted average of the forms,

\[ |x_i-m|\le\sqrt{2D_G E_{\max}} \le R,\qquad R=\sqrt{2(n-1)E_{\max}}. \]

Thus every state in the slab satisfies the explicit coordinate bound

\[ \boxed{ m_{\rm lo}-R\le x_i\le m_{\rm hi}+R\quad\text{for every node}.} \]

The slab is closed inside this bounded form box times a compact phase torus, so it is compact. Its product volume is finite; with the stated Haar normalization it is at most (m_hi-m_lo+2R)^n. It also has positive volume: choose uniform form with mean strictly inside the interval and phase consensus. There E=0<E_max, and a sufficiently small open neighborhood stays within both strict inequalities.

The smooth vector field preserves the two defining quantities in both time directions. A local solution starting in the slab cannot leave it. On this compact set the smooth field is bounded and locally Lipschitz, excluding finite-time escape and supplying continuation for all positive and negative times. Hence the flow Phi_t is defined on the whole slab and maps it bijectively onto itself.

Exact volume preservation

In local phase charts the full divergence vanishes directly:

\[ \operatorname{div}F =\sum_i\frac{\partial\dot x_i}{\partial x_i} +\sum_i\frac{\partial\dot\theta_i}{\partial\theta_i}=0. \]

Each form rate depends on phase rather than form; each phase rate depends on form rather than phase. The smooth periodic phase dependence makes this calculation compatible across torus charts. The flow Jacobian obeys

\[ \frac d{dt}\det D\Phi_t =(\operatorname{div}F)\circ\Phi_t\;\det D\Phi_t=0, \]

with initial determinant one. Thus the full product volume, and its restriction to the invariant slab, is preserved. This proof uses the nonlinear sine field itself, not a tangent matrix or a statistical assumption of random mixing.

Recurrence from finite measure: an elementary argument

Fix any positive time increment h and write T=Phi_h. This is an invertible measure-preserving map of a finite-measure space. For a measurable subset A of the slab, let B subset A consist of the states that never revisit A at any strictly positive integer iterate. The sets B,T(B),T^2(B),... are pairwise disjoint: an intersection would give one point of B a later return to A. They all have the same measure. Finite total measure therefore forces B to have measure zero.

States in A that return only finitely often are also a null set: their last return lies in B, so they belong to the countable union union_{j>=0} T^(-j)(B). Almost every point of A consequently revisits A infinitely many times.

Apply this result to a countable basis of neighborhoods in the compact metric slab, and remove the countable union of exceptional null sets. For each remaining state, use successively smaller basis neighborhoods containing that state and choose increasing return iterates. This gives

\[ \boxed{ \Phi_{n_jh}(z)\longrightarrow z,\qquad n_j\longrightarrow\infty, \quad\text{for almost every }z\in\mathcal S.} \]

This is recurrence of the full nonlinear state on the phase torus. The full-measure set may depend on the fixed increment h; the proof does not require one set chosen simultaneously for every positive real increment. A single declared increment already proves continuous-time recurrence along a sequence of future times.

Almost all these recurrent states are nonstationary. Indeed, strict positivity of K makes thetadot=0 imply Lx=0. Connectivity then gives x=c*1, a one-dimensional subspace of R^n with zero Lebesgue measure for n>=2. The equilibrium set is contained in that form subspace times the phase torus, and therefore has zero ambient product measure. This does not require classification of the additional condition S(theta)=0.

One can exhaust the whole finite-state manifold by the countable invariant slabs E_max=j and -j<=m<=j, with positive integers j. Thus the same almost-everywhere conclusion holds in the full product measure as well. The declared finite-volume slab remains useful for a bounded, inspectable family admission.

An exact exceptional preparation prevents a chosen-state claim

Almost everywhere is essential. On the same reversible P2 law, take u=x_0-x_1=2*sqrt(beta) and delta=theta_1-theta_0=0 initially, with sigma=nu_0+nu_1>0. Its storage is exactly 2*beta, the separatrix value. Put

\[ \Omega=\frac{w\sigma}{\pi\sqrt\beta}. \]

The complete relative equations of Section 8 admit the explicit nonstationary solution

\[ u(t)=2\sqrt\beta\,\operatorname{sech}(\Omega t),\qquad \delta(t)=\pi-4\arctan\!\bigl(e^{\Omega t}\bigr). \]

At t=0 these are the stated form contrast and equal phases. Direct differentiation gives udot=a*sigma*sin(delta) and deltadot=-b*sigma*u. As t->infinity, form contrast tends to zero and relative phase tends to -pi. The full reconstruction from Section 8 approaches an antipodal equilibrium and does not return near its initial form contrast.

This exception needs no irrational stored phase. For example, beta=1,nu_0=nu_1=1,x=(1,-1),theta=(0,0) is a wholly rational preparation with E=2 and weighted form mean zero. It lies inside the slab E_max=3,-1<=m<=1 but is nonstationary and nonrecurrent. Slab membership and the almost-everywhere theorem therefore cannot certify recurrence of that selected state.

What this establishes for a pulse investigation

The selected conservative nonlinear law has a finite-volume mechanism for nonstationary recurrence that is not confined to the exact P2 librations or tangent modes. It does not follow that all states recur, that every node is active, or that a nontrivial observable exhibits a periodic signal. An exact common period, a bounded return time, a return of real phase lifts, persistent winding identity and a universal frequency require their own hypotheses and proofs.

The measure is the declared ambient product measure. Its almost-everywhere statement is not automatically a theorem on a chosen fixed-energy surface, fixed-mean hyperplane, synchronized submanifold or specially prepared family. An absolutely continuous preparation distribution supported in the slab inherits recurrence with probability one. No such distribution is supplied by a finite binary grid, a list of random seeds, a singular prior or the existence of a numerical sampler. None makes a captured point recurrent by itself.

Equilibria remain stationary by uniqueness, and the exceptional separatrix remains a valid counterexample. The theorem neither creates initial activity nor selects microscopic e=0. It also supplies no attracting limit cycle: volume and storage remain conserved. The separate acute identity barrier provides invariant phase-identity neighborhoods that support this recurrence. It does not assign an exact periodic waveform to their members. Recurrence, geometric trapping and periodic-orbit stability retain distinct hypotheses.

With a separate open acquisition mechanism, the same theorem also implies repeated finite joint-matching episodes on the fixed doubled-C5 law. Strict nonmutual returns separate those episodes; the almost-everywhere conclusion concerns its full-dimensional preparation neighborhood, not a recurrence verdict for the exact crossing source.

Persistence does not establish capture from outside

There is a precise formation boundary under the same complete conservative flow. If a family U is invariant in both time directions, then

\[ \Phi_T(z)\in\mathcal U\quad\Longleftrightarrow\quad z\in\mathcal U. \]

The reverse implication is invariance; the forward implication follows by applying Phi_(-T) to the endpoint. In particular, a certificate that a trajectory lies in such a protected identity family cannot establish that it entered that family from outside. This statement requires two-sided invariance; a merely forward-invariant dissipative capture region need not have this property.

Recurrence supplies a further obstruction. Let A be a closed target in an admitted conservative slab, and let z be a recurrent state outside A. If dist(Phi_t(z),A) tended to zero as t->infinity, a returning sequence Phi_(t_j)(z)->z and continuity of distance would give dist(z,A)=0, contradicting closedness and z not in A. Consequently, the basin converging to A from outside it is confined to the recurrence-null exceptional set. This uses the ambient measure above, not a measure on an arbitrarily selected preparation surface.

Neither result excludes temporary organization, entry into a noninvariant observational region, exceptional separatrix convergence, or exchange between a region and its retained environment. They do require a formation study to declare its target, initial set and lifetime separately from a maintenance certificate. Apparent regional settling must keep the hidden state and the finite-memory boundary; eliminating constituents does not turn conservative exchange into exact irreversible loss.

These conservative obstructions do not invalidate the separate validated native-law transit from a winding-zero preparation into a maintained winding-one pattern. That result uses a different complete dissipative law and a forward capture region. Its retained continuous proof, frozen finite-executor verdict and supplied initial preparation keep their separate evidence scopes.

11. Counterexamples prevent an unconditional claim

The theorem has concrete boundaries.

• An exact unforced equilibrium remains still. A susceptibility to a probe does not supply that probe or start motion.
• A pure common-origin input has F=0 and zero conjugate response. Other noncollocated observations can hide excited modes or have identically zero transfer.
• A phase-output response can be low-pass even when the conjugate-work response has an interior maximum.
• Zero capacity, an indefinite or additionally degenerate phase Hessian, disconnected support or a different pressure/phase law fall outside the stated positive-definite quotient theorem.
• At e=0 the finite stable-gain proof does not apply. At vanishing reciprocal coupling the same second-order mechanism disappears; pure diffusion is a different closure.

The permanent-pulse admission above strengthens the unforced e>0 sine boundary to approach toward the equilibrium set, including held zero capacities. Its native-law recurrence extension retains the separate regular-domain hypotheses. The exact e=0 periodic branch has different conservation and attraction properties; it does not repair a failed positive-loss claim without changing the declared constitutive premise.

A periodic input may balance loss in a driven response; the required input and structural work must be retained. A hidden TNFR environment can supply a history-dependent influence, but its state, feedback and initial condition must be carried by the memory derivation. It cannot silently be replaced by a freely specified periodic signal.

These are conditional mathematical mechanisms compatible with the repository's research aim. They do not uniquely select the fundamental pressure law, identify physical matter or establish that every TNFR state must oscillate. The existence of this representation or response class alone establishes no novelty relative to established network dynamics.

12. Evidence and integration boundaries

A resonance report must retain the complete law and coefficients, support and node order, positive capacities, exact target geometry, quotient convention, input vector, output definition and clock. It must distinguish:

1. An exact identity or a theorem applying to an admitted class.
2. A numerical pole calculation, which is not a certified root enclosure.
3. A tangent transfer and gain classification with an independently chosen probe and observation.
4. A finite nonlinear response with its own preparation, horizon, error budget and unchanged reserved-evaluation protocol.

The sine comparison owner already owns the consumed sine rates and storage work. The resonance assessment owner and its independent controls connect the admitted mathematical results to detached reports. Their execution and numerical boundaries belong to the resonance contract; the regional guide owns public usage. The scale owner and its independent fine-state controls retain exact paired-state observations, prepared internal pulses and their complete variation. Its splitting report encloses asymptotic coefficients with a proved existential instability interval; it computes no finite-amplitude return multiplier or numerical interval endpoint. The coordinate-memory owner retains exact hidden-block memory and finite observation tests. The native relational owner and RA realization owner have separate laws and execution contracts.

Auxiliary graph-wave rhythm() spectra, resonance-named diagnostics and operator labels cannot substitute for the full generator and declared transfer in this document. Response gain, exact prepared periodic exchange, almost-everywhere recurrence and conditional transverse instability are distinct results of their declared complete laws. Autonomous excitation, selection of the constitutive boundary and physical measurements retain their separate evidence obligations.