Retained phase records and finite contact readout¶

Native isolated-pattern recovery, retained phase offset, finite contact and removal budgets and the receiver mean record.

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

13. A retained collective phase offset after isolated-pattern recovery¶

An isolated ring can recover the same internal winding geometry while retaining a different common phase relative to a separately declared reference. This result concerns the selected relational law on a supplied C5, not a new state variable or a physical interpretation of phase. It uses the isolated-ring capture theorem and keeps the offset rows that the earlier relative-state memory studies explicitly removed from their observations.

State, reference and exact mean identities¶

Use the oriented unit cycle i -> i+1 modulo 5, one common held capacity nu>0, e,w,beta>0, the reference phase completion theta_dot=(nu*w/beta)*H^-1*Bx, and no input or event during relaxation. Set

\[ \kappa=2\pi/5,\qquad C=\cos\kappa>0,\qquad \theta_i^*=i\kappa. \]

Choose continuous lifts in the acute winding-one component and write

\[ x=\bar x\mathbf1+u,\qquad \theta=\theta^*+m\mathbf1+v,\qquad \mathbf1^Tu=\mathbf1^Tv=0. \]

The scalar m is the arithmetic mean of the lifted deviation from this reference, not an angle reconstructed from form. A second, untouched ring at the same twist and common phase defines the comparison frame. Independently resetting the two component phase frames would erase the chosen comparison and is not part of the protocol. A common rotation of both rings leaves the final relative offset unchanged.

Let (Sz)_i=z_(i+1), B=2I-S-S^-1, and J=S-S^-1. Here J denotes the skew cyclic difference, not the earlier full-state Jacobian. The actual wrapped edge gaps are

\[ \delta_i=\kappa+v_{i+1}-v_i. \]

On this acute branch the native relative resultant is

\[ z_i=e^{i\delta_i}+e^{-i\delta_{i-1}} =2\cos\!\left(\frac{\delta_i+\delta_{i-1}}2\right) e^{i(\delta_i-\delta_{i-1})/2}. \]

The cosine factor is positive and the displayed angle belongs to the principal regular branch. Thus the phase source is exactly linear in the centered phase deviation on this particular cycle component:

\[ g=-\frac1{2\pi}Bv,\qquad \dot u=-aBu-bBv,\qquad a=\frac{\nu e}{2},\quad b=\frac{\nu w}{2\pi},\qquad \dot{\bar x}=0. \]

This exact form-mean conservation uses both degree two and common capacity; it is not imported into an irregular or heterogeneous-capacity graph. The phase metric still contains nonlinear internal geometry:

\[ H_i=2\pi\cos\!\left(\kappa+\frac{(Jv)_i}{2}\right) \operatorname{sinc}\!\left(\frac{(Bv)_i}{2}\right), \qquad \dot m=\frac{\nu w}{5\beta}\sum_i\frac{(Bu)_i}{H_i}. \]

Although sum(H_i*theta_dot_i)=0 exactly, the weights depend on the evolving phase geometry. That instantaneous weighted identity cannot be integrated as conservation of sum(theta_i) or of sum(H_i*theta_i). The calculation below shows that the arithmetic phase mean actually changes.

The first nonzero retained offset¶

Put c=nu*w/(2*beta*pi*C). Expansion at the twist gives

\[ H_i^{-1}=\frac1{2\pi C} \left[1+\frac{\tan\kappa}{2}(Jv)_i+O(\|v\|^2)\right], \]

and hence

\[ \dot m=\frac{c\tan\kappa}{10}(Bu)^TJv +O(\|u\|\,\|v\|^2). \]

The term linear in form vanishes because sum(Bu)=0. For centered preparations u(0)=epsilon*u_0, v(0)=epsilon*v_0, their first variations obey

\[ \dot u_1=-aBu_1-bBv_1,\qquad \dot v_1=cBu_1. \]

The commuting identities B^T=B, J^T=-J, BJ=JB imply

\[ \frac{d}{dt}(u_1^TJv_1)=-a\,u_1^TBJv_1. \]

The other two terms are zero quadratic forms of the skew matrix BJ. On every nonconstant Fourier mode the two-row tangent matrix has negative trace -a*lambda and positive determinant b*c*lambda^2; all centered first variations decay. Integrating the preceding identity therefore gives

\[ \int_0^\infty u_1^TBJv_1\,dt=\frac1a u_0^TJv_0. \]

Consequently the limiting phase offset satisfies the conditional asymptotic law

\[ \boxed{\quad m_\infty-m(0)= \frac{w\tan\kappa}{10\beta\pi e\cos\kappa} \,\epsilon^2u_0^TJv_0+O(\epsilon^3). \quad} \]

This is an integrated nonlinear effect of the supplied nodal phase metric, not a fitted trajectory or a new phase equation. Common capacity cancels from its coefficient. In fact changing the common positive capacity multiplies both evolution rows by the same factor, so it changes the relaxation clock but not the limiting offset from a fixed preparation. No claim uniform in e -> 0, or independent of the selected phase law, follows.

For the negative-winding reference, use kappa=-2*pi/5 throughout. The sine and skew coefficient change sign; cosine, the positive metric at the twist and the recovery barrier remain the same. This is the second sector admitted by the shared coefficient helper, not another law.

The infinite-time remainder is justified by local stability, not by integrating a fixed-time expansion without control. For fixed model coefficients, the smooth quotient field and its Hurwitz linearization give constants independent of sufficiently small epsilon such that ||z(t,epsilon)||<=K*|epsilon|*exp(-lambda*t) and z(t,epsilon)=epsilon*z_1(t)+O(epsilon^2*exp(-lambda_1*t)), with positive decay exponents. Variation of constants supplies the second bound after shrinking the neighborhood if needed. The quadratic mean-rate expression then differs from its first-variation value by an integrable O(epsilon^3) bound; its cubic remainder is integrable as well. This proves the displayed asymptotic coefficient and finite phase limit. This general derivation supplies no numerical remainder constant at a chosen nonzero amplitude; Section 14 closes that separate obligation for one fixed pair and model.

This result does not exclude invariants involving the internal coordinates. Indeed, the limiting phase defines a local asymptotic-phase projection that is constant along each recovering trajectory. Its quadratic expansion is the expression above, including the internal skew product. What fails to be conserved is the bare arithmetic phase mean, not every possible state-dependent phase coordinate.

The general local mechanism clarifies the distinction between a drifting mean and a conserved origin coordinate. In a declared smooth relative chart write y'=f(y) and m'=h(y), with h(0)=0. Assume a forward-invariant neighborhood on which the relative flow Phi_t and its initial-state derivative satisfy uniform exponential decay bounds

\[ \|\Phi_t(y)\|\le C e^{-\lambda t}\|y\|,\qquad \|D_y\Phi_t(y)\|\le C e^{-\lambda t},\qquad \lambda>0, \]

and let h be C^1 with bounded derivative there. These are local recovery and regularity hypotheses, not consequences of common-origin symmetry alone. They make the following integral and its first derivative converge uniformly on smaller neighborhoods:

\[ \psi(y)=\int_0^\infty h(\Phi_t(y))\,dt,\qquad D\psi(y)f(y)=-h(y),\qquad I=m+\psi(y),\qquad \dot I=0. \]

The derivative identity follows by shifting the lower integration limit along the flow. The map (y,m)->(y,m+psi(y)) is an invertible local chart, and I is the limiting origin of the recovering trajectory. Conversely, such a time-independent triangular chart removes mean drift only if its correction solves that derivative equation. On any larger proposed domain, a closed relative orbit of period T necessarily satisfies integral_0^T h(Phi_t(y)) dt=0 for a single-valued correction to exist. The local recovery result does not supply a global chart through other basins or recurrent relative motion.

For an exact scalar control, y'=-lambda*y, m'=c*y^2, lambda>0, has psi=c*y^2/(2*lambda): its bare mean drifts while this nonlinear origin is constant. The static controls verify both transformed rows without evolving a graph. This construction depends on the selected complete law and the retained relative state; eliminated coordinates can carry information needed by psi. It is neither the bare linear form charge Q nor, by default, a primitive-local chart or an independently justified constitutive law. Circular phase origins retain their chosen lift and local domain throughout.

Two equal-mean preparations with opposite retained offsets¶

For a concrete Fourier pair, use

\[ u_{0,i}^{\pm}=\pm\cos(i\kappa),\qquad v_{0,i}=\sin(i\kappa). \]

Choose one shared origin bar(x)=m(0)=0 for both rings; any common initial offsets are recovered by the exact shift symmetries. This choice does not recenter the components independently after they evolve.

Both have zero means, the same initial primitive phases and winding, and identical form/phase storage. Since J*sin(i*kappa)=2*sin(kappa)*cos(i*kappa) and sum(cos(i*kappa)^2)=5/2, the preceding result becomes

\[ m_\infty^{\pm}-m(0) =\pm A\epsilon^2+O(\epsilon^3),\qquad A=\frac{w\tan^2\kappa}{2\beta\pi e}>0. \]

The signs are nonzero and opposite for all sufficiently small positive amplitudes. There is also an exact comparison symmetry: the map (x,theta) -> (-R*x,-R*theta), where (Rz)_i=z_(-i), preserves the positive-winding reference modulo fixed nodewise turns. It maps the positive form preparation to the negative one while leaving the sine phase preparation unchanged. Relabeling and simultaneous form/phase reversal are symmetries of this law with common capacity. With m(0)=0, uniqueness and the same continuous lift give m_infinity^-=-m_infinity^+ exactly within their admitted recovery family.

Capture can be admitted separately from the offset's asymptotic error. For general centered u_0,v_0, it is sufficient that

\[ |\epsilon|\max_i|v_{0,i+1}-v_{0,i}|<\pi/10, \qquad \frac{\epsilon^2}{2} \left(u_0^TBu_0+\beta v_0^TBv_0\right) <\beta(V_{\rm face}-V_*). \]

The first condition retains strict acute gaps. The second bounds storage above its critical twist value, using the zero first variation and abs(cos(delta))<=1 in the second variation; it places the state inside the isolated-ring basin. For the displayed Fourier pair, beta=1 and |epsilon|<=1/32 suffice: the storage excess is at most 5*epsilon^2, while

\[ V_{\rm face}-V_*=5\cos(2\pi/5)-4\cos(3\pi/8)>\frac{13}{1000}. \]

For example, the exact radical formulas give cos(2*pi/5)>309/1000 and cos(3*pi/8)<383/1000, proving this lower bound. The gap perturbation is at most 2*|epsilon|<pi/10. These explicit basin conditions prove recovery; they do not certify the size or sign of the truncated offset formula at epsilon=1/32.

A signed operational readout, separate from attachment work¶

An isolated component's common phase is not an intrinsic gauge-independent memory label. Keep the untouched reference ring, the inherited common frame and a declared matching port on each ring. After ideal recovery, the two forms have the same conserved mean and their relative port phase is delta=m_infinity-m(0), on the small continuous branch. A hypothetical unit attachment at those ports has storage cost

\[ \Delta E_{\rm attach}=\beta[1-\cos\delta] =\frac{\beta A^2}{2}\epsilon^4+O(\epsilon^5). \]

That cost is even in delta. In particular it is exactly equal for the opposite-offset preparations and cannot distinguish their signs.

The fresh signed port response does distinguish them. Put the recovered pattern on the left, with phase offset delta, and the untouched reference on the right. Their relative port resultants after the hypothetical attachment are 2*C+exp(-i*delta) and 2*C+exp(i*delta). All form gradients are zero at this prepared instant. Therefore

\[ \begin{aligned} g_L&=\pi^{-1}\operatorname{Arg}(2C+e^{-i\delta}) =-\frac{\delta}{\pi(1+2C)}+O(\delta^3),& g_R&=-g_L,\\ \dot x_L&=\nu w g_L =\mp\frac{\nu w A}{\pi(1+2C)}\epsilon^2+O(\epsilon^3),& \dot x_R&=-\dot x_L. \end{aligned} \]

The immediate phase rates remain zero because q=0; the form response is the nodal phase-pressure channel. The already established attachment observation evaluates such candidate state/support data without executing an edge event. An actual attachment with nonzero cost still requires declared work or another admitted state/event budget. Its occurrence and timing are not derived by this readout.

The retained information is thus a relation between the recovered pattern and the declared reference, accessible through a signed port observation even when winding, internal geometry, mean form and attachment cost agree. It is not an absolute phase, a selected physical clock, a complete coarse state, or a demonstration of physical memory or spin. Preparation, reference, selected law and readout remain explicit. No finite-amplitude trajectory, finite-time completion of relaxation or numerical asymptotic-error bound has been evaluated by this general derivation. The following fixed-case proof supplies a finite-amplitude bound without evaluating a trajectory.

The shared bound_relational_cycle_memory kernel retains exact centered preparation directions, the sufficient capture-amplitude family and the bounded leading coefficient as separate evidence. Its RelationalCycleMemoryBounds report does not assign an uncomputed finite-amplitude remainder or evolve a graph. The coefficient controls include the exact tangent skew integral and the direction/basin contracts. The mechanism controls check the native mean-rate cross term and the existing hypothetical attachment's signed response versus even storage cost.

14. A finite-amplitude retained-phase certificate without a trajectory¶

Fix e=w=1/2, beta=nu=1, the isolated winding-one C5 and untouched reference of Section 13. The two supplied centered directions are now the exact rational vectors

\[ U=(1,-1,0,0,0),\qquad V=(0,1,-1,0,0),\qquad u(0)=\pm\epsilon U,\quad v(0)=\epsilon V,\quad m(0)=0. \]

They have identical initial means, phases and storage, and U^T*J*V=2. They are not the Fourier mirror pair from Section 13; no exact opposite-offset or equal-attachment-cost symmetry is assumed for their subsequent responses. The following prior inequalities bound the entire infinite recovery. A dyadic amplitude is selected only after those inequalities have been established, without sampling either response.

Storage controls the state and the integrated form response¶

Let z=(u,v) have the Euclidean norm and use the ball of radius r=1/100. In this ball every edge differs from its twist gap by at most 2*r; thus its cosine exceeds C-2*r>7/25, with 3/10<C=cos(kappa)<1/2. The centered C5 Laplacian satisfies lambda_min(B)>1, ||B||<=4, and ||J||<=2. Taylor's integral formula at the critical twist consequently gives

\[ \mathcal E:=E-V_*\ge\frac18\|z\|^2. \]

For the specified directions, U^T*B*U=V^T*B*V=6, so the initial excess obeys mathcal E(0)<=6*epsilon^2. If 0<epsilon<=1/1024, this is strictly below r^2/8; the initial state is inside the ball. Storage nonincrease prevents a first exit and yields

\[ \sup_{t\ge0}\|z(t)\|\le7\epsilon,\qquad \dot{\mathcal E}=-\frac14\|Bu\|^2, \qquad \int_0^\infty\|u\|^2dt\le24\epsilon^2. \]

The ball stays strictly acute and regular. The isolated-ring capture theorem gives u,v -> 0; no quantitative exponential constant is needed in the estimates below. Common offsets are retained as in Section 13.

A cross term controls the integrated phase deformation¶

Let B^+ be the inverse on the centered subspace, extended by zero on constants, and let Pi be the orthogonal centering projection. Write

\[ a=\tfrac14,\qquad b=\frac1{4\pi}>\tfrac1{16},\qquad M=\operatorname{diag}\left(\frac1{2H_i}\right),\qquad \dot v=\Pi MBu. \]

The explicit two-neighbor metric gives 0<M_i<1/2 throughout the ball. For F=u^T*B^+*v, differentiation of the actual rows gives

\[ \dot F=-a\,u^Tv-b\|v\|^2+u^TB^+\Pi MBu \le-\frac1{32}\|v\|^2+\frac52\|u\|^2. \]

Here ||B^+||<1 bounds the last term by 2*||u||^2, and |u^T*v|/4<=||u||^2/2+||v||^2/32 supplies the remaining constants. Since |F(0)|<=2*epsilon^2 and capture gives F(infinity)=0, integration and Cauchy--Schwarz imply

\[ \int_0^\infty\|v\|^2dt\le1984\epsilon^2<2048\epsilon^2, \qquad \int_0^\infty\|u\|\,\|v\|dt\le224\epsilon^2. \]

These are whole-recovery integral estimates. They are neither sampled losses nor a claim that capacity, storage or a phase mean can be omitted.

An integrated cubic remainder for the approximate phase invariant¶

Put d=1/(4*pi*C)<1/3, tau=tan(kappa)<10/3, and K=d*tau/(10*a)<1/2. For each node define

\[ t_i=(Jv)_i/2,\qquad s_i=(Bv)_i/2,\qquad h_i=\frac{C}{\cos(\kappa+t_i)\operatorname{sinc}s_i}. \]

The denominator after dividing by C is D_i=(cos(t_i)-tau*sin(t_i))*sinc(s_i). Since |t_i|<=||v||<=1/100 and |s_i|<=2*||v||, the elementary sine/cosine remainders give

\[ D_i>\frac9{10},\qquad D_i=1-\tau t_i+d_{2,i},\quad |d_{2,i}|\le2\|v\|^2. \]

For example, the three contributions to d_2 are cos(t)-1, tau*(t-sin(t)) and (cos(t)-tau*sin(t))*(sinc(s)-1); the bounds t^2/2, tau*|t|^3/6 and (1+tau*|t|)*s^2/6 suffice. Dividing by the positive D_i now proves

\[ |h_i-1|\le4\|v\|,\qquad |h_i-1-\tau t_i|\le17\|v\|^2. \]

These are bounds on the actual inverse metric, including its zero-source limit, rather than a substituted polynomial phase law.

Use the quadratic approximation to the asymptotic phase,

\[ I_2=m+K u^TJv. \]

The exact linear form row and the quadratic mean-rate term cancel in I_2_dot. The remaining mean-rate term is bounded by 14*||u||*||v||^2; the nonlinear correction in K*u^T*J*v_dot is bounded by 6*||u||^2*||v||. These follow directly from the two inverse-metric inequalities, ||B||<=4, ||J||<=2, d<1/3, K<1/2 and ||Pi||=1. Consequently

\[ \begin{aligned} |\dot I_2|&\le14\|u\|\|v\|^2+6\|u\|^2\|v\|,\\ \left|m_\infty-I_2(0)\right| &\le\left(14\cdot7\cdot224+6\cdot7\cdot24\right)\epsilon^3\\ &=22960\epsilon^3<2^{15}\epsilon^3. \end{aligned} \]

The limiting skew product is zero because both centered coordinates decay. Thus this is an explicit error bound on the limiting phase itself, obtained without an infinite-time numerical integration or an assumed decay rate.

One declared nonzero amplitude and its signed readout¶

Select

\[ \epsilon=2^{-20} \]

after the preceding uniform proof. It is inside the stated trapping family. Let

\[ \mathcal C=2K=\frac{\sin\kappa}{5\pi\cos^2\kappa}. \]

Certified trigonometric bounds give 3/5<mathcal C<1. For instance, cos(kappa)<31/100 implies cos(kappa)^2<1/10 and sin(kappa)>19/20; together with pi<22/7, these give mathcal C>133/220>3/5. The prior remainder bound now yields

\[ \boxed{\qquad \left|m_\infty^\pm\mp\mathcal C\,2^{-40}\right|<2^{-45}. \qquad} \]

In particular,

\[ m_\infty^+>\frac{91}{160}\,2^{-40}>0,\qquad m_\infty^-<-\frac{91}{160}\,2^{-40}<0,\qquad |m_\infty^\pm|<\frac{33}{32}\,2^{-40}<\frac14. \]

These symmetric enclosures do not assert that the two actual offsets are exact negatives. They prove nonzero offsets of opposite signs for the two finite, rationally directed preparations.

For either offset delta, the same hypothetical matching-port attachment to the untouched reference has

\[ \dot x_L=\frac1{2\pi}\operatorname{Arg}(2C+e^{-i\delta}), \qquad \dot x_R=-\dot x_L. \]

Because |delta|<1/4 and 2C+cos(delta)>0, the left response has the opposite sign to delta and cannot be zero. Interval evaluation of this same expression preserves the two separated readouts. The nonnegative cost 1-cos(delta) remains even; unlike the Fourier mirror case, equal costs for these rational preparations have not been proved. Any actual attachment still needs its independently declared event budget and timing.

The shared certify_relational_cycle_memory returns this fixed-case RelationalFiniteMemoryCertificate, with capture, integrated-remainder and signed readout bounds kept separate. The finite-memory controls check admission and the nonlinear interval consequences. The separate exact proof controls verify the derivative cancellation, integral budgets and centered inverse without advancing a trajectory. This does not replace the general coefficient report's unavailable remainder by a bound for arbitrary models or directions.

The result is an analytic finite-amplitude distinction in the limiting relative state. It supplies no finite recovery duration or finite-time readout guarantee, and does not identify physical memory, spin or a unique law of nature. The small amplitude is a conservative mathematical witness, not a fitted physical scale or an optimal robustness radius.

15. A finite-time contact readout with residual-state and event budgets¶

Retain exactly the law, two rational preparation directions, untouched reference and amplitude epsilon=2^-20 of Section 14. The clock is the declared structural clock with nu=1; the numbers below are not laboratory seconds. The proposed contact joins position zero of the recovering left ring to position zero of the untouched right ring, preserving both states. No edge is executed in this calculation.

Convergence alone already implies that sufficiently late readouts have the limiting signs. Here the additional result is a prior quantitative time and a full-state error budget. In particular, the remaining form deformation is not set to zero, and the finite-time phase is not replaced by its limit.

An explicit decay estimate on the already admitted ball¶

Use the same centered state z=(u,v), storage excess mathcal E and cross term F=u^T*B^+*v as in Section 14. In the trapped radius-1/100 ball, Taylor's formula and ||B||<=4 give

\[ \frac18\|z\|^2\le\mathcal E\le2\|z\|^2, \qquad |F|\le\frac12\|z\|^2. \]

The auxiliary proof functional

\[ \mathcal H=\mathcal E+\frac1{32}F \]

is not a new physical storage or a change to the dynamics. It satisfies

\[ \begin{aligned} \frac7{64}\|z\|^2&\le\mathcal H\le\frac{129}{64}\|z\|^2,\\ \dot{\mathcal H} &\le-\frac{11}{64}\|u\|^2-\frac1{1024}\|v\|^2 \le-\frac1{1024}\|z\|^2 \le-\frac1{2064}\mathcal H. \end{aligned} \]

The first differential inequality uses exactly mathcal E_dot=-||Bu||^2/4, lambda_min(B)>1, and the previously proved cross inequality F_dot<=-||v||^2/32+(5/2)*||u||^2. Since mathcal H(0)<=(97/16)*epsilon^2, integration yields

\[ \|z(t)\|\le8\epsilon e^{-t/4128}=:R(t). \]

The existing first-exit argument keeps the trajectory in the ball on which these inequalities hold. This estimate therefore covers the entire continuous recovery, rather than assuming that a finite numerical endpoint has entered a local regime.

A phase-tail bound in the retained reference frame¶

The exact mean phase rate is m_dot=(d/5)*sum_i(h_i-1)*(Bu)_i, because sum_i(Bu)_i=0. Here d<1/3 and |h_i-1|<=4*||v|| on the same ball. Consequently,

\[ |\dot m| \le\frac{16\sqrt5}{15}\|u\|\|v\| \le\frac32\|z\|^2. \]

Integrating the squared decay envelope from the observation time gives

\[ |m_\infty-m(t)| \le3096 R(t)^2<4096R(t)^2=:Q(t). \]

Thus each recovering-ring phase differs from its limiting twist-plus-offset by at most R(t)+Q(t). Its form differs from the common zero form by at most R(t). The untouched reference remains exactly at its supplied equilibrium.

For a nonnegative integer n, choose the declared observation time T_n=4128*n. Since exp(1)>2, the exact rational envelopes

\[ R_n=8\epsilon\,2^{-n},\qquad Q_n=4096R_n^2 \]

bound the residual state and phase tail at that time. They also bound all later times provided recovery continues without an event. No transcendental time inversion or trajectory evaluation is required.

Fresh contact rates, including the no-contact comparison¶

At the declared time write delta_t=m(t)+v_0(t) for the relative port phase and abbreviate R=R_n, Q=Q_n. The two relative neighbor resultants after the hypothetical contact are exactly

\[ \begin{aligned} Z_L={}&e^{i(\kappa+v_1-v_0)} +e^{i(-\kappa+v_4-v_0)}+e^{-i\delta_t},\\ Z_R={}&2C+e^{i\delta_t}. \end{aligned} \]

The state-dependent form gradients and degree-three rates are

\[ \begin{aligned} q_L&=3u_0-u_1-u_4,& \dot x_L^{\rm contact}&=-\frac{q_L}{6} +\frac{\operatorname{Arg}Z_L}{2\pi},\\ q_R&=-u_0,& \dot x_R^{\rm contact}&=\frac{u_0}{6} +\frac{\operatorname{Arg}Z_R}{2\pi}. \end{aligned} \]

These are the actual native rows on the proposed support. They need not be opposites at finite time. The residual q also permits nonzero immediate phase rates; the zero-phase-rate statement for the ideal recovered contact in Section 13 does not apply here.

For comparison, on the unchanged isolated support the left rate is

\[ \dot x_L^{\rm free}=-\frac{(Bu)_0}{4} -\frac{(Bv)_0}{4\pi}, \qquad |\dot x_L^{\rm free}|\le\frac43R, \qquad \dot x_R^{\rm free}=0. \]

This baseline distinguishes the effect of the proposed contact from motion that would already have occurred without it.

There are two equivalent ways to enclose the readout. Direct rational interval evaluation uses u_i,v_i in [-R,R], the Section 14 enclosure for m_infinity, and m(t)-m_infinity in [-Q,Q] in the exact formulas above. Discarding the correlations among those intervals only enlarges the bounds. The contact increment is enclosed separately by subtracting the no-contact baseline. The following simpler estimates also prove sign admission before that evaluation.

Let r_L^infinity and r_R^infinity be the ideal limiting contact rates from Section 14. The resultant differences from that ideal contact satisfy

\[ |Z_L-(2C+e^{-im_\infty})|\le5R+Q,\qquad |Z_R-(2C+e^{im_\infty})|\le R+Q. \]

For |m_infinity|<1/4, both ideal real parts exceed 3/2. If 5R+Q<1/2, the straight segments to the actual resultants stay in Re Z>1; therefore |d Arg Z|<=|dZ| along them. Combining this fact with the form terms and pi>3 gives

\[ \begin{aligned} |\dot x_L^{\rm contact}-r_L^\infty| &\le\frac53R+\frac16Q<2R+Q,\\ |\dot x_R^{\rm contact}-r_R^\infty| &\le\frac13R+\frac16Q<2R+Q,\\ |\dot x_L^{\rm contact}-\dot x_L^{\rm free}-r_L^\infty| &\le3R+\frac16Q<4R+Q. \end{aligned} \]

The right contact increment equals its contact rate, since its baseline is zero. These are bounds on one-sided hypothetical rates at the supplied state, not on a trajectory following the event.

One finite horizon and its separate event-work budget¶

Select one sufficient horizon from those prior inequalities:

\[ n=40,\qquad T=165120,\qquad R=2^{-57},\qquad Q=2^{-102}. \]

There is no search for the first or fastest readable time. Both the internal ring edges and the proposed contact are strictly acute at this horizon; 5R+Q<1/2 also certifies the resultant branches used above.

\[ |r_L^\infty|=|r_R^\infty| >\frac{|m_\infty|}{64}>2^{-47}. \]

At the chosen horizon 4R+Q<2^-54. Hence the finite-time left and right contact rates, and their respective contact-minus-no-contact increments, retain the certified limiting signs in each preparation. The two preparations give opposite signs. Each arm keeps its own enclosure; their actual magnitudes are not asserted to be equal.

The storage jump for the same state-preserving unit contact is separately

\[ \Delta E_{\rm attach}(T) =\frac12u_0(T)^2+1-\cos\delta_T =\frac12u_0(T)^2+2\sin^2\!\left(\frac{\delta_T}{2}\right). \]

Here |u_0|<=R and delta_T in [m_infinity_lower-R-Q,m_infinity_upper+R+Q] give a direct outward enclosure. At this horizon the port-phase intervals exclude zero, so the jump is strictly positive. The form contribution is bounded by R^2/2; it is not silently discarded. The earlier continuous loss is not an event reservoir. Executing this contact would require a supplied event and its positive work or another independently admitted budget.

The shared certify_relational_cycle_memory_readout returns a RelationalMemoryReadoutCertificate for nonnegative integer decay_blocks, retaining the fixed amplitude and preparation. Its default 40 blocks give the witness above. Earlier inconclusive bounds remain explicitly unavailable instead of reporting a zero signal or an executed measurement. The reader encloses the full contact, no-contact and event expressions; it neither advances a graph nor inserts an edge. The readout controls check the exact report and unavailable-result contracts. The proof controls verify the cross-storage decay, tail and horizon budgets; the native mechanism controls check finite residual contact/no-contact rates, their support degrees and the state-preserving storage jump.

This closes the finite-time readout obligation for this supplied experiment within the native model. It does not certify post-contact maintenance, nondestructive readout, an autonomous connection schedule, optimal timing, laboratory units, physical memory or a uniquely selected microscopic law.

16. A finite contact transmits a signed form response¶

Keep the two preparations, amplitude, reference and observation time T=165120 from Section 15. Supply the state-preserving matching-port unit connection at that time and retain it for one declared duration h>0. The joined graph consists of two C5 rings and one bridge, with unit held capacities, e=w=1/2, beta=1, no forcing and the same native relational law. The comparison without contact continues the two original isolated flows from exactly the same state at T.

The result below concerns continuous model solutions. The certificate does not execute the connection, sample a trajectory, choose the event or supply its work. Its new observation is an accumulated form change over a nonzero duration, including the receiver's response, rather than an instantaneous rate at the proposed event.

A state-scaled continuous-flow bound¶

Use the full twenty-coordinate state deviation z from the joined equilibrium with zero forms and two aligned winding-one twists. In particular, retain both common phases and all form coordinates; no centering or regional mean constraint is imposed on the joined flow. Write rho=||z||_infinity. For each preparation a sufficient initial bound is

\[ \rho_0=\max\{|m_{\infty,\rm lower}|,|m_{\infty,\rm upper}|\} +Q+R<2^{-39}, \qquad R=2^{-57},\quad Q=2^{-102}. \]

Consider the fixed sup-norm ball of radius r=1/100. Each node has degree d_i equal to two or three. If its relative neighbor resultant is C_i+i*S_i, the equilibrium has S_i=0 and each incident equilibrium cosine is at least C=cos(2*pi/5)>3/10. The unit Lipschitz bounds for sine and cosine therefore give, throughout the ball,

\[ C_i\ge\frac7{25}d_i>0,\qquad |S_i|\le2d_i\rho,\qquad |q_i|\le2d_i\rho. \]

With c_0=7/25, the phase metric and its zero-source extension satisfy

\[ H_i=\frac{\pi S_i}{\arctan(S_i/C_i)}\ge\pi C_i, \qquad H_i\big|_{S_i=0}=\pi C_i. \]

There is no singular phase row at S_i=0: its mobility can also be written as the smooth function

\[ \frac1{2H_i} =\frac1{2\pi}\int_0^1\frac{C_i}{C_i^2+s^2S_i^2}\,ds. \]

The actual native form and phase rows consequently obey

\[ |\dot\theta_i|\le\frac{\rho}{\pi c_0}\le\frac{25}{21}\rho, \qquad |\dot x_i|\le\left(1+\frac1{\pi c_0}\right)\rho\le3\rho. \]

The form row alone is also uniformly Lipschitz with constant three in this norm. Its form-coordinate derivative row sum is one. The phase derivative of the resultant argument has row sum at most 2*d_i/|C_i+i*S_i|, so its contribution after multiplying by 1/(2*pi) is at most 1/(pi*c_0). The ball is convex, and this derivative bound therefore compares any two states in it. A bound on a phase-row derivative is not needed below.

Let local time s start at contact. Gronwall's inequality and a first-exit argument now give, whenever rho_0*exp(3*h)<r,

\[ \begin{aligned} \|z(s)\|_\infty&\le\rho_0e^{3s},\\ \|z(s)-z(0)\|_\infty&\le\rho_0(e^{3s}-1),\\ |\dot x_i(s)-\dot x_i(0)|&\le3\rho_0(e^{3s}-1), \qquad 0\le s\le h. \end{aligned} \]

Integration yields the state-scaled remainder

\[ \left|x_i(h)-x_i(0)-h\dot x_i(0)\right| \le\rho_0(e^{3h}-1-3h). \]

This error vanishes with the actual prepared displacement. It does not replace the small memory response by an amplitude-independent acceleration bound.

One duration with accumulated contact and no-contact responses¶

For 0<h<=1/3, the shared exact-clock exponential bounds apply to 3*h. If E_hi encloses exp(3*h) from above, use

\[ \rho_{\rm tube}=\rho_0E_{\rm hi},\qquad D=\rho_0(E_{\rm hi}-1),\qquad B=\rho_0(E_{\rm hi}-1-3h). \]

These respectively enclose the state tube, total state disturbance and form-response remainder. The contact rate intervals [a_i,b_i] from Section 15 give accumulated form changes in [h*a_i-B,h*b_i+B]. Without contact, the continued isolated decay keeps the left form rate in [-4*R/3,4*R/3] at every later time, while the reference rate stays zero. Thus the no-contact form changes lie in h*[-4*R/3,4*R/3] and {0}. Subtract these enclosures to retain the contact-induced changes as a separate comparison; no second joined-flow remainder is required.

Choose the single duration

\[ h=2^{-12}. \]

The inequalities already prove that this duration is sufficient before evaluating any trajectory. Since exp(3*h)<2, rho_tube<2*rho_0<r, D<6*rho_0*h, and B<9*rho_0*h^2. Section 15 gives each initial contact-rate magnitude greater than (255/256)*2^-47. Meanwhile,

\[ \frac Bh<\frac9{16}\,2^{-47},\qquad \frac43R<\frac1{512}\,2^{-47}. \]

Therefore both accumulated contact form changes, and their respective contact-minus-no-contact changes, keep their predicted signs. Even after both error budgets, the per-duration sign margin exceeds (221/512)*2^-47. The two preparations produce opposite signs. In particular, the initially unperturbed right ring acquires a nonzero signed form change during the contact; its no-contact change is exactly zero. The two arms have independent bounds, not an assumed equality of response magnitudes.

Every original edge remains strictly acute because its phase gap differs from +/-kappa by at most 2*rho_tube<2*r. The bridge gap is at most 2*rho_tube. Continuous motion in this domain preserves the winding-one identity of each original cycle throughout the declared contact duration. This is not a claim of unchanged internal shape: the bounded form/phase disturbance is the interaction being measured.

Event work and continuous loss retain different accounts¶

The initial attachment cost is exactly the finite-time cost from Section 15; its enclosure includes the residual form mismatch. The maximum of the two cost upper bounds is a sufficient prospective work budget for either preparation; it is not asserted to be the minimal required work. This is a conditional budget bound, not an inferred reservoir or an executed supply of work.

On the joined support the continuous loss is nonnegative and satisfies

\[ L=\frac12\sum_i\frac{q_i^2}{d_i} \le2\rho_{\rm tube}^2\sum_i d_i =44\rho_{\rm tube}^2, \qquad 0\le\int_0^h L\,ds\le44\rho_{\rm tube}^2h, \]

because the eleven-edge graph has total degree twenty-two. This optional loss bound is separate from the event jump; it is not work available to pay for insertion and does not make the addition passive. No removal event or post-contact continuation is included in this protocol.

The shared certify_relational_memory_contact returns a RelationalMemoryContactCertificate and per-preparation RelationalMemoryContactCase reports. It reuses the fixed 40-block readout certificate and the shared exponential enclosure, with default duration=1/4096. Other admitted durations 0<h<=1/3 retain conservative bounds and explicit sign availability; they do not select an optimal contact time. The contact controls, continuous proof controls and native mechanism controls keep report arithmetic, analytic premises and the engine's nodal rows distinct.

This establishes finite-duration information transfer under one supplied connection while both cycle windings persist. It does not derive the event's occurrence, certify nondestructive readout or arbitrarily long contact, or identify a physical memory carrier, physical time or a unique constitutive law.

17. A receiver mean survives contact removal and isolated recovery¶

Keep the complete fixed protocol of Section 16, including h=1/4096. At T+h, supply one state-preserving removal of the matching-port bridge. Then evolve each isolated unit C5 independently under its native unforced law with held unit capacities and the same structural clock. Recompute each component's degree, pressure and phase metric on its actual support; the connected relational executor is not thereby extended to a disconnected union. No further event, input or capacity change is part of this continuation.

The observation is the receiver's arithmetic mean form relative to its declared initial mean zero. This differs from the port form change proved in Section 16. It also differs from winding, internal shape or an absolute form-origin claim. A common translation of all initial and final forms leaves this mean change invariant.

Bound the regional mean before invoking conservation¶

At the start of contact the entire receiver is still at its exact supplied equilibrium. Only its port acquires a nonzero form rate when the bridge is added; the other four receiver nodes keep the same neighbor states and support. If the initial right-port rate is enclosed by [a_R,b_R], the sum of initial receiver form rates is therefore enclosed by that same interval.

Let B=rho_0*(E_hi-1-3*h) be the per-node continuous remainder from Section 16. Summing all five form changes and dividing by five gives

\[ \boxed{\qquad \mu_R(T+h)\in \left[\frac h5 a_R-B,\ \frac h5 b_R+B\right]. \qquad} \]

The error is B, not B/5: all five nodes may contribute a remainder. The no-contact receiver mean remains exactly zero. No equal-and-opposite regional transfer is assumed on the irregular joined graph.

The regional intervals have opposite nonzero signs for the two fixed preparations. This also follows from conservative inequalities independent of any sampled response. For 0<|delta|<1/4, use |sin(delta)|>(3/4)*|delta|, 2C+cos(delta)<2, and atan(y)>(2/3)*y for the relevant 0<y<1. With pi<4, the ideal contact-rate magnitude exceeds |delta|/32. Section 14 gives |m_infinity|>(91/160)*2^-40, and Section 15 changes that rate by less than 2R+Q<2^-55.

The already fixed preparation also obeys rho_0<(17/16)*2^-40. At h=2^-12, the bound B<9*rho_0*h^2 consequently gives

\[ \frac{h}{5}|\dot x_R^{\rm contact}(0)|-B > h\left(\frac{91}{25600}-\frac1{163840}-\frac{153}{65536}\right)2^{-40} =\frac{1989}{1638400}\,2^{-52}>0. \]

Thus the positive-form preparation writes a positive receiver mean, and the negative-form preparation writes a negative one. Exact outward evaluation of the existing intervals provides tighter bounds; it does not fit or evaluate a new trajectory.

The cut has its own storage jump¶

Removal leaves every nodal coordinate, and hence each regional mean, unchanged. Its exact storage jump is

\[ \Delta E_{\rm cut} =-\left\{\frac12(x_{L0}-x_{R0})^2 +1-\cos(\theta_{L0}-\theta_{R0})\right\}\le0. \]

For example, the whole-contact state radius rho_tube immediately gives -4*rho_tube^2<=Delta E_cut<=0. A sharper endpoint enclosure reuses the already certified all-coordinate disturbance D: the final bridge phase lies in its initial interval enlarged by 2D, and the final form mismatch lies in [-R-2D,R+2D]. Evaluate the nonnegative edge cost with these intervals and negate its endpoints to enclose the jump. The phase intervals still exclude zero in this fixed case, so this sharper jump is strictly negative.

Deletion therefore adds no positive storage requirement. It does not retroactively fund the positive insertion work, select when either event occurs or identify the removed edge storage with an accessible reservoir.

Both isolated rings remain in their recovery basins¶

For either ring at the cut, every form and lifted phase deviation from its original aligned twist is at most rho_tube. Each form-edge difference is at most 2*rho_tube, giving at most 10*rho_tube^2 of form storage over the five edges. For phase storage, write the oriented edge gaps as kappa+v_(i+1)-v_i. The linear Taylor terms around the twist sum to zero; the second derivative of 1-cos is at most one. Thus another 10*rho_tube^2 bounds the phase excess, and

\[ 0\le E_R-V_*\le20\rho_{\rm tube}^2 <V_{\rm face}-V_*. \]

The last strict inequality follows already from rho_tube<2^-38 and the previously proved V_face-V_*>13/1000. Section 16 preserves the strict acute winding-one sector, and the state-preserving cut changes none of its internal gaps. Both post-cut components therefore satisfy the isolated-ring capture theorem. Each remains regular, retains its winding and converges to a uniform form and uniform twist, with its own common phase offset.

The retained mean is exactly conserved during that recovery¶

On a strictly acute winding-one C5 let B_5 be the cycle Laplacian and v any consistent lifted phase deformation from the uniform twist. The two-neighbor circular resultant gives exactly

\[ g=-\frac1{2\pi}B_5v, \qquad \dot x=-\frac14B_5x-\frac1{4\pi}B_5v. \]

The pair of relative neighbor angles stays within an interval of width less than pi, so its argument is their arithmetic midpoint on this branch; this is the domain needed for the first identity. Since the Laplacian has zero column sums,

\[ \frac d{dt}\left(\frac15\sum_{i\in R}x_i\right)=0. \]

Combining conservation with capture identifies the receiver's limiting uniform form with mu_R(T+h). Its certified nonzero interval is therefore retained for every later time as a regional mean, even while the individual node forms continue to reorganize. The isolated no-contact receiver keeps uniform form zero. The signs distinguish the preparations after the connection that transmitted the signal has been removed.

This conservation step uses common held capacity and degree-two native pressure; it is not a generic arithmetic-mean conservation theorem for irregular networks or heterogeneous capacities. It does not depend on the particular post-cut phase row beyond remaining in the admitted acute domain. Accordingly, the same retained mean follows if the post-cut phase row is replaced by any law already admitted by the isolated capture theorem, with the same native form row and common capacity. That conditional continuation does not transfer the preceding preparation or contact-response calculation to another law.

The shared certify_relational_memory_retention reuses the fixed contact report and separately reports the receiver-mean interval, component capture margins and removal jump. The retention controls, exact proof controls and native regional controls check these distinct obligations without advancing a recovery trajectory.

The result is a conditional persistent receiver record in the declared form coordinate. It does not equate that coordinate with a measured physical quantity, demonstrate immunity to arbitrary future forcing or topology changes, select an autonomous contact/cut mechanism, or claim that the record is encoded by a different winding or internal geometry.