Successor motion: sufficient state and finite principal confinement

The principal-chamber theorem excludes stationary storage minima on the proposed maintained-parent UM support. Its source-specific comparison path also reaches a principal cut while preserving storage and both means. Neither result describes the source's motion. This owner admits the complete state and analytical bounds for nonstationary retention under the unchanged law. The fixed conditional certificate retains all 53 branches through structural time 1/2 and proves a child-parent relative phase turn within that window.

The research program owns gate status and subsequent scope. The discussion below introduces no new pressure, event selector, primitive coordinate or physical identification. This certificate was computed for a hypothetical UM family from the retained THOL source. A separate committed event and primitive inclusion review now admits the actual UM endpoint into that family without repeating the motion.

1. Complete state, law and identity

Keep all 21 nodes and 53 edges of the re-admitted successor chamber. Use its exact node order, positive symmetric conductances W, positive held capacities nu, bare degrees d, original structural clock and zero forcing. Signed form x and declared real phase lifts theta supply the 42 continuous coordinates. Support, W and nu have zero derivative on this segment.

For the selected beta=1 instance of the weighted complete law, write

\[ M=\operatorname{diag}\frac{\nu_i}{\pi d_i},\qquad S_i(\theta)=\sum_{j\in N(i)}\sin(\theta_j-\theta_i),\qquad \dot x=MS(\theta),\qquad\dot\theta=ML_Wx. \]

Pressure is independently specified as p_i=S_i/(pi*d_i). The stored post-relief UM pressure is not substituted into this continuous row. Conductances enter the form gradient in the phase row, whereas phase pressure and degree normalization retain bare support. This is the declared conservative completion, not the general operator runtime or native resultant-pressure flow.

The storage and two conserved weighted means are

\[ H=Q_W(x)+V(\theta),\qquad Q_W=\frac12x^TL_Wx,\qquad V=\sum_e(1-\cos\delta_e),\qquad \rho_i=\frac{d_i}{\nu_i}. \]

Both means use rho; neither is replaced by arithmetic centering. Each conditional source member carries its own fixed W, nu, H and means. These structural quantities require their own measurement and clock bridge before any laboratory interpretation.

Orient every edge (i,j) as i<j and let B have row e_j^T-e_i^T. The retained branch vector k gives

\[ \delta=B\theta-2\pi k,\qquad |\delta_e|<\pi. \]

All 53 inequalities define the proposed principal identity. Its 33 fundamental cycle rows C obey C*B=0, so C*delta=-2*pi*C*k. Retaining just the two older constituent winding numbers would discard part of this identity. A principal cut is not a singularity of the smooth sine vector field.

2. Phase velocity retains form information; phase alone does not

The existing form/phase/memory representation provides the method of differentiating the complete phase row and retaining its initial velocity. Its unit-support coefficients cannot be copied to the present weighted attachment. With the actual held matrices here,

\[ \boxed{\ddot\theta=A S(\theta),\qquad A=ML_WM,\qquad \dot\theta(0)=ML_Wx_0.} \]

The constraint rho^T*dot(theta)=0 is essential. A freely initialized second-order equation would otherwise admit an extra mean-phase velocity absent from the original complete law. On the rho-mean-zero subspace, ML_W is invertible: its kernel consists of constants, and its image has zero rho mean. Thus the original form mean and phase velocity reconstruct the full form. This is a representation with the same information, rather than a first-order reduction to phase alone.

For clarity, A above denotes the symmetric acceleration matrix of this weighted model; it is not the differently defined A in the linked unit-law owner. It is positive semidefinite with rank 20 and null vector rho. This property does not establish attraction or a minimum of V.

Integrating the displayed equation gives the exact history identity

\[ \theta(t)=\theta_0+tML_Wx_0+ \int_0^t(t-s)A S(\theta(s))\,ds. \]

The initial form remains an independent source through ML_W*x_0. Eliminating it from a displayed state does not remove it from the dynamics. This is the same hidden-initialization obligation identified by the general memory owner.

There is a direct counterexample at the proposed source's own law and phase state. Let mu_x=(rho^T*x_0)/(rho^T*1) and define

\[ x_+=x_0,\qquad x_-=2\mu_x1-x_0. \]

These admitted signed forms have the same form mean and the same Q. Keep the same theta, W and nu: both full states then have the same phase, phase pressure, storage, both means and all principal branch labels. Yet their phase velocities are opposite. The source storage admission has Q_W(x_0)>0, so connected positive conductances ensure that these velocities are nonzero. At least one edge velocity also differs: a common phase velocity with zero rho mean would have to vanish.

Consequently the instantaneous phases, their pressure and these conserved values do not determine even the direction of every branch's motion. This is a sufficient-state obstruction, not a second preparation to execute or a change to the reserved source. It does not exclude a separately proved restricted invariant family.

3. Exact edge acceleration and an energy velocity bound

Since S=-B^T*sin(delta), the same complete law gives

\[ \boxed{\ddot\delta=-J\sin\delta,\qquad J=BML_WMB^T, \qquad\dot\delta(0)=BML_Wx_0.} \]

J is symmetric positive semidefinite of rank 20. The edge state retains C*delta=-2*pi*C*k and C*dot(delta)=0; 53 independently initialized angles and velocities would be a different model. Retaining these constraints and the original two means gives another full-information representation. It neither closes the stationary four-coordinate map nor forgets the initial form. For interval calculations, each parameter member keeps its own constant J throughout the window.

Let b_e=e_j-e_i for edge (i,j) and define

\[ a_e=(Mb_e)^TL_W(Mb_e)=J_{ee} =M_i^2\sigma_i+M_j^2\sigma_j+2M_iM_jW_{ij}, \qquad\sigma_i=\sum_jW_{ij}. \]

The semidefinite Cauchy inequality in the actual form quadratic gives

\[ |\dot\delta_e|^2 =|(Mb_e)^TL_Wx|^2\leq2Q_W(x)a_e\leq2Ha_e. \]

The last step uses V>=0 and the exact conserved H. It does not assume prior principal confinement: the complete sine law and the nonnegative potential continue across cuts. Therefore the condition

\[ \pi-|\delta_e(0)|-T\sqrt{2Ha_e}>0 \]

is sufficient to retain that original edge branch throughout [0,T]. Whole-source interval upper bounds for H and a_e, and a lower bound for the initial margin, make it a uniform conditional certificate. They cannot mix one endpoint's means or energy with another endpoint's law.

A failed inequality is only an insufficient bound. It is not an observed crossing or evidence of inconsistent dynamics. Similarly, |ddot(delta_e)|<=sum_f |J_ef| is a valid global acceleration bound, but can discard substantial cancellations between correlated edge currents. The full field, cycle constraints and source correlations can therefore matter when a scalar bound does not decide the complete identity.

The static motion owner rebuilds the hypothetical reset primitives before evaluating these bounds. predict_successor_motion returns separate ideal and conditional reports, including signed initial edge rates and accelerations, source storage, energy coefficients, speed bounds and horizon margins. Its alternative global acceleration enclosure uses sparse rows of A:

\[ |\ddot\delta_{ij}|\leq\sum_k|A_{jk}-A_{ik}|d_k, \]

because |S_k|<=d_k. This global magnitude bound differs from the signed instantaneous acceleration. The owner does not evolve a source or certify a cut from a failed margin. Its independent controls check the complete rows, energy inequality, domains and source immutability.

3.1 Static admission of the unchanged proposed source

Fresh interval evaluation of the retained THOL source and hypothetical UM reset, at T=1/2, certifies the energy inequality for exactly 41 original edges in both the ideal and conditional reports. Every retained conditional margin is strictly greater than 593/1000. The 12 remaining edges are exactly those incident to parent node 2. Their bounds do not certify this horizon; the statement is not that these edges cross a cut.

For the oriented child-minus-parent edge (2,20), the entire conditional source interval satisfies the strict rational enclosures

\[ -\frac1{6000}<\delta_{2,20}(0)<-\frac1{7000},\qquad -\frac{269}{50}<\dot\delta_{2,20}(0)<-\frac{537}{100}, \]

\[ -\frac{77}{100}<\ddot\delta_{2,20}(0)<-\frac34, \qquad1475<J_{ee}<1476. \]

These bounds use the actual unchanged preparation, held coefficients and complete field; they are static calculations, not a motion response. The large diagonal edge-acceleration coefficient also makes a straight-line extrapolation of the initial negative velocity insufficient. The remaining currents contribute to the same row, so this coefficient alone supplies neither a frequency nor a restoring-sign theorem. The half-unit attempt below tests a resolved direction change on the full coupled law instead of merely certifying an arbitrarily short continuity window.

4. Which reductions can describe motion

The stationary reduction imposes equal true-twin phases, equal return-chain gaps, aligned middle nodes and equality of the leaf and parent phases. Their role in classifying minima does not make all these conditions invariant. For example, equal consecutive chain gaps require both 2*theta_16-theta_14-theta_18=0 and the corresponding rate equality 2*dot(theta_16)-dot(theta_14)-dot(theta_18)=0 if that constraint is to persist. Middle alignment and a leaf's stationary equality have their own tangency conditions. The complete phase row consumes form contrasts that the four-coordinate map does not retain.

True twins provide a more limited useful symmetry. For any transposition preserving the actual bare support, W and nu, both complete rows commute with the transposition. Uniqueness then makes its simultaneous equal-form, equal-phase fixed-point subspace invariant. Nonuniform capacities across different groups do not invalidate this argument; unequal capacities or consumed weights within a proposed symmetry do.

The nine listed twin pairs in the principal owner give a possible exact 12-group synchronized description of the ideal source, with parent 2, node 3 and child 20 still distinct. Admission needs the exact primitive pair equalities and coefficient symmetries: the ideal UM map gives paired phase updates and paired contact scores from equal inputs. Equality of cached interval outputs alone would not prove that premise. This uses the method of the weighted pair quotient, with new group multiplicities and actual conductances, not its unchanged unit-interpair coefficients.

The existing conditional envelope allows independent errors in changed phase outputs and new conductances. It therefore does not impose those exact pair equalities on every member. An ideal quotient cannot silently replace that full uncertainty family. A validated quotient plus transverse error analysis would need its own proof; it is not assumed in the fixed attempt below. The ontology likewise distinguishes a sufficient effective description from the existence of the organization being described.

5. Fixed prospective confinement and turning test

The analytical attempt retained the unchanged source and froze one structural-time horizon and one computational budget before evaluating its reserved response:

• Horizon T=1/2, divided into exactly 256 steps of 1/512.
• Validated Taylor order 12 on the full 42-component form/lifted-phase state.
• A 900-second wall budget checked between steps; an already started final step may finish after the budget. Wall time is a resource limit, not the structural clock or a mathematical confinement horizon.
• The complete 21-node, 53-edge conditional source box, with held parameter uncertainty and all original branch integers; the ideal endpoint remains a separate object rather than a replacement for the box.
• A strict positive margin for every original principal edge on every complete step, combining the full tube and the independent conserved-energy bounds where available, rather than checking only stored endpoints.
• At least one positive child-minus-parent phase-velocity enclosure at the fixed endpoints t_n=n/512, 1<=n<=256, following its strictly negative initial enclosure. The successful endpoint, if any, must certify that sign for the whole admitted source family.

The two requested outcomes are joint: retained principal identity through the whole window and a direction change of the child-parent relative phase inside it. The latter distinguishes the attempt from arbitrarily short retention guaranteed by continuity. It would establish a finite turning motion, not a period, attraction, formation, all-time maintenance or an autonomous event. No laboratory clock is inferred.

The endpoint set is fixed in advance. Checking the stated existential sign criterion on that set does not authorize searching new horizons, selecting a different observable or retuning the source after seeing the response. Whole-tube inclusion and remainder admission are required; small numerical residuals or endpoint margins alone do not establish them. A rectangular Taylor tube may extend beyond a cut when the independent energy inequality still proves that the actual trajectory cannot reach it; raw tube and combined retention margins retain distinct provenance. An enclosure that proves neither requested result remains explicitly unresolved under this attempt. A subsequent calculation would require a separate admission rather than rewriting the evaluated protocol.

This section records the prospective protocol and its mathematical rationale; the retained outcome is reported separately below. The execution plan remains the only queue for any subsequent work.

The fixed-attempt consumer uses the shared held sine field and validated Picard/Taylor owner. It retains all 42 source coordinates and every original branch, records each attempted step before continuing, and distinguishes unavailable enclosures, unresolved branch bounds and resource exhaustion from actual escape. It does not itself authenticate protocol-freeze chronology or execute a structural operator.

6. Retained finite confinement and turning result

The protocol, primitive source and complete Python source archive were frozen before the single attempt. The retained result records all 256 accepted steps, reaching exactly T=1/2 within the fixed 900-second wall budget. No step-size, source, order, horizon or criterion was retuned.

Every one of the 53 original principal branches remains strict over every whole step. The minimum certified combined retention margin obeys the exact rational lower bound

\[ \min_{e,\,0\leq t\leq1/2} \bigl(\pi-|\delta_e(t)|\bigr)>\frac{158}{100}. \]

The initial child-parent relative velocity is strictly negative as admitted in section 3.1. The first predetermined endpoint with a uniformly positive velocity enclosure is t=21/512; its step record gives the coarser strict containment

\[ \frac9{50}<\dot\delta_{2,20}(21/512)<\frac{19}{100}. \]

Continuity therefore gives a relative-phase direction change in (0,21/512) for every member of the conditional family. This endpoint is the first certified positive member of the frozen observation set, not an exact turning time shared by all family members. The full-window branch certificate and this sign change satisfy the joint prospective target.

The statement covers the complete independent 2^-40 conditional source envelope, including the ideal mathematical reset, under each member's own held W and nu and the declared unforced complete law. It is a validated mathematical continuation. That certificate computation executed no UM event, represented runtime trajectory or SHA closure. Applying it to an actual endpoint requires separate arithmetic, support, grammar and primitive-source inclusion admission.

The later single runtime attempt retains an included raw reset but rolls back after a diagnosed pressure-guard false rejection. Its detached refreshed copy is not a committed successor. That failure and its guard correction alone establish no runtime transfer. The separately frozen corrected UM transaction now supplies a committed endpoint. Its independent primitive review admits all continuous coordinates, held parameters and branch labels into this same family. This establishes theorem applicability to that endpoint under the declared complete law and clock, without executing a new trajectory or changing any frozen motion evidence. A separate SHA event and changed-law admission now completes the selected word; this certificate does not extend through that capacity reset or certify the new continuation. The separate actual post-SHA result retains all branches through 1/2 after rebuilding its changed field and source. Both adapters now share the fixed principal-retention loop, while this study preserves its original conditional family, reversal target, record schema and immutable source archive.

The result separates two previously unresolved possibilities: absence of a stationary storage minimum does not prevent finite principal confinement, and an invariant shell's access to a cut does not force the actual flow to take that path during this window. It establishes neither periodic motion nor all-time maintenance, a newly formed stationary minimum, autonomous organization or a physical particle interpretation.

The retained-evidence controls check frozen byte associations and recompute consumed field, branch and sign readouts from primitive step data without rerunning the producer. They check Picard inclusion and endpoint arithmetic; the recorded derivative jets and remainder bounds retain their frozen-producer provenance and are not regenerated by this audit. Those checks remain distinct from the shared kernel's mathematical enclosure obligations. Hashes associate evidence bytes rather than authenticate their acquisition or chronology.

7. Effect of a single-parent IL phase reset

The complete IL action and its single-target work identity supply a reusable mechanism for one public or staged IL action at parent 2. That mechanism preserves x, W, support and nu while moving the parent's phase toward its bare-neighbor resultant. Consequently M, rho and J remain unchanged, as does the complete instantaneous phase velocity ML_W*x, including the child-parent relative velocity. Its subsequent acceleration can change because the sine-current vector changes.

Conditional on a nonzero resultant R*exp(i*mu), an admitted shortest displacement a=wrap(mu-theta_2) and 0<=lambda<=1, the continuous event lift changes only theta_2 by lambda*a. The existing single-target identity gives

\[ \Delta H=R\,[\cos a-\cos((1-\lambda)a)]\leq0. \]

The inequality follows on |a|<=pi; a regular source admits its chosen shortest-arc branch before using the map. It supplies no numerical result for this successor until the actual resultant, branches, default coefficient, live preconditions and grammar context are re-admitted. The form mean is unchanged, while the lifted-phase mean has the event increment rho_2*lambda*a/sum(rho) in this declared lift. A changed mean is not a failure of the between-event conservation law.

For this one-target map, the 41 non-parent initial edge gaps and their energy coefficients are unchanged, and H cannot increase. Their existing half-unit energy certificates therefore transfer whenever the same original branch identity is retained. The 12 parent edges need fresh initial margins. This argument does not transfer the full frozen motion response to a new phase source outside its admitted box. It also cannot introduce a stationary storage minimum inside the original principal chamber: the global minimum exclusion depends on the unchanged support, not on which source reached that chamber.

The pressure-only IL glyph would leave the consumed continuous state unchanged; immediate constitutive refresh would remove its stored-pressure contraction. The public/staged IL phase reset is therefore essential to this proposed comparison. Multiple simultaneous adjacent targets require the existing shared-edge work correction and are not covered by the single-target inequality above. These are analytical admission conditions, not an evaluated IL event, completed word or new motion certificate.

These identities are not needed for the finite confinement certificate in section 6, which was obtained without an IL action. They supply no new reserved comparison or automatic requirement to add that operator.