Derived form phase and inherited closure

Directed diffusion, sufficient amplitude/phase observations, wave-coordinate scope and regular continuation.

Section numbers are stable locators across this document family. The parameter reference owns the reading map; the execution plan alone owns active tasks. Each result retains its stated model and scope.

21. Derived form phase, wave coordinates and genuine continuation

A phase observed in form, independently stored primitive phase, and a phase law chosen to realize a desired motion are different objects. The following reuse of transport, memory and source geometry separates them. No result below establishes sustained identity or physical particle emergence.

Variable contract for the derived form family

This contract instantiates the parameter properties for the directed-triangle family. It does not redefine the engine's primitive variables. Use form unit X, declared clock unit T, and the orthonormal regional frames displayed below.

Quantity	Type, domain and units	Status and necessary boundary
Fine EPI x_i	Signed real form coordinate, units X	Evolving fine state; zero is a coordinate value, not deletion of a node or absence of all structure
Fine capacity nu_i	Positive real, units T^-1 in this family	Supplied and held, with the stated regional-constancy conditions for Gram closure; no new capacity law is derived
Fine pressure p_i=P_i(x)	Real evaluated response, units X	Pure-EPI -L_rw*x on full support; evaluated before the predicted rate, never reconstructed from a measured derivative
Primitive phase theta_i	Circle coordinate, radians	Held at zero in the controls and unused by this law; it is not the rotating form orientation below
Support, weights and t	Declared oriented graph, nonnegative conductance, common ordered clock	Held premises; row normalization uses the full graph. Absolute conductance scale cancels from this pressure; neither laboratory time nor metric distance is inferred
Mean mu_a and contrast z_a=u_a+i*v_a	Real mean and two real contrast coordinates, units X	Derived from three fine values in a fixed frame; together they reconstruct the region. Complex notation adds no complex fine EPI
Amplitude r_a and orientation psi_a	r_a=abs(z_a)>=0, units X; circular psi_a=arg(z_a) where r_a>0	Derived, frame dependent; psi_a is unavailable at zero amplitude while Cartesian evolution remains defined. Amplitude alone is not complete form
Joint Gram H=zz^dagger	Hermitian positive semidefinite, rank at most one, units X^2	Derived constrained observation; discards common contrast orientation. Its closure depends on the interface and capacity domain
Cross relation K_AB=z_A*z_B^dagger	Complex matrix, units X^2, K_BA=K_AB^dagger	A block of the same joint Gram, not independent freely selectable state or an extra force; joint and zero-contrast constraints are given below

Any pressure/capacity factorization of an effective rate requires its own normalization and inheritance evidence. Neither psi_dot nor the angular mobility from a polar metric can silently replace primitive nu_f. The phase distinctions separate primitive phase, form orientation and a reporting-frame angle. The regional partition and its frames are supplied; this construction does not derive their autonomous selection. Physical identification of these collective quantities remains open and need not map each primitive to a sensor.

Notation is local to each owner: this H is a Gram observation, not the weighted transport metric also called H elsewhere. This K_AB is a cross relation, not tetrad curvature K_phi, a path-potential kernel or an eliminated-state memory kernel. Equal symbols do not identify these objects.

Shared engine observation of the form coordinates

The form geometry owner now exposes these coordinates and their instantaneous nodal pushforward through observe_regional_form(graph, regions) and the SDK's Network.regional_form(regions). The ordered triples are supplied reporting frames. Observation itself requires no directed-cycle topology, common capacity or pure-EPI pressure; those are additional premises of the autonomous reductions below.

For each triple, it retains the rational raw contrasts a=x_0-x_1, b=x_0+x_1-2*x_2, representing z=a/sqrt(2)+i*b/sqrt(6), and the mean. Squared amplitude is exactly a^2/2+b^2/6 for the materialized fine inputs. The full Gram retains Re(Q_ab)=a_a*a_b/2+b_a*b_b/6 and sqrt(12)*Im(Q_ab)=b_a*a_b-a_a*b_b. The corresponding derivatives project the actual shared rounded nodal product with its retained arithmetic defect. No derivative is inferred from an assigned phase or a pressure fitted to a later response. Numerical polar estimates remain separate from this exact algebra; zero contrast makes angle and polar rates unavailable without erasing the Cartesian or Gram rate.

This is a read-only integration of the derived observation, not a new constitutive pressure or replacement primitive phase law. Full-state closure, future evolution, source provenance and hypothesis admission remain separate. The production controls bind the observation to the shared nodal integrator and its finite Euler arithmetic. The SDK guide owns usage; the pressure admission result explains why the observed angle cannot automatically become a global source.

21.1 Directed form rotation and exact eliminated-state memory

Take outgoing unit support i -> i+1 (mod 3), common fixed capacity nu>0, and pure-EPI pressure. For the existing orthonormal basis U=[(1,-1,0)/sqrt(2), (1,1,-2)/sqrt(6)], write the two real form coordinates as z=u+i*v. The outgoing Laplacian and exact Cayley owner give

\[ \dot z=\nu\left(-\frac32-i\frac{\sqrt3}{2}\right)z,\qquad \dot{\arg z}=-\frac{\sqrt3\nu}{2},\qquad \frac{d}{dt}|z|^2=-3\nu|z|^2. \]

The angle requires z!=0; the Cartesian law is defined everywhere. Reversing the edges reverses the rotation; reciprocal symmetrization removes it while preserving contrast decay. Directed support supplies orientation, so the form-only reflection obstruction in section 17 is not contradicted. No frequency is fitted, but the origin of that support remains a premise. The radius contracts by exp(-2*pi*sqrt(3)) per turn, independently of capacity. This is transient rotation, not a maintained pattern.

Put a=3*nu/2, b=sqrt(3)*nu/2. Eliminating the second coordinate gives

\[ \dot u=-au+be^{-at}v(0)-b^2\int_0^t e^{-a(t-s)}u(s)\,ds, \qquad \ddot u+3\nu\dot u+3\nu^2u=0. \]

Both the negative memory kernel and its hidden initial source are derived. A second-order observable equation thus follows from first-order nodal evolution without added microscopic inertia. The general elimination algebra is reusable; epi_memory's reversible graph adapter is not extended to this directed example. Actual fresh-pressure controls give different u_dot for equal observed u and different hidden v, while primitive phase stays zero. Controls: directed form phase and memory.

21.2 A coupled amplitude and phase law derived from fine diffusion

Join two equally oriented unit directed triangles by reciprocal unit links between corresponding vertices. Each node has two outgoing neighbors. This declared support differs from the reciprocal-prism model. With pure-EPI pressure and common fixed capacity nu, the exact complete scalar-state coordinates are two means and two complex contrasts:

\[ \dot\mu_a=\frac\nu2(\mu_b-\mu_a),\qquad \dot z_a=\left(-\frac{5\nu}{4}-i\frac{\sqrt3\nu}{4}\right)z_a +\frac\nu2 z_b. \]

For z_a=r_a exp(i psi_a) with positive amplitudes, these imply

\[ \dot r_a=-\frac{5\nu}{4}r_a+\frac\nu2r_b\cos(\psi_b-\psi_a),\qquad \dot\psi_a=-\frac{\sqrt3\nu}{4} +\frac\nu2\frac{r_b}{r_a}\sin(\psi_b-\psi_a). \]

The sine interaction, its coefficient and amplitude ratio follow from the fine generator; no synchronization gain is supplied. For delta=psi_1-psi_0, delta_dot=-(nu/2)*(r_0/r_1+r_1/r_0)*sin(delta). Equal amplitudes form an invariant restricted family with delta_dot=-nu*sin(delta). Outside it, equal angles can have different future rates. Angles alone are not sufficient state; at zero amplitude the Cartesian coordinates remain the continuation variables.

The complete contrast budget is

\[ \frac{d}{dt}(|z_0|^2+|z_1|^2) =-\frac{3\nu}{2}(|z_0|^2+|z_1|^2)-\nu|z_0-z_1|^2. \]

Derived interaction, synchronization and rotation therefore coexist with strict loss. This is a constructive emergent observable phase law. Identifying it with primitive phase or canonical phasor-argument pressure still requires full vector-field matching; a sine term alone is insufficient. Controls: coupled reduction and phase-only obstruction.

Inherited observation identity and source work

The fine model above now admits an exact observation-specific closure, rather than only a phase-only counterexample. Retain the same supplied directed support, fixed common positive capacity, orthonormal regional frames and pure-EPI law. Primitive phase is held and unused, Gamma is absent, and no operator event, clipping or numerical time step enters this continuous result. The selected observations are regional contrast squares, relative orientation and their cross-region contributions to the contrast budget. They are not the complete fine state or its tetrad.

On the positive-amplitude chart, put q=r_1/r_0, delta=psi_1-psi_0. The polar equations already derived from the fine generator give

\[ \dot q=\frac{\nu}{2}(1-q^2)\cos\delta,\qquad \dot\delta=-\frac{\nu}{2}(q+q^{-1})\sin\delta. \]

Thus the two angles together with the amplitude ratio have a closed evolution while both amplitudes stay positive. An absolute amplitude is additionally needed to predict form magnitude or its quadratic budget: common scaling preserves all angles and q, but multiplies that budget by the scale squared. Equal amplitudes supply a restricted invariant family, not a reason to discard the ratio for arbitrary preparations.

A division-free description also covers zero regional amplitude. Define

\[ I_a=|z_a|^2,\qquad c=\operatorname{Re}(\overline z_0z_1),\qquad s=\operatorname{Im}(\overline z_0z_1). \]

These quantities obey I_0,I_1>=0 and c^2+s^2=I_0 I_1. Direct differentiation of the inherited Cartesian law gives the closed system

\[ \begin{aligned} \dot I_0&=-\frac{5\nu}{2}I_0+\nu c,& \dot I_1&=-\frac{5\nu}{2}I_1+\nu c,\\ \dot c&=-\frac{5\nu}{2}c+\frac\nu2(I_0+I_1),& \dot s&=-\frac{5\nu}{2}s. \end{aligned} \]

The common internal rotation cancels in this observation. The constraint defect has derivative -5*nu*(c^2+s^2-I_0*I_1); realizable initial data remain realizable under the inherited linear Cartesian flow. Conversely, any such data with I_0>0 reconstruct a representative z_0=sqrt(I_0), z_1=(c+i*s)/sqrt(I_0). If I_0=0<I_1, take z_0=0, z_1=sqrt(I_1); at total zero both vanish. The common orientation is discarded. This proves sufficiency for the selected observations without claiming a globally unique minimal state for TNFR.

Write E=I_0+I_1, D=I_0-I_1. Its exact finite-time modes are

\[ \begin{aligned} (E+2c)(t)&=e^{-3\nu t/2}(E+2c)(0),\\ (E-2c)(t)&=e^{-7\nu t/2}(E-2c)(0),\\ D(t)&=e^{-5\nu t/2}D(0),\qquad s(t)=e^{-5\nu t/2}s(0). \end{aligned} \]

Here nu*c is the cross-region contribution to each dI_a/dt under the displayed split. This is an inherited contrast-square budget, not laboratory energy, primitive unit-phasor resultant amplitude, external Gamma or a separately supplied sustaining source. Positive contributions at both regions do not imply conservation of that contribution: the complete balance still satisfies E_dot=-(3*nu/2)*E-nu*|z_0-z_1|^2. The model cannot maintain nonzero contrast indefinitely. At z_0=0, z_1!=0, it nevertheless has I_0_dot=0 and I_0_ddot=nu^2*I_1/2>0: local contrast can start growing by transfer while total contrast decreases. No angle or event is invented at the zero-amplitude boundary.

Same angular history, different radial response. With both fine regional means initialized to 1/2, compare real positive contrasts (z_0,z_1)=(5,5)/128 and (1,7)/128. Both have E=25/8192 and relative phase zero. Their continuously lifted observed phases follow the identical law psi_a(t)=-sqrt(3)*nu*t/4 for all t>=0 (modulo 2*pi for circular phases): after removing this common rotation, the real linear coupling preserves positive amplitudes. Their initial c values are 25/16384 and 7/16384, however, so their initial E_dot values are respectively -75*nu/16384 and -111*nu/16384. Even the full angular trace and present total contrast do not identify its radial response. These are two bounded fine preparations, not fitted trajectories or a new pressure recipe.

The retained information depends on the promised observation:

Prediction	Sufficient inherited information	Boundary
Angular evolution	Both angles and q	Positive-amplitude chart only; no absolute form scale
Total contrast budget	E,c	Closed two-coordinate system; no regional allocation or orientation
Regional contrast, relative angle and cross-region budget	I_0,I_1,c,s on the constraint	Relative angle is available only when both amplitudes are nonzero
Fine EPI, pressure and general field observations	Both means and both Cartesian contrasts	The common contrast rotation is not generally a node relabeling or a tetrad-preserving symmetry

This result supplies a sufficient inherited observation and an exact missing-information obstruction. It does not select the primitive Arg source over J_phi/pi, or conversely. That identification would still need the independently specified observation map and full radial, angular and mean vector-field matching from section 15. Feeding an angle read from form back as an extra pressure can change the fine law; the existing source-closure control already checks that boundary. The coupled-reduction controls derive these identities from the fine graph generator and check the finite modal evolution, information-loss witnesses and zero-amplitude continuation. They introduce no independent solver or parameter sweep.

Admitting a form-derived angle into the primitive pressure law

This is the first F2 admission control following the S/D state cards. Keep the two equally oriented directed triangles and matched reciprocal links above, common held capacity nu>0, fixed unit conductance, and aligned supplied regional frames. Set the EPI and phase pressure coefficients to e,w>0; capacity/degree source readings vanish here. No events, Gamma or clipping are included. Consider the additional identification theta_ai=psi_a=arg(z_a) at positive regional amplitudes. It is a hypothesis to test, not a consequence of the nodal equation or of observing an angle.

For delta=psi_B-psi_A in the regular branch abs(delta)<pi, each node sees one own-region phase and one other-region phase. The centered resultant is 2*cos(delta/2)>0, so the actual Arg-based pressure gives

\[ g_{\phi,A}=\frac{\delta}{2\pi},\qquad g_{\phi,B}=-\frac{\delta}{2\pi}. \]

For delta!=0, this source is nonzero but constant within each region. Its projection on each zero-mean contrast plane vanishes, whereas the complete mean rows are

\[ \dot\mu_A=\frac{e\nu}{2}(\mu_B-\mu_A)+\frac{\nu w\delta}{2\pi},\qquad \dot\mu_B=\frac{e\nu}{2}(\mu_A-\mu_B)-\frac{\nu w\delta}{2\pi}. \]

The contrast row remains

\[ \dot z_A=e\nu\left[\left(-\frac54-i\frac{\sqrt3}{4}\right)z_A +\frac12z_B\right], \]

with A/B exchanged for the other region. Thus the original contrast reduction survives this source, but the full pure-EPI law does not: its mean response changed. The previously derived budget becomes E_dot=-(3*e*nu/2)*E-e*nu*abs(z_A-z_B)^2; adding this source does not maintain internal contrast. This specializes the existing source-closure distinction to the directed S/D reference instead of treating mean and contrast matching as interchangeable.

Tangency is a separate obligation. If a prospective pressure P and an independent phase law Omega are supplied, the constraint theta=Psi(x) can be invariant only if, on that constraint,

\[ \Omega(x,\Psi(x))=D\Psi(x)\,N P(x,\Psi(x)). \]

Matching an earlier fine model additionally requires its complete form row to agree. Defining Omega by this identity constructs a conditional constrained model; it does not independently justify the identification. Here direct differentiation of the inherited Cartesian row gives, with q=r_B/r_A>0,

\[ \dot\psi_A=-\frac{\sqrt3 e\nu}{4}+\frac{e\nu q}{2}\sin\delta,\qquad \dot\psi_B=-\frac{\sqrt3 e\nu}{4}-\frac{e\nu}{2q}\sin\delta, \qquad \dot\delta_\psi=-\frac{e\nu}{2}(q+q^{-1})\sin\delta. \]

The existing phase proposal owner reads supplied frequency, primitive phase and support, not EPI. With common frequency nu and both neighbors admitted, its continuous relative row is delta_theta_dot=-k*sin(delta), where k is the declared coupling strength. Two states with the same phase/capacity/support therefore receive identical proposals even when their form amplitude ratios differ. For e=w=1/2, nu=1, delta=pi/6, ratios q=1,2 require respectively delta_psi_dot=-1/4,-5/16. No fixed k matches both; a common moving frame cannot remove a relative-rate difference. This is an admission obstruction, not an error in the independently declared phase writer.

There is a restricted positive bridge. The contrast rows imply q_dot=(e*nu/2)*(1-q^2)*cos(delta), so q=1 is invariant. On that manifold, the relative rows coincide for declared k=e*nu. The primitive and derived common angular speeds still differ by nu+sqrt(3)*e*nu/4; equality requires an explicit moving common-phase frame, or comparison modulo common phase. With a fixed effective gate 0<gamma<=pi/2 and abs(delta)<gamma, the relative flow remains admitted and r_dot=e*nu*(-5/4+cos(delta)/2)*r preserves positive amplitude at finite times while dissipating it. This is a conditional continuous-law bridge with changed mean dynamics, not proof that finite Euler proposals preserve it exactly, a derivation of k, or a maintenance mechanism.

Zero and frame boundaries. The original Cartesian transport extends through zero contrast; the proposed angle-fed pressure generally does not have a unique continuous extension as a function of form alone. Fix z_B>0 real and compare z_A=epsilon with z_A=epsilon*exp(-i*pi/6). As epsilon -> 0+ their fine forms approach the same state, but their phase source contributions to mu_A_dot approach respectively zero and nu*w/12. Both paths remain within the controls' fixed gate pi/2. An arbitrary angle assignment at zero cannot repair these incompatible limits. Retaining primitive phase, retaining justified history, or changing the source law would define different model choices. This excludes the proposed global continuous form-only law, not every possible derived-phase law.

Regional frame rotation also changes the reported angle. Identifying it with primitive phase requires a transported frame convention; a node relabeling cannot silently reset that convention. These supplied data remain premises. The two admission controls derive the complete rows from the existing directed generator and compare them with graph-facing fresh pressure and actual phase proposals. They check instantaneous represented residuals, not a newly reserved trajectory or physical identification.

Exact capacity domain of the inherited observation

Keep the same supplied six-node support, pure-EPI outgoing pressure and fixed positive capacities, but allow all six capacities to differ. Regional frames are unchanged, and both means can vary independently of the contrasts. The observation (I_0,I_1,c,s) has an autonomous evolution for every fine state if and only if capacities are constant within each triangle. The two regional constants nu_0,nu_1 need not equal one another. This statement is about the specified observation and all preparations; it does not exclude additional special invariant preparations outside this capacity family.

For necessity, let U be the two-column orthonormal zero-mean basis of one triangle and let n_a be its three capacities. Changing only the hidden regional mean difference by d changes the unscaled fine diffusion rates by opposite constants d/2 in the two triangles. After capacity multiplication, the internal rates change by +/- (d/2)*U^T*n_a. Independence of each I_a_dot=2*y_a^T*y_a_dot from this hidden mean for every real contrast y_a requires U^T*n_a=0. The kernel of U^T consists exactly of constant vectors. This argument also works on an arbitrarily small open neighborhood of a uniform fine form inside its admissible band; it needs no unbounded EPI.

For sufficiency, put the constant capacity nu_a on each triangle. The shared fine generator then gives the closed mean and complex-contrast rows

\[ \dot\mu_a=\frac{\nu_a}{2}(\mu_b-\mu_a),\qquad \dot z_a=\nu_a\left(-\frac54-i\frac{\sqrt3}{4}\right)z_a +\frac{\nu_a}{2}z_b. \]

Writing w=c+i*s=conj(z_0)*z_1 gives

\[ \dot I_a=-\frac52\nu_a I_a+\nu_a c,\qquad \dot w=\left[-\frac54(\nu_0+\nu_1) +i\frac{\sqrt3}{4}(\nu_0-\nu_1)\right]w +\frac{\nu_0 I_1+\nu_1 I_0}{2}. \]

Regional frequency detuning is therefore inherited from these capacities; it does not destroy this observation's closure. The realizability constraint is preserved by the Cartesian flow. The appropriate fixed positive metric weights contrast by inverse capacity:

\[ \frac{d}{dt}\left(\frac{I_0}{\nu_0}+\frac{I_1}{\nu_1}\right) =-\frac32(I_0+I_1)-|z_0-z_1|^2. \]

Including both means adds -3*(mu_0-mu_1)^2 to the derivative of sum_a((3*mu_a^2+I_a)/nu_a). These are exact fixed-model quadratic budgets, not a capacity evolution law or physical energy identification. In particular, allowing two different regional capacities supplies no sustained nonzero contrast in this passive model.

There is also an independent loss of orientation outside the admitted family. Set capacities to (1+epsilon,1,1,1,1,1), epsilon>0, with both means 1/2. Preparations (z_0,z_1)=(r,r) and (i*r,i*r) have identical Gram observations and means, but their unweighted total-contrast rates are -(3+epsilon)*r^2 and -3*r^2. At epsilon=1, r=1/16, the difference is exactly -1/256. Thus retaining means alone cannot repair the reduction: the discarded common modal orientation can become relevant. This sharp domain result goes beyond the previous isolated hidden-mean obstruction.

The existing coupled-reduction controls verify the exact coefficient kernel, both inherited laws, the weighted budgets and the orientation witness. A production pressure control uses fine EPI (10,6,8,13,9,8)/16, capacities (3/2,3/2,3/2,5/4,5/4,5/4), zero primitive phases and Gamma absent. Its refreshed binary64 pressures and nodal products equal the rational fine-generator values exactly; the weighted contrast and full-form rates are -39/256 and -51/256. This binds one represented case to the exact family without introducing a solver or an asymptotic runtime claim.

Unequal-capacity mode selection and a finite phase-only discriminator

This continues the already admitted unequal-regional-capacity family, rather than introducing an oscillator or a pressure law. Keep the supplied equally oriented directed unit triangles, reciprocal unit inter-region links, fixed positive capacities a=nu_0, b=nu_1, and pure-EPI pressure. Primitive phase is held and unused, Gamma is absent, and there are no operator events. Unlike the common-capacity episode below, unequal capacities generally activate capacity pressure in a full mixture; this result does not include that mixture.

Candidate and identity. The full model retains both regional means and complex contrasts. Its proposed observed identity is the contrast direction up to a common nonzero complex scale, together with the separately retained absolute contrast budget. This is an observation-specific mode identity, not a sufficient description of the complete fine EPI, tetrad or a physical NFR. The initial family is every finite fine scalar state with nonzero contrast; small common scaling about a uniform mean places these preparations inside an admitted bounded chart. The conclusion below classifies this whole family, including its exceptional preparations and undefined-angle events.

Set

\[ c_0=-\frac54-i\frac{\sqrt3}{4},\quad D=a-b,\quad S=a+b,\quad A=\begin{pmatrix}a c_0&a/2\\b/2&b c_0\end{pmatrix}. \]

The inherited row is z_dot=A*z; c_0 here is a matrix coefficient, not the previous real cross-correlation. With the positive-real-part square root,

\[ \Delta=\sqrt{ab+c_0^2D^2},\qquad \lambda_\pm=\frac{c_0 S\pm\Delta}{2},\qquad R_\pm=\frac{-c_0D\pm\Delta}{a}. \]

These eigenvectors are (1,R_plus) and (1,R_minus); they are not assumed orthogonal. Since c_0^2=11/8+i*5*sqrt(3)/8, the radicand has strictly positive real part. Thus Re(Delta)>0, the eigenvalues are distinct for all a,b>0, and lambda_plus has the larger real part. The roots are nonzero and obey R_plus*R_minus=-b/a. For initial contrasts (z_0,z_1), set

\[ A_+=\frac{z_1-R_-z_0}{R_+-R_-},\qquad A_-=\frac{R_+z_0-z_1}{R_+-R_-}. \]

The exact Cartesian solution is

\[ z(t)=A_+\binom{1}{R_+}e^{\lambda_+t} +A_-\binom{1}{R_-}e^{\lambda_-t}. \]

For A_plus!=0, write eta=A_minus/A_plus. Wherever z_0(t)!=0,

\[ \frac{z_1(t)}{z_0(t)}= \frac{R_++\eta R_-e^{-\Delta t}}{1+\eta e^{-\Delta t}} \longrightarrow R_+. \]

Therefore the generic limiting relative phase is arg(R_plus), the amplitude ratio tends to abs(R_plus), and both observed phase rates tend to Im(lambda_plus). Projective errors are O(exp(-Re(Delta)*t)) as t tends to infinity; a pure slow-mode preparation has zero error throughout. The nonzero exceptional preparations A_plus=0 remain on the fast eigenline with ratio R_minus. Total zero has no observed phase. Arbitrarily small A_plus can delay approach arbitrarily, so the theorem supplies no uniform finite locking time across all preparations.

There is a sharp geometric lag bound. Put h=-c_0, d=(a-b)/sqrt(ab). The principal inverse hyperbolic sine gives

\[ R_+=\sqrt{b/a}\,e^{\operatorname{asinh}(hd)},\qquad \delta_+=\operatorname{Im}\operatorname{asinh}(hd),\qquad |\delta_+|<\arctan\frac{\sqrt3}{5}\simeq0.3334731723. \]

For d>0, the argument of sqrt(1+h^2*d^2) is strictly between zero and arg(h). Consequently Im(h/sqrt(1+h^2*d^2))>0: the lag increases strictly with d, hence with the capacity ratio. Oddness treats d<0; the large-d limit asinh(h*d)~log(2*h*d) gives the strict bound and its limiting sharpness. Common scaling of both capacities changes rates, not R_plus. Exchanging capacities inverts that ratio and reverses the limiting lag. At a=b, the two ratios are +1 and -1, recovering the existing equal-capacity modes.

No positive-capacity locking transition or sustained contrast follows. The real-part gap never closes in this family, and the generic lag varies smoothly with positive capacity ratio. The existing budget gives

\[ V=\frac{|z_0|^2}{a}+\frac{|z_1|^2}{b},\qquad \dot V=-\frac32(|z_0|^2+|z_1|^2)-|z_0-z_1|^2 \le-\frac32\min(a,b)V. \]

This is phase locking of a decaying contrast. At its limiting zero state, the angles are undefined; finite measurement resolution or binary64 rounding can lose them earlier. Zero capacity is an excluded boundary. No physical critical point, maintained oscillator or autonomous formation is inferred.

Zero crossings are regular in the full state. Each regional contrast in a nonzero two-mode solution can have at most one zero for t>=0: cancellation requires a fixed modulus of exp(-Delta*t), which is strictly decreasing. The two contrasts cannot vanish simultaneously unless the contrast solution is identically zero. At z_0=0, z_0_dot=a*z_1/2!=0, and conversely for the other region. These are simple chart crossings. Continue the Cartesian nodal state, rather than assigning a phase at zero or inventing an operator event.

A nonredundant finite prediction. On the positive-amplitude chart set q=r_1/r_0, delta=psi_1-psi_0. The full inherited equations are

\[ \dot q=\frac54Dq+\frac12(b-aq^2)\cos\delta,\qquad \dot\delta=\frac{\sqrt3}{4}D-\frac12(b/q+aq)\sin\delta. \]

Compare with the explicitly declared approximation that freezes q=1 and keeps only its angular row. This is not another canonical law. If D!=0, q_dot at q=1 is D*(5/4-cos(delta)/2), always nonzero, so initially equal amplitudes do not justify keeping that ratio fixed. At q=1,delta=pi/2, both models have the same initial relative phase and relative angular velocity, but

\[ \ddot\delta_{\rm full}(0)=-\frac58D^2,\qquad \ddot\delta_{\rm frozen\ ratio}(0)=0. \]

Thus equal initialization and instantaneous relative-phase rates still fail to close the subsequent angular response. This strengthens the earlier comparison of different hidden amplitudes at the initial time: here the two models initially agree on those amplitudes too, and their evolution creates the discrepancy.

Bounded control, 2026-09-26. The saved specification uses (a,b)=(1,2), (2,1) and the equal-capacity control (1,1), means (1/2,1/2), contrasts (1,i)/16, and times (0,0.02,0.1,0.5,2) in the declared structural clock. The fine six-node real generator comes from structural_diffusion_operator; the detached shared matrix exponential is checked against the analytic two-mode formula and an independent SciPy real-matrix propagation. The scalar comparator uses its own exact half-angle Riccati realization. No coefficient is fitted.

Capacities (a,b)	Full / frozen-ratio initial relative-phase acceleration	Full relative phase at t=0.5	Frozen-ratio phase at t=0.5	Generic limiting lag
(1,2)	-0.625 / 0	0.643272	0.702739	-0.228726
(2,1)	-0.625 / 0	0.984147	1.069849	0.228726
(1,1)	0 / 0	1.090415	1.090415	0

Angles are radians; accelerations are radians per squared declared time unit. Maximum complex-contrast discrepancy from the mode formula is below 2e-16; the independent fine-propagation difference is below 1.4e-15. These are finite numerical agreement checks, not rigorous enclosures or asymptotic runtime certificates. The local bundle artifacts/research/phase_capacity_lock_2026_09_26/ retains the specification, script hash, source/version provenance and complete results. The shared test owner contains the mathematical and engine-path controls. This is synthetic evidence; physical measurement, primitive-phase identification and novelty relative to external physics remain unestablished.

Reciprocity boundary for a physical realization

The directed candidate above is dissipative, but that does not make it a passive reciprocal RC/thermal model. A first-order scalar realization dx/ds=-S^(-1)*K*x, with positive diagonal storage S and symmetric positive-semidefinite conductance Laplacian K, is similar to -S^(-1/2)*K*S^(-1/2). Its eigenvalues are real and nonpositive. This is the reversible scope distinguished by the pressure foundation, not a statement about every physical system that can dissipate energy.

For the coupled triangles, the complex contrast block A above instead has, with T=diag(sqrt(a),sqrt(b)),

\[ T^{-1}AT= \begin{pmatrix}-5a/4&\sqrt{ab}/2\\\sqrt{ab}/2&-5b/4\end{pmatrix} -i\frac{\sqrt3}{4}\operatorname{diag}(a,b). \]

The first matrix is Hermitian. Multiplying an eigenvector equation by its conjugate transpose therefore gives, for every contrast eigenvalue,

\[ \operatorname{Im}\lambda=-\frac{\sqrt3}{4} \frac{a|v_0|^2+b|v_1|^2}{|v_0|^2+|v_1|^2}<0. \]

The real generator contains these eigenvalues and their conjugates. No invertible linear state chart, node relabeling or positive constant clock rescaling converts this complex spectrum into the reciprocal real spectrum. This excludes that exact closed physical identification for every a,b>0; it does not exclude directed transport, nonreciprocal devices or other models with additional state. A finite resemblance of two traces is also weaker than generator equivalence. The measurement contract retains the required independent bridge and the unresolved source admission.

Regional appearance and decay under one unchanged law

The zero-amplitude continuation above gives a complete transient episode, without requiring indefinite maintenance or another capacity law. Retain the same connected support, common held capacity kappa>0, equal zero primitive phases and absent Gamma. With any fixed positive four-channel mixture, capacity and phase gradients vanish, as does topology pressure because every node has two outgoing support neighbors. Put rho=kappa*w_epi. The preceding pure-EPI formulas apply with nu=rho; no active source channel has been removed or renormalized after observation.

Prepare both regional means at m, with z_0(0)=0 and z_1(0)=R>0. Region zero has no initial internal form contrast. The complete solution is

\[ z_{0,1}(t)=\frac R2e^{-i\sqrt3\rho t/4} \left(e^{-3\rho t/4}\mp e^{-7\rho t/4}\right),\qquad \mu_0(t)=\mu_1(t)=m. \]

The minus sign belongs to region zero. Its inherited angle becomes defined for t>0, rotating with region one's angle at -sqrt(3)*rho/4; no primitive phase is written at the initial zero. With I_env=R^2 and y=exp(-rho*t),

\[ I_0(t)=\frac{I_{env}}4 y^{3/2}(1-y)^2,\qquad I_1(t)=\frac{I_{env}}4 y^{3/2}(1+y)^2. \]

I_0 grows strictly until its unique maximum and then decreases strictly:

\[ t_* = \frac{\log(7/3)}\rho,\qquad I_{max}=\frac{4I_{env}}{49}\left(\frac37\right)^{3/2}. \]

Indeed its logarithmic derivative for t>0 is rho*(-3/2+2*y/(1-y)), which changes sign exactly at y=3/7. The existing budget I_0_dot=-(5*rho/2)*I_0+rho*c identifies the cause: incoming cross-region contribution exceeds local loss before the peak, equals it there and becomes insufficient afterward. Throughout the episode,

\[ I_0+I_1=\frac{I_{env}}2 \left(e^{-3\rho t/2}+e^{-7\rho t/2}\right) \]

strictly decreases. This is redistribution of prepared environmental form, not growth of the complete contrast budget or creation of new nodes.

A fixed observation criterion. For a threshold I_cut>0 fixed before evaluation, the equation I_0(t)=I_cut has two positive finite crossings if I_cut<I_max, one tangency if equal, and none if greater. A positive-duration episode defined by I_0>I_cut exists only in the first case. For I_cut=0, contrast is positive for every finite t>0; it disappears only asymptotically. Threshold exit therefore means loss of this declared observation, not exact extinction, a topology event or physical death. The threshold never enters the evolution law. Amplitude scaling across it is not a dynamical bifurcation.

Reserved three-preparation control. Fix node order (0,1,2,3,4,5), regions ((0,1,2),(3,4,5)), kappa=1, weights (phase,epi,vf,topo)=(1/4,1/2,1/8,1/8), m=1/2, I_cut=1/8192 and sample times (0,1/2,1,2,5). Region zero starts uniform; region one has fine form (m+a,m-a,m), hence I_env=2*a^2.

Preparation	Fixed input difference	Continuous prediction
Connected, stronger input	a=1/16	Peak/threshold about 1.465813; below at t=1/2, above at 1,2, below at 5
Connected, weaker input	a=1/32	Peak/threshold about 0.366453; never reaches the threshold
Disconnected control	a=1/16; omit the reciprocal interregional edges at initialization	Recipient remains uniform, with exactly zero contrast

All three use the same pressure recipe and capacity/phase laws. The last graph has constant outgoing degree one, so its other pressure channels also vanish. Its different normalization is computed from its actual support, not copied from the connected generator. All continuous forms stay inside their initial convex hull. The strong entry is in (1/2,1) and exit in (2,5); no numerical root search is needed to predict those reserved brackets.

The existing coupled-reduction test owner checks the exact solution and crossing classification, then executes shared pressure refresh and nodal Euler steps with dt=1/32 through time five. The independent exact Euler reference is frozen before each run. With its row-stochastic matrix M, the difference obeys delta_next=M*delta+d, where d retains separate pressure-realization and nodal-rounding defects. Thus the cumulative sum of ||d||_infinity bounds every represented-state error against that reference. This finite Euler comparison is separate from the continuous curve and does not certify its exact crossing times in binary64. Regional means, the complete fine form, primitive phases, capacities and the admitted form hull remain part of the check.

Finite outcome. All three runs matched the reserved classifications. For the stronger input, measured I_0 at times (1/2,1,2,5) was approximately (6.789873e-5,1.468016e-4,1.771180e-4,3.847178e-5), compared with the fixed threshold 1.220703125e-4. The weaker run remained below that threshold at the reserved samples, and the disconnected recipient remained exactly uniform at every executed step. Captured pressure-realization defects were zero in these runs; the largest nodal-rounding defect was 5.56e-17 or less. The cumulative represented-state bound was below 6.85e-15, and the final error against exact Euler was below 2.63e-16. These finite arithmetic observations do not absorb Euler discretization error into a claim of exact continuous flow.

The result is a predicted regional contrast episode on supplied support and preparation. It does not identify the scalar observation with a complete NFR identity, derive spontaneous region selection or establish physical emergence. Earlier finite identity windows and phase-driven cycle response remain distinct results. The new peak and crossing classification uses the existing fine form law without supplying an independent phase oscillator.

Collective interaction closure and relational state

The previous Gram result concerns one specified isolated six-node component. Its ability to interact is a separate obligation: equal local observations must predict equal selected responses in the same surrounding preparation. This section derives that boundary from the same pure-EPI nodal law, without adding primitive phase motion, a sustaining source or an operator policy. Means have form units; Gram entries have squared-form units. Rates refer to the declared structural clock, with no physical identification assumed.

All-state linear criterion. For a fixed real linear fine law on a collection of triangles, the means and complex contrasts form complete coordinates. Write its unique real-linear decomposition as

\[ \dot\mu=D\mu+\operatorname{Re}(Ez),\qquad \dot z=Az+B\overline z+C\mu,\qquad H=zz^\dagger. \]

Here D is real, the other coefficient matrices may be complex, and the state domain allows independent means and contrasts. The observation (mu,H) is autonomous for every fine preparation if and only if E=B=C=0. Its closed law is then

\[ \dot\mu=D\mu,\qquad \dot H=AH+HA^\dagger. \]

To see necessity, the same Gram fiber consists of exp(i*theta)*z, including its common sign reversal. Equality of all mean rates forces E=0. Differentiating H gives, in addition to its displayed closed terms, B*conj(z)*z^dagger + C*mu*z^dagger and their adjoints. Under the rotation, these have distinct Fourier weights minus two and minus one. Their vanishing for every angle, mean and contrast forces B=0 and C=0. Sufficiency follows by substitution. Positive-semidefinite H of rank at most one is preserved because the underlying solution is z(t)=exp(A*t)*z(0), including total zero. In the older two-region convention, H_10=c+i*s and H_01=c-i*s.

This necessity concerns homogeneous linear laws on the full stated domain. For a general nonlinear law, projectability means equality of the projected rates on every fiber; full equivariance is sufficient, not necessary. For example z_dot=i*omega(mu,z)*z, with any real omega, has H_dot=0 even when omega is not rotation invariant. Restricted preparations and promised outputs can also admit a weaker closure. No all-model criterion is inferred.

The linear criterion is directly testable on the actual nodal generator. With mean lift R, mean projection M, block contrast basis U and J=diag([[0,-1],[1,0]],...), it is equivalent to

\[ MGU=0,\qquad U^TGR=0,\qquad (U^TGU)J=J(U^TGU). \]

In these equally oriented triangle frames, this says that every three-by-three block of G is real circulant: a mean scalar and a complex-linear contrast block supply exactly its three real coefficients. A useful sufficient support class therefore has nonnegative circulant conductance blocks and capacity constant within each triangle. Outgoing normalization must be performed on the whole graph; the circulant blocks make its row strength constant within each triangle. This is a declared interface class, not a derivation of the graph, its coupling weights or a universal material interaction.

A fully normalized interaction. Take components A and B, each the existing pair of unit directed triangles, and add reciprocal weight-w links between corresponding vertices, with w>0. All capacities are held at one; primitive phase is held and unused, Gamma is absent, and no event occurs. Every outgoing strength is now 2+w. If G_6 is the earlier six-node generator, put alpha=2/(2+w) and kappa=w/(2+w). The exact full law is

\[ G_{12}=\begin{pmatrix} \alpha G_6-\kappa I_6&\kappa I_6\\ \kappa I_6&\alpha G_6-\kappa I_6 \end{pmatrix}. \]

Keeping the old internal rates unchanged after attaching these links would describe a different pressure law. In contrast coordinates let

\[ A_6=\begin{pmatrix}-(5+i\sqrt3)/4&1/2\\1/2&-(5+i\sqrt3)/4\end{pmatrix}, \quad A_{12}=\begin{pmatrix} \alpha A_6-\kappa I_2&\kappa I_2\\ \kappa I_2&\alpha A_6-\kappa I_2 \end{pmatrix}. \]

Then z=(z_A,z_B) obeys z_dot=A_12*z. The four means evolve by the same block construction applied to the earlier mean generator. Consequently four means and the full four-by-four Gram matrix close exactly. This is a positive effective-composition result for the declared observation and interface.

Local descriptions alone fail. Write H_A=z_A*z_A^dagger, H_B=z_B*z_B^dagger and the cross relation K=z_A*z_B^dagger. Differentiation from the fine generator, rather than a new constitutive choice, gives

\[ \begin{aligned} \dot H_A&=\alpha(A_6H_A+H_AA_6^\dagger)-2\kappa H_A +\kappa(K+K^\dagger),\\ \dot H_B&=\alpha(A_6H_B+H_BA_6^\dagger)-2\kappa H_B +\kappa(K+K^\dagger),\\ \dot K&=\alpha(A_6K+KA_6^\dagger)-2\kappa K+\kappa(H_A+H_B). \end{aligned} \]

Thus the product of the two separate component descriptions discards necessary relational information. When both component contrast vectors are nonzero, exactly one additional relative common angle is missing generically; the cross block is a redundant, division-free way of retaining it through zeros. These correlations are calculated from the same real EPI values. They are not new primitive coordinates, a fitted coupling, or a quantum density matrix.

For the selected component contrast E_A=trace(H_A), the interface contribution is 2*kappa*(Re(trace(K))-E_A). The corresponding two contributions sum to -2*kappa*||z_A-z_B||^2; neither is a separately conserved energy transfer. Regional mean exchange kappa*(mu_B-mu_A) already factors through the means. In contrast, individual vertex currents need the oriented fine values and do not in general factor through the common-rotation quotient.

An exact prospective witness fixes all four means at 1/2, sets z_B=(r,r), and compares z_A=(r,r) with z_A=(-r,-r). The surrounding component is unchanged; both preparations have the same separate means and Gram matrices. Their initial contrast rates are nevertheless

\[ \dot E_A^{+}=-3\alpha r^2,\qquad \dot E_A^{-}=-3\alpha r^2-8\kappa r^2. \]

At w=2 and r=sqrt(2)/16, the actual fine forms are dyadic: every triangle is 1/2 + (1,-1,0)/16, except for the reversed deviations in component A in the second preparation. The predicted rates are respectively -3/256 and -11/256, with interface contributions zero and -1/32. The cross block changes sign. Rotating both components together instead preserves the entire Gram observation and its selected responses; rotating one with its environment fixed is not that symmetry. Boundary geometry decides what can be discarded.

A single vertex-specific attachment generally falls outside this class. Its port value contains u/sqrt(2)+v/sqrt(6), which the common-rotation quotient omits, and the full row normalization can also mix means with contrasts. The available exact memory/complete-coordinate owners remain the fallback; the matched-port result does not establish closure for arbitrary connections.

Evidence and scope. The existing coupled-form controls derive the twelve-node projection and cross-Gram evolution, check the exact equal-local-state witness and bind it to fresh production pressure and a shared Euler step. The fixed finite control uses w=2, capacities one, the stated dyadic preparations, node order 0,...,11, binary64 and dt=1/32 on the standard unforced scalar path. It distinguishes continuous instantaneous rates from that held-pressure Euler endpoint. No physical record, random preparation, fitted parameter or additional solver enters the result.

This supplies a conditional interacting effective state and locates the information that its separate parts omit. The existing passive dissipation boundary still applies: it neither produces sustained contrast nor establishes autonomous formation, material constituents, physical spin or universal TNFR closure. The active continuation belongs only to the execution plan.

Held affine sources and the S0-to-D consistency verdict

The homogeneous criterion above does not by itself admit S0's non-form pressure channels. Retain the same ordered triangles, mean lift R, mean projection M, orthonormal contrast basis U and complex structure J. Thus x=R*mu+U*y, z_a=y_(2a)+i*y_(2a+1), M*R=I and U^T*R=0. For a fixed affine fine law

\[ \dot x=Gx+b, \]

hold both G and b fixed when comparing representatives of (mu,H). The full real-form domain allows independent means and contrasts and complete common-rotation fibers. Its exact all-state projectability criterion is

\[ MGU=0,\qquad U^TGR=0,\qquad [U^TGU,J]=0,\qquad U^Tb=0. \tag{A} \]

The last condition says that the supplied rate source is uniform within each region: b=R*h, where h=M*b. Different regions may have different sources. The resulting autonomous observation law is

\[ \dot\mu=D\mu+h,\qquad \dot H=AH+HA^\dagger,\qquad D=MGR, \]

where A is the complex representation of U^T*G*U. Hence H(t)=exp(A*t)*H(0)*exp(A^dagger*t) preserves the realizable rank-at-most-one positive-semidefinite domain, including total zero. A retained mean source need not preserve a global form mean or give bounded absolute form.

Necessity and sufficiency. The general affine coordinate law is

\[ \dot\mu=D\mu+\operatorname{Re}(Ez)+h,\qquad \dot z=Az+B\overline z+C\mu+c. \]

Equal mean rates along z -> exp(i*theta)*z force E=0. In the Gram rate, B*conj(z)*z^dagger carries Fourier weight minus two and (C*mu+c)*z^dagger carries weight minus one; their adjoints carry the opposite weights. Equality for every angle forces each coefficient to vanish. For every contrast and independently variable mean this implies B=0 and C*mu+c=0, hence C=0 and c=0. These are exactly (A). Substitution proves sufficiency. One region is no exception: its intensity still distinguishes a nonzero fixed contrast source on different orientation representatives. An imaginary scalar multiple of the identity in A is invisible to H, but already commutes with J; that common angular-generator freedom cannot cancel an additive source. The result is for fixed affine laws on the stated full domain, not a necessity theorem for arbitrary nonlinear or restricted models.

A source fixes an orientation reference. Suppose the homogeneous three conditions hold but the complex contrast source c is nonzero. At equal means, choose z_+=tau*c and z_-=-tau*c, where tau>0 has clock units. Both preparations have H=tau^2*c*c^dagger, yet

\[ \dot H_+-\dot H_-=4\tau cc^\dagger\ne0. \]

This excludes the proposed observation while leaving the complete affine fine law well defined. Rotating the source with the preparation would change the experiment and would not repair this fixed-source quotient. At total zero, the instantaneous Gram derivative is zero even when c is nonzero; with the homogeneous conditions still in force, H(t)=t^2*c*c^dagger+O(t^3). A zero-only first-derivative check therefore misses the obstruction. Retaining the oriented Cartesian contrasts, or an appropriate source-relative observation, is possible without changing the fine pressure law. For example z*c^dagger reconstructs z when this fixed c is nonzero, but then it retains the orientation that the original quotient discarded. The existing memory owner treats other explicit choices of incomplete observations.

Apply the criterion to the admitted reference. In S0, G=e*N*G_W and b=N*F, with held phase, capacity, support, conductances and coefficients. The source condition is consequently U^T*N*F=0, not U^T*F=0 in general. It admits precisely the pressure sources whose capacity-weighted rates are regional means. For positive capacity constant within each triangle, this is equivalent to F being constant within each triangle. With zero capacity, only its product with pressure enters the criterion; no division by capacity is used. Arbitrary within-region capacity variation must still pass all three generator conditions. Neither the source nor the generator condition can compensate for a failure of the other on the full domain.

Joint obligation	Scoped verdict and existing owner
Complete fine law and zero capacity	S0 is a globally defined affine reference at finite times; its state card holds phase, capacity and support and supplies its pressure realization
Collective evolution	Proved exactly when (A) holds; a nonzero contrast source has the explicit equal-observation witness above
Derived angle and zero contrast	Cartesian/Gram flow stays defined; an angle at zero is unavailable and need not be invented
Units, clock and reporting frames	Apply the existing joint covariance rules to the full law; rotating only a representative with fixed source is an active state change
Form and contrast budgets	Retain the mean source; under (A) the prior homogeneous contrast balance is unchanged. A source-free diffusion energy theorem does not silently apply to the complete affine form law
Changing capacity or operator events	Outside held S0 unless separately admitted; the capacity-law review and API contracts own the continuous/event distinction
Represented execution	Exact-real projectability is distinct from pressure realization, rounded nodal products, clipping and executor endpoints; the shared form observer retains rate defects rather than turning them into a theorem
Generative or physical interpretation	No unique pressure, maintained pattern, autonomous substrate or physical constituent is selected by this reduction

This closes the held affine-source compatibility question without adding a law to enforce it. It does not extend the S0 reference to the default hybrid runtime or enlarge D's observables to fine pressures and general tetrad values. Any narrower bounded preparation domain needs its own admission; the full-domain necessity proof cannot be inferred merely from a few represented trajectories.

The shared derive_regional_affine_closure checks explicitly supplied node order, ordered partition, generator G and rate source b. In these frames, each three-by-three generator block must be real circulant and each regional source triple must be constant. This is equivalent to (A) and permits exact coefficient residuals without introducing irrational basis roundoff. For a block with first row (c0,c1,c2), the mean coefficient is c0+c1+c2 and its complex contrast coefficient is c0-(c1+c2)/2 + i*sqrt(3)*(c2-c1)/2. The report separates the rational real and scaled imaginary parts. This detached algebraic admission does not infer G or b from a graph snapshot, verify a caller's held-law declaration, advance the graph or certify an executed trajectory. Use b in nodal-rate units; passing the pressure F in place of N*F would test a different law.

Production controls. The existing source-closure tests now exercise this admission and the shared form observer on two preparations of the unit undirected prism. In node order (region,vertex), with regions 0,1 and vertices 0,1,2, compare x_+=(1/2+(1,-1,0)/16) in both triangles with reversed contrasts x_-. Both have means (1/2,1/2) and every real Gram entry 1/128; imaginary Gram entries vanish. Support, coefficients, capacities and primitive phases are identical across each pair. No random seed, time integration or fit is used.

For the positive control, capacities are 1/4 in region 0 and 1 in region 1, with pressure weights (phase,EPI,capacity,topology)=(0,1/2,1/2,0) and equal primitive phases. The native capacity channel gives pressure source F=(1/8,1/8,1/8,-1/8,-1/8,-1/8), hence mean-rate source (1/32,-1/8). The admitted contrast generator is real, A=((-1/6,1/24),(1/6,-2/3)). Both actual refreshed preparations have the same Gram-rate matrix ((-1/512,-5/1024),(-5/1024,-1/128)) and mean rates (1/32,-1/8). All pressure-realization and canonical-product defects are exactly zero in this binary64 dyadic control.

For the excluded control, all capacities are the binary64 value 0.1, primitive phases are (0,pi/3,pi/6) in both triangles and pressure weights are (1/4,1/2,1/4,0). The captured represented phase source is F=(f,-f,0) repeated, with f=6004799503160661/2^57 in this execution. The exact reference Gram rate in every real entry is -nu/128 +/- nu*f/4: the signs are opposite despite identical means and Gram. Fresh production pressure and the shared nodal product retain that sign distinction. The tests independently assemble the ordered phasor source and account exactly for pressure-assembly and product-rounding defects; they do not silently identify binary64 operations with the exact-real affine flow. This single pair excludes the proposed quotient for that held law. It does not reject the fine dynamics or demonstrate physical formation.

Source-relative future response with identical initial Gram rates

F3's sign witness already distinguishes the initial Gram derivatives. The following F4 protocol asks the stronger information question: can the same means, Gram and its initial derivative determine later contrast under one fixed admitted fine law? It reuses the directed-triangle generator and the radial/tangential source decomposition. The law, inputs, horizon and bounds below are fixed before the production trajectory is evaluated. This is a prospective implementation control, not reserved physical data or selection of a new pressure law.

Exact witness. Keep equal regional means and contrasts on the two equally oriented directed triangles, common capacity nu>0, EPI pressure weight e>0, and a fixed identical regional rate source with complex contrast c. Its regional mean can be nonzero, but must be the same for both preparations. The existing fine generator induces

\[ \dot z=a z+c,\qquad a=-\alpha-i\beta,\qquad \alpha=\frac{3e\nu}{4},\quad \beta=\frac{\sqrt3e\nu}{4}>0. \]

For any nonzero c choose z_+(0)=i*k*c, z_-(0)=-i*k*c, where k=sqrt(3)*tau and tau>0 has clock units. Every entry of the two-region Gram is the same real intensity h=abs(z)^2. At the preparation time,

\[ h_+(0)=h_-(0)=k^2|c|^2,\qquad \dot h_+(0)=\dot h_-(0)=-2\alpha k^2|c|^2, \qquad \ddot h_+(0)-\ddot h_-(0)=4k\beta|c|^2>0. \]

Thus retaining the first Gram derivative does not generally close the observation. No comparison of different sources is needed. The direction of rotation is inherited from the supplied support, not fitted to the response. For an arbitrary real regional source triple (f0,f1,f2), the exact source-relative fine contrast tau*(f2-f1,f0-f2,f1-f0) realizes i*sqrt(3)*tau*c; this identity does not require c to be real. The concrete protocol below fixes ideal dyadic preparations and retains their defects against the represented source, rather than changing them after source materialization.

Finite prediction. Put E(t)=exp(a*t) and q(t)=(E(t)-1)/a. Then

\[ z_\pm(t)=c\{q(t)\pm ikE(t)\},\qquad \Delta h(t):=h_+(t)-h_-(t) =4k|c|^2\operatorname{Im}\{q(t)\overline{E(t)}\}. \]

Equivalently,

\[ \Delta h(t)=4k|c|^2\int_0^t e^{-\alpha(s+t)}\sin\{\beta(t-s)\}\,ds. \]

This is strictly positive for 0<beta*t<pi. In particular, for beta*t<=pi/2, use sin(u)>=u*(1-u^2/6) and exp(-alpha*(s+t))>=exp(-2*alpha*t) to obtain

\[ \Delta h(t)\ge 2k\beta|c|^2t^2 e^{-2\alpha t} \left(1-\frac{\beta^2t^2}{6}\right). \]

The formula and sign precede the trajectory. They follow from the same affine solution already used by the response owner, rather than a fitted response curve.

What information is missing? Define the derived source-relative coordinate w=conj(c)*z=R+i*I. Its exact rows are

\[ \dot R=-\alpha R+\beta I+|c|^2,\qquad \dot I=-\beta R-\alpha I,\qquad \dot h=-2\alpha h+2R. \]

The constraint R^2+I^2=abs(c)^2*h still holds. Initial h and its derivative fix R, but not the sign of the transverse component I. The two preparations have R=0 and opposite I. Their inherited rotation subsequently changes radial source work differently. Retaining oriented Cartesian contrast, or this source-relative coordinate for fixed nonzero c, restores the missing information. These are observations of existing form and the declared source, not new primitive degrees of freedom, an autonomous source or a maintaining mechanism. If the inherited rotation is absent (beta=0), this reflected pair has identical intensity for all times; the discriminator needs its stated law.

Shared source-relative observation. For several supplied regions, the production source-relative owner and SDK Network.source_relative_form retain the full matrix W=z*c^dagger, reusing the existing form observer and its represented-rate evidence. Here c is the contrast projection of an independently supplied held fine rate source b, not a pressure or an inferred measured derivative. The F4 coordinate w is a diagonal entry of W. Local diagonal entries alone lose an unforced region's contrast when its c component is zero, whereas the complete matrix gives z=W*c/(c^dagger*c) whenever the known vector c is nonzero. The source-relative description then retains orientation relative to that source; it does not make orientation an irrelevant symmetry of a fixed source-driven law.

Writing c_j=u_j/sqrt(2)+i*v_j/sqrt(6), the exact scaled coordinates are Re(W_ij)=a_i*u_j/2+b_i*v_j/6 and sqrt(12)*Im(W_ij)=b_i*u_j-a_i*v_j. The held-source pushforward replaces a_i,b_i by their nodal rates. If c varies, the complete chain rule is W_dot=z_dot*c^dagger+z*c_dot^dagger; the second term needs its own justified source law. No polar coordinate or division is used by the observer, so zero contrast and c=0 remain valid observations. If c=0, W alone carries no form orientation. Source means remain separate and may still drive regional means. The report is a derived observation of supplied state/law data, not an autonomous evolution, primitive phase identification or physical bridge. The SDK guide owns exact field names, source admission and usage.

Frozen production protocol. The maintained entry point is source_relative_form_response.py, with contract controls.

Item	Fixed declaration before evaluation
Support and order	Nodes 0,...,5, with label 3*region+vertex, in lexicographic region/vertex order; two triangles, directed edges vertex -> vertex+1 mod 3 and reciprocal links between matching vertices; unit conductances
Complete held law	Scalar EPI; nu=1; primitive phases (0,pi/3,pi/6) repeated; pressure weights (phase,EPI,capacity,topology)=(1/2,1/2,0,0); no phase/capacity evolution, operator events or Gamma forcing
Native ideal source	Each node has its successor and its matching cross-region vertex as outgoing neighbors; the regular midpoint phase channel gives F=(1/12,-1/24,-1/24) repeated
Preparations	x_+=(1/2,5/8,3/8) repeated and x_-=(1/2,3/8,5/8) repeated; both means 1/2, tau=1, c=1/(8*sqrt(2))+i/(8*sqrt(6)) and abs(c)^2=1/96
Clock and execution	Declared structural time, T=1, 128 steps of dt=1/128; refresh native pressure before every shared nodal Euler update; hold all other state and configuration
Observations	Shared regional-form observer; retain exact represented means, Gram, Gram rates, source/pressure/product defects and endpoint EPI; compare the first regional intensity, with the second region as a repeated consistency control
Exact initial prediction	Every Gram entry 1/32; every initial Gram-rate entry -3/128; initial intensity second-derivative difference 1/64
Prospective finite prediction	At T=1, Delta h >= 127/65536; this uses exp(-3/4)>=1/4 in the analytic bound, not an evaluated endpoint
Numerical admission	Separate source-model and accumulated arithmetic allowances, each 1e-12 in fine-EPI infinity norm; include the analytic Euler error below; fail rather than refit if an allowance is exceeded
Decision	Each trajectory must agree with its predeclared analytic intensity within its error budget; the observed positive intensity gap must exceed 1/1024
Ablation	Keep the same topology, EPI coefficient, capacity, preparations, clock and pressure weights, but set primitive phase uniformly to zero so the native phase source vanishes; do not renormalize channel weights

The positive-source preparation is independent of the future response. Capture its represented non-EPI source prospectively and compare it with an independently assembled phase calculation. The ideal and represented source are different reference objects: exact initial Gram-rate equality belongs to the ideal affine model, while the observer retains any realized preparation, pressure and product defects. Equality must not be manufactured by replacing measured rates with the analytic prediction. No random seed or parameter fit is used. The run report records source and runtime provenance.

Error budget fixed before execution. Let B=(1/2)*G_W be the complete six-node rate generator and b the repeated ideal source above. Both exp(B*t) and I+dt*B are contractions in the fine infinity norm for these step sizes: their entries are nonnegative and every row sums to one. Since x''(t)=exp(B*t)*B*(B*x(0)+b), local truncation and contraction give

\[ \delta_{\mathrm{Euler},\pm} \le \frac{T\,dt}{2}\|B(Bx_\pm(0)+b)\|_\infty. \]

In the repeated regional rows these second derivatives are respectively (-7/128,3/128,1/32) and (-1/128,-3/128,1/32). Therefore the fixed fine-EPI bounds are delta_Euler,+=7/32768 and delta_Euler,-=1/8192. A held source discrepancy contributes at most T*norm(b_represented-b,inf). The exact represented residual of each implemented step against its affine Euler row is accumulated separately; contraction bounds its final contribution by the sum of the residual norms. The protocol allocates 1e-12 to each of these two contributions, so epsilon_+=7/32768+2e-12 and epsilon_-=1/8192+2e-12 before evaluation. An initial materialization discrepancy, if present, must be added explicitly; the declared dyadic EPI preparations have none in binary64.

Throughout the ideal interval, each centered regional component is bounded by r_max=1/8+1/12=5/24. Centering is an orthogonal projection in Euclidean norm, so an infinity-norm fine error epsilon changes regional intensity by at most 6*r_max*epsilon+3*epsilon^2. Apply this separately to both signs. The sum of these intensity budgets is smaller than the margin between 127/65536 and the fixed decision threshold 1/1024. Thus the decision does not rely on a sign smaller than the admitted discretization error. The zero-source ablation has exact equal intensity trajectories, although represented execution may differ within its separately retained error budget.

Frozen evaluation and result. The prediction record was written before the first production evaluation. Its SHA-256 is 1a08e804f2ce17b0457eac26f4196e5cb2fd26e1b4374b96ee730938ec1fc022. The result record retains exact rational observations, configuration, runtime versions and fingerprints of the producer and relevant engine owners; its SHA-256 is d1a37558f4e5386470b291b648eff0cfa4b2b9a9559a267ef46de76d6be7b94a. Subsequent tests are regression reproductions of that evaluated protocol, not new reserved observations. The original producer bytes preserve the source fingerprint of this first result. The maintained producer now enforces scientific admission even under optimized Python and rejects replacement of an existing response record. Its later source fingerprint therefore differs; new-source regression pairs must use distinct output paths. Neither this hardening nor the new observation API changes the first frozen prediction or converts a replay into independent reserved evidence. The current runner also records W and its held-source rate through the shared source-relative observer, using its prospectively captured source and embedded form report. These additional read-outs bind the derived coordinate to actual engine execution without a second pressure refresh or a duplicated chart.

Quantity at T=1	Plus preparation	Minus preparation
Observed regional intensity	0.02437996979119982	0.01958940432108876
Continuous analytic estimate	0.024345599871311546	0.01960831302470951
Prospective intensity-error budget	2.67166e-4	1.52633e-4
Accumulated runtime arithmetic bound	4.65812e-15	4.59263e-15
Actual endpoint error against exact ideal Euler	4.76235e-16	3.23218e-16

The observed gap was 0.004790565470111058, above the frozen 1/1024 threshold and the theoretical lower bound after both error budgets. The same-weight uniform-phase ablation gave a gap of about -1.016e-16, within its retained arithmetic bound. The native source's infinity-norm discrepancy from the ideal rational source was 4.626e-18; both initial observed intensity rates were equal in this execution, with common defect 1.735e-18 against -3/128. No clipping acted, and capacity, primitive phase and support remained fixed. The complex-exponential entries above are numerical estimates; the exact rational prediction and error inequalities own the decision.

F4 therefore supports the scoped prediction: an intensity and its first derivative need not be a sufficient state, even for one fixed admitted law. Its omitted source-relative quadrature is already available from the fine form, frame and source; no new microscopic variable or pressure term is needed to describe this response. This completes the four-stage reference exercise, not the derivation of a physical constituent, unique pressure realization, autonomous source, emergent substrate or indefinite persistence.

Joint realizability of collective relations

The interaction law uses correlations of one deterministic fine preparation. Their admissible values are constrained before any evolution is considered. For components with complex contrast vectors z_A, assemble all blocks H_A=z_A*z_A^dagger and K_AB=z_A*z_B^dagger into one matrix H. It is Hermitian positive semidefinite of rank at most one. Conversely every such matrix has a factor H=z*z^dagger: if a diagonal entry H_qq>0, choose z_i=H_iq/sqrt(H_qq); the only rank-zero case is H=0, with z=0. Nonzero factors differ by one common phase. With the retained means and regional lifts this constructs real fine EPI. Any extra numerical clipping rails must still be checked on that lift; they are not a Gram axiom.

Put E_A=trace(H_A)=z_A^dagger*z_A. Multiplication gives

\[ K_{AB}K_{AB}^\dagger=E_BH_A,\qquad K_{AB}^\dagger K_{AB}=E_AH_B,\qquad K_{AB}K_{BC}=E_BK_{AC}. \]

These are consequences of the factorization, not additional dynamics. If E_A=0, then z_A=0 and every relation incident to A is zero; a phase at that zero is unavailable. If E_B>0, two complete relations through B determine the third by the displayed identity. No division is admitted at E_B=0: relations to that zero component say nothing about K_AC. The full positive-semidefinite/rank condition is the admission criterion; these convenient identities are not claimed to be a complete independent set for arbitrary partially recorded block data.

Pairwise admissibility alone is insufficient. Unit scalar intensities and relations K_AB=K_BC=1, K_AC=-1 give

\[ H_{\rm proposed}=\begin{pmatrix}1&1&-1\\1&1&1\\-1&1&1\end{pmatrix}. \]

Each two-component principal block is positive semidefinite and rank one. But v=(1,-1,1) gives v^T H_proposed v=-3, so the whole matrix cannot arise from any single fine preparation. It also violates K_AB*K_BC=E_B*K_AC. The example embeds into the existing two-contrast components by giving each a local direction (1,0); introducing more coordinates does not repair the incompatible relations.

For m nonzero components of two complex contrasts each, their separate local Gram descriptions have 3*m real degrees of freedom. The full joint Gram has 4*m-1, leaving m-1 independent relative common orientations. Thus relational properties matter but are not arbitrary pairwise additions. This count is on the nonzero stratum with fixed component frames; zero components have no orientation to retain. Means are additional coordinates.

The collective controls check reconstruction from actual fine-coordinate preparations, cross-relation identities, a zero-contrast boundary and the incompatible-pairs witness. They test realizability, not temporal stability or a new engine trajectory. An ensemble-averaged Gram can have higher rank, but that would be a different statistical observation with explicitly declared preparation and evolution. No ensemble rule, quantum interpretation, phase-winding theorem or new constitutive law is supplied by this algebraic result.

Imperfect interface: ambiguity and a finite predictive bound

The exact collective law above assumes a circulant interface. Its loss need not make every reduced prediction useless, but the discarded orientation can then matter. This section keeps the same twelve-node pure-EPI law, unit held capacity, fixed support, unused primitive phase and absent Gamma. Only the outgoing conductance 0 -> 6 changes from 2 to 2+epsilon, with 0<=epsilon<=1; the reverse edge and every other conductance stay fixed. The affected outgoing strength is 4+epsilon, not four. Define

\[ \eta=\frac{\epsilon}{4+\epsilon},\qquad r^T=\tfrac12 e_6^T-\tfrac14 e_1^T-\tfrac14 e_3^T. \]

Directly normalizing that row gives the exact rank-one update

\[ G_\epsilon=G_0+\eta e_0r^T,\qquad r^T\mathbf1=0, \qquad \|e_0r^T\|_\infty=1. \]

Both generators have diagonal minus one, nonnegative off-diagonal entries and zero row sums. Their positive stochastic semigroups contract the infinity norm. This is a change of a supplied structural interaction, not an additional force, a new phase law or a derivation of why that conductance changes.

A prospective ambiguity, including finite time. Set a>0, v=(1,-1,0) repeated four times, and x^+(0)=b*1+a*v, x^-(0)=b*1-a*v. These preparations have identical four means and identical full joint Gram, including all cross-component relations. The lost coordinate is now the common orientation of the whole pair, rather than the relative orientation already retained by its cross block. Their initial first-region mean rates are

\[ \dot\mu_0^+(0)=\eta a/6,\qquad \dot\mu_0^-(0)=-\eta a/6. \]

Consequently exact all-state mean/Gram closure fails for every epsilon>0. This obstruction is not a numerical defect. Linear sign symmetry keeps the two exact Gram trajectories equal while their means can separate.

Let M_0=(e_0^T+e_1^T+e_2^T)/3 denote the first mean row. The old unperturbed mean stays at b. In its equal-region contrast sector,

\[ r^Te^{G_0s}v=\tfrac12 e^{-3s/8}\cos(\sqrt3 s/8). \]

This follows from the already derived directed-triangle generator (S-I)/4 on that sector. Duhamel's identity therefore gives

\[ q(t):=M_0e^{G_\epsilon t}v =\frac\eta2\int_0^t M_0e^{G_\epsilon(t-s)}e_0\, e^{-3s/8}\cos(\sqrt3 s/8)\,ds. \]

For u>=0, the nonnegative exponential series for G_epsilon=P_epsilon-I gives M_0 exp(G_epsilon*u)e_0>=exp(-u)/3. On 0<t<=1/2, the cosine is positive and decreasing. Thus

\[ q(t)\ge\frac{\eta t}{6}e^{-t}\cos(\sqrt3 t/8),\qquad \mu_0^+(t)-\mu_0^-(t)=2a q(t). \]

Any deterministic single-valued predictor receiving only that initial joint observation and the same perturbed generator must err on at least one preparation by at least

\[ \frac{a\eta t}{6}e^{-t}\cos(\sqrt3 t/8). \]

This is an information-loss lower bound, not a claimed physical uncertainty principle. Retaining the omitted orientation restores the complete Cartesian state; retaining only the observation permits a set-valued prediction instead.

An upper bound available from the observation alone. For arbitrary admissible initial means and joint Gram, put

\[ L_Q=\min_j(\mu_j-\sqrt{2/3}\sqrt{H_{jj}}),\quad U_Q=\max_j(\mu_j+\sqrt{2/3}\sqrt{H_{jj}}),\quad B_Q=(U_Q-L_Q)/2. \]

Each row of the orthonormal triangle contrast basis has norm sqrt(2/3). Every fine representative of this observation therefore lies in [L_Q,U_Q]; this estimate does not choose its hidden common orientation. Center at b_Q=(L_Q+U_Q)/2. Since the perturbation annihilates constants, Duhamel and stochastic contraction give, for the same initial fine state,

\[ \|x_\epsilon(t)-x_0(t)\|_\infty\le\eta t B_Q, \qquad \|\mu_\epsilon(t)-\mu_0(t)\|_\infty\le\eta t B_Q. \]

The unperturbed mean/Gram solution is independent of the chosen representative and hence supplies one computable reference for the whole fiber. Orthogonal contrast projection gives ||z||_2<=sqrt(12)*B_Q for both trajectories and ||z_epsilon-z_0||_2<=sqrt(12)*eta*t*B_Q. Applying ||zz^dagger-ww^dagger||_F<=(||z||_2+||w||_2)*||z-w||_2 yields

\[ \|H_\epsilon(t)-H_0(t)\|_F\le24\eta t B_Q^2. \]

These absolute bounds have respectively form and squared-form units, apply to every representative, and remain defined at zero regional amplitude. At uniform global form, B_Q=0 and both responses vanish exactly. Unequal regional means can give B_Q>0 even with initially zero contrasts; that case must not be silently declared frozen. No division by amplitude or angular accuracy guarantee is used. The proof reuses the finite-semigroup defect method of structural morphisms and the projection/memory framework, without applying its reversible graph adapter to this directed support.

The outcome is a conditional finite predictive enclosure despite failure of exact observation closure. A requested error below the lower ambiguity bound cannot be guaranteed without more information; the upper bounds give sufficient absolute error budgets for the unchanged collective reference. No claim of relative accuracy at vanishing amplitude, universal lifetime or maintenance follows, and increasingly tight passive bounds are not a separate campaign.

Bounded control fixed before execution. Use epsilon=1/4 (eta=1/17), b=1/2, a=1/16, the two global-sign preparations above, node order 0,...,11 and the existing binary64 scalar engine path (nonvectorized pressure and the scalar shared-integrator branch). Evaluate at t_star=1/32 with 32 fresh-pressure shared Euler steps of h=1/1024. Primitive phases stay zero, capacities stay one, no events or Gamma are active, and the preparations stay inside the configured [0,1] form rails. The analytical bounds hold over 0<=t<=1/2; this control samples only t_star, not a parameter or horizon sweep.

The independent references are the exact rational Euler recurrence and the continuous matrix exponential. For these preparations, ||x(0)-b*1||_infty=a, ||G_epsilon||_infty=2 and h<=1 make the Euler transition stochastic. The integral local remainder and contraction give the continuum discretization bound 2*a*t_star*h=1/262144 per trace. Actual pressure and integration rounding defects are captured separately as exact differences of represented values and propagated with stochastic gain at most one. Their sum is an arithmetic bound, not a replacement for the time-discretization bound. The matrix exponential is a floating reference, not a formal transcendental certificate; the displayed lower/upper bounds are proved independently.

The three bounded controls passed with the frozen preparation and method. The whole affected research module passed 36 tests. The retained endpoint metrics are:

Quantity at t_star	Value and interpretation
Executed first-region mean separation	3.7651500332e-5 form units
Analytical continuum separation lower bound	Approximately 3.7117457513e-5 form units
Combined numerical uncertainty for the pair	At most 7.6293945342e-6 form units from truncation plus captured arithmetic
Observation-only mean error upper bound	Approximately 1.3266320524e-4 form units
Observation-only Gram Frobenius error upper bound	Approximately 2.2977941176e-4 squared-form units
Continuous exponential-reference mean / Gram errors	Approximately 1.8815585843e-5 / 1.2017643258e-5, each within its respective bound
Largest accumulated arithmetic bound	Below 1.55e-15 form units; separate from the Euler bound

The displayed transcendental bounds are rounded summaries of the exact expressions above. The pair's numerical uncertainty is smaller than its separation; the two responses are resolved without changing a coefficient, preparation or error budget after execution. The finite results confirm the implemented realization, while the analytic argument establishes the stated all-preparation upper bounds and selected-family lower bound. Compact run metrics are recorded as JUnit properties by the controls; local output is artifacts/research/imperfect_interface_2026_09_26.xml (ignored evidence). No physical data were evaluated, and this completes the bounded robustness gate rather than opening further passive-transport optimization.

21.3 Canonical phase geometry has directed sensitivity on reciprocal support

Return to the unchanged reciprocal prism and its retained nonrepeated phase preparation theta=(-a,a,0,-a,a,a), a=pi/6. The shared derivative Dg=(R-I)/pi uses R_ij=1[j in N_i]*sum_{k in N_i}cos(theta_j-theta_k)/|S_i|^2. Every supported entry is positive. Put A=(3-sqrt(3))/4, B=(sqrt(3)-1)/2. On triangle zero,

\[ \frac{R_{01}R_{12}R_{20}}{R_{02}R_{21}R_{10}} =\frac{AB(2/7)}{AB(5/14)}=\frac45. \]

Positive diagonal detailed balance would require this ratio to equal one. Thus reciprocal graph support does not guarantee a reversible phase-source response. The repeated control (-a,a,0) on both triangles instead has R_ij=1[i~j]*c_j/(1+sqrt(3)), c=(sqrt(3)/2,sqrt(3)/2,1) repeated, and obeys detailed balance with weights c_i. This supplies directional sensitivity from existing phase geometry, without replacing the graph by an imposed directed topology.

It does not select a velocity: reversing every phase leaves R unchanged and reverses g. Nor does it contradict the phase metric in variational section 13.6: differentiating a state-dependent mobility contributes another term to Dg away from equilibrium. Cycle imbalance is neither oriented source work nor an autonomous oscillation. The complete metric differential now accounts for every weighted antisymmetric entry. The joint source/work classification separately retains the phase velocity and mean: identical response geometry can give opposite source work. The matrix-asymmetry branch is closed in scope; the missing maintenance law cannot be replaced by this diagnostic. At the retained point, the mobility-derivative contribution to entry (0,1) is exactly (3-sqrt(3))/(4*pi)-1/(6*(1+sqrt(3)))>0; the held-mobility Hessian term alone misses it. This cross-entry check is already complete. Controls: source-response cycle balance.

21.4 Local wave realizability does not select a wave law

On fixed reciprocal unit-prism support, retain unit capacities and mixed pressure p=-e*L*x+w*g(theta) with fixed coefficients e,w>0. Let Pi center a field, y=Pi*x, v=Pi*p. In a lifted chart with global phase spread below pi/2, R is irreducible, nonnegative and row stochastic. Its positive left stationary vector l proves that Pi*Dg is invertible on centered phase space: Pi*Dg*h=0 implies Dg*h=c*1; multiplying by l^T gives c=0, and the centered kernel is zero. Hence (y,theta modulo rotation)->(y,v) is a local coordinate change. Per-edge U3 alone is not this global-spread premise.

Any supplied smooth centered acceleration F(y,v) has the local lift

\[ w\Pi Dg\,\dot\theta=F(y,v)+eLv,\qquad \dot\mu=w\,\overline{g(\theta)}. \]

Common phase rotation remains free. Choosing the existing graph wave F=-Ly conserves its centered degree-metric energy H=3*(||v||^2+y^T L y)/2 while the chart is valid; its coordinate pullback has a nondegenerate ten-dimensional two-form. This uses phase contrast as the wave coordinate. Choosing that acceleration is still a constitutive premise: the same invertibility permits other accelerations and proves no global continuation. At uniform phase, k=w/pi gives theta_dot=(y-e*v)/k modulo rotation; eigenvalue-one modes require a=(1+e^2)/k, b=e in section 17.4, its active neutral boundary.

Mean motion is essential. At the retained prepared phase theta=(0,pi/3,pi/6) repeated, with y=a*(1,1,-2) repeated, a>0, the full requirement x_ddot=-Lx is outside the source derivative's image. The centered wave is liftable but forces

\[ \ddot\mu=(1+e^2)a\frac{\sqrt3-2}{1+\sqrt3}\ne0, \]

even though mu_dot=0 there. This is different from the rejected harmonic motion of the extracted tetrad fields. Controls: centered wave lift and mean.

The stronger constitutive-condition classification distinguishes power balance, a Poisson tensor and a chosen Hamiltonian. In this selected centered symplectic chart, ydot=v restricts the Hamiltonian to 3||v||^2/2+U(y) but leaves U undetermined. This is stronger than arbitrary-acceleration realizability and weaker than a derived wave law. The exact tests retain the original form and distinguish the alternatives by its existing transverse mode, not a new trajectory. The subsequent locality/transport classification separates one-hop acceleration from one-hop primitive-phase dependence. On the retained phase, a common-phase gauge cannot make the selected -kappa*L*y lift primitive-local. Exact diffusion embedding also differs from a leading overdamped balance. These fixed-capacity restrictions do not select the missing complete-triad relation.

21.5 A form-coordinate boundary need not require an operator

The reciprocal prism already gives a finite-time autonomous control. Keep pure-EPI weight e>0, unit capacity, means 1/2, primitive phases zero, and the unnormalized coefficients of section 12:

\[ q=e^{-et/3},\qquad u_0=\frac{q^3(1-4q^2)}{32},\quad u_1=\frac{q^3(1+4q^2)}{32},\quad v_0=v_1=0. \]

They satisfy the shared induced generator. At t_*=3*log(2)/e, q=1/2, the first regional contrast crosses zero: (u_0,u_1)=(0,1/128), with rates (e/384,-e/96). Its polar angle loses its domain and reappears on the opposite ray. For t>=0, EPI remains in [11/32,21/32]; this is not node creation from vacuum. Full Cartesian evolution, fresh source and continuation remain regular, with unchanged graph/source ranks and nonzero primitive phasor resultants. Boundary transfer can regrow local contrast while global contrast decreases.

Before treating a geometric transition as compulsory, distinguish loss of an observer's chart from loss of full-state continuation or admissibility. The polar singularity alone does not require Mutation. Existing THOL gates compare measured absolute acceleration against configured thresholds on an admitted invocation; operator names alone do not establish a mathematical bifurcation. Their source docstrings now state that boundary correctly. Controls: exact form-zero continuation.

Historical source-bound validation record: artifacts/research/phase_law_reuse_validation_2026_09_19.json.