Pressure premises and constitutive response

Locality, covariance, phase-domain boundaries and the prospective pressure comparison.

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.

4. What locality and symmetry can derive

Consider only an affine local EPI pressure on a finite loopless support:

\[ P_i(x)=b_i+\sum_j A_{ij}x_j, \qquad A_{ij}=0\ \text{off support for }i\ne j. \]

Uniform-shift invariance gives A*1=0. Requiring every uniform form to be an equilibrium gives b=0. Consequently

\[ P_i(x)=\sum_{j\ne i}a_{ij}(x_j-x_i). \]

The local maximum principle for every x is equivalent to a_ij>=0: the sufficiency follows term by term; for necessity set x_i=0, one selected neighbor x_j=-1, and all other coordinates zero. This derives a possibly directed diffusive generator under those premises. It does not select its gains.

Reciprocity needs an additional positive measure h satisfying h_i*a_ij=h_j*a_ji. Then W_ij=h_i*a_ij is symmetric. With unit off-diagonal row sums, d_i=h_i and P=-L_rw*x. Undirected support is insufficient: a triangle with forward rates 2/3 and reverse rates 1/3 has symmetric support but violates detailed balance, because the two cycle products are 8/27 and 1/27.

There is a restricted way to remove conductance ratios without fitting them. If reciprocal conductances depend only on support and are invariant under every support automorphism, they are constant on each undirected edge orbit. On a connected edge-transitive graph with nonzero conductance all conductances are one common positive value; row normalization removes it and yields the unweighted random walk. Multiple edge orbits leave ratios undetermined. This is a genuine consequence of the added symmetry premises, not a derivation of why a network must have that symmetry or why it must emerge.

Fixed phase, capacity and topology sources instead give b=F; the full multichannel pressure need not annihilate a uniform EPI field or satisfy a maximum principle. Removing those sources would change the model. Self-loops also change normalization without contributing an EPI difference and require their own convention. The portable covariance controls test these restricted matrix identities alongside the existing support owner.

4.1 Exact low-degree reduction and its nonlinear boundary

On a loopless support of maximum degree two, suppose all phases have a common real lift of width strictly below pi. A singleton neighbor has its own phase; the circular mean of two neighbors is their lifted arithmetic midpoint. The displacement from the node remains in the same regular branch. Consequently g_phi=G_U theta/pi exactly, without a small-angle approximation. If EPI uses the same row-normalized support, then

\[ p=G_U\left(e x+\frac{a}{\pi}\theta+b\nu+t k\right). \]

This explains exact compensation families on paths and common-chart cycles. It does not extend to arbitrary degree, nonmatching transport, or winding configurations without a common chart. With undirected support the displayed reduction has sum_i k_i*p_i=0; the nonlinear phase law has no such general support-degree conservation identity.

For an explicit boundary, take unit-conductance K4, phases (0,0,0,pi/3), uniform EPI and common positive capacity. Every edge is strictly U3-compatible at the configured pi/2 threshold, and every resultant is nonzero. Put beta=atan(sqrt(3)/5). Direct phasor addition gives

\[ g_\phi=\frac1\pi(\beta,\beta,\beta,-\pi/3),\qquad \sum_i g_{\phi,i}=\frac{3\beta-\pi/3}{\pi}<0. \]

Indeed tan(3*beta)=9*sqrt(3)/10<sqrt(3)=tan(pi/3), with both angles in (0,pi/2). The other three gradients vanish, so any positive phase coefficient gives a negative instantaneous EPI mean rate. Reflecting all phases reverses this rate. Oddness under reflection therefore does not make the phase channel an antisymmetric pairwise flux. The compact pressure controls check this analytic result and the low-degree reduction against actual materialized pressure, with explicit floating-point tolerance.

This is a constitutive boundary, not a demonstrated conservation bug. The regular phase metric already relates this phase reading to a cosine energy with a state-dependent diagonal metric; that metric generally differs from the transport degree metric. Neither identity supplies phase motion or a conserved EPI total. Replacing the phasor law by a conservative edge flux would require an independent conservation premise and would change the model. Unit covariance, joint dynamical closure and the identity intended to persist must be assessed before selecting such a replacement.

4.2 When scale covariance and regularity force linear form response

The affine premise in section 4 can itself be derived under stronger, explicit assumptions. Fix the graph and all non-EPI state. For a source-free pressure Q on a real form chart, suppose

\[ Q(x+c\mathbf1)=Q(x),\qquad Q(a x)=a Q(x)\quad(a>0), \]

whenever the arguments are admitted, and suppose Q has a Frechet derivative at an interior uniform state. Work in a translated chart containing an open neighborhood of zero and the segment from zero to each form being considered; shift invariance transfers the derivative to zero. Homogeneity gives Q(0)=0. For every admitted v,

\[ Q(v)=\lim_{\epsilon\downarrow0}\frac{Q(\epsilon v)-Q(0)}{\epsilon} =DQ(0)v. \]

Thus Q is exactly linear on that domain, not merely approximately linear near equilibrium. Locality then restricts its matrix support, and the maximum principle from section 4 fixes the off-diagonal signs. The resulting family is Q_i=sum_j a_ij*(x_j-x_i), a_ij>=0. This does not select those coefficients, reciprocity, row normalization or the active support. Differentiability means one linear total derivative; separate directional derivatives are insufficient.

Row normalization can be justified by another explicit premise. Suppose this linear row is anonymous and unweighted: only the center and neighbor multiset are available, with no other structural attributes distinguishing neighbors. Permutation symmetry then gives Q_k=c_k*sum_j(x_j-x_i). Require invariance when every neighbor observation is repeated m times. It follows that m*c_(mk)=c_k; taking k=1 yields c_k=c_1/k. The arithmetic mean is thereby selected up to the common single-neighbor gain c_1. Fixing that gain defines a normalization, not a newly derived physical rate. Repetition of observations is an added premise about this row, not a theorem about cloning physical nodes, adding parallel transport channels or changing topology. Weighted rows still need a meaning and aggregation rule for their conductances.

Exact amplitude covariance is an added scale-free constitutive premise. It is stronger than correctly converting units while also transforming dimensional parameters. EPI-offset invariance is justified only for a channel whose chosen form origin is redundant; it is not a theorem about every operator or a physically distinguished vacuum. For the mixed pressure, this argument can apply to Q(x)=P(x)-P(0) at held non-EPI state if its premises hold. It does not determine the independent source P(0) or eliminate that source.

Regularity is essential. With gaps d_ij=x_j-x_i, the comparison response

\[ Q_i^{\rm cmp}(x)= \begin{cases} \displaystyle\frac{\sum_{j\in N_i}d_{ij}^3}{\sum_{j\in N_i}d_{ij}^2}, &\sum_jd_{ij}^2>0,\\ 0,&\text{otherwise} \end{cases} \]

uses only existing form differences, has no new dimensional parameter, and satisfies locality, relabeling, affine form-chart covariance and the maximum principle. It is continuous at uniform form since its magnitude is bounded by the largest neighbor gap. However, at the center of a two-leaf star, directions with gaps (1,0) and (0,1) each give one, while their sum (1,1) also gives one. Its directional derivative at uniform form is not additive, so no total derivative exists there. This is a countermodel to uniqueness from covariance and continuity alone, not an installed pressure law. The existing covariance controls compare it with the shared exact EPI owner.

There is also a useful stationary boundary independent of linearity. On fixed finite connected undirected support with positive capacities, assume every local form maximum with a strictly lower neighbor has strictly negative source-free pressure. A nonuniform stationary form would have a maximum plateau with a boundary node adjacent to a lower value, contradicting its zero pressure. Hence only uniform stationary form is possible. Nonlinearity alone cannot evade this obstruction while keeping that strict response premise. Zero capacity, disconnected support, zero response on unequal neighbors or an additional source changes the hypotheses. This is not a convergence theorem or an exclusion of phase identity, moving patterns or finite-lived regions. For directed support the same plateau argument needs strong connectivity; weak connectivity alone permits distinct terminal values.

4.3 Phase domain, orientation and a discriminating structural response

The circle supplies periodicity, not a unique real-valued pressure. On regular branches the implemented g_phi respects common rotation and node relabeling. It is odd under internal phase conjugation theta->-theta. That conjugation is different from a graph permutation representing a spatial reflection. Holding x,nu,G,W fixed gives P(-theta)-P(theta)=-2*a*g_phi(theta). Consequently it is not a redundancy of the full scalar-form law when a>0. If phase orientation is declared merely a coordinate convention with unchanged scalar form and pressure, its coefficient must also transform as a->-a. Otherwise conjugation describes a distinct model state. No symmetry claim is complete without the transformation of every state, output and coefficient.

Two separate global-domain obstructions are explicit:

• On P2, the ideal phase pressure is wrap(delta)/pi. Its limits at an antipodal gap are +1 and -1, although the neighbor resultant is nonzero. No continuous real-valued extension matches both limits. Moreover a single-valued periodic odd response must satisfy f(pi)=0, since pi and -pi denote the same circle point. Signed binary64 ties are a representation convention, not a globally odd circular law.
• With center phase zero and two neighbors at 0 and pi±epsilon, the neighbor resultant tends to zero. The pressure tends to -1/2 and +1/2 respectively. Here the limiting target directions are not antipodal to the center. No value assigned at zero resultant makes this response continuous.

These controls do not obey a strict U3 edge domain and are not evidence against the regular-domain theorem. A model may restrict its domain, retain a lift or extra state, define an event rule, or select a different response; each is an additional obligation. The pressure reader's numerical fallback supplies none of those dynamical justifications. The existing regular phase metric and its current/curvature identity remain the owner of local geometry:

\[ J_{\phi,i}=\frac{|S_i|}{k_i}\sin(\pi g_{\phi,i}). \]

On the regular non-antipodal reciprocal domain, both readings have the same alignment potential V_phi, but different metrics: g_phi=-H_phi^-1*grad V_phi with H_phi,ii=pi*|S_i|*sinc(pi*g_phi,i), while J_phi/pi=-(pi*diag(k))^-1*grad V_phi. The common potential therefore does not select the response metric or a phase-to-form coupling. Equivalently, J_phi/pi=(|S_i|/k_i)*sinc(pi*g_phi,i)*g_phi,i is a nodewise attenuation on that domain. This does not order total EPI source work: nodal work factors can have different signs and the attenuation is not spatially uniform. The existing forcing/Dirichlet balance remains necessary for that comparison.

This identity exposes a concrete selection question: should form pressure read only the resultant direction, as the present phase channel does, or also its amplitude? Both readings already exist in the repository; neither is a phase clock. Matching their coherent linear response does not answer the question. For neighbor gaps epsilon*a_j, let m, v and c3 be respectively the mean, variance and central third moment of the unscaled a_j. On the regular branch near consensus,

\[ g_\phi=\frac{\epsilon m-\epsilon^3c_3/6}{\pi}+O(\epsilon^5),\qquad \frac{J_\phi}{\pi}= \frac{\epsilon m-\epsilon^3(c_3+3mv+m^3)/6}{\pi}+O(\epsilon^5). \]

The first possibly nonzero discriminating coefficient is (m^3+3*m*v)/(6*pi) at cubic order; it vanishes for m=0. For two neighbors with gaps m±d, the comparison is exact: phase pressure is m/pi while the matched-gain current is sin(m)*cos(d)/pi, for |d|<pi/2 and an admitted principal mean m. The production controls use m=pi/8, spreads pi/16 and 3*pi/16, and center phase pi/4. All edge gaps are strictly acute. The phase pressure stays fixed while the existing current changes; no coupling coefficient is fitted and no alternative dynamics is installed. These are model-discrimination controls, not laboratory evidence selecting either response. Reflection symmetry or agreement to first order likewise does not select a unique smooth periodic response function. The complete exchange comparison now derives the associated phase row, global continuation and work balance. It also shows why a bounded phase-only repair cannot preserve global passivity with native Arg pressure and cosine storage. Normalized pairwise superposition selects the sine response within an explicit comparison class; it is not a consequence of circular phase alone or a default pressure change.

Prospective finite-response discriminator

The same P3 preparation gives an exact-real prediction for this bounded comparison. Keep unit conductances, common fixed capacity kappa>0, common initial scalar form, no Gamma and held phases (c+m-d, c, c+m+d). Require abs(m)+abs(d)<pi/2. Let a>0 and e>0 be the same effective phase and EPI coefficients in both models. The comparison source replaces only g_phi by the already defined J_phi/pi; doing so is a declared constitutive hypothesis, not an existing default engine law.

For spreads d_1,d_2, the topology source is identical and the capacity gradient is zero. Thus those full-mixture channels cancel in the difference of the two form responses, even though the P3 topology source itself need not vanish. The present Arg source has center m/pi and endpoint differences that are antisymmetric under path reflection. This odd subspace is invariant under L_rw, so the center difference is zero. For the comparison source, the even part is proportional to (-1,1,-1), a L_rw eigenvector of eigenvalue two. Integration of that forced mode yields

x_center^Arg(t; d_2) - x_center^Arg(t; d_1) = 0,

x_center^J(t; d_2) - x_center^J(t; d_1)
    = a*sin(m)*(cos(d_2)-cos(d_1))/(2*pi*e)
      * (1-exp(-2*kappa*e*t)).

The odd component of the current source likewise cannot reach the center in this fixed linear transport model. Nonzero sin(m) and distinct cosines therefore supply a finite-response discriminator at every positive horizon. This is a paired exact-real prediction with held supporting coordinates, before clipping or numerical stepping. It is not an absolute-response formula with the topology channel silently omitted. A numerical comparison must check the declared unclipped interval and separate pressure realization, integration and rounding defects. A frozen second spread is a numerical prospective control, not an independent physical observation.

Executed fixed control. The shared-engine experiment now evaluates four trajectories with common initial EPI 1/2, capacity one, c=pi/4, m=pi/8, spreads pi/16 and 3*pi/16, and the complete normalized mixture (phase, EPI, capacity, topology)=(1/4,1/2,1/8,1/8). It refreshes pressure before each of 256 shared Euler calls with h=1/256, holds phase and support, and declares Gamma absent. Only the alternative phase source is substituted. The nonzero topology contribution remains in each trajectory; the initial total central Arg pressure is -3/32, despite its positive phase contribution. This prevents a phase-only interpretation of the absolute motion.

The declaration and predictions are written and hashed before evolution. For the even mode, Euler replaces exp(-2*kappa*e*t) by (1-2*kappa*e*h)^256. The signed central response below is wide minus narrow:

Phase source	Continuous prediction	Euler prediction	Engine observation
Arg direction	0	0	-5.55e-17
J_phi/pi	-0.002874319847547102	-0.002877592338102031	-0.002877592338101642

For the current source the analytic integration defect is about -3.27249e-6, whereas engine minus discrete prediction is about 3.89e-16. Thus the measured difference is resolved independently of the known Euler error. The fixed budgets are 2e-14 for pressure realization, 2e-15 for one represented-step rounding defect, 1e-11 for the discrete trajectory and 5e-6 for the paired continuous response. All pass. Positive averaging with source bound 0.203125 gives the analytic enclosure [0.296875,0.703125]; raw represented proposals and accepted values also remain within the predeclared open interval (0.25,0.75), inside the hard clip [0,1].

The regression controls also compare against an independent augmented matrix power and check evidence hashes, full-mixture retention and rejection of changed declarations. Run python -m tnfr.research.phase_form_response --output <fresh-directory> to produce the declaration, trajectories and evidence sidecar. This is local prospective recording with binary64 analytical evaluations and selected error budgets, not an interval certificate or external preregistration. It establishes the predicted finite implementation distinction; both constitutive hypotheses remain possible until an independent reduction or physical measurement criterion selects the information their pressure must preserve.

4.4 Constitutive admission ledger and remaining choices

The nodal equation fixes a typed rate product. The following ledger prevents extra constitutive conditions from being promoted to consequences of that product. Its mathematical owners above replace parallel research campaigns.

Requirement	What it justifies	What remains open or conditional
Relabeling covariance when node names are bookkeeping	Permuting all nodal fields and support must permute pressure	It does not select an aggregation, graph, coefficient or actual state symmetry.
Phase periodicity and common rotation with no supplied phase reference	Dependence on circular relative configuration on the admitted domain	Neither the Arg rule, phase orientation, behavior at cancellation nor a phase clock is selected.
Form-chart and time-unit covariance	The coefficient transformation in section 3	Fixed normalized numeric weights are not a physical unit law; a vacuum may invalidate form-offset redundancy.
Source-free exact amplitude covariance plus regularity at uniform form	The linear family proved in section 4.2	These are added premises; they do not constrain an independently supplied source or forbid nonlinear models with another scale/domain.
Locality plus a local maximum principle in that linear family	Nonnegative neighbor-difference coefficients	Nonlinearity, reciprocity, row speed and the support must be addressed separately. The full sourced pressure does not obey this form-only maximum principle.
Reciprocal edge response with a declared measure	Detailed balance and the corresponding Dirichlet identities	Undirected support alone does not suffice. A weighted EPI charge is not thereby conserved in the presence of other channels.
Anonymous linear row plus whole-neighborhood observation replication	Selects the arithmetic mean up to a common gain	Replication is an additional premise, not graph cloning. More general row gains can be absorbed into effective capacity for isolated EPI diffusion; replacing primitive capacity everywhere changes its other laws.
Regular phase-domain closure or a declared boundary rule	A defined continuation when a trajectory reaches branch/cancellation boundaries	Current-state U3 admission and a numerical fallback do not establish future domain invariance.
Full-state identity, source evolution and event laws	A complete candidate for maintenance or formation	Well-posedness still needs a domain and regularity argument. A pressure formula, grammar admission or tetrad snapshot alone does not supply the complete laws.

For reciprocal transport, positive capacity and held coefficients, the actual mean balance is d/dt sum_i(d_i/nu_i)*x_i=sum_i d_i*F_i, with the metric also held fixed. It is an accounting identity, not a requirement that the right side vanish. With W,nu,e,F fixed and a compatible stationary target x*, the difference x-x* obeys homogeneous diffusion even though absolute EPI does not. Reuse the forced-support owner, rather than rejecting the full pressure merely because a source-free property fails.

The present pressure is therefore retained as a configured, regular-domain map, with a well-characterized EPI sector and explicit source choices. Neither its normalized coefficients nor its use of phase direction alone is uniquely derived. An unresolved constitutive distinction is what information a derived phase-to-form response must preserve: direction, resultant amplitude and its source work, with a declared phase domain and orientation. This must be fixed before promoting an alternative to dynamics. The existing metric, joint derivative and discrimination controls provide the route; another arbitrary potential, numerical trajectory or fitted force cannot remove the freedom.

4.5 Regular pressure on a state with derived phase

The S/D reference admission distinguishes independently retained primitive phase from the orientation of a regional form contrast. This distinction matters at zero contrast even when every positive-amplitude phase lies well inside an acute chart. The directed two-region lift already gives a concrete obstruction and complete mean/contrast budgets. The following admission results classify the obstruction and the remaining constitutive freedom; they do not install another pressure or select a mechanism to sustain a pattern.

State and equivalence. Write the regional contrasts as z in C^r, with means mu, capacities, support, interfaces and reporting frames retained. Complex notation represents real form coordinates. In this section these other data are fixed parameters. A real scalar source q(z;mu,metadata) is an additional regional mean-pressure contribution, not an angular velocity or a complete nodal law. Its contribution to mean velocity also carries the declared capacity. If q is to be an observation on the admitted Gram state Q=z*z^dagger, it must satisfy

\[ q(e^{i\alpha}z;\mu,\mathrm{metadata}) =q(z;\mu,\mathrm{metadata}). \]

This is an active common modal rotation at fixed frames, expressing equality of the selected observations. It is not a symmetry imposed on every TNFR form law. A passive reporting-frame change instead transforms the complex coordinates and interface matrices together. A regional relabeling likewise permutes the whole source vector and its metadata; it does not leave each indexed scalar unchanged.

An amplitude-independent phase source cannot retain directional response at a vanishing contrast. Consider two regions with r_A,r_B>0 and a regular relative-phase interval I. Suppose the proposed scalar source is f(delta) throughout this domain, independently of the two amplitudes. Fix nonzero z_B=R*exp(i*beta) and choose any delta_1,delta_2 in I. The paths

\[ z_A^{(j)}(\epsilon)=\epsilon e^{i(\beta-\delta_j)},\qquad \epsilon\downarrow0, \]

have the same limiting real form and retained data. Their source limits are f(delta_1) and f(delta_2). A continuous extension at (0,z_B) therefore requires f to be constant on I. If I is symmetric and f is additionally odd, that constant is zero. The conclusion requires both paths to belong to the admitted domain; a model excluding this boundary has a narrower obligation.

For the existing two-triangle lift, f(delta)=delta/(2*pi): the choices delta=0,pi/6 yield limits 0,1/12, or mean-rate contributions 0,nu*w/12. The incompatibility occurs away from antipodal or zero-resultant primitive phase configurations. Replacing Arg by a nonconstant amplitude-independent sine response retains this zero-contrast obstruction. An assigned angle at zero changes neither limiting value. Amplitude attenuation would change the source away from zero; it is a new constitutive choice, not a continuous completion agreeing with the original rule.

There is no corresponding contradiction for reference S with independently stored theta: these paths approach different retained phase states there. Its own primitive-resultant and wrap boundaries still require the section 4.3 admission. Retained justified history can also distinguish approaches, but then the history belongs to the state; it is not a form-only closure.

Differentiability plus scale-free contrast response excludes an extra rotation-invariant mean source. Let the contrast domain contain an open neighborhood of zero and the radial segments to all considered states, and be invariant under common modal rotation. At fixed means and metadata, assume q has a real Frechet derivative at zero and

\[ q(\lambda z)=\lambda q(z)\quad(\lambda>0). \]

The section 4.2 argument gives q(0)=0 and q(z)=Dq(0)z wherever these radial scalings are admitted. Invariance under the rotation alpha=pi also gives q(-z)=q(z), whereas a real linear derivative gives Dq(0)(-z)=-Dq(0)z. Hence

\[ \boxed{q(z)=0.} \]

This is a theorem about this additional scalar source and these premises. It does not set the contrast-vector dynamics to zero: complex-linear transport obeys F(e^{i alpha}z)=e^{i alpha}F(z), not scalar invariance. The existing all-state linear criterion already determines when mean/contrast separation and a complex-linear generator yield a closed Gram law, including zero. Its inherited transport and complete budgets remain available without an added mean source.

The homogeneity assumption here scales contrasts while holding means fixed. It is not automatically the homogeneity of the entire fine form. Applying the section 4.2 argument to a complete mean/contrast state can leave linear mean-dependent terms; they must not be erased by this contrast-only result. For a source with nonzero value at zero contrast, the result applies to its contrast-dependent difference only if that difference meets the premises. Smoothness at zero, exact homogeneity and the chosen Gram equivalence are additional admissibility decisions, not consequences of the nodal product.

Two countermodels expose the freedom. They are mathematical comparisons, not proposed engine mechanisms. Let a supplied dimensionless interface orientation tau_AB have unit magnitude, and put s_AB=Im(conjugate(z_A)*tau_AB*z_B). With the reference D convention U_a_new=U_a*R(beta_a), reporting coordinates transform as z_a_new=exp(-i*beta_a)*z_a. Transform the interface as tau_AB_new=exp(i*(beta_B-beta_A))*tau_AB; s_AB is then unchanged. For aligned matched frames the existing interface convention gives tau_AB=1. This datum must not be silently reset after changing frames.

• q=kappa*s_AB is smooth, common-rotation invariant and zero when either contrast vanishes. It is quadratic, not degree-one homogeneous. For pressure units X, kappa has units X^-1. A form-unit conversion z_new=lambda*z transforms kappa_new=kappa/lambda, so the pressure converts correctly. Correct unit covariance does not imply fixed-kappa homogeneity or derive kappa's value.
• q=s_AB/sqrt(abs(z_A)^2+abs(z_B)^2) away from total zero, with value zero there, is degree-one homogeneous and continuous on the whole contrast space: abs(q)<=sqrt(abs(z_A)^2+abs(z_B)^2)/2. It is smooth when total contrast is nonzero, including either single-zero stratum. It has no real total derivative at total zero. For aligned frames, q at (1,i) is 1/sqrt(2) and at (-1,-i) is the same; its homogeneous directional response cannot be a real linear derivative. Continuity alone therefore does not imply the vanishing theorem.

The second comparison has a stronger positive regularity property: its extension is globally 1-Lipschitz in the real contrast norm. In aligned frames put y=(u_A,v_A,u_B,v_B), r=norm(y) and h=u_A*v_B-v_A*u_B. Then norm(grad h)^2=r^2 and y dot grad h=2h, so for r>0,

\[ \left\|\nabla\frac{h}{r}\right\|^2 =1-\frac{3h^2}{r^4}\le1. \]

The same identity holds after the orthogonal coordinate change representing a unit interface tau. Integration along a line segment proves the Lipschitz bound; if the segment crosses zero, split it there and use continuity. Consequently a finite constant-coefficient Cartesian mean/contrast ODE with this source inserted through a fixed linear map is globally Lipschitz and has a unique solution for every finite time. Failure of differentiability at zero does not invalidate this well-posed countermodel. The nodal equation does not require the optional total-derivative premise of the vanishing theorem. Continuity by itself would not have established this stronger claim.

These comparisons require no new microscopic EPI component, but neither is derived by naming its ingredients structural. For example, on the retained two-region model with common fixed capacity, inserting opposite mean-pressure sources (q,-q) preserves the total mean. With the original linear contrast row retained, its complete contrast budget remains E_dot=-(3*e*nu/2)*E-e*nu*abs(z_A-z_B)^2: a source uniform within each region has zero contrast projection. The Lipschitz example therefore provides a well-defined nonlinear mean response through zero contrast, not internal contrast maintenance. Its source choice is still independent; capacity, support and any changed contrast row need their own laws and budgets.

Explicit premise	Consequence	Remaining admission
Phase-only source, amplitude independence and continuous single-zero continuation	Relative-phase response must be constant on the admitted interval	Nonconstant response needs amplitude dependence, retained state/history or a restricted domain
Gram-state scalar source, positive degree-one contrast homogeneity and a total derivative at zero	The extra contrast-dependent scalar mean source vanishes	Mean-dependent terms and covariant contrast-vector dynamics remain; none of these premises is universal
Existing complex-linear transport and its admitted interface/capacity domain	Cartesian and Gram evolution continue through zero with inherited budgets	Fine pressure, graph and held-capacity premises remain supplied
Reporting-frame/relabeling covariance	State, interfaces, observations and indexed sources transform together	This does not choose a response function, gain or a physical phase reference
Smoothness without fixed-parameter homogeneity, or homogeneity without a derivative	Nontrivial invariant sources remain possible; the norm-normalized comparison is even globally Lipschitz	The displayed comparison can define a well-posed Cartesian law; coefficient/source selection, complete budgets and physical justification remain independent

The admission result separates a valid derived mechanism from an unjustified additional feedback. The existing Cartesian interaction is regular at zero; the attempted amplitude-independent angle-fed pressure is not. No selected gain, artificial sustaining term, controller threshold or new default law follows from that distinction.

The compact pressure controls check the countermodels' scale behavior, the exact gradient bound and the nonlinear directional response. They do not execute the alternative law or promote its existence to constitutive selection.