Joint form, phase and capacity response

Signed representation, pressure derivatives and finite phase/capacity compatibility.

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.

9. Signed EPI and phase: an explicit representation test

Consider the proposed observation z_i=x_i*exp(i*theta_i), retaining capacity, support and coefficients. It might seem to package form and phase in one complex coordinate. A necessary condition for a closed differentiable law on this observation is that two underlying states with the same z give the same derivative of every smooth function of z. In particular,

\[ |z_i|^2=x_i^2,\qquad \frac{d|z_i|^2}{dt}=2x_i\nu_i p_i. \]

The angular velocity cancels from this identity. We can therefore test projectability without assuming any missing phase law.

9.1 A sign ambiguity with different radial responses

At fixed support write p=-w_e*L*x+F(theta,nu,G). Under the common change T:(x,theta)->(-x,theta+pi*1), exact circle algebra gives the same z. The non-EPI source F is unchanged under a common phase rotation on its regular branch, while the EPI difference changes sign. Thus

\[ p(Ts)=-p(s)+2F(s),\qquad \left.\frac{d|z_i|^2}{dt}\right|_{Ts} -\left.\frac{d|z_i|^2}{dt}\right|_s=-4x_i\nu_iF_i. \]

This supplies an obstruction whenever the displayed product is nonzero. It excludes a closed law on this observation for the stated domain, regardless of any real angular velocities one might subsequently propose.

A bounded two-node witness uses all four configured channel weights 1/4, capacities (1,2), unit reciprocal conductance and phase consensus:

State	EPI	Phase	Non-EPI source	Pressure	Squared-modulus rate
A	(1/4,1/2)	(0,0)	(1/4,-1/4)	(5/16,-5/16)	(5/32,-5/8)
B	(-1/4,-1/2)	(pi,pi)	(1/4,-1/4)	(3/16,-3/16)	(-3/32,3/8)

Both exact observations are (1/4,1/2), both phase gaps satisfy strict U3, and all form values lie inside the default scalar interval. No clipping, selected operator word or trajectory is needed for the obstruction. The pure-EPI global-flip control has F=0 and no such radial defect; that control does not prove full complex-state closure or specify angular motion.

The proof uses exact circle components (1,0) and (-1,0). Executable pressure controls separately materialize uniform binary64 phase values and verify zero relative-phase pressure; they do not assert that evaluating exp(i*float(pi)) gives exactly (-1,0).

9.2 Zero form does not erase a node's phase

For x=(0,1), nu=(1,1) and the same weights, change the zero-form node's phase from zero to alpha, where 0<alpha<pi/2, holding the other phase zero. The complex observation remains (0,1). On the singleton-neighbor phase branch, the positive node's pressure changes by w_phi*alpha/pi; its squared-modulus rate changes by 2*w_phi*alpha/pi. Phase at a zero EPI coordinate thus affects an observable neighbor response.

The production represented quarter-pi control has pressures (1/4,-1/4) versus (3/16,-3/16) and radial rates (0,-1/2) versus (0,-3/8). Those exact rational outputs describe the retained binary64 calculation; the symbolic proof above does not depend on its transcendental rounding.

This also locates a conceptual choice. If a new theory declares phase absent at zero form, its pressure law must respect that equivalence. The current phase channel does not. Calling the zero coordinate a vacuum cannot resolve the conflict; the node, its neighbors and its other attributes still exist.

9.3 A faithful encoding, without a new law

Away from zero and on a fixed sign sector, the polar map is locally invertible: its Jacobian determinant is x. Globally it identifies opposite signed forms with a half-turn of phase; at zero it loses an entire phase circle. Restricting to positive form needs a separate invariant-domain argument. Indeed x=(0,1/2), nu=(2,1) at phase consensus gives xdot=(-1/4,1/8) in the same pressure model: the nonnegative boundary points outward. A clipping rule would be an additional map, not a proof of invariance of the unclipped law.

A faithful node representation is (x,u,nu) with x real and |u|=1. It retains phase at zero form and is the cylinder embedding (x,cos(theta),sin(theta)) with the existing capacity coordinate. Its two form/phase tangent columns have identity Gram matrix, including at x=0. Equivalently retain (z,u,nu) with z*conj(u) real and recover x=Re(z*conj(u)). This is an encoding of the existing state, not an assertion that extra physical degrees of freedom have emerged.

For a differentiable unit-circle path, u_dot=i*omega*u for some real omega; that kinematic identity does not determine omega. The EPI law still supplies only xdot=nu*P. Capacity, phase, changing support and history require their own justified closure or reduction. The linear epi_memory and operator_quotient owners supply the established closure principles; their linear certificates cannot certify this nonlinear observation without the displayed fiber test.

Five detached controls in test_epi_phase_representation_scope.py cover these statements. They reject this particular lossy packaging, not every complex-valued formulation of TNFR, and do not derive autonomous NFR formation.

10. Joint pressure response and the capacity product rule

10.1 One identity, with three distinct neighborhood responses

Fix unique support, active conductance edges and effective channel coefficients on a differentiable segment; positive symmetric conductances may vary smoothly. Write U for the unweighted unique-neighbor averaging matrix, L_U=I-U, and L_W for the weighted EPI random-walk Laplacian. On a regular phase branch with nonzero neighbor resultants, let R be the circular-mean derivative from phase_response and J=R-I. The current pressure realization is

p = -e L_W x + w_phi g(theta) - v L_U nu - w_topo L_U k,
Dg = J/pi,               x_dot = nu*p.

Here k is the fixed unique-support degree. Products between nodal vectors are componentwise. Set q=theta_dot/pi and a=nu_dot; these are supplied velocities, not new state primitives or selected laws. Differentiation gives

p_dot = -e L_W (nu*p) - e L_W_dot x + w_phi J q - v L_U a,
x_ddot = a*p + nu*p_dot.

The first term transports the current EPI rate. The second differentiates the normalized transport geometry. For d_i=sum_j W_ij and g_E_i=sum_j W_ij*(x_j-x_i)/d_i, it is

(-L_W_dot x)_i = [sum_j W_dot_ij*(x_j-x_i) - d_dot_i*g_E_i]/d_i.

Its zero-strength rows are zero within this fixed active-edge domain. The row-strength correction is necessary; differentiating only the weighted numerator changes the law. The phase derivative uses R, not the final IL/UM stage Jacobian, whose merge and gain have a different meaning. Capacity uses L_U, not L_W: zero-weight edges still participate in the capacity and phase channels, parallel edges count once in unique support, and self-neighbors count once. Fixed topology has zero derivative; this does not say its baseline pressure contribution vanishes. The product term a*p is essential even when capacity differences remain unchanged.

With [x]=[p]=X, [nu]=T^-1, [q]=T^-1, [a]=T^-2, the coefficient units are [e]=1, [w_phi]=X and [v]=X*T. Thus all pressure-rate terms have units X/T and both acceleration terms have units X/T^2. Expressing angular velocity in pi units preserves exact rational controls without pretending that represented float(pi) is exact pi or identifying phase speed with nu.

derive_joint_nodal_response centralizes this conditional identity. Optional conductance_rates align with the materialized effective-edge entries, including both directions; omission retains the fixed-conductance behavior. The returned transport is the shared derivative record. The joint result's computed epi_flow_pressure_rate and epi_geometry_pressure_rate sum to the complete epi_pressure_rate, without duplicating the transport formula. Input edges and their rates are associated before canonical ordering; both may be reordered together without changing the result. The shared transport derivative materializes them once and rebuilds the snapshot's untrusted cached fields. The joint observer verifies that the declared phase-reference neighborhoods match the unique support. Its first domain requires nonempty neighborhoods; isolates are rejected, not assigned a fictitious phasor mean. Coefficients are never renormalized. The supplied stored pressure is declared input. The observer does not verify that it is a refreshed canonical pressure, that the ideal cosine Gram belongs to the captured phases, or that the live wrap branch is regular. Tests obtain compatible control states through forcing_realization and separately check its fresh/stored/represented residuals. A returned identity is neither a derivative of binary64 arithmetic nor a runtime execution certificate.

The effective channel coefficients are held fixed after configuration; this observer never normalizes them. Changing coefficients adds their derivatives times the corresponding channels; support changes, operator jumps and REMESH history require their existing reset/event descriptions. Finite sampled capacity secants are not silently substituted for a smooth instantaneous a. A zero-conductance support edge cannot gain conductance through this fixed-active-edge derivative. Changing transport weights may also change tetrad metric distances when explicit lengths are absent; no fixed-metric tetrad claim follows from this identity.

10.2 Pressure-invisible motion can change form acceleration

At the same initial state and identical supplied conductance rates, differences between two supplied phase/capacity velocity pairs obey

delta(p_dot) = w_phi J delta(q) - v L_U delta(a),
delta(x_ddot) = nu*delta(p_dot) + p*delta(a).

Common rotation lies in ker J. A common capacity rate delta(a)=c*1 also leaves p_dot unchanged, but changes acceleration by c*p. Thus observing only pressure response does not identify the full response of form. If all p_i are nonzero, equality of both pressure rate and EPI acceleration forces delta(a)=0; for positive phase weight the remaining freedom lies in ker J. At zero-pressure coordinates that inference degenerates. In particular, at p=0, pressure tangency implies zero instantaneous EPI acceleration but does not prove that the trajectory remains on the zero-pressure set.

The existing two-completion P2 control in test_constitutive_capacity_scope.py now uses the shared owner instead of a separate acceleration calculation. Its identical initial state, tetrad and EPI rate still permit different accelerations. These are countermodels to a claimed implication, not two outcomes of one fully specified autonomous law.

10.3 Which phase-source motions can capacity compensate?

On connected undirected unique support with positive degrees d_U, range(L_U)={b: d_U^T b=0}. This follows from diag(d_U)L_U being the connected symmetric combinatorial Laplacian. For positive v and w_phi, a capacity velocity can cancel a supplied phase-source velocity precisely when

d_U^T J q = 0.

If solvable, v L_U a=w_phi J q determines a only up to a uniform vector. This is an algebraic instantaneous criterion; positivity boundaries and a physical evolution law add independent requirements. The joint source map [w_phi J, -v L_U] has rank n-1 if d_U^T J=0, and rank n otherwise: the capacity image already fills the codimension-one hyperplane, and a phase column outside it completes the range. Its tangent kernel therefore has dimension n+1 or n, respectively. Weighted conductance strengths cannot replace the unique-support degrees in this statement.

The undirected-support premise must be checked independently. Symmetric effective conductance alone is insufficient: reciprocal unit edges 0<->1<->2 plus a zero-weight arc 0->2 retain symmetric EPI transport, but their directed support has d_U=(2,2,1) and d_U^T(U-I)=(-1,0,1). The shared joint derivative correctly covers that snapshot; this undirected capacity-range theorem does not. Its portable regression prevents transferring the theorem merely from conductance admission. Likewise the existing derive_forced_support_balance solves a weighted EPI Poisson problem. A capacity reconstruction must reuse the exact algebra with the support operator and its own gauge, not pass the live weighted EPI problem unchanged.

At consensus R=U; cancellation is equivalent to w_phi*q+v*a being spatially constant. Away from consensus it can fail even strictly inside U3. On a three-leaf star with phases (0,0,0,alpha), cos(alpha)=3/5, sin(alpha)=4/5, the center row is R_0=(0,13/37,13/37,11/37) and each leaf row points to the center. Therefore

d_U=(3,1,1,1),        d_U^T J=(0,2/37,2/37,-4/37).

All edge gaps are below pi/2 and all resultants are nonzero. For q=(0,0,0,1) the weighted sum is -4/37, so no capacity velocity cancels this phase-source motion. This extends the same strict-star geometry used by the reciprocal-closure audit without assuming a variational law or adding a controller.

There are also exact finite compatible families. In one fixed normalized structural unit on unit P2, choose e=1/2, w_phi=v=1/4, w_topo=0, and

x=(1/4,0),  theta(t)=pi*t*(1,-1),  nu(t)=(1-t,3/2+t),  |t|<1/4.

Singleton means give g=(-2t,2t) and all pressure channels sum to zero for the whole interval. Capacity remains positive and U3 strict, so the constant nonuniform form satisfies the nodal equation along this supplied family. Holding capacity fixed was essential when applying the earlier phase-only rigidity results to a held total source; joint compensation supplies another possibility. The family does not select its own angular speed or capacity law, establish stability, or explain its preparation. It is a finite compatibility control, not autonomous emergence.

10.4 Full tetrad and the remaining closure obligation

The same primitive dependencies must be carried into the tetrad. In the table, delta_ij=wrap(theta_j-theta_i) retains the oriented separation:

Field	Conditional response and additional domain
Phi_s	At fixed metric, Phi_s_dot=B_G*p_dot. Actual telemetry reads stored pressure; refresh and metric provenance remain necessary. Moving metric adds B_G_dot*p.
Phase gradient	At nonzero wrapped gaps away from the cut, its derivative is the neighbor mean of sign(delta_ij)*(theta_dot_j-theta_dot_i). At zero gaps, the absolute value generally has only directional derivatives.
K_phi	On its regular branch, K_phi_dot=(I-R)*theta_dot=-pi*J*q. Resultant and wrap boundaries remain explicit.
xi_C	The implemented fit uses static pressure-coherence products and metric distances, not a phase autocorrelation. Differentiating it needs absolute-pressure, sample/bin and estimator-branch conditions; the spectral fallback is a different observation.

Consequently a smooth pressure derivative is not automatically a smooth whole-tetrad evolution, and pressure-invisible phase motion can still change local phase observations. The source tangency identity and reciprocal variational requirements constrain a proposed completion; neither selects the missing velocities. The shared exact controls are in test_joint_nodal_response.py. The single execution plan records the open closure obligation.

10.5 Geometry response is an input to closure, not its selection law

The integrated smooth-conductance response reuses the transport derivative already present in the repository. It closes an accounting omission in the joint observer, not the missing evolution law. Positive weights, fixed incidence, a regular phase chart and declared differentiable rates remain premises. No connection intensity is chosen to create a desired response.

A common instantaneous conductance scaling W_dot=c*W gives L_W_dot=0: numerator and row-strength effects cancel. Yet its explicit Dirichlet work is c*E_D, which can be positive. Thus a rise in this geometry-dependent energy need not alter the EPI pressure or mean acceleration. Nonuniform edge changes can instead alter both, even when phase and capacity velocities vanish. Full pressure and acceleration, not an energy sign alone, identify which channel supplies the response.

One exact control uses unit P3, x=(3/2,1,0), nu=(1,2,4), consensus phase, e=1/2, w_phi=v=1/4 and zero topology weight. Initially p=0 because x=2*1-nu/2. A declared unit rate on edge 0--1, zero on 1--2, with zero phase/capacity rates gives p_dot=(0,3/16,0) and x_ddot=(0,3/8,0), so the ordinary mean acceleration is 1/8. This separates instantaneous zero pressure from invariance under changing conductance. It supplies no law causing that edge rate.

The joint geometry controls differentiate the pressure expression independently, retain the full mean and product rule, and check compatibility with the zero-rate default. The fixed-conductance binary64 Dirichlet observer separately rejects any represented asymmetry. Its former approximate symmetry admission could report a zero identity residual although its displayed energy derivative was different; the boundary regression now preserves that distinction. Numerical energy diagnostics remain different from the exact rational transport budgets.

The conservation, variational and scale-reduction owners provide no missing rate selection here. Their residuals, supplied Hamiltonians and projected fine-model rates depend on an already specified evolution. Using their outputs to invent that evolution would reverse the dependency. The single G3 plan records their ownership and keeps autonomous maintenance unresolved.

Historical source-bound validation record: artifacts/research/joint_geometry_admission_validation_2026_09_19.json.

10.6 Pressure as an observation or a sufficient state coordinate

The previous controls distinguish possible completions and supplied changes. There is also a pressure-only closure obstruction within one fixed complete law. Use the native relational exchange model on unit P2 with equal held capacity nu>0, effective pressure coefficients e,w>0, storage scale beta>0, and a lifted acute phase difference abs(delta)<pi/2. There are no inputs, events, clipping or support changes. With the two nodes numbered 1 and 2, write

\[ d=x_2-x_1,\qquad \delta=\theta_2-\theta_1,\qquad a=\frac w\pi,\qquad H(\delta)=\pi\operatorname{sinc}\delta>0, \qquad p=p_1=-p_2=ed+a\delta. \]

Here sinc(delta)=sin(delta)/delta, extended by one at zero. The full law gives the exact relative-state equations

\[ \dot d=-2\nu p,\qquad \dot\delta=\frac{2\nu w d}{\beta H(\delta)},\qquad \dot p=-2e\nu p+\frac{\kappa d}{H(\delta)},\qquad \kappa=\frac{2\nu w^2}{\beta\pi}. \]

Choose any nonzero acute delta and d=-a*delta/e. Both nodal pressures and both form rates then vanish, but

\[ \dot p=-\frac{2\nu w^3\delta}{\beta\pi^2eH(\delta)}\ne0. \]

At consensus d=delta=0, the same pressure vector (0,0) has zero pressure rate and remains stationary. Therefore no autonomous first-order map of the instantaneous pressure vector alone can reproduce both states, even with support, capacity, coefficients and clock fixed. The difference comes from retained phase/form geometry, not an unspecified external phase velocity. The compensated state has positive joint storage E=d^2/2+beta*(1-cos(delta)) and instantaneous loss E_dot=-2*e*nu*d^2<0: phase changes while form is instantaneously stationary. Zero net pressure is a cancellation of channels, not absence of structure.

This does not exclude pressure coordinates. Keeping one additional relative coordinate gives an exact alternative chart:

\[ (d,\delta)\longleftrightarrow(p,d),\qquad \delta=\frac{p-ed}{a},\qquad \dot d=-2\nu p,\qquad \dot p=-2e\nu p+ \frac{\kappa d}{H((p-ed)/a)}. \]

The transformation is invertible on the admitted domain abs((p-ed)/a)<pi/2, since a>0. These are the existing equations in different coordinates, not a new pressure law or primitive variable. On this equal- capacity P2 the common form and lifted phase means are constant; retain their initial values too when reconstructing absolute nodal state. The two displayed coordinates describe only the relative state.

Alternatively, eliminating the retained form difference gives exact pressure memory. Set

\[ d_p(t)=d(0)-2\nu\int_0^t p(s)\,ds. \]

Substituting d=d_p(t) into the displayed pressure row produces a closed history-dependent equation on its admitted domain. The initial hidden value d(0) must remain supplied: p(0) cannot determine it. This is the same hidden-state obligation as in the joint memory construction, here with an explicit one-coordinate elimination. It is not a memory-free pressure law, a claim that one hidden scalar suffices on arbitrary networks, or a reconstruction of pressure from an evaluated form derivative.

The substrate and scale distinction still applies. A uniform form origin, pressure cancellation and an empty support are different statements. This calculation neither identifies a physical vacuum nor derives nodes, support or an external source from pressure alone. The focused native controls check the cancellation and sufficient-coordinate identities against the shared evaluator, with represented arithmetic kept separate from this ideal continuous derivation.

11. Finite joint phase-capacity source compatibility

The instantaneous condition in section 10.3 has a finite algebraic counterpart without choosing velocities. Fix a connected reciprocal unique support, with nonempty neighbor rows, and fixed channel coefficients in one declared unit system. Let C_ij=1 when j is in row i, d_i=|N(i)|, D=diag(d), B=D-C and L_U=D^-1 B. Self-neighbors count once and parallel neighbors once; zero-conductance support edges remain present. Symmetry of positive EPI conductance does not establish reciprocity of this support.

For an oriented phase-pressure vector g, held non-EPI source f, w_phi,w_topo>=0 and v=w_vf>0, the finite level equation is

\[ f=w_\phi g-vL_U\nu-w_{\rm topo}L_Ud. \]

This is a source equation, not the nodal time-evolution law. A zero total pressure additionally requires f=w_epi*L_W*x. The existing weighted EPI balance owner solves a different problem with a different invariant measure.

11.1 Necessary and sufficient condition and all capacities

Define r=w_phi*g-w_topo*L_U*d-f. Then

\[ \exists\nu:\ vL_U\nu=r \quad\Longleftrightarrow\quad d^\top r=0 \quad\Longleftrightarrow\quad d^\top(w_\phi g-f)=0. \]

Proof. Reciprocal connected support makes B a symmetric positive semidefinite Laplacian with kernel span{1}. Thus B*nu=D*r/v is solvable exactly when its right-hand side sums to zero. Also d^T L_U=0, so the fixed topology channel changes the solution profile but cannot remove an incompatible weighted phase-source mean. This includes a self-supported singleton; an isolated node is outside the stated domain.

There is exactly one support-degree-centered solution z, and all solutions are

\[ z=(vB+dd^\top)^{-1}Dr,\qquad d^\top z=0, \qquad \nu=z+c\mathbf1. \]

The displayed rank-one term fixes a computational coordinate in the declared unit system; it is not a new physical interaction or source coefficient. Its matrix is positive definite because the only null direction of B is not orthogonal to d. Multiplication by 1^T proves d^T z=0, after which the solve reduces to the original equation. An incompatible residual is reported without a profile. Subtracting its mean would change the held source and would answer a different question.

The uniform coordinate c leaves this source unchanged, but it is not a physical equivalence: away from zero pressure it changes xdot=nu*p by c*p. Neither the centered representative nor a minimum-norm solution is a derived capacity-selection law.

11.2 Positive capacity and declared bands

Strict positivity requires c>-min(z). For optional finite closed nodewise bands 0<=ell_i<=nu_i<=u_i, put

\[ L=\max_i(\ell_i-z_i),\qquad U=\min_i(u_i-z_i),\qquad a=-\min_i z_i. \]

The allowed shifts are [L,U] intersected with (a,infinity). Since ell_i>=0, L>=a: the lower endpoint is included precisely when L>a. The intersection is nonempty exactly when L<=U and U>a. Missing upper bounds mean no upper limit; every compatible finite profile then admits positive capacities. A closed singleton above a is allowed; the singleton at a is not. These are supplied admissibility bands, not inferred runtime clipping rules or forward-invariant bounds.

11.3 Exact compatible P2 family and strict-U3 obstruction

The P2 family in section 10.3 supplies a compatible control for the entire interval |t|<1/4. Here t labels configurations; assigning it a physical clock would be an additional choice. Its data and complete reconstruction are

w_epi=1/2,  w_phi=v=1/4,  w_topo=0,
x=(1/4,0),  theta/pi=(t,-t),  g=(-2t,2t),  f=(1/8,-1/8),
z=(-1/4-t,1/4+t),  nu=z+c*1.

The choice c=5/4 recovers the previously supplied nu=(1-t,3/2+t); it is not selected by the theorem. Bands [3/4,7/4] on both nodes give c in [1+t,3/2-t], which contains 5/4 throughout the strict domain. Singleton neighbor means give the exact displayed g, all edge gaps are below pi/2, and f=w_epi*L_W*x, so total pressure is zero. The theorem describes a finite set of possible states, not their preparation or motion.

A three-leaf star gives a contrasting exact obstruction with rational phase-pressure coordinates. Set

\[ \theta/\pi=(0,1/3,1/3,-1/3),\qquad d=(3,1,1,1). \]

The center's neighbor sum is 3/2+i*sqrt(3)/2, whose argument is pi/6; each leaf sees the center's zero phase. Therefore

\[ g=(1/6,-1/3,-1/3,1/3),\qquad d^\top g=1/6. \]

Every edge gap is pi/3<pi/2, all resultants are nonzero, and the oriented wrap charts are regular. Yet for f=0 and w_phi>0, the residual is w_phi/6, so no capacity vector exists. Changing the positive capacity weight, uniform capacity, positivity bands or topology weight cannot fix it. U3 admission is therefore insufficient for finite source compensation. This does not rule out other source levels or other constitutive models.

The phase realizations above are analytic constructions, not rounded trigonometric tests. Reflection preserves a cosine Gram and its response matrix while reversing g; a Gram alone cannot authenticate the oriented source used in this theorem.

11.4 Shared implementation, tetrad and remaining freedom

derive_phase_capacity_balance in phase_response.py reuses snapshot rebuilding, the unweighted support-gradient owner, exact scalar/vector admission and the existing rational inverse. It explicitly checks reciprocal connected support, reconstructs derived snapshot fields, returns exact compatibility/equation/centering residuals, and reports the full shift interval. Declared rational inputs remain rational; represented real inputs are treated as their exact binary64 values. The theorem also holds for exact real data, but this executable owner does not manipulate arbitrary transcendental phase sources. It neither certifies a phase chart nor writes a graph or chooses a trajectory.

The existing capture_non_epi_forcing / decompose_non_epi_forcing path can supply its represented source and coefficients. The resulting centered profile recovers captured capacity minus its unique-support weighted mean. That integration check is exact algebra on captured data; phase realization, stored-pressure freshness and binary64 kernel defects keep their existing separate scope. The portable controls are in test_phase_capacity_balance.py.

All tetrad fields retain the dependencies in section 10.4. In particular, on the exact zero-pressure P2 family with fixed graph-distance metric, Phi_s=0 while the phase gradient is 2*pi*abs(t) and curvature is (2*pi*t,-2*pi*t). The static pressure-coherence inputs and distances used by xi_C do not change; availability and fit/fallback semantics still belong to that observer. Thus constant pressure does not imply constant full tetrad.

Finite level membership is stronger than tangent cancellation at one point. The existing higher-order escape example in source tangency and the phase-only double-star obstruction remain necessary boundaries. Even finite compatible families leave angular motion, uniform capacity, source evolution and support evolution unselected. They do not prove autonomous generation, attraction, stability or a physical particle. The single execution plan carries the next constitutive-closure gate.

The restoring-contribution test now separates this finite compatibility from a conditional joint energy minimum. It derives the missing capacity slope and curvature requirements without choosing their physical origin. A source-compatible P2 perturbation can leave all four tetrad read-outs unchanged while changing that potential; neither identical diagnostics nor zero pressure closes the autonomous laws.