Finite registration and collective identity after weighted-star contact¶
This owner tests necessary conditions for finite common registration under the actual canonical contact. Registration is a supplied geometric observation. Calling an episode a new collective NFR additionally requires the identity, retained state and response obligations in the ontology. The research program alone queues work.
The nonlinear continuation is the declared held weighted sine model. Its full real-lift tangent at consensus is a separate mathematical control, not a globally equivalent circular field or a second proposed canonical law. Canonical describes the registered AL/UM/SHA preparation; the law-admission owner retains the additional premises needed to justify a complete continuation.
1. Analytical question and fixed static-control contract¶
Retain the complete ten-node source, word and held laws of the contact owner.
There are no new operator invocations, changed weights, sources, forcing,
capacity laws, physical clocks or event selectors. Each nominal post-word
law keeps its own support, degrees, capacities and conductances. The existing
independent form/radian boxes have radius zeta=2^-20; they do not rerun the
word or replace perturbed state by an exact collective family.
Use the previously fixed observation
Q=||P*x||^2/X0^2 + min_phi sum_i d_circle(theta_i,phi)^2/Theta0^2,
with P=I-11^T/10, one common circle center, X0 one form unit,
Theta0 one radian and target Q<=1/4. Its absence at every nominal
preparation and throughout the full-dimensional boxes is already proved.
The two original marked constituent identities remain part of the question.
A change of the observation or its scales would be a different study.
The permanent-registration obstruction does not exclude finite entry. Before committing to an episode calculation, compare the following actual laws and distinguish their conserved quantities:
| Alternative | State and dynamics | Purpose |
|---|---|---|
| Joined nonlinear law | All ten forms and declared continuous radian lifts; weighted form gradient, bare sine incidence and actual held capacities | Test whether the displayed storage and two means forbid the target |
| Matched disconnected law | The existing functional-links-disabled word output, with its own degrees, capacities and two components | Test whether separate component invariants exclude the same circular target for all time |
| Full consensus tangent | The same joined initial fine state and declared real lifts, with sine differences replaced by those real differences in the full twenty-coordinate law | Test geometric compatibility of its own quadratic storage and means; later compare whether nonlinear terms are needed for a proposed episode |
The tangent is a separately declared real-lift model. Its linear force does not define a periodic vector field on the phase torus. The common circular observation can nevertheless be applied to its projected state. The previous small-consensus-box error theorem does not cover this post-word preparation. Neither the two regional outputs nor their eight-dimensional tangent realization determine the full nonlinear registration observation.
Analytical obligations. For the joined law, construct a target configuration with the same full storage and both conserved lifted means, and, if possible, a continuous configuration path preserving them. Retain the exact five-block family only for its admitted nominal initial state. This is a compatibility test for the displayed invariants, not a proof that they are all the invariants, an integral curve or a reached state. For the disconnected law, prove a lower bound for the global circular phase cost from independently conserved component means and retained edge branches. Any passage from circular observations to lifted means needs its own proof. Check the nominal sources and the already admitted independent boxes.
Static controls. Reuse the three existing source and matched-control fixtures. Independently check actual own-law mean weights, per-component storage and branch margins, and the Cauchy bound against the unchanged target. Check a correlated joined target construction through exact quadratic identities and outward storage intervals, without replacing it by an independent uncertain-state box. Re-admit the tangent's quadratic storage on the same fine initial state. Preserve source/configuration hashes and record these observation and measure-independent proof obligations. No trajectory, eigenvalue, new word configuration, empirical response, entry time, residence duration or later interaction is evaluated here.
The admissible result is a scoped obstruction or compatibility result, with remaining dynamical and identity premises explicit. If the tangent also permits registration, this limits mechanism discrimination; it does not introduce nonlinearity as a universal NFR definition. No numerical horizon or identity threshold is selected from an inspected response.
2. The displayed joined invariants permit the fixed target¶
Write the nominal inserted conductances as \(u\) on edges \(4\)--\(5,\ldots,8\) and \(v\) on edge \(4\)--\(9\), with \(9/10<u,v<1\). All old conductances, degrees and held capacities remain those of the actual successor. For arbitrary fine state use
The actual weights of both conserved lifted means are
For a selected admitted initial state \(z^0=(x^0,\theta^0)\), put \(\bar x=\mu^\mathsf T x^0/M\) and \(\bar\theta=\mu^\mathsf T\theta^0/M\). The second expression uses its declared continuous lifts. It is not asserted to be a globally defined mean on the phase torus. Let \(e_4\) be the fourth-index coordinate vector and define
Then
The first centering is the actual conserved weighted mean; the last is the arithmetic centering required by \(Q\). They have not been substituted for one another.
Given the actual source storage \(H_0>0\), take the correlated configuration
It has exactly the same \(H_0,\bar x,\bar\theta\), while its phase cost is zero. With Section 1's fixed observation units,
For every nominal source \(H_0<9/20\), so
For every point of the admitted independent \(\zeta\)-box, the previously proved \(H_0<2025/4096<1/2\) instead gives
All four left A forms are equal in (2.5), as are all four right A forms; all phases agree. Both marked center-leaf identities therefore hold exactly. The configuration also belongs to the nominal five-block family \((0,1,2,3),(4),(5),(6,7,8),(9)\). It is a comparison configuration, not a newly supplied source, a reached state or a prediction of entry.
2.1 A protected continuous path with the same displayed invariants¶
The compatibility is not limited to a disconnected point of the constant-storage set. Use the source's retained lifts and, for \(0\le s\le1\), set
The factor \(X_0\), equal to one declared form unit, keeps the interpolated form directions in the same units. Both directions in \(v\) have zero \(\mu\)-weighted mean. The actual edge \(4\)--\(9\) gives an explicit uniform lower bound. Put \(b=1/1024\) in form units. Since \(q_4-q_9=1\) and \(x^0_4-x^0_9\ge b-2\zeta>0\),
Here the scalar \(v=W_{49}>9/10\) is the retained edge conductance; the vector \(v(s)\) is the interpolated form direction. This bound includes both endpoints and all independent source errors without sampling the path.
Every source edge lift gap is acute, including the whole independent box: old gaps have magnitude at most \(2\zeta\), and inserted gaps at most \(2/5+2\zeta<\pi/2\). Equation (2.9) multiplies every such gap by \(1-s\). Since \(1-\cos t\) is increasing with \(|t|\) for \(|t|<\pi\),
The last strict inequality follows also from (2.10) at \(s=0\). Define
This is a continuous configuration path, starts at the actual initial state because \(\lambda(0)=1\), and ends at (2.5). Its storage and both lifted means are constant by direct substitution. Its unchanged storage is below \(35/64\), so the existing pointwise protected-star barrier retains both marked identities along the whole path; the phase branches also remain in their original acute chamber.
For the exact nominal source, (2.9)-(2.12) preserve equality inside each of the five admitted blocks. For a general point of the twenty-dimensional source box no such equality is imposed: its path retains the full fine state. The held law and its weights do not vary along either construction.
This removes an obstruction from the displayed storage, two means, protected chamber and nominal block constraints. It does not establish that these are all first integrals, that the complete flow is ergodic on this set, or that it follows (2.9)-(2.12). A continuous path in a conserved set is not an integral curve of its vector field.
2.2 The full circular observation has an exact admitted lift formula¶
A useful geometric identity avoids a generic circle minimization on this admitted domain. Suppose real representatives \(\vartheta_i\) have \(\max_i\vartheta_i-\min_i\vartheta_i<\pi\). For any candidate common center, let \(\delta_i\) be its closest real phase differences. The variance identity and the definition of shortest circle distance give
The strict span assumption justifies the middle equality. Taking the common center to be the arithmetic mean \(\bar\vartheta\) modulo \(2\pi\) attains the last expression, since every \(|\vartheta_i-\bar\vartheta|<\pi\). Thus this is the global circle minimum.
In the joined support, node 4 is adjacent to all nine other nodes. Every admitted initial hub gap is less than \(2/5+2\zeta<\pi/3\) in magnitude. Conservation of \(H<1/2\) prevents any continued hub edge gap from first reaching \(\pi/3\) or \(-\pi/3\), whose single bare phase cost would already be \(1/2\). Consequently, for all time,
The same gap condition holds along the constructed path, where every source gap shrinks. Therefore (2.13) gives
on the actual joined flow and on that path, using their retained common lifts. This is an exact formula for the full-state observation, not a reduction to regional means or to the eight tangent coordinates. Individually wrapped stored representatives cannot simply be inserted into the raw variance. Their association with the common hub chart, or with the continued lifts, remains necessary. No assertion about arbitrary phase configurations outside this domain has changed.
3. The matched disconnected law cannot enter the target¶
The disconnected alternative retains its own degrees and capacity \(117/32\) at node 4. Its normalized component mean weights, in each star's center/leaves/boundary order, are
Pairwise cancellation preserves each component's form mean and, on continued lifts, each component's phase mean. Their phase-mean difference \(\Delta\) is initially \(\alpha\) for the nominal source and satisfies
on the matched box. It remains constant. This does not imply that the individual phases remain equal to their component mean.
3.1 Associate the circular observation with the retained means¶
The existing full-coordinate storage bound, Section 20.2, gives each disconnected component the independent all-time storage budget
Its original internal edge lift gaps have magnitude at most \(2\zeta\). A first crossing of \(\pi/2\) or \(-\pi/2\) would put at least one bare edge phase cost at one, contrary to (3.3). Every such continued edge gap is therefore acute at every time.
Let
be the full phase part of \(Q\) in the fixed radian unit. Suppose first that \(J\le1\), and choose a minimizing common center. Its nearest phase lifts are \(\varphi+\delta_i\), with \(|\delta_i|\le1\). Write each continued phase as \(\theta_i=\varphi+\delta_i+2\pi n_i\) for integers \(n_i\). Along an old edge the difference between its continued acute gap and \(\delta_j-\delta_i\) has magnitude less than
It is also an integer multiple of \(2\pi\), so it is zero. Connectivity of each old star makes \(n_i\) one constant integer on the left and one on the right. Consequently, for
the difference of the retained component means gives
The last inequality uses (3.2), not an assumed globally single-valued phase mean. Cauchy--Schwarz now yields \(J=\sum_i\delta_i^2\ge\Delta^2/c\).
The same bound holds when \(J>1\). Indeed each normalized five-node weight vector has squared norm at least \(1/5\), so \(c\ge2/5\); by (3.2), \(\Delta^2/c<5/8<1\). Thus for every time and without a raw-variance substitution,
Here \(c<4/9\), since both summands in (3.4) are less than \(2/9\). The final strict bound follows from \(17/256-27\zeta/8+9\zeta^2>17/256-4\zeta\) at the frozen \(\zeta=2^{-20}\). For the exact nominal source, the stronger \(\zeta=0\) lower bound is available.
Therefore the full same-observation disconnected alternative never enters \(Q\le1/4\), for any nominal source or any member of its admitted box. This is a full-state, all-time control obstruction. It does not follow merely from target absence initially, from a component being disconnected, or from independently recentering the two stars.
4. The full tangent has its own compatible quadratic geometry¶
The declared tangent keeps the actual joined support, conductances, capacities and initial fine coordinates. On their selected real phase lifts it obeys
where \(K=\operatorname{diag}(\nu_i/d_i)\) is the actual held mobility and \(L_N\) is the bare Laplacian. The two cross terms in \(\dot H_2\) cancel, and both \(\mu\)-weighted means are conserved. The nonlinear storage \(H\) is not assigned this conservation law. Nor does (4.1) define a periodic phase field: changing one initial phase lift by \(2\pi\) generally changes the tangent evolution.
For \(a=1/16,\ b=1/1024,\ c_0=a-b=63/1024\), the nominal source has
Only the five contact edges carry the initial phase contrast. Hence
Replace \(H_0,V\) by \(H_{2,0},V_2\) in (2.5) and (2.9)-(2.12). Now \(V_2(\theta(s))=(1-s)^2V_2(\theta^0)\), so the same strict normalization argument applies. It supplies a constant-\(H_2\), constant-means configuration path and target point with
The nominal path stays in the five-block family.
The whole-box extension is also explicit. Represent \(H_2\) as the squared norm of edge components \(\sqrt{W_{ij}/2}(x_i-x_j)\) and \((\theta_j-\theta_i)/\sqrt2\). Independent coordinate errors bounded by \(\zeta\) change this vector by less than \(\sqrt{116}\zeta<11\zeta\), because \(\sum W_{ij}<39\) and there are nineteen bare edges. Using (4.2),
There is also an exact observation formula along this alternative. Node 4 is still universal. Conservation of \(H_2<1/2\) implies \(|\theta_i-\theta_4|<1\), so the whole real phase span is less than \(2<\pi\). The lemma (2.13) therefore gives the same full-state formula (2.15) throughout the tangent evolution and its constructed path, with its independently retained real lifts.
Thus every admitted box source has the same displayed-invariant path construction and a target point with \(Q<9/125\), retaining all fine coordinates. Since \(1-\cos t\le t^2/2\), the pointwise nonlinear storage satisfies \(H\le H_2\). Conservation of \(H_2\) and (4.4) also retain both original geometric identities along the tangent flow under the same pointwise \(35/64\) barrier. This does not make \(H\) a tangent invariant or establish registration.
The construction matches only the explicitly displayed quadratic storage and means. Additional tangent invariants, including modal actions when separately admitted, may restrict an actual orbit. No eigenvalue, propagator, trajectory, entry time or dwell interval has been evaluated. In particular, (4.3) is not evidence that the tangent reaches the target, just as (2.7) is not evidence that the nonlinear flow does.
5. Static controls and the remaining dynamical question¶
The canonical contact controls reuse the original joined and disconnected source reports. These static regressions execute the unchanged source-fixture words; they introduce no new research action configuration and regenerate no archived asset producer. They check the actual bare degrees and capacities in (2.2) and (3.1), the component storage/branch premises, the exact quadratic identities (2.4), and the unchanged circular target bounds. The correlated construction \(r^2=2H_0/s_4\) keeps the same source energy in its target: an outward interval for \(H_0\) and an independent interval for \(r\) would not by themselves certify this equality. The tangent's quadratic storage is rebuilt from the same initial fine forms and retained lifts, not copied from its nonlinear counterpart.
These checks accompany three distinct conclusions:
- The joined source and a target point share the displayed invariants and a continuous protected configuration path.
- The matched disconnected source is excluded from the same circular target for all time by its separate means.
- The full tangent also has geometrically compatible target configurations under its displayed invariants.
The connected and tangent paths are not acquired histories. Neither conclusion supplies finite entry, a residence interval, or a subsequent response from an actual reached state. The permanent-registration obstruction remains unchanged, and the previous near-consensus approximation budget is not transferred to these preparations.
A future entry result would therefore distinguish the joined interaction from the disconnected alternative under this observation. Establishing the role of nonlinear terms would additionally require the independently admitted full tangent comparison. If that alternative also satisfies an eventual finite-episode protocol, the observation does not isolate a nonlinear mechanism. This is not a definition that an NFR must be nonlinear. The ontology still requires an independently justified common identity, retained constituent/boundary state, nontrivial lifetime and continued response; crossing a registration threshold alone supplies none of those remaining obligations.
6. A conditional metric certificate for whole-time observations¶
This section supplies an error theorem for the two admitted joined laws. It does not evaluate either response or assert entry, residence, a reached state or a subsequent response. A numerical protocol must retain its own preparation, horizon, work limits and decision rules. The certificate uses the shared metric Picard/Taylor owner and its whole-bin observations; a numerical trace from the shared integrator is separate evidence, not the enclosure's error premise.
6.1. The fixed metric and its complete-law scope¶
Choose any orientation of the actual nineteen edges. Let \(B\) be its incidence matrix, \(L_N=BB^\mathsf T\) its bare Laplacian, and \(L_W=B\operatorname{diag}(W_e)B^\mathsf T\) its conductance Laplacian. Keep the exact represented conductances and the actual degrees and capacities; here
The original clock and both complete evolution rows are
The tangent alternative replaces the sine by its argument. Define the constant symmetric positive definite metric
Connectivity makes each Laplacian positive away from its constant kernel, and \(\mu^\mathsf T{\bf1}=M>0\) removes that kernel. The two added terms retain errors in both common origins; they do not assume that rounded states conserve either mean exactly.
For the nominal source, let \(R_0\) repeat the five block coordinates on \((0,1,2,3),(4),(5),(6,7,8),(9)\), and put \(R=\operatorname{diag}(R_0,R_0)\). The already proved invariant-family identity \(F_{\mathrm{fine}}(Ry)=R f(y)\) applies to both laws. Consequently the reduced metric
is positive definite and measures the exact full-state error of the injected chart. Both fine Laplacians, including the inherited bare multiplicities, enter (6.4). This is not a five-node substitution with one unweighted sine edge per block pair.
The restriction is available only after source membership and the held-law closure conditions have been admitted. Arbitrary independent fine perturbations and unequal inserted weights cannot be projected onto this chart without a separate error argument. A full-state numerical implementation that loses equality within a block also needs to retain that error; the existence of the invariant family does not erase numerical residuals outside it.
6.2. A whole-tube growth bound without spectral sampling¶
At any real lifted phase state, set \(a_e=1-\cos(b_e^\mathsf T\theta)\ge0\). Then \(\sum_e a_e=V(\theta)\). The Jacobian of (6.2) is
For an arbitrary variation \(z=(z_x,z_\theta)\), the mean terms in (6.3) annihilate both Jacobian rows because \(\mu^\mathsf TK={\bf1}^\mathsf T\). Cancellation of the constant-cosine terms gives the exact scalar identity
Put \(X^2=z_x^\mathsf TL_Wz_x\) and \(\Theta^2=z_\theta^\mathsf TL_Nz_\theta\). For every actual edge, Cauchy--Schwarz in the conductance quadratic form and the corresponding term of the bare quadratic form imply
where \(s_i=\sum_jW_{ij}\). The second inequality can equivalently be expressed as the bare effective-resistance bound at an existing unit edge; no estimated eigenvalue is needed.
For \(9/10<u,v<1\), use the exact strength bounds \((11,5,5,5,13,12,6,6,6,9)\), with strict bounds where inserted weights occur. The nine edge types give the following convenient rational upper bounds:
| Edge type | Upper bound for \(g_e^2\) |
|---|---|
| Left center--leaf | \(22\) |
| Left center--boundary 4 | \(11+2197/1024+13/8<15\) |
| Left leaf--boundary 4 | \(5+2197/1024+13/8<9\) |
| Right center--leaf | \(192/25+8/3+16/5<14\) |
| Right center--boundary 9 | \(896/25<36\) |
| Right leaf--boundary 9 | \(2248/75<30\) |
| Inserted 4--right center | \(2197/1024+192/25+13/20<11\) |
| Inserted 4--right leaf | \(2197/1024+8/3+13/24<6\) |
| Inserted 4--9 | \(2197/1024+576/25+13/10<27\) |
Thus \(g_e<6\) for all edges, and (6.6) implies
This is the logarithmic growth rate of the norm, rather than of its square: along a variation, \(\tfrac12(d/dt)\|z\|_{\mathcal W}^2=z^\mathsf T\mathcal WJz\). For the nonlinear law, a convex Picard tube satisfying \(H=F+V<1\) everywhere therefore admits
for solutions that remain in that tube. Indeed \(3V/\pi\le V<1\). The line segment needed for the two-solution argument lies in the admitted convex tube. There is no assumption that \(1-\cos\) is globally convex, or that its value at interpolated endpoints bounds its value in the interior.
For the tangent law \(L_{\cos}=L_N\), so (6.6) vanishes identically and \(\gamma=0\) is exact on the entire state space. Both conclusions transfer to \(W_r\) by (6.4) and exact invariance. The true source's stronger \(H<1/2\) bound does not establish \(H<1\) for a numerical rectangular tube: the latter remains a separate whole-tube admission check.
6.3. Carrying the metric ball through Taylor steps¶
Suppose the incoming state is enclosed by \(\|y(t_0)-c\|_{W_r}\le r\). Its coordinate projection has radii \(r\sqrt{(W_r^{-1})_{ii}}\). Admit a convex Picard tube containing the center solution and all solutions from that projected initial box for the entire declared step \(0\le t\le h\). For the nonlinear branch it must also satisfy the domain in Section 6.2. The coordinate projection is an enclosure used for derivatives and domain checks; it does not replace the retained initial ball.
Let \(p(t)=\sum_{j=0}^p c_jt^j\) be the center's order-\(p\) Taylor polynomial. Interval jets enclose every coefficient, and a whole-tube derivative bound gives a coordinate remainder \(D_i t^{p+1}\), where \(D_i\) includes the factorial divisor. Consequently every time, not just the endpoint, satisfies
All arithmetic and coefficient enclosures must be outward. At \(h\), select an exact represented center \(c_+\) and enclose the center solution's coordinate errors by \(d_i\). A sufficient metric error bound is
The next step retains \((c_+,r_+,W_r)\). Replacing this ball by an independent-coordinate box, intersecting marginal boxes to reduce its radius, or treating its center as a new exact source would change the error argument. Local interval arithmetic, Taylor remainders and center representation error belong in \(\eta\), including when the chosen center is nondyadic.
For an observation bin \(t\in[a,b]\subseteq[0,h]\), evaluate the polynomial with interval time and scale the retained whole-tube remainder \(D_ih^{p+1}\) by \([a/h,b/h]^{p+1}\). Add the incoming-ball projection with \(r\max(1,e^{\gamma h})q_i\). This gives a whole-bin enclosure without starting a new solution or reinterpreting a final remainder as an endpoint-only error. The shared metric_taylor_observation_boxes helper performs this arithmetic on an already admitted step and the same complete field. Its input association, Picard/domain premises and retained remainder provenance must be established by the evidence consumer; the observation helper does not authenticate them.
6.4. Exact lifted observation and episode decisions¶
The continuing true nonlinear solution has \(H<1/2\), and the true tangent solution has its own \(H_2<1/2\). The hub and short-arc proof therefore makes the fixed circular observation exactly
along each actual solution, with the declared observation units. In the exact five-block chart, with multiplicities \(n=(4,1,1,3,1)\), this is
The multiplicities are fine-node counts, not inverse-capacity mean weights. Direct outward evaluation of (6.13) on a whole-bin enclosure from (6.10) encloses the actual circular \(Q(t)\) at every time in that bin. The rectangular observation box may contain unrelated phase states outside the short arc: the equality is required for the true enclosed solution, whose branch association has already been proved, not for every artificial corner of the box. This does not permit applying raw lifted variance to an arbitrary captured source.
There is also a useful independent relation between metric and observation error. The universal bare hub star gives \(L_N\succeq P\), and \(W_e>9/10\) gives \(L_W\succeq(9/10)L_N\). Hence
The true nonlinear flow has \(Q<2\). For example, its continued hub gaps have absolute value below \(\pi/3\); on the larger interval \([-\pi/2,\pi/2]\), \(t^2\le(\pi^2/4)(1-\cos t)\), by concavity of the sine on the corresponding half interval. Summing hub terms and using (6.14) for form gives \(Q\le(5/2)H<5/4<2\). The tangent similarly satisfies \(Q\le(20/9)H_2<2\).
If a center at a specified time has metric error at most \(r\), (6.14) and the quadratic difference identity imply
In particular \(r\le1/4096\) suffices for absolute observation error below \(1/1024\). This is a conditional pointwise error allowance. An endpoint radius alone does not bound the full bin's temporal variation or replace (6.10).
A certified residence interval requires consecutive whole bins with \(Q_{\mathrm{upper}}\le1/4\), covering the entire declared minimum duration. A bin with \(Q_{\mathrm{lower}}>1/4\) excludes membership throughout that bin. Overlap with the threshold is unresolved; it is neither entry nor exclusion. An initial absence certificate and a later certified residence imply that entry occurred between them, without identifying its exact time. To certify no entry over a fixed horizon, every bin covering that horizon must be excluded. A failed tube, excessive carried error or exhausted work budget leaves the uncovered interval unavailable.
These statements supply a method and its sufficient premises. They do not report an evaluated episode, transfer a result between the nonlinear and tangent branches, or turn registration into a complete collective NFR identity.
7. Frozen central-source finite-horizon exclusion¶
The protocol, preparation and source freeze were retained before either continuation was evaluated. The single attempt completed both branches without a solver, domain or resource failure. The raw response and result receipt retain exact rational bounds and the separately continued numerical endpoints. The source archive contains the pre-evaluation proof and implementation, rather than this later interpretation. Content receipts associate bytes; they do not independently authenticate execution or establish a mathematical premise.
7.1. Evaluated scope and result¶
Only the central nominal represented successor, with \((m,\alpha)=(1/4,201/512)\), was continued. Both laws used the same twenty-coordinate source and its exact five-block lift, actual held conductances and capacities, original structural clock, and no further inputs, events or Gamma. Elapsed time \([0,8]\) corresponds to graph-clock time \([7,15]\). No independent preparation box was projected into the nominal family, and the other two nominal sources were not evaluated.
The unchanged observation was \(Q\le1/4\), with minimum residence \(1/16\). Each branch used 8,192 shared RK4 steps of size \(2^{-10}\) and an independent carried metric certificate: 128 steps of size \(1/16\), Taylor order twelve and sixteen whole-time bins per certificate step. Thus the 2,048 bins per branch cover every point of the full horizon. The fixed error allowances were \(2^{-12}\) for the certificate radius and numerical endpoint metric error, with a total 900-second wall budget and no retries or mesh changes. Evaluation finished in about 204 seconds on the recorded runtime; this is an observation, not a performance promise.
Every whole-bin lower bound exceeded \(1/4\) for both laws. Therefore neither joined trajectory enters the target anywhere on this horizon; in particular neither supplies the declared residence episode. The following outward-rounded intervals enclose the actual minimum of \(Q(t)\) on \([0,8]\), using the smallest lower and smallest upper whole-bin bounds:
| Complete law | Enclosure of the minimum Q | Largest endpoint metric radius, rounded up |
|---|---|---|
| Actual weighted nonlinear sine | \([0.33559,0.33670]\) | \(2.345\times10^{-12}\) |
| Same-state full real-lift tangent | \([0.33991,0.34097]\) | \(1.712\times10^{-17}\) |
The RK4 endpoint errors relative to the actual solutions, including the independent metric-ball radii, were below \(1.086\times10^{-11}\) and \(8.645\times10^{-12}\), respectively. Numerical samples were not used to infer whole-window absence. Nonlinear Picard tubes retained a storage domain margin greater than 0.31475 below \(H=1\); the actual trajectory's stronger conserved \(H<1/2\) still supplies its separate circular-chart association. The existing storage theorem retains both constituent identities throughout these trajectories.
7.2. Interpretation and implementation boundary¶
This finite exclusion strengthens the earlier initial absence result for one exact preparation and one horizon. It is not all-time exclusion for the joined law, exclusion of other collective identities, or a rejection of all TNFR completions. The displayed energy/mean-compatible target and path remain valid configuration comparisons: this calculation shows why they cannot be promoted to actual dynamical access. The two finite exclusions do not establish equivalence of their nonlinear and linear trajectories or transfer conserved quantities between their laws.
The shared static source now serves both the original independent operator controls and the episode owner. The latter re-admits primitive source, support, actual weights, capacity, family membership and growth domains. The canonical event association remains the retained static producer and its independent word tests; a matching coordinate tuple does not authenticate that history.
Whole-time observation construction is centralized in
metric_taylor_observation_boxes, also consumed by the existing directional
maintenance adapter. Its tests include an interior excursion invisible to
endpoint-only reasoning, prescribed clocks and negative growth. Retained
episode summaries rebuild Q from primitive boxes and check clock/state carry;
stored labels and cached Q values cannot decide the verdict. Those summaries
are not independent rederivations of all archived Taylor derivatives.
No new collective NFR, autonomous birth or physical identification follows. The execution plan owns the next task and its reopening conditions.
8. An exact all-time tangent obstruction with a nonlinear drift boundary¶
The reachability question has an exact sufficient certificate for the retained
central tangent model, derived below.
It reuses the unchanged source, support, capacity, structural clock and
Q<=1/4 target. It evaluates no new trajectory, event, source or horizon.
The polynomial below is a constructed mathematical witness, not a fitted
evolution law or a claim that the sufficient lower bound is optimal.
8.1. Complete relative state and the actual observation¶
Use the admitted blocks A=(0,1,2,3), H=(4), U=(5), V=(6,7,8),
D=(9). Relative to A, write
These are all eight coordinates of the exact nominal family after fixing
its two conserved origins. They are not the separate eight-dimensional
minimal realization of two relative outputs for arbitrary fine perturbations.
Block conserved-mean weights are
rho=(4,32/13,5/4,9/2,5/8), with total M=1335/104. For either channel,
its fixed weighted mean and four relative coordinates reconstruct the base
value as Y_A=mean-sum_{a!=A} rho_a*z_a/M, and Y_a=Y_A+z_a.
No internal coordinate is discarded from this nominal family.
The central source in this chart is exactly
The actual cross conductances, not ideal affinity estimates, are
Grounding the base A in the weighted and bare block Laplacians gives
Set
The relative tangent row is z_dot=A*z/pi in original structural time.
The 1/4 term retains the base block's motion; deleting the base derivative
would give a different model. E is positive definite because both grounded
supports are connected with positive weights. Its energy is
H2=z^T E z/2, and A^T E+E A=0 exactly.
For the fixed observation, use fine-node multiplicities, not rho. With
n=(1,1,3,1) for the four nonbase blocks,
The last equality applies to the continued true solutions by the
circle-to-lift proof: sine storage
H<1/2 bounds hub gaps by pi/3; tangent storage H2<1/2 bounds them by
one radian. Each retained phase span is below pi for all time. This proof
does not license independent wrapping or a global linear vector field on
the phase torus. The two observation units remain the fixed form unit and
radian from Section 1.
8.2. Four exact quadratic invariants and a sufficient witness¶
Put X=-A^2. Each matrix G_k=E X^k is symmetric and obeys
A^T G_k+G_k A=0: use A^T E=-E A and the commutation of A with X.
For the exact values (8.3), rational elimination gives rank four for
G_0,...,G_3. The map G -> A^T G+G A has rank 32 on the 36-dimensional
space of symmetric eight-by-eight matrices. These four matrices therefore
span the complete quadratic-invariant space on this mean-fixed chart.
No eigenvalue sampling or unproved nonresonance assumption supplies that rank.
One simple exact certificate in this space is
At (8.3), exact rational LDL elimination certifies O-G positive definite,
and exact source evaluation gives
The strict inequalities, not the displayed decimal, certify the result. The matrix, rank, LDL and source-value checks use exact rational operations in the shared algebra owner and rank owner. A floating candidate search served only to find a witness; neither its optimizer status nor its approximate eigenvalues are premises of this certificate. G need not be a positive energy or the maximal minorant of O.
On the tangent trajectory the quadratic I is constant. Consequently
The fixed common registration is thus unreachable in the central joined
tangent model for all time. This is stronger than its frozen [0,8]
exclusion and has no numerical integration horizon. It does not cover the
other nominal sources, independent preparation boxes or a different kernel.
The earlier same-energy/means target and configuration path remain valid;
they cannot retain this additional invariant. Their geometric compatibility
never implied that the tangent flow could follow the path.
8.3. The actual sine law does not conserve this certificate¶
Let r(z)=F_sine(z)-A*z/pi in the same relative chart. Its phase half is
zero. For each block representative, replace every bare phase difference
delta in the form row by sin(delta)-delta, multiply by its actual
capacity/degree and 1/pi, then subtract the same base-A row. This defines
the full remainder with the original support and time; it is not a
two-output reduction or a pressure reconstructed from measured velocity.
For all states in the declared chart,
At the retained source all active gaps are alpha or minus alpha. Thus
The exact rational coefficient k=2*z0^T*G*c satisfies
Here 0<alpha<pi implies sin(alpha)<alpha; the strict sign does not
depend on floating trigonometry. Its outward interval evaluation agrees
with approximately 0.00014143252563073545 per structural time unit.
This is an actual-source counterexample to conservation of (8.7) under
sine. The nonlinear law initially increases I, so the result does not
demonstrate transport toward entry. It also does not establish that I can
never later decrease or that some other nonlinear obstruction is absent.
No nonzero homogeneous quadratic first integral on an open sine-state domain.
The complete rank characterization gives a stronger stopping boundary for
this method. If a homogeneous quadratic were conserved by the analytic sine
field on an open mean-fixed lifted domain, its derivative would vanish as an
analytic identity. Its lowest-degree part must then satisfy the tangent
Lyapunov equation. Every matrix in the characterized space is block diagonal,
so write it as diag(S,T), and put B=N E_theta, C=N E_x.
Both B and C are invertible; in particular det B=801/5 exactly.
At relative phase theta=t*e_j, all edge gaps are zero or plus/minus t, so
the exact form remainder is B*e_j*(t-sin(t))/pi. Conservation for arbitrary
relative forms x would require
Hence SB=0, giving S=0. The tangent equation then gives C^T T=0, so
T=0. Analyticity permits this identity test even if the originally proposed
open domain did not contain consensus. This excludes another nonzero
homogeneous quadratic first integral on an open domain of this sine model.
It does not exclude special-trajectory constancy, restrictions to a single
energy surface or nonquadratic invariants; the known nonlinear H is conserved.
The constitutive countermodel shares the consensus tangent, but this source-specific sine drift is not its drift certificate. Shared tangents do not transfer nonlinear invariants, transport coefficients or the original finite exclusion.
An exact necessary condition for any later sine entry follows without a
new experiment. Since Q>=I on its admitted continued lifts, reaching
Q(T)<=1/4 requires
This is a required signed drift budget for an indefinite quadratic, not physical energy or event work. It is not evidence that the transfer occurs or that it is sufficient for registration. Initial positive drift and the frozen finite no-entry result remain separate observations of its scope.
8.4. Reusable certificate and verification boundary¶
The research reader re-admits the fixed tagged preparation through the existing handoff owner, rebuilds the fine-to-block state and exact matrices, and verifies the invariant and observation inequalities using shared rational kernels. It adds no evolution law or integrator. Its controls independently check fine rows, weighted-mean reconstruction, arithmetic Q, invariant-space rank, strict PSD, source separation and the nonzero sine derivative. Invalid primitive associations and unsupported candidate certificates are distinct from successful exclusion. The original producer and its frozen evidence remain unchanged.
The exact linear obstruction is now closed. All-time sine reachability, formation of a new collective NFR and physical identification remain open. The execution plan alone queues any subsequent transfer investigation.
9. Exact quartic cancellation and its actual-source limitation¶
This bounded quartic comparison keeps Section 8's source, actual support, capacities, structural clock, quadratic G and common-registration target. It asks whether a homogeneous quartic correction removes the leading sine drift, then whether the resulting bounds inform this source. No event, trajectory, preparation change or higher-order sequence is evaluated. The answer separates exact formal cancellation from an unsuccessful source-domain certificate.
9.1. The fixed polynomial question¶
Write z=(x,theta) in the eight relative coordinates (8.1), and set
R(z)=pi*r(z). Thus the full sine law is pi*z_dot=A*z+R(z).
Let m_ij count neighbors in block j of a representative in block i,
and mu_i be that representative's actual capacity divided by its fine
unweighted degree. Blocks are ordered A,H,U,V,D; set theta_A=0 when
computing differences. The cubic part of R is exactly
Bare neighbor counts, rather than conductances, enter this form row.
The phase row is already linear. Subtracting the moving base row preserves
the actual relative law; r_3 includes the factor pi relative to a rate
in structural time. For I=z^T Gz, define J=I+P_4 and require
Use homogeneous degree-four monomials z^a, ordered by descending
lexicographic exponent tuples in (8.1). The domain contains even total
form degree, 0, 2 or 4: its dimension is 35+100+35=170. The codomain
contains odd form degree, 1 or 3: its dimension is 80+80=160.
A's off-diagonal block form maps the first sector into the second;
G is block diagonal and (9.1) is phase-cubic, so the right side belongs
to that codomain. The exact matrix entries are
Terms with a_i=0 contribute zero. This is a specified 160-by-170
rational system, not a fit to sampled dynamics.
All denominators in L are invertible modulo the verified prime 65521. Its first 160 columns satisfy
Consequently L has rational rank 160, its leftmost pivots are columns
0 through 159, and the fixed sector's kernel has dimension ten.
The ten products (z^T G_i z)(z^T G_j z), 0<=i<=j<=3, lie in that
kernel by Section 8. Their exact coefficient rank is ten, so they span it.
This statement concerns the declared parity sector; it does not classify
the entire 330-dimensional space of homogeneous quartics.
Setting coefficients 160 through 169 to zero therefore gives a unique rational P4 in the declared gauge. Exact sparse differentiation verifies (9.2) independently of the elimination used to obtain it. Full row rank also means every right side in this odd-form sector is removable here. The result is not a sine-specific solvability condition or a selection of the sine constitutive law. Changing the gauge adds tangent invariant products and can change actual-source correction and remainder bounds.
9.2. Cancellation does not improve the actual initial drift¶
At the unchanged source (8.2), exact rational evaluation gives
with approximate values 0.029345935642430904 and
0.32860219558033166. Retain alpha, c and k from (8.11)-(8.12), and
put ell=DP_4(z0)c. The independently differentiated polynomial gives
ell approximately -0.9125382876424339. At this source,
r_3(z0)=-alpha^3*c/6, so (9.2) gives DP_4(z0)Az0=k*alpha^3/6.
The complete sine derivative, including the correction's nonlinear drift, is
Outward rational trigonometric evaluation certifies
The value is approximately 0.0029054773480158866 per structural time
unit. Thus this prescribed correction increases the initial signed drift
by more than a factor of twenty. Formal cancellation near consensus is
not a smaller remainder at the retained finite-amplitude source.
Neither I nor J is identified with physical energy or support-event work.
For every state in this lifted family, the exact remaining derivative is
Its Taylor expansion starts at total degree six: R-r_3 starts at degree
five, and DP_4 has degree three. Equation (9.8), rather than degree counting
alone, supplies the actual-source bound. No finite-amplitude smallness follows
from that local order statement.
9.3. A continued source domain and explicit remainder bounds¶
The actual conserved sine storage is
h=H(z0) approximately 0.4180275409303929, with h<21/50<1/2.
Each initial bare-edge phase gap has absolute value below pi/3. Nonnegative
edge storage prevents any such continued gap from reaching pi/3, where
that edge alone would contribute 1/2. On this continued component,
1-cos(delta)>=9*delta^2/(2*pi^2) first gives V>=(9/10)*V2.
For every coarse edge direction d, exact inversion of (8.4) gives
d^T E_theta^(-1)d<=1/3. Hence delta^2<14/45.
The Taylor inequality 1-cos(delta)>=delta^2*(1-delta^2/12)/2
improves coercivity to 97/100; applying the same resistance bound gives
delta^2<28/97<3/10, and a second Taylor use gives 39/40.
Consequently, throughout the actual continued source domain,
These are bounds for the exact nonlinear law, not its tangent energy.
Let h_+ be the rational upper endpoint of the admitted storage enclosure.
Choose outward rational radii b_i>=sqrt(2*h_+*(D^(-1))_ii) and edge
bounds g_ij>=sqrt(2*h_+*d_ij^T D^(-1)d_ij), with d embedded in the
phase coordinates. These correlated edge bounds retain the grounded
Laplacian; subtracting unrelated coordinate bounds is unnecessary.
For P_4=sum_a p_a z^a, define
For s=3,5, put v_(s,i)=mu_i*sum_j m_ij*g_ij^s/s!, with g_ii=0,
in each of the five absolute block rows. Let b_(s) have relative form entries
v_(s,i)+v_(s,A) for nonbase blocks and four zero phase entries.
The scalar sine inequalities bound |R| by b_(3) and |R-r_3|
by b_(5). Equation (9.8) therefore gives the rational certificate
where pi_- is the rational lower endpoint of pi. The implementation
returns B4 approximately 97.41432691428348 and epsilon6
approximately 10.559048741729947; the retained rational values,
not rounded decimals, are the bounds. They imply, for structural t>=0,
This lower bound is already below 1/4 at time zero. The initial value of Q itself is known separately; (9.12) fails as a uniform transfer certificate.
9.4. The symmetric-bound obstruction is not merely loose intervals¶
Reuse the same-storage/means target construction from Section 2. Set all
relative phases to zero and only relative form x_H=a, where
a^2=2h/(E_x)_HH. Reconstruct the two origins using the unchanged weighted
means. This configuration has exactly H=h, lies in the continued short-arc
component and retains both marked constituent identities. Its observations
and the fixed polynomial are
The values are approximately 0.05982303634503962 and
-0.43208384351250384. Thus any symmetric absolute bound for P4 on
this same-storage/means domain must exceed 0.43, whereas
J(z0)-1/4<0.079. Even a sharp symmetric bound cannot make (9.12)
certify separation at time zero. Refining the enclosing coordinate boxes
alone cannot fix this particular method. The target is a static comparison;
no claim is made that the actual source reaches it or follows the earlier
configuration path. A smaller reachable set would need its own proof.
9.5. Implementation, independent checks and stopping boundary¶
The quartic owner re-admits the frozen primitive preparation through Section 8's reader, assembles (9.1)-(9.3) and checks rank and kernel products exactly. Its optional SymPy elimination proposes coefficients; separately differentiated rational identities and the fixed gauge decide admission. A supplied candidate undergoes those same checks. Missing backend support is not nonexistence. The independent controls check fine-row cubic assembly, the modular determinant, a known energy correction, the actual full sine derivative, storage and edge bounds, the same-H target, invalid candidates and absence of producer replay.
This closes the prescribed one-order calculation: leading cancellation exists, but this fixed correction does not certify actual-source transfer. Other gauges, higher orders, asymmetric bounds and smaller invariant sets remain unassessed; none is activated automatically. The original frozen finite exclusion and all-time tangent obstruction retain their scope. All-time sine entry, acquisition of a new collective identity, constitutive selection and physical identification remain open. The execution plan alone selects any further investigation; the result and its limits remain here.