Source-local capture and a transient-winding obstruction on an energy-matched path¶
One continuous family of preparations, under one fixed complete law, has source intervals that provably acquire different winding classes. An interior interval has actual winding two at the declared observation time but cannot retain that winding. Thus the finite-transit winding is not a sufficient selector of the eventual organized state, even when initial phase, means and storage are held exactly equal for each correlated source realization.
This result extends the matched preparation pair. It supplies two positive capture corridors and one negative selector result; it does not locate a basin boundary or identify the eventual class in the negative corridor. All assessments below are analytic scalar bounds. No trajectory, reverse-flow source, time-coefficient producer or parameter sweep is used.
Fixed complete law and source budgets¶
Use the isolated unit C9 with nodes \(0,\ldots,8\), cycle edges \(j--(j+1\bmod9)\), unnormalized Laplacian \(L\), degree matrix \(2I\), held capacities one, signed form \(x\), and primitive circular phase with continued lifts \(\theta\). The complete law, clocks and storage are
Both original-time rows are \(e\) times their structural-time rows. Both ordinary means, equivalently the degree-weighted means on this regular support, are conserved. Support, pressure, capacity and clock are supplied; there are no events, controllers or state resets during acquisition. The sine law is fixed throughout: no constitutive parameter is varied.
Retain the exact preparation constants
The original time is \(204800/1023\). The earlier reference radius \(r_c=1/12\) remains a possible eventual recovery radius, but the broad corridors below use an acute-cell capture criterion at \(T\); they do not assert entry into that small Euclidean ball at \(T\).
The graph identities
are the algebraic basis for exact storage matching.
A continuous exact energy isometry, including the errors¶
Put
Direct multiplication gives \(\mathcal Ku=v\), \(\mathcal Kv=-\kappa u\), \(\mathcal K^3=-\kappa\mathcal K\), \(\mathcal K^TL=-L\mathcal K\), and \(\mathcal K\mathbf1=\mathbf0=\mathbf1^T\mathcal K\). For the preparation parameter \(s\in[0,1]\), define
The polynomial formula proves existence without a numerical matrix inverse. It implies \(Q_s^TLQ_s=L\), preservation of both mean functionals, \(Q_0=I\), and \(Q_1a=b\). The map is continuous and invertible. It is an isometry of preparation energy, not a symmetry of the complete form-and-phase flow.
For every single latent pair
the entire source path is
Thus initial phase, both means and complete storage are exactly the same for every \(s\) in that realization. Each fixed-\(s\) source family retains the full sixteen-dimensional relative interior of its form/phase error domain. Cartesian marginal bounds do not replace the correlations in (7). In particular \(Q_1\) need not equal the previous pair's metric reflection on errors; its uncertainty is re-admitted here.
Uniform admission follows from
The last inequality uses the Neumann expansion of \(I-s\mathcal K\). Hence every mapped form error has magnitude at most \(C_Q\epsilon\); phase errors still have magnitude at most \(\epsilon\).
The common source cost is bounded once, at \(s=0\):
Equation (5), not a second box estimate, transfers this same cost to all \(s\). Every source has initial winding zero.
Source-local predictors with complete transient errors¶
Write \(Q_sa=p(s)h+q(s)r\). With \(D_s=1+\kappa s^2\), \(P_0=9M+M^{-1}\) and \(C=18M\kappa-2/M=-4/(3M)\), the exact scalar formulas are
Here \(p\) decreases from \(9M+M^{-1}\) to \(9M-M^{-1}\); \(q\) increases from one to two. Indeed,
For the latter, the numerator is at least \(1-2\kappa-3\kappa^2>0\).
The isolated acquisition estimate gives \(\|e^{-A\tau}P_9\|_{2I}\le e^{-\tau/5}\), \(\|f\|_{2I}\le F=\sqrt{18}\), and \(\lambda_{\max}(L)\le4\). Let \(\rho\ge e^{-40}\). A single bound valid over the whole path is
The source norm uses \(\|ph+qr\|_2^2=2p^2-2pq+60q^2\le2P_0^2+240\). An implementation may replace this global upper bound by the corresponding interval's endpoint bounds, retaining the same inequality.
Variation of constants for \(z=\gamma x\) and integration of both complete rows give, for every admitted source,
The large source lift is retained in (12); it is not removed from the semigroup or uncertainty calculation. Outward scalar arithmetic gives
These decimal constants are exact rational upper bounds.
For circular phase and potential calculations alone write the proxy as
No integer-turn shift is applied to the actual trajectory. Since \(\nabla U(tr)\cdot h=0\), \(\nabla^2U\preceq L\) globally, and \(\|\nabla U\|_2\le6\),
This correlated estimate avoids replacing the nine cosine arguments by unrelated extrema. The maximum uses the two endpoints and only the analytic stationary points in the admitted \(t\)-range:
The possible interior candidates over this path are \(2\pi/9,3\pi/7,4\pi/9\). Candidate inclusion is checked against the interval; in particular the saddle value at \(3\pi/7\) is not charged to an interval that excludes it. This is a finite extrema theorem, not a grid or response search.
The oriented proxy gaps consist of five copies of \(t\), two copies of \(t+\delta\), one \(t-2\delta\), and the closing gap \(-8t\). For a winding-\(k\) chart replace the latter by \(-8t+2\pi k\). Every actual gap differs by at most \(\sqrt2E\). Strict inclusion in \((-\pi,\pi)\) certifies actual winding \(k\) at \(T\); inclusion in \((-\pi/2,\pi/2)\) additionally certifies acute-cell entry.
Positive capture corridors and their energy barriers¶
In the acute winding-\(k\) cell the nine principal gaps have sum \(2\pi k\). At a boundary with one gap \(+\pi/2\), Jensen's inequality on the other eight gives the phase-energy barrier
A gap \(-\pi/2\) gives a larger bound for \(k=1\); it is incompatible with the remaining acute gaps' required sum for \(k=2\). The nonnegative form energy cannot lower this boundary barrier.
If a complete endpoint is in the acute cell and has \(H(T)<B_k\), storage nonincrease prevents leaving that cell. The fixed form mean and the bounded phase-gap chart make the relevant storage sublevel compact. Zero loss requires \(x=0\); invariance then requires \(f(\theta)=0\). In an acute cell all equal sine currents have equal gaps, so the only relative equilibrium is the regular winding-\(k\) target. LaSalle therefore gives convergence to that target, with the common phase origin determined by the retained lift mean and chart. It also gives eventual entry into any positive target-neighborhood radius, without supplying a uniform additional entry time.
The three parameter intervals below were declared before the scalar assessment. Shared outward dyadic128/Machin arithmetic applied to (10)--(18) yields the following conservative rational bounds.
| Source interval | Certified geometry at \(T=200\) | Complete \(H(T)\), upper | Relevant threshold, lower | Strict energy margin, lower |
|---|---|---|---|---|
| \(0\le s\le1/8\) | Acute, winding one | \(2.343558461\) | \(B_1>2.348243101\) | \(0.004684640\) |
| \(3/4\le s\le4/5\) | Winding two | \(7.337282704\) | \(H_2^*>7.437166400\) | \(0.099883696\) |
| \(63/64\le s\le1\) | Acute, winding two | \(7.438205301\) | \(B_2>7.439277423\) | \(0.001072122\) |
Here \(H_2^*=9(1-\cos(4\pi/9))\). The actual acute-gap margins exceed \(0.783403918\) and \(0.087184187\) in the first and third rows. The interior row's actual principal-gap margin exceeds \(0.348983859\), so its winding assertion is separated strictly from an antipodal ambiguity. All margins apply uniformly to the complete correlated error family, including the transported form errors. Equations (18) and (1) prove permanent winding-one and winding-two capture in the first and third corridors, respectively.
Winding two at the transit time cannot persist in the interior corridor¶
The middle row has energy below every critical point in the closure of the winding-two sector. To see this, a cycle critical point has a common sine current on all oriented edges. For positive current, write every non-antipodal gap as \(\delta\) or \(\pi-\delta\), \(0<\delta<\pi/2\). If \(j\) gaps are obtuse, winding two requires
Its potential is
Indeed \(n\sin(c/n)\) increases with \(n>0\) on this range, since \(\sin z-z\cos z>0\) for \(0<z<\pi/2\). Negative current gives negative principal gaps and cannot have winding two. The limiting all-\(\pi/2\) case is incompatible with the C9 winding sum. At an antipodal critical point the common current is zero; all gaps are \(0\) or \(\pm\pi\). A limiting sum \(4\pi\) requires at least four antipodal edges, so \(U\ge8>H_2^*\).
Now fix any source in \(s\in[3/4,4/5]\). Its actual winding at \(T\) is two, while \(H(T)<H_2^*\). On the form-mean-zero leaf the storage sublevel bounds \(x\); circular phases lie in a compact torus. LaSalle implies that every omega-limit point has \(x=0\) and \(f(\theta)=0\). By (20), no such point can lie in the closed winding-two sector. Arbitrarily late visits to that closed sector would have a convergent subsequence there and would create such an omega-limit point. Therefore there is a finite later time after which the trajectory stays outside the entire closed winding-two sector.
This is a phase-passage obstruction for actual complete trajectories, not merely a proxy observation. It concerns the sector itself, not one chosen lift cell: simultaneous cancelling branch-cut crossings need not change winding. A phase passage must occur after \(T\), but the certificate does not determine its time, the last passage, or the eventual class.
Information consequence and implementation boundary¶
The source parameter and the complete source formulas give sufficient local selectors on two nontrivial intervals. In contrast, the claim that the actual finite-transit winding, or its correctly certified proxy winding, always specifies the acquired persistent class is false on the interior interval. Equal initial phase, means and storage still hold throughout this comparison. The fine form information in (7) is retained; it is not an inferred pressure or a controller.
The intervals \((1/8,3/4)\) and \((4/5,63/64)\) remain undecided by these bounds. Their endpoints are certificate boundaries, not located basin boundaries. Nor does the interior result show convergence to winding one. The two positive corridors establish acute-cell capture at the fixed time, not the earlier small-ball radius condition at that time. The complete law permits antipodal passage during formation; no acute constraint is imposed on the whole transit.
The result is conditional on the supplied isolated support, complete sine law, clock and correlated preparation. It derives neither autonomous preparation nor a uniquely selected constitutive law, substrate, size or physical identification. No old acquired-source certificate is transferred through the new map without re-admitting its errors.
The terminal-limit theorem continues these same histories. It proves complete-state convergence and narrows the interior corridor's energy-compatible critical configurations, while retaining saddle exceptions. The additional source-specific capture proof uses the actual mixed state to select winding one for this corridor and exclude those exceptions; the other path intervals retain their own scope.
The private path calculator admits a closed parameter interval, structural time, and separate latent form and phase error budgets. It rebuilds the exact generator and source identities, uses the shared semigroup radius and acute-sector boundary owners, and retains all phase-gap, storage and source-policy margins. Its source-budget and initial-winding flags are separate from the sufficient capture or exit classification. A supplied interval with unresolved bounds is not classified as a failed formation experiment.
The independent controls check the exact Cayley isometry, correlated error transport, canonical cycle offsets, analytic potential extrema, all-face energy bounds and the critical energy comparison. The three fixed corridor assessments evaluate scalar inequalities only. The code supplies neither a trajectory nor a crossing history, and it does not consume an earlier acquisition verdict.