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Internal replica pulses and their perturbations

The prepared internal pulse, complete variational law, small-amplitude splitting and persistent constituent activity under the same conservative doubled-cycle model.

Part of Coarse-graining, coherence geometry and bridge results. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

9. A finite-amplitude internal pulse with persistent collective identity

Keep the complete double-replica C5, the same conservative sine law e=0,w,beta>0, no inputs or events, and now a common strictly positive held capacity nu at all ten fine nodes. This additional capacity premise is needed for the invariant family below; paired equality alone was enough for sections 7 and 8.

9.1. The exact invariant preparation

Let alpha=2*pi/5 and c_alpha=cos(alpha)>0. Supply the exact collective state

\[ X_i=\bar X,\qquad \Theta_i=\bar\Theta+i\alpha\pmod{2\pi}, \qquad i=0,\ldots,4, \]

and identical signed internal coordinates

\[ u_i=u,\qquad \delta_i=\delta \]

for all five pairs. The constants bar X and bar Theta specify the common form and phase origins. The target angles are exact turns i/5; replacing them by rounded radian values and accepting a small balance residual would not establish this invariant family.

At each coarse node the two phase differences are +alpha and -alpha modulo 2*pi. Their sine currents cancel, while their cosines add. The exact retained-coordinate rows of section 7 therefore give

\[ \dot X_i=0,\qquad \dot\Theta_i=0,\qquad \dot u_i=-a\nu c_\alpha\cos\delta\sin\delta,\qquad \dot\delta_i=b\nu u. \]

The internal derivatives are identical at all five pairs, so smooth uniqueness proves that the full preparation remains in this family. Its complete nonlinear evolution reduces exactly to

\[ \boxed{\quad \dot u=-a\nu c_\alpha\cos\delta\sin\delta,\qquad \dot\delta=b\nu u . \quad} \]

The ten constituent coordinates remain x_i,plus=bar X+u, x_i,minus=bar X-u and theta_i,plus/minus=bar Theta+i*alpha+/-delta. Static collective means do not mean that the constituents are stationary. Neither an external periodic input nor an additional oscillator equation has been installed: the two displayed rows are an invariant restriction of the already supplied complete law.

9.2. Conserved amplitude and exact periods

Because a=beta*b, the internal quantity

\[ \boxed{\quad H=u^2+\beta c_\alpha\sin^2\delta\quad} \]

is conserved:

\[ \dot H =-2a\nu c_\alpha u\cos\delta\sin\delta +2\beta b\nu c_\alpha u\sin\delta\cos\delta=0. \]

It is part of the existing full storage, not a second independently postulated energy. Indeed, section 7 gives exactly

\[ E_f=4E_{*,C5}+20H,\qquad E_{*,C5}=5\beta(1-c_\alpha). \]

Define

\[ \Omega=\nu\sqrt{ab c_\alpha} =\frac{w\nu\sqrt{c_\alpha}}{\pi\sqrt\beta}, \qquad m=\frac{H}{\beta c_\alpha}. \]

For 0<m<1 and a preparation in |delta|<pi/2, the level set is a nonstationary closed curve around (u,delta)=(0,0). Its amplitudes are

\[ |\delta|_{\max}=\arcsin\sqrt m<\pi/2,\qquad |u|_{\max}=\sqrt H . \]

The pair chart consequently remains valid for the entire motion. Set eta=2*delta; differentiating the actual phase row gives

\[ \ddot\eta+\Omega^2\sin\eta=0 . \]

This pendulum equation is a derived form of the retained internal system. The conserved level already proves that the orbit is a libration, and direct quadrature gives its period. With the elliptic parameter m, define

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

Since sin(delta)=sqrt(m)*sin(psi) on a quarter orbit,

\[ \boxed{\quad T_{\rm labeled}=\frac{4}{\Omega}K(m) =\frac{4\pi\sqrt\beta}{w\nu\sqrt{c_\alpha}}K(m). \quad} \]

This is the period of the full labeled circular fine state, not merely of a tangent mode. On a nonzero libration the half-period map is (u,delta)->(-u,-delta). It swaps the two members of every pair while leaving X,Theta fixed. The unordered state of section 8 therefore has the smaller fundamental period

\[ \boxed{\quad T_{\rm unordered}=T_{\rm labeled}/2.\quad} \]

To see why it is the fundamental unordered period, the nonstationary closed energy curve is traversed once in T_labeled and each unordered state has exactly two fine lifts on it. Those lifts are related by the half-period swap; there is no fixed lift on a positive-energy orbit. The invariants R,U,Q change along this orbit, so the unordered state is not stationary despite the fixed collective means.

The same elementary integral bounds used for the two-node pulse give

\[ T_0=\frac{2\pi}{\Omega},\qquad T_0<T_{\rm labeled}\le\frac{T_0}{\sqrt{1-m}}, \]

and half of each bound for the unordered period. The strictly positive amplitude is essential: T_0 is the small-amplitude limiting period, not a period assigned to a stationary state. In particular, the pulse frequency depends on the supplied capacity, coefficients, geometry and amplitude. As m approaches one, K(m) diverges. This is a persistent periodic internal motion under a declared law, not a universal constant frequency.

9.3. Collective identity and the stronger fine-edge condition

Throughout every admitted libration, the collective means remain exactly the winding-one target. The inherited phase-current factor R_i*R_j=cos(delta)^2 varies periodically on every coarse edge, while its two incident sine currents still cancel at each coarse node. Consequently a static collective mean can contain changing internal interaction strength. The fine phase differences along an oriented base edge are

\[ \alpha,\qquad \alpha+2\delta,\qquad \alpha-2\delta \pmod{2\pi}. \]

Preserving the nonantipodal pair chart is therefore weaker than keeping every fine edge acute. The latter holds for the complete period under the sharper condition

\[ |\delta|_{\max}<\frac{\pi/2-\alpha}{2} \quad\Longleftrightarrow\quad \boxed{\quad m<\sin^2(\pi/20).\quad} \]

This nonempty finite-amplitude subfamily has fixed collective winding, moving constituents and strictly acute fine edges at all times. Its fine cycle windings also retain those of the duplicated target: continuously varying strictly acute edges cannot cross an antipodal wrapping boundary. This conclusion follows from the exact orbit and its amplitude bound; it does not require a numerical trajectory.

For larger librations with 0<m<1, the static collective target and valid pair chart still follow, but the all-fine-edge acute certificate does not. A loss of that sufficient condition is not by itself a formation, instability or winding-change verdict. General transverse perturbations away from the common-internal-state family remain governed by the full retained law and its independent trapping conditions. Exact period, common internal phase and the half-period swap cannot be inferred for those perturbations from an energy bound alone.

9.4. Boundaries and scientific scope

Within the admitted pair chart:

  • H=0 forces u=delta=0 and is stationary. No pulse starts spontaneously from that exact equilibrium.
  • H=beta*c_alpha is the separatrix level. Nonstationary preparations inside the chart approach its boundary |delta|=pi/2 asymptotically and do not have a finite libration period. The boundary states u=0,delta=+/-pi/2 are stationary states of the full fine law, outside this pair-midpoint chart.
  • H>beta*c_alpha gives u^2>=H-beta*c_alpha>0, so delta moves monotonically and leaves the admitted pair chart in finite time. This excludes this libration certificate, not every periodic or rotating full circular-state solution. The full fine law continues smoothly and would need another observation chart.

The result establishes a family of finite-amplitude internal pulses inside a persistent collective organization. The common capacity, exact twist, zero loss and identical signed internal preparation are explicit premises. This invariant family has lower dimension than the ambient fine state space; its exact periodic behavior is not an almost-everywhere or generic-attraction theorem.

Nothing here selects microscopic zero loss, supplies the initial amplitude, forms the paired support, forces all internal modes to align, or identifies a material particle. The pulse is sustained by conservative exchange already present in the chosen nodal law, and the distinction between labeled and unordered periods follows from its exact symmetry.

The replica pulse owner exposes assess_sine_replica_pulse for this declared exact family, with the common signed internal preparation and capacity supplied explicitly. It retains symbolic target turns and marks captured-graph membership as uncertified. Interval bounds classify the internal energy, enclose the two periods and independently test the stronger all-fine-edge acute condition; an unresolved boundary does not become a passing certificate. The period enclosure reuses the shared elliptic-integrand bound. The replica tests compare full nodal rows, exact storage identities and independent elliptic-function values without an ODE trajectory or a rounded-target membership assertion.

10. Complete variation around the finite internal pulse

Retain the exact doubled-C5 pulse and all hypotheses of section 9. The instantaneous variational identities hold at any declared point of its invariant family while the pair chart is valid. The finite-period return statements additionally require the nonstationary libration condition 0<m<1. Write the reference internal phase as d(t) to distinguish it from a perturbation, and put

\[ \alpha=\frac{2\pi}{5},\quad c=\cos\alpha,\quad s=\sin\alpha,\quad C(t)=\cos d(t),\quad S(t)=\sin d(t). \]

The perturbations below keep the supplied graph, common held capacity and law fixed. They include all twenty real nodal-state directions, not just perturbations that preserve the common signed internal preparation. They do not include perturbations of the support or constitutive laws.

10.1. Direct differentiation of the full retained law

Perturb (X_i,Theta_i,u_i,delta_i) by (xi_i,eta_i,v_i,zeta_i). On the ordered cycle define

\[ (Lf)_i=2f_i-f_{i+1}-f_{i-1},\qquad (Df)_i=f_{i+1}-f_{i-1}, \]

with indices modulo five. Differentiating the complete rows of section 7, before restricting a spatial mode, gives

\[ \boxed{\begin{aligned} \dot\xi&=-\frac{a\nu}{2} [cC^2L\eta+sCS\,D\zeta],\\ \dot\eta&=\frac{b\nu}{2}L\xi,\\ \dot v&=-a\nu c [\cos(2d)I+\tfrac12S^2L]\zeta +\frac{a\nu sCS}{2}D\eta,\\ \dot\zeta&=b\nu v . \end{aligned}} \]

For example, the derivative of the mean form row contains the neighbor internal-phase term -(a*nu/2)*C*S*sum_j sin(Theta_j-Theta_i)*zeta_j. The two target sine values are +s,-s, producing D*zeta. The term from differentiating the local cos(delta_i) vanishes because the unperturbed incident sine sum is zero. Differentiating the internal form row gives the opposite signed D*eta term and the coefficient C^2*zeta_i-(S^2/2)*(zeta_(i+1)+zeta_(i-1)), equal to the displayed cos(2d)I+(S^2/2)L expression.

Thus transverse perturbations couple internal and collective coordinates. Treating the five pairs as independent pendula would omit the two D terms and generally change the variation being tested.

10.2. Exact spatial decomposition without discarding directions

Use the Fourier convention f_i=hat f_k*exp(i*q_k*i), with q_k=2*pi*k/5. Then

\[ \lambda_k=2-2\cos q_k,\qquad L\mapsto\lambda_k,\qquad D\mapsto2i\sin q_k . \]

In the order (hat xi,hat eta,hat v,hat zeta), the complex block is

\[ \mathcal A_k(t)= \begin{pmatrix} 0&-a\nu cC^2\lambda_k/2&0&-ia\nu sCS\sin q_k\\ b\nu\lambda_k/2&0&0&0\\ 0&ia\nu sCS\sin q_k&0& -a\nu c[\cos(2d)+S^2\lambda_k/2]\\ 0&0&b\nu&0 \end{pmatrix}. \]

Real nodal perturbations obey hat f_(5-k)=conjugate(hat f_k). For each representative k=1,2, change only the analysis basis to

\[ (\hat\xi_k,\hat\eta_k,i\hat v_k,i\hat\zeta_k). \]

The resulting real-coefficient matrix is

\[ \boxed{ A_k(t)= \begin{pmatrix} 0&-a\nu cC^2\lambda_k/2&0&-a\nu sCS\sin q_k\\ b\nu\lambda_k/2&0&0&0\\ 0&-a\nu sCS\sin q_k&0& -a\nu c[\cos(2d)+S^2\lambda_k/2]\\ 0&0&b\nu&0 \end{pmatrix}.} \]

Its real and imaginary parts follow two identical four-real-dimensional systems. This is a basis transformation, not complexification of the scalar physical form coordinate. Mode k=0 uses the original four real coordinates and the same displayed matrix with lambda_0=sin(q_0)=0. The dimensions are exactly

\[ 4+2\cdot4+2\cdot4=20 . \]

No relative mean, hidden capacity, fine coordinate or spatial perturbation has been dropped. Capacities are held parameters of the stated model, so they are not additional tangent state directions.

The common mode contains two constant origin perturbations xi_0,eta_0 and the pendulum variation

\[ \dot v_0=-a\nu c\cos(2d)\zeta_0,\qquad \dot\zeta_0=b\nu v_0 . \]

The nonzero spatial modes are the sixteen real directions transverse to the common-preparation family. They are the relevant blocks when asking whether arbitrary fine perturbations preserve internal coordination.

10.3. Zero-amplitude control and dimensionless parameters

At u=d=0, C=1,S=0 and the internal-collective cross terms vanish. The exact spectral control is

\[ \text{mean sector: }\quad \pm i\Omega\,\lambda_k/2,\qquad \text{internal sector: }\quad \pm i\Omega, \qquad \Omega=\nu\sqrt{ab c}. \]

Mode zero contributes two zero origin modes instead of a nonzero mean frequency. The two nonzero mean-frequency ratios are

\[ \frac{\lambda_1}{2}=\frac{5-\sqrt5}{4},\qquad \frac{\lambda_2}{2}=\frac{5+\sqrt5}{4}. \]

Including Fourier multiplicities, there are two zero directions, ten internal oscillator directions and eight nonzero mean oscillator directions. This agrees with the full fine-graph decomposition in section 7. All are a stationary-target control or a small-amplitude limit. A nonstationary finite-amplitude period is not assigned to the zero-amplitude equilibrium, and this limiting spectrum does not decide finite-amplitude stability.

There is also an exact reduction of the apparent parameter freedom. Divide both form perturbations by sqrt(beta*c), leave phase perturbations unchanged, and use tau=Omega*t. The block becomes

\[ \boxed{ \widetilde A_k(\tau)= \begin{pmatrix} 0&-C^2\lambda_k/2&0&-\tan\alpha\,CS\sin q_k\\ \lambda_k/2&0&0&0\\ 0&-\tan\alpha\,CS\sin q_k&0& -[\cos(2d)+S^2\lambda_k/2]\\ 0&0&1&0 \end{pmatrix}.} \]

The base pulse itself satisfies

\[ \frac{dd}{d\tau}=\widehat u,\qquad \frac{d\widehat u}{d\tau}=-\cos d\sin d,\qquad \widehat u=\frac{u}{\sqrt{\beta c}},\qquad m=\widehat u^2+\sin^2d . \]

Consequently the return multipliers depend only on m and the fixed geometry. Changing the reference position on the same orbit conjugates the return matrix and does not change its spectrum. At fixed m, nu,w,beta change the time or coordinate scales, not independent stability parameters. Changing beta while holding raw u,d fixed usually changes m, so it is not that fixed-amplitude comparison.

10.4. Symplectic blocks and the correct half-period return

Let

\[ J_0=\operatorname{diag} \left( \begin{pmatrix}0&-1\\1&0\end{pmatrix}, \begin{pmatrix}0&-1\\1&0\end{pmatrix} \right). \]

For every displayed real block, -J_0*A_k(t) is symmetric. Equivalently

\[ A_k(t)^TJ_0+J_0A_k(t)=0 . \]

Thus a fundamental matrix Phi_k(t) initialized by Phi_k(0)=I satisfies Phi_k(t)^T*J_0*Phi_k(t)=J_0. This is a tangent consequence of the same conservative law, not an auxiliary Hamiltonian added to the fine evolution.

Write T=T_labeled and h=T/2. At the half period the base pulse has (u,d)->(-u,-d). The induced member-swap derivative is

\[ S_*=\operatorname{diag}(1,1,-1,-1),\qquad S_*^2=I . \]

Since C is unchanged while S changes sign,

\[ A_k(t+h)=S_*A_k(t)S_* . \]

The raw matrix is therefore generally not half-periodic: its two cross terms reverse sign. They happen to vanish in mode zero, which does not remove the swap identification of the base state.

The full labeled return and the symmetry-correct unordered return are

\[ \boxed{\qquad M_k=\Phi_k(T),\qquad B_k=S_*\Phi_k(h),\qquad M_k=B_k^2 . \qquad} \]

Indeed the second-half propagator is S_*Phi_k(h)S_*, and multiplying it by the first-half propagator proves the square identity. Applying S_* after the first half is essential: it identifies the tangent space at the swapped endpoint with the original one. A spectrum of the raw Phi_k(h) alone is not the unordered return spectrum.

Both S_* and Phi_k(h) are symplectic, hence so are B_k and M_k. Their determinant is one and their multipliers occur in reciprocal and complex-conjugate pairs. These restrictions are exact; they supply no numerical value for a nontrivial finite-amplitude multiplier.

10.5. Amplitude-dependent phase shear is a neutral-family effect

The period is strictly increasing with the conserved internal amplitude:

\[ \frac{dT}{dH} =\frac{4}{\Omega\beta c}K'(m)>0,\qquad K'(m)=\frac12\int_0^{\pi/2} \frac{\sin^2\psi}{(1-m\sin^2\psi)^{3/2}}\,d\psi . \]

This follows by differentiation under the integral for 0<m<1. For example,

\[ \frac{\pi}{2\Omega\beta c} <\frac{dT}{dH} \le\frac{\pi}{2\Omega\beta c(1-m)^{3/2}} . \]

To identify its tangent effect, choose the smooth initial section d(0)=0,u(0)=sqrt(H) with the common origins fixed. Let v_t be the flow tangent there and v_H the derivative of that initial state with respect to H. They are independent for H>0. Differentiate the exact identity z(H,T(H))=z(H,0) to obtain

\[ M_0v_t=v_t,\qquad M_0v_H=v_H-T'(H)v_t. \]

Differentiating the corresponding half-period swap gives

\[ B_0v_t=v_t,\qquad B_0v_H=v_H-\tfrac12T'(H)v_t. \]

The two common origins have identity returns as well. Thus the common mode has a nontrivial neutral time-amplitude shear. Repeated comparison at the same clock phase can accumulate a linear-in-cycle phase shift between neighboring amplitudes, even though each preparation remains on its own conserved libration. This is not an exponentially growing transverse mode or proof of nonlinear orbital instability. Fixing H removes that amplitude direction; comparing only equally timed waveforms does not.

10.6. A discriminating stability question, not a stability verdict

After the common-mode directions have been identified, the prospective linear test is confined to the k=1,2 representative blocks:

  • A certified return multiplier |lambda|>1 yields exponential transverse growth. For the smooth periodic orbit here, an unstable multiplier of its transverse return map also certifies nonlinear orbital instability. The criterion can be checked through B_k or M_k=B_k^2, with the correct return convention.
  • If all transverse multipliers lie on the unit circle and their return matrices are semisimple, the transverse linear solutions are bounded: powers of the return matrices are bounded, and the continuous propagator over one compact period is bounded.
  • Unit-circle eigenvalues with nontrivial Jordan blocks can yield polynomial growth and do not pass that bounded-linear criterion. Neutral amplitude shear in the separately identified common family must not be counted as a transverse instability.

Bounded linear variation alone does not prove nonlinear orbital stability, attracting synchronization or retention of one exact frequency after a general perturbation. Nonlinear resonances and higher-order terms remain separate obligations. In particular, even when every fine edge stays acute along the pulse, positivity of an instantaneous storage Hessian does not by itself solve this problem. The quadratic variation has a time-dependent coefficient:

\[ \frac{d}{dt}\frac12 z^T H_2(t)z =\frac12 z^T\dot H_2(t)z \]

when the Hamiltonian tangent terms cancel. It is not generally a conserved positive quadratic norm. The earlier all-state energy barrier keeps admitted solutions near the static winding target, but need not keep them near this particular periodic orbit or phase-locked to it.

This reduction supplies all perturbation directions, exact symmetry constraints and a one-amplitude stability question. It does not evaluate a monodromy matrix, prove a finite-amplitude instability, establish a stability interval, or run a trajectory or Floquet sweep. The following small-amplitude calculation resolves the leading transverse signs with a separately proved existential amplitude range.

The replica pulse owner exposes assess_sine_replica_pulse_variation with the exact family premises reused from the pulse assessment. It reports the three real instantaneous block types, their multiplicities, the dimensionless versions, fixed symplectic form, swap matrix and stationary-limit frequencies. Instantaneous coefficients are not transition matrices. Periodic-reference and half-return claims remain conditional on the separate libration certificate, and orbital stability is explicitly unassessed. The replica tests compare these blocks with the full fine nodal Jacobian and check the coordinate transformations, symmetry and zero-amplitude controls.

10.7. Internal motion produces a finite same-type work-response asymmetry

This calculation uses the existing pulse and full variation, with no added force or constitutive row. Its declared preparation is positive twist alpha=2*pi/5, e=0,w=beta=nu=1, u(0)=delta(0)=1/32, and zero common origins. Fix real Fourier representative k=1, clock tau=t/pi, and readout horizon h=1/16. The computational check uses Taylor order ten; the proof below is independent of a numerical trajectory. These choices are fixed before response evaluation, not selected by a sign search.

Matched form inputs and storage-work outputs

For pair index j, define the two real fine form masks

\[ f_C(j,\pm)=\cos(2\pi j/5),\qquad f_I(j,\pm)=\mathord\pm\sin(2\pi j/5). \]

These are distributed perturbations on existing nodes, not new point contacts or new edges. An infinitesimal initial form increment along either mask leaves the initial phases unchanged. Subtract the unperturbed pulse and differentiate with respect to that increment at zero. In the real block of section 10.2 this supplies the two initial form columns p(0)=I, with phase columns q(0)=0. Here p=(xi,i*v) and q=(eta,i*zeta) denote real mode amplitudes, not the primitive pressure. The fine reconstruction uses cosine collective amplitudes and sine internal amplitudes.

Put c=cos(alpha), g=sin(alpha)^2, l=1-c and D=diag(l,1). The inherited equations in this clock are

\[ p'=-C(\tau)q,\qquad q'=Dp,\qquad C(\tau)=\begin{pmatrix} c\,l\cos^2\delta&g\cos\delta\sin\delta\\ g\cos\delta\sin\delta&c[1-(1+c)\sin^2\delta] \end{pmatrix}, \quad u'=-c\cos\delta\sin\delta,\quad\delta'=u. \]

If P(tau) is the form-to-form transition block, the actual fine conjugate work outputs give H(tau)=20 D P(tau). Indeed each mask has squared norm five, with fine form Laplacian eigenvalues 2*lambda_1 and four, respectively; thus (f_C^T L delta_x,f_I^T L delta_x)=(20*l*xi,20*i*v). The same normalization is used for both input/output roles. The quantity tested is Delta(tau)=H_CI(tau)-H_IC(tau). It is a finite-time linear response about a moving, finite-amplitude reference, not a finite-amplitude probe experiment or a stationary frequency transfer.

Time ordering supplies the first nonzero antisymmetric term

For the normalized work kernel K=D P, differentiating the full block gives

\[ K_{12}-K_{21} =\frac{l g c^2 u_0[1+l\cos^2\delta_0]}{60}\,\tau^5 +O(\tau^6). \]

All lower antisymmetric coefficients vanish. Equivalently the coefficient matrix is D(C'(0) D C(0)-C(0) D C'(0))D/60. Each instantaneous C is symmetric, but its successive values need not commute in this metric. Freezing C would erase that information and restore symmetric work transfer. This is why the equilibrium reciprocity theorem does not decide this moving-background experiment.

A finite positive bound at the declared horizon

Energy conservation gives u^2+c*sin(delta)^2<1/768, hence abs(u)<1/24. For 0<=tau<=1/4, integration first gives 1/48<=delta<=1/24, and then u>=1/32-1/288=1/36. These are bounds on the actual moving pulse. Throughout this interval ||C D||_infinity<=3/8. For s>=v, direct differentiation of the displayed C, using 3/10<c<1/3, g>9/10, l>2/3 and the small-angle cosine lower bound 99/100, gives

\[ \left[D(C(s)D C(v)-C(v)D C(s))D\right]_{12} \ge\frac{99}{40000}(s-v). \]

For completeness the derivative of the inner commutator's 12 entry with respect to s is g*c^2*u(s)*((3-c)*cos(2*(delta(s)-delta(v)))+l*cos(2*delta(s)))/2. The undifferentiated commutator vanishes at s=v, so the derivative's lower bound integrates to the preceding inequality.

Write the convergent Volterra expansion as P=sum_n P_n, where

\[ P_0=I,\qquad P_{n+1}(\tau)=-\int_0^\tau C(s)D\int_0^s P_n(v)\,dv\,ds. \]

D P_0 and D P_1 are symmetric. The antisymmetric contribution of D P_2 has weight v(s-v) multiplying the commutator above, hence is at least 33*h^5/800000. The terms n>=3 have total two-entry error at most

\[ \frac{2(3/8)^3h^6}{720[1-(3/8)h^2/56]}. \]

This follows from ||P_n||_infinity <= (3/8)^n h^(2*n)/(2*n)! and the geometric bound on successive tail terms. At the predeclared h=1/16,

\[ \boxed{\Delta(h)\ge \frac{588921}{962047508480000}>6\times10^{-10}>0.} \]

Thus the finite sign is proved, rather than inferred from the fifth-order term alone. The pulse's energy also lies below its existing all-fine-edge acute threshold. This retains the prepared identity; it does not assert orbital stability under arbitrary disturbances.

Reversal controls and scope

  • Reversing initial internal motion to u_0=-1/32 at the same delta_0 gives a strictly negative response by the same interval argument. Opposite signs do not require equal finite magnitudes.
  • Reversing the base twist while retaining the positive Fourier orientation changes the sign of the off-diagonal C entries; the response contrast reverses exactly. Their general factor is sin(alpha)*sin(q_1); the abbreviation g=sin(alpha)^2 applies to the fixed positive-twist reference. Changing the reporting orientation is a different operation.
  • Swapping both members in every pair sends (u,delta) to (-u,-delta) and reverses the internal input/output mask. Transforming the ports with the state preserves the physical comparison; a signed matrix entry alone is not a label-independent intrinsic property.
  • A stationary reference, or the explicit control that freezes C, has reciprocal same-type response. A turning point u_0=0 alone does not establish a stationary reference or zero finite contrast.

The replica pulse owner exposes assess_sine_replica_pulse_work_response. It rebuilds the pulse and uses the common variational block in the joint ten-dimensional pulse/two-column field. The shared validated Taylor kernel encloses its finite response, tube and remainder in tau=t/pi; unavailable enclosures remain explicit. The reader retains general admitted pulse coefficients, while the finite rational lower bound above is only for the declared unit-parameter preparation. The response controls and SDK export preserve those boundaries. Its derivative report does not itself certify a finite probe radius. The continuation below bounds finite kicks separately; no spatial Hall observation, physical unit bridge or magnetic law has been derived.

For the frozen order-ten evaluation the outward-rounded response enclosure is 9.9053976622818e-10 < Delta(h) < 9.9053976623020e-10, consistent with the independent lower bound above. This numerical value uses structural units and the declared port normalization, not measured physical units.

10.8. A controlled finite probe of the complete fine law

Retain every preparation, port and clock premise of section 10.7. Before evaluating any finite kicked response, fix epsilon=2^-20, h=1/16 and Taylor order ten. For each mask f_C,f_I, prepare the two fine states x_0+epsilon*f and x_0-epsilon*f, with the same initial phases. All four trajectories follow the complete twenty-coordinate sine field, including the modes and backreaction that a finite kick can generate. No restriction to the background's invariant family or its linear Fourier block is made.

In the scaled clock the degree-four fine rows are

\[ x_i'=\tfrac14\sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad \theta_i'=\tfrac14\sum_{j\sim i}(x_i-x_j). \]

Let Phi_h denote this full flow. Define the centered work response H_epsilon,ij=f_i^T L[x_h(+epsilon*f_j)-x_h(-epsilon*f_j)]/(2*epsilon) and Delta_epsilon=H_epsilon,CI-H_epsilon,IC. The common unperturbed background cancels; no measured derivative is used to reconstruct pressure.

Uniform nonlinear error, before evaluating a finite response

In the maximum norm, globally ||DF||<=2, ||D^2F||<=4 and ||D^3F||<=8. These follow from the four normalized edge contributions, the gap bound abs(v_j-v_i)<=2||v||, and bounded sine derivatives. The phase row is linear. For any initial direction b of norm at most one, put E=exp(2*tau). Differentiating the complete flow and applying the variational integral inequalities yields

\[ \|D\Phi_\tau b\|\le E,\qquad \|D^2\Phi_\tau[b,b]\|\le2E(E-1),\qquad \|D^3\Phi_\tau[b,b,b]\|\le4E(E-1)(2E-1). \]

For example the third bound solves w'<=2w+12*v*z+8*v^3, where v<=E and z<=2E(E-1). At h=1/16, the positive exponential series gives E=exp(1/8)<=1/(1-1/8)=8/7, so the third bound is 288/343. It holds throughout the kicked initial-state segments, not only at their center. The actual port rows satisfy ||f_C^T L||_1<=40 and ||f_I^T L||_1<=40: their eigenvalues are 4*l<4 and four and each ten-node mask has entries bounded by one. Taylor's centered-difference remainder therefore gives

\[ |\Delta_\epsilon-\Delta|\le \frac{80}{6}\frac{288}{343}\epsilon^2 =\frac{3840}{343}\epsilon^2. \]

This is uniform for every 0<epsilon<=2^-20. Combining it with the independent finite-time lower bound of section 10.7, at the declared radius,

\[ \boxed{\Delta_\epsilon(h)\ge \frac{12712959417}{21118866906152960000}>6\times10^{-10}>0.} \]

Domain, preparation and interpretation

The reference fine-edge acute margin exceeds pi/10-1/12>13/60 on this window. Each kicked flow differs from the reference by at most 8*epsilon/7 per coordinate, so every edge margin remains larger than 13/60-16*epsilon/7>0. The support, capacities and lifted circular phase information are retained throughout. Preparation changes the stored energy; thereafter each trajectory follows the same unforced conservative law.

The reversed-motion reference has the opposite finite sign by the same error bound. General stationary nonlinear reciprocity does not follow from the linear theorem. For the particular zero-pulse reference, global member swap leaves the source and collective mask fixed and reverses the internal mask: that separate symmetry makes its centered cross responses zero.

The raw four-reading contrast is 2*epsilon*Delta_epsilon, much smaller than the normalized response. Its proved lower margin is about 1.15e-15 in these structural units. This is a mathematically nonzero finite signal, not a practical measurement/noise budget or an experimentally selected scale. The radius is a sufficient error-controlled choice, not a fundamental TNFR constant, optimal amplitude or inferred physical limit.

The fixed-template reader assess_sine_replica_pulse_finite_work_response reuses the shared full sine rows and validated Taylor kernel. Its four order-ten evaluations at the predeclared radius all admit strict whole-time tubes. The outward-rounded normalized contrast enclosure is 9.9051618144648e-10 < Delta_epsilon < 9.9056335170602e-10, contained in the earlier tangent enclosure widened by the proved nonlinear error. Analytic sign, finite numerical sign and failed-trial availability remain separate fields; the full-flow controls also check reversal, port normalization and the SDK projection. No numerical retry or amplitude adjustment was used to obtain this verdict.

10.9. Parametric response and complete-history reversal

The same moving-background tangent has an exact representation useful for comparison with physical oscillator response. In section 10.7 set y=D^(-1/2)q; then

\[ y''+S(\tau)y=0,\qquad S(\tau)=D^{1/2}C(\tau)D^{1/2}=S(\tau)^T. \]

For the matrix solution G(0)=0,G'(0)=I, the matched work kernel is H(tau)=20*D^(1/2)*G'(tau)*D^(1/2). It is therefore a weighted initial-velocity-to-velocity block of a parametric oscillator. This exact tangent coordinate transformation does not identify primitive form with a laboratory displacement or supply physical masses. Its varying stiffness is determined by the autonomous internal preparation, not an installed external modulation. The quadratic oscillator expression E_tan=(||y'||^2+y^T S y)/2 has derivative y^T S' y/2; treating it as an isolated conserved energy would discard its evolving background.

For the full fine conservative flow, R(x,theta)=(-x,theta) satisfies F(Rz)=-R F(z). The reversed history on a window is z_R(tau)=R z(h-tau), which starts at R z(h), not generally R z(0). If M=D Phi_h(z(0)), its reversed-history propagator satisfies exactly

\[ M_R=R M^{-1}R. \]

The real Fourier block is also symplectic. In (p,q) ordering, write M=[[A,B],[C_*,E_*]]; the constant symplectic form gives M^-1=[[E_*^T,-B^T],[-C_*^T,A^T]]. Consequently the reversed-history form/form block is E_*^T, not generally A^T. The asymmetric forward work block thus does not demonstrate failure of microscopic reversibility. Reversing the initial internal velocity at fixed phase, exchanging two probe roles, and reversing a complete history are distinct experiments.

The physical comparison uses these identities to separate transient response, driven steady-state nonreciprocity and possible effective geometric forces. A time-ordered commutator alone is neither a Berry curvature nor a magnetic field.

There is a sharper boundary for a proposed adiabatic stiffness-mode phase. Here S depends on the single real coordinate delta. A full libration traverses and retraces an interval of delta. Its eigenvalues remain distinct: S_22-S_11=c^2*((2-c)-(3-c)*sin(delta)^2)>0, since the fixed pulse has sin(delta)^2<=m<1/200 throughout its orbit. For any smooth single-valued connection depending only on delta, integral_closed A(delta) ddelta=0; reverse path transport is the inverse even for a matrix-valued connection. Thus this particular adiabatic stiffness-mode construction has trivial holonomy. The finite response propagator integrates its generator against time, not a connection against d(delta), so its asymmetry is not removed. This does not exclude nonadiabatic phases, Floquet geometry or a connection on the retained full phase space. It prevents promoting the present finite-delay asymmetry to a geometric magnetic flux merely because the stiffness changes over a pulse.

10.10. A clock-independent stiffness-family discriminator

An oscillator resemblance does not establish equality of its stiffness family. Keep the doubled-C5, real k=1, conservative pulse of section 10.9. With c=cos(2*pi/5), l=1-c and X=cos(delta)^2, its whitened stiffness has trace T and determinant V

\[ T=c(2-c+c^2)X-c^2,\qquad V=l^2X[(1+c)X-(1+c-c^2)]. \]

Eliminating X gives a quadratic V=a_* T^2+b T+d_0, with

\[ \boxed{a_*= \frac{l^2(1+c)}{c^2(2-c+c^2)^2} =\frac{1160+480\sqrt5}{1089}>2.} \]

This coefficient belongs to the declared geometry and mode, not a universal TNFR or physical constant. Multiplying stiffness by a constant clock factor r sends (T,V) to (r*T,r^2*V) and leaves a_* unchanged. For a physical candidate M y''+K(t)y=0 with known constant positive-definite mass M, use T=tr(M^-1 K) and V=det(M^-1 K). Fixed congruent changes of state and work coordinates preserve these invariants. Using K alone while discarding M does not. A time-dependent chart introduces velocity and additional stiffness terms and cannot silently repair a mismatch.

Exact obstruction for one affine stiffness control

Suppose an admitted two-mode candidate instead has symmetric whitened J(t)=J_0+rho(t)*J_1, with both matrices fixed. If tr(J_1) is nonzero, eliminating rho makes determinant a quadratic in trace with coefficient det(J_1)/tr(J_1)^2<=1/4, because

\[ \operatorname{tr}(J_1)^2-4\det(J_1) =(J_{1,11}-J_{1,22})^2+4J_{1,12}^2\ge0. \]

If tr(J_1)=0, the trace is constant. Hence no such family can coincide with a varying segment of the specified TNFR pulse under a constant clock and fixed work-compatible coordinates. Arbitrarily nonlinear timing rho(t) does not change this obstruction. Uniform scaling J(t)=rho(t)*J_0 is included. This concerns the curvature of a family, not the instantaneous ratio det(J)/tr(J)^2, which retains the usual symmetric-matrix bound.

This does not require adding a primitive coordinate: the existing delta already changes several matrix entries through different nonlinear functions. One scalar parameter can trace a non-affine matrix curve. The obstruction concerns one fixed matrix direction, not the number of scalar TNFR variables.

For three samples, define without division

\[ N=(V_3-V_2)(T_2-T_1)-(V_2-V_1)(T_3-T_2),\qquad Q=(T_3-T_2)(T_2-T_1)(T_3-T_1). \]

When traces are distinct, the TNFR template requires N-a_*Q=0, whereas every affine symmetric family obeys (4N-Q)Q<=0. These exclusion decisions are independent of sample ordering. Shared outward interval arithmetic can therefore exclude a template or an affine candidate without dividing by tiny gaps. Intervals containing zero do not prove an identity; overlapping traces leave the three-point curvature unresolved. Correlated uncertainty may be conservatively enclosed, never silently discarded.

assess_sine_replica_stiffness_trace_curve implements these necessary screens from three supplied trace/determinant observations. It uses shared scalar/interval admission, supplies no fitted clock or phase, and retains not_excluded as distinct from a successful admission. The controls construct stiffness from the full formula, exercise clock and coordinate changes, and include affine and uncertain counterexamples.

Passing this screen is insufficient: it neither checks the full stiffness matrix nor autonomous pulse timing, forcing, damping, preparation or physical readout. Several degrees of freedom, independently varying stiffness channels, retained circuit state and eliminated-mode memory must be evaluated using their actual complete laws. The mechanical candidate admission uses this restriction only for its declared ideal two-mode reduction.

11. Small-amplitude transverse splitting with collective response retained

This result continues the exact conservative doubled-C5 pulse and the complete real mode blocks of section 10. It resolves a local small-amplitude question; it does not search over amplitudes, integrate a variational trajectory or introduce a different law.

Write epsilon=sqrt(m) and choose the reference pulse section d(0)=0,u(0)/sqrt(beta*c)=epsilon, with c=cos(alpha) and alpha=2*pi/5. The two representatives have

\[ \ell_k=\lambda_k/2=1-\cos q_k,\qquad g_k=\tan\alpha\sin q_k,\qquad q_k=2\pi k/5,\quad k=1,2 . \]

In particular 0<ell_k<2, and neither value is an integer. Their collective limiting multipliers are exp(+/-2*pi*i*ell_k) for the full return and exp(+/-pi*i*ell_k) for the swap-correct half return. They are distinct from the internal limiting multiplier +1. This separation permits a local two-dimensional internal spectral reduction without discarding its coupling to the collective coordinates.

11.1. Fix the period before expanding the coefficients

The dimensionless pulse period is 4*K(m) in tau=Omega*t. Use the period-normalized coordinate

\[ \sigma=\omega(m)\tau,\qquad \omega(m)=\frac{\pi}{2K(m)} =1-\frac m4+O(m^2). \]

The full period is now exactly 2*pi. The analytic pulse equations and analytic period give, uniformly on this fixed interval,

\[ d(\sigma,\epsilon)=\epsilon\sin\sigma+O(\epsilon^3), \quad \cos^2d=1-\epsilon^2\sin^2\sigma+O(\epsilon^4), \quad \cos d\sin d=\epsilon\sin\sigma+O(\epsilon^3). \]

Oddness of the initial section and vector field makes d odd in epsilon. These are analytic Taylor expansions on a fixed compact interval, with bounded remainders for sufficiently small amplitude; they do not use a simulated periodic response.

For one representative mode suppress the index k, put gamma=sqrt(ell)*g and use y=eta/sqrt(ell), z=zeta in its real block. Eliminating the two form derivatives yields the symmetric second-order equations. After the time change they read

\[ \begin{aligned} y_{\sigma\sigma} &+\ell^2[1+\epsilon^2(1/2-\sin^2\sigma)]y +\gamma\epsilon\sin\sigma\,z +O(\epsilon^4)y+O(\epsilon^3)z=0,\\ z_{\sigma\sigma} &+[1+\epsilon^2 f(\sigma)]z +\gamma\epsilon\sin\sigma\,y +O(\epsilon^4)z+O(\epsilon^3)y=0,\\ f(\sigma) &=\frac12-(2-\ell)\sin^2\sigma . \end{aligned} \]

The 1/2 terms come from omega(m)^(-2)=1+m/2+O(m^2). Leaving the reference period uncorrected would change the internal resonant coefficients.

11.2. The induced collective motion changes the splitting

At zero amplitude the internal solution is z_0=A*cos(sigma)+B*sin(sigma). The internal spectral subspace induces a collective correction y=epsilon*y_1+O(epsilon^2). Its leading 2*pi-periodic solution satisfies

\[ y_1''+\ell^2y_1=-\gamma\sin\sigma\,z_0 . \]

There is a unique periodic solution because ell is not an integer. It is

\[ \boxed{ y_1= -\frac{\gamma A}{2(\ell^2-4)}\sin2\sigma -\frac{\gamma B}{2\ell^2} +\frac{\gamma B}{2(\ell^2-4)}\cos2\sigma .} \]

The denominators ell^2 and ell^2-4 retain respectively the constant and second-harmonic collective responses. Setting y_1=0 would not be the full fine variational problem.

Substitution in the internal equation, followed by projection onto cos(sigma),sin(sigma), gives the resonant coefficients

\[ \boxed{\begin{aligned} P&=\frac{\ell}{4}-\frac{\gamma^2}{4(\ell^2-4)},\\ Q&=\frac{3\ell-4}{4} -\frac{\gamma^2}{2\ell^2} -\frac{\gamma^2}{4(\ell^2-4)} . \end{aligned}} \]

Here P,Q are coefficient names local to this perturbation calculation; Q is not the retained internal correlation of section 8. The diagonal part alone would give ell/4 and (3*ell-4)/4; the remaining terms are the calculated feedback from the collective response.

To see their return-map meaning, use rotating internal amplitudes z=A(sigma)*cos(sigma)+B(sigma)*sin(sigma) with the usual variation-of-constants condition. The resonant part of their equation is

\[ \binom{A}{B}'= mG\binom{A}{B}+\text{higher-order and removable periodic terms}, \qquad G=\begin{pmatrix}0&Q/2\\-P/2&0\end{pmatrix}. \]

For example, forcing -m*(P*A*cos(sigma)+Q*B*sin(sigma)) gives the averaged rows A'=m*Q*B/2 and B'=-m*P*A/2. The nonresonant harmonics can be removed by a periodic near-identity change of variables; integrating their zero-mean part over one fixed period gives no additional leading return term. Equivalently this is the leading analytic reduction of the isolated internal spectral subspace. Its full return matrix in such a basis is

\[ M_{\rm int}=I+2\pi mG+O(m^{3/2}),\qquad \sigma_*^2=-PQ/4 . \]

The symbol sigma_* denotes a splitting coefficient, not the period-normalized time sigma.

11.3. Exact algebraic signs for the two spatial modes

Using tan(alpha)^2=5+2*sqrt(5) and the exact cycle angles gives:

Representative P Q sigma_*^2=-P*Q/4
k=1 (95+sqrt(5))/164 -(479+245*sqrt(5))/164 (23365+11877*sqrt(5))/53792 > 0
k=2 (55+21*sqrt(5))/41 (14+21*sqrt(5))/41 -(2975+1449*sqrt(5))/6724 < 0

All signs follow directly from positive integers and sqrt(5)>0. They are exact identities, not signs extracted from a numerical monodromy. The first internal pair therefore has a real leading splitting and the second an imaginary one.

11.4. Analytic remainder and the actual sufficiently-small conclusion

The return matrices are analytic in epsilon: their coefficients, the fixed-period pulse and the finite-interval linear initial-value problem are analytic in that parameter. At epsilon=0 the collective pair is separated from +1. A sufficiently small fixed contour around +1 therefore defines an analytic real two-dimensional spectral subspace of the full return. The displayed leading matrix is the restriction to that subspace in an analytic basis.

That subspace is symplectic for sufficiently small amplitude. At zero it is the nondegenerate internal oscillator plane, and nondegeneracy persists by continuity. Since the full return is symplectic, the restricted return has determinant one.

There is also an exact parity restriction. Replacing epsilon by -epsilon changes the reference pulse by the member swap, so the full return matrices are conjugate by S_*. Let P_epsilon be the analytic internal spectral projector and let V_0 be a fixed basis for the limiting internal plane, ordered as the rotating amplitudes. Then

\[ P_{-\epsilon}=S_*P_\epsilon S_*, \qquad S_*V_0=-V_0,\qquad V_\epsilon=P_\epsilon V_0,\qquad V_{-\epsilon}=-S_*V_\epsilon . \]

The columns of V_epsilon remain independent for sufficiently small amplitude. In this basis the restricted return matrix is exactly even in epsilon, because M(-epsilon)=S_*M(epsilon)S_* intertwines the displayed bases. It is therefore analytic in m=epsilon^2 and satisfies

\[ M_{\rm int}=I+2\pi mG+O(m^2),\qquad \frac{\log M_{\rm int}}{2\pi}=mG+O(m^2). \]

The logarithm is the real branch near the identity. It is an effective return generator, not the raw instantaneous rotating-amplitude row. Trace and determinant are basis-independent, so their resulting expansions hold even when another analytic basis is used.

Let t_int be the trace. Determinant one implies

\[ t_{\rm int} =2+4\pi^2\sigma_*^2m^2+O(m^3). \]

Indeed det(G)=-sigma_*^2, so the order-two trace coefficient is fixed by the determinant identity. In particular,

\[ \boxed{\quad t_{\rm int}=2+4\pi^2\sigma_*^2m^2+O(m^3),\qquad t_{\rm int}^2-4=16\pi^2\sigma_*^2m^2+O(m^3). \quad} \]

This establishes a remainder-controlled existence statement. For each mode there are positive constants C,m_0 bounding the remainder by C*m^3 on 0<m<m_0. Because the exact leading coefficient is nonzero, shrinking that interval makes it dominate the remainder. No numerical values of those constants or an explicit amplitude radius are obtained here.

For k=1, put

\[ \sigma_1= \sqrt{\frac{23365+11877\sqrt5}{53792}}>0 . \]

The internal labeled multipliers then satisfy

\[ \Lambda_\pm=1\pm2\pi\sigma_1m+O(m^2), \]

and for every sufficiently small positive m they are a real reciprocal pair with one member strictly greater than one. The swap-correct half-return multipliers satisfy 1+/-pi*sigma_1*m+O(m^2), consistently with M=B^2. This is an exponentially unstable transverse mode of the prepared periodic orbit. Both real spatial copies of this representative have the same conclusion.

For k=2, put

\[ \omega_2= \sqrt{\frac{2975+1449\sqrt5}{6724}}>0 . \]

Its internal labeled multipliers satisfy

\[ \Lambda_\pm=1\pm2\pi i\omega_2m+O(m^2). \]

For every sufficiently small positive m they are nonreal and, because their product is one, lie exactly on the unit circle. They are distinct and semisimple. The corresponding collective pair starts as a simple isolated nonreal unit-circle pair and remains so for a sufficiently small amplitude interval: leaving the circle would require a reciprocal-conjugate collision, excluded locally by its initial spectral separation. Thus this representative passes the bounded linear transverse criterion locally. It is not a nonlinear orbital-stability theorem for that mode, and it cannot stabilize the whole pulse against the unstable k=1 directions.

11.5. Interpretation and what remains unmeasured

The same full nonlinear law admits the exact periodic family and also makes its sufficiently small nonzero members transversely unstable. There is no contradiction: invariant preparation proves that the orbit exists, whereas the return splitting asks what nearby states do. An unstable transverse return multiplier gives nonlinear orbital instability of this smooth periodic orbit; it does not imply that every perturbation grows or that the exact orbit ceases to exist.

The amplification is produced by the periodic internal coefficients and their coupled collective response. It does not require an external periodic source or a newly installed pressure law. In the complete conservative system, departure from that particular internal waveform can transfer storage among its retained modes without changing total storage. This is distinct from the neutral time-amplitude shear in mode zero.

The harmonic mechanism is explicit. The sin(sigma)^2 stiffness term contains a constant part and a second harmonic. The order- epsilon*sin(sigma) cross term sends the internal first harmonic into the collective constant and second-harmonic responses; those feed back into the internal first harmonic through the same cross term. This is parametric feedback in the derived tangent law, not an externally prescribed oscillatory drive, a claim about all states, or invocation of the named Resonance operator.

Nor does orbital instability mean unbounded fine form, loss of every coherent geometry or unavoidable winding change. For sufficiently small pulses, the earlier acute-target trapping domain contains open neighborhoods of their initial states: both their centered norm and excess storage tend to zero with amplitude. Solutions from an admitted neighborhood remain near that static phase identity, even while some depart from the common-internal-state periodic orbit. A robust geometric organization and a fragile synchronized waveform can therefore coexist under the same storage balance.

The result supplies no certified numerical interval of unstable amplitudes, no verdict for a chosen nonzero represented amplitude solely because it is called small, no growth time at that amplitude, and no finite-amplitude continuation of either mode class. It does not justify changing the constitutive law to preserve a preferred pulse, or identify this prepared organization with physical matter.

The replica pulse owner exposes assess_sine_replica_pulse_splitting. It retains the exact coefficients as a+b*sqrt(5), encloses their signs and the leading return-logarithm slopes, and reuses the same declared model, capacity and symbolic target through a stationary reference template. That reference is not itself labeled unstable. The API accepts no finite amplitude and supplies no numerical amplitude radius, remainder constant, computed multiplier or selected-preparation verdict. The replica tests independently check the forced collective harmonics, resonant projections, algebraic coefficients and return normalization.

12. Collective identity with independently active constituents

This theorem combines the full-state barrier of Section 7.5, the retained internal law of Section 8, and nonlinear recurrence. It uses the complete doubled C5 support, e=0, common held capacity nu>0, w,beta>0 and one structural clock. There are no inputs, events, clipping, discarded constituents or changing edges. The support, pair partition and zero-loss law remain supplied premises. No attracting pulse or new constitutive mechanism is introduced.

12.1. An admitted open full-state family

Let a=w/pi, b=w/(beta*pi) and alpha=2*pi/5. For each pair i=0,...,4 the exact target phase is theta_(i,+),*=theta_(i,-),*=i*alpha, with uniform target form. The fine graph has ten nodes, degree four and twenty edges. Its target storage and spectral gap are

\[ E_*=20\beta(1-\cos\alpha),\qquad \lambda_{2,f}=5-\sqrt5 . \]

Reversing the base cycle orientation gives the same conclusions for winding minus one, with |alpha| in the radius condition.

Write P_f=I-11^T/10, v=P_f x and use the phase chart

\[ \theta=\theta_*+c\mathbf1+h\pmod{2\pi}, \qquad h\perp\mathbf1,\qquad c\in\mathbb R/2\pi\mathbb Z . \]

The common circular phase origin is retained. In the same fixed model coordinates as the preceding barrier, define

\[ Z_f^2=\|v\|^2+\|h\|^2,\qquad 0<r,\quad \alpha+\sqrt2r<\frac\pi2, \]
\[ c_r=\cos(\alpha+\sqrt2r)>0,\qquad \kappa_f=\frac{5-\sqrt5}{2}\min(1,\beta c_r)>0 . \]

This is a genuine chart throughout the closed radius ball and a slightly larger neighborhood. Indeed two representations of the same circular state would have (h_i-h_j)-(h'_i-h'_j)=2*pi*(n_i-n_j). Its magnitude is at most 2*sqrt(2)*r<2*pi, forcing all integers equal; centering then gives h=h' and the same c on its circle. The chart differential is invertible. In particular there is no hidden wrapping boundary inside the ball.

For Z_f<=r every fine edge has its target-compatible increment +/-alpha+h_j-h_i, strictly inside (-pi/2,pi/2). The target is critical. Its vanishing first variation, the phase Hessian bound H_phase>=c_r*L_f along the segment to the target, and the fine spectral gap give

\[ \mathcal E:=E_f-E_* \ge\frac{\lambda_{2,f}}2\|v\|^2 +\frac{\beta c_r\lambda_{2,f}}2\|h\|^2 \ge\kappa_f Z_f^2 . \]

Choose an excess ceiling and finite positive-width mean interval,

\[ 0<\varepsilon_*<\kappa_f r^2,\qquad m_{\rm lo}<m_{\rm hi}. \]

Here epsilon_* is a family ceiling, not the small-amplitude parameter of Section 11. Because fine degree and capacity are common, the conserved weighted form mean is the ordinary mean m(x)=sum(x)/10. Define

\[ \boxed{\mathcal U= \{Z_f^2<r^2,\quad \mathcal E<\varepsilon_*, \quad m_{\rm lo}<m(x)<m_{\rm hi}\}.} \]

Energy and mean conservation prevent a first radius exit in either time direction: an exit would require mathcal E>=kappa_f*r^2>epsilon_*. Consequently

\[ Z_f(t)^2\le\frac{\mathcal E(0)}{\kappa_f} <\frac{\varepsilon_*}{\kappa_f}<r^2 \qquad(t\in\mathbb R). \]

The full smooth flow is complete on the bounding energy/mean slab by the recurrence owner's compactness argument. The chart, both strict inequalities and the mean interval are preserved, so Phi_t(U)=U for every real t. Fine edges remain acute and every cycle retains its target winding. In particular every five-edge loop following the positive base orientation and choosing one constituent from each consecutive pair retains winding one.

The family is open in the full twenty-dimensional state manifold, not just in a synchronized subspace. Form coordinates are x=m*1+v with v perpendicular 1. Together with the phase chart these are locally invertible coordinates, and all inequalities are strict. A uniform-form target with its mean inside the interval has an open neighborhood in U, proving positive volume. Its closure is compact: |x_i-m|<=||v||<=r bounds form, and phases belong to a compact torus. Thus U has finite positive volume for fine-form Lebesgue measure times phase-torus Haar measure.

12.2. Every nontip pair remains internally active

In the retained pair coordinates the fine norm splits exactly as

\[ Z_f^2=2\|P_5X\|^2+ 2\|P_5(\Theta-\Theta_*)\|^2+ 2\sum_i(u_i^2+\delta_i^2). \]

Hence |delta_i|<D:=r/sqrt(2)<pi/2, so the nonantipodal pair chart remains valid for all time. Let

\[ \mathcal T_i=\{u_i=\delta_i=0\},\qquad \mathcal U_{\rm active} =\mathcal U\setminus\bigcup_{i=0}^4\mathcal T_i . \]

Each synchronized tip T_i is invariant under the full smooth flow: u_i_dot=-A_i*sin(delta_i) and delta_i_dot=b*nu*u_i vanish there, irrespective of the motion in other pairs. Two-sided uniqueness then makes its complement invariant. A nontip pair cannot arrive at that tip at a finite time. This conclusion uses realizable fine coordinates, rather than only the polynomial constraint among R,U,Q.

Each T_i has codimension two in the fine chart and zero ambient volume. Therefore U_active is still open, of positive finite volume, and invariant for every real time. Removing the five tips does not remove any positive-volume portion of U.

The acute full-state geometry supplies a uniform restoring sign. Define

\[ F_i=\frac12\sum_{j\sim i}R_j\cos(\Theta_j-\Theta_i), \qquad A_i=a\nu F_i,\qquad c_i=b\nu . \]

For a base edge {i,j}, averaging the cosines of its four actual fine edge increments gives

\[ R_iR_j\cos(\Theta_j-\Theta_i) =\frac14\sum_{s,t=\pm1} \cos(\Theta_j+t\delta_j-\Theta_i-s\delta_i) \ge c_r . \]

Since 0<R_i<=1, each summand R_j*cos(Theta_j-Theta_i)>=c_r. Its upper bound is one. Consequently

\[ \boxed{c_r\le F_i\le1,\qquad A_i\ge a\nu c_r>0.} \]

At any nontip state the unordered internal velocity is nonzero. If Q_i!=0, then R_i_dot=-c_i*Q_i!=0. If Q_i=0, the remaining nontip possibilities in this chart are:

  • R_i=1,U_i>0, where Q_i_dot=c_i*U_i>0;
  • U_i=0,R_i<1, where Q_i_dot=-A_i*(1-R_i^2)<0.

Thus every pair of every state in U_active has a moving internal state (R_i,U_i,Q_i) at every finite time. This does not require each separate scalar coordinate or each fine node rate to be nonzero. It also does not give a uniform positive lower bound on the velocity norm: the open family contains states arbitrarily close to a tip.

12.3. Bounded internal circulation without a common period

There is a stronger all-state consequence of the same retained rows. Normalize internal form only for this calculation:

\[ y_i=\frac{u_i}{\sqrt\beta},\qquad \Omega=\frac{w\nu}{\pi\sqrt\beta}>0. \]

Then along the full interacting trajectory

\[ \dot\delta_i=\Omega y_i,\qquad \dot y_i=-\Omega F_i(t)\sin\delta_i,\qquad c_r\le F_i(t)\le1 . \]

The coefficient F_i(t) is generated by the other retained coordinates; it is not an imposed periodic drive. Since (delta_i,y_i) never equals zero, its argument has a continuous real lift psi_i=arg(delta_i+i*y_i) for all real time. Direct differentiation yields

\[ -\dot\psi_i =\Omega\frac{F_i(t)\delta_i\sin\delta_i+y_i^2} {\delta_i^2+y_i^2}. \]

For |delta_i|<D<pi/2, sinc(D)<=sin(delta_i)/delta_i<=1, taking the continuous value one at zero. The quadratic quotient therefore satisfies

\[ \boxed{\Omega c_r\,\operatorname{sinc}(D) \le-\dot\psi_i\le\Omega,\qquad \operatorname{sinc}(D)=\frac{\sin D}{D}>0.} \]

Each constituent pair undergoes endlessly repeated turns in this internal phase plane in forward time, and oppositely in backward time. From any initial angular value, the unique next full-turn crossing has elapsed time Delta t_i bounded by

\[ \frac{2\pi}{\Omega}\le\Delta t_i \le\frac{2\pi}{\Omega c_r\operatorname{sinc}(D)}. \]

Successive axis crossings have analogous quarter-turn bounds. These are bounds on angular traversal under the declared structural clock, not on recurrence of the internal amplitude or the full state. Different pairs may change amplitude, modulate their traversal times and exchange storage; no exact shared frequency or phase locking has been assumed or proved. Each pair is active, but their dynamics are still coupled; no statistical independence is claimed. A pair swap shifts psi_i by pi and leaves its derivative unchanged. A projective advance of pi likewise need not return the unordered state unless its amplitude also returns. No positive minimum amplitude follows from these angular bounds.

12.4. Almost-everywhere recurrence in the same family

The full conservative sine flow has zero divergence in fine form and circular phase, as proved by the recurrence owner. Restrict that preserved measure to the finite-volume invariant U_active. The same finite-measure recurrence argument gives, for every fixed sampling increment s>0,

\[ \Phi_{n_js}(z)\longrightarrow z,\qquad n_j\longrightarrow\infty, \quad\text{for almost every }z\in\mathcal U_{\rm active}. \]

All these recurrent states are nonstationary by Section 12.2. Here recurrence is measured on the full fine state with circular phases. It is not inferred using ambient Lebesgue measure in the twenty-five constrained invariant coordinates (X,Theta,R,U,Q). An absolutely continuous preparation distribution on the fine family inherits the almost-sure conclusion. A selected state, finite grid, fixed-energy surface or other singular preparation does not receive a recurrence certificate from this ambient-measure theorem.

The quantifiers are distinct: every admitted state preserves its geometry and its five active internal constituents, with the angular traversal bounds above; almost every such state also returns arbitrarily near its full initial state. There is no full-state return deadline, generic exact period or numerical chosen-state guarantee.

12.5. Overlap with the prepared pulse and its instability

Take a pulse preparation from Section 9 at a phase crossing: X_i=m inside the mean interval, Theta_i=i*alpha+c, delta_i=0 and the same u_i=sqrt(H)>0 in all five pairs. At that instant

\[ Z_f^2=10H,\qquad \mathcal E=20H. \]

Thus the explicit sufficient conditions

\[ 0<H< \min\left(\frac{r^2}{10}, \frac{\varepsilon_*}{20},\beta\cos\alpha\right) \]

put the preparation in U_active and in the nonlinear libration family. Every bound on the right is strictly positive. The entire pulse and an open full-state neighborhood of its preparation consequently remain in the admitted collective family, with all constituents active.

Section 11 supplies an existential m_*>0 such that the prepared pulse is transversely orbitally unstable for 0<m=H/(beta*cos(alpha))<m_*. Intersecting this interval with the preceding positive interval proves a nonempty overlap without computing m_*. There is no contradiction: the larger phase organization and internal circulation remain protected while an exact common waveform can be fragile. Nearby states are not thereby proved periodic, synchronized, or convergent to another selected pulse.

Finally, two-sided invariance itself prevents capture into U_active from its complement. The formation boundary still applies. This is a conditional joint maintenance result for a supplied organization, not autonomous construction of its support or partition, selection of microscopic zero loss, universal fractality, or identification of a material constituent.

12.6. Captured-source admission and evidence

The existing replica pulse owner exposes assess_sine_replica_persistence and SineReplicaPersistenceAssessment. It checks the complete ordered doubled-cycle support, common positive capacity and conservative model on one retained capture. Exact target turns are separate from represented source phases. The shared geometry owner supplies whole-source norm, canceled excess-storage and first-exit bounds; its general rational spectral lower bound can be weaker than the exact 5-sqrt(5) used above without invalidating an admitted certificate.

The reader distinguishes the declared finite-volume family's admission, source trapping, each pair's exact tip status, and source membership in the stricter excess/absolute-mean family. Its captured form values supply the actual mean; a relative observation alone would not provide that common origin. Trapping and nontip activity may be certified even when the source is outside the smaller declared mean/excess family. Family almost-everywhere recurrence never becomes selected-source recurrence through these checks.

For numerical angular bounds the reader uses the safe lower factor c_r*cos(D)<=c_r*sinc(D). The inequality follows from sin(D)-D*cos(D)>=0 for 0<=D<pi/2. An outward speed enclosure may touch zero for an extremely small exact positive capacity; that numerical limitation does not negate strict angular circulation under the admitted theorem. A finite upper traversal-time bound remains an angular bound rather than a state-return deadline.

The replica tests check the complete source rows and geometric prerequisites independently. The reader evaluates no trajectory, waits for no recurrence, changes no law, and assigns no finite-amplitude instability radius.