Collective phase families and live contact feedback

Exact phase-offset partitions, nearby full-state moving windows, contact/storage exchange and complete receiver-rigidity conditions.

Part of Sine pattern geometry and dissipative capture. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

39. Compatible internal geometry and collective phase dynamics

A region can retain phase differences while its common form and phase continue to evolve. The full nodal rows determine when this is possible; an instantaneous equality of phase rates does not suffice. This section classifies exact collective families under Section 35's conservative unit law, fixed finite connected simple support, unit held capacities and tau=t/pi. No input, event or change of support is introduced.

A declared partition with fixed internal phase offsets

Partition every fine node into nonempty blocks B_a. Supply fixed circular offsets psi_i; a consistent real lift can be used for differentiation. Consider the whole family

\[ \mathcal M_\psi: \qquad x_i=X_a,\qquad \theta_i=\Theta_a+\psi_i \quad(i\in B_a), \]

with freely varying collective forms X_a and phases Theta_a. Internal phase differences are fixed, but different nodes need not share a phase. The block partition and offsets are supplied geometry, not inferred membership or a newly created node. Define, using full graph degrees,

\[ q_{ib}=\frac{|N(i)\cap B_b|}{d_i},\qquad Z_{ib}=\frac1{d_i} \sum_{j\in N(i)\cap B_b}e^{\mathrm i(\psi_j-\psi_i)}. \]

The family is invariant under the complete law if and only if, in each block a, the following conditions hold:

1. For every external block b!=a, q_ib=q_ab is independent of i.
2. For every external block b!=a, the full complex value Z_ib=Z_ab is independent of i.
3. Every internal sine sum vanishes: Im Z_ia=0 for all i in B_a.

These are conditions on support and phase offsets for all collective states. Failure does not exclude an accidental special trajectory with additional relations between its collective coordinates. In particular, equal neighbor counts alone do not settle the nonlinear form row.

Necessity and sufficiency use both consumed rows

On the proposed family the phase row at i in B_a is

\[ \theta_i'=\sum_{b\ne a}q_{ib}(X_a-X_b). \]

For its value to be independent of i for every independent choice of X, each external coefficient must agree. The form row is

\[ x_i'=\operatorname{Im}Z_{ia} +\sum_{b\ne a}\operatorname{Im} \left[Z_{ib}e^{\mathrm i(\Theta_b-\Theta_a)}\right]. \]

Independent variation of the collective phase differences makes the external sine and cosine coefficients independent Fourier components. Their equality forces equality of the external complex values Z_ib. The remaining internal imaginary parts must be one common constant c_a. Undirected internal edges give

\[ \sum_{i\in B_a}d_i\operatorname{Im}Z_{ia} =\sum_{\substack{i,j\in B_a\\j\sim i}} \sin(\psi_j-\psi_i)=0. \]

Since sum_(i in B_a) d_i>0, that common constant is zero. There is no additional condition on Re Z_ia: the form row does not consume it and the phase row has zero internal form differences. For example a one-block P3 with offsets (0,0,pi) has internal real values (1,0,-1), zero internal sine current and uniform-form equilibria. Requiring the complete internal complex sums to agree would wrongly reject this family.

Conversely, the three conditions make both fine rows constant within their respective blocks. The field is tangent to M_psi everywhere; smooth uniqueness therefore preserves the family. Its exact closed law is

\[ \boxed{\begin{aligned} X_a'&=\sum_{b\ne a}\operatorname{Im} \left[Z_{ab}e^{\mathrm i(\Theta_b-\Theta_a)}\right],\\ \Theta_a'&=\sum_{b\ne a}q_{ab}(X_a-X_b). \end{aligned}} \]

Thus phase-offset geometry and support determine whether all constituents maintain compatible rates while the collective coordinates move. This is an invariant-family reduction, not a quotient for every fine state.

Reciprocity, inherited storage and conserved means

Let m_a=sum_(i in B_a) d_i, N_ab=m_a q_ab and P_ab=m_a Z_ab. Counting the same undirected cross edges in reverse gives

\[ N_{ab}=N_{ba},\qquad P_{ab}=\overline{P_{ba}},\qquad |Z_{ab}|\le q_{ab}. \]

The full fine storage restricted to the family is exactly

\[ \begin{aligned} E_{\mathcal M} &=E_{\mathrm{int}}(\psi) +\sum_{a<b}\left[ \frac{N_{ab}}2(X_a-X_b)^2+N_{ab} -\operatorname{Re}\left( P_{ab}e^{\mathrm i(\Theta_b-\Theta_a)}\right)\right],\\ E_{\mathrm{int}}(\psi) &=\sum_a\sum_{\{i,j\}\subset B_a,\ i\sim j} [1-\cos(\psi_j-\psi_i)]. \end{aligned} \]

The closed rows consequently satisfy

\[ X_a'=-\frac1{m_a}\frac{\partial E_{\mathcal M}}{\partial\Theta_a}, \qquad \Theta_a'=\frac1{m_a}\frac{\partial E_{\mathcal M}}{\partial X_a}. \]

This is a restriction of the existing storage and reciprocal field, not an independently supplied energy. It preserves that storage and the collective weighted sums sum_a m_a X_a and sum_a m_a Theta_a in continuous lifts. The fixed contribution sum_i d_i psi_i restores the full lifted phase sum.

In general the inherited form coupling Z_ab and phase coupling q_ab are different. Phase cancellation can give |Z_ab|<q_ab, and complex offsets can remain in the interaction. Replacing both by the weights of a bare normalized-sine graph would then change the complete law. This is the same retained-information boundary as in nonlinear replica inheritance.

For each block separately, internal form storage is zero and internal phase storage is constant. In fact q_R=S_R=0 at every point of this family, not only initially. Section 36's internal channels and the net regional power P_R therefore vanish throughout its motion. This does not remove interaction: the collective forms and phases, cross-edge storage and common regional rates can change. Internal regional storage does not count that cross-edge organization.

One moving realization on the retained private-leaf support

Use the same C5 plus five private leaves as Section 37, with blocks R=(0,...,4) and Q=(5,...,9). Supply offsets psi_i=psi_(5+i)=sigma*2pi*i/5, for sigma=+1 or -1. The external coefficients are

\[ q_{RQ}=Z_{RQ}=\frac13,\qquad q_{QR}=Z_{QR}=1. \]

Every receiver internal sine sum cancels between the two cycle directions; the leaves have no internal edges. Write the block forms as a,b and phases as Theta,Psi. The complete reduced rows are

\[ a'=\tfrac13\sin(\Psi-\Theta),\qquad b'=-\sin(\Psi-\Theta),\qquad \Theta'=\tfrac13(a-b),\qquad \Psi'=b-a. \]

Putting u=a-b, phi=Psi-Theta gives

\[ u'=\tfrac43\sin\phi,\qquad \phi'=-\tfrac43u,\qquad \tfrac12u^2+1-\cos\phi=\text{constant}. \]

The receiver retains its exact acute winding and internal storage V_5 while interacting with its moving environment. Its collective weighted form sum is 15a+5b; the analogous lifted phase sum has the same weights. The common regional phase motion is compensated by the environment, in accord with Section 38's global phase balance. The inherited pendulum reuses the reciprocal exchange mechanism; it is not another fundamental oscillator, frequency identification or universal permanent-pulse postulate.

Exact compatibility is not finite-time formation into the family

The complete smooth field has a unique flow in both time directions. If a trajectory met this invariant family at a finite time, the unique backward solution through that point would also lie in the family. Therefore a preparation outside M_psi cannot enter it exactly at finite time under this unchanged autonomous law. In particular, globally flat phases cannot enter the displayed nonzero-twist family exactly.

This does not exclude entry into a neighborhood with nonzero width, nor finite retention of a less restrictive identity. Those require an actual entry state, declared finite error bounds and a comparison under the same complete law. The existing full-state finite-window argument illustrates that separate obligation on its own support; it is not automatically a certificate for these arbitrary partitions. No acquisition trajectory or attraction theorem is supplied by the exact classification.

Shared admission distinguishes proof from undecided equality

The partition owner provides assess_sine_phase_offset_partition(source, blocks, phase_offset_turns) and SinePhaseOffsetPartition. Declared rational-turn offsets specify the phase geometry independently of a response. Its .evaluate(...) returns a SinePhaseOffsetState for admitted collective coordinates, retaining fine reconstruction, the inherited rows and storage. It does not install a coarse runtime, select a partition or certify that the source already belongs to the supplied family.

The all-real mathematical criterion is stronger than any particular finite equality recognizer. Exact cancellations admitted by the implementation, including sine oddness and angle reflection, can prove the required identities. Certified interval exclusion of zero can prove a failed equality. An unrecognized symbolic cancellation or an interval containing zero remains unavailable; it is not evidence of inequality and must not be converted into a failed mathematical criterion.

The held native joint quotient, the replica family and the independent-swap criterion retain their different models and quantifiers. This theorem concerns supplied fixed offsets and all collective states on their invariant family; it does not claim closure of arbitrary fine states, autonomous formation, support selection or physical identification.

40. A finite-width window around compatible moving geometry

Section 39's exact family is not an acquisition target that can be entered from outside at finite time. A neighborhood with nonzero width has a different obligation: errors in both the receiver and its environment must remain controlled while the reference moves. The following sufficient certificate uses the actual storage and complete conservative field. It proves finite retention for an independently supplied entry neighborhood; it does not produce that neighborhood from an earlier preparation.

Reference motion, error chart and declared duration

Retain the C5 with five private leaves, unit capacities, e=0, w=beta=1, no input or events, and tau=t/pi. Use Section 39's certified two-block family with receiver winding sigma=+1 or -1, matching receiver/leaf phase offsets and alpha=2pi/5. Reference block forms are a,b, their phases are Theta,Psi, and

\[ u=a-b,\qquad \phi=\Psi-\Theta,\qquad h=\tfrac12u^2+1-\cos\phi. \]

The inherited pendulum preserves h and satisfies |phi'|<=4sqrt(2h)/3. Choose a radian contact envelope 0<Gamma<pi/2, with |phi(0)|<Gamma and h<1-cos(Gamma). Energy and continuity then keep the reference contact phase in (-Gamma,Gamma) for all positive and negative times. This includes the zero-pulse reference; a nonzero pulse is not required.

Fix a finite duration T>0 in tau and two radian error radii:

\[ 0<\rho_R<\frac\pi{10},\qquad 0<\rho_C<\frac\pi2-\Gamma. \]

Set kappa_R=cos(alpha+rho_R)>0 and kappa_C=cos(Gamma+rho_C)>0. The internal receiver and contact errors use different margins. Contact convexity is a sufficient premise of this certificate, not a defining property of a regional identity. Exact compatible families can also move through nonacute contacts; this estimate does not assess their transverse robustness.

For an actual full solution and the reference, use continuously matched phase lifts. On oriented edge i->j, write f_e=x_i-x_j, delta_e=theta_j-theta_i, xi_e=f_e-f_e^* and eta_e=delta_e-delta_e^*. Admit the initial chart separately: |eta_e(0)|<rho_R on receiver edges and <rho_C on contacts. A small scalar storage value alone does not establish these phase-chart premises.

The moving-reference relative-storage identity

For U(delta)=1-cos(delta), define

\[ D=\frac12\sum_e\xi_e^2 +\sum_e\left[U(\delta_e^*+\eta_e)-U(\delta_e^*) -\sin(\delta_e^*)\eta_e\right]. \]

This is the Bregman remainder of the existing full storage, not another physical energy or a new evolution rule. The complete field has the constant reciprocal matrix with blocks 0,-K;K,0. Differentiating the remainder along both actual solutions cancels the quadratic form terms and gives exactly

\[ \boxed{D' =\sum_e(\delta_e^*)' [\sin(\delta_e^*+\eta_e)-\sin(\delta_e^*) -\cos(\delta_e^*)\eta_e].} \]

Internal reference gaps are fixed at sigma*alpha; only the five contacts contribute, each with reference gap phi. While the admitted chart holds, convexity and Taylor's remainder imply

\[ D\ge\frac12\sum_e\xi_e^2 +\frac{\kappa_R}2\sum_{e\in R}\eta_e^2 +\frac{\kappa_C}2\sum_{e\in C}\eta_e^2, \qquad |D'|\le\frac{|\phi'|}{\kappa_C}D\le\lambda D, \]

where

\[ \lambda=\frac{4\sqrt{2h}}{3\kappa_C},\qquad d_* =\frac12\min\{\kappa_R\rho_R^2,\kappa_C\rho_C^2\}. \]

Thus a proved initial upper bound D(0)<=D_0, together with the initial chart and the prospective strict inequality

\[ \boxed{D_0e^{\lambda T}<d_*} \]

retains the complete error chart for every |tau|<=T. Indeed a first chart exit would require D>=d_*, contradicting the integrated bound. This is a full-interval proof, not a claim inferred from sampled endpoints.

For independent initial nodal form/phase error radii r_i,p_i>=0, the shared sufficient initial bound is

\[ D_0=\frac12\sum_{\{i,j\}\in E} [(r_i+r_j)^2+(p_i+p_j)^2]. \]

The same component sums must pass the separate initial edge-chart tests. During the window, every edge form error is at most sqrt(2D_max); receiver and contact phase errors are at most sqrt(2D_max/kappa_R) and sqrt(2D_max/kappa_C), respectively, where D_max=D_0 exp(lambda T). Initial weighted form/phase mean errors remain their conserved values. Retaining those origins with the edge bounds controls the relative full state on this connected support; no hidden environmental coordinate has been replaced by an imposed boundary drive.

Internal identity and actual work remain compatible

The reference receiver has uniform form and equal oriented principal gaps sigma*alpha. Its internal linear phase term cancels because sum_(e in R) eta_e=0. The receiver's contribution to D is therefore exactly E_R-V_5. Nonnegative contact contributions give

\[ 0\le E_R-V_5\le D\le D_{\max}. \]

The receiver retains acute winding sigma throughout the window, with bounded internal form as well as phase. The environment continues its own nodal motion. The regional work identity still determines its actual net work; zero work for the exact reference does not imply zero work for every nearby trajectory. This result supplies a positive-width continuation criterion without assuming that an arbitrary environment is harmless.

Conserved storage and backward time restrict acquisition claims

The full reference storage is H^*=V_5+5h. At the declared entry instant, the exact relation to an actual state is

\[ H-H^*=D+u\sum_{e\in C}\xi_e +\sin\phi\sum_{e\in C}\eta_e. \]

With the component error radii above, put

\[ L_0=|u|\sum_{\{i,j\}\in C}(r_i+r_j) +|\sin\phi|\sum_{\{i,j\}\in C}(p_i+p_j). \]

The whole declared entry box has the conserved-storage enclosure H in [H^*-L_0,H^*+L_0+D_0]. It stays valid along every covered trajectory because the full law conserves H.

If its upper bound is below 7/2, the phase-flat acquisition barrier excludes every phase-flat source from reaching this entry box. In the vanishing-width limit, a necessary compatibility condition is

\[ h\ge\frac{7/2-V_5}{5} =\frac{5\sqrt5-11}{20}. \]

This floor neither proves entry nor uniquely selects a pulse. The larger sector proof also excludes any initial zero-winding receiver from a target box whose conserved full storage is strictly below 7/2. Initial winding and the actual full budget remain separate admitted observations; a nonacute unit-winding source is not automatically covered by that particular zero-winding test.

A named acquisition source must also have a conserved storage compatible with the declared box. For example Section 38's private-leaf ramp x_(5+i)=-3omega(i-2) has exact full storage 45omega^2. Its large-rate acquisition cannot be joined to a small-error reference with h=1/50 and full storage near V_5+1/10: the budgets are disjoint. Discarding that discrepancy would remove the actual environmental state or silently introduce loss. Budget overlap is necessary, not sufficient, for a new favorable preparation to acquire into this neighborhood.

The error estimate also holds backward for the same finite duration. Every admitted entry-box state already had the acute identity over its certified preceding window. Hence this certificate cannot describe first appearance at the central instant; an initially identity-absent preparation must precede that window if the same support and law apply throughout. Formation followed by later stabilization remains a separate possibility.

The example u=1/5, phi=0, Gamma=1/4, rho_R=rho_C=1/10, T=5 and nodal form/phase error radii 1/1000 gives a nonzero-width finite retention certificate with a moving reference. These declared values illustrate the sufficient inequalities without optimizing a threshold or executing a new trajectory. Setting the reference pulse to zero gives a complementary low-storage control: maintenance can be certified while phase-flat acquisition is excluded.

assess_sine_moving_pattern_window rebuilds the declared reference and applies these independent chart, reference-motion, storage-growth and acquisition-budget checks. It also encloses full storage directly from the primitive edge uncertainty boxes. When the initial receiver chart is certified, the lower bound additionally retains its correlated cycle minimum V_5, rather than independently minimizing all five phase gaps. The primitive enclosure remains valid when the relative-storage certificate is unavailable. The contract and usage retain the distinction between a certified finite neighborhood and an independently demonstrated formation-to-retention handoff.

41. Actual contact motion can exchange storage with internal structure

The contact pendulum of Section 39 conserves its own storage only on the exact compatible family. Away from that family, the same full nodal law supplies a correction from unequal contact phases. This identifies an actual exchange mechanism to examine, rather than adding a controller or assuming that a prepared compatible pattern has already formed.

Mean-contact rows from the complete state

Keep the unchanged conservative unit C5/private-leaf support of Sections 39–40. Match receiver node r_i with its actual private leaf l_i. No uniformity of the observed forms or phases is assumed. Define

\[ a=\frac15\sum_i x_{r_i},\qquad b=\frac15\sum_i x_{l_i},\qquad u=a-b, \]

and, in declared continuous phase lifts,

\[ \phi_i=\theta_{l_i}-\theta_{r_i},\qquad \bar\phi=\frac15\sum_i\phi_i,\qquad \eta_i=\phi_i-\bar\phi,\qquad \sum_i\eta_i=0. \]

Internal cycle currents cancel in the receiver mean. Summing the complete phase rows cancels its internal form Laplacian as well. In tau=t/pi, the exact mean-contact laws are therefore

\[ \boxed{u'=\frac43\left\langle\sin\phi_i\right\rangle, \qquad \bar\phi'=-\frac43u,} \]

where brackets denote the mean over the five contacts. Equivalently, if Z=<exp(i eta_i)>, then u'=(4/3) Im[exp(i bar_phi) Z]. The contact deviations are live fine-state coordinates. Replacing Z by one recovers the exact compatible family, but it is not justified for an arbitrary acquired state.

Define the chart-dependent collective contact storage

\[ h_c=\tfrac12u^2+1-\cos\bar\phi. \]

Its exact derivative is

\[ \boxed{h_c' =\frac43u\left[ \left\langle\sin(\bar\phi+\eta_i)\right\rangle -\sin\bar\phi\right].} \]

Thus collective contact storage is not a second general invariant. Its exchange with the remaining structure can have either sign. The formula does not select an initial source, a direction of long-time transfer or a new constitutive law.

The storage split retains its phase-lift dependence

Let d_i=x_(r_i)-x_(l_i), and use F_R,V_R for the actual internal receiver stores. Direct edge summation gives

\[ \begin{aligned} H&=5h_c+\mathcal R,\\ \mathcal R &=F_R+V_R+\frac12\sum_i(d_i-u)^2\\ &\quad+\sum_i\left[ U(\bar\phi+\eta_i)-U(\bar\phi) -\sin\bar\phi\,\eta_i\right], \qquad U(s)=1-\cos s. \end{aligned} \]

Consequently R'=-5h_c' because the full storage H is conserved. The remainder is signed in general. If all segments joining bar_phi to the five contact phases lie in one convex cosine-potential chart, its contact terms are nonnegative. Even in that chart, nonnegativity does not make the remainder an independently controlled reservoir or specify when it feeds the receiver. Its actual transfer remains fixed by the full rows.

The contact mean uses retained real lifts. Independently wrapping each contact and then averaging can change bar_phi, h_c and the signed remainder, while leaving the actual full storage unchanged. A single snapshot can declare a starting lift, but it does not reconstruct discarded phase history. These collective coordinates are not new globally intrinsic physical energies. The common-real-phase preparation below fixes their initial lift unambiguously, and the complete smooth phase rows continue it.

A phase-flat preparation has a discriminating fourth derivative

At a preparation with one common real phase, set v_i=phi_i'(0), m=<v_i>=-4u/3, and delta_i=v_i-m. All form rates vanish initially. The first three derivatives of h_c are zero. Differentiating the exact transfer row gives

\[ \boxed{h_c^{(4)}(0) =\frac{16u^2}{3}\left\langle\delta_i^2\right\rangle -\frac{4u}{3}\left\langle\delta_i^3\right\rangle.} \]

To check the cancellation, write g=<sin(phi_i)>-sin(bar_phi). At zero phase, g=g'=g''=0, and g'''=-(<v_i^3>-m^3); mean higher phase jets cancel because bar_phi=<phi_i> exactly. Expanding <(m+delta_i)^3> gives the displayed variance and third-moment terms. This is a local derivative of the full law, not a finite-time approximation substituted for it.

One phase-flat preparation above the necessary phase-path budget

On receiver nodes (0,...,4) and private leaves (5,...,9), supply

\[ u=\frac{119}{100},\qquad \epsilon=\frac1{20},\qquad x_{r_i}=\frac u4+\epsilon(\mathbf1_{i=0}-\mathbf1_{i=1}), \qquad x_{l_i}=-\frac{3u}{4}, \]

with all phases zero. The weighted form and phase means are zero. The initial receiver has zero winding, so the proposed acute unit-winding identity is absent. Exact edge summation gives

\[ H=\frac52u^2+4\epsilon^2 =\frac{14201}{4000}=3.55025. \]

This is above the necessary 7/2 phase-path barrier. It does not make Section 40's particular small-error reference box accessible. That box retains acute receiver winding, hence V_R>=V_5; its five contact form gaps are each at least 1/5-2/1000=99/500. Consequently every state in that box satisfies the sharper, correlation-aware lower bound

\[ H\ge V_5+\frac52\left(\frac{99}{500}\right)^2 =V_5+\frac{9801}{100000} >\frac{14201}{4000}. \]

An independent componentwise edge enclosure can overlap the source budget while forgetting this exact cycle constraint. Such outer-interval overlap is not accessibility. Conservation excludes a handoff to that fixed box; the control tests finite acute identity only. Any later maintenance neighborhood requires its own source-compatible storage admission, without post hoc damping or removal of environmental state.

Removing the transverse dipole gives an exactly invariant zero-winding contact-pulse control. The control is not claimed to have the same energy as the nonzero-dipole source. Its invariant geometry supplies the analytic comparison without a second numerical response or amplitude search.

For the nonzero dipole, the actual centered contact velocities are

\[ (\delta_0,\ldots,\delta_4) =\frac1{60}(-7,7,-1,0,1),\qquad \langle\delta_i^2\rangle=\frac1{180},\qquad \langle\delta_i^3\rangle=0. \]

Therefore h_c^(4)(0)=4u^2/135>0. Time reversal at common initial phase makes form even and phase odd, so locally

\[ h_c(\tau)=h_c(0)+\frac{u^2}{810}\tau^4+O(\tau^6). \]

The first nonzero transfer is toward the collective contact pulse. It does not give immediate transfer into the desired organized receiver. This also does not mean every receiver field is stationary: internal form and phase can exchange while the aggregate remainder decreases. That local result does not decide whether later modulation produces the target identity; the separate finite response below evaluates that question.

Retained full-law control excludes acute winding on its declared horizon

One bounded control used exactly this preparation, unchanged law and support. Its declared original-clock horizon was t in [0,24], time step 3/32, Taylor order 12 and at most 256 steps. The observation asked whether acute receiver winding +1 or -1 holds for an original-clock duration of at least 1/2, with strict positive acute margin; the configured margin threshold is zero.

The declaration and frozen protocol preceded evaluation. Outward dyadic intervals contain the declared exact rational preparation; their centers do not replace it. The retained response completed all 256 steps without an evaluation error, covering the full original-clock interval [0,24]. Every whole-time tube certifies receiver winding zero. The verdict is therefore acute_winding_excluded_on_horizon, not incomplete or unresolved evidence. The source archive retains the evaluated implementation and preparation.

Initial regional storage is 3/400=0.0075; its final enclosure is contained in [0.0061989419859282,0.0061989419859330]. The endpoint full-storage enclosure contains the conserved exact value 14201/4000=3.55025. Thus exceeding the necessary phase-path budget and breaking the unseeded family's reflection symmetry did not produce the target identity in this control. The local fourth derivative did not supply that finite verdict; the whole-time proof did.

The exclusion concerns this source, complete law, support and horizon. It excludes neither later acquisition nor other admitted preparations. There is no response-driven retry, extended horizon, retuning or added loss. Even a positive acute-entry response would still need source-compatible full-state evidence for Section 40's subsequent maintenance window; this negative response supplies no such handoff.

observe_sine_collective_pulse in the shared partition owner rebuilds these actual-state means, feedback and optional flat-phase jet. It retains explicit integer contact-turn offsets and the signed remainder; it does not simulate the declared finite response. The contract and usage example describe its detached evidence and availability.

A structural alternative separates storage release from acquisition

The same law admits a different initial transfer sign without changing its coefficients or waiting longer. Keep all receiver forms equal to a, all leaf forms equal to b, and all receiver phases equal. Write u=a-b>0 and supply contact lifts phi_i=gamma+eta_i, with mean(eta_i)=0. For a symmetric, nonconstant deviation multiset with abs(eta_i)<min(gamma,pi/2-gamma) and 0<gamma<pi/2, S=mean(sin eta_i)=0 and C=mean(cos eta_i)<1. Then

\[ h_c'(0)=\frac43u(C-1)\sin\gamma<0. \]

This is an exact release from collective contact storage into its signed remainder. It is not merely a different scalar bookkeeping observation: the full law also forces an internal receiver response. Put s_i=sin(phi_i) and let indices follow the C5 cycle. Initially,

\[ x_{r_i}'=\frac{s_i}{3},\qquad \theta_{r_i}'=\frac u3,\qquad \theta_{r_i}''=\frac{6s_i-s_{i-1}-s_{i+1}}9. \]

The receiver starts with zero internal form and phase storage. Direct differentiation of its edge sums gives

\[ F_R''(0)=\frac19\sum_i(s_{i+1}-s_i)^2,\qquad V_R^{(4)}(0)=3\sum_i (\theta_{r_{i+1}}''-\theta_{r_i}'')^2. \]

For phase storage all lower derivatives vanish: the initial relative phases and their first rates are zero. Both displayed quantities are strictly positive when s is nonconstant, since 6I-A_C5 has strictly positive eigenvalues and preserves the constant/nonconstant decomposition. Thus internal form grows at order tau^2 and internal phase storage at order tau^4 for a sufficiently short nonzero interval. No acute winding, directed full-cycle passage or finite-width handoff follows from that local onset.

In particular, choose 0<rho<min(gamma,pi/2-gamma) and the cyclic deviations (rho,-rho,-rho,rho,0). Reflection i -> 3-i (mod 5), including the matched private leaves, preserves this source. The complete law preserves that symmetry by the same equivariance and uniqueness argument, so the receiver cannot acquire nonzero winding whenever no receiver edge is antipodal; in particular, acute unit winding is excluded. Nevertheless this source has h_c'(0)<0 and both positive internal derivatives above. Permuting the same multiset to (rho,rho,-rho,-rho,0) removes that particular reflection obstruction while leaving the initial contact resultant, transfer rate and total storage unchanged:

\[ H=\frac52u^2+5-(1+4\cos\rho)\cos\gamma. \]

Removing a symmetry obstruction is not a formation theorem. This exact comparison explains the next admission question: which ordered contact geometry transfers into a directed receiver phase configuration while retaining a source-compatible full-state maintenance budget? A negative collective transfer rate or a growing internal store cannot answer it alone. These conditional preparations are not an additional scheduled response, parameter search or modification of the frozen control.

45. Complete contact feedback distinguishes cancellation from rigidity

The relative-rate identity at the end of Section 44 gives an instantaneous condition. Its persistence depends on both the receiver's internal current and the actual contact response. The same conservative unit C5/private-leaf law supplies the needed derivatives and admits a precise classification of exactly rigid receiver geometry. A finite retained identity need not be rigid, so this classification does not replace the broader acquisition and retention question.

The complete centered rows and their derivatives

Use L=L_R, the five-node cycle Laplacian, and let P remove the receiver mean. In matched receiver/leaf order put

\[ \vartheta=P\theta_R,\qquad q=Px_R,\qquad r=x_R-x_Q, \qquad u=\langle r\rangle,\qquad \phi=\theta_Q-\theta_R, \]

in continuous phase lifts. Let s=S_C5(vartheta) be the incoming internal sine current and f=sin(phi) componentwise. The full normalized rows give

\[ q'=\frac13(s+Pf),\qquad r'=\frac13s+\frac43f,\qquad u'=\frac43\langle f\rangle, \]

\[ \boxed{\vartheta'=\frac13(Lq+Pr),\qquad \phi'=-\frac13(Lq+4r).} \]

All derivatives in this section use tau=t/pi. Differentiating the first phase row and retaining the actual contact current gives

\[ \boxed{\vartheta''=\frac19\bigl[(L+I_5)s+(L+4I_5)Pf\bigr].} \]

Define L_cos(vartheta) as the cycle Laplacian whose edge weights are cos(vartheta_j-vartheta_i). Since s'=-L_cos(vartheta)vartheta', one more derivative is

\[ \boxed{\vartheta'''=\frac19\left[-(L+I_5)L_{\cos}(\vartheta)\vartheta' +(L+4I_5)P\bigl(\cos\phi\mathbin{\odot}\phi'\bigr)\right],} \]

where odot denotes componentwise multiplication. None of these rows assumes that a small regional storage, a zero measured rate or a collective mean determines the environmental coordinates.

Vanishing first and second rates do not imply a fixed geometry

Prepare receiver phases at a uniform unit twist, copy each phase to its private leaf, and choose any nonzero centered form vector q. Supply r=u*1-Lq; receiver and leaf forms can then be reconstructed with any declared common form origin. Initially s=0, phi=0, and

\[ \vartheta'=0,\qquad \vartheta''=0,\qquad \boxed{\vartheta'''=\frac19(L+4I_5)Lq\ne0.} \]

Both factors on the centered subspace are invertible. In particular, for q=kappa*(1,-1,0,0,0), the exact third derivative is

\[ \vartheta'''=\kappa(22/9,-22/9,1,0,-1). \]

The contact phase differences have rates Lq-4u*1/3 at that instant (the absolute leaf phase rates are Lq-u*1). Their subsequent sine currents break the initial compensation and produce the cubic receiver response. This is an initial derivative of the complete field, not a finite-time trajectory estimate. It excludes the inference that even simultaneous zero first and second relative rates prove rigidity.

Exact rigidity has no additional moving compensation family

Suppose the receiver has a fixed relative phase geometry on a nonempty open time interval, so vartheta'=0 throughout that interval. The full finite sine vector field is real analytic, its solutions exist for all real time, and their storage is finite and conserved. The global bound |x_i'|<=1 and the linear phase row permit at most linear form growth and quadratic lifted-phase growth on finite intervals, precluding finite-time blowup in either direction. Consequently the analytic identity vartheta'=0 extends to the entire solution. In particular vartheta, and hence s, are constant.

The first two vanishing phase rows imply

\[ Pr=-Lq,\qquad Pf=-(L+4I_5)^{-1}(L+I_5)s, \qquad q'=(L+4I_5)^{-1}s. \]

Thus q' is constant. But the receiver form storage q^T Lq/2 is bounded above by the conserved full storage H; since L is positive definite on centered vectors, q is bounded for all real time. A nonzero constant q' is impossible. Therefore

\[ s=0,\qquad Pf=0,\qquad q=\text{constant},\qquad \phi_i'=a_i-4u/3,\quad a=Lq,\quad\sum_i a_i=0. \]

Every contact sine equals one common analytic function m(tau). For any pair of contacts the analytic identity

\[ 0=\sin\phi_i-\sin\phi_j =2\sin\frac{\phi_i-\phi_j}{2} \cos\frac{\phi_i+\phi_j}{2} \]

forces at least one factor to vanish identically: if one analytic factor is nonzero somewhere, the other vanishes on an open interval and hence everywhere. The corresponding continuous lifts either differ by a fixed multiple of 2pi, or sum to a fixed odd multiple of pi.

In the supplementary case, summing their phase rates gives a_i+a_j-8u/3=0, forcing u to be constant. Since u'=4m/3, this forces m=0 identically. If m is not identically zero, all contacts must instead agree modulo 2pi. Their equal derivatives imply that all a_i agree, hence a=0 and q=0. Both form blocks are uniform and all contact phases agree: this is precisely the moving compatible family of Section 39, with its actual contact pendulum.

If m=0 identically, each contact phase is a fixed integer multiple of pi. Its zero derivative gives a_i=4u/3 at every node, hence u=0, a=0 and q=0. All forms are equal throughout the full graph, and every phase is stationary. Individual contacts can be aligned or antialigned; these mixed-contact states are equilibria, not another moving compensation mechanism.

Finally, at an acute cycle critical point s=0 forces all oriented edge sines to agree. Sine is injective on the acute interval, so unit winding forces the uniform gaps +2pi/5 or -2pi/5. Thus an exactly rigid acute unit-winding receiver belongs either to the known uniform-block moving family or to a static equilibrium with zero contact torques. This classifies individual rigid trajectories, rather than only families required to close for every supplied collective state.

A retained second-order description exposes the same feedback

The contact row also gives the exact equation

\[ \boxed{\phi''=-\frac19 [(L+4I_5)s+(L+16I_5)\sin\phi].} \]

Together with the displayed equation for vartheta'', this is a closed second-order description of four centered receiver phase coordinates and five contact phases. It retains nine initial velocities, rather than claiming a first-order closure from instantaneous phases alone. If v=vartheta' and c=phi', the original relative forms reconstruct exactly by

\[ Lq=Pc+4v,\qquad Pr=-Pc-v,\qquad u=-\frac34\langle c\rangle. \]

The inverse of L is restricted to the centered subspace. The two conserved degree-weighted form and lifted-phase means recover the common origins: if they are m_x,m_theta, the receiver form mean is m_x+u/4 and its phase mean is m_theta-<phi>/4. The leaf state then follows from x_Q=x_R-r and theta_Q=theta_R+phi.

These eighteen relative coordinates plus the two origins carry the same twenty fine form/phase coordinates. This is a support-specific instance of the exact form/phase representation, not an information loss, a new microscopic law or an additional research branch. It makes explicit where the environmental response re-enters a putatively compensated receiver.

Consequence for the longer retention question

Exact entry from outside either rigid invariant family at a finite time remains excluded by uniqueness. That fact does not rule out acute identity with bounded internal evolution. In the compensated example with u=0, the full storage is V_5+13kappa^2; sufficiently small nonzero kappa gives H<B, where B=5-4cos(3pi/8) is the acute-face barrier. The regional barrier then protects its already prepared acute identity even though the nonzero third derivative proves it is not rigid. This is maintenance of an initially present identity, not formation from zero winding.

The prospective formation-to-retention requirement of a scaled duration at least one is not closed by these derivative or rigidity results. Its permissible targets can evolve internally. They still require actual full-state entry, bounded accumulated relative transport and compatible regional work over that duration. Instantaneous compensation is a useful quantity to inspect, but it is not a substitute for those interval bounds.

In particular, each target edge i->j needs a continuous lift satisfying

\[ \left|\delta_{ij}(0)+\int_0^s[v_j(\tau)-v_i(\tau)]\,d\tau\right| <\frac\pi2\qquad\text{for every }0\le s\le1, \]

with the declared winding sector and full-state uncertainty retained. Cancellation only at the endpoint cannot establish this prefix condition. The generic gap-acceleration bound of four gives a phase-error radius of two at this horizon, already greater than pi/2; that estimate alone cannot certify the required acute interval, even from zero initial relative rate. The actual restoring and contact feedback, or existing validated full-state enclosures, must supply tighter state-dependent control. These are consequences of the selected complete sine law, not a proof that it is the uniquely emerging law of TNFR.