Canonical contact between maintained weighted stars

This study admits an actual registered action on the state-complete weighted stars. The earlier unit-bridge comparison is a control, not an operator output specification. This is an analytical admission with declared current-engine static controls, not an acquired-source response, autonomous event selection or continuous trajectory campaign.

Canonical here identifies the registered preparation and its admitted grammar. The held weighted sine continuation is a separately declared complete model; executing the word does not derive that continuation uniquely. Its full real-lift tangent is a comparison instrument with its own domain and storage. The law-admission owner separates execution conformance, constitutive justification and linearization.

1. Prospective source, word and discriminators

The following source, action and checks are fixed before evaluating the static word. Preserve their scope if a precondition or prediction fails.

• Ordered nodes are 0,...,9. Each block (0,...,4) and (5,...,9) is the original five-node star: three center-leaf edges of conductance three and four member-boundary edges of conductance two. Supply the first block's edges before the second's, in the shared star owner's order. Original edge lengths are one. There are initially no cross edges.
• For each block, forms are (m+a,m+a,m+a,m+a,m) with a=1/16, and capacities are (4,2,2,2,8). In the ideal family, 15/64<=m<=17/64; left phases are zero and all right phases are the same declared radian lift 3/8<=alpha<=2/5. This common form origin is a newly supplied preparation; continuous translation invariance does not establish invariance of UM's absolute-form admission or affinity policy.
• Every node has auxiliary Si=1/2, stored pressure zero and an empty glyph history; no prior emission or latency metadata is supplied. The isolated continuous law is conservative weighted-form/bare-sine exchange with original structural time, zero form loss and unit exchange/storage coefficients. Its source pressure is zero. Capacities are held between events. Stored event pressure and refreshed continuation pressure remain separate channels; no independent pressure-memory law is introduced.
• Select only boundary node 4. Use the registered tokens emission,coupling,silence, validate the complete word with the core compatibility profile and pass its contexts to execute_network_operator_stage. SHA is this word's selected closure, not a compulsory ending. Set strict grammar rejection and live operator precondition validation. There is no all-node dispatch.
• Set AL_boost=UM_theta_push=1/1024, UM_vf_sync=UM_dnfr_reduction=1/8, SHA_vf_factor=1/2. Enable all four UM flags: bidirectional, capacity synchronization, stored-pressure stabilization and functional links. Declare UM_COMPAT_THRESHOLD=3/4, UM_CANDIDATE_COUNT=0 and proximity mode, with the full node inventory and no restricted sample. Both U3 limits are the represented math.pi/2.
• Declare EPI_LATENT_MAX=1, VF_BASAL_THRESHOLD=SHA_MIN_VF=1/8, UM_MIN_EPI=UM_MIN_VF=1/8, minimum coupling degree one, form rails [-1,1] and hard clipping. The predicted states do not reach a rail. Use seed 17, history window eight and structural time _t=7 throughout. Static controls supply the fixed UTC event timestamp 2030-01-02T03:04:05+00:00; it is provenance, not elapsed structural time. Optional diagnostics, callbacks and user extensions are absent.
• The finite represented probes are (m,alpha)=(1/4,201/512), (15/64,3/8) and (17/64,409/1024). These are exact binary64 dyadics inside the declared domain, not rounded replacements for exact pi/8. Forms, capacities and these phases are materialized in binary64; backend and actual input/source configuration are retained by the static tests. Independent high-precision score checks use the represented affinity regularizer float(1e-9) and tolerance 2**-46. The proof and actual represented outputs have separate arithmetic obligations.
• Predict the complete support and primary reset from the registered maps, then inspect grammar/lifecycle effects, strict rejection controls and stage atomicity. Check the actual weighted continuation and all storage work, including old-edge coordinate changes. Rebuild both complete rows, their degree normalization and pressure from the actual successor. Retention must use the original positive star subgraphs plus the actual nonnegative added-edge storage, not the fixed unit-bridge reader.
• The discriminators are an admitted complete word, a strictly positive joint identity margin, nonzero initial component exchange and the presence or absence of exact collective-family tangency after contact. Check a blocked UM after committed AL against that post-AL snapshot; a word has no promised whole-word rollback. No failed case authorizes a parameter retuning, silent glyph substitution or bypass of U3.

No reserved physical observation or historical response is evaluated here. An actual acquired-source campaign would require its own source archive, protocol, numerical/measurement budget and retained response.

2. Word admission and complete primary reset

Write b=1/1024 and c=a-b=63/1024. The following algebra first describes the ideal-real maps on the source family. The frozen represented probes retain their separate arithmetic checks in Section 7.

AL admits the selected source form m<1 and capacity 8>1/8. Its only primary coordinate change is

\[ x_4:m\longmapsto m+b. \]

The form rail is inactive. UM then reads the four existing neighbors 0,1,2,3, whose phases and the target phase are all zero. Their consensus and all bidirectional phase proposals are zero. No right-node phase contributes to this reset: those nodes are new-link candidates, not existing alignment neighbors. Thus every phase is unchanged by the complete word.

The target form after AL exceeds 1/8, its capacity exceeds 1/8 and its four existing neighbors satisfy U3 exactly. Their capacity mean is (4+2+2+2)/4=5/2. The configured synchronization and silence maps therefore give

\[ \nu_4^{\mathrm{UM}} =8+\tfrac18(\tfrac52-8)=\tfrac{117}{16},\qquad \nu_4^+=\tfrac12\nu_4^{\mathrm{UM}}=\tfrac{117}{32}. \]

All other capacities are unchanged. Stored pressure starts at zero; UM's multiplier is 1-1/8=7/8, so its target stored-pressure write is also zero. AL and SHA do not supply a nonzero stored-pressure change. SHA sees capacity 117/16>1/8 and retains the post-AL form.

The ordered core word is AL,UM,SHA: AL supplies the opening, UM retains its independent live U3 obligation, and SHA is this word's admitted closure. Only node 4 is a target. Its resulting glyph history is (AL,UM,SHA) and its operator step count is three; old neighbors receiving phase writes do not thereby execute those glyphs. The complete stage implementation also owns activation, latency, preserved-form and context metadata. Those records are checked through the shared lifecycle, not inferred from the primary reset equations. Structural time remains 7; the three events are not three elapsed units of the subsequent continuous law.

The execution boundary is stage-wise. A rejected UM must leave its own post-AL input snapshot unchanged, but it does not undo the already committed AL. Neither a valid word nor its successful primary reset implies whole-word rollback or autonomous occurrence.

3. Actual affinity, candidate inventory and five-edge contact

The live phase-affinity owner uses 1-|wrap(theta_j-theta_i)|/pi, not cosine. Its score is a configured operator policy, distinct from the sine dynamics and cosine storage. With equal Si=1/2 and positive forms, the ideal-real functional-link formula reduces to

\[ W_{4j} =1-\frac{\alpha}{2\pi} -\frac{|x_4-x_j|}{4(x_4+x_j+\epsilon_E)}, \qquad \epsilon_E=\operatorname{float}(10^{-9})>0 . \]

The regularizer is the represented implementation constant. Runtime evaluation normalizes the two forms before the subtractions and divisions; this real formula does not prescribe a different binary64 operation order.

The target's only nonneighbors are the five right nodes. All have phase gap alpha<=2/5<pi/2, so both the candidate and final phase gates admit them. With the full candidate inventory, exactly five new edges are proposed. Their two ideal weight classes are

\[ \begin{aligned} W_A&=1-\frac{\alpha}{2\pi} -\frac{c}{4(2m+a+b+\epsilon_E)} &&\text{on }(4,5),(4,6),(4,7),(4,8),\\ W_B&=1-\frac{\alpha}{2\pi} -\frac{b}{4(2m+b+\epsilon_E)} &&\text{on }(4,9). \end{aligned} \]

Uniformly on the declared ideal family, every new form difference is strictly less than a=1/16, and its positive endpoint sum exceeds 2m>=15/32. Using pi>3 gives

\[ \frac{\alpha}{2\pi}<\frac1{15},\qquad \frac{|x_4-x_j|}{4(x_4+x_j+\epsilon_E)}<\frac1{30}, \qquad \frac9{10}<W_A,W_B<1 . \]

Both weight classes therefore exceed the declared 3/4 threshold with a strict margin. No edge is replaced by a unit weight; all original edge attributes remain associated with their old support.

The actual successor has nineteen edges and ordered bare degrees and capacities

\[ \begin{gathered} d=(4,2,2,2,9,\ 5,3,3,3,5),\\ \nu=(4,2,2,2,117/32,\ 4,2,2,2,8). \end{gathered} \]

Its mobilities nu_i/d_i are

\[ (1,1,1,1,13/32,\ 4/5,2/3,2/3,2/3,8/5). \]

This is neither the unit-bridge model nor a copy of either isolated star law. The positive held capacities are the actual post-SHA values. The right center and leaves, in particular, no longer have equal mobility.

4. Complete event work and a uniform joint storage margin

For the actual weighted support use

\[ H=\tfrac12\sum_{\{i,j\}}W_{ij}(x_i-x_j)^2 +\sum_{\{i,j\}}[1-\cos(\theta_j-\theta_i)]. \]

Before the word the two isolated star storages are 4a^2 each, so H^-=8a^2=1/32. AL changes the four old left boundary-edge form differences from a to c. Its complete old-edge storage jump is

\[ \Delta H_{\mathrm{AL}} =4(c^2-a^2)=-8ab+4b^2=-\frac{127}{262144}. \]

A positive form increment need not increase this storage: here it reduces the target's contrast with its four old neighbors. This is an accounted hybrid jump, not continuous dissipation.

UM leaves every phase and form unchanged on its old support, so its old-edge coordinate work is zero. Its capacity reset adds no storage term. The five actual new edges supply the entire remaining jump:

\[ \Delta H_{\mathrm{UM}} =2W_Ac^2+\tfrac12W_Bb^2+5(1-\cos\alpha)>0. \]

SHA changes only capacity and associated lifecycle state, hence Delta H_SHA=0. The exact ideal endpoint storage is consequently

\[ \boxed{\qquad H^+=4c^2+4a^2+2W_Ac^2+\tfrac12W_Bb^2 +5(1-\cos\alpha). \qquad} \]

The actual represented endpoint must use its actual forms, phases and weights in this sum, rather than an ideal weight substituted after execution. The following generous ideal bound uses only positivity, W_A,W_B<=1, b,c<a and 1-cos alpha<=alpha^2/2:

\[ \begin{aligned} H^+ &<8a^2+\tfrac52a^2+\tfrac52(2/5)^2\\ &=\frac1{32}+\frac5{512}+\frac25 =\frac{1129}{2560} <\frac9{20}<\frac{35}{64}. \end{aligned} \]

Thus 9/20 is a convenient prospective upper target, not an estimated response. It leaves margin 31/320 to the identity barrier. The admitted ideal domain has a nonempty open interior in the two source parameters m,alpha. It does not assert robustness to independent perturbations of all twenty coordinates through the runtime word.

5. Continuous completion and retention on the actual successor

After the completed word select the explicit full-support law

\[ S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad p_i=\frac{S_i}{\pi d_i},\qquad \dot x_i=\frac{\nu_i}{\pi d_i}S_i,\qquad \dot\theta_i=\frac{\nu_i}{\pi d_i}(L_Wx)_i . \]

Its support, actual conductances and positive capacities are held; all twenty form/phase coordinates and both boundaries remain dynamic. There is no forcing or loss, and both rates use original structural time. The selected pressure is recomputed from the actual support and phase state. Terminal stored event pressure zero is not a claim that this new constitutive pressure vanishes. In fact at the nominal endpoint it is nonzero on the target and all new neighbors.

The complete law conserves H by the pointwise cancellation sum_i (nu_i/(pi d_i))[(L_Wx)_i S_i-S_i(L_Wx)_i]=0. The actual inverse-capacity weights d_i/nu_i likewise give its conserved form and registered phase means. Neither old isolated normalization nor the unit-bridge mean weights transfer.

Each original positive star remains an intact subgraph. Write H_r for its full original seven-edge storage, evaluated at the evolving state. Every added edge contributes nonnegative storage. The star Dirichlet inequality therefore gives, for either center and each of its three leaves,

\[ H\geq H_r\geq\frac{35}{16}(x_{r0}-x_{rj})^2 . \]

An offending internal circular phase gap at least pi/2 contributes at least one. Consequently the strict endpoint certificate H<35/64 retains both declared center-leaf identities for all real times under the selected law. Global existence follows as before from the globally Lipschitz lifted field. This argument does not require equal mobilities, exact internal synchronization or a collective quotient law.

The general protected-subgraph criterion is sufficient; extra support can raise a particular constrained form minimum. For this actual five-edge footprint the joint threshold is nevertheless sharp. Set the left forms to (1/2,0,7/16,7/16,11/32) and every right form to 11/32, with all phases equal. On the same held conductances the right star and all five new edges have zero storage; the left arm reaches the boundary with H=35/64. This is a comparison state of the held continuation, not another invocation of the affinity map.

6. Initial exchange coexists with loss of exact-family tangency

The word leaves each region's four A forms and phases internally equal. This is instantaneous family membership. Under the actual successor law all old internal sine inputs initially vanish, whereas the target receives five copies of sin alpha and each right node receives -sin alpha. In particular,

\[ \dot x_4=\frac{65}{32\pi}\sin\alpha,\quad \dot x_5=-\frac4{5\pi}\sin\alpha,\quad \dot x_j=-\frac2{3\pi}\sin\alpha\ (j=6,7,8),\quad \dot x_9=-\frac8{5\pi}\sin\alpha . \]

The right center-leaf difference has exact initial derivative

\[ \boxed{\qquad \frac{d}{dt}(x_5-x_j)=-\frac2{15\pi}\sin\alpha\ne0, \qquad j=6,7,8 . \qquad} \]

Its phase contrast also initially changes:

\[ \frac{d}{dt}(\theta_5-\theta_j) =\frac2{15\pi}(2a+W_Ac)>0 . \]

Thus the right exact internally coherent family is not tangent to the new field. Finite-width identity retention is stronger for this purpose than insisting that every internal deviation remain zero. The event itself does not write these right-node contrasts; they are initial derivatives of the admitted subsequent law.

The component storage supplies can be computed independently by differentiating only the original seven edges of each component along the full successor field. Their old sine gradients initially vanish. The left boundary's old form gradient is -8c, giving

\[ \dot H_1=-\frac{65c}{4\pi}\sin\alpha =-\frac{4095}{4096\pi}\sin\alpha<0 . \]

The right old form gradients are 2a on its four A members and -8a at its boundary. Using its actual unequal mobilities yields

\[ \begin{aligned} \dot H_2 &=-\frac{\sin\alpha}{\pi} \left[2a\left(\frac45+3\frac23\right)-8a\frac85\right]\\ &=\frac{36a}{5\pi}\sin\alpha =\frac9{20\pi}\sin\alpha>0 . \end{aligned} \]

The five-edge storage absorbs the unequal supplies:

\[ \dot H_{\mathrm{new}} =-\dot H_1-\dot H_2 =\frac{11259}{20480\pi}\sin\alpha>0 . \]

These are strict initial exchange signs throughout the ideal parameter domain, not a prediction of monotonic component transfer for all time. Only full H is conserved. The new edge weights do not appear in these particular initial component formulas because the own-component phase gradients vanish; weights remain essential to the full phase field, new-edge storage and subsequent response.

7. Shared protection reader, represented controls and scope

The protected-star owner provides assess_sine_weighted_star_support_retention(source,stars=...) and SineWeightedStarSupportRetention. Its inputs identify two disjoint marked five-node stars inside the actual weighted source. It re-admits the complete primitive state and law, checks the retained positive star subgraphs and includes every actual added edge and positive held capacity. The certificate consumes fresh full storage; it does not copy isolated verdicts or use the fixed unit-bridge observer on a different support.

The static canonical controls use the frozen probes and word through the shared stage dispatcher. Their independent expected calculations must include the full primary reset, target-only history and lifecycle, actual nineteen-edge incidence, all complete rate and pressure rows, every storage jump, strict retention margin, exchange and tangency discriminator. Invalid domains and a blocked stage are distinct from an unavailable sufficient storage bound.

All three frozen forms and boosts have exact dyadic sums; the capacity mean, blend and SHA product above are also binary64-exact. The phase consensus is the zero special case. The right radian lifts are the frozen dyadics, not pi/8, and they remain unchanged by the primary word. Cross-gap and score arithmetic still have separate numerical obligations: the shared phase-difference reader uses atan2(sin(diff),cos(diff)) with its active backend, and the score uses its production normalized arithmetic. The predeclared independent high-precision checks and tolerance in Section 1 cover those finite comparisons; they are not a universal exactness theorem for libm or all source parameters.

For a general represented m in the ideal interval, the sum m+b can round, especially across a binary64 binade. One must retain that actual endpoint and its actual work instead of silently applying the nominal exact-addition claim. The ideal open-domain inequalities and the three finite represented controls supply different evidence.

The result is a conditional canonical contact followed by an admitted identity-preserving continuous completion. It supplies no automatic operator timing, candidate-access law, larger NFR identity, attraction, spontaneous organization or physical identification. Auxiliary Si, the selected word, absolute-form affinity and the continuation law remain declared inputs under the ontology and collective action contract. Other preparations, actions or future events need their own complete admission and storage accounting.

8. Prospective finite internal-response comparison

This extension fixes one analytical comparison before evaluating its static controls. Sections 1-7 and their original protocol remain unchanged.

• Retain the complete ideal source family, word, target, policies and nineteen-edge successor from Sections 1-6. The three represented probes remain the same dyadics, with the same event clock and numerical allowance. Reuse their current-engine fixture endpoints; do not acquire a trajectory or replay historical producers.
• The matched alternative changes only UM_FUNCTIONAL_LINKS=False before the word. Admit its complete actual reset independently. Do not require edge-dependent caches or diagnostics to equal the attached graph. Capture its two connected components separately; the connected-support reader cannot consume a disconnected graph as if its normalization were unchanged.
• After either word, hold its actual support, conductances and capacities. Use the conservative weighted-form/bare-sine law of Section 5, original structural clock, no forcing, loss, inputs or further events. Duration h=1/128 means elapsed time after _t=7, not operator steps or seconds.
• Observe the right center-leaf form contrast R=x_5-x_6, with the same labels and signed form coordinates in both branches. It is initially zero. Predict R_attached(h)<-1/32768, while the detached right-star invariant family gives R_control(t)=0 for every time. Retain both original identity obligations; a distinct internal response does not redefine either NFR.
• Bound the full nonlinear remainder analytically. The global form-speed bound supplies a whole-window centered-form radius; the existing complete conditional acceleration envelope then bounds the Taylor remainder. Initial derivatives alone, a single Euler step or a reference-family tangent calculation cannot establish the endpoint claim.
• The shared calculation must retain exact rational duration, outward rational interval arithmetic, actual radian phases and rebuilt primitive law/state. Its endpoint enclosure has no integration discretization error; interval width and analytical remainder are retained explicitly. Independent trigonometric comparisons continue to use 150 decimal digits and the predeclared 2**-46 score allowance for represented event weights.
• Static controls check all three matched actual words, unchanged node primitives, component support/normalization, the attached strict endpoint separation and both retention arguments. General kernel controls cover zero capacity, zero-conductance bare neighbors, invalid duration/state/law, forged cached fields and a mathematically exact independent solution.

The fixed horizon or separation threshold is not adjusted after controls. A failure closes only its specified sufficient test or remains unresolved; it does not authorize another source, event or finer trajectory campaign. This is a conditional finite-response theorem, not sampled motion, laboratory evidence, autonomous occurrence or formation of a larger organization.

9. A full-law finite form enclosure from a derived radius

The following shared estimate applies beyond the particular stars. Retain a finite connected simple undirected bare support, fixed finite nonnegative conductances, held nonnegative capacities, positive exchange coefficient w and phase-storage scale beta, zero loss, no inputs and no events. Both original-clock rows are

\[ \dot x_i=\frac{w\nu_i}{\pi d_i} \sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad \dot\theta_i=\frac{w\nu_i}{\beta\pi d_i}(L_Wx)_i . \]

The phase lifts evolve under the second row; they are not frozen for the estimate. The field is globally Lipschitz, hence supplies a unique solution on every finite time interval. A zero capacity freezes the corresponding rows of this complete law. A supported zero-conductance edge still contributes to bare degree and sine incidence.

Let \(P=I-\mathbf1\mathbf1^T/n\) be the fixed orthogonal arithmetic-centering projector, and let \(\mathcal R_0=\|Px(0)\|_2\). Bare incidence bounds each form speed globally by w nu_i/pi, giving

\[ \|\dot x(t)\|_2\leq V:=\frac{w}{\pi}\|\nu\|_2,\qquad \|Px(t)\|_2 \leq\mathcal R_0+\int_0^t\|P\dot x(s)\|_2\,ds \leq\mathcal R_0+Vh=:\mathcal R_h \quad(0\leq t\leq h). \]

This derives confinement on the whole requested window. It does not assume that the arithmetic mean is conserved: the projection has norm one even when the mean moves. Positive h is elapsed original structural time. The estimate has no imposed small-time domain, although its resulting interval can become uninformative for a large duration.

Write the weighted strength, bare-normalized mobility and their product as

\[ s_i=\sum_{j\sim i}W_{ij},\qquad \mu_i=\frac{\nu_i}{d_i},\qquad g_i=\mu_i s_i . \]

Since every form contrast is at most sqrt(2) ||Px||_2 in magnitude, the actual weighted phase row satisfies

\[ |(L_Wx)_i|\leq\sqrt2\,\mathcal R_h s_i,\qquad |\dot\theta_i|\leq \frac{\sqrt2\,w\mathcal R_h}{\beta\pi}g_i . \]

Differentiate the complete form row, retaining the phase evolution:

\[ \ddot x_i=\frac{w\mu_i}{\pi} \sum_{j\sim i}\cos(\theta_j-\theta_i) (\dot\theta_j-\dot\theta_i). \]

Consequently the shared conditional acceleration estimate has its whole-window confinement premise established by the preceding proof:

\[ |\ddot x_i(t)|\leq A_i:= \frac{\sqrt2\,w^2\mathcal R_h}{\beta\pi^2} \mu_i\left(\sum_{j\sim i}g_j+d_i g_i\right), \qquad 0\leq t\leq h . \]

Actual conductances enter g, while bare neighbors and degree enter the outer sum. They cannot be interchanged. Integrating the second derivative gives an exact nonlinear remainder bound,

\[ \left|x_i(t)-x_i(0)-t\dot x_i(0)\right| \leq \frac12 A_i t^2,\qquad 0\leq t\leq h. \]

For a contrast, the sum of its two acceleration bounds is sufficient. No constant-Jacobian model, sampled trajectory or Euler update is used. The numerical implementation can take outward upper enclosures of \(\mathcal R_0,V,A_i\) and outward intervals of the independent initial constitutive rates. Its endpoint interval is x_i(0)+h*initial_rate_i+[-A_i h^2/2,A_i h^2/2]. Interval width and the analytic nonlinear remainder remain separate; there is no numerical-integration discretization error to estimate.

10. Uniform separation at the fixed horizon

Apply the complete estimate with w=beta=1 to the actual nineteen-edge successor from Sections 2-6. All quantities below use elapsed time after the completed word; the structural clock starts at _t=7 and is not rescaled.

The post-word forms comprise eight copies of m+a, one m+b and one m. Their arithmetic mean is m+(8a+b)/10, so the exact initial centered radius satisfies

\[ \mathcal R_0^2 =\frac85a^2-\frac85ab+\frac9{10}b^2 =\frac{64521}{10485760}<\frac1{144}. \]

This bound is independent of the common form origin. The actual capacities from Section 3 give

\[ \|\nu\|_2^2 =2(4^2+3\cdot2^2)+(117/32)^2+8^2 =\frac{136569}{1024}<144 . \]

Thus V<4, using pi>3. At the prospectively fixed h=1/128,

\[ \|Px(t)\|_2 <\frac1{12}+4h =\frac{11}{96}<\frac18 \qquad(0\leq t\leq h). \]

This is the needed confinement of the full form vector. It includes all ten forms; every phase continues to evolve without a small-angle restriction. The bound does not presume that the right four-member family remains coherent.

All five new conductances are at most one. In node order 0,...,9, weighted strengths and mobility-weighted strengths therefore obey

\[ \begin{aligned} s&\leq(11,5,5,5,13,\ 12,6,6,6,9),\\ g&\leq(11,5,5,5,169/32,\ 48/5,4,4,4,72/5). \end{aligned} \]

The observed contrast is R=x_5-x_6. Node 5 has neighbors 6,7,8,9,4 and node 6 has neighbors 5,9,4. Their acceleration coefficients from Section 9 add to at most

\[ \begin{aligned} C={}&\frac45 \left(3\cdot4+\frac{72}{5}+\frac{169}{32} +5\frac{48}{5}\right)\\ &+\frac23 \left(\frac{48}{5}+\frac{72}{5}+\frac{169}{32} +3\cdot4\right)\\ ={}&\frac{109519}{1200}<96 . \end{aligned} \]

With \(\mathcal R_h<1/8\), sqrt(2)<3/2 and pi>3,

\[ \frac{\sqrt2\,\mathcal R_h}{\pi^2}<\frac1{48}, \qquad |\ddot R(t)|<2 \quad(0\leq t\leq h). \]

The initial contrast is zero. Section 6 gives its exact initial rate, and the declared phase domain supplies a uniform strictly negative bound:

\[ \dot R(0)=-\frac{2\sin\alpha}{15\pi}<-\frac7{495}. \]

Indeed sine is increasing on [3/8,2/5], with sin(3/8)>=3/8-(3/8)^3/6=375/1024>1/3; also pi<22/7. The integrated full-law bound therefore yields

\[ R(t)<-\frac7{495}t+t^2<0 \qquad(0<t\leq1/128), \]

and at the fixed endpoint,

\[ \boxed{\qquad R(1/128)<-\frac7{495\cdot128}+\frac1{16384} =-\frac{401}{8110080}<-\frac1{32768}. \qquad} \]

The last strict margin is 307/16220160. This closes the specified finite-response discriminator uniformly on the ideal two-parameter source family. It does not certify monotonicity of R or of either component's storage supply throughout the window. The right center-leaf form difference is nonzero, while its absolute value remains below the declared identity limit by the independent all-time storage certificate.

11. Matched disconnected control and both identity obligations

Disabling only UM_FUNCTIONAL_LINKS removes the candidate-link branch. AL, UM's old-neighbor consensus, capacity blend, stored-pressure relief and SHA retain the same primary resets. Hence the matched endpoint has exactly the same node forms, phases and capacities, including x_4=m+b and nu_4=117/32, but no cross edges. Equal primary coordinates do not mean equal continuous laws: the actual disconnected support and its two sets of bare degrees must be retained.

Capture the two connected components separately. The right one is the original star with capacities (4,2,2,2,8), common A form m+a, boundary form m and all phases initially alpha. Its exact internal family xi=eta=0 is invariant under the full isolated law from Section 39c. Its common form and relative boundary phase can move, but every right center-leaf form difference stays zero:

\[ R_{\mathrm{control}}(t)=0\qquad\text{for all real }t . \]

The changed left boundary capacity has no path into this disconnected component. It must still be admitted in the left component's own law; it cannot be silently reset to eight to fit an old reader.

Both control identities are retained independently. Their complete component storages after the word are 4c^2 and 4a^2=1/64, respectively. Each is strictly below 35/64 and remains conserved under its own positive held-capacity law. The protected-star barrier does not require the left capacity to belong to the original one-parameter family. The connected attached branch instead uses its full storage including all five links, as proved in Section 5.

The two branches thus keep the same declared identities while their right internal responses are separated at the specified time. This is a difference caused by the complete supplied contact law under the matched intervention, not evidence of an autonomous contact selector, new primitive operator or physical interaction identification.

12. Shared finite observer and limits of the evidence

The weighted support owner provides SineSupportExchangeComparison.finite_form_response_bounds(duration=...), returning SineFiniteFormResponse. It re-admits primitive state, support, conductances, coefficients and capacity; admits an exact positive duration; derives the global speed and whole-window centered radius; then reuses conditional_form_acceleration_bounds. It returns initial constitutive rate intervals, full acceleration intervals, separate componentwise remainder bounds and endpoint form intervals. All nodal tuples retain source.nodes order.

The result's status="enclosed" means that these conditional-law intervals have been constructed. A contrast interval containing zero does not prove zero response, and this generic status does not itself certify the weighted-star separation. The actual contrast must satisfy its independently declared strict bound. Export rebuilds the result from its primitive source and duration; cached radii, rates, endpoints and verdict-like fields cannot replace the premises.

The canonical static controls apply this shared observer to the three predeclared actual endpoints and independently admit their matched controls. The general finite observer also retains its zero-capacity and zero-conductance domains. These calculations are mathematical enclosures of an ideal solution starting at the supplied represented coordinates, not observations of a measured or numerically integrated trajectory.

No primitive-source uncertainty, acquired-history error or future event is implicitly covered. A later acquisition needs its own association and error budget. The finite theorem strengthens the initial tangency obstruction into a strict response interval while preserving both identities; it does not prove minimal collective state, nonlinear attraction, autonomous organization or a laboratory interpretation.

13. Prospective exact collective-state comparison

This bounded extension keeps Sections 1-12 and their frozen comparisons. Its question is exact continuous-state closure on the actual canonical successor, not a new word, trajectory, time horizon or formation claim.

• Retain all ten nodes, nineteen actual edges, actual conductances, positive post-SHA capacities, conservative complete law and original structural clock. No constituent or dynamical boundary is removed from the fine law.
• Compare the five ordered blocks (0,1,2,3), (4), (5), (6,7,8), (9) with the older four blocks (0,1,2,3), (4), (5,6,7,8), (9). Use the shared exact weighted partition criterion with zero phase offsets. Predict an invariant five-block family and an obstruction to the four-block family from the unequal right center/leaf bare and weighted coefficients.
• Derive both reduced rows, actual cross-edge multiplicities, conductance sums, inherited storage and inverse-capacity mean weights. The coarse phase channel must retain multiplicities; do not replace it by an unweighted simple five-node graph or identify its mean weights with physical masses.
• Keep every collective phase as a declared real radian lift. The actual endpoint's equal coordinates within each block must be admitted exactly; a family certificate alone does not establish source membership. No conversion of alpha/pi into an invented rational turn is permitted. The existing rational-turn evaluator retains its separate input contract.
• Reuse the three represented endpoints from Section 1 through their shared current-engine regression fixtures. Additional consumers introduce no new word or response acquisition. At each endpoint, compare every lifted fine row with an independent nineteen-edge calculation, complete storage and conserved means; retain the Section 8 finite-response observation unchanged.
• Exercise the admitted nonlinear family at one additional detached state: collective forms (3/8,1/4,5/16,9/32,7/32) and radian lifts (1/16,-1/32,3/8,11/32,5/16), on the central probe's actual held support and conductances. It deliberately separates the right center and leaves. This is a supplied family state, not a later point on a trajectory.
• Admission controls retain independent within-block form/phase perturbations, nonzero-offset rejection for the new zero-offset radian interface, invalid scalars and forged cached state/partition claims. Independent numerical comparisons use 150 decimal digits and outward rational intervals; exact coefficient equalities require algebraic equality, not interval overlap.

Success is a complete exact state for this family, a scoped obstruction to the older grouping, and a faithful reconstruction of the declared observable. It does not assert minimality among all coordinates, an off-family closure, automatic operator descent or a newly acquired collective identity.

14. Exact zero-offset state in declared radian lifts

The weighted partition theorem already characterizes invariant fixed-offset families. The present state bridge specializes its offsets to zero, retaining arbitrary real collective radian lifts. It neither restricts those lifts to rational multiples of pi nor changes the original clock.

Consider a finite connected simple undirected support, positive held conductances \(W_{ij}\), positive held capacities \(\nu_i\), bare degrees \(d_i\), and the complete law of Section 5. Set

\[ \mu_i=\frac{\nu_i}{d_i},\qquad \dot x_i=\frac{\mu_i}{\pi}\sum_{j\sim i}\sin(\theta_j-\theta_i), \qquad \dot\theta_i=\frac{\mu_i}{\pi} \sum_{j\sim i}W_{ij}(x_i-x_j). \]

For an ordered partition \(\mathcal B=(B_0,\ldots,B_{k-1})\), define the linear lift \(\iota\) by

\[ x_i=X_a,\qquad \theta_i=\Theta_a\quad(i\in B_a),\qquad (X,\Theta)\in\mathbb R^{2k}. \]

These are independent form and phase coordinates. They are not circular phase averages or a projection of an arbitrary fine state onto the family. For \(b\ne a\), write

\[ n_{ib}=\#\{j\in B_b:j\sim i\},\qquad c_{ib}=\sum_{\substack{j\in B_b\\j\sim i}}W_{ij}. \]

The family is invariant for every collective state if and only if both \(\mu_i n_{ib}\) and \(\mu_i c_{ib}\) are independent of \(i\in B_a\), for every distinct pair of blocks. Internal edges contribute zero to both rows on this zero-offset family. Sufficiency follows by substitution. For necessity, vary the independent external phases, and separately the independent external forms, while comparing the rows of two nodes in the same block. Each external sine coefficient and each external linear coefficient must agree. Equality at a single source is insufficient.

When these conditions hold, define

\[ M_a=\sum_{i\in B_a}\frac{d_i}{\nu_i},\qquad N_{ab}=\sum_{i\in B_a}n_{ib},\qquad C_{ab}=\sum_{i\in B_a}c_{ib}\quad(a\ne b), \]

with zero diagonal. The undirected fine support gives symmetric \(N,C\). Multiplying a common fine row coefficient by \(1/\mu_i\) and summing within its block proves that its value is \(N_{ab}/M_a\) or \(C_{ab}/M_a\). Consequently the exact descended law is

\[ \boxed{\displaystyle \dot X_a=\frac1{\pi M_a}\sum_{b\ne a}N_{ab} \sin(\Theta_b-\Theta_a),\qquad \dot\Theta_a=\frac1{\pi M_a}\sum_{b\ne a}C_{ab}(X_a-X_b).} \tag{14.1} \]

The fine vector field composed with \(\iota\) equals the lift of this vector field for every \((X,\Theta)\). Both fields are globally Lipschitz: their sine derivatives are bounded and their form dependence is linear, with finite fixed coefficients. Uniqueness and global existence therefore give an exact correspondence of solutions for every initial state in the family and every real elapsed time. No Taylor truncation or numerical trajectory enters this conclusion.

This correspondence includes the actual fine pressure law

\[ p_i=\frac1{\pi d_i} \sum_{b\ne a}n_{ib}\sin(\Theta_b-\Theta_a), \qquad \dot x_i=\nu_i p_i. \]

Constituent pressures can differ within a block even though their form rates agree. Thus \(M_a\) is an inherited inverse-capacity weight, not an asserted physical mass or a complete canonical macro-capacity/pressure contract. Equation (14.1) is derived from the specified fine driver; it does not infer that driver by dividing an observed response by a capacity.

Membership remains a separate premise. A supplied fine source lies in this declared chart exactly when its forms and its real radian lifts each agree within every block. Circular equality modulo full turns alone does not establish equality of those declared lifts. A different lift chart or nonzero-offset family requires its own admission; no silent wrapping or averaging supplies it.

15. The canonical contact has an exact five-block state

Use the ordered labels

\[ U=\{0,1,2,3\},\quad V=\{4\},\quad C=\{5\},\quad L=\{6,7,8\},\quad B=\{9\}. \]

Here \(C\) names the right center block; it is not the conductance matrix entry \(C_{ab}\). Every boundary node remains dynamical. The actual support and post-SHA capacities from Sections 2-3 give, in fine node order,

\[ d=(4,2,2,2,9,5,3,3,3,5),\qquad \nu=(4,2,2,2,117/32,4,2,2,2,8), \]

\[ \mu=(1,1,1,1,13/32,4/5,2/3,2/3,2/3,8/5). \]

Let \(w_A\) be the common actual held conductance of the four edges from node 4 to nodes 5 through 8, and \(w_B\) that of edge \(\{4,9\}\). The general ideal source family and the three separately admitted represented successors retain the respective coefficient obligations in Sections 3 and 7. After contact these weights are held: UM affinity is not recomputed as the continuous coordinates move.

All nodes of \(U\) have mobility one and the same external neighbor \(V\), with one bare edge and conductance two. The center's three internal edges and the leaves' different internal incidence have zero current on the family. This equality of complete external rows, despite unequal graph roles, proves closure of \(U\); graph automorphism is not its premise. All three nodes of \(L\) have mobility \(2/3\), with one edge each to \(C,V,B\) of weights \(3,w_A,2\). Their rows also agree. Singleton blocks require no within-block comparison. Section 14 thus proves full nonlinear family invariance for arbitrary collective forms and radian lifts.

The inherited data, in order \((U,V,C,L,B)\), are

\[ M=(4,\;32/13,\;5/4,\;9/2,\;5/8),\qquad M_{\rm tot}=\frac{1335}{104}. \tag{15.1} \]

Coarse pair	Bare multiplicity \(N_{ab}\)	Conductance sum \(C_{ab}\)
\(U,V\)	\(4\)	\(8\)
\(V,C\)	\(1\)	\(w_A\)
\(V,L\)	\(3\)	\(3w_A\)
\(V,B\)	\(1\)	\(w_B\)
\(C,L\)	\(3\)	\(9\)
\(C,B\)	\(1\)	\(2\)
\(L,B\)	\(3\)	\(6\)

All other off-diagonal entries vanish. There are seven coarse pairs but nineteen fine edges; the three internal \(U\) edges account for the difference between nineteen and the sixteen cross-edge incidences counted once in this table. Their zero storage on this family does not remove their contribution to the fine degrees and hence to (15.1).

For explicit complete rows, set \(S_{ab}=\sin(\Theta_b-\Theta_a)\) and \(D_{ab}=X_a-X_b\). Equation (14.1) becomes

\[ \pi\dot X= \begin{pmatrix} S_{UV}\\ \frac{13}{32}(4S_{VU}+S_{VC}+3S_{VL}+S_{VB})\\ \frac45(3S_{CL}+S_{CB}+S_{CV})\\ \frac23(S_{LC}+S_{LB}+S_{LV})\\ \frac85(S_{BC}+3S_{BL}+S_{BV}) \end{pmatrix}, \tag{15.2} \]

\[ \pi\dot\Theta= \begin{pmatrix} 2D_{UV}\\ \frac{13}{32}(8D_{VU}+w_A D_{VC}+3w_A D_{VL}+w_B D_{VB})\\ \frac45(9D_{CL}+2D_{CB}+w_A D_{CV})\\ \frac23(3D_{LC}+2D_{LB}+w_A D_{LV})\\ \frac85(2D_{BC}+6D_{BL}+w_B D_{BV}) \end{pmatrix}. \tag{15.3} \]

All phase rates in (15.3) are radians per original structural time, not turn rates. The division by \(M_L=9/2\) in (14.1) already accounts for the three copies in \(L\); inserting an extra factor of three in its per-member rows would change the law.

At each actual admitted successor, the five coordinates are exactly

\[ X=(m+a,\;m+b,\;m+a,\;m+a,\;m),\qquad \Theta=(0,\;0,\;\alpha,\;\alpha,\;\alpha). \tag{15.4} \]

For the frozen probes these are actual represented values with exact within-block equality. In particular, \(\alpha\) remains the supplied dyadic radian, not an assumed rational multiple of pi. With \(c=a-b\), the initial row numerators are

\[ \pi\dot X=\sin\alpha (0,\;65/32,\;-4/5,\;-2/3,\;-8/5), \tag{15.5} \]

\[ \pi\dot\Theta= \left(2c,\; \frac{13}{32}(-8c-4w_Ac+w_Bb),\; \frac45(w_Ac+2a),\; \frac23(w_Ac+2a),\; \frac85(-8a-w_Bb)\right). \tag{15.6} \]

These source rows are checks of the all-state result, not its substitute. The additional detached state in Section 13 evaluates the same family away from the equality \(X_C=X_L,\Theta_C=\Theta_L\), without asserting that a solution has reached that state.

15.1 Why the earlier four-block family fails

Merge \(C\) and \(L\) into the former right \(A\) block. The center and a leaf have unequal external bare coefficients \(4/5\) and \(2/3\) toward \(V\), with difference \(2/15\); their weighted coefficients toward \(V\) differ by \(2w_A/15\). They also have unequal external coefficients toward \(B\). The necessary all-state condition in Section 14 therefore fails, independently of numerical tolerances.

The source (15.4) belongs instantaneously to that four-block equality set, but (15.5)-(15.6) give

\[ \frac{d}{dt}(x_5-x_6)=-\frac{2\sin\alpha}{15\pi}\ne0,\qquad \frac{d}{dt}(\theta_5-\theta_6) =\frac{2(w_Ac+2a)}{15\pi}>0. \tag{15.7} \]

Thus family membership at one time does not imply tangency. Isolated equilibria or other special points inside an excluded family do not repair its all-state closure. The five-block description retains this specific internal distinction; it is not a minimality theorem among all linear or nonlinear coordinate choices.

15.2 Why an unweighted simple five-node sine channel is insufficient

The descended law is not obtained by putting one ordinary node of the same normalized bare-sine comparison law at each block and using the seven coarse pairs as a simple support. Such a model would give \(V\), of coarse degree four, one common sine coefficient \(\widetilde\nu_V/4\). The actual four coefficients toward \(U,C,L,B\) are

\[ \frac{13}{32}(4,\;1,\;3,\;1). \tag{15.8} \]

The four phase arguments can vary independently. Equality of the two rows throughout the family would require their four coefficients to agree term by term, which (15.8) excludes for every scalar \(\widetilde\nu_V\). A common clock rescaling cannot change these ratios. Choosing conductances on the seven coarse edges does not repair a bare sine channel, because those conductances enter the other row.

The multiplicity-aware law (14.1) retains precisely the inherited \(N,C,M\) data needed here. Its two coupling arrays are derived restrictions of the original law, not newly chosen forces. This obstruction concerns the stated same-coordinate, same-clock simple graph representation; it is not a claim that every possible change of variables or richer coarse model is impossible.

16. Inherited storage, means and finite-response observation

On the invariant family, the full fine storage restricts exactly to

\[ H_{\rm col}(X,\Theta)= \frac12\sum_{\{a,b\}}C_{ab}(X_a-X_b)^2+ \sum_{\{a,b\}}N_{ab}\bigl[1-\cos(\Theta_b-\Theta_a)\bigr]. \tag{16.1} \]

The pair sums use the seven pairs in Section 15 once each. Zero-offset internal edges have identically zero storage. This is equality with full storage on the family, not a neglect of internal energy for arbitrary nearby states. With \(J_M=\operatorname{diag}(1/(\pi M_a))\),

\[ \dot X=-J_M\nabla_\Theta H_{\rm col},\qquad \dot\Theta=J_M\nabla_X H_{\rm col}. \]

Hence \(H_{\rm col}\) is conserved. Symmetry of \(N\) and \(C\) also gives

\[ Q_x=\sum_a M_aX_a,\qquad Q_\theta=\sum_a M_a\Theta_a,\qquad \dot Q_x=\dot Q_\theta=0. \tag{16.2} \]

These are exactly the fine sums with weights \(d_i/\nu_i\), pulled back through \(\iota\). Their normalized means use \(M_{\rm tot}\) from (15.1). For source (15.4),

\[ \overline x=m+\frac{338}{445}a+\frac{256}{1335}b,\qquad \overline\theta=\frac{221}{445}\alpha. \tag{16.3} \]

The phase mean is a mean of the declared real lifts. Independent full-turn changes within a block can leave circular fine observations unchanged while changing this chart's membership or lifted mean; neither conservation nor the reader removes that dependence. Block-common full-turn changes preserve membership and change the mean by their explicitly weighted lift shift.

Substituting (15.4) into (16.1) returns the complete post-word storage from Section 4:

\[ H_{\rm col}=4c^2+4a^2+2w_Ac^2+\frac12w_Bb^2+ 5(1-\cos\alpha). \]

Thus the same strict full-storage retention certificate applies to the reconstructed source and its exact lifted solution. It proves the two specified internal identities under this law. The five blocks are sufficient continuous coordinates on the invariant family, not five newly formed NFRs or an additional identity policy.

The fine observable from Section 8 pulls back without approximation:

\[ (x_5-x_6)\circ\iota=X_C-X_L. \tag{16.4} \]

By solution correspondence, the existing finite theorem is therefore also a theorem of this collective law:

\[ X_C(1/128)-X_L(1/128)<-1/32768 \]

for the same admitted source and held law. Its matched disconnected control remains exactly zero as proved in Section 11; the attached five-block coefficients are not substituted for that other support. This transfers a previously proved full nonlinear bound through an exact state map. It acquires no new trajectory, changes no horizon or numerical budget, and asserts no bound for off-family errors.

17. Shared radian-state reader and canonical boundaries

The weighted partition owner extends the existing report with two operations returning SineWeightedRadianState:

• observe_radian_state() re-admits the captured primitive source, complete law and partition, requires a certified zero-offset family, and checks exact within-block equality of both signed forms and declared radian lifts before deriving its collective state.
• evaluate_radians(collective_form, collective_phase_radians) admits supplied collective coordinates and constructs their fine lift. It evaluates the same complete held law there, and separately reports whether those exact coordinates equal the captured source. A supplied family state need not be that source, nor a later trajectory state.

The pre-existing rational-turn evaluator keeps its separate contract. The new interface accepts exact or represented finite real radians and requires literal zero offsets; it does not approximate alpha/pi by a rational or replace missing state by an average.

Every consuming calculation rebuilds primitive premises. The report retains the independent fine source evaluation, block form and radian phase rates, inherited bare rows, storage, mean weights, lifted means and source membership. Exact coefficient identities prove row equality; overlapping outward intervals alone cannot do so. The interval residuals independently check the numerical evaluation. Export re-admits the source and supplied coordinates instead of trusting cached row, storage or membership flags.

The canonical contact controls reuse the three frozen source fixtures, checking the five-block certificate, four-block obstruction, actual membership, all lifted fine rows, storage, means and the finite observable. The separate detached source in Section 13 checks the full nonlinear family beyond its initial right-member equality. The radian-state controls exercise source membership, radian/turn separation, lift covariance, independent full rows and rebuilt exports. The existing partition controls retain their primitive admission, offset and cached-report boundaries.

A complete state for these continuous solutions is not a complete canonical operator-admission state. The lift does not supply target selection, history, U1-U6 context, event timing, secondary writes or lifecycle data. Those remain with the operator contracts and actual execution owners. Conversely, the already admitted fine word does not establish a descended collective operator or its autonomous occurrence.

The scope matches the ontology: the result derives exact effective dynamics on a declared family, retaining both dynamical boundaries and the internal contrast induced by contact. It establishes no off-family closure, transverse attraction, minimum state dimension, autonomous assembly, repeated-scale closure or physical identification.

18. Prospective full-state perturbation comparison

This gate fixes its inputs and discriminator before evaluating the new controls. It reuses Sections 8 and 13; neither a new word nor a continuous trajectory is acquired. Earlier protocols and results remain unchanged.

• Reuse the three represented attached/control source pairs with (m,alpha)=(1/4,201/512),(15/64,3/8),(17/64,409/1024) and their actual held support, conductances and post-word capacities. Keep both complete normalized-sine laws, zero forcing/loss, w=beta=1 and original clock. The control consists of its two admitted connected components, not an invented connected support or a copy of the attached degrees.
• At elapsed zero, add independent signed errors to all ten forms and all ten declared radian lifts. Each form error has magnitude at most 2^-20 in the declared form units; each phase error has magnitude at most 2^-20 radians. The same twenty errors are supplied to the matched branches. No equality, zero mean or within-block correlation is imposed. Capacities, support and conductances are held, including affinity-derived weights; the word is not rerun on perturbed inputs.
• Retain h=1/128 and R=x_5-x_6. The prospective discriminator is R_attached(h)-R_control(h) < -1/32768 for every admitted perturbation. The control need not remain null. Bound each full solution's deviation from its unperturbed reference through the whole interval, including every form and phase coordinate. The attached reference is exactly the five-block lifted solution from Sections 14-16; no averaging replaces a perturbed fine state by that reference.
• Require the same two marked identities to persist under each branch's own complete law, using the protected old-star storage barrier 35/64 and a bound on the full perturbed initial storage. Means and their possible perturbations remain part of the fine state. No decay or indefinite tracking of the reference is predicted.
• Reuse the existing static word fixtures once per selected test session. Generic arithmetic controls may use an independent weighted path, exact frozen/zero-conductance cases, invalid inputs, cached-field corruption and the declared short-window boundary. Those controls test the envelope and admission, not an acquired scientific response. Compare nontrivial coefficients with independent 150-digit arithmetic and retain outward rational bounds; interval overlap is not a proof.
• Stop with the uniform certificate or an explicit insufficient bound or scoped obstruction. Do not enlarge the budget, change the horizon or select a favorable perturbed sample after seeing a result. This is post-word preparation robustness under held laws: it does not prove that UM produces this entire box, preserve an event's live admission, create a collective event, or identify a physical phenomenon.

19. Whole-window error from two solutions of the same held law

This comparison extends the complete-state envelope of Section 9 to initial uncertainty. In each application, both solutions use the same support, conductances, capacities, coefficients and original clock. The attached and disconnected laws require separate applications.

For the shared conservative comparison, let \(w,\beta>0\), let held capacities and conductances be nonnegative, and retain a finite connected simple bare support. Write \(s_i=\sum_{j\sim i}W_{ij}\), \(d_i=\#\{j:j\sim i\}\), and \(\mu_i=\nu_i/d_i\). The complete rows are

\[ \dot x_i=\frac{w\mu_i}{\pi} \sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad \dot\theta_i=\frac{w\mu_i}{\beta\pi} \sum_{j\sim i}W_{ij}(x_i-x_j). \tag{19.1} \]

Keep real phase lifts for both solutions. Suppose their initial coordinate differences obey supplied component bounds

\[ |\widetilde x_i(0)-x_i(0)|\le e_i,\qquad |\widetilde\theta_i(0)-\theta_i(0)|\le f_i,\qquad e_i,f_i\ge0. \]

These are independent coordinate bounds; no zero-mean constraint, synchronization or equality within a block is assumed. The global inequality \(|\sin u-\sin v|\le|u-v|\) gives

\[ |\dot{\widetilde x}_i-\dot x_i| \le a_i\|\widetilde\theta-\theta\|_\infty,\qquad a_i=\frac{2w\nu_i}{\pi}, \]

\[ |\dot{\widetilde\theta}_i-\dot\theta_i| \le b_i\|\widetilde x-x\|_\infty,\qquad b_i=\frac{2w\mu_i s_i}{\beta\pi}. \tag{19.2} \]

Bare degree cancels in \(a_i\), whereas the weighted strength remains in \(b_i\). A zero-conductance edge therefore still contributes to the sine law; it does not supply the same contribution to the phase-gradient law. Zero capacity sets both coefficients to zero in this particular complete model.

Set \(e=\max_i e_i\), \(f=\max_i f_i\), and choose finite upper bounds \(A\ge\max_i a_i\), \(B\ge\max_i b_i\). For a positive duration \(h\), define the actual whole-window suprema

\[ E_x=\sup_{0\le t\le h}\|\widetilde x(t)-x(t)\|_\infty,\qquad E_\theta=\sup_{0\le t\le h} \|\widetilde\theta(t)-\theta(t)\|_\infty. \]

The globally Lipschitz fields ensure that both solutions exist and these suprema are finite. Integrating (19.2), then taking the suprema, gives \(E_x\le e+AhE_\theta\) and \(E_\theta\le f+BhE_x\). Consequently, if

\[ D=1-ABh^2>0, \]

the entirely rational comparison in \(A,B,e,f,h\) is

\[ \boxed{\displaystyle E_x\le X=\frac{e+Ahf}{D},\qquad E_\theta\le T=\frac{f+Bhe}{D}.} \tag{19.3} \]

Using those global bounds once more in each component gives

\[ |\widetilde x_i(t)-x_i(t)|\le e_i+a_i hT,\qquad |\widetilde\theta_i(t)-\theta_i(t)| \le f_i+b_i hX \quad(0\le t\le h). \tag{19.4} \]

Replacing \(a_i,b_i,A,B\) throughout by rigorously outward upper bounds preserves all conclusions. This proves the numerical enclosure without replacing either solution by an Euler step, linearization or independently chosen trajectory.

The condition \(D>0\) is sufficient for this particular comparison, not a dynamical instability threshold. When it fails, a nonzero initial box receives an unavailable error envelope rather than a claim of divergence or a division by a nonpositive denominator. If every \(e_i=f_i=0\), uniqueness gives exact zero differences for any finite \(h\), independently of that comparison condition. An unavailable temporal envelope can still retain a separately valid initial storage bound.

19.1 A complete initial storage box

For the same law the conserved full storage is

\[ H_\beta(x,\theta)=\frac12\sum_{\{i,j\}}W_{ij}(x_i-x_j)^2+ \beta\sum_{\{i,j\}}[1-\cos(\theta_j-\theta_i)]. \]

At the supplied reference source define \(q_{ij}=x_i(0)-x_j(0)\), \(\gamma_{ij}=\theta_j(0)-\theta_i(0)\), \(r_{ij}=e_i+e_j\), and \(z_{ij}=f_i+f_j\). The cosine is globally one-Lipschitz, and its potential lies in \([0,2]\). Therefore every initial state in the coordinate box satisfies

\[ H_\beta(\widetilde x(0),\widetilde\theta(0)) \le \frac12\sum_{\{i,j\}}W_{ij}(|q_{ij}|+r_{ij})^2+ \beta\sum_{\{i,j\}} \min\{2,\;1-\cos\gamma_{ij}+z_{ij}\}. \tag{19.5} \]

An outward upper bound for the reference cosine potential can replace its exact value. Equation (19.5) includes every held edge and the actual \(\beta\); it is not an average of component budgets. It needs no temporal small-gain condition. Under a matching positive-storage identity theorem, conservation can transport this source budget for all time even though the solution-difference envelope is only finite.

20. A twenty-coordinate domain preserves separation and both identities

Fix the source, actual held laws, horizon and matched initial errors of Section 18. Write

\[ \epsilon=2^{-20},\qquad h=1/128. \]

Every one of the ten forms and ten declared radian lifts may vary independently by at most \(\epsilon\). In particular, the actual perturbed states generally do not belong to the five-block family. The unperturbed attached reference does, and Section 15 supplies its exact nonlinear lift. No projection replaces a perturbed state by that reference.

For the attached law, the degree/capacity values in Section 15 and the held weights \(0<w_A,w_B<1\) imply

\[ \max_i\nu_i=8,\qquad \max_i\frac{\nu_i s_i}{d_i}<\frac{72}{5}. \]

Thus \(A<16/3\) and \(B<48/5\), using \(\pi>3\). For the disconnected control the right boundary has instead \(d_9=4,s_9=8,\nu_9=8\), hence \(\nu_9s_9/d_9=16\). Its coefficient cannot be replaced by the attached boundary value. The left boundary retains its actual post-SHA capacity \(117/32\); there its corresponding product is \(117/16\). All other control products are at most eleven. Both connected control components and the attached branch are therefore covered by the common upper bounds

\[ A_*=\frac{16}{3},\qquad B_*=\frac{32}{3},\qquad 1-A_*B_*h^2=\frac{287}{288}>0. \tag{20.1} \]

Applying (19.3) independently under each law gives, through the whole interval,

\[ \|\widetilde x-x\|_\infty \le\frac{300}{287}\epsilon<\frac{21}{20}\epsilon,\qquad \|\widetilde\theta-\theta\|_\infty \le\frac{312}{287}\epsilon<\frac{11}{10}\epsilon. \tag{20.2} \]

The disconnected law is handled on its two actual connected components. Taking the maximum of their coordinate bounds gives (20.2) for that whole branch; adding an artificial edge or borrowing the attached degrees would not.

20.1 The matched control need not remain null

Let \(R=x_5-x_6\), and distinguish reference values \(R_{\rm on}\), \(R_{\rm off}\) from perturbed values \(\widetilde R_{\rm on},\widetilde R_{\rm off}\). At the fixed horizon, Sections 10-11 prove

\[ R_{\rm on}(h)<-\frac{401}{8110080},\qquad R_{\rm off}(h)=0. \]

For either branch, (20.2) bounds its contrast error by \(2(21/20)\epsilon\). Therefore

\[ \widetilde R_{\rm on}(h)-\widetilde R_{\rm off}(h) < -\frac{401}{8110080}+\frac{21}{5}\epsilon = -\frac1{32768}-\frac{1549}{103809024} < -\frac1{32768}. \tag{20.3} \]

This is the predeclared strict discriminator for every point in the twenty-dimensional initial error box. The matched initial errors are preserved in the statement; the conservative sum of the two error budgets does not rely on a favorable cancellation between branches. In particular, the perturbed disconnected contrast is only bounded, not assigned zero.

The conclusion compares two different complete held laws. It is not an error estimate obtained by applying (19.3) directly between those laws: that theorem was applied separately to each perturbed branch and its own unperturbed solution. Its exact hypotheses, including the capacity and degree change at the boundary, are therefore retained.

20.2 Full initial storage protects both marked identities

A useful independent bound is stronger than the direct edge box (19.5) in this particular source class. For \(\beta=1\), write the full storage as the squared Euclidean norm of the edge vector whose components, for each edge, are

\[ \sqrt{W_{ij}/2}\,(x_i-x_j),\qquad \sqrt2\,\sin\bigl((\theta_j-\theta_i)/2\bigr). \]

Errors bounded by \(\epsilon\) in every form and radian lift change that vector by at most

\[ \epsilon\sqrt{\,2\sum_{\{i,j\}}W_{ij}+2|E|\,}. \tag{20.4} \]

This follows from the \(2\epsilon\) bound on each difference and the Lipschitz bound for the half-angle sine. It is valid for independent errors in all fine coordinates and every choice of their signs, with no small-angle approximation.

The attached support has nineteen edges and

\[ \sum_{\{i,j\}}W_{ij}=34+4w_A+w_B<39. \]

The vector error in (20.4) is consequently less than \(\sqrt{116}\epsilon<11\epsilon\). Section 4 gives \(H_{\rm on}(0)<9/20\), so

\[ \sqrt{\widetilde H_{\rm on}(0)} <\frac7{10}+11\epsilon<\frac{45}{64}, \qquad \widetilde H_{\rm on}(0) <\frac{2025}{4096}<\frac12<\frac{35}{64}. \tag{20.5} \]

For each disconnected original star, the seven edges have conductance sum seventeen, so (20.4) gives vector error less than \(7\epsilon\). The nominal component storages are \(4c^2<1/64\) on the left and \(4a^2=1/64\) on the right. Separately,

\[ \widetilde H_{{\rm off},r}(0) \le(1/8+7\epsilon)^2<1/32<35/64, \qquad r\in\{\mathrm{left},\mathrm{right}\}. \tag{20.6} \]

All inequalities in (20.5)-(20.6) are strict at the declared \(\epsilon=2^{-20}\). The shared numerical reader can certify the same sufficient source margins with its direct edge bound (19.5); it need not reproduce this independent norm argument.

The held positive capacities enter each complete evolution but not this conserved storage. Applying the protected-star barrier of Section 5 to the attached support, and the original-star barrier to each disconnected component, proves that all three center-leaf form gaps of each marked star stay strictly below \(1/2\) in magnitude and their circular phase gaps stay strictly below \(\pi/2\) for all time. The support and capacity differences between branches have not been suppressed.

This all-time identity statement and the finite-time comparison statement are different conclusions. The source is not reset during either proof. No forever-small tracking error, convergence to the five-block family, attraction, or persistence of one exchange sign follows.

20.3 What the finite approximation retains

Equation (20.2) bounds every form and phase deviation from the reference fine solution; the attached reference is exactly the ten-coordinate solution lifted by Section 15. Thus that collective description has a quantified finite domain of predictive accuracy even when its defining equalities fail in the actual fine state. The exact closed collective equations themselves have not become an autonomous description of arbitrary off-family states.

The perturbations may change both conserved lifted means. With positive fine weights \(d_i/\nu_i\), each normalized initial mean changes by at most \(\epsilon\), because it is a positive weighted average of the coordinate errors. Each law preserves its own two means, so the corresponding mean differences remain constant. No centering operation or zero-mean restriction discards them. The attached and control weights differ and must remain separate.

The source cube has nonzero width in all twenty coordinates. This proves robustness of a supplied post-word preparation under the actual held laws. It does not prove that a perturbation of the pre-word source is mapped into this cube by AL, UM and SHA, that the live word remains admitted there, or that any controller selects this action.

21. Shared perturbation reader and evidence boundary

The weighted support owner provides

SineSupportExchangeComparison.initial_state_perturbation_bounds( duration=..., form_error_bounds=..., phase_error_bounds=...),

returning SineInitialStatePerturbation. It re-admits primitive source, complete law, support, conductances and capacities; accepts a positive exact or represented duration and finite nonnegative per-node form and radian-error bounds; then builds outward bounds for (19.2). Node-indexed fields retain the captured source order.

The report exposes both coefficient vectors and their maxima, the comparison denominator, the two global error suprema and the refined whole-window component bounds (19.4). It separately retains initial form, phase and total storage upper bounds from (19.5), with the actual storage coefficient. In particular, initial_phase_storage_upper_bound already includes the factor beta, so the two reported storage contributions sum to the reported full bound. A nonpositive comparison denominator returns explicit temporal unavailability for a nonzero error box. Exact zero errors use uniqueness for zero temporal bounds even at a duration outside that sufficient comparison regime.

These are envelopes around the exact solution from the supplied source. No nominal trajectory is constructed, no perturbed sample is selected, and no extrema are inferred from a finite sample of box corners. An available generic error envelope does not by itself certify a nonzero contrast or a star identity. Those consumers must combine it with their own reference observation and matching complete storage theorem.

The canonical contact controls reuse the three frozen actual source/control pairs at the unchanged horizon and error budget. They retain both control components, rebuild the coefficients independently, apply the two contrast-error budgets, and require every appropriate initial storage bound to lie strictly below \(35/64\). The mathematical result is the all-box theorem above; static numerical controls check implementation and represented-source admission rather than observing a trajectory. The separate perturbation controls exercise independent weighted-path coefficients, nonuniform error budgets, zero-capacity/zero-conductance cases, the unavailable denominator boundary, zero-error uniqueness and rebuilt exports.

Reports and SDK exports rebuild the calculation from primitive source, duration and initial budgets; cached coefficients, denominators, temporal bounds and storage claims are not premises. Phase uncertainty is in the declared real radian lifts. Conductance/capacity uncertainty, support changes, event errors, measurement authentication and acquired-history error are not included. The canonical operator contract and ontology retain their separate admission, occurrence, formation and physical-identification obligations.

22. Prospective preparation-to-flow admission

This protocol fixes the next analytical domain and static controls before any new perturbed word evaluation. All earlier protocols and responses remain unchanged. There is no new continuous trajectory or horizon scan.

• Retain the complete preparation, configuration, clock, target4 and registered atomic AL,UM,SHA path from Section 1. Vary all ten initial forms and all ten initial radian phases independently by at most epsilon=2^-24; hold capacities, stored pressure, Si, auxiliary state, history and grammar context fixed. Reuse the same three central/lower/upper dyadic nominal probes. The control changes only UM_FUNCTIONAL_LINKS=False and receives the same perturbed initial coordinates.
• Prove live admission and support margins from the authoritative maps, including the old-neighbor and candidate U3 gates, circular consensus, merged phases, affinity and the full uncapped candidate inventory. Do not assume the nominal five-edge footprint or equal produced weights. Account separately for AL work, old-edge UM phase work, inserted-edge storage and SHA's zero storage jump.
• Distinguish exact real arithmetic of these supplied maps from numerical implementation. The conditional arithmetic allowance is sigma=2^-42 for each consumed circular separation/consensus result, final primary form/lifted-phase output and affinity output relative to its exact-map value. Capacities and stored zero pressure retain their exact represented identities. This allowance is a hypothesis for a uniform implementation statement, not a claimed universal error theorem for libm or NumPy. Static controls must check their own numerical associations independently.
• The prospective continuation budgets are 2*epsilon for each form and phase lift relative to its nominal post-word source, and 4*epsilon for each of the five inserted conductances; old conductances have zero error budget. Admit actual node order, support, capacities and both complete laws before applying those budgets. Do not transfer a same-law certificate to changed weights or silently restore the five-block family.
• Retain original elapsed h=1/128 and R_attached(h)-R_control(h) < -1/32768, with both marked identities retained. Prove a full nonlinear state-and-coefficient comparison or name the insufficient bound or admission/selector obstruction. The perturbed control receives its own error allowance.
• For represented static controls use exactly one fixed sign pattern at each of the three nominal probes. In node order0..9, form signs are (1,-1,1,-1,1,-1,1,-1,1,-1) and phase signs are (-1,1,-1,1,-1,1,-1,1,-1,1), each multiplied by epsilon. Run each attached/control word once per selected test session through the shared executor. These finite controls are not an exhaustive box sample.
• Preserve raw final phases. Their declared comparison chart subtracts exact 2*pi at nodes0,2,4 and zero turns elsewhere, for this sign pattern. Use outward pi intervals to admit those lift differences; subtracting represented math.tau is not an exact symmetry of the continuous law. Check ideal stage maps, actual work and affinities with independent 150-digit arithmetic. Record backend, source and configuration association.
• Generic continuation tests cover nonuniform conductance uncertainty, zero-capacity/zero-weight limits, invalid budgets, unavailability and fresh export. No new operator semantics, acquisition, autonomous selector or physical identity is asserted. Do not tune these budgets or signs after seeing the reserved static outcomes.

23. Admission and the ideal full preparation map

Retain Section 22's fixed capacities, auxiliary values, target, history, configuration and registered stages. Write the twenty initial errors as \(\delta_i,\eta_i\), with \(|\delta_i|,|\eta_i|\le\epsilon=2^{-24}\). The perturbed left phases are \(\eta_i\), \(0\le i\le4\); the right phases are \(\alpha+\eta_i\), \(5\le i\le9\). This section first studies the exact real maps, using exact circular geometry. Their numerical association is a separate hypothesis in Section 24.

The AL/SHA proposal owner and UM proposal/merge owner supply the maps. AL adds \(b=1/1024\) only to node 4's form. Every form stays between \(7/32\) and \(1/2\); neither hard rail is reached. Node 4's initial form is strictly below the configured latent maximum one, and its post-AL form is strictly above UM's minimum \(1/8\). Its initial capacity eight is above the AL and UM thresholds \(1/8\). The original degree is four. These are live-state conditions, not inferences from a token.

All four old neighbors pass U3 because their phase separations from node 4 are at most \(2\epsilon\). With bidirectional UM, the actual proposal includes the target and those four old neighbors in its circular consensus. Define

\[ q=1/1024,\qquad \gamma=\arg\sum_{i=0}^4 e^{\,i\eta_i}. \]

The resultant has positive real part, and its direction lies between the smallest and largest input phases in this acute interval. Thus \(|\gamma|\le\epsilon\). There is only one target proposal, so the ideal shortest-arc merge gives the following declared lifts:

\[ \theta_i^+=(1-q)\eta_i+q\gamma\quad(0\le i\le4), \qquad \theta_i^+=\alpha+\eta_i\quad(5\le i\le9). \tag{23.1} \]

All left lift errors remain at most \(\epsilon\). Old left phase differences become \((1-q)(\eta_j-\eta_i)\); the final old-neighbor U3 condition still holds. The right nodes are not old neighbors of the target and receive no phase write.

The uncapped candidate inventory contains exactly nodes 5 through 9. Before and after the phase write, their separations from the target are bounded by \(\alpha+2\epsilon<1/2\), well inside the configured U3 limit. No nearest-candidate ordering or random subselection is needed when the count is zero. Their final separations are also greater than \(1/3\), so the relevant shortest arcs have a fixed sign and never approach the cut.

23.1 Uniform affinity and support margins

Let \(u=x_4^+\), \(v=x_j^+\), and let \(\varepsilon_{\rm EPI}\) be the admitted represented regularizer float(1e-9). Since all forms are positive and all Si values remain \(1/2\), the ideal affinity is

\[ w_{4j}=1-\frac{\theta_j^+-\theta_4^+}{2\pi} -\frac{|u-v|}{4(u+v+\varepsilon_{\rm EPI})}. \tag{23.2} \]

The shared phase affinity is linear in the circular separation, not its cosine. Formula (23.2) is the real-arithmetic value of the normalized implementation; the implementation's rounding is not silently omitted from its subsequent admission.

Both \(u,v\) change by at most \(\epsilon\) from their nominal post-AL values. Their denominator is at least \(7/16\). For \(F(u,v)=|u-v|/(u+v+\varepsilon_{\rm EPI})\), the numerator changes by at most \(2\epsilon\), the denominator by at most \(2\epsilon\), and \(0\le F\le1\). Consequently

\[ |F(u,v)-F(u_0,v_0)|\le\frac{64}{7}\epsilon. \]

The phase separation changes by at most \(2\epsilon\). With \(\pi>3\), each ideal affinity therefore satisfies

\[ |w_{4j}-w_{4j}^0| <\left(\frac13+\frac{16}{7}\right)\epsilon =\frac{55}{21}\epsilon<3\epsilon. \tag{23.3} \]

Section 3 gives \(w_{4j}^0>9/10\) throughout the nominal domain. Thus all five perturbed candidates still exceed \(3/4\). Their positive separation above \(1/3\) also gives a uniform upper margin below one in (23.2). All five edges are inserted, and no existing edge is modified. The weights need not be equal across the four right-A targets.

Capacity synchronization reads the same four old compatible capacities \((4,2,2,2)\), independent of the coordinate errors. The production arithmetic evaluates their mean as \(5/2\) and the target update as

\[ 8+\frac18(5/2-8)=117/16. \]

Every operation here has its exact binary64 result. SHA then gives \(117/32\), also exactly represented and positive; its live input is above \(1/8\). No other capacity changes. The held stored pressure is zero, and its UM multiplication leaves zero exactly. Stored event pressure is distinct from the independently specified pressure used in the subsequent continuous law.

The supplied word and its actual ordered grammar context remain the ones in Section 1. No source-history error, additional target, callback or changed grammar configuration is supplied. The live AL/UM/SHA and U3 margins above supplement that context; they do not replace it or turn a valid word into an autonomous invocation rule.

23.2 Work includes the old-edge phase change

Before AL, the old left boundary gaps are \(-a+\delta_4-\delta_i\), \(0\le i\le3\). Its ideal old-edge work is

\[ \Delta H_{\rm AL} =-\frac{127}{262144} +2b\left(4\delta_4-\sum_{i=0}^3\delta_i\right), \qquad \left|\Delta H_{\rm AL}+\frac{127}{262144}\right| \le\frac{\epsilon}{64}. \tag{23.4} \]

UM changes no form. Its left old-edge phase differences contract by \(1-q\) in (23.1), so its ideal old phase work is nonpositive, with magnitude at most \(14\epsilon^2\) from the seven edges and \(|\eta_j-\eta_i|\le2\epsilon\). The right old phase storage is unchanged. To this old-edge work one must add all five new-edge stores

\[ \sum_{j=5}^9 \left[\frac{w_{4j}}2(x_4^+-x_j^+)^2+ 1-\cos(\theta_j^+-\theta_4^+)\right]. \tag{23.5} \]

SHA has zero storage jump because this storage does not consume capacity. Thus imperfect preparation does not inherit the earlier zero old-edge UM phase-work identity. Numerical work is computed from the actual before/after states; a finite arithmetic allowance alone does not turn an ideal nonpositive value into an exact runtime sign guarantee.

24. Exact circular association and conditional numerical scope

The real-map result covers the entire twenty-coordinate preparation box. To transfer it to a particular numerical execution, Section 22 separately requires the arithmetic allowance \(\sigma=2^{-42}\) for the consumed circular separations/consensus, final primary form and lifted-phase outputs, and affinity outputs. This includes discrepancies from represented pi/tau constants as well as the specified arithmetic and backend. It is not a universal error theorem for math, NumPy or every future environment.

Because \(\sigma/\epsilon=2^{-18}\), these allowances are much smaller than every live and selector margin above. For example, old/final U3 separations are below \(2\epsilon+3\sigma<1/16\), and candidate separations are below \(\alpha+2\epsilon+3\sigma<1/2\). Affinity remains above \(9/10-3\epsilon-\sigma>3/4\) and strictly below one. Under the specified arithmetic hypothesis, the unchanged inventory therefore has the same five-edge footprint. A runtime failure of that hypothesis is not repaired by this conditional statement.

Raw circular phases must be preserved. If a numerical left phase is negative before wrapping, the stage may store a value close to represented math.tau rather than a small negative number. For an actual stored binary64 value \(r_i\), a declared comparison lift is

\[ \widehat\theta_i=r_i-2\pi k_i,\qquad k_i\in\mathbb Z. \tag{24.1} \]

The scalar \(r_i\) is admitted as its exact represented rational value, while \(2\pi\) is exact mathematical pi. Its error from the reference lift is enclosed with outward pi intervals. Subtracting represented math.tau instead would change the exact circular state by math.tau minus \(2\pi\), and cannot be called an exact symmetry.

For the frozen alternating-sign controls the declared integers are \(k=(1,0,1,0,1,0,0,0,0,0)\). For the uniform conditional statement the chart is admitted from the actual near-zero/near-full-turn output, rather than imposing that one sign pattern. Subtracting a fixed integer multiple of exact \(2\pi\) from a continuous phase solution preserves both sine rows and storage. Hence (24.1) associates the actual raw state with a small-error lifted solution without materializing an irrational lift as an invented rational source coordinate. Lifted means retain the corresponding constant offset.

Compared with the independently admitted nominal post-word source, the ideal reset and two arithmetic allowances give the following conservative envelopes:

\[ |\widehat x_i-x_i^0|,\ |\widehat\theta_i-\theta_i^0|<\epsilon+2\sigma<2\epsilon, \qquad |\widehat W_{4j}-W_{4j}^0|<3\epsilon+2\sigma<4\epsilon. \tag{24.2} \]

Here the form and radian quantities retain their separate units. Every old conductance is unchanged, and capacities and bare support are admitted exactly as above. The nominal and perturbed affinity associations are both needed when using the last bound. This is not a return to the equal-weight five-block family: the actual produced weights and coordinates can break it.

A finite static control checks its own associations and actual stage outcome. Such checks neither verify the arithmetic hypothesis at every point of the preparation box nor authenticate another execution. Conversely, the ideal all-box theorem is not based on sampling those three control points.

25. Complete-flow error when held conductances also vary

The same-law theorem of Section 19 does not cover (24.2), because the weighted phase row changes. Here is the needed extension. Keep the same bare support, held capacities, positive \(w,\beta\), zero loss and original clock. Let \(W\) be the reference conductances and \(\widetilde W\ge0\) the second set, with

\[ |\widetilde W_{ij}-W_{ij}|\le\rho_{ij}. \]

Retain initial form/lift errors \(e_i,f_i\) as in Section 19. For the reference solution alone,

\[ |\dot x_i|\le v_i=\frac{w\nu_i}{\pi},\qquad |x_i(t)-x_j(t)|\le Q_{ij}=|x_i(0)-x_j(0)|+h(v_i+v_j) \quad(0\le t\le h). \tag{25.1} \]

No reference trajectory is evaluated. With \(\overline W_{ij}=W_{ij}+\rho_{ij}\), choose outward upper bounds for

\[ a_i=\frac{2w\nu_i}{\pi},\qquad b_i=\frac{2w\nu_i}{\beta\pi d_i} \sum_{j\sim i}\overline W_{ij},\qquad G_i=\frac{w\nu_i}{\beta\pi d_i} \sum_{j\sim i}\rho_{ij}Q_{ij}. \tag{25.2} \]

Split the phase-row difference as \(L_{\widetilde W}(\widetilde x-x) +(L_{\widetilde W}-L_W)x\). The second term is an independent coefficient forcing, bounded by \(G_i\) using (25.1). Thus

\[ |\dot{\widetilde x}_i-\dot x_i| \le a_i\|\widetilde\theta-\theta\|_\infty,\qquad |\dot{\widetilde\theta}_i-\dot\theta_i| \le b_i\|\widetilde x-x\|_\infty+G_i. \]

Let \(A=\max a_i\), \(B=\max b_i\), \(G=\max G_i\), \(e=\max e_i\), \(f=\max f_i\). If \(D=1-ABh^2>0\), the same integrated comparison gives

\[ \boxed{\displaystyle X=\frac{e+Ahf+Ah^2G}{D},\qquad T=\frac{f+Bhe+hG}{D},} \tag{25.3} \]

\[ |\widetilde x_i(t)-x_i(t)|\le e_i+a_i hT,\qquad |\widetilde\theta_i(t)-\theta_i(t)| \le f_i+b_i hX+hG_i. \tag{25.4} \]

These are full nonlinear error bounds under two admitted held laws, not a substitution of nominal weights into the perturbed law. The initial storage box (19.5) extends by replacing every \(W_{ij}\) by \(\overline W_{ij}\) in its form contribution; the bare phase contribution is unchanged, including its \(\beta\) factor.

A failed denominator is again unavailability of this comparison. Exact zero state and conductance budgets give the same-law zero-error uniqueness case for any finite horizon. Zero initial state error alone cannot justify that conclusion when conductances can differ.

26. The admitted preparation domain reaches a separated continuation

Apply Section 25 to the post-word envelopes (24.2): \(e_i=f_i=2\epsilon\), \(\rho_{4j}=4\epsilon\) on the five inserted edges, and zero uncertainty on old edges. The disconnected control has zero conductance uncertainty. All actual capacities retain Section 15's values.

For the attached branch, even the enlarged conductance upper bounds give \(\max_i\nu_i\overline s_i/d_i<16\). The control's maximum is sixteen as in Section 20. Thus both branches admit the common constants \(A_*=16/3\), \(B_*=32/3\), \(h=1/128\) and \(D_*=287/288\).

On each uncertain edge the reference initial form gap is at most \(1/16\), and \(\nu_4+\nu_j<12\). Equation (25.1) and \(\pi>3\) give \(Q_{4j}<3/32<1/8\). The largest mobility-weighted sum of the five conductance budgets is at node 4:

\[ \max_i\frac{\nu_i}{d_i}\sum_j\rho_{ij} =\frac{65}{32}(4\epsilon). \]

Consequently \(G<65\epsilon/192<\epsilon\); the disconnected control has \(G=0\). Using \(G\le\epsilon\) for a common conservative comparison yields

\[ X\le\frac{19203}{9184}\epsilon <\frac{21}{10}\epsilon,\qquad T\le\frac{2505}{1148}\epsilon <\frac{11}{5}\epsilon. \tag{26.1} \]

These bounds retain every fine coordinate and every admitted weight variation. No exact five-block closure of the perturbed law is asserted.

For the nominal attached/control reference pair, Sections 10-11 still supply \(R_{\rm on}(h)<-401/8110080\) and \(R_{\rm off}(h)=0\). Applying the two separate contrast-error bounds in (26.1) therefore gives

\[ \widetilde R_{\rm on}(h)-\widetilde R_{\rm off}(h) < -\frac{401}{8110080}+\frac{42}{5}\epsilon = -\frac1{32768}-\frac{76513}{4152360960} < -\frac1{32768}. \tag{26.2} \]

The actual perturbed control is not assumed null. This establishes the ideal preparation-to-response theorem and, under Section 24's independently admitted arithmetic association, its numerical-successor version at the same fixed horizon.

26.1 Both identities survive the changed law

At the nominal post-word coordinates, changing the five weights by at most \(4\epsilon\) can increase the full storage by no more than

\[ \frac12\sum_{j=5}^9 4\epsilon(x_4^0-x_j^0)^2 \le\frac{5}{128}\epsilon<\epsilon. \]

The reference-coordinate storage under each candidate attached law is therefore below \(9/20+\epsilon<49/100\). Its conductance sum is below \(39+20\epsilon<40\), with nineteen bare edges. The storage-vector argument (20.4), now with coordinate errors \(2\epsilon\), gives norm error less than \(22\epsilon\). Hence every admitted attached successor satisfies

\[ H_{\rm on}^{\rm candidate}(0) <(7/10+22\epsilon)^2 <(45/64)^2<1/2<35/64. \tag{26.3} \]

For each disconnected star, all conductances are unchanged and its nominal storage is at most \(1/64\). The same reasoning gives

\[ H_{{\rm off},r}^{\rm candidate}(0) \le(1/8+14\epsilon)^2<1/32<35/64. \tag{26.4} \]

The actual positive old-star conductances and held capacities satisfy the same protected-subgraph premises. Conservation under each candidate's own law therefore retains both specified internal identities for all time. This does not extend the finite tracking bounds indefinitely, select the word's occurrence, or prove a new collective identity.

27. Reusable coefficient envelope and the execution boundary

The weighted support owner provides state_and_conductance_perturbation_bounds( duration=..., form_error_bounds=..., phase_error_bounds=..., conductance_error_bounds=...), returning SineStateConductancePerturbation. It rebuilds the reference primitive state, law, support and capacity; admits all node/edge budgets before arithmetic; and derives the reference node speeds, edge contrast bounds, enlarged strengths and coefficient-forcing terms in (25.1)-(25.2). Node budgets follow source-node order, and conductance budgets follow its admitted edge order.

The report retains both global error suprema, refined component errors, the strict comparison denominator and the independent initial-storage box, including the actual \(\beta\) factor. It explicitly distinguishes unavailable temporal bounds from valid source storage information. Export rebuilds the consumed quantities from reference source, duration and budgets.

This generic envelope consumes a reference source and a declared uncertainty box. It does not inspect an executed candidate, certify an event's output, or choose the lift integers in (24.1). Before applying it to an actual successor, its consumer must admit the complete support and node order, exact capacities, old-edge preservation, each produced conductance, and the actual form/circular-lift association. Those independent obligations cannot be replaced by a favorable generic verdict or cached bound.

The canonical controls execute only Section 22's frozen alternating-sign preparations through the shared stage owner. They retain the raw states, stage/lifecycle and grammar evidence, the actual U3/consensus and affinity calculations, separate event work and declared exact-turn associations. Independent high-precision checks evaluate their own arithmetic allowance. Both attached and control continuation envelopes then consume their nominal represented reference law and declared budgets, after the actual successor's uncertainty association has been checked. These finite controls do not prove a global backend accuracy guarantee or exhaust the preparation box.

The separate coefficient-perturbation controls check nonuniform weights/budgets, independent coefficients, zero-weight and zero-capacity limits, unavailable windows and rebuilt exports, without running the scientific word or a trajectory.

The result closes a conditional preparation, registered event and finite-flow association. It does not identify arbitrary continuous coordinates with a full canonical operator state, supply an autonomous event selector, derive a physical force, or establish hierarchical formation. Changed capacities, topology uncertainty and unadmitted arithmetic remain outside this certificate.

28. Prospective relative-origin tangent question

The next bounded question concerns the state needed to predict relative regional motion under the already admitted held contact law. It does not continue the finite post-word source to equilibrium or evaluate a trajectory. The following comparison is fixed before any new rank or modal calculation.

• Retain the original node order, nineteen-edge contact, old conductances, post-SHA capacities, zero loss/forcing and w=beta=1. Analyze the nominal weight family with four equal W_A cross conductances and one W_B, with 9/10<W_A,W_B<1. This is a declared containing domain for the ideal affinities; actual represented values require their own membership check. Unequal preparation-generated weights are outside this family theorem.
• Use the separately declared common-form/common-phase equilibrium family x_i=x_star, theta_i=theta_star. These hypothetical equilibrium coordinates do not replace or reinterpret the actual stored post-word state. Retain all twenty infinitesimal coordinates in (delta x,delta theta) order, with local real phase lifts and the original structural clock t.
• The observations are left-minus-right means of form and phase, each using its own held law's inverse-mobility weights mu_i=d_i/nu_i and normalized separately over the original regions (0,...,4) and (5,...,9). Do not use the attached degrees or means in the disconnected control.
• Compare with the already admitted functional-links-off word's fourteen-edge law and unchanged post-word capacities. Its full twenty-coordinate tangent is required; a zero-conductance supported contact is a different law.
• Derive both complete tangent rows and an exact obstruction or closure for the two observed coordinates. Use the existing exact linear-observation owner for minimal all-initial-state linear realizations. Factor only the common positive w/pi from the generator for rational algebra, recording the corresponding factor in every original-time evolution row. No rounded pi, numerical rank threshold or fitted oscillator is admitted.
• The finite exact-rational controls reuse the three original represented attached laws and their matched disconnected laws from Sections 1 and 8. Reuse the current module fixtures; execute no additional word. Freeze a maximum of 128 rank calls per realization, no eigenvalue evaluations and no trajectory integration. Exhaustion is unresolved, with no approximate fallback. Exact symbolic derivations over the declared weight domain are separate from these represented controls.
• Check the full generator, own-law means, common-origin null directions, any admitted five-block output quotient and its invisible fine directions. Discriminate a two-coordinate autonomous response from a response needing additional internal coordinates. Minimality concerns arbitrary fine infinitesimal initial states; one selected eigenvector or a special regional preparation cannot establish it. Hidden initialization remains part of any causal memory description.

Stop with a sufficient minimal tangent realization or a named domain/budget obstruction. No nonlinear periodic orbit, attraction, autonomous contact, new NFR identity or physical identification is inferred from this calculation.

29. The complete tangent and its actual regional observations

Fix the nominal held law of Section 28 and write \(u=W_A\), \(v=W_B\). The following dimension theorem holds for every real \(9/10<u,v<1\), independently of the affinity formula. A represented law belongs to this theorem only after its four \(u\) entries, its single \(v\) entry and the remaining primitive premises have been checked. The unequal-weight preparations of Sections 22-27 are not silently included.

Choose any separately declared equilibrium \(x_i=x_*\), \(\theta_i=\theta_*\). Let \(\xi_i,\eta_i\) denote infinitesimal form and real radian phase increments about it. This reference equilibrium is not the captured post-word preparation, which need not be an equilibrium. With bare and conductance Laplacians \(L_N,L_W\), and \(D=\operatorname{diag}(\nu_i/d_i)\), direct differentiation of the complete law gives

\[ \dot\xi=-\frac1\pi D L_N\eta,\qquad \dot\eta=\frac1\pi D L_W\xi,\qquad J=\frac1\pi K,\quad K=\begin{pmatrix}0&-D L_N\\D L_W&0\end{pmatrix}. \tag{29.1} \]

Both channels and all twenty coordinates use the original clock \(t\). The pressure tangent is independently obtained from the supplied pressure map as \(-(L_N\eta)_i/(\pi d_i)\); multiplication by \(\nu_i\) then gives the first row. No stored-pressure value or observed rate is substituted for that constitutive calculation. The positive common \(1/\pi\) may be factored out for exact row-space algebra, but it remains in every full, reduced and memory evolution equation.

The common form and phase origins are in \(\ker K\). The quadratic storage and the two full inverse-mobility means satisfy

\[ H_2=\frac12\xi^T L_W\xi+\frac12\eta^T L_N\eta,\qquad \dot H_2=0,\qquad \frac d{dt}\sum_i\frac{d_i}{\nu_i}\xi_i =\frac d{dt}\sum_i\frac{d_i}{\nu_i}\eta_i=0. \tag{29.2} \]

These identities retain their declared held-law premises. In particular, the quadratic storage is not the full nonlinear storage evaluated away from equilibrium. The two channel Laplacians are different; replacing both by the conductance Laplacian would change the tangent.

Let \(L=\{0,\ldots,4\}\), \(R=\{5,\ldots,9\}\), put \(m_i=d_i/\nu_i\), and define

\[ M_L=\sum_{i\in L}m_i,\quad M_R=\sum_{i\in R}m_i,\qquad \ell_i= \begin{cases}m_i/M_L&i\in L,\\-m_i/M_R&i\in R.\end{cases} \qquad y_x=\ell\xi,\quad y_\theta=\ell\eta. \tag{29.3} \]

The normalization is separate in each region. These are differences of declared structural observations, not inferred laboratory coordinates or physical masses. For the attached law,

\[ (m_i)_{i=0}^9= (1,1,1,1,32/13,5/4,3/2,3/2,3/2,5/8),\qquad M_L=84/13,\quad M_R=51/8. \tag{29.4} \]

For the functional-links-off control, the fourteen-edge support gives

\[ (m_i^{\rm off})_{i=0}^9= (1,1,1,1,128/117,1,1,1,1,1/2),\qquad M_L^{\rm off}=596/117,\quad M_R^{\rm off}=9/2. \tag{29.5} \]

The changed target capacity remains \(117/32\) in both laws. The different target and right-hand degrees must not be transferred from the attached law to its control. Each disconnected region conserves its own two weighted sums, so its relative outputs obey exactly

\[ \dot y_x^{\rm off}=\dot y_\theta^{\rm off}=0. \tag{29.6} \]

The two output rows are independent. Consequently their minimal all-initial-state tangent realization has dimension two, with zero generator. This is not a statement that the control's internal tangent is frozen: only the declared two observations are constant. Indeed the regional weighted means are conserved by its full nonlinear held law too.

29.1 The contact reads internal boundary information immediately

Write \(\kappa=M_L^{-1}+M_R^{-1}=445/1428\). Cancellation of internal edge fluxes in the attached law gives, for arbitrary fine increments,

\[ \dot y_x=\frac{\kappa}{\pi} \left(\eta_5+\eta_6+\eta_7+\eta_8+\eta_9-5\eta_4\right), \qquad \dot y_\theta=\frac{\kappa}{\pi} \left((4u+v)\xi_4-u\sum_{j=5}^8\xi_j-v\xi_9\right). \tag{29.7} \]

Neither boundary expression is determined by the two regional means. For example take all forms zero, all right-hand phases zero, and

\[ \eta_i=-\frac8{13}a\ (i=0,\ldots,3),\qquad \eta_4=a. \]

For every real \(a\), both initial outputs vanish, just as at the zero tangent state. Nevertheless

\[ \dot y_x=-\frac{2225}{1428\pi}a,\qquad \dot y_\theta=0. \tag{29.8} \]

Arbitrarily small nonzero \(a\) therefore excludes an autonomous two-coordinate tangent law for arbitrary fine preparation. The obstruction does not rely on selecting an oscillatory eigenvector, estimating a frequency, or exciting a particular numerical trajectory.

30. Exactly eight coordinates are necessary and sufficient

The five blocks of Section 15 are \(U=\{0,1,2,3\}\), \(V=\{4\}\), \(C=\{5\}\), \(L_3=\{6,7,8\}\), \(B=\{9\}\), in that order. To avoid confusing the block \(L_3\) with the original left region, subscripts below refer to this declared five-block order. Let \(P\) map a fine vector to the inverse-mobility-weighted mean in each block, and \(E\) lift a block vector uniformly to its constituents. Here \(P\) is simply the arithmetic average on \(U,L_3\) and the identity on the singleton blocks; \(PE=I_5\).

The inherited block mass matrix is

\[ M=\operatorname{diag}(4,32/13,5/4,9/2,5/8). \]

Let \(L_{\mathcal N},L_{\mathcal C}\) be the five-node Laplacians with the inherited bare counts and conductance totals from Section 15:

Pair	\(\mathcal N\)	\(\mathcal C\)
\(U,V\)	\(4\)	\(8\)
\(V,C\)	\(1\)	\(u\)
\(V,L_3\)	\(3\)	\(3u\)
\(V,B\)	\(1\)	\(v\)
\(C,L_3\)	\(3\)	\(9\)
\(C,B\)	\(1\)	\(2\)
\(L_3,B\)	\(3\)	\(6\)

Set \(A=M^{-1}L_{\mathcal N}\), \(B_*=M^{-1}L_{\mathcal C}\). The star's internal fluxes cancel under \(P\), and each constituent has the same external coefficient towards each other block. Consequently

\[ P D L_N=A P,\qquad P D L_W=B_*P. \tag{30.1} \]

Thus \(\mathcal P=\operatorname{diag}(P,P)\) satisfies

\[ \mathcal P K=K_5\mathcal P,\qquad K_5=\begin{pmatrix}0&-A\\B_*&0\end{pmatrix}. \tag{30.2} \]

This is an output quotient of the full tangent for arbitrary fine increments, not an assumption that the fine perturbation lies in the exact nonlinear five-block family. The ten-dimensional fine kernel \(\ker\mathcal P\) is invariant and invisible to these linear observations. The same projection need not close the full sine dynamics away from the family.

On block coordinates the common regional row is

\[ e=\left(\frac{13}{21},\frac8{21}, -\frac{10}{51},-\frac{12}{17},-\frac5{51}\right), \qquad e\mathbf1=0. \tag{30.3} \]

Both block common origins are invisible as well. Hence the tangent output needs at most eight independent coordinates. One explicit sufficient choice is

\[ q_a=(P\xi)_a-(P\xi)_B,\qquad r_a=(P\eta)_a-(P\eta)_B,\qquad a=U,V,C,L_3. \tag{30.4} \]

If \(Tz=(z_U-z_B,z_V-z_B,z_C-z_B,z_{L_3}-z_B)\), and \(E_0q=(q_U,q_V,q_C,q_{L_3},0)^T\), then \(TE_0=I_4\) and these eight coordinates obey

\[ \frac d{dt}\begin{pmatrix}q\\r\end{pmatrix} =\frac1\pi \begin{pmatrix}0&-T A E_0\\T B_*E_0&0\end{pmatrix} \begin{pmatrix}q\\r\end{pmatrix},\qquad (y_x,y_\theta)=(eE_0q,eE_0r). \tag{30.5} \]

This gives an exact sufficient realization without computing a mode.

30.1 A uniform lower bound on observable dimension

Consider four form-channel output rows appearing, up to signs, among the first four output-Krylov levels of \(K_5\):

\[ \mathcal R= \begin{pmatrix} e\\ eB_*\\ eAB_*\\ eB_*AB_* \end{pmatrix}. \tag{30.6} \]

A direct exact determinant expansion in columns \(U,V,C,L_3\) gives

\[ \det\mathcal R_{[:,U,V,C,L_3]} =\frac{9164597}{16633085543424}\,F(u,v), \tag{30.7} \]

\[ \begin{split} F(u,v)={}&147968u^3v+1300644u^3-208896u^2v^2-484895u^2v\\ &+11594856u^2-1763766uv^2-14815038uv-330078v^2 . \end{split} \tag{30.8} \]

The nonvanishing is uniform, not an inference from finite probes. On the larger closed rectangle \(9/10\le u,v\le1\),

\[ \begin{split} \partial_u F &=443904u^2v+3901932u^2-417792uv^2-969790uv +23189712u-1763766v^2-14815038v\\ &\ge 443904(9/10)^3+3901932(9/10)^2+23189712(9/10)\\ &\hspace{1cm}-417792-969790-1763766-14815038=\frac{798565717}{125}>0, \tag{30.9}\\ \partial_v F &=147968u^3-417792u^2v-484895u^2-3527532uv -14815038u-660156v\\ &\le147968-14815038(9/10)<0. \end{split} \]

It follows that

\[ F(u,v)\le F(1,9/10)=-\frac{26064879}{10}<0. \tag{30.10} \]

All four rows in \(\mathcal R\) belong to \(Z=\{z\in\mathbb R^{1\times5}:z\mathbf1=0\}\), which has dimension four. Equation (30.7) therefore proves that the observable form rows span all of \(Z\).

Right multiplication by \(A\) is injective on \(Z\). Indeed the connected bare quotient has \(\ker_{\rm left}A=\operatorname{span}\{\mathbf1^T M\}\); that positive row has nonzero sum, so it has no nonzero intersection with \(Z\). Applying one further \(K_5\) multiplication to each observable form row gives four independent phase rows \(-\mathcal R A\). They are observable by power four at the latest. Together the two channels provide eight independent output rows. Combined with the upper bound, this proves

\[ \boxed{\dim_{\rm obs}J_{\rm attached}=8,\qquad \dim_{\rm obs}J_{\rm disconnected}=2.} \tag{30.11} \]

Multiplication by the common nonzero \(1/\pi\) cannot change these row-space dimensions. This is an exact theorem throughout the declared weight domain, including represented weights admitted to that domain. It does not require the same row-selection sequence or rank progression at every parameter value. The full twenty-coordinate tangent has a twelve-dimensional invisible subspace for this observation: the ten fine block-zero-mean directions, together with the two common block origins.

31. Internal state and the boundary of the result

The contact therefore changes more than the number of constant regional origins. Its relative regional response needs six further coordinates beyond the two requested observations. Four relative block form coordinates and four relative block phase coordinates provide one minimal realization. They need not be the coordinate rows selected by the shared exact algorithm.

This is observation-specific minimality among linear autonomous realizations valid for arbitrary infinitesimal initial state. It is not a claim that eight coordinates reconstruct all fine state, that every fine mode affects this observation, or that eight coordinates define a complete canonical NFR. Neither an unobserved internal initialization nor a specially correlated preparation may be set to zero when applying (30.11).

Equivalently, complete the visible pair to coordinates \((y,h)\) of this eight-dimensional realization, with six hidden coordinates. If its factored generator has blocks \(G_{yy},G_{yh},G_{hy},G_{hh}\), the original-time equation after eliminating \(h\) is

\[ \begin{split} \dot y(t)={}&\frac1\pi G_{yy}y(t) +\frac1\pi G_{yh}e^{G_{hh}t/\pi}h(0)\\ &+\frac1{\pi^2}\int_0^t G_{yh}e^{G_{hh}(t-s)/\pi}G_{hy}y(s)\,ds. \tag{31.1} \end{split} \]

This reuses the existing coordinate-memory construction. The initial hidden contribution is part of the prediction; the convolution is not a newly supplied pressure law or irreversible loss. No matrix exponential or trajectory needs to be evaluated to establish the identity or the minimal dimension.

The already proved nonlinear five-block family does carry an exact eight-coordinate relative description after both common origins are discarded. That separate on-family fact does not promote this all-fine-state tangent quotient to a nonlinear quotient for arbitrary fine preparation. The ten invisible fine directions are invisible to the tangent observation; finite-amplitude invisibility would require a further proof. The complete-state and event distinctions remain those of the NFR ontology and operator foundation: a law for continuous observations does not supply grammar context, operator admission, a contact occurrence rule or a new maintained identity.

32. Shared consensus reader and represented controls

The shared owner is assess_sine_consensus_observation, returning SineConsensusObservation. It rebuilds the primitive held support, coefficients and state through the corresponding source owner, then derives both full Laplacians, actual \(d_i/\nu_i\) means, the full generator and the output rows. Positive capacities and no isolated nodes are required for this observation. A disconnected source uses its admitted component reader; supported zero-conductance edges remain in the bare Laplacian and the degrees.

The generic source supports conservative exchange \(w>0,\beta>0\) with the same structural distinction: \(J=(w/\pi)K\), \(K_{x\theta}=-D L_N\), \(K_{\theta x}=D L_W/\beta\). It calls the existing exact observation owner and rational matrix kernels. The resulting reduced generator retains the same \(w/\pi\) in original time. Rational rows are coordinates of the factored model, not automatically observed derivatives in physical time.

The exact reported realization is for the normalized rational or represented primitive law. No rounded \(\pi\), approximate rank, eigenvalue fit or replacement equilibrium state enters it. source_in_consensus_family separately checks literal equality of the captured normalized forms and radian lifts; a false flag does not invalidate a tangent about the separately declared consensus family or exclude other equilibria. Export rebuilds the result rather than trusting cached generators or rank verdicts. A rank-call budget exhaustion raises an explicit obstruction; it supplies no partial or approximate certificate.

The generic consensus-observation controls check complete generators, invalid premises, component observations, source rebuilding and exact realizations. The canonical contact controls reuse the three original attached/control fixtures specified in Section 28. Their actual weight membership, own-law means, full twenty-coordinate tangents, quotient intertwiners, invisible directions and dimensions are checked independently. The uniform real-parameter proof in Section 30 is separate from those finite represented controls. There is no new word, source trajectory, eigenvalue evaluation, equilibrium acquisition or claim of physical identification in this gate.

33. Prospective finite-amplitude validity domain

This gate compares the complete nonlinear held law with the consensus tangent and its minimal observable realization, initialized from the same full fine preparation. It does not assume that a tangent-invisible coordinate is zero. The following domain and checks precede new finite evaluations.

• Retain Section 28's nominal nineteen-edge law, positive post-SHA capacities, original clock, w=beta=1, zero forcing/loss and four equal W_A plus one W_B, with 9/10<W_A,W_B<1. The matched disconnected law retains its own degrees and regional observations. No new event or weight selection occurs.
• Supply a separate near-consensus preparation domain: relative to declared common form and real phase origins, all ten form and phase increments are independently bounded by epsilon=1/16. This box does not contain the old finite post-word phase separation. The old captured endpoints remain unchanged; a new detached preparation on their held law is not an acquired successor.
• Fix the whole window 0<=t<=h=1/16. The target error allowances for each left-minus-right regional observation are 5*epsilon**3 in form and 3*epsilon**3 in radian phase. Derive, rather than assume, both nonlinear amplitude confinement and accumulated nonlinear-versus-tangent error. Account for every fine coordinate and the actual two coupling channels.
• Identify whether the kernel of the complete eight-state tangent observation remains invisible to the nonlinear outputs. The fixed witness has all form increments zero, phase increments (eta_6,eta_7,eta_8)=(a,a,-2a) and every other phase increment zero, with a=epsilon/4=1/64. Compare its sign-reversed partner. Preserve full storage and global means in this comparison; equal instantaneous regional means alone would not test the complete tangent kernel.
• Finite controls reuse the three original nominal held contact laws and their disconnected counterparts. Evaluate only the two static witness preparations per nominal law, their exact initial rates/storage and the declared box bounds. Use exact represented/rational primitives, outward shared intervals and independent 150-digit arithmetic. Source/box membership and the inherited law are re-admitted; raw declared phase lifts are not wrapped or projected.
• Reuse existing source, exact observation and interval owners. Retain the 128-rank-call budget per rebuilt realization, with no numerical rank threshold, eigenvalue computation, response fitting, operator word or trajectory. Generic independent boundary and analytic-solution controls are separate from these six scientific preparations. A failed sufficient window is unavailable, not a measured failure of the dynamics or an invitation to retune this protocol.

Close with a quantitative validity domain and a named nonlinear closure mechanism or obstruction. Initial derivative separation alone is not a measured fixed-horizon response. The error theorem does not establish formation, attraction, autonomous occurrence or physical identification.

34. A full-state nonlinear error bound about consensus

Retain Section 33's held law and a declared common origin \((x_*,\theta_*)\), with real continuous phase lifts. Write \(\xi_i=x_i-x_*\), \(\eta_i=\theta_i-\theta_*\). The initial box bounds every fine coordinate independently:

\[ |\xi_i(0)|\le e_i,\qquad |\eta_i(0)|\le f_i. \]

The full nonlinear solution and the full consensus tangent start from the same \((\xi(0),\eta(0))\). Initializing the minimal tangent realization with its complete observation of this vector then reproduces the full tangent outputs exactly. Discarded fine coordinates are not assumed to vanish.

The following reusable estimate allows the conservative coefficients \(w,\beta>0\); the frozen star comparison has \(w=\beta=1\). The complete lifted field is globally Lipschitz on a finite held support, so its solution and the linear comparison exist uniquely for every finite time. Define the original-clock row bounds

\[ a_i=\frac{2w\nu_i}{\pi},\qquad b_i=\frac{2w\nu_i}{\beta\pi d_i}\sum_{j\sim i}W_{ij}, \qquad A=\max_i a_i,\quad B=\max_i b_i. \tag{34.1} \]

The bare neighbor count belongs in the first channel; the conductances belong in the second. Since \(|\sin z|\le|z|\), the component inequalities are

\[ |\dot\xi_i|\le a_i\|\eta\|_\infty,\qquad |\dot\eta_i|\le b_i\|\xi\|_\infty. \tag{34.2} \]

Put \(e=\max_i e_i\), \(f=\max_i f_i\). For a positive duration \(h\) with \(D_h=1-ABh^2>0\), the coupled supremum argument of Section 19 gives, throughout \(0\le t\le h\),

\[ \|\xi(t)\|_\infty\le X=\frac{e+Ahf}{D_h},\qquad \|\eta(t)\|_\infty\le T=\frac{f+Bhe}{D_h}, \tag{34.3} \]

\[ |\xi_i(t)|\le X_i=e_i+a_i hT,\qquad |\eta_i(t)|\le T_i=f_i+b_i hX. \tag{34.4} \]

These are derived amplitude bounds for the full nonlinear solution. They do not presume either the nonlinear five-block family or closeness to a chosen tangent trajectory. A nonpositive comparison denominator is failure of this sufficient finite-window estimate, not failure of the dynamics. The exactly zero initial box is separately settled by uniqueness at equilibrium.

34.1 The missing sine terms are cubic and enter only the form row

For every real \(z\), Taylor's integral remainder gives

\[ |\sin z-z|\le |z|^3/6. \]

Subtract the full tangent initialized at the same point. With \(\delta\xi=\xi-\xi_{\rm lin}\) and \(\delta\eta=\eta-\eta_{\rm lin}\), the exact error equations are

\[ \delta\dot\xi=-\frac w\pi D_\nu L_N\delta\eta+r(t), \qquad \delta\dot\eta=\frac w{\beta\pi}D_\nu L_W\delta\xi, \qquad \delta\xi(0)=\delta\eta(0)=0, \tag{34.5} \]

where \(D_\nu=\operatorname{diag}(\nu_i/d_i)\), and

\[ r_i(t)=\frac{w\nu_i}{\pi d_i} \sum_{j\sim i} \left[\sin(\eta_j-\eta_i)-(\eta_j-\eta_i)\right]. \]

Thus (34.4) supplies the uniform residual bounds

\[ |r_i(t)|\le R_i= \frac{w\nu_i}{6\pi d_i}\sum_{j\sim i}(T_i+T_j)^3, \qquad R=\max_i R_i. \tag{34.6} \]

No conductance factor is inserted into this bare sine remainder. The phase row is already linear in form, so it has no separate nonlinear residual. Applying the same two-channel supremum inequalities to (34.5) yields

\[ E_x:=\sup_{[0,h]}\|\delta\xi\|_\infty \le\frac{hR}{D_h},\qquad E_\theta:=\sup_{[0,h]}\|\delta\eta\|_\infty \le\frac{Bh^2R}{D_h}, \tag{34.7} \]

\[ |\delta\xi_i(t)|\le hR_i+a_i hE_\theta,\qquad |\delta\eta_i(t)|\le b_i hE_x. \tag{34.8} \]

Every fine coordinate is included. In particular, a fine direction in the kernel of the tangent observation can contribute to \(r\) and then to the observed nonlinear error.

For a fixed linear observation row \(c\), its error is bounded by the sum of the absolute row coefficients times the corresponding component bounds in (34.8). The two regional rows have \(\|\ell\|_1=2\), so

\[ |y_x-y_{x,\rm lin}|\le2E_x,\qquad |y_\theta-y_{\theta,\rm lin}|\le2E_\theta. \tag{34.9} \]

The initialization of the minimal realization remains the full \(C(\xi(0),\eta(0))\). An arbitrary selected Krylov chart uses its own \(|C|\) error projection; its coordinate scaling is not silently replaced by the normalization of these regional rows.

35. A finite twenty-coordinate domain meets the frozen allowances

Return to the nominal nineteen-edge family \(9/10<u,v<1\). The largest capacity is eight. The maximum of \((\nu_i/d_i)\sum_jW_{ij}\) is less than \(72/5\); the right boundary row attains that limiting value as \(v\) tends to one. Therefore the rational original-time bounds

\[ A\le\frac{16}{3},\qquad B\le\frac{48}{5} \tag{35.1} \]

hold uniformly using \(\pi>3\). With \(e_i=f_i=\epsilon=1/16\) and \(h=1/16\), these larger coefficients give

\[ D_h\ge\frac45,\qquad X\le\frac53\epsilon,\qquad T\le2\epsilon. \tag{35.2} \]

Consequently every phase difference is at most \(4\epsilon\) on the full interval. Equation (34.6), or its global version, then gives

\[ R\le\frac{8}{3}\frac{(4\epsilon)^3}{6} =\frac{256}{9}\epsilon^3. \]

Substitution in (34.7)-(34.9) proves

\[ E_x\le\frac{20}{9}\epsilon^3,\qquad E_\theta\le\frac43\epsilon^3, \tag{35.3} \]

\[ \boxed{ |y_x-y_{x,\rm lin}|\le\frac{40}{9}\epsilon^3 =\frac5{4608}<5\epsilon^3,\qquad |y_\theta-y_{\theta,\rm lin}|\le\frac83\epsilon^3 =\frac1{1536}<3\epsilon^3.} \tag{35.4} \]

These are whole-window absolute error bounds for arbitrary admitted fine preparation, not relative errors divided by a possibly vanishing output. They concern a specified amplitude and time range; they are not an all-time approximation theorem.

There is also a direct statement for the explicit minimal chart of (30.4). Each block mean has row one-norm one, and subtracting the boundary block raises it to at most two. Accordingly every one of its four relative-form coordinates satisfies the first bound in (35.4), and every relative-phase coordinate satisfies the second. The eight-state tangent predictor is therefore a quantitatively controlled approximation to that declared chart. A differently normalized minimal realization retains the \(|C|\) projection rule instead.

35.1 Both marked identities persist beyond the approximation window

The full initial nonlinear storage has the general box bound

\[ H(0)\le\frac12\sum_{\{i,j\}}W_{ij}(e_i+e_j)^2 +\frac\beta2\sum_{\{i,j\}}(f_i+f_j)^2, \tag{35.5} \]

using \(1-\cos z\le z^2/2\). For the attached star support, there are nineteen bare edges and

\[ \sum_{\{i,j\}}W_{ij}=34+4u+v<39. \]

Thus every point in the frozen box satisfies

\[ H(0)<116\epsilon^2=\frac{29}{64}<\frac{35}{64}. \tag{35.6} \]

The added terms are nonnegative and the original two weighted stars remain as subgraphs. The previously proved \(35/64\) barrier therefore protects every marked center-leaf form difference below \(1/2\) and every corresponding circular phase gap below \(\pi/2\) for all time under this unchanged held law. The approximation window in (35.4) and the all-time identity conclusion have different proofs and different scopes.

For the disconnected comparison, each star has seven bare edges and conductance sum seventeen, so its initial storage is at most

\[ H_{\rm star}(0)\le48\epsilon^2=\frac3{16}<\frac{35}{64}. \tag{35.7} \]

The changed capacity of the left boundary does not enter storage and is still retained in its actual dynamics. Both control identities are protected, and each control region's own weighted means remain exactly constant under the full nonlinear law. Hence the nonlinear error of its two constant tangent outputs is exactly zero, even though internal states can move.

36. A tangent-invisible preparation produces nonlinear relative motion

Use the frozen signed preparations \(z_+\) and \(z_-=-z_+\) relative to the declared common origins. All form increments vanish; the only nonzero phase increments of \(z_+\) are

\[ (\eta_6,\eta_7,\eta_8)=(a,a,-2a),\qquad a=\epsilon/4=1/64. \tag{36.1} \]

The weighted average of this three-leaf block is zero. Every other block coordinate is zero, so \(\mathcal Pz_\pm=0\). By Section 30, every coordinate of the complete minimal tangent realization vanishes:

\[ Cz_+=Cz_-=0. \]

This tests the full tangent-invisible kernel, not just equality of the two regional means. Both global weighted means and both initial regional outputs also vanish. All six nonzero phase magnitudes across the two preparations lie inside the frozen box; no fine coordinate is projected away in the nonlinear law.

Each perturbed right leaf has three incident bare phase edges, to its center, its boundary and the left boundary. Thus their complete attached initial storage is

\[ H(z_+)=H(z_-) =3\left[2(1-\cos a)+(1-\cos2a)\right] =9-6\cos a-3\cos2a. \tag{36.2} \]

The form storage is zero. The equality is exact for every \(u,v\) in the declared domain because the phase storage counts bare supported edges. In the disconnected control the same bracket has coefficient two, and remains equal between the two signs there; storage is not equated across different supports.

Internal edge fluxes cancel from the regional means. The attached cross-edge sine sum gives exactly

\[ \dot y_x(z_+)= \frac{\kappa}{\pi}(2\sin a-\sin2a) =\frac{2\kappa}{\pi}\sin a(1-\cos a)>0,\qquad \kappa=\frac{445}{1428}, \tag{36.3} \]

\[ \dot y_x(z_-)=-\dot y_x(z_+),\qquad \dot y_\theta(z_\pm)=0. \]

The sign follows from \(0<a<\pi\), not from numerical roundoff. Its leading amplitude dependence is cubic:

\[ 2\sin a-\sin2a=a^3+O(a^5). \]

Consequently the first nonzero relative output jet already appears in the form row at first order in time, with cubic amplitude dependence. At fixed nonzero \(a\), \(y_x(t)=\dot y_x(0)t+O(t^2)\). The eight-state tangent initialized by either preparation is identically zero, whereas the nonlinear outputs differ immediately. Initial derivative separation is not a measured response at \(h\), and no claim about its sign throughout the whole interval is needed for this obstruction.

The two signs also have identical initial storage and global means. Their different nonlinear output derivatives exclude an exact deterministic nonlinear closure using this same eight-coordinate map, even if initial full storage is appended as an additional coordinate. This does not exclude every possible nonlinear state representation of dimension eight, nor establish a minimal nonlinear dimension. The failure is the specific loss of phase information inside a tangent-invisible leaf block: cancellation of the phase sum does not imply cancellation of the sine sum.

Equations (35.4) and (36.3) are compatible. The tangent remains a certified approximation on the declared box and interval, while its zero-error extension to finite hidden preparations fails. No additional pressure law, operator or physical interpretation has been introduced.

37. A shared nonlinear validity reader with actual box admission

The shared consensus observation owner provides SineConsensusObservation.nonlinear_response_bounds with the keyword inputs duration, consensus_form, consensus_phase, form_amplitude_bounds and phase_amplitude_bounds. Its result is SineConsensusNonlinearResponse.

The consumer freshly rebuilds the complete source law and exact observation realization. The positive duration, common origins and nonnegative nodal radii use exact-or-represented scalar admission. It verifies that every captured form and phase increment actually lies inside the declared independent box. Real radian lifts are retained without wrapping, projection or replacement of hidden coordinates. Thus a bound for a hypothetical box is never mistaken for evidence that an unrelated captured preparation belongs to it.

The captured initial output and selected state are computed as \(Oz_0\) and \(Cz_0\). The report retains these initial coordinates separately from the whole-box bounds. Every solution in that box is compared with the tangent initialized from its own complete initial state; the captured point does not replace the other independent preparations in the theorem.

The implementation obtains its row coefficients from the freshly derived full rational generator and the outward original-time \(w/\pi\) interval. The shared coupled-supremum arithmetic in the support owner is used first for nonlinear amplitudes and then for the two error channels with the cubic form forcing. The report retains:

• The whole-window fine amplitudes and edge phase-difference bounds.
• Each nodal cubic form remainder and the propagated form and phase errors.
• Absolute error projections for both regional outputs and every selected minimal-state coordinate.
• The comparison denominator, availability status and explicit unavailable reasons.

An invalid source, origin or box rejects admission. A valid box with an unsupported sufficient window returns unavailable error bounds; these are not replaced by zeros. The exactly zero box is the equilibrium case and uses uniqueness at any positive duration. Export re-admits the source, box and observation and rebuilds the bounds, rather than trusting cached amplitudes, coordinates or error verdicts. The finite rank-call budget of the exact observation owner remains in force.

The generic nonlinear-response controls exercise the shared amplitude and error bounds, source membership, invalid inputs, unavailable windows and independent analytic controls. The six frozen signed preparations reuse the three already admitted nominal held laws. Their support, conductances, capacities and clock are preserved, while their forms and phases are explicitly supplied detached near-consensus preparations. They are not the original post-word endpoints, which lie outside this small consensus box. Affinities are not re-evaluated at the newly supplied phases; no new word or successor acquisition is asserted. The old source evidence remains unchanged.

The canonical contact controls check both signs against independent fine edge sums and high-precision static arithmetic, including the complete tangent-kernel initialization, equal signed storage, source membership, the nonlinear output derivative and the prospective whole-box allowances. These are finite controls of represented laws alongside the uniform analytical theorem. No nonlinear trajectory, matrix-exponential response, fitted coefficient or numerical rank threshold is used to turn a static check into a temporal observation.

The NFR ontology still separates sufficient predictive state, identity maintenance and formation. This gate supplies an approximation with a quantified validity domain and an exact obstruction to extending the same state map without error. The all-time identity bounds do not supply formation from a nonmember source, an event selector, attraction or a physical identification.

38. Preparation contract for a future common-identity mechanism

This section consolidates the premises needed before a formation response can be evaluated. It chooses no target, source, controller, event schedule or new numerical protocol. The execution plan owns the current task and status. The distinctions below apply the existing NFR ontology.

Keep the actual preparation paths distinct

The original nominal preparation consists of two disconnected supplied stars. The explicitly invoked AL,UM,SHA word changes form, support and capacity; its nominal phases remain unchanged. The actual five generated conductances and complete event work remain retained. Its nineteen-edge successor is an admitted source for the held conservative law; it already has the internal maintenance certificate. A formation claim must identify what further common identity is absent, rather than rename that maintenance result.

The independently perturbed pre-word domain in Sections 22-27 includes unequal generated conductances and explicit numerical-association premises. It is not the equal-four-weight nominal family of Sections 28-36. A formation claim over that whole domain must carry its actual law uncertainty, phase lifts and source correlations.

The near-consensus sources of Section 33 are separately supplied detached preparations on retained nominal held laws. They are not reached versions of the original post-word endpoints. The eight-state tangent and its finite-error bounds neither acquire those sources nor retain all nonlinear information needed by a future identity: Section 36 supplies an exact counterexample to that stronger interpretation.

Required admission before any reserved response

1. Select one source family and its scope. Retain actual fine forms, continuous phase lifts, support, conductances, capacities, clock and preparation correlations. The ten-node state includes both marked constituents and their dynamic boundaries. A reduced observation needs its own sufficiency or error proof; the eight-state tangent alone is not a nonlinear formation state.

2. Define a common identity independently of its hoped-for result. Specify its positive-width geometric or dynamical conditions, retained constituents, relative registration, symmetry convention and lifetime. Separate a definition from a sufficient storage certificate. Failure of that certificate is unavailable evidence, not proof of nonmembership in the geometric identity. Separate normalization of the two constituents must not erase the relative relation whose formation is being tested.

3. Prove absence on the source domain. Give a strict exclusion margin for every claimed source member, retaining its correlated errors. If the predicate is defined only on connected support, evaluating it on disconnected stars is unavailable unless a common-domain observation or an explicit support-transition interpretation has been supplied. Installing an edge may satisfy a support condition without establishing a newly organized dynamical relation.

4. Declare the complete causal path. Choose the already admitted held flow or a specified hybrid path. Holding weights fixed differs from recomputing affinity during motion. Support, degree normalization and actual d/nu means must follow the selected path. If an event is used, its live admission, ordered history, target/context, pressure provenance, complete reset and support map all remain premises. Admission does not select occurrence; no controller is inserted merely to produce entry.

5. Keep all work channels. Use the actual full storage before and after each event, including old-edge changes and every added-edge cost. Account for continuous regional exchange with its retained environment and cross-boundary storage. Positive incoming work is not a lost-energy reservoir. Any supplied work, storage coordinate or occurrence law must be stated separately, with matching structural units.

6. Freeze absence, entry, retention and later discrimination together. Before a new response, declare an entry window or bounded hitting rule, the required subsequent duration, source/reading errors and a meaningful same-source alternative. Retain the actual reached state into the maintenance and later response checks. The earlier 1/16 tangent-error horizon and the 1/128 internal-contrast comparison are not automatically formation deadlines. Prior inspected endpoints cannot become reserved evidence for a newly selected identity.

Apply the existing obstructions without overstating them

The conserved full-storage sublevel cannot be entered from outside under the same held conservative law. Likewise a two-sided invariant exact family cannot be reached from a nonmember in finite time by a unique unchanged flow. The conservative formation boundary does not exclude entry into a different noninvariant observational identity with a separately bounded lifetime.

For the original nominal successor, the retained source lifts take only the values zero and \(\alpha\in[3/8,2/5]\), so every actual edge difference is initially in \((-\pi/2,\pi/2)\) and every oriented cycle has winding zero. The separately admitted near-consensus box has one globally consistent lift with every edge difference at most \(2\epsilon=1/8\), and has the same initial zero-winding property. Their proved full-storage budgets, \(H<9/20\) and \(H<29/64\), respectively, are below one. Each bare phase-edge term is nonnegative, so conservation prevents any absolute principal edge gap from reaching \(\pi/2\); the original cycle windings cannot change on these held supports. This is an admission consequence of those source certificates, not a new evaluated formation response. The inequality \(H<1\) alone would not establish zero winding for an arbitrary source on a larger cycle. Nor does this restriction exclude a common identity that is not defined by acquiring nonzero winding.

Reuse mechanisms with their actual premises

The common-pattern formation contract already separates source absence, entry, retention and continued response. Its observation-domain and correlation requirements transfer as a method; its localized cycle geometry, carrier bands, preparation and loss do not.

The phase-nearest contact supplies a full-source barrier for a jointly registered identity under an actual grammatical contact, and its work ledger keeps old-edge resets and attachment distinct. Its doubled-C5 support, three-candidate selector and prepared registration cannot become a star formation certificate. The current star's event work instead remains owned by Section 4.

The outer-winding acquisition is a genuine source-to-new-identity construction with carried child residuals and a paired control. Its product support, dissipative complete law, clock, large supplied form preparation and positive attachment work remain essential. Transfer its source-to-entry-to-retention architecture, not its winding verdict or capture coefficients.

The conservative regional balance and retention budget show how a retained environment can exchange both form and phase work. The handoff obstruction shows why low regional entry storage alone is insufficient. Their unit-support coefficients and winding barriers must be re-derived for any proposed weighted star region.

The same-orbit conservative construction proves conditional finite formation followed by finite retention without loss. It retains a different supplied support, flow-defined preparation and extremely small positive uncertainty width; it is not an operational star source or evidence that every conservative identity is attainable. The assembly-work obstruction adds reusable endpoint and telescoping work accounting, while its specific recovered C9 sources, contact costs and acute endpoint class do not transfer.

These owners already supply the required distinctions and proof methods. A subsequent star mechanism must match them to its declared source and target before execution. No new formation catalogue or parallel queue, unexamined target, or autonomous-occurrence claim follows from this audit.

39. Prospective admission of robust permanent common registration

This bounded analytical question tests permanent capture under the existing nominal held law. It does not evaluate a finite entry trajectory, add an event or claim that every possible common identity must have this lifetime. The candidate is a common full-state registration, retaining both marked constituents and all ten fine form/phase coordinates.

Declare the observation scales X0 equal to one existing form unit and Theta0 equal to one radian. These are supplied scales, not physical constants or a dynamical identification of form with phase. With the arithmetic centering projection P=I-11^T/10, define the dimensionless observation

\[ Q(x,\theta)=\frac{\|Px\|_2^2}{X_0^2} +\frac{1}{\Theta_0^2}\min_{\varphi\in\mathbb S^1} \sum_{i=0}^9 d_{\mathbb S^1}(\theta_i,\varphi)^2. \]

Here circle distance is the absolute shortest angular distance. A separately chosen phase origin for each star would erase the relation being tested; there is one common origin in the minimization. The closed target is Q<=1/4, with the existing two constituent identities retained. Its radius is an explicit geometric policy. The full circle minimum, rather than an arbitrary unwrapped phase variance, defines this observation globally.

Permanent capture means entry after some finite time followed by membership at every later time. Asymptotic approach to the same closed target is also considered. Robustness here requires a full-dimensional open source family under each fixed law, not only the original two-parameter nominal surface or a list of represented probes. A finite residence episode remains a different question.

Reuse the nominal successor family of Section 1 and, under each fixed law, the already admitted independent form/radian error box zeta=2^-20 of Sections 18-21. The source must be excluded from the target with a strict uniform margin before any entry assertion. Keep every perturbed fine state; the five-block family and eight tangent coordinates do not replace it. For the recurrence admission, freeze the full-storage ceiling 3/4 and own-law weighted form-mean slab [-1,1]. These specify an invariant family, not an input or a physical preparation distribution.

The proof must re-admit the actual conductance form gradient, bare sine incidence, strictly positive capacities and original clock. Apply the existing nonlinear recurrence argument only after proving bounded form on the connected positive-conductance subgraph, conservation of storage and the d/nu form mean, and exact zero divergence of the full nonlinear rows. Circular phases are compact without imposing a phase-lift bound. Ambient Lebesgue-form/circle-Haar measure does not become a measure on the nominal surface, an exact energy level or a selected point.

Static implementation controls reuse only the three nominal source reports and existing full-state boxes. Check the complete-law hypotheses, compact coordinate bound, actual source inclusion, target exclusion and export rebuilding. Independent generic controls must include supported zero weights whose positive edges still span, and a disconnected positive-conductance case where boundedness fails. No trajectory, eigenvalue, new operator word, new source pattern, fitted response or reserved endpoint is evaluated.

The admissible outcomes are a robust permanent-entry mechanism under these premises or a precise obstruction with its measure and lifetime scope. A negative result must retain finite-lived formation and exceptional preparations as separate questions. It cannot be repaired by interpreting hidden conservative state as irreversible loss or invoking a supplied operator as an autonomous occurrence law.

40. A source-excluding common registration on the full circle state

Retain Section 39's observation scales, full ten-node state and closed registration class \(\mathcal C=\{Q\le1/4\}\). The two strict constituent identities are additional conditions along the motion; they do not need to make their intersection with \(\mathcal C\) closed. The obstruction below already holds for the larger closed class \(\mathcal C\).

Both terms of \(Q\) remove only a common origin. The form term is invariant under a common form translation, and the phase term under a common circle rotation and any change of phase representatives by integer turns. The two supplied scales make the sum dimensionless; they do not identify form units with radians or derive the radius from a fundamental constant. This is a positive-width registration policy, not a small response threshold or a new universal definition of an NFR. Configurations near full form and phase consensus lie in its interior.

40.1 Absence at every nominal preparation and successor

For any two circle points \(p,q\) and any common center \(\varphi\), the metric triangle inequality and the real squared-sum inequality give

\[ d(p,\varphi)^2+d(q,\varphi)^2 \ge \frac{[d(p,\varphi)+d(q,\varphi)]^2}{2} \ge \frac{d(p,q)^2}{2}. \tag{40.1} \]

Pair the five left nodes with the five right nodes. At a nominal source every cross-pair circle distance is \(\alpha<\pi\). Summing (40.1) over these disjoint pairs gives a phase cost at least \(5\alpha^2/2\). The common center \(\varphi=\alpha/2\) attains it, so this is the exact global circle minimum, not a minimum over a chosen unwrapped chart. The nonnegative form term then yields

\[ Q(z^0)\ge\frac52\alpha^2\ge\frac{45}{128} =\frac14+\frac{13}{128}. \tag{40.2} \]

The nominal AL, UM and SHA stages leave these phases unchanged. Consequently (40.2) excludes the pre-word preparation, each nominal stage and its post-word successor, irrespective of their different form, support, capacity and storage values. The invoked word does not itself jump into this registration class. This statement uses the already admitted nominal reset, not a claim that every possible UM execution leaves phase unchanged.

40.2 Full-dimensional source exclusion under each held law

Now fix one admitted nominal post-word law and perturb all ten forms and all ten radian phases independently by at most \(\zeta=2^{-20}\) in their respective declared units. These are the existing source boxes of Sections 18-21, not a new word execution or a reset onto the five-block family. Each cross-pair distance is at least \(\alpha-2\zeta>0\), even if a stored phase crosses a representative's zero cut. Hence (40.1) gives

\[ \begin{split} Q(z)&\ge\frac52(\alpha-2\zeta)^2\\ &=\frac52\alpha^2-10\alpha\zeta+10\zeta^2 >\frac{45}{128}-4\zeta\\ &=\frac14+\frac{26623}{262144}. \end{split} \tag{40.3} \]

Here \(3/8\le\alpha\le2/5\) was used, with the fixed observation scales of Section 39. This is a strict uniform absence margin on the full source box. The proof never substitutes a separately centered phase for either constituent and requires no calculation of a numerical circle minimizer.

The independent all-coordinate storage bound (20.5) remains

\[ H(z)<\frac{2025}{4096}<\frac12<\frac{35}{64}. \tag{40.4} \]

All forms lie in \((0,1)\): their nominal extremes are bounded by \(15/64\) and \(21/64\), with an additional error at most \(\zeta\). Every positive-weight form average is therefore strictly inside \([-1,1]\). In particular, the whole source box lies inside the frozen energy/mean family of Section 39. Its interior has positive twenty-dimensional product volume. The phases are circles; the small local boxes provide positive-volume charts without restricting the full recurrence theorem to real phase lifts.

41. Weighted recurrence admitted on the actual complete law

The nonlinear recurrence owner extends the existing unit-conductance proof to held nonnegative conductances whose positive-edge subgraph is connected. It keeps the bare sine incidence distinct from the weighted form gradient:

\[ \dot x=\frac1\pi KS(\theta),\qquad \dot\theta=\frac1\pi KL_Wx,\qquad K=\operatorname{diag}(\nu_i/d_i). \tag{41.1} \]

For the current source, all nineteen conductances are positive, \(W_{\min}>9/10\), and all ten held capacities are positive. The actual inverse mobility weights are

\[ \mu=(1,1,1,1,\;32/13,\;5/4,\;3/2,3/2,3/2,\;5/8). \tag{41.2} \]

Thus the required invariants are the full storage \(H\) already used above and \(m_\mu(x)=\sum_i\mu_i x_i/\sum_i\mu_i\). Pairwise bare current cancellation preserves the latter. Each form row depends only on phase, while each phase row depends only on form, so the full twenty-dimensional divergence is exactly zero. Conductance heterogeneity and the degree change at node 4 have not been discarded.

Apply that theorem with the frozen family

\[ \mathcal S=\{H\le3/4,\;-1\le m_\mu(x)\le1\} \subset\mathbb R^{10}\times(\mathbb S^1)^{10}. \tag{41.3} \]

Its positive-conductance path bound gives

\[ |x_i-m_\mu(x)|\le R,\qquad R=\sqrt{\frac{2(10-1)(3/4)}{W_{\min}}} <\sqrt{15}<4. \tag{41.4} \]

All real form coordinates consequently lie in \((-5,5)\); the phase torus is already compact. The closed family \(\mathcal S\) has finite positive product volume, is invariant in both time directions, and supports the smooth volume-preserving complete flow. The cited proof therefore gives recurrence for almost every point of \(\mathcal S\).

No conserved phase-lift mean is used for compactness, and this argument does not infer return of real lifts from circle return. The stronger source bound (40.4) separately retains both original center-leaf identities for all time by Section 5. It is not needed to replace the torus by an acute chart in the recurrence proof. For these particular sources the previously established zero windings also persist; that is a distinct identity consequence.

42. Robust permanent acquisition is excluded under the held law

Let \(z\) be a recurrent point of the source box. It has \(Q(z)>1/4\) with the margin (40.3). The circle minimum in \(Q\) is continuous: it is the minimum of a continuous function over the compact common-center circle. Consequently, for any return sequence \(\Phi_{t_j}(z)\to z\) with \(t_j\to\infty\),

\[ Q(\Phi_{t_j}(z))\longrightarrow Q(z)>\frac14. \tag{42.1} \]

Arbitrarily late returned states are outside \(\mathcal C\). No finite entry time can therefore be followed by membership in \(\mathcal C\) at every later time. Nor can the distance to this closed class tend to zero: continuity of distance along the same return sequence would contradict \(z\notin\mathcal C\). This is the conservative capture boundary applied after admitting its actual weighted-law premises.

Almost every point of the full-dimensional source box is recurrent. Thus the subset permanently captured in \(\mathcal C\), or asymptotically approaching it, has ambient product measure zero. In particular, no nonempty open subfamily of this box can have permanent capture for every preparation. The requirement of robust permanent common registration in Section 39 is excluded, even though both original constituents remain protected.

The quantifiers are essential. This is a statement for each fixed admitted law and its full-dimensional preparation box. It supplies no recurrence verdict for one selected represented point, the original two-parameter nominal surface, a fixed-energy surface, or a finite list of seeds. Probability one requires an absolutely continuous preparation distribution; none is inferred from a numerical sampler. The exact nonrecurrent separatrix example in the recurrence owner remains a counterexample to a stronger chosen-state assertion.

What the obstruction does and does not require next

There is no finite-time no-entry theorem for \(\mathcal C\) here. A recurrent orbit can enter it, remain for a nonzero interval, respond while there, and later leave. Neither the existence of such an episode for the present source nor its lifetime has been evaluated. The negative permanent result does not invalidate the separate conservative finite-formation constructions or the dissipative formation results linked in Section 38.

The nominal word does not enter this target by (40.2), and no additional event or occurrence law is present in (41.1). A robust permanent-capture mechanism would have to remove a proved premise of the obstruction and justify the resulting complete model. A declared non-volume-preserving event path, an open environment or another constitutive law has its own state, work, admission and occurrence obligations. Selecting one merely to force capture would not derive it. Eliminating hidden conservative coordinates likewise does not supply an irreversible loss mechanism.

For a finite-lived common identity the appropriate missing evidence is different: independently bounded entry, a declared nontrivial residence interval, and a continued response that distinguishes the acquired relation from ordinary tangent or separate-component passage. That question can retain the present full-state target and existing fine law; it must not replace this scoped permanent obstruction by either universal impossibility or an unproved formation claim.

43. Shared recurrence admission and static evidence boundary

The weighted support owner provides SineSupportExchangeComparison.recurrence_bounds( energy_ceiling=..., form_mean_bounds=...), returning SineSupportRecurrence. This is a detached theorem-admission report, not a trajectory or a classifier for individual recurrence.

The consumer re-admits primitive state, support, coefficients and capacities before rebuilding the actual d/nu weights, weighted form mean, full storage and mean-rate residual. Strictly positive capacities and a connected positive-conductance subgraph are required. Zero-conductance bare edges remain neighbors in degree, sine incidence and phase storage; their presence is compatible with the theorem when the positive edges still connect all nodes. Mere bare connectivity does not certify the weighted form bound.

The report retains the positive-edge indices and weight floor, the conservative n-1 path-length bound, outward form-radius and coordinate bounds, and exact zero divergence. Positive energy ceiling and a positive-width mean interval define the invariant family. Captured-state membership is assessed separately as inside, outside or unresolved; failure to include that particular snapshot does not negate the admitted family theorem. Equal captured phases permit exact rational form-only storage, while general phases use the existing outward storage interval.

individual_recurrence_status remains unavailable_for_chosen_state, including when captured family membership is inside. The family-level recurrence flag does not promote any of the three original probes to a recurrent orbit. Export through to_dict() rebuilds consumed fields and flags from the primitive source and the supplied family bounds.

The independent weighted recurrence controls cover actual weights and bare degrees, preserved mean, compact bounds, source membership boundaries, strict scalar admission, supported zero weights, disconnected positive-edge support and rebuilding. The canonical contact controls reuse the original three source reports and the existing twenty-coordinate boxes. Their target-exclusion checks retain the circular pair inequality (40.1), not a raw-lift substitute for \(Q\). No new word, eigenvalue, selected return, trajectory or post hoc entry threshold is used. The uniform proof, finite static implementation checks and unavailable individual recurrence remain separate evidence statements.

The separate finite-registration owner compares the actual source storage and lifted means with the fixed target, proves all-time target exclusion in the matched disconnected control, and retains the distinction between a configuration path, a dynamical episode and a collective NFR. It supplies no additional event or formation claim.

44. The actual canonical word leaves constitutive freedom

This source-specific check answers the prospectively fixed question of which complete continuous laws can follow the actual canonical word. It reuses the central represented preparation, all four retained static snapshots and the existing periodic counterfamily, with the previously fixed coefficient epsilon=1/16. No new word, trajectory, source scan, calibration or physical observation is evaluated. The general nonselection result was already known; the result here is admission at this particular executed boundary, including retained constituent identities.

Actual prefix consumers and pressure ordering

The shared source invokes execute_network_operator_stage with target 4, the admitted core sequence context and no compute_delta_nfr callback. The word dispatcher and atomic stage owner therefore perform no constitutive-pressure refresh between these events. Generic stage footprint labels describing an optional refresh do not establish its occurrence.

Prefix	Consumed state and resulting restriction	Retained output
Source	Signed form, positive capacities, primitive phases, old support, supplied parameters and empty actor history	Fourteen edges; all stored pressures zero; structural time 7
AL	Active capacity, form admission/rails, configured boost and lifecycle/context	x_4=257/1024, actor history [AL], unchanged phases, support and stored pressure
UM	Live old neighbors 0,...,3, wrapped U3 admission, form/capacity thresholds, stored pressure and configured policies; candidate affinities additionally inspect right-node form, phase and Si	Five actual new conductances; nu_4=117/16; unchanged phases; history [AL,UM]; zero stored pressure remains zero
SHA	Live capacity, selected closure context, factor/floor and lifecycle state	nu_4=117/32; same nineteen edges and form; history [AL,UM,SHA]; zero stored pressure

The represented target form after AL is uniform-real BEPI, admitted by the shared signed-scalar contract. It cannot be replaced by a magnitude projection or unchecked float. Every prefix retains its recorded context, target/history association and lifecycle metadata; none advances structural time. The actor's steps are 0,1,2,3, with zero grammar debt and no supplied prior IL. Neither a scalar equality nor a restored record authenticates the original execution; the frozen source archive and static execution controls retain that provenance.

UM aligns only the old neighbors, whose phases are all zero. The right nodes are candidates for new links, not alignment neighbors. Their common phase alpha=201/512 affects the existing affinity policy. Changing a subsequent constitutive current does not change that policy, the stored-pressure multiplier acting on zero, or any input consumed by these three stages. In particular, the initial and post-AL constitutive pressures vanish for both currents because all existing edge gaps vanish. Their reciprocal phase rates are nevertheless nonzero: zero pressure is not complete-state equilibrium.

After UM, the new support gives nonzero fresh constitutive pressure while the raw record still stores zero. This is the declared event-to-flow handoff, not a pressure-memory equation. Requiring an interstage refresh or persistent independent pressure would define a different complete execution model.

Two admitted complete continuations on that same successor

Retain actual conductances W, bare degrees d, held capacities nu and original time. Put K=diag(nu_i/d_i) and q=L_W*x. For either epsilon=0 or 1/16, declare pressure before evaluating rates:

\[ j_\epsilon(\delta)=\sin\delta+\epsilon\sin^3\delta,\quad S_{\epsilon,i}=\sum_{j\sim i}j_\epsilon(\theta_j-\theta_i),\quad p_{\epsilon,i}=\frac{S_{\epsilon,i}}{\pi d_i}, \qquad \dot x=\frac{K S_\epsilon}{\pi},\quad \dot\theta=\frac{Kq}{\pi}. \tag{44.1} \]

There is no forcing, capacity/support evolution or later event selector. Pressure consumes bare incidence; q consumes actual conductance. The current is smooth, odd and periodic, so both rows define a globally Lipschitz lifted field and respect circular phase and common phase/form origin symmetries. The choice of reciprocal phase row, held channels, coefficients and clock remains a premise. A successful word does not derive them. These continuous symmetries do not extend automatically through UM's absolute-form affinity policy.

With the same finite signed form chart, define

\[ U_\epsilon(\delta)=1-\cos\delta+ \epsilon\left(\frac23-\cos\delta+\frac{\cos^3\delta}{3}\right),\quad H_\epsilon=\frac12 x^\mathsf T L_Wx+ \sum_{\{i,j\}}U_\epsilon(\theta_j-\theta_i). \tag{44.2} \]

Because U_epsilon'=j_epsilon, the gradients are (q,-S_epsilon). Thus dot H_epsilon=(q^T K S_epsilon-S_epsilon^T K q)/pi=0. With positive held capacities, mu_i=d_i/nu_i gives mu^T x_dot=sum_i S_epsilon,i/pi=0 and mu^T theta_dot=sum_i q_i/pi=0. The phase mean uses the declared continuous lifts, not a global mean on the torus. These are each model's continuous balances; event jumps change support and capacities and require separate work. At consensus j_epsilon'(0)=1, so the full tangents coincide, while the nonlinear currents and storages differ. The Jacobians at the nonconsensus successor need not coincide, and equal cancellation equilibria are not claimed.

Exact instantaneous discriminator and event work

Write s=sin(alpha)>0, a=1/16, b=1/1024, c=a-b=63/1024. Only the five inserted edges have a nonzero phase gap at the successor. Subtracting the sine continuation from the fixed comparison gives

\[ p_{1/16}-p_0=\frac{s^3}{16\pi} (0,0,0,0,5/9,-1/5,-1/3,-1/3,-1/3,-1/5), \tag{44.3} \]

\[ \dot x_{1/16}-\dot x_0=\frac{s^3}{16\pi} (0,0,0,0,65/32,-4/5,-2/3,-2/3,-2/3,-8/5), \qquad \dot\theta_{1/16}-\dot\theta_0=0. \tag{44.4} \]

This is a strict difference at the same full state and original clock, not a numerical trajectory claim. Since q_0=2c and K_0=1, both models have theta_dot_0=2c/pi>0. Any common time-rescaling factor matching the complete vectors there would be one, contradicting the nonzero form-rate difference at node 4. The discrepancy therefore cannot be removed by a common clock change, including a state-dependent factor at this state.

Writing the actual inserted weights as u on 4--5,...,8 and v on 4--9, the own-law storage jumps on the unchanged raw word are

\[ \Delta H_\epsilon^{AL}=-8ab+4b^2,\qquad \Delta H_\epsilon^{UM}=2u c^2+\frac{v b^2}{2}+5U_\epsilon(\alpha), \qquad \Delta H_\epsilon^{SHA}=0. \tag{44.5} \]

The UM work is strictly positive. Continuous conservation does not remove that supplied assembly work or derive its reservoir or occurrence. The added potential changes that work by 5*epsilon*(1-cos(alpha))^2*(cos(alpha)+2)/3>0. Raw event storage and fresh pressure are evaluated separately, with no artificial event duration.

Own-law maintenance and information synergies

The alternative is not merely a field defined at one point. Put v_delta=1-cos(delta) in [0,2]. Then

\[ U_{1/16}=v_\delta+\frac1{16}v_\delta^2(1-v_\delta/3),\qquad v_\delta\le U_{1/16}\le\frac{67}{64}v_\delta, \tag{44.6} \]

because 0<=v_delta-v_delta^2/3<=3/4. The actual source bound of Section 4 therefore implies, using the newly proved conservation of H_1/16,

\[ H_{1/16}(t)=H_{1/16}(0) \le\frac{67}{64}H_0(0)<\frac{603}{1280}<\frac12<\frac{35}{64}. \tag{44.7} \]

Nonnegative added-edge storage and U_1/16>=U_0 retain the same old-star Dirichlet and circular-gap barriers used in Section 5. Thus both marked center-leaf identities persist for all time under this comparison's own law. The hub edges also retain their initially short gaps below pi/3 in magnitude. This is a re-derived sufficient certificate, not a transfer of sine dynamics. It establishes no new collective identity or entry/no-entry result for Q; the frozen sine/tangent episode remains confined to its original laws.

The five-block closure also survives this particular odd pairwise current: on the first block the only possibly nonzero incident phase gap has degree-normalized capacity factor one; the right leaves have identical retained inputs. Replacing each sine in the same multiplicity equations by j_1/16 preserves equality of both fine rows within each block. This is exact closure on the retained nominal family, not robustness to arbitrary off-family state or closure of the full operator context.

Finally, j_1/16=(67*sin(delta)-sin(3*delta))/64. For arbitrary incident phase lists, its pressure depends on third-harmonic information not determined by degree and the first moment. Both moments are computable from already retained primitive phases. UM's use of a neighbor resultant for its alignment reset does not impose a first-moment restriction on continuous pressure. The existing all-degree information theorem would select a first-harmonic current only with its independent information premise and normalization. This word supplies neither necessity.

Shared comparison and evidence scope

The detached handoff reader reuses the retained preparation, shared signed-scalar/graph admission, complete sine support reader and exact interval arithmetic. It rebuilds consumed fields at every prefix and adds the explicitly declared current/potential correction; it installs no engine default, pressure callback or trajectory. Its source-specific API additionally requires the fixed canonical fingerprint of the complete tagged preparation record after primitive admission. This keeps all configuration/history/lifecycle associations, including their scalar and container types, without creating a parallel operator-admission catalog. The record fingerprint and original-file byte receipt are separate associations; neither authenticates execution. Its independent controls check fine edge sums, both rate rows, own storage and event work, strict instantaneous separation, nontrivial five-block closure and invalid primitive/context associations. The frozen scientific producer is not replayed or modified.

The result is source-specific canonical-word compatibility with distinct complete continued fields and retained constituent identities. It proves neither compatibility with every operator/history nor a selected fundamental law. The sole execution plan owns the next question; additional variants under unchanged premises would not strengthen the missing justification.

45. A clock-independent response relation is not closed geometry

Retain the actual central represented successor with m=1/4, alpha=201/512, its held conductances/capacities and the complete sine law of Section 5. No event or source producer is replayed. For the original left/right regions define D=bar_x_right-bar_x_left using their own normalized weights d_i/nu_i, and retain R=x_5-x_6. Sections 15 and 29.1 give the actual joined masses and their reciprocal sum:

\[ M_{\rm left}=84/13,\qquad M_{\rm right}=51/8,\qquad k=M_{\rm left}^{-1}+M_{\rm right}^{-1}=445/1428. \tag{45.1} \]

Internal bare sine currents cancel in each weighted regional sum. If J=sum_{i=5}^9 sin(theta_i-theta_4), the complete fine rows therefore give D_dot=-k*J/pi, independent of a coarse closure assumption. At the actual source, D=113/6528, R=0, and Section 6 implies exactly

\[ \dot D=-\frac{5k}{\pi}\sin\alpha<0,\qquad \dot R=-\frac{2}{15\pi}\sin\alpha,\qquad \frac{\dot R}{\dot D}=\frac{952}{11125}. \tag{45.2} \]

The source slope does not identify alpha or the held conductances. Smooth flow and D_dot!=0 give a local graph R(D) along this particular trajectory by the inverse function theorem. Under a common positive clock change both rates acquire the same factor, so their ratio is unchanged. Here D decreases: -D, rather than D, is an increasing local trajectory parameter. Neither construction supplies elapsed laboratory time or a spatial metric. The whole-law clock contract still transforms all consumed rows, inputs and schedules.

There is no common local slope function of (D,R) on a full neighboring state domain, even after allowing positive time reparameterization. To see this within the already proved five-block family, hold every form, support entry, conductance and capacity fixed, and change only

\[ \theta_5=\alpha+\varepsilon,\qquad \theta_9=\alpha-2\varepsilon . \tag{45.3} \]

All other phases keep their actual source values. This is a separately supplied nearby state, not a new word endpoint or a reached trajectory state. It preserves D, R, all form information and the weighted lifted phase mean: the changed phase weights are 5/4 and 5/8. In the notation of Section 15 only Theta_C and Theta_B change. Direct fine-edge sums give

\[ \begin{aligned} J_\varepsilon&=\sin(\alpha+\varepsilon)+3\sin\alpha +\sin(\alpha-2\varepsilon),\\ N_\varepsilon&=\frac{46}{15}\sin\varepsilon +\frac45\sin(3\varepsilon)-\frac23\sin(2\varepsilon) +\frac45\sin(\alpha+\varepsilon)-\frac23\sin\alpha,\\ \dot D_\varepsilon&=-kJ_\varepsilon/\pi,\qquad \dot R_\varepsilon=-N_\varepsilon/\pi,\\ \left.\frac{d}{d\varepsilon} \frac{\dot R_\varepsilon}{\dot D_\varepsilon}\right|_0 &=\frac{62(5+\cos\alpha)}{375k\sin\alpha}>0. \end{aligned} \tag{45.4} \]

Thus identical (D,R) can have distinct projected orbit directions, not merely different speeds. This excludes a closed two-observation direction field on that neighborhood. It does not exclude a local graph on a single trajectory, a restricted preparation, or a description retaining more state.

The perturbations preserve the declared identities under their own continued flow. For |epsilon|<=1/64, the changed cross gaps remain positive, hence J_epsilon>0. Form storage is unchanged. The seven right internal phase terms add at most 12*epsilon^2 using 1-cos z<=z^2/2; the two changed cross terms add at most 3*|epsilon| using the Lipschitz bound for cosine. The actual source bound from Section 4 then gives

\[ H_\varepsilon\le H_0+3|\varepsilon|+12\varepsilon^2 <\frac{71}{160}+\frac3{64}+\frac{12}{4096} =\frac{2527}{5120}<\frac12<\frac{35}{64}. \tag{45.5} \]

Here H_0<1129/2560<71/160; the perturbed energy need not equal H_0. The existing full-storage theorem retains both old-star identities. It does not establish access to the perturbed state or a new common identity.

Independent static controls rebuild the fine rows from the associated retained source and check the phase perturbation without trajectories. The operational identification boundary separates this projective nonclosure from unconsumed-length ambiguity, relabeling covariance, physical distance and causal propagation. The useful result is a retained-internal-state obligation; an invariant instantaneous ratio alone does not derive geometry or select a constitutive law.