Primitive phase closure and source selection

Symmetry obstructions, nonlinear orientation, source work and actual event occurrence.

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.

17. Primitive phase origin: symmetry, retained state and the missing row

Status. The supplied response in section 16 is not promoted to an autonomous law. The following admission tests identify what a proposed origin must retain and reject several tempting replacements. They concern the same unit prism, repeated triples, fixed unit capacity, positive coefficients e=w_epi, w=w_phase, and the regular strict phase chart. Write k=w/pi. The complete repeated EPI rows remain

\[ \dot z=-ez-k\zeta,\qquad \dot\mu=w c(\zeta). \]

Here z observes EPI contrast and zeta observes the contrast of primitive nodal phases. Keeping both does not add a new nodal attribute. The missing object is a justified evolution of the existing phase state, not another diagnostic name for its current value.

17.1 Existing phase writers do not select the supplied rotating contrast

Every local neighbor set on the repeated prism sees the same phase triple; the global phasor mean has the same argument. For an ideal simultaneous ordinary coordination step with uniform gains k_G,k_L, put K=k_G+k_L. Inside the common lift, before wrapping,

\[ \theta_i^+=\beta+\eta_i+K\{\pi c(\eta)-\eta_i\},\qquad \beta^+=\beta+K\pi c(\eta),\quad \eta^+=(1-K)\eta,\quad \zeta^+=(1-K)\zeta. \]

Thus this policy can shrink, erase or reverse the contrast; it cannot continuously turn its orientation. A negative multiplier is a discrete reversal, not a resolved nonzero-radius rotation. Adaptive scalar gains change the multiplier without supplying an additional direction, provided the repeated state and uniform-gain hypotheses remain valid. The nonzero common shift must be kept. This is an exact ideal per-invocation identity, not a continuous-time law, future schedule certificate or bit-exact claim about trigonometric/wrap arithmetic.

Existing owner	Reusable mechanism	Boundary for the phase-origin question
Ordinary phase coordination	Global/local phasor relaxation and scalar gain history	The repeated contrast has the scalar map above; gain adaptation does not select a circulating source
Public/staged Coherence phase proposal	Snapshot phasor relaxation	A common simultaneous coefficient gives the same contrast scaling; ordinary native apply_glyph(IL) instead calls the pressure-only primitive
Configured engine phase model	Supplied capacity-as-angular-rate plus gated sine coupling	Uniform free advance changes the common phase only; its sine term is an alignment model, not a derived persistent contrast clock
Optional extended system	Pressure-dependent phase response	Its independent pressure row fails the fresh-pressure comparison already audited; section 17.4 also classifies the fresh-pressure linear comparison
Exact memory elimination	Retain the effect of discarded state and its initial condition	Elimination requires a complete fine law; it cannot select the missing phase row
Joint nodal response	Pressure derivative and EPI acceleration from supplied phase/capacity rates	Supplies the correct accounting interface, not those rates' constitutive origin

In the configured sine comparison with common free advance and positive coupling, the repeated lifted phase variance dissipates on the strict chart: pair contributions are proportional to -(eta_i-eta_j)*sin(eta_i-eta_j), strictly negative for a nonzero gap. This continuous comparison is distinct from finite Euler steps. Common phase rotation can coexist with decay of the relative source; an absolute phase rhythm alone does not maintain EPI contrast. Heterogeneous capacity, additional state, nonuniform stages and changed support require their own analysis. These restrictions do not classify the complete operator catalog.

Full-phase extension: the configured coordinator consumes contrast. The repeated-triple restriction can be removed for a diameter bound. Assume one common open-semicircle lift, with width D<pi. Every nonempty local phasor mean and the global mean have representatives inside that interval; an isolate uses its own phase. With shared finite gains k_G,k_L>=0, k_G+k_L<=1, the ideal simultaneous coordinator is

\[ \theta_i^+=(1-k_G-k_L)\theta_i+k_G b+k_L\ell_i. \]

The global target b cancels from pair differences. Hence D^+<= (1-k_G)D, even for all six independent phases. This bounds phase contrast; it neither excludes transient angular motion nor defines a continuous clock. The default enabled adaptive policy clamps its gains between configured finite endpoints with a positive global floor and sum of maxima below one. For executable accounting use the exact represented floor m read from DEFAULTS['PHASE_ADAPT']['kG_min'] and q=1-m<1. Its nominal expression 1/(8*pi^2) is not an exact identity for the stored binary64 coefficient. Custom overrides, disabled adaptation and altered configuration do not automatically inherit that bound.

For a finite sequence let D_j be the post-coordinator diameter and r_j=max(0,D_before_next-D_j) the observed increase from all intervening phase writers in compatible lifts. If epsilon_j>=0 bounds the next map's represented realization defect, then

\[ D_{j+1}\le qD_j+qr_j+\epsilon_j,\qquad D_N\le q^N D_0+ \sum_{j=0}^{N-1}q^{N-1-j}(qr_j+\epsilon_j). \]

These are comparison observations, not added forcing or an event policy. No unobserved phase write may be assigned r_j=0. Uniform common phase advance alone contributes no diameter increase. For the ideal no-increase case, repeated coordination exhausts phase contrast; the finite defect ledger does not by itself supply a uniform asymptotic binary64 bound. The interval must remain admissible at every invocation, and its coherent lift must be verified rather than inferred from wrapped maxima/minima.

A detached coordinator control uses the existing evidence-returning version to retain the actual gains, local targets, proposals and phase writes. Its exact rational endpoint-minus-convex-proposal error is measured relative to captured represented targets, whose interval membership is checked. This does not certify their transcendental accuracy. This public evidence is not a sealed full-runtime certificate. The native runtime calls the legacy coordinator; its invocation is identified from the existing implementation, without replaying the frozen native experiment. The result proves neither that every other writer contracts nor that all capacities remain equal. Positive phase contrast is also insufficient for positive EPI work. That separate test is the joint transfer budget. Controls: configured diameter and finite ledger.

Native writer audit: no ideal contrast renewal in the retained class. Operator names alone do not identify an evolution map. Ordinary runtime._update_nodes calls selectors._apply_glyphs, then operators.apply_glyph, which dispatches through GLYPH_OPERATIONS. It does not invoke the corresponding public operator class or all-target stage. In particular, native _op_IL contracts stored pressure and leaves phase unchanged; public Coherence() can additionally lock phase. The following atlas identifies the actual default path between coordinators.

Owner in chronological order	Primitive phase effect and required boundary
Capacity adaptation	No phase write. Equal represented capacities inside the common rails remain fixed regardless of eligibility: the shared proposal retains its input when the represented neighbor mean equals it
Auxiliary math step, history and automatic REMESH	Auxiliary state has no inverse node-phase projection. Automatic REMESH calls the protected delayed-EPI owner, not the phase/capacity structural-memory interpolation
Validators and callbacks	Built-in validators do not write phase. Arbitrary callbacks receive the mutable graph and are excluded unless their writes are accounted for; suppressed callback errors do not undo partial writes
Fresh pressure and default selection	Fresh default Si with equal positive capacity chooses IL, except forced AL/EN lag branches. These native primitives write no phase, capacity or support
Default integrator	Scalar and NumPy paths advance EPI and time, holding primitive phase and capacity fixed. The extended-wrapper flag alone does not replace this runtime integrator
Canonical clamps	Already centered phases are retained exactly. Outside [-math.pi,math.pi), represented normalization still needs a coherent lift and a signed error; ideal wrapping preserves circular phase
Next coordinator	The preceding default-gain theorem gives ideal diameter factor at most q<1

The selector statement uses initialized nodes: nonzero EPI or retained nonempty glyph history. IL is then grammar-admissible and is the first fallback for a rejected forced candidate. The default selector does not apply the parametric selector's soft repetition filter. Consequently the reachable native glyph set is {IL, AL, EN} in this class, including when lag counters force AL before EN. Later entries in the fallback list are not evidence of their reachability. Fresh Si and the unchanged default configuration are hypotheses, not facts about arbitrary selectors.

On the retained fixed unit prism, valid built-in steps preserve unit capacity under these maps, even with different adaptation eligibility. The former two-product blend could increase an eligible nu=0.3 by 2^-54 while leaving an ineligible node fixed. The shared fixed-point guard now removes that numerical defect, extending represented equal-capacity preservation beyond the unit input. Nonuniform arithmetic and out-of-rail clamps keep their separate effects. The whole-step integration records this extension and finite ordinary-runtime controls. Phase-capable public operators, custom integrators/selectors, generic callbacks, structural-memory REMESH and the optimizer/FFT free-advance model are separate execution paths.

In exact arithmetic, fixed support, initialized nodes, unit capacity, fresh default selection, the default integrator, no additional graph writers, and the common open-semicircle domain therefore give r_j=0 between coordinators. The phase hull is invariant and

\[ D_N\le q^N D_0. \]

This is a conditional ideal composition theorem, indexed by successful coordinator events. Infinite event count, physical-time scheduling and non-Zeno behavior are separate premises. It does not prove repeated binary64 convergence or disappearance of all EPI patterns: native AL/EN can still change EPI directly, and held pressure need not be fresh pressure. The theorem excludes phase-contrast renewal in the declared route, not every possible TNFR maintenance mechanism.

For binary64, sum the oscillations of all intervening signed phase errors in compatible lifts into E_j>=0. If their ideal maps preserve the hull, r_j<=E_j and the finite bound becomes D_(j+1)<=q*D_j+q*E_j+epsilon_j. A common error contributes zero oscillation. Neither omitted writers nor unchecked lifts have a zero error by default. A uniform nonzero error bound gives a possible residual floor, not convergence to zero. Public/staged IL is also ideally hull-preserving for coefficients in [0,1], including sequential proposals, but it must retain its own proposal error: its certified two-neighbor displacement and displayed mean are rounded independently.

The same interval gives |g_i|<=D/pi for the ideal fresh phase pressure. On six nodes with fixed phase coefficient w, ||w*g||_2<=w*sqrt(6)*D/pi and |w*y^T*g|<=w*sqrt(6)*||y||_2*D/pi. Thus bounded form contrast cannot receive persistent nonzero phase-source work from this ideal shrinking interval. Other source channels and finite glyph jumps keep their own balances; this is not an all-channel work theorem.

The frozen native record supplies only one coordinator invocation, whose before/after phases are all zero. It contains neither adjacent coordinator endpoints nor effective gains/targets/proposals. It therefore supports one consensus-preservation observation, not a consecutive-cycle error ledger. An authenticated adjacent pair would suffice for aggregate r_j; attribution to individual writers additionally requires intermediate chronology. No old history is reconstructed and no frozen Mutation decision is rerun.

Controls: native selection and dispatch, integrator, capacity and clamp boundaries, and the preceding diameter/finite-ledger controls. These component checks and the source audit are not a sealed full-runtime execution certificate.

17.2 A form-only equivariant phase map cannot wind around the origin

The actual graph automorphisms simultaneously permute the three indices in both triangles. In the normalized internal basis, a cyclic permutation and the exchange of indices 0 and 1 act as

\[ Cz=e^{2\pi i/3}z,\qquad Sz=-\bar z,\qquad SCS=C^{-1}. \]

The same actions apply to zeta. The three reflection axes have angles pi/6, pi/2, 5pi/6 modulo pi. They are distinct from the three c(zeta)=0 source lines in section 16.4, which have angles 0, pi/3, 2pi/3. In particular the common source c is invariant under permutations; it is not an orientation-odd quantity.

Suppose a proposed memoryless phase observation is zeta=F(mu,z), with no other symmetry-breaking state, and respects these graph actions: F(mu,Oz)=O F(mu,z) for every generated permutation matrix O. Assume an invariant open regular domain and a locally Lipschitz complete reduced vector field, so local solutions are unique. For any reflection R, Rz=z implies R F(mu,z)=F(mu,z). Hence z_dot also lies on that reflection axis. The plane consisting of that axis and arbitrary mu is invariant. Local uniqueness forbids a trajectory from crossing it at a finite time while it remains in the regular domain.

Consequently any trajectory with |z|>0 remains in one open sector of width pi/3, or on one mirror ray. Its continuously lifted angle cannot complete a winding. This excludes the circle of section 16 as an autonomous response of this particular class, even with the moving mean retained. It does not exclude local angular motion, loops contained inside a sector, nonunique/nonsmooth models, oriented boundary inputs, independent phase state, or another support symmetry. Passage through an undefined zero-form phase leaves the nonzero-form domain. If the equivariant locally Lipschitz law also extends to z=0, then F(mu,0)=0 makes that set invariant and excludes finite-time passage through it as well. A discontinuous label-based choice would leave these hypotheses, rather than derive a canonical orientation.

At uniform form, equivariance additionally gives F(mu,0)=0. If F is differentiable there, its real derivative must commute with both C and S, so D_z F=a(mu) I. Commutation with rotations alone would allow aI+bJ, where Jz=iz; reflection forces b=0. Inserting a fixed quarter turn therefore supplies a handedness absent from this graph/form state. The polynomial control F(z)=Re(z^3) Jz does respect both actions and allows turning inside sectors, but its angular contribution vanishes on every mirror axis. This algebraic control is not an installed phase law.

The proof uses the existing permutation owner and equivariance framework, with the shared exact prism lift; it does not assume the full nonlinear phase model has the continuous O(2) symmetry of the isolated linear EPI sector.

17.3 Relative oriented area is already available in the joint state

With independent primitive phase contrast, define the observation

\[ \ell=\operatorname{Im}(\bar z\zeta). \]

Under a common orthogonal basis change, ell is multiplied by its determinant: its magnitude is frame-independent and its sign reverses under reflection. It is distinct from c, graph winding, and the auxiliary field named chirality. The nodal EPI row gives, for r=|z|>0,

\[ r^2\dot\psi=-k\ell,\qquad \dot\ell=-e\ell+\operatorname{Im}(\bar z\dot\zeta). \]

The Cartesian angular numerator Im(conj(z)*z_dot) and the area balance remain defined at zero form, where psi itself is undefined. The second identity identifies the missing contribution: sustaining oriented area requires a phase response to offset its EPI damping. It is not an independent evolution law or a conserved charge.

The complete reflection-fixed state now requires both z and zeta to lie on the same axis. For example, at z=ir, zeta=s with positive r,s, Re(z_dot)=-k*s is nonzero although z lies on a mirror axis. The joint state can therefore carry a direction unavailable to an instantaneous form-only closure. This instantaneous crossing witness supplies neither a repeated orbit nor the missing phase derivative. In the prescribed circular control, ell=-omega*r^2/k; its sign is inherited from the supplied phase motion.

An independent mode in the other triangle, a capacity contrast or an actual retained history vector can also change the state stabilizer. Their evolution and sustaining work still need justification; a scalar common mean does not remove the mirror constraint. The existing observe_exact_map_symmetries already distinguishes support/operator symmetries from the stabilizer of several supplied fields. Reuse it when admitting such a richer state rather than inferring symmetry breaking from an angle or adding a second audit API.

17.4 Admission conditions for a joint linear response

At a regular uniform equilibrium, a differentiable, permutation-equivariant joint phase law has a contrast linearization of the form

\[ \dot z=-ez-k\zeta,\qquad \dot\zeta=a z+b\zeta, \]

where a,b are real derivatives of the still unspecified law. Symmetry fixes the matrix form, not either coefficient. These symbols classify a possible response; they are not new engine settings or inferred physical constants. For a declared constant linear law the following identities are exact; for a nonlinear law they concern its linearization only:

\[ \dot\ell=(b-e)\ell,\qquad \lambda^2+(e-b)\lambda+(ka-eb)=0,\qquad \ddot z+(e-b)\dot z+(ka-eb)z=0. \]

Each temporal eigenvalue occurs in both spatial components. The linear contrast block is asymptotically stable precisely when b<e and ka>eb. Its eigenvalues are non-real precisely when (e+b)^2<4ka. Nonzero purely imaginary eigenvalues require both b=e and ka>e^2; the conditional angular rate would then be sqrt(ka-e^2). This does not determine a TNFR clock. At that boundary,

\[ Q=a|z|^2+2e\operatorname{Re}(\bar z\zeta)+k|\zeta|^2 =k|\zeta+(e/k)z|^2+(a-e^2/k)|z|^2 \]

is positive definite and conserved for the declared linear model. It is a derived comparison quadratic, not the tetrad energy or a new canonical Hamiltonian. Neutral linear oscillation is not amplitude selection, attraction, nonlinear stability or source generation; the complete mean must still solve mu_dot=w*c(zeta) and stay in its allowed band.

There is a direct test against an existing candidate. The optional phase formula at unit capacity is theta_dot=A*sin(pi*p)+B*p+G*kappa_graph*J_phi, with its configured positive coefficients. In the hypothetical fresh-pressure comparison near uniform phase, the repeated internal pressure is -e*z-k*zeta and the mean-sine current linearizes to -zeta. Put h=A*pi+B>0 and g=G*kappa_graph>0. Then

\[ a=-he,\quad b=-hk-g,\quad \operatorname{tr}M=-(e+hk+g),\quad \det M=eg,\quad \operatorname{disc}M=(e-hk-g)^2+4khe>0. \]

Both eigenvalues are real and negative. Thus this restricted linear comparison cannot supply a local oscillatory contrast mode. It complements the nonlinear one-direction work result in the variational audit. It is not the implemented independent-pressure system: its different pressure row and previously recorded inconsistency remain unchanged. No sign reversal or replacement coefficient is installed to manufacture the missing oscillation.

17.5 Memory preserves omitted phase information; it does not derive its law

For the declared constant linear comparison, eliminating zeta exactly gives

\[ \zeta(t)=e^{bt}\zeta(0)+a\int_0^t e^{b(t-s)}z(s)\,ds, \]

\[ \dot z(t)=-ez(t)-k e^{bt}\zeta(0) -ka\int_0^t e^{b(t-s)}z(s)\,ds. \]

This is the same variation-of-constants mechanism as derived EPI memory, applied to a different, conditionally specified complete law. The initial phase term is required. Eliminating it or choosing a memory kernel to force a desired rhythm would replace the model. The existing passive repeated EPI mode instead has z(t)=exp(-e*t)z(0): its retained lag vectors are parallel, with Im(conj(z(t))*z(t-tau))=0 for 0<=tau<=t within that same passive evolution, or with consistently continued prehistory. Arbitrary stored initial history need not satisfy this identity. That passive history alone cannot create the missing oriented area on this family. This is not a theorem against memory in other multichannel, nonlinear or changing-support dynamics.

Even full instantaneous nodal agreement does not select the phase row. At the same z!=0, zeta=0, both logical completions zeta_dot=0 and zeta_dot=a*z give z_dot=-e*z, but their EPI accelerations differ by -k*a*z. Both respect the graph permutations. The distinction is an exact underdetermination witness, not a proposed pair of engine laws.

17.6 Consequence for the single research queue

A mechanism intended to evade this family's obstruction must retain actual directional state or a justified boundary interaction and derive its evolution from the declared fine TNFR mechanism. An internal angle alone, common phase advance, a scalar synchronization gain, or passive single-mode history is insufficient on this family. First establish the actual phase row and its projection, then audit the oriented-area budget, source/form work, full mean and phase chart. If the law is differentiable at uniform state, calculate its a,b from that law; do not choose them from a desired period. Nonlinear, finite-amplitude and hybrid mechanisms need their own tests rather than being rejected by this local linear classification.

The tetrad remains a required read-out of primitive state. Neither ell nor the auxiliary quadratic is promoted to a selector or added evolution term. No new production dynamics, trajectory, parameter sweep or second research queue is introduced. Exact controls and finite configured-map comparisons have separate owners:

• Permutation and form-only obstruction controls.
• Existing coordination/phase-model controls.
• Joint-state, spectrum and memory controls.
• Current G3 execution gate.

17.7 Contact obstructions identify closure premises, not a universal limit

The two-C5 contact results use held unit capacity, fixed reciprocal support, fresh form/phase pressure and the supplied U3-gated sine phase row. Their continuous causal structure is triangular: theta_dot=G(theta) and x_dot=F(x,theta). At the same initial phases, changing EPI can change the form response but not the phase equation. Wherever the phase initial-value problem is unique, its trajectory is consequently unchanged. Without uniqueness, the set of admitted phase solutions still does not depend on EPI; an extra solution-selection rule would be another premise. The phase cutoffs make this distinction relevant. Neither the tetrad nor refreshed EPI pressure supplies a return term to this phase row.

The negative formation results therefore classify this completion. They do not show that all TNFR closures prohibit formation. Conversely, conserving or recovering an already prepared winding does not demonstrate its generation. Full-support pressure continues to read edges excluded from phase coupling; an interaction cutoff does not delete structural support.

Operator admission and continuous interaction are separate choices. U3 compares circular gaps with a configured ceiling before UM/RA execution. The continuous comparison additionally uses that ceiling to discard sine interactions and renormalize the remaining neighbor count. The nodal EPI identity requires neither that reuse nor the value pi/2, the sine profile, or capacity as free angular rate. Revising a continuous closure need not silently redefine the operator contract. The phase-model owner and grammar scope retain those boundaries.

Positive alignment mobility does not supply potential work. Consider a conditional comparison on fixed simple undirected support. Let a be a fixed, bounded, nonnegative, even, 2*pi-periodic function, continuous or piecewise continuous with finitely many breakpoints per period. For circular gaps set

\[ S_i=\sum_{j\sim i}a(\theta_j-\theta_i)\sin(\theta_j-\theta_i),\qquad \dot\theta_i=c(t)+m_i S_i,\quad m_i>0, \]

\[ \psi_a(\delta)=\int_0^\delta a(s)\sin s\,ds,\qquad V_a=\sum_{\{i,j\}\in E}\psi_a(\theta_j-\theta_i). \]

The edge primitive is even, periodic, nonnegative and Lipschitz, so the potential is independent of chosen phase representatives. It is a potential of this comparison law, not physical or tetrad energy. For absolutely continuous solutions satisfying the row almost everywhere, with integrable rates and common drift and finite mobilities along the retained interval, edge pairing gives

\[ \frac{dV_a}{dt}=-\sum_i S_i\dot\theta_i =-\sum_i m_i S_i^2\le0 \quad\text{almost everywhere}. \]

At regular points the ambient gradient is -S. At a breakpoint level set the corresponding gap derivative vanishes almost everywhere on that set; its assigned finite interaction value therefore does not change the chain-rule balance. Common drift cancels because sum_i S_i=0. The claim concerns admitted continuous solutions, not global existence or uniqueness, finite Euler nonincrease or binary64 execution. Positive mobility may depend on EPI, phase or capacity without adding a derivative-of-mobility term: V_a itself has no such dependence. Feedback only through this mobility cannot replenish its potential, although it can change trajectories and redistribute supplied contrast. The hard gate uses a=1 where abs(wrap(delta))<=pi/2 and zero outside, with m_i=1/n_i when the admitted count is positive; choose any positive mobility at an isolate of the admitted support, where S_i=0.

Replacing the common free rate by node rates omega_i instead gives V_a_dot=-sum_i S_i*omega_i-sum_i m_i*S_i^2. Calling omega_i=nu_i does not derive that additional work or its sign; a capacity law and its joint budget remain necessary. Making a depend on EPI or time also leaves the fixed-potential class and requires the resulting additional work terms.

Removing the gate changes the preparation bound. The preceding dissipation structure does not preserve every quantitative exclusion. Keep rings (0,1,2,3,4) and (5,6,7,8,9) with contacts (0,5),(1,6). Prepare the recipient uniformly at phase zero and the donor at theta_(5+k)=gamma+k*q, modulo 2*pi, where q=2*pi/5 and gamma=pi-q/2. The donor has winding one; the bridge gaps have representatives +(pi-q/2) and -(pi-q/2). Under the explicitly different ungated law a=1, its cosine potential is

\[ V_1(0)=5(1-\cos q)+2+2\cos(q/2) =\frac{35-3\sqrt5}{4} >Q=10(1-\cos q)=\frac{25-5\sqrt5}{2}. \]

Here Q is the same necessary ring contribution for two strictly acute wound rings; final bridge contributions are nonnegative. The gated preparation instead has V_g(0)=5(1-cos q)+2<Q. Thus removing the cap removes this particular insufficient-budget exclusion. It neither proves a formation trajectory nor selects the ungated law. Both comparisons dissipate their own potential, and their respective basins and domain boundaries remain separate. No revised law or trajectory is installed by this exact static comparison.

Reuse the existing closure tests before adding feedback. The mixed-derivative result already fixes a reciprocal EPI response under its declared block-gradient premise. The independent potential, auxiliary mobilities and time scale remain free; stronger structural constraints do not remove that freedom. A feedback term alone is not a generation result: the audited pressure feedback is locally dissipative, while the capacity response-slope classification separates decay, instability and mere cancellation under an independently supplied form-capacity relation. The derived form phase does inherit an amplitude-dependent phase law from fine transport, but its contrasts decay; it cannot silently replace primitive phase or establish maintained identity.

The unresolved object is an independently justified complete constitutive relation, not an extra diagnostic or a gain chosen to obtain formation. Before a candidate trajectory, derive its actual mixed response, complete joint work and mean balance, and local spectrum from that relation. A candidate intended to generate structure outside the linear regime needs the corresponding finite-amplitude argument instead. A chosen relation remains a declared model unless independently selected; the same-state, different-acceleration controls above expose that distinction. The single execution queue owns the next obligation; these admission conditions create no parallel contact, capacity or phase-law campaign.

18. Nonlinear phase response, oriented area and tetrad reuse

Section 17's negative linear eigenvalues do not exclude a nonlinear instantaneous angular response. This section examines the existing optional phase formula on the same repeated prism, retaining its common pressure. The formula and its coefficients remain configured premises. Detached phase-row evaluations are distinguished from the hypothetical continuous model that substitutes fresh canonical pressure at every time. The latter is not the implemented independent-pressure extension.

18.1 The complete nonlinear projection retains the common pressure

Let U=[P/sqrt(2),Q/sqrt(6)] be the orthonormal three-node lift and Pi=I-11^T/3. Use real two-vectors for the complex coordinates z,zeta when multiplying by U. Work in the common regular lift max(eta)-min(eta)<pi/2; node-dependent wrap branches outside it do not inherit the displayed linear centered phase source. Put

\[ y=U z,\quad \eta=U\zeta,\quad k=w/\pi,\quad h=-ey-k\eta,\quad q=w c(\eta),\quad p=h+q\mathbf1. \]

Thus h is centered pressure and q is its common component, not the absolute EPI mean. The EPI rows are y_dot=h, mu_dot=q. Write the inspected optional phase formula as theta_dot=f(p)+g J(eta), with f(s)=A*sin(pi*s)+B*s and a held common positive coupling g=G*kappa_graph. Its exact repeated-state projection is

\[ \dot\eta=\Pi f(h+q\mathbf1)+gJ(\eta),\qquad \dot\beta=\frac13\sum_i f(h_i+q),\qquad \dot\zeta=U^T\dot\eta. \]

The sum of the mean-sine current J is zero. The pressure mean cannot be removed before applying the sine. Explicitly,

\[ \Pi f(h+q\mathbf1)=B h+ A\{\sin(\pi q)\Pi\cos(\pi h)+\cos(\pi q)\Pi\sin(\pi h)\}. \]

Absolute translations of mu and beta leave these contrast equations unchanged; their rates still belong to the complete state. With det(u,v)=u_1 v_2-u_2 v_1, the oriented-area row becomes

\[ \dot\ell=-(e+Bk)\ell +A\det\{z,U^T\sin(\pi p)\} +g\det\{z,U^TJ(\eta)\}. \]

This is a projection of a supplied formula, not a new law derived from the nodal product. It identifies nonlinear angular contributions missed by the linear comparison while retaining their provenance.

18.2 Nonzero oriented-area production from initially uniform phase

Take e=1/2, w=1/4, unit capacity, the inactive capacity-channel weight 1/4, and repeat the following triple in the two triangles:

\[ \mu=3/5,\quad y=(1/6,1/3,-1/2),\quad \theta=\pi\mathbf1. \]

The EPI entries are (23/30,14/15,1/10), all inside (0,1). The phase contrast, current, common pressure and oriented area initially vanish: zeta=0, J=0, q=0, ell=0. Fresh pressure is p=(-1/12,-1/6,1/4). Evaluating the existing phase row with its declared real coefficients A=1/2, B=3/20 gives

\[ \dot\ell= \frac{5\sqrt6-3\sqrt2-8}{48\sqrt3}>0,\qquad \dot\beta=\frac{-\sqrt6+3\sqrt2-2}{24}\ne0. \]

The first numerator is positive because 17>12*sqrt(2), or 289>288 after squaring positive sides. Numerically ell_dot is about 5.78316e-5; the linear B*p term contributes no oriented area here. The common phase rate is retained even though the initial EPI mean rate is zero. The nonuniform EPI preparation already breaks some graph symmetries; this is not spontaneous handedness from a fully symmetric initial state.

Production pressure and phase-row readers reproduce this detached instantaneous comparison within binary64 rounding. No trajectory, persistent orbit or consistency of the optional independent-pressure evolution follows. In particular the future identity z_dot=-e*z-k*zeta requires fresh pressure along the comparison, not only at its initial point.

The small-amplitude origin of the effect is explicit. At zeta=0, the projected linear and cubic sine terms have zero determinant with z. The first possible nonzero term is

\[ \dot\ell=-\frac{A(\pi e)^5}{4320} \operatorname{Im}(z^6)+O(|z|^8). \]

This follows from det(z,U^T(Uz)^3)=0 and det(z,U^T(Uz)^5)=Im(z^6)/36. Thus a linear or cubic truncation misses this particular transverse production mechanism. It remains part of the configured nonlinear response; the expansion does not select that response or an angular clock.

18.3 The same oriented area is readable through the tetrad

On this fixed unit-distance prism, let T_G be the canonical inverse-square source-to-potential kernel. Directly from the graph distances,

\[ T_G\mathbf1=\tfrac72\mathbf1,\qquad T_G U_{\rm repeated}=-\tfrac14 U_{\rm repeated}. \]

For fresh repeated pressure, the one-triangle field projections satisfy

\[ U^T K_\phi=\zeta,\qquad U^T\Phi_s=\tfrac14(ez+k\zeta),\qquad \boxed{\ \ell=\frac4e\det(U^T\Phi_s,U^T K_\phi)\ }. \]

This connects the joint orientation to two existing tetrad read-outs; it requires no additional field API or telemetry-driven action. It is restricted to the known support, explicit unit lengths, repeated lift, positive known e and fresh canonical pressure. The remaining tetrad fields retain their usual diagnostic roles. Neither absolute EPI mean nor absolute common phase is reconstructed, and no complete-state or generic-graph theorem follows.

Stored pressure must not silently replace fresh pressure. For a repeated stored-minus-ideal pressure defect delta, its internal projection delta_z=U^T delta changes the inferred area to

\[ \ell_{\rm fields}=\ell-\frac1e\det(\delta_z,\zeta). \]

The shared forcing capture already separates stored-pressure and numerical kernel defects. Finite field-reader comparisons retain those residuals; the displayed exact identity is not an equality of separately rounded binary64 operations.

18.4 Production of orientation is not sustained identity

The full nonlinear work balance, including the mean-pressure term, is centralized in the variational owner, section 13.15. It proves a sufficient finite-pressure-domain dissipation criterion for the hypothetical fresh-pressure model. The state in section 18.2 can generate oriented area while that comparison energy decreases. This separates a nonlinear angular response from a self-maintained pattern.

The auxiliary harmonic substrate supplies no shortcut: at the current prepared nonuniform prism phase state its proposed geometric velocity has no primitive-phase lift, even allowing all six phase velocities. The exact rank obstruction is in section 3.8. Passive reciprocal exchange likewise transfers energy without supplying the loss; its scoped balance is retained there.

Controls reuse the existing normalized lift, graph pressure, configured phase row and tetrad readers: nonlinear orientation, tetrad projection and pressure defect, and full mean-work balance. Any proposed maintenance mechanism on this domain must account for sustained source work as well as angular response; none of these calculations installs a new phase law or controller. This is an admissibility condition for that mechanism, not an independent research queue.

19. Nonrepeated neighbors and local oriented transfer

The repeated-triple family suppressed one actual geometric input. Retain the fixed unit prism and unit capacities, but now give its two triangles independent form and primitive phase. Use the same normalized three-by-two lift U: x_a=mu_a*1+U z_a, theta_a=beta_a*1+U zeta_a. For b=1-a, let

\[ \gamma_{ai}=\arg\left(\sum_{j\ne i}e^{i\theta_{aj}}+e^{i\theta_{bi}}\right), \qquad r_a=U^T\gamma_a, \]

in one regular common lift of the phases and neighbor directions. The canonical source is (gamma_a-theta_a)/pi, not an arithmetic mean of phase differences. With EPI and phase weights e,w>0, k=w/pi, the nodal projection is exactly

\[ \dot z_a=-\frac{4e}{3}z_a+\frac e3z_b-k\zeta_a+kr_a, \qquad \dot\mu_a=\frac e3(\mu_b-\mu_a)+k(\overline\gamma_a-\beta_a). \]

Consequently the existing relative area ell_a=det(z_a,zeta_a) satisfies

\[ \dot\ell_a=-\frac{4e}{3}\ell_a +\frac e3\det(z_b,\zeta_a)+k\det(r_a,\zeta_a) +\det(z_a,\dot\zeta_a). \]

The first boundary term comes from the neighboring form; the second comes from the actual neighbor-phasor directions. Repeated phases make r_a=0 because every node sees the same three phases. Nonrepeated phases need not. Neither term specifies the missing primitive-phase evolution. If the existing optional phase row is used as a fresh-pressure comparison, keep zeta_a_dot=U^T f(p_a)+g U^T J_a and its pressure mean inside the sine. The regional phase mean also has boundary exchange:

\[ \dot\beta_a=\overline{f(p_a)}+ \frac g9\sum_i\sin(\theta_{bi}-\theta_{ai}). \]

Only the full-graph sine-current sum vanishes. Freezing either regional mean would change the model. These projections reuse the full-graph normalization and regional balance owner.

An actual boundary-source witness. Start with equal uniform positive EPI in both triangles, and choose a=pi/6, theta_0=beta+(-a,a,0), theta_1=beta+(-a,a,a). The global phase spread is pi/3, strictly within the regular chart. Put gamma=atan(1/(3*sqrt(3)))>0. Triangle zero's neighbor directions are beta+(0,0,gamma), so its source is (1/6,-1/6,gamma/pi). At this instant,

\[ z_0=z_1=0,\quad \zeta_0=(-\sqrt2a,0),\qquad \dot\ell_0=-\frac{w\gamma}{3\sqrt3}\ne0,\qquad \dot\mu_0=\frac{w\gamma}{3\pi}>0. \]

The area derivative is independent of the phase velocity at this instant, since det(z_0,zeta_0_dot)=0. Replacing the second triangle by the repeated phase triple makes the area derivative zero. Detached production captures reproduce the difference and keep stored-pressure defects separate. Because the initial internal form is zero, its angle is undefined: this proves an instantaneous nonparallel form response, not a finite rotation. The prepared phase pattern already contains structure; no creation from a uniform full state is claimed.

The interaction is real within the declared nodal source, but its ongoing supply is a separate question. The full-prism storage theorem now bounds all six independent phase/form directions in a declared domain; the local nonlinear result also excludes a missing small-amplitude oscillatory mode. Thus a regional change must be accounted for together with its environment. An instantaneous boundary contribution is not evidence of indefinite replenishment. Controls: regional area, means and actual source.

20. Phase-reset source work and actual occurrence

The existing Mutation proposal provides a concrete phase map to audit. Its default branch proposes theta_k^+=theta_k+s modulo 2*pi, where s=factor*copysign(1,p_k_stored)*pi/4; the default factor yields the represented shift |s|=1/4. For nonzero pressure this uses its sign; the implementation's signed-zero behavior is an additional map convention, not sign(0)=0. The stored sign need not match fresh pressure. The detached control below stores the fresh pressure before obtaining its proposal. This is a configured operator map, not a derived occurrence law. An admitted Mutation retains EPI, capacity, support and stored pressure. Consequently it has zero instantaneous EPI-energy jump and leaves the stored nodal rate unchanged. Its potentially useful effect is a different phase source at a later pressure refresh. The proposal alone certifies neither temporal eligibility, grammar, selection nor that refresh.

20.1 Exact source-work criterion for a finite phase proposal

Retain fixed unit capacity and the prism's pressure coefficients. For a fixed symmetric quadratic observable S=x^T Q x, let b=Qx. On a regular phase path theta(t)=theta+t*s*e_k, 0<=t<=1, the shared source derivative Dg_phi=(R-I)/pi gives

\[ \Delta\dot S_{\rm phase} =\frac{2ws}{\pi}\left[ \sum_{i\sim k}b_i\int_0^1R_{ik}(\theta+tse_k)\,dt-b_k\right]. \]

The path parameter is not physical time. The formula requires nonzero neighbor resultants and a fixed wrap branch along that path. Equivalently, each affected resultant changes exactly by Z_i^+=Z_i+(exp(i*s)-1)*exp(i*theta_k); the finite phase source must use those actual directions, not an arithmetic midpoint or linearized mean. If the path changes branch, its jump must be accounted for separately. For Dirichlet energy E_D=x^T Bx/2, use b=Bx and omit the factor two. Dirichlet rate, internal-amplitude rate and energy jump are distinct quantities.

At initial phase consensus, each of the target's three neighbors changes its mean direction by

\[ a(s)=\arg(2+e^{is}),\qquad a'(s)=\frac{1+2\cos s}{5+4\cos s}. \]

For 0<s<=1/4, 0<a(s)<s/3. If the target is a positive fresh-pressure maximum under pure EPI pressure p=-e Bx/3, then (Bx)_k is a negative minimum, and

\[ \Delta\dot E_{D,\rm phase} =\frac w\pi\left[-s(Bx)_k+a(s)\sum_{i\sim k}(Bx)_i\right]>0. \]

This is a positive increment of a prospective rate, not proof that its total rate becomes nonnegative. For the unchanged shared preparation x=(3/4,1/4,1/2) on each triangle, phase zero, e=1/2, w=1/4, and one default proposal at the first positive-pressure maximum, exact-real accounting gives

\[ \Delta\dot E_D=\frac{3}{64\pi},\qquad \Delta\dot S=\frac{1}{32\pi},\qquad \dot S^+=-\frac14+\frac{1}{32\pi}<0, \]

where S=sum_a ||x_a-mean(x_a)||^2. The proposal improves the amplitude rate but still does not hold this shape. Tests evaluate the pure native proposal and its copied phase-only endpoint through existing forcing and regional readers. They keep kernel arithmetic and stale-pressure defects distinct; they do not apply a Mutation operator or manufacture event history. Controls: finite proposal work.

20.2 Eligibility, selection and the missing sustaining law

Mutation's signed sampled growth gate uses actual history, not the current product nu*p. U4b context and final selector decisions are additional requirements. The default-policy reachability result shows that uniform positive capacities and fresh default Si force the default base choice to Coherence, even when Mutation's growth and grammar gates pass. Increasing source work in a copied proposal cannot bypass this obstruction.

This yields three separate questions: can the state admit Mutation, does the actual policy select it, and would its subsequent fresh source offset loss? Neither a favorable proposal nor an admissible operator label answers all three. The finite native control is recorded in the single execution plan; it does not promote the configured Si consumer to an emergent physical law. An absent Mutation also does not imply absent phase evolution in general: Coherence and the later phase-coordination step have their own phase maps.

20.3 Geometric admissibility does not require a reset

On this fixed support, consider only regular phasor geometry and the U3 edge condition. In a regular chart with signed difference delta_ij=theta_i-theta_j, the active boundary |delta_ij|=pi/2 gives the first-order necessary viability condition delta_ij*(omega_i-omega_j)<=0. It restricts a supplied phase velocity; when equality holds, that condition alone is not a finite-time viability proof. At a strict interior state, every fixed finite velocity is admissible for a sufficiently short interval by continuity.

There is a stronger exact non-necessity control: held phases and arbitrary common rotations preserve every phase difference and every neighbor-resultant magnitude for finite time. Neither regularity nor U3 alone therefore requires a nonzero phase reset, selects its magnitude, or fixes an event time. A zero neighbor resultant instead makes its direction undefined; evaluating that undefined direction cannot supply a unique continuation. A separate continuation rule and evidence would be required. No crossing is simulated or asserted here, and additional coupled constraints could change the problem.

Thus the failure of the tested configured mechanism cannot be repaired by calling its reset geometrically inevitable. The current constitutive gap is a justified joint phase/event law; another diagnostic threshold or supplied operator sequence would not close it.