Mutation action: phase work, contextual descent and geometric identity

ZHIR has a compatible conditional event map on the selected complete sine law. Its observed-growth gate does not determine the sign of its storage work, preserve every acute phase sector or select an occurrence time. The actual phase jump changes fresh pressure and form rates while retaining the phase-rate row. Associated histories, target marks, pressure branch and grammar remain necessary parts of collective execution state.

The admission owner establishes general temporal compatibility separately from one restricted NAV successor. This owner completes the event calculation on general admissible states. No retained research source is mutated and no continuous response campaign is evaluated. Runtime histories below are explicitly manufactured reader inputs, not observed responses of this law.

1. Complete law, event domain and actual action

Use a finite simple loop-free connected undirected graph with at least two nodes, positive conductances, signed real form x, circular phase theta and held positive capacities nu. On each continuous segment the selected law is

\[ q=L_Wx,\quad S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i),\quad M=\operatorname{diag}\frac{\nu_i}{\pi d_i},\quad p_i=\frac{S_i}{\pi d_i},\quad \dot x=MS,\qquad\dot\theta=Mq. \tag{1} \]

Here d counts distinct bare neighbors. Conductances enter q, not S or d. Pressure is independently specified by (1); it is not reconstructed from an observed response. The structural clock, unforced flow and fixed support are premises. The weighted-law owner provides its storage and support scope.

At a supplied event, retain target set A and the pre-event runtime pressure r, which can differ from the freshly evaluated p. The actual Mutation proposal chooses one of two phase branches:

\[ x_i^+=x_i,\qquad\nu_i^+=\nu_i,\qquad r_i^+=r_i,\qquad \theta_i^+=\theta_i+\delta_i\pmod {2\pi}, \tag{2} \]

with delta_i=0 off A. On A the default branch uses delta_i=f*copysign(1,r_i)*pi/4; the canonical ideal f=1/pi gives magnitude 1/4 radian. An explicit ZHIR_theta_shift instead supplies the fixed shift. The factor owner admits the active finite factor and excludes a full-turn no-op. Its runtime arithmetic and modulo operations remain distinct from the ideal equations.

The default branch retains the sign bit at zero: positive and negative zero select opposite rotations. The fixed branch does not consume the dynamic direction or its inactive factor, but still requires the temporal gate. ZHIR consumes no random draw. It preserves support, conductances and EPI_kind; the latter is a semantic identity field, not a geometric proof.

Both branches require active capacity and an accepted live signed two-sample EPI secant strictly above xi, plus independent grammar context. Prior IL and a recent destabilizer are required; an instantaneous nu*r prediction cannot replace the secant. Histories and thresholds retain their time basis. Timestamped evidence ends at the current form, but the pure reader alone does not authenticate its last timestamp against an execution clock.

The atomic stage proposes every target from one immutable snapshot and returns target-associated decision observations, including admitted evidence, the actual phase map and recent context. It records grammar/lifecycle changes after commit. Quarter-circle regime labels and an acceleration-threshold flag are diagnostics. Two EPI samples suffice for the growth gate but not for a three-sample acceleration observation: its legacy numeric zero must not be interpreted as measured zero acceleration. Detection creates no child or support edge and proves no dynamical bifurcation.

2. Complete work and both refreshed evolution rows

For any declared real lifts, use

\[ H(x,\theta)=\frac12x^TL_Wx+ \sum_{\{i,j\}\in E}[1-\cos(\theta_j-\theta_i)]. \tag{3} \]

Equation (1) conserves H between events because its two contributions are q^T MS and -S^T Mq. This auxiliary storage is not identified with a physical energy. Let g_ij=theta_j-theta_i and h_ij=delta_j-delta_i. Substitution in (2), with each undirected edge counted once, gives the exact ideal work

\[ \Delta H=\sum_{\{i,j\}\in E} [\cos g_{ij}-\cos(g_{ij}+h_{ij})]. \tag{4} \]

There is no form-storage jump, conductance factor in the phase term or capacity-storage term. Nevertheless the unchanged form contribution still uses the actual weighted Laplacian. Different source or target conductances cannot be replaced by the old unit graph when evaluating complete H.

For adjacent targets, (4) includes their common edge once. With equal shifts, that edge has zero work. Adding two isolated-target costs generally counts a change that never occurs in the joint event. At g_ij=0 with both shifts sigma, joint work minus the sum of isolated work is -2*(1-cos(sigma)). This is independent of the edge's conductance.

After constitutive refresh the exact changed rows are

\[ \Delta p_i=\frac1{\pi d_i}\sum_{j\sim i} [\sin(g_{ij}+h_{ij})-\sin g_{ij}],\qquad \Delta\dot x_i=\nu_i\Delta p_i,\qquad \Delta\dot\theta_i=0. \tag{5} \]

The last identity follows from unchanged x, capacity and support, hence unchanged Mq. Even untargeted neighbors can change pressure and form rate. Stored pressure is unchanged by the event itself; a refresh is a separate constitutive handoff. Continuing with held r instead of the new p is a different complete evolution model.

Set rho_i=d_i/nu_i and R=sum rho_i. The two continuous mean leaves use these weights. With the event lift explicitly chosen as theta^+=theta+delta,

\[ \Delta\bar x_\rho=0,\qquad \Delta\bar\theta_\rho=\frac1R\sum_i\rho_i\delta_i. \tag{6} \]

A principal circular coordinate alone does not define this lifted mean. Adding a full turn to a chosen event lift changes (6), although it does not change (4) or (5). The retained lift therefore belongs to any mean claim. A common shift of every node is an ideal global phase symmetry: storage, fresh pressure and both rows remain unchanged while the lifted phase mean changes by that shift. Runtime modulo rounding requires endpoint assessment.

Under an affine clock change s=c*t+b with c>0, scaling every capacity by 1/c scales both rows of (1) by 1/c while leaving p unchanged. Associated timestamps and xi must transform with their declared units, as in the admission owner. The angular shift, H work and normalized event means then stay the same. Keeping the numeric growth threshold while changing its time unit is a different admission policy.

3. Growth eligibility imposes no passivity theorem

Consider a target with phase zero, every neighbor at epsilon in (0,1/8), and the default positive shift sigma=1/4. Fresh pressure is positive and

\[ \Delta H=d_i[\cos\epsilon-\cos(\epsilon-1/4)]>0. \tag{7} \]

The event overshoots the neighbors: 1/4-epsilon>epsilon, and cosine strictly decreases with the magnitude of these small gaps. Take epsilon=1/10, nu_i=4 and xi=1/10. Then

\[ \dot x_i=\frac4\pi\sin(1/10)>1/10. \tag{8} \]

For example, sin(1/10)>1/10-(1/10)^3/6 and pi<22/7 prove the strict inequality. Any finite form is allowed. Smooth local backward flow exists, and continuity keeps the rate above xi on some short preceding interval. Integrating proves that genuine same-law positive secants can coexist with positive event work. No numerical trajectory is needed for this existence argument; no particular manufactured runtime record is authenticated by it.

Conversely, with every neighbor at 1/4 the same positive shift exactly aligns the target and gives Delta H=d_i*(cos(1/4)-1)<0. Capacity 4 also gives a strict rate margin above xi. Thus both signs occur inside regular acute zero-winding preparations with true short-secant compatibility. The observed growth test neither chooses a storage sign nor supplies an event reservoir.

At an exactly synchronized phase state, a default signed-zero direction instead yields positive work d_i*(1-cos(1/4)) at a single target. This additional implementation witness concerns the branch convention; it does not claim an accepted supplied history was generated from a stationary law.

4. Sufficient geometric preservation and an allowed obstruction

Fix principal edge representatives g_ij in an acute source sector. If

\[ |g_{ij}|+|\delta_j-\delta_i|<\pi/2\quad\hbox{for every edge}, \tag{9} \]

then the straight event interpolation theta+s*delta, 0<=s<=1, keeps every edge in the same principal chart and acute sector. The reference integers do not change, so each oriented cycle winding is preserved. This is a sufficient event condition; it establishes neither later maintenance nor confinement to an older joint form/phase ball.

One positive-width general domain has all phases in a common lift interval of width less than 1/4. Under arbitrary targeted default shifts of magnitude 1/4, the new interval has width less than 3/4<pi/2. This preserves the acute zero-winding class. During the actual phase-aligning IL stages, a defined neighbor resultant within an interval of width less than pi lies in that same interval; interpolation by lambda in [0,1] remains there. Consequently this phase-domain argument extends through admitted surrounding IL/OZ/SHA stages with the usual IL alignment factor: OZ and SHA do not change phase. All other state, history, capacity, grammar and U6 hypotheses still require their own admission; this phase argument does not replace them.

No unconditional acute-preservation claim holds even for default ZHIR and fresh pre-event pressure. A star with center phase 0, two leaves at 7/5 and one at -7/5 has all gaps acute. The center has

\[ p_0=\frac{\sin(7/5)}{3\pi}>0, \qquad\delta_0=1/4. \tag{10} \]

Its trailing gap becomes -7/5-1/4=-33/20, whose magnitude exceeds pi/2. At center capacity 1, the rate is also greater than xi=1/10, so the same local-flow argument supplies short genuine compatible secants. Strict source, rate and exit margins persist on a neighborhood. Valid runtime grammar and unchanged semantic kind therefore do not guarantee acute geometric identity. The star has no cycles: this example proves sector exit, not winding change. For the rate margin, the alternating lower bound through degree seven gives sin(7/5)>33/35, while 3*pi/10<33/35 follows from pi<22/7.

5. Collective descent and complete-word context

For a permutation-preserving support and capacity profile, the fine phase map is equivariant when targets, pressure directions/fixed shifts, configurations and consumed histories transform with their members. A retained contextual collective state containing that association therefore determines the event. On a twin-invariant stratum, an orbit-closed target set with equal shifts on each pair preserves the stratum. Targeting one member can leave it even while remaining acute. It then requires the larger fine state or a retained mark; the old reduced dynamics cannot be asserted for the new state.

The marked contact's eleven-cell reduction has two singleton ports and nine double cells. Acting on either singleton or on an entire double cell with equal shifts preserves its physical symmetry, conditional on the same associated context. Equal physical coordinates alone need not descend: one history can pass the live secant gate and another fail; a recent destabilizer can be present or missing. Even numerically equal +0/-0 stored pressure can choose opposite runtime shifts. The continuous mean or an unordered inventory of histories loses these distinctions.

The complete control word is

IL -> OZ -> IL -> ZHIR -> IL -> SHA.

Starting with initialized form and zero debt, its successive U2 debts are 0,1,0,1,0,0. The pre-ZHIR IL discharges OZ's debt while leaving OZ recent enough for U4b; the next IL handles ZHIR, and SHA closes the word. Every stage still needs live preconditions and its own actual observation. The shorter IL,OZ,ZHIR,IL,SHA is a valid grammar word but retains one unit of debt, so it does not satisfy the selected zero-terminal-debt condition. No initial stabilizer can prepay future destabilizations.

The synthetic complete-word control supplies timestamped EPI reader records before execution. All its chosen stages preserve form, so they do not create an EPI endpoint mismatch. They do not manufacture a new continuous response or turn operator steps into elapsed structural time. Local action controls use ZHIR's genuine validated-word step with an explicitly supplied matching live prefix context; they do not claim that such a prefix ran in that probe.

6. Execution evidence and remaining scope

The collective Mutation action controls reuse the actual public atomic stage and detached sine field owner. They check independent complete work, adjacent-target corrections, both work signs with strict nodal rate margins, raw versus refreshed pressure, both rows and lifted means, signed-zero/fixed branches, associated decision records, complete-word debt and finite U6, symmetry loss and the acute-exit counterexample. One invalid target rejects the whole phase stage before graph-owned commit. Actual represented endpoints are assessed separately from ideal arithmetic, with no second executor or new event policy.

These results complete ZHIR's bounded action compatibility under the stated law and execution premises. They do not derive occurrence, a preferred phase shift, universal collective closure, permanent identity or a physical particle. Other registered operators keep their own event and history contracts; the existing mechanism map and execution plan own cross-operator scope and subsequent work.