Return-path geometry and collective response

Native return-path equilibrium and causal oscillatory response, with separately declared sine/cubic storage-family geometry and its inverse response.

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

11. Return-path geometry and the formation boundary

The present 11-node, 12-edge graph has cycle rank two: its independent cycles are the two supplied C5 rings. Both edges along the mediator path are graph bridges. At an equilibrium, sum grad(V_phi)=0 over one side of either bridge. Internal sine terms cancel and the only remaining term is the sine of its phase gap. Strict acute admission then forces that gap to be zero. Thus the current equilibrium has inherited ring periods and aligned connecting phases, but no additional independent inter-region circulation period.

This does not exclude a composite NFR or stable binding on this topology. An additional cycle is only one possible geometric discriminator, not a necessary condition for every collective relation. Conversely, repeating the local recovery test cannot manufacture that extra cycle invariant.

A supplied return path changes the cycle space. For example, adding 1--6 to the current graph supplies a five-edge mixed cycle, whereas the older two-direct-bridge geometry has a four-edge mixed cycle. Strictly acute gaps force the latter's integer period to zero; a longer cycle merely removes that elementary length obstruction. It does not prove a nonzero period is compatible with the two existing ring periods or with the critical sine balance. The following admission solves those simultaneous constraints without a trajectory search. Supplying a return path still does not explain the origin or timing of a primitive edge.

11.1 Complete state, support and equilibrium equations

Keep both oriented C5 rings 0->1->2->3->4->0 and 5->6->7->8->9->5, the path 0->10->5, and add the supplied unit edge 1--6. This connected 11-node, 13-edge support has cycle rank three. Use the same relational law, held strictly positive capacities, e,w,beta>0, no forcing or events, and strictly acute gaps. These are constitutive and support premises, not a derived rule for creating the return edge.

The phase row at equilibrium implies Bx=0, hence uniform form. The form row then requires g=0. In the acute domain, every neighbor resultant has positive real part relative to its node; therefore g=0 is equivalent to zero neighbor sine sum. This reuses the algebraic circulation and integral period criterion of support balance, Section 30. Its separately supplied sine phase evolution is not imported into this relational model. Positive capacities affect recovery rates, not these equilibrium equations.

Choose the two ring cycles and the mixed cycle 0->10->5->6->1->0. They form an integer cycle basis: the edges 1->2, 6->7 and 0->10 give an identity submatrix, and removing them leaves a spanning tree. Write their sine-circulation coordinates as a,b,q. The special ring edges 0->1 and 5->6 carry a-q and b+q; the other four edges of each ring carry a and b. The three connecting edges, oriented 0->10, 10->5 and 6->1, carry q. Thus the necessary and sufficient equations are

\[ \begin{aligned} \arcsin(a-q)+4\arcsin a&=2\pi,\\ \arcsin(b+q)+4\arcsin b&=2\pi\sigma,\\ 3\arcsin q+\arcsin(b+q)-\arcsin(a-q)&=2\pi m, \end{aligned} \]

where sigma=+1 or -1, m is an integer, all currents have magnitude strictly below one, and arcsines use their acute principal branch. The possible mixed period must be solved jointly, not assigned by cycle length.

11.2 Existence, uniqueness and the forced zero mixed period

For a positive ring period, write its special angle as t. The period equation forces 0<t<pi/2, bulk angle A=pi/2-t/4, and bulk sine cos(t/4). Define

\[ Q(t)=\cos(t/4)-\sin t,\qquad Q'(t)=-\tfrac14\sin(t/4)-\cos t<0. \]

Its range is (-d,1), where d=1-cos(pi/8)<1/2.

Equal ring periods (1,1). If the other special angle is s, then q=Q(t)=-Q(s), so abs(q)<d. The mixed angle sum 3*arcsin(q)+s-t has magnitude below pi/2+3*arcsin(d)<pi. Consequently m=0. For nonzero q, monotonicity makes s-t have the same sign as q, so their sum cannot vanish. The unique solution is

\[ q=0,\qquad t=s=2\pi/5,\qquad a=b=\sin(2\pi/5). \]

Both original twists survive unchanged; all three connecting gaps vanish.

Opposite ring periods (1,-1). Write the right special angle as -s. Then q=Q(t)=Q(s), hence s=t and b=-a. The mixed sum 3*arcsin(Q(t))-2*t lies strictly between -3*pi/2 and 3*pi/2. Again only m=0 is possible. Therefore the connecting angle is u=2*t/3 and the remaining scalar equation is

\[ \boxed{F(t)=\cos(t/4)-\sin t-\sin(2t/3)=0.} \]

Here F'(t)=-sin(t/4)/4-cos(t)-(2/3)*cos(2t/3)<0 throughout (0,pi/2). Its signs bracket a unique root t_*:

\[ \pi/6<t_*<\pi/4. \]

For an elementary endpoint proof, pi<4, cos(z)>=1-z*z/2 and sin(z)<z give F(pi/6)>71/72-1/2-4/9=1/24>0. At the other endpoint, F(pi/4)<1-1/sqrt(2)-1/2<0. Continuity and strict monotonicity establish existence and uniqueness independently of a numerical solver.

Set u=2*t_*/3 and A=pi/2-t_*/4. A nodal reconstruction, modulo 2*pi, is

\[ \begin{aligned} \theta_L&=(0,t_*,t_*+A,t_*+2A,t_*+3A),\\ \theta_R&=(2u,2u-t_*,2u-t_*-A,2u-t_*-2A,2u-t_*-3A),\\ \theta_{10}&=u. \end{aligned} \]

It has periods (1,-1,0) and nonzero connecting sine circulation q=sin(u)>0. The largest absolute gap is A, so its minimum acute margin is exactly t_*/4, between pi/24 and pi/16. A zero mixed period therefore does not imply zero connecting circulation.

11.3 Recovery, observable distinction and storage cost

For either exact equilibrium, the existing local recovery theorem applies: a sufficiently small joint form/phase perturbation recovers exponentially modulo one common form offset and one common phase rotation. There are twenty stable quotient directions. The statement admits unequal strictly positive held capacities. It supplies neither a quantified basin here nor a capture certificate for attachment from the previous support.

The independent reflection of one ring used in Section 9 is no longer a symmetry: it sends the supplied return edge to an absent edge. Only the identity and whole-ring exchange preserve this graph. In particular, the mediator is its unique degree-two node adjacent to two degree-three nodes; fixing it fixes or swaps both ring ports, leaving no independent reflection. The connecting gap distinguishes the two equilibria: zero versus 2*t_*/3. This is an actual geometric distinction, not merely the loss of a symmetry argument. Neither equilibrium has nonzero pressure or nodal rates. Its sine circulation is an algebraic balance quantity, not dissipative transport or an identified magnetic current.

Connection interpreted as a causally shared pulse remains a dynamical hypothesis. These equilibria provide reference geometries, not persistent oscillations. The existing pulse boundary distinguishes damped collective motion from a nontrivial recurrent full state under strict unforced dissipation. Moreover, freezing the mediator on this new support leaves the direct return edge active: the null of Section 10 cannot be transferred to this graph as a no-influence control.

Their phase storage is

\[ \begin{aligned} V_+&=10(1-\cos(2\pi/5)),\\ V_-&=8(1-\sin(t_*/4))+2(1-\cos t_*)+3(1-\cos(2t_*/3)),\\ V_-&>V_+. \end{aligned} \]

The strict inequality needs no numerical fit: 1-cos(z) is strictly convex on the acute interval. Each ring's five signed gaps have fixed sum +2*pi or -2*pi, so Jensen's inequality bounds its storage below by that of the uniform twist. The opposite solution has nonuniform ring gaps and positive connecting storage. This comparison does not select a winding or imply that the dynamics can change sectors while remaining acute.

It also gives a formation obstruction. The exact opposite-winding equilibrium on the original mediator-only support has uniform form and storage beta*V_+. A zero-supply passive event, including a simultaneous nodal reset, followed by unforced fixed-support relational evolution cannot converge to the new opposite equilibrium with greater storage beta*V_-. Even immediate formation of that target requires excess initial storage or declared supply of at least beta*(V_--V_+); dissipation adds its own nonnegative cost. An available budget is only necessary, not a selector or a sufficient capture condition. Passivity is an additional event premise throughout. For equal windings, inserting the zero-gap return edge is storage-neutral and preserves the old equilibrium, but no occurrence time follows. Simply adding the edge to the old opposite equilibrium fails the acute and positive-resultant options: its gap is -4*pi/5, and each new port resultant has relative real part 2*cos(2*pi/5)+cos(4*pi/5)=(sqrt(5)-3)/4<0. The nonzero imaginary parts instead permit the full regular law and its explicit regular-domain executor. This removes that particular chart restriction, not the preceding passive-storage obstruction to the final target.

11.4 Static computational contract

The return-geometry instrument reuses the engine's exact cycle reconstruction, one-chord topology extension, rational interval trigonometry and detached relational field. It encloses the scalar root in turns, with certified opposite endpoint signs, and constructs a rational midpoint witness for represented native checks. Exact periods and strict acute admission hold for that witness; its sine balance remains explicitly unresolved by the rational reconstruction owner. The exact equilibrium belongs to the analytic root above. Small binary64 rates at the midpoint are reported residuals, not a zero-rate proof. With forty bisections, the exact turn bracket is [2619154484885/26388279066624, 436525747481/4398046511104]. Its midpoint gives approximately t=0.6236341875541443 radians, connecting gap 0.4157561250360962 and acute margin 0.1559085468885361. The ideal storage-excess interval is contained in [0.47999183896,0.47999183898]; these are structural units with beta=1, not laboratory energy measurements. The instrument also reports native midpoint residuals without subtracting represented drift. Run it with python -m benchmarks.relational_return_geometry; the default preparation fixes unit capacities, e=w=1/2, beta=1, ascending node order and no randomness. The analytic equilibrium and storage comparison retain the wider parameter hypotheses above. The static controls also reject simply copying the old opposite twists onto the added edge. No trajectory, automatic connection, new pressure law or physical bridge is introduced by this instrument.

11.5 Coupled-cycle geometry changes with admitted phase storage

This section changes the complete law explicitly. Sections 11.1-11.4 above retain their native pressure, positive loss and original evidence. Here reuse only their supplied eleven-node support, cycle basis and geometric reconstruction. Choose the sine/cubic storage family, unit capacities, e=0,w=beta=1, no inputs or events, and tau=t/pi:

\[ x'=K S_\epsilon(\theta),\qquad \theta'=KLx,\qquad K=\operatorname{diag}(1/d_i),\qquad S_{\epsilon,i}=\sum_{j\sim i}j_\epsilon(\theta_j-\theta_i), \]

\[ j_\epsilon(\delta)=\sin\delta+\epsilon\sin^3\delta, \qquad U_\epsilon'=j_\epsilon,\quad U_\epsilon(0)=0,\qquad E_\epsilon=\tfrac12x^{\mathsf T}Lx+\sum_e U_\epsilon(\delta_e), \qquad E_\epsilon'=0. \]

Each finite epsilon>=0 specifies one law with the same clock, held support and capacity. The prescribed sector is (1,-1,0) in the two oriented ring cycles and the mixed cycle 0->10->5->6->1->0. Changing storage does not select these periods or create the return edge.

The scalar equation describes every acute critical state in this sector

At any equilibrium, the positive phase mobility implies uniform form and the form row then requires zero current divergence. On the acute interval,

\[ j_\epsilon'(\delta)=\cos\delta(1+3\epsilon\sin^2\delta)>0. \]

Consecutive bulk edges along each degree-two path therefore have equal gaps. Write the left special gap as v and the right one as -s. Their ring periods and strict acuity force v,s in (0,pi/2) and bulk magnitudes A(v)=pi/2-v/4, A(s)=pi/2-s/4. The connecting current is both Q_epsilon(v) and Q_epsilon(s), where

\[ Q_\epsilon(v)=j_\epsilon(A(v))-j_\epsilon(v),\qquad Q_\epsilon'(v)=-\tfrac14j_\epsilon'(A(v))-j_\epsilon'(v)<0. \]

Hence v=s. All three oriented connecting edges have the same current, so strict monotonicity gives a common gap u. The zero mixed period forces 3u-2v=0. Thus every acute critical geometry in this sector must satisfy

\[ \boxed{F_\epsilon(v):= j_\epsilon(\pi/2-v/4)-j_\epsilon(v)-j_\epsilon(2v/3)=0.} \]

This is not a restriction to a guessed reflection-symmetric ansatz: the equal gaps follow from full nodal balance, the supplied periods and acute current injectivity. Conversely the reconstruction below gives a complete critical state from any root.

Existence, uniqueness and constitutive dependence for every finite coefficient

Put a=cos(v/4), b=sin(v) and c=sin(2v/3), so F_epsilon=F_0+epsilon*F_3, where F_0=a-b-c and F_3=a^3-b^3-c^3. The scalar derivative is strictly negative on 0<v<pi/2:

\[ \partial_vF_\epsilon= -\tfrac14j_\epsilon'(\pi/2-v/4) -j_\epsilon'(v)-\tfrac23j_\epsilon'(2v/3)<0. \]

At v=pi/6, the elementary bounds used above give a>71/72 and b+c<17/18, hence F_0>1/24 and F_3>0. At v=2*pi/5, the bulk and special angles coincide, so F_epsilon=-j_epsilon(4*pi/15)<0. Continuity and strict monotonicity therefore prove exactly one root v_epsilon, with the uniform bracket

\[ \boxed{\pi/6<v_\epsilon<2\pi/5\qquad(\epsilon\ge0).} \]

At zero coefficient this is the retained root t_* from Section 11.2, whose sharper bracket pi/6<t_*<pi/4 remains valid. That sharper upper endpoint is not a bound asserted for the whole storage family.

The root varies smoothly, and strictly increases, for all finite epsilon>=0. At a root, F_3 must be positive: otherwise a< b+c follows from a^3<=b^3+c^3 and b,c>0, making F_0+epsilon*F_3<0. Implicit differentiation now gives

\[ \boxed{\frac{dv_\epsilon}{d\epsilon} =\frac{F_3(v_\epsilon)}{ \tfrac14j_\epsilon'(A(v_\epsilon))+ j_\epsilon'(v_\epsilon)+\tfrac23j_\epsilon'(2v_\epsilon/3)}>0.} \]

This distinguishes actual critical geometries, rather than only changing a response around an unchanged pattern. Their bulk gaps decrease while the special and connecting gaps increase. In particular, for epsilon>0, F_0(v_epsilon)=-epsilon*F_3(v_epsilon)<0: the new geometry is not an equilibrium of the unchanged sine or native law.

For interval arithmetic, the same root equation can be evaluated as (1-lambda)*F_0+lambda*F_3=0, where lambda=epsilon/(1+epsilon). This is multiplication of the stationary equation by a positive number. It does not rescale one dynamical row, change the structural clock, or install an additional parameter or limiting constitutive law.

Reconstruction, identity and local protection

Set v=v_epsilon, u=2v/3 and A=pi/2-v/4. The same nodal reconstruction as before gives

\[ \theta_L=(0,v,v+A,v+2A,v+3A),\qquad \theta_R=(2u,2u-v,2u-v-A,2u-v-2A,2u-v-3A),\qquad \theta_{10}=u, \]

with phases taken modulo 2*pi and uniform form. The oriented cycle periods are exactly (1,-1,0). Its bulk current is a_epsilon=j_epsilon(A), right bulk current is -a_epsilon, and all three connecting currents are q_epsilon=j_epsilon(u)>0. The equation a_epsilon=j_epsilon(v)+q_epsilon verifies every port balance; internal path nodes cancel pairwise. These are balanced stationary edge currents, not a claim of a persistent oscillation.

Because v<2*pi/5, the largest gap is A>v>u. Its acute margin is pi/2-A=v/4>pi/24, uniformly over the family. The full phase Hessian Z diag(j_epsilon'(delta_e)) Z^T is therefore positive definite modulo common rotation. Together with the form Hessian L, it gives a strict local minimum of the conserved storage on the relative-state quotient. Its target value is

\[ E_\epsilon^*=8U_\epsilon(A)+2U_\epsilon(v)+3U_\epsilon(u). \]

The local conserved-storage argument then supplies nonlinear Lyapunov protection: for each fixed coefficient, sufficiently small joint perturbations remain near this geometry, after retaining the degree-weighted form mean and a consistent local lifted phase mean. The neighborhood and excess storage must use this law and this target. No numerical basin or acceptance of a captured preparation follows merely from a positive Hessian. This is local protection under zero loss, not the native positive-loss attraction result in Section 11.3 and not formation from an initial state outside the sector.

The cycle-geometry owner provides assess_return_path_storage_geometry and its ReturnPathStorageGeometryAssessment with a PhaseRootBracket. It retains the supplied support and coefficient while enclosing the scalar root and reconstructing target geometry. A rational midpoint is an approximate target with a residual, not an exactly critical state; source association does not mean the captured network occupies this target or follows the alternative law. Exact periods, root enclosures and nonlinear evolution remain different kinds of evidence.

The transferable result is a persistent winding identity with a uniquely deforming acute critical geometry under this explicit storage freedom. Neither topology alone nor the common consensus tangent fixes that geometry. The selected support, winding preparation and constitutive family remain premises; no event, universal phase potential or physical identity is derived.

11.6 A geometric interval constrains a law and a separate nodal response

Retain exactly the support, periods, unit capacities, conservative j_epsilon, w=beta=1 and tau=t/pi of Section 11.5. The new input is an independently supplied closed interval for the special edge angle v, including its origin and uncertainty. It is a constraint on an equilibrium of this supplied family, not an assertion that an arbitrary captured state is at equilibrium. No response, pressure reconstructed from a response, or root midpoint is used to choose the coefficient.

The inverse exists on a proper, half-open geometric branch

Write v_0 for the unique zero of F_0 and v_infinity for the unique zero of F_3 in (pi/6,2*pi/5). The latter root exists because F_3 is positive at pi/6, negative at 2*pi/5, and strictly decreasing:

\[ F_3'(v)=-\tfrac34\cos^2(v/4)\sin(v/4) -3\sin^2(v)\cos(v) -2\sin^2(2v/3)\cos(2v/3)<0. \]

At v_0, the identity a=b+c gives F_3(v_0)=3bc(b+c)>0, so v_0<v_infinity. At v_infinity, a^3=b^3+c^3 with positive b,c implies a<b+c and hence F_0(v_infinity)<0. It follows that

\[ \boxed{\epsilon(v)=-\frac{F_0(v)}{F_3(v)}, \qquad v\in[v_0,v_\infty)} \]

is nonnegative, continuous, zero only at v_0, and unbounded as v approaches v_infinity from below. On that interval,

\[ \frac{d\epsilon}{dv} =-\frac{F_0'(v)+\epsilon(v)F_3'(v)}{F_3(v)}>0. \]

Thus it is a bijection from [v_0,v_infinity) to all finite epsilon>=0. In particular, the limiting angle is not an equilibrium of any finite member of the family. The equation F_3=0 describes a geometric limit; it does not silently install an infinite coefficient or a rescaled pure-cubic time-evolution law.

For an exact observation interval I=[v_lo,v_hi], intersect it with this half-open branch before inferring any coefficient. The compatible set is empty if v_hi<v_0 or v_lo>=v_infinity. Otherwise its lower endpoint is zero when v_lo<=v_0, and epsilon(v_lo) when v_lo>v_0. Its upper endpoint is epsilon(v_hi) if v_hi<v_infinity; if v_hi>=v_infinity, the set is unbounded above. Equality at v_0 is admissible, whereas equality at v_infinity is excluded for every finite law. These distinctions cannot be replaced by clipping the coefficient to a convenient numerical search range.

In turn coordinates s=v/(2*pi), the same statements use the prior enclosure (1/12,1/5) and the two roots in that interval. A numerical reader can certify endpoint signs with outward intervals. A sign interval containing zero supplies neither a strict comparison nor an exact root equality; a finite arithmetic budget may therefore leave classification unresolved. This is an arithmetic limitation, distinct from an incompatible geometric observation.

Finite geometric accuracy does not imply uniform coefficient accuracy

Let

\[ C_\infty=\frac{F_0(v_\infty)}{F_3'(v_\infty)}>0. \]

Taylor expansion at the simple zero of F_3 gives

\[ \epsilon(v)=\frac{C_\infty}{v_\infty-v}+O(1),\qquad v_\infty-v_\epsilon=\frac{C_\infty}{\epsilon} +O(\epsilon^{-2}), \]

\[ \frac{d\epsilon}{dv}(v_\epsilon) \sim\frac{\epsilon^2}{C_\infty}. \]

The geometry saturates while the unscaled law becomes stronger. A fixed angle error therefore becomes increasingly uninformative about the coefficient near the limiting root. If the geometric interval reaches that limit, the absence of a finite upper coefficient bound is intrinsic to the observation, even with exact arithmetic. This does not invalidate the forward equilibrium or its local stability for any fixed finite coefficient.

The full nodal response probes curvature rather than only current balance

Orient the thirteen edges consistently with the shared geometry owner, and let Z be the incidence matrix with -1 at the tail and +1 at the head. At the exact equilibrium reconstructed from v, define

\[ H_\epsilon(v)=Z\operatorname{diag} \left[\cos\delta_e(v) \left(1+3\epsilon\sin^2\delta_e(v)\right)\right]Z^{\mathsf T}. \]

Here S_epsilon=-Z j_epsilon(Z^T theta) and its phase derivative is -H_epsilon; fixed integer lift offsets do not change the derivative. Every edge curvature is positive on the admitted acute branch. With y=delta x and eta=delta theta, the complete twenty-two-coordinate linearization in the declared clock is

\[ \begin{pmatrix}y'\\\eta'\end{pmatrix} =\begin{pmatrix}0&-K H_\epsilon\\KL&0\end{pmatrix} \begin{pmatrix}y\\\eta\end{pmatrix}. \]

For a prescribed form preparation y(0)=h, eta(0)=0, it follows that

\[ y'(0)=0,\qquad \eta'(0)=KLh,\qquad \boxed{y''(0)=-K H_\epsilon(v) K Lh.} \]

The first zero form rate does not imply equilibrium: phase initially moves and subsequently changes the form-driving current. The new observation is this full nodal acceleration, in units of form per tau squared. The geometric constraint uses j_epsilon; the reserved response tests j_epsilon' together with the complete support and mobility.

For the specified mediator preparation h=e_10, set

\[ c_A=j_\epsilon'(A),\quad c_v=j_\epsilon'(v),\quad c_u=j_\epsilon'(u),\qquad B=\frac{c_A+c_v+4c_u}{9}. \]

The actual degrees give (KLh)_10=1, (KLh)_0=(KLh)_5=-1/3, with all other coordinates zero. Therefore, in node order 0,...,10,

\[ \boxed{y''(0)= (B,-c_v/9,0,0,-c_A/6, B,-c_v/9,0,0,-c_A/6,-4c_u/3).} \]

This expression contains all three geometric curvature classes. Its degree-weighted sum is exactly zero, as required by conservation of sum_i d_i*x_i. No projection of the mediator preparation onto a chosen mode is required. For the full nonlinear preparation x(0)=x_*+alpha*h, theta(0)=theta_*, its initial second form derivative is exactly alpha times the displayed expression. That local derivative identity still gives no certified finite-time or finite-amplitude trajectory.

Outward bounds retain the same implicit geometry and law

Suppose the geometric observation implies a finite coefficient enclosure [epsilon_lo,epsilon_hi]. For every compatible angle retain the exact relations A=pi/2-v/4, u=2v/3 and the nodal affine reconstruction from Section 11.5. Enclose the three edge curvatures by outward trigonometric and arithmetic evaluation over this set, and propagate them through the fixed matrix product -K H K Lh. This yields a componentwise enclosure of the complete acceleration for every compatible equilibrium and coefficient. Separate angle and coefficient intervals can widen the answer by discarding correlations for interval evaluation, but they cannot create additional admitted equilibria. In particular, choosing unrelated angles from different edge boxes is not a new geometry.

Endpoint evaluation of the curvatures alone is insufficient without a monotonicity proof for those complete expressions. The inverse coefficient map is monotone; this does not imply monotonicity of every product of cos(delta), sin(delta)^2 and the changing coefficient. Strict endpoint sign bounds also remain necessary before dividing by an enclosure of F_3.

An unbounded compatible coefficient set does not generally permit a finite response bound. For this mediator preparation, c_u=cos(2v/3)*(1+3*epsilon*sin(2v/3)^2) tends to positive infinity as v approaches v_infinity. The mediator acceleration consequently tends to negative infinity. A finite enclosure cannot be manufactured from the geometric limit. Special preparations such as constant form may still have an identically zero response and must be distinguished from this case.

The shared phase-cycle owner exposes assess_return_path_geometry_response and ReturnPathGeometryResponseAssessment. It re-admits the source premises, two ordered cycles, mediator, supplied special_turn_bounds, form_direction and observation_origin. Its coefficient status separates bounded, unbounded_above, incompatible and unresolved_interval_arithmetic; a missing finite bound is not a point fit. Only a bounded compatible input supplies the reader's finite full response enclosure. The supplied observation-origin label is provenance metadata, not independent authentication or physical validation. The captured source association does not certify that its stored phases occupy the inferred equilibrium.

Geometry does not determine the clock or the missing dynamical scales

Multiplying every phase current by a positive constant leaves the zero divergence condition and equilibrium geometry unchanged. More generally, the supplied laws x'=a*K*S_epsilon, theta'=b*K*Lx, with independent positive a,b, have the same equilibria and replace the predicted acceleration by -a*b*K*H_epsilon*K*Lh. Positive held capacities also affect K without changing this zero-current equilibrium condition. Changing from tau to r*tau multiplies accelerations by r^-2. The angle observation cannot recover these freedoms.

Consequently the result is conditional on the independently fixed clock, capacities, current normalization and phase mobility. A frozen mathematical control can infer the coefficient interval from geometry, then compare a separately evaluated full nodal response without changing that interval. Success checks joint consistency and the implementation of this declared family. It does not select the family among arbitrary potentials, infer microscopic formation, or validate a physical correspondence.

Frozen known-source control

The declaration fixes coefficient 1 for a mathematical source, a 10^-6 turn observation grid, the mediator form direction, all eleven response coordinates and 95-decimal independent arithmetic. The source coefficient is excluded from the predictor's inputs. The independently quantized special angle is s in [0.137625,0.137626]; strict outward signs of the source root equation certify that this interval contains its exact equilibrium angle.

Separate producer invocations retained the source protocol, prediction and response. The exact predicted coefficient bounds are contained in the conservative decimal display [0.99999715,1.00005518]. Thus geometry excludes the sine baseline within this family. All eleven independently differentiated nodal responses lie in their saved intervals. For example the mediator acceleration is approximately -2.1142183971, inside the predicted interval, whose outward decimal display is [-2.11428951,-2.11420582] in form per tau squared. The coefficient was not refitted after evaluating that response.

The producer reuses the existing write-once evidence transport and retains a source archive. It re-admits the frozen geometric premises before consuming the prediction; hashes alone do not authenticate its mathematical content or chronology. The independent high-precision response is a finite cross-check, not an interval-certified oracle or physical measurement. The enclosure proof above and the finite implementation check have different evidential roles. The bounded geometry-to-response gate is complete; coefficient sweeps or finite-amplitude forecasts are not required by this result.

12. A causally shared oscillatory response, with a dissipation limit

Question and scope. Can the admitted form/phase dynamics itself generate an oscillatory response shared by the two regions? Here a local collective oscillation means a nonreal mode of the native equilibrium derivative that contributes to a donor-to-receiver response. Matching frequencies or an eigenvector drawn across both rings is insufficient. A maintained pulse would add a separate persistence obligation. Neither meaning is a definition of every possible NFR or a physical identification.

Fix the native opposite-winding equilibrium of Sections 11.1-11.3, unit capacities, e=w=1/2, beta=1, its existing structural clock and no inputs or events. The graph, constitutive law and preparation remain explicit premises. An initial donor perturbation probes the response; it is not a periodic forcing or an autonomous explanation of how that perturbation arose.

12.1 The oscillatory mechanism uses the existing two rows

Let B be the unweighted Laplacian, D the degree diagonal, and K the Laplacian weighted by the equilibrium edge cosines. At this equilibrium H_i=pi*sum_j cos(delta_ij)>0. With u=delta x, v=delta theta, the existing recovery derivative is

\[ \frac{d}{dt}\begin{pmatrix}u\\v\end{pmatrix} =J\begin{pmatrix}u\\v\end{pmatrix},\qquad J=\begin{pmatrix} -eD^{-1}B&-wH^{-1}K\\ (w/\beta)H^{-1}B&0 \end{pmatrix}. \]

Phase deformation changes pressure and hence form; form contrast drives phase motion back. This is a derived feedback within the supplied nodal law, not an additional oscillator equation or a prescribed frequency. Its restoring and damping rates depend on geometry through B,K,H,D and on the declared model coefficients and clock. The same coefficients can be overdamped on P2; oscillation is not guaranteed merely by naming a phase coordinate.

The shared engine now provides evaluate_relational_uniform_tangent and Network.relational_uniform_tangent. It admits exactly uniform represented form even away from phase equilibrium, retaining the native field and the general Arg derivative. At an ideal equilibrium that derivative reduces to -H^-1 K. The observer does not declare equilibrium from a small residual, repair the numerical common-offset defect, or change the time-evolution law. Its contract separates materialized coefficients from exact derivatives and spectral proof.

12.2 A nonzero transfer is stronger than a shared eigenvector

Let a be a declared initial perturbation and O_R an observation of the receiving ring. The Laplace transform of its tangent response is

\[ T_R(s)=O_R(sI-J)^{-1}a. \]

For a simple eigenvalue, a right vector v_k and a left row l_k normalized by l_k v_k=1 give residue O_R v_k (l_k a). Both preparation and observation must participate. This expression is invariant to eigenvector rescaling; using a Hermitian Laplacian projector for this generally nonnormal J would give a different calculation. Degenerate independent oscillators permit extended eigenvectors while their cross-region transfer is exactly zero.

There is also an exact causal result for the ideal return graph. Order the two coordinates by node. Every off-diagonal edge block from node j to i is

\[ J_{ij}=\begin{pmatrix} e/d_i&w\cos\delta_{ij}/H_i\\ -w/(\beta H_i)&0 \end{pmatrix},\qquad \det J_{ij}=\frac{w^2\cos\delta_{ij}}{\beta H_i^2}>0. \]

Suppose a right eigenvector vanishes on the entire receiving ring. Its equations at nodes 5 and 6 force respectively node 10 and node 1 to vanish; the equation at 10 then forces node 0. The equations at 0, 1 and 2 force nodes 4, 2 and 3 in turn. No nonzero eigenvector can be invisible on that ring. The transpose argument starting with a left eigenvector zero on the donor forces nodes 10, 6, 5, 9, 7 and 8 to vanish. Thus complete donor form/phase preparation and complete receiver observation meet the eigenvector controllability/observability criteria: no mode is absent from the full donor-to-receiver transfer. This does not assert that every single-node preparation or scalar observation detects every mode, nor that the mediator moves in every mode.

For the numerical comparison, fix a=e_x0 before evaluation. The primary receiver outputs are x_5-mean_R(x) and v_5-mean_R(v). Mediator outputs are its two rates, avoiding a spurious apparent motion from subtracting a changing common reference. No phase angle derived from a mode is identified with primitive nodal phase.

The causal null sets cross-region blocks of the full tangent to zero, retaining its three diagonal blocks for donor, receiver and mediator. This is an explicit linear ablation with held local coefficients, not a newly admitted disconnected nonlinear geometry. Its donor-only subspace is invariant, so receiver response is identically zero for every time and every resolvent point outside the spectrum, even if eigenvalues are degenerate. Its offsets need not have the native quotient symmetry; use the complete 22-coordinate null. Merely freezing node 10 leaves edge 1--6 active and cannot serve as that null.

12.3 Certified existence of an ideal damped complex mode

A scalar spectral moment proves presence without certifying individual numerical eigenvalues. Write L=D^-1 B and M=H^-1. Block multiplication at the exact equilibrium gives

\[ \operatorname{tr}J^3=-e^3\operatorname{tr}L^3 +\frac{3ew^2}{\beta}\operatorname{tr}(LMKMB). \]

The graph is triangle-free. Its degrees give sum_edges 1/(d_i*d_j)=7/3, so tr(L^3)=11+6*(7/3)=25. Set r=cos(t_*), z=sin(t_*/4), b=cos(2*t_*/3) and s=r+z+b. The cosine strengths s_i=H_i/pi are s on four ports, 2*z on the six internal ring nodes and 2*b on the mediator. Expanding the remaining trace into diagonal and two-edge walks gives

\[ \begin{aligned} T:=\pi^2\operatorname{tr}(LMKMB) &=\sum_i\frac{d_i}{s_i} +\sum_{\{i,j\}\in E}\left(\frac1{d_i s_j}+\frac1{d_j s_i} +\frac{4\cos\delta_{ij}}{s_i s_j}\right)\\ &=\frac{29}{s}+\frac{38}{3z}+\frac4{3b} +\frac{4(2r+b)}{s^2},\\ \operatorname{tr}J^3&=-\frac{25}{8}+\frac{3T}{8\pi^2}. \end{aligned} \]

The exact root bracket of Section 11 and shared outward rational trigonometry enclose this ideal trace strictly inside [0.72428869449,0.72428869451]. No numerical eigensolver or trajectory enters that sign certificate. The two common-offset eigenvalues are zero; all twenty quotient eigenvalues have negative real part by local recovery. If they were all real, the sum of their cubes would be negative. The certified positive trace contradicts that possibility. Therefore at least one damped complex-conjugate pair exists for this exact geometry and model. Section 12.2 further proves that some donor preparation and receiver observation retain such a pair causally.

This theorem does not count all complex pairs, certify individual frequencies, or prove that this same pair moves the mediator. Those more specific statements retain their separate numerical evidence below. The certificate is conditional on the stated geometry and coefficients, not a universal pulse law for TNFR.

12.4 Persistence is a separate claim

All nonneutral modes of the ideal acute equilibrium have negative real part by the existing recovery theorem. More strongly, the storage recurrence argument excludes nonstationary full-state periodic or relative-periodic motion in the specified regular chamber under strictly positive dissipation and capacity. These statements do not exclude every recurrent lossy diagnostic or all other TNFR completions. They do exclude using this model as evidence of a permanent unforced full-state pulse.

A mode with lambda=-alpha+i*omega contributes a damped sinusoid to the infinitesimal response. Its period 2*pi/abs(omega) and amplitude decay time 1/alpha have units of the declared structural clock. Neither is a calibrated physical time. A sum of such modes need not visibly complete a cycle; a nonzero complex residue alone is not a nonlinear limit cycle or phase-locking theorem. For a fixed finite horizon, smooth dependence supplies the usual first-order response epsilon*exp(J*t)*a about the exact equilibrium, with a local higher-order remainder; no numerical remainder bound is certified here.

12.5 Retained static evidence and numerical boundaries

The collective-pulse instrument uses the shared uniform-form tangent and the same forty-bisection rational midpoint as Section 11. Its static record retains the complete state, matrices, declared observations, initial direction, all poles and residues, controls, root enclosure and ideal trace certificate. There is no finite-amplitude evolution or fitted frequency in this calculation.

Common form and phase offsets are removed by the exact difference map D_0 z=(x_i-x_10, v_i-v_10), i=0,...,9, with lift L_0 setting the mediator coordinates to zero. Here D_0 is an observation map, distinct from the degree diagonal D above. D_0 L_0=I; the ideal law induces J_q=D_0 J L_0. The materialized generator's nonzero row sums and the defect D_0 J-J_q D_0 are retained, not silently corrected. Their maximum magnitude was 4.17e-17. This numerical quotient is not asserted to close the represented generator exactly.

The numerical quotient has six conjugate pairs and eight real poles. The positive-imaginary representatives are listed below; the report retains their conjugates and the real modes as well. "Unresolved" means the mediator-rate residue magnitude is below the stated numerical resolution, not a missing coordinate or a proof of zero motion.

Real part	Positive imaginary part	Period	Decay time	Mediator-rate response
-0.438279	0.530782	11.8376	2.28165	Unresolved
-0.438120	0.534008	11.7661	2.28248	Resolved
-0.265023	0.269281	23.3332	3.77326	Resolved
-0.242613	0.272205	23.0825	4.12179	Unresolved
-0.134737	0.050636	124.0860	7.42187	Resolved
-0.045952	0.058180	107.9960	21.76184	Unresolved

All six have resolved primary receiver residues for the declared donor form direction. For example, the -0.265023+0.269281i component has receiver form/phase residue magnitudes approximately (0.0438520,0.0273921) and mediator-rate magnitudes (0.0195353,0.00876979). Its amplitude falls to exp(-1) in only 0.161712 cycles; after one period its modal envelope is approximately 0.002063 of its initial value. The existence of an oscillatory component therefore should not be presented as a long train of shared pulses.

Numerical pole and residue zero policies were 5.71e-14 and 6.93e-13, respectively, based on binary64 precision, matrix/observation scale and eigenbasis conditioning. They are resolution policies, not physical thresholds. The eigenbasis condition number was 24.35, the smallest distinct-pole separation 0.00323028, and the maximum eigenvector residual below 3.44e-15. Reassembling the full transfer at the declared s=1+2i differed from direct solution by 1.06e-16; full and quotient resolvents differed by 1.56e-17. These are consistency checks, not rigorous error bounds on individual ideal poles or residues. The certified trace sign remains separate evidence.

The all-route block-diagonal control has identically zero receiver transfer. Freezing only the mediator does not: the receiver's second derivative per unit donor direction is approximately (-0.00424688,0.00281912) through the remaining return route. This is why apparent shared timing and intervention on just one route cannot settle connection.

Execution used Python 3.13.6, NumPy 2.5.3, NetworkX 3.6.1, Windows AMD64, binary64 native coefficients and eigensystems, exact rational observation maps/defects, the shared interval kernel, and no randomness. Listed owner fingerprints are a partial source manifest, not a complete dependency archive or provenance authentication. The stored matrices permit independent checks without regenerating the producer. Its tests also use an analytic oscillator transfer and disconnected degenerate oscillators as independent controls; those test matrices are not added to TNFR dynamics.

This closes local pulse admission: oscillatory causal participation follows under the specified law and geometry, while the individual residues and strong damping have the stated numerical scope. A visibly oscillating finite nonlinear response and a sustained pulse remain distinct further claims.