Phase-to-form source matching and exchange

Causal source admission, regional frames, directed exchange and the prescribed-input response.

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.

15. When an observed phase can be a causal source

Section 14 establishes possible instantaneous compensation under a prepared primitive phase. It does not establish that an angle observed in another model can be inserted as that source. This admission question uses the existing morphism, projection and memory, phase response and regional-balance owners. No additional phase equation is selected here.

15.1 Source matching precedes phase tangency

Let X_dot=f(X) be an already closed fine law and let a proposed complete macro state be J(X)=(rho(X),Theta(X),V(X),...). On a regular chart, an autonomous macro law Z_dot=F(Z) represents that fine evolution only if

\[ F(J(X))=DJ(X)f(X). \]

The right side must also agree at fine states with identical J(X); otherwise the observation is not a closed state. For a proposed canonical macro EPI channel, one necessary component of this identity is

\[ D\rho(X)f(X) =\operatorname{diag}(V(X))\, \Delta\mathrm{NFR}_{\rm macro}(J(X)). \]

The phase component separately requires F_theta(J(X))=DTheta(X)f(X). Tangency of the phase assignment cannot repair a failure of the EPI row. An inherited decomposition can have a nonzero source if it is already present in the projected fine vector field. It cannot count that field once as passive transport and again as an independently added force. The internal vector-form observation used below is not asserted to have the canonical scalar macro pressure formula rejected in section 13.

For the retained unit prism with fine pure-EPI coefficient e, the existing exact identity is c_dot=e*INDUCED*c. Adding canonical phase pressure F_phi on those same fine nodes changes this projection by R_int*F_phi at unit capacity. Preserving the same internal law requires that projection to vanish. This does not require the entire source vector to vanish: a fiberwise-constant source affects the omitted mean coordinates instead. Preserving the full EPI state requires the full EPI row of the identity.

At the preparation of section 14.3, e=1/2 and each nonzero u=1/16. The original pure-EPI model gives u_dot=-1/32; the added ideal phase source contributes 1/24, producing u_dot=1/96. The actual captured projection is retained exactly and compared with an independent represented phase-source evaluation. Its difference from the ideal is recorded rather than fixed to one platform's binary64 literal. The contribution is nonzero and reverses the sign of the retained rate. Thus this pressure changes fine dynamics, not the pushforward of the original passive model. No choice of a phase velocity can remove that instantaneous EPI mismatch.

A regular observation makes the same issue explicit without polar singularities. On the common internal eigenray x=mu*1+u*(1,-1,0,1,-1,0), define, only as a conditional observation, theta_i=beta-k*(x_i-mu). Take k>0 and a local chart with 0<k*u<pi/4. All edges satisfy strict U3, the mean direction is beta, and the ideal canonical phase source equals (w_phi*k*u/pi)*p in each triangle. The observed passive form still has u_dot=-e*u. Feeding this source back instead gives

\[ \dot u=(-e+w_\phi k/\pi)u. \]

Choosing w_phi*k/pi=e would cancel decay, but that is a selected new interaction and not an identity derived by observing the old law. The example introduces no installed coefficient or phase rule.

Changing the constant EPI coefficient cannot generally rescue an amplitude-independent modal angle either. For fixed capacity/support, a linear EPI observation with inherited rate -e_0*L*x would need w_phi*g(Theta(x))=(e-e_0)*L*x to represent the same evolution with a new passive coefficient e. Along positive rescalings of a centered field, modal arguments and the left side are unchanged, while the right side scales with amplitude. Matching two distinct positive amplitudes therefore forces both sides to vanish. For internal-only matching the same statement applies to the projected source, not necessarily the full pressure vector. This excludes a fixed-coefficient interpretation on that amplitude family; it does not exclude separately derived amplitude-dependent interactions.

15.2 Regular observation and the zero-amplitude boundary

For a fixed finite connected reversible pure-EPI model with strictly positive fixed capacities and e>0, the fine state tends to its conserved weighted consensus mu*1. A continuous phase-source observation F_obs therefore vanishes asymptotically if it satisfies F_obs(mu*1)=0. Local Lipschitz regularity also transfers the exponential fine-state bound to that source. The same statement holds for just its internal projection when that projection vanishes at consensus.

Consensus compatibility is an essential hypothesis. A constant map Theta(X)=theta_prepared can be smooth while prescribing a nonuniform phase source forever. It encodes the prepared pattern and does not derive it from EPI. On the vertex-transitive prism, a nodewise phase observation equivariant under graph automorphisms must be constant across nodes at uniform EPI, provided no other state breaks that symmetry. Its available canonical phase source is then zero. For a non-transitive graph symmetry only enforces equality within vertex orbits; the stronger conclusion does not follow automatically. Equivariance here is equality of the nodewise phasor vector under permutation, not equality only up to an extra rotation.

Polar orientation avoids continuity at consensus rather than this argument. The internal rays z=r and z=i*r, r>0, approach the same zero form with different angles. Both can be decaying exact modes of the retained fine law. Their angle difference remains finite while their amplitudes vanish, so no continuous angle extension at the common origin exists. An amplitude-independent angle can be a useful observation on the punctured domain, but it does not by itself provide a finite sustaining mechanism. The normalized-shape term +kappa*q from section 12 is likewise a chain-rule term, not a source that may be added back to EPI.

There is also a bound that does not assume angle continuity. Along the unchanged passive unit-prism trajectory, S(t)<=S(0)*exp(-2*e*t). An available canonical phase channel obeys |g_i|<=1; for fixed finite w_phi, its six-node vector satisfies ||F_phi||_2<=w_phi*sqrt(6). Consequently its diagnostic work on that passive trajectory obeys

\[ |2\langle y,F_\phi\rangle| \le 2w_\phi\sqrt{6S(0)}e^{-et},\qquad \int_0^\infty |2\langle y,F_\phi\rangle|\,dt \le \frac{2w_\phi\sqrt{6S(0)}}e. \]

This bounds a measured comparison, not an executed source contribution. It does not apply unchanged after feeding the source back and thereby changing the fine law. Phase availability, wrap branches and binary64 realization remain separate from this exact-real bound.

15.3 The current supporting phase has only common-rotation freedom

For the prepared phases (0,pi/3,pi/6) in each triangle, every node's neighbor resultant has direction pi/6 and magnitude 1+sqrt(3). The canonical phase-source derivative uses the existing mean-response matrix, not an operator-stage Jacobian:

\[ R_{ij}=\mathbf1_{j\in N(i)} \frac{\cos(\theta_j-\pi/6)}{1+\sqrt3},\qquad Dg=(R-I)/\pi. \]

All three nonzero row entries are strictly positive and sum to one. Connected support makes this stochastic matrix irreducible, so ker(R-I)=span(1) and rank(R-I)=5. This property persists in a regular neighborhood where all these entries stay positive. Thus a differentiable path in that neighborhood that preserves the full phase source can only rotate all phases together; its relative phase profile is fixed. This specialization reuses the general source-tangency result in support balance, section 22. The exact coefficients contain sqrt(3); symbolic controls do not pass rounded surds into the owner's exact rational Gram validator.

With fixed unit capacity, fixed EPI, held conductance/support and channel coefficients, nonzero w_phi and no other changing source, stationarity requires constant g. The result gives a precise necessary phase-motion condition for that candidate. It does not impose constant source on every moving or periodic pattern, and does not select the common rotation speed or derive a law preserving the relative phases. Known higher-dimensional source tangents on other graphs are unaffected.

15.4 Persistent support requires persistent directed work

For a separately justified coupled law on this same unit-capacity, unit-conductance prism, with fixed e>0, exact canonical pressure and no pressure residual, section 14 gives

\[ \dot S=-2eS-\frac{2e}{3}D+W_F,\qquad D=|z_0-z_1|^2,\quad W_F=2\langle y,F_{\rm int}\rangle. \]

Here F_int is the orthogonal internal projection of the true source. At S>0, nondecrease requires and is equivalent to

\[ \frac{\langle y,F_{\rm int}\rangle}{S} \ge e\left(1+\frac{D}{3S}\right). \]

Large source magnitude alone is insufficient: its sign and direction matter. Cauchy-Schwarz gives the necessary norm bound ||F_int||>=e*sqrt(S)+e*D/(3*sqrt(S)). In particular, a proposed local source with ||F_int||<=L*sqrt(S) and L<e cannot compensate loss there. This condition tests a given source law; it does not prescribe a gain.

If a solution maintains S(t)>=s_*>0 throughout [0,T], integration requires

\[ \int_0^T W_F(t)\,dt \ge 2e s_*T+s_*-S(0). \]

The omitted disagreement integral is nonnegative. Indefinite active maintenance therefore needs sustained directed source work. One positive snapshot or a finite passive transfer does not prove it. A supplied fixed phase can algebraically support forced equilibrium, but assuming it remains fixed does not explain the source's autonomous origin or preservation. Actual pressure defects, changing capacity/metric and events require their own retained terms before this inequality is used.

The bounded admission result excludes promoting passive internal angles to a new autonomous source without a matching vector-field identity. It does not exclude an independently derived nonlinear, non-gradient or hybrid TNFR completion. Existing memory, directed transport, configured phase maps and auxiliary wave models supply their declared dynamics, but none of the reviewed owners derives the missing primitive phase law without an extra premise. The constitutive origin remains open; no source-preserving velocity or compensating gain is installed as its answer.

Portable controls are centralized in source matching, observation regularity and work and prepared-source tangency. They reuse the previous prism fixture and physics owners without changing runtime dynamics. Exact conditional identities, actual binary64 captures and supplied counterexamples retain distinct evidence scopes.

The joint-law analysis establishes the following conditional representation: canonical phase pressure has an exact state-dependent positive diagonal metric on its regular reciprocal-support domain. Its alignment cost and connection to current/curvature are derived once in variational sections 13.6-13.7. This supplies a geometric representation, not the missing phase clock or reciprocal EPI response. A separately supplied phase-only relaxation has a finite cost budget and cannot maintain the prism's internal amplitude under the stated uniform regularity assumptions. The earlier constant-metric and passive-source obstructions remain valid in their respective scopes.

16. Phase and form: directed exchange, frames and the moving mean

Consolidated result: a restricted driven phase/form response. On the fixed unit triangular prism, canonical phase pressure from a prescribed rotating phase contrast has an exact circular particular EPI response with nonzero internal amplitude and a periodic, nonconstant mean. This result requires unit capacity, fixed positive EPI/phase weights, the regular strict-U3 chart below, and no other active source, clipping, event or pressure defect. It is a theorem of the stated ideal model, not an autonomous NFR or a physical measurement. The normalized structural-time input is part of the prescription.

Claim	Evidence and scope
Directed phase/form coupling and complete mean	Exact nodal projection, sections 16.1 and 16.3
Circular response and threefold mean modulation	Prescribed input and its derived particular response, sections 16.4-16.5
Convergence from another EPI preparation	Exact comparison under the same entire prescribed input, with a free mean offset, section 16.7
Reproducible numerical illustration	Example 179 and its declared rounding/quadrature comparisons, section 16.8
Autonomous source generation, phase-clock selection and source robustness	Open; not supplied by the displayed response or the same-input comparison

The phase/form question has three distinct objects. Their existing owners remain authoritative; none is renamed into another state variable.

Object	What it represents	What is needed to evolve it
Primitive nodal theta_i	The circular coordinate consumed by canonical phase pressure and U3	A specified phase law or a justified relation to a complete evolving state
Internal form angle psi_a=arg(z_a)	Orientation of two nonzero internal EPI coordinates in a declared basis	The pushforward of the fine nodal rate, including amplitude ratios and source projections
Local frame angle chi_a	Choice of basis used to report the same internal form	The chosen coordinate transformation; its derivative is not a physical source

Section 9 and section 15 already delimit signed scalar EPI, primitive phase and causal closure. This section sharpens their relation through the retained prism without introducing a phase law or another observation API.

16.1 Radial and angular effects of an actual phase source

Keep the fixed unit prism, unit capacities and EPI weight e>0, using the same normalized structural-time convention as the earlier cycle controls. For each triangle write x_a=m_a*1+u_a*P+v_a*Q, where P=(1,-1,0), Q=(1,1,-2), and z_a=sqrt(2)*u_a+i*sqrt(6)*v_a. Let F_a be an independently computed fine forcing vector; pressure defects, if present, must be retained as separate source terms. Its internal projection is

\[ f_a=\frac{P^TF_a}{\sqrt2}+i\frac{Q^TF_a}{\sqrt6},\qquad \dot z_a=\frac e3(z_b-4z_a)+f_a. \]

For r_a=|z_a|>0, put psi_a=arg(z_a) and exp(-i psi_a)f_a=s_a+i t_a. The same nodal rate gives

\[ \dot r_a=\frac e3\{r_b\cos(\psi_b-\psi_a)-4r_a\}+s_a, \qquad \dot\psi_a=\frac e3\frac{r_b}{r_a}\sin(\psi_b-\psi_a)+\frac{t_a}{r_a}. \]

The displayed neighbor-angle expressions require r_b>0. At z_b=0, use the real and imaginary parts of exp(-i psi_a)z_b instead of assigning that neighbor an angle. Thus angular alignment is generally not closed on angles alone: neighbor amplitudes and projected sources matter. For S=|z_0|^2+|z_1|^2, the shared regional budget becomes

\[ \dot S=-2eS-\frac{2e}{3}|z_0-z_1|^2+2\sum_a r_a s_a. \]

Only the radial source projection contributes to instantaneous amplitude maintenance. A source tangent to the nonzero modes can turn the form while the amplitude still dissipates. Conversely a primitive phase channel can have a radial projection: its name does not restrict it to changing the internal angle. At z_a=0, the Cartesian rate remains defined while the angle and its rate are unavailable. A nonzero prepared source can seed an internal mode there; it does not prove spontaneous generation of that source.

16.2 Internal phase requires a frame and transported comparisons

Let B=[P/sqrt(2),Q/sqrt(6)] be the orthonormal internal basis. Change only coordinates using B'_a=B O_a, O_a orthogonal, and let y_a=O_a^T z_a in two-real-component notation. The fine EPI is unchanged. The inherited edge transport is T_ab=O_a^T O_b, so fixed frames give

\[ \dot y_a=\frac e3(T_{ab}y_b-4y_a)+O_a^Tf_a. \]

Alignment and the interaction energy use y_a^T T_ab y_b and ||y_a-T_ab y_b||^2, not the untransported coordinate difference. For rotations O_a=R(chi_a), the meaningful angular difference is psi'_b-psi'_a+chi_b-chi_a. Reflections additionally reverse orientation. A common active O(2) transformation of form is a symmetry of this pure-EPI generator; independent active rotations need not be. An active change of form and a passive change of basis are different operations.

For moving frames put Omega_a=O_a^T O_dot_a. Then

\[ D_t y_a:=\dot y_a+\Omega_a y_a =\frac e3(T_{ab}y_b-4y_a)+O_a^Tf_a, \qquad \dot\psi'_a=\dot\psi_a-\dot\chi_a. \]

The displayed angular formula assumes SO(2); a reflected frame also reverses angular orientation. The matrix covariant derivative covers both. A rotating chart can create an arbitrary displayed angular speed without changing the fine field. Source work must likewise use the physical tangent, represented by dy_a+Omega_a y_a dt. Omitting the frame term can fabricate work. These matrices are induced by basis changes; they are not the auxiliary arg(K_phi+i J_phi) connection in physics/gauge.py, a new fundamental gauge field, or a derivation of the primitive phase clock.

16.3 Exact primitive-phase to internal-form coupling on repeated triples

Now let both triangles have the same form (mu,u,v) and the same primitive phase triple. On a common regular lift write

\[ \theta=\beta\mathbf1+\eta,\quad \eta=s_\theta P+t_\theta Q=(s_\theta+t_\theta,-s_\theta+t_\theta,-2t_\theta),\quad \zeta=\sqrt2s_\theta+i\sqrt6t_\theta. \]

Assume max(eta)-min(eta)<pi/2, fixed unit capacity, and only EPI and phase pressure with weights e>0,w>0. Each node sees one copy of each triple phase. Since sum eta=0, the strict range condition implies |eta_i|<pi/3; the common resultant Z(eta)=sum_i exp(i eta_i) has positive real part. Define c(eta)=Arg Z(eta)/pi. Without a branch jump, the canonical ideal phase-pressure channel is

\[ g=-\frac{\eta}{\pi}+c(\eta)\mathbf1,\qquad \boxed{\ \dot z=-ez-\frac w\pi\zeta,\quad \dot\mu=w c(\eta)\ }. \]

This is the fine canonical source projected by the existing internal and mean rows, not the different block-constant quotient of phase_quotient.py. No primitive phase evolution is supplied by this projection. In particular, the internal coordinate zeta describes contrasts of primitive phase; its argument is neither any one theta_i nor an independently derived clock. The real and imaginary parts of zeta are not new free physical coefficients.

Writing z=r exp(i psi), zeta=rho exp(i alpha) gives

\[ \dot r=-er-\frac w\pi\rho\cos(\alpha-\psi),\qquad \dot\psi=-\frac{w\rho}{\pi r}\sin(\alpha-\psi). \]

Antiparallel phase contrast can support amplitude without turning it; quadrature contrast can turn it without paying the radial loss. Constant nonzero r requires the independently realized contrast to satisfy

\[ \zeta=-\frac\pi w(e+i\dot\psi)z. \]

This is a necessary balance for a proposed curve, not a method for fitting pressure after seeing the desired motion. It specifies what a missing source law would have to generate. At z=0, z_dot=-(w/pi)zeta remains regular without inventing an angle for the zero form.

16.4 The common source reveals nonlinear threefold geometry

The mean row cannot generally be discarded. Since eta_0+eta_1+eta_2=0,

\[ \operatorname{Im}Z=-4\prod_i\sin(\eta_i/2). \]

On the stated strict chart this vanishes exactly when one eta_i is zero. Consequently mu_dot=0 requires t_theta=0 or t_theta=s_theta or t_theta=-s_theta: the contrast zeta lies on three lines, not on a full circle. This is a nonlinear geometric restriction of the actual phasor mean, not a new phase threshold.

For small rho=|zeta|, with a fixed orientation alpha,

\[ \sum_i\eta_i^3=\frac{\rho^3}{\sqrt6}\sin(3\alpha),\qquad c(\eta)=-\frac{\rho^3}{18\sqrt6\pi}\sin(3\alpha)+O(\rho^5). \]

The internal source projection is isotropic and linear in zeta, whereas the omitted mean first sees a cubic, threefold angular dependence. A linear mode calculation alone misses it. This does not break passive coordinate covariance; it limits promotion of a pure-EPI active O(2) symmetry to the full canonical phase source.

For fixed mu and constant positive r, the required nonzero zeta must remain on one of the three lines along a continuous regular curve. A uniform nonzero rotation would rotate zeta off that line and is therefore impossible under these restrictions. Transient turning remains possible: for fixed line angle gamma, the radial balance gives psi_dot=e*tan(gamma-psi) wherever the balance is admissible. More generally a closed constant-radius form cycle on that line has zero perpendicular component, since that component obeys b_dot=-e*b; its orientation cannot recur nontrivially. These statements do not exclude moving means, varying amplitudes, other phase families or graphs.

16.5 Prescribed rotating phase contrast: periodic particular response

Allowing the mean to respond supplies a constructive comparison. Prescribe the primitive phase contrast first, with rho>0, omega!=0 and fixed beta. The internal EPI row then gives its periodic particular response:

\[ \zeta(t)=\rho e^{i(\omega t+\alpha_0)},\qquad z(t)=-\frac{w}{\pi(e+i\omega)}\zeta(t),\qquad \mu(t)=\mu_0+w\int_0^t c(\eta(\tau))d\tau. \]

These expressions solve both displayed EPI rows exactly. The form is calculated from a declared phase source; pressure is not reconstructed from a measured desired trajectory. The phase curve and angular clock are supplied, making this a conditional analytic control, not an autonomous emergence mechanism or a numerical maintenance campaign. Other internal initial values also contribute their decaying homogeneous term under this supplied source; the displayed circular response is the periodic particular solution. The form radius is r=w*rho/(pi*sqrt(e^2+omega^2)). The sufficient bound rho<pi/(2*sqrt(2)) keeps every phase gap strictly below pi/2 through the full rotation, since every pair difference is at most sqrt(2)*rho.

Cyclic permutation of the triple leaves Z unchanged, while negating the triple conjugates Z. Thus exactly c(alpha+2pi/3)=c(alpha) and c(alpha+pi/3)=-c(alpha) on this rotating family. For constant omega the mean source has zero integral over each T/3, T=2pi/|omega|; mu is periodic rather than drifting. Its leading small-contrast modulation is at 3|omega|. Distinguish the source from its integral: c has leading amplitude rho^3/(18*sqrt(6)*pi), whereas the mean's oscillatory component has leading amplitude w*rho^3/(54*sqrt(6)*pi*|omega|). More precisely, for fixed nonzero omega,

\[ \mu(t)-\mu_0= \frac{w\rho^3}{54\sqrt6\pi\omega} \{\cos(3\alpha(t))-\cos(3\alpha_0)\}+O(\rho^5) \]

uniformly over a fixed number of cycles as rho -> 0. The exact response can contain higher harmonics permitted by the same symmetry. This is a prediction within the supplied family, not an observed physical frequency or an independently selected TNFR clock. The complete EPI returns while its internal amplitude never vanishes. If a runtime EPI band is needed, the whole supplied curve, including the mean modulation, must fit that band.

The full work retains this moving mean. With D=H=3I, the Dirichlet energy is constant and

\[ F^TD\dot x=\|\dot x\|_H^2 =6r^2\omega^2+18\dot\mu^2. \]

The time integral of the mean source is zero, but its work is not. Discarding the mean would both violate its EPI equation and lose that work. This reuses the oriented cycle identity rather than inventing a second energy definition.

16.6 What the result does and does not close

The derived relations identify how primitive phase geometry can maintain amplitude, turn internal form, and modulate the common mean together. They also identify which apparent phase changes depend on the observer's frame. The missing equation is still the evolution or justified constitutive origin of zeta and any other primitive channels. Defining it retrospectively from the required circular motion would prescribe the answer. The source's clock, formation, perturbation response and autonomous selection remain G3 gates.

Controls reuse the shared fine generator, phase support and regional budgets: frame covariance and phase/form exchange. No new production phase rule, force, topology or gauge API is introduced.

16.7 Same-input response theorem and an all-time band bound

Fix the graph, capacity and coefficients of section 16.3, and supply the same regular phase contrast to both triangles. For any continuous prescribed contrast zeta(t), the repeated-form sector has the exact response

\[ z(t)=e^{-et}z(0)-\frac w\pi\int_0^t e^{-e(t-s)}\zeta(s)ds, \qquad \mu(t)=\mu(0)+w\int_0^t c(\eta(s))ds. \]

This is the solution of the projected nodal equation, not a fitted kernel. The preceding circular expression is its periodic particular solution. Under that same phase input, two repeated-form preparations satisfy delta z(t)=exp(-e*t)*delta z(0) and delta mu(t)=delta mu(0). In the actual six-node metric H=3I,

\[ \|\delta x(t)\|_H^2= 6e^{-2et}|\delta z(0)|^2+18\delta\mu(0)^2. \]

The result also admits a full EPI preparation that is initially different in the two triangles. Put mu=(m_0+m_1)/2, d=(m_0-m_1)/2, Z=(z_0+z_1)/2 and D=(z_0-z_1)/2. Here D denotes the difference mode only; the degree metric elsewhere remains 3I. The inherited rows are

\[ \dot\mu=wc,\quad \dot d=-\frac{2e}{3}d,\quad \dot Z=-eZ-\frac w\pi\zeta,\quad \dot D=-\frac{5e}{3}D. \]

For two full EPI preparations under the same complete phase input,

\[ \|\delta x(t)\|_H^2= 18\delta\mu(0)^2+18e^{-4et/3}\delta d(0)^2 +6e^{-2et}|\delta Z(0)|^2+6e^{-10et/3}|\delta D(0)|^2. \]

After removing the neutral common-mean offset, the norm contracts at least as exp(-2*e*t/3). Thus a rotating prescribed contrast has a unique periodic EPI response per mean offset, approached under that fixed input. This is conditional attraction of the EPI response, not an isolated attractor of the full state, stability to phase/capacity/support perturbations, or a binary64 runtime convergence theorem. When a phase-from-form rule changes the input after perturbing EPI, the same-input hypothesis must be checked rather than assumed. If the prescribed contrast is instead held at zero, the common mean stops moving and all nonuniform modes decay; the source remains necessary for the displayed maintained response.

A sufficient band bound for the periodic particular solution. For rho<pi/(2*sqrt(2)), the exact mean-source magnitude obeys

\[ |c|\le C_0:=\frac{\rho^3}{6\pi(3-\rho^2/2)}. \]

Indeed, sum eta=0 cancels the linear sine term, |sum sin eta|<=sum |eta|^3/6<=rho^3/6, and Re Z>=3-rho^2/2>0; use |atan y|<=|y| on the regular branch. The mean-source period is P=2pi/(3|omega|), with zero integral over that period. Reducing time modulo P bounds the partial integral by C_0*min(t_mod,P-t_mod)<=C_0*P/2. Hence, for the particular response,

\[ |\mu(t)-\mu_0|\le \frac{w\rho^3}{18|\omega|(3-\rho^2/2)}=:M_0, \qquad |x_i(t)-\mu_0|\le M_0+\sqrt{\frac23}\,r. \]

The last term uses the norm of each row of the orthonormal internal lift. This is an all-time exact-model bound, not merely the range of plotted samples. A declared open EPI band containing that whole interval needs no clipping for this particular solution. General initial data add the explicitly decaying transient above and need their own band check. The band bound follows from the inequalities above. The same-input response controls are in response theorem tests.

16.8 Reproduce and interpret the documented result

From the repository root, with its Python dependencies installed, run

python examples/08_emergent_geometry/179_phase_form_driven_response.py --output-dir artifacts/phase_form_driven_response

The example declares its phase input first, reconstructs the analytic EPI response and evaluates it at fixed sample points. It compares nodal rates with shared pressure readers and numerically integrates only the explicit scalar mean-source integral. It does not execute an engine/glyph trajectory, derive a primitive phase law or reset the mean to force endpoint closure. The command records the parameters, sample grid, finite rounding residuals, mean quadrature comparison and numerical endpoint residuals. These finite readings illustrate the exact results above; they are not their proof.

The default demonstration uses e=1/2, w=1/4, unit capacity, rho=1/4, omega=1/2, beta=pi and mu_0=1/2. The inactive capacity channel has weight 1/4; its gradient vanishes on this uniform-capacity preparation. These are declared illustration values, not fitted parameters or selected physical constants. The source definition, regular branch and entire nodal state must be retained when reproducing the result.

The recorded JSON and CSV contain 97 snapshots over one supplied cycle. Composite Simpson integration uses 16 and 32 subdivisions per output interval; their maximum mean-offset difference is 4.48e-15, a refinement comparison rather than a certified quadrature bound. The radius is 0.02813488488; the sampled mean variation is 3.789703416e-5 peak-to-peak. The maximum shared nodal-rate residual is 8.33e-17. The uncorrected accumulated endpoint offsets are about 2.34e-19 and 1.20e-19; adding the initial mean rounds these differences to zero, so the raw offsets are retained separately. No endpoint reset is applied.

The middle panel magnifies the mean offset by 10^6; the right panel shows the internal radial balance, distinct from the full velocity-work identity above. The optional plot requires Matplotlib (viz-basic extra). Displaying the endpoint tetrad does not certify closure of these reduced coordinates; its primitive-state and representation dependencies remain those of section 13.4.

The controls have distinct responsibilities:

Owner	Reproducible obligation
Internal projection fixture and exchange controls	Actual graph support, signed EPI projection, radial/tangential budgets, nonlinear mean and full work
Frame controls	Passive basis covariance versus active changes of form; physical tangent in moving frames
Response theorem controls	Same-input solution and decay of full EPI preparation differences
Example 179	Finite numerical reproduction with explicit source, quadrature and represented-pressure residuals
Single G3 gate	Still-missing origin of the primitive source, clock and autonomous response