Conservative regional formation and its barriers¶

Finite winding entry, regional channel balance, distributed source geometry, retention barriers and a robust finite passage under the full conservative environment.

Part of Sine pattern geometry and dissipative capture. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

35. Conservative exchange can produce finite regional winding¶

Section 27's positive-loss capture theorem does not extend by setting its loss coefficient to zero: its preparation scaling divides by that coefficient. There is instead a direct finite-time mechanism under the conservative complete law. It concerns a receiver's cycle winding, an observation of part of the full network. It does not claim entry into a two-sided invariant full-state trapping set, which remains excluded by the conservative formation boundary.

A global full-law phase enclosure¶

Let the support be finite, connected, simple, undirected and unit-weighted, with at least two nodes. Keep unit held capacities, e=0, w=beta=1, no forcing and no support events. In the declared clock tau=t/pi, the normalized-sine rows are

\[ x'=KS(\theta),\qquad \theta'=KLx,\qquad K=\operatorname{diag}(1/d_i),\quad S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i). \]

Here and below a prime denotes d/dtau. Smooth sine and the linear form row give a unique global solution in real phase lifts; wrapping remains an observation, even through an antipodal crossing. Put v0=KLx(0). Since |S_i|<=d_i and ||KL||_infinity=2, direct integration gives

\[ \boxed{\begin{aligned} \|x(\tau)-x(0)\|_\infty&\le\tau,\\ \|\theta(\tau)-\theta(0)-\tau v_0\|_\infty&\le\tau^2. \end{aligned}} \]

Indeed the phase remainder is the integral of KL(x(s)-x(0)), whose norm is at most 2s. Consequently every oriented edge satisfies the full nonlinear bound

\[ \left|\delta_{ij}(\tau)-\delta_{ij}(0) -\tau(v_{0j}-v_{0i})\right|\le2\tau^2. \]

No linearized trajectory replaces either consumed row. On a supplied interval [a,b] with 0<=a<=b, take the range of the affine edge expression over that interval and add [-2b^2,2b^2]. If each resulting interval lies strictly inside one branch ((2h_ij-1)pi,(2h_ij+1)pi) with declared integer h_ij, the entire solution has those edge offsets throughout the interval. For an oriented cycle, telescoping the actual lifted differences then gives W=-sum(h_ij). A failed enclosure is inconclusive; it does not prove that the actual solution leaves the branch. This is a whole-interval result, not a collection of sampled winding observations.

One frozen two-port preparation¶

Reuse the supplied return-path support: the two cycles (0,1,2,3,4) and (5,6,7,8,9), mediator 10, edges 0-10, 10-5, and the return edge 1-6. This section uses the smooth conservative sine law above; the older native-law equilibrium and pulse certificates on this support are not transferred. Declare

\[ R=\{5,6,7,8,9\},\quad Q=V\setminus R,\qquad x_{10}(0)=24,\quad x_1(0)=-24, \]

\[ x_i(0)=0\ (i\notin\{1,10\}),\qquad \theta_i(0)=0\quad\text{for all }i. \]

Thus the receiver starts with uniform form, equal phases and winding zero. Its initial phase velocities are derived from the full nodal law:

\[ (v_{05},v_{06},v_{07},v_{08},v_{09})=(-8,8,0,0,0). \]

The two port degrees remain three. These rates are not independently prescribed receiver inputs: they follow from the retained environmental forms, supplied support and complete KLx row. Freeze the retention interval [1/4,1/3] in tau, equivalently [pi/4,pi/3] in t. Along the oriented receiver cycle the affine gap slopes are (16,-8,0,0,-8). The global enclosure gives, simultaneously throughout that interval,

\[ \begin{aligned} \delta_{56}&\in[34/9,50/9]\subset(\pi,3\pi),\\ \delta_{67},\delta_{95}&\in[-26/9,-16/9]\subset(-\pi,-\pi/2),\\ \delta_{78},\delta_{89}&\in[-2/9,2/9]\subset(-\pi,\pi). \end{aligned} \]

The fixed offsets are therefore (1,0,0,0,0) and the receiver winding is exactly -1 for every time in the frozen interval. Its strict branch margin is at least pi-26/9>0; the other faces are farther away. Since it begins with winding zero, at least one receiver edge crosses an antipodal branch before or at tau=1/4. The smooth dynamics are uninterrupted there. The proof does not locate a unique crossing or certify the receiver's winding at every earlier time.

This is an intentionally limited identity claim. In particular the two outer gaps above are obtuse throughout the interval, so the result is not an acute-sector capture certificate. No attraction, asymptotic winding, indefinite maintenance or complete NFR identity follows.

Regional work retains the cross-boundary storage¶

For any node partition R,Q, define internal storage from edges lying entirely within the region, denoted by E(R):

\[ E_R=\frac12x_R^{\mathsf T}L_Rx_R +\sum_{\{i,j\}\in E(R)}(1-\cos(\theta_j-\theta_i)), \]

and similarly E_Q. Let E_cross sum the same edge storage over the cross-boundary edges exactly once. The actual rates still use full-support q=Lx, S and K, not the isolated regional Laplacians. For i in R put

\[ f_i=\sum_{j\in Q:j\sim i}(x_j-x_i),\qquad s_i=\sum_{j\in Q:j\sim i}\sin(\theta_j-\theta_i). \]

The regional gradients are q_i+f_i and -S_i+s_i. Nodewise cancellation of q_i x_i'-S_i theta_i'=0 under the conservative rows proves

\[ \boxed{E_R'=P_R:=\sum_{i\in R}(f_i x_i'+s_i\theta_i'),\qquad E_Q'=P_Q,\qquad E_{\rm cross}'=-P_R-P_Q.} \]

This is the multi-port specialization of the existing receiver work convention. Both form and phase work are needed. The global storage E=E_R+E_Q+E_cross is conserved; incoming work can later return and is not an irreversible loss term or a reusable event allowance.

For the frozen preparation the initial ledger is exactly

\[ (E_R(0),E_Q(0),E_{\rm cross}(0))=(0,864,576),\qquad E(0)=1440. \]

The environment supplies all nonzero initial form, but the energy stored on its contacts must not be counted again as environmental internal storage. Because both delta_67 and delta_95 lie in (-pi,-pi/2), their phase storage contributions each exceed one. Therefore throughout the retention interval

\[ \boxed{\int_0^\tau P_R(s)\,ds=E_R(\tau)>2.} \]

At the initial instant all sine currents vanish, so x'(0)=0 and P_R(0)=0. The strictly positive integrated work is a conclusion about the subsequent full-network exchange, not an initial power score relabelled as stored energy.

Acute winding requires internal phase redistribution¶

Keep the same law, support and unit capacities, globally equal initial phases, and uniform initial receiver form; environmental form amplitudes may be arbitrary. In a fully acute C5 with winding +1, every principal gap is positive: otherwise the other four, each smaller than pi/2, could not sum to 2pi. The negative-winding case has the reversed signs. The sum of the two inner gaps 7->8->9 therefore has magnitude strictly between pi/2 and pi, because the other three contribute less than 3pi/2. In particular

\[ \boxed{\cos(\theta_9-\theta_7)<0} \]

is necessary for fully acute winding at any time. This is a correlation of existing phases, not a new coordinate or an additional law. The relevant organization is an oriented phase pattern, not equality of all phases.

The shared inner path yields a stronger time restriction than treating its edges as independent. Write z=theta_9-theta_7. The full rows give

\[ z'=x_9-x_7+\frac{x_6-x_5}{2},\qquad z(0)=z'(0)=0, \]

\[ |z''|\le3,\qquad |z(\tau)|\le\frac32\tau^2. \]

The derivative bound follows from |x_i'|<=1. Equivalently the exact pair-current coefficients in z''=c^T S are c_9=1/2,c_7=-1/2,c_6=1/6,c_5=-1/6, with all others zero; sum_edges |c_i-c_j|=3. This retains the telescoping correlation instead of adding two independent edge-error radii.

If tau^2<=pi/3, the individual inner raw gaps have magnitude at most 2tau^2<=2pi/3<pi by the global enclosure, so their principal sum is exactly z. Acute winding would require |z|>pi/2, contradicting the bound. Thus tau^2>pi/3 is necessary, independently of environmental amplitude. The existing [1/4,1/3] winding interval lies entirely inside this exclusion. The inequality is not an optimal entry time or an obstruction after its cutoff. Additional ports, nonuniform receiver form, nonflat preparation or different capacities require their own calculation.

The complete rows already transmit phase motion to the inner path¶

There is an explicit local propagation mechanism, although it does not prove finite acute entry. Put A=KL. At globally flat phase, x'(0)=0 and theta''(0)=0, while differentiation of the same complete sine rows gives

\[ v_0=Ax(0),\qquad x''(0)=-A^2x(0),\qquad \boxed{\theta'''(0)=-A^3x(0).} \]

For the supplied preparation x_10=H,x_1=-H, all other forms zero, the inner phase third derivatives at nodes (7,8,9) are (5H/9,0,-5H/9). For example, the receiver form second derivatives are (7H/9,-7H/9,H/6,0,-H/6). Hence

\[ z'''(0)=-10H/9,\qquad z(\tau)=-5H\tau^3/27+O(\tau^5). \]

The odd phase powers follow from the exact time-reversal symmetry about this flat-phase preparation: after removing its common phase origin, theta(-tau)=-theta(tau) and x(-tau)=x(tau). At H=24, the leading inner spread is -40tau^3/9, in the same orientation as the certified negative winding. This local expansion has no supplied finite-time remainder; extrapolating it past the necessary-time cutoff would not certify entry. analyze_sine_conservative_phase_transport retains the admitted source, full-row derivatives and correlated transport bounds without replacing the nonlinear law by its Taylor polynomial.

A finite regional work budget can retain an admitted acute state¶

There is a sufficient continuation condition that retains form as well as phase. On an acute winding-one C5, convexity of 1-cos(delta) gives the minimum phase storage

\[ V_5=5\bigl(1-\cos(2\pi/5)\bigr). \]

At an acute-sector boundary at least one gap is pi/2. The other four sum to 3pi/2; convexity minimizes their storage at 3pi/8 each. A -pi/2 boundary gap cannot carry winding one with the remaining four gaps in [-pi/2,pi/2]. Reflection gives the negative-winding case. Thus the exact minimum phase storage on either sector's boundary is

\[ \boxed{B=1+4\bigl(1-\cos(3\pi/8)\bigr) =5-4\cos(3\pi/8).} \]

Suppose an independently admitted full state at time T has acute receiver winding +1 or -1. For a declared finite horizon h, suppose a proved bound on its actual boundary work satisfies

\[ E_R(T)+\sup_{0\le s\le h}\int_T^{T+s}P_R(u)\,du \le E_{\max}<B. \]

The exact regional balance then prevents a first acute-face exit before T+h: at such an exit phase storage alone would be at least B. Throughout this interval the internal form storage also satisfies F_R<=E_max-V_5<B-V_5, where B-V_5 is approximately 0.01435 in the declared units. A strict budget supplies a positive acute margin and retains the full regional form differences; it is not just a winding sample. The actual full-support phase rates and environmental work remain part of the state and balance. This uses the phase-barrier geometry of the sector theorem, not its positive-loss capture conclusion. It supplies no entry from a flat receiver by itself.

For the frozen H=24 preparation, entry into this low-storage region would require genuine return of storage to the environment. At tau=1/4, the existing phase bound gives

\[ \delta_{56}\in[31/8,33/8]\subset(\pi,4\pi/3),\qquad \delta_{67},\delta_{95}\in[-17/8,-15/8]. \]

The first edge has phase storage greater than 3/2. Each outer edge has storage greater than 5/4, since 15/8>7pi/12 and cos(7pi/12)<-1/4. Consequently E_R(1/4)>4. Since B<7/2, any later entry time T with E_R(T)<B must satisfy

\[ \boxed{\int_{1/4}^{T}P_R(u)\,du =E_R(T)-E_R(1/4)<-1/2.} \]

This necessary net return is compatible with global conservation and is not dissipation. It identifies internal redistribution and export, beyond the already proved incoming transfer, as obligations of a low-storage entry claim. It does not exclude an acute high-energy passage and does not prove that the required return or entry occurs. No new trajectory is inferred from these analytic restrictions.

A bounded full-state organization control¶

The prospective declaration retains the same support and conservative complete law, with H=3, all initial phases zero and unit capacities. Receiver initial phase velocities are (-1,+1,0,0,0) in tau; total initial storage is 45/2, partitioned as receiver 0, environment 27/2 and cross-boundary 9. This normalizes one preparation for study. It is not an optimal amplitude or a selected law.

The horizon is t in [0,12], using original structural time, with 256 fixed steps of 3/64 and Taylor order 10. The target is a contiguous whole-time acute receiver winding interval of at least 1/4 in t, with fixed edge branches and acute margin at least 1/16 radian, certified by a strict lower bound greater than that threshold. These numerical and observation choices are supplied policies. Full regional form, phase rates, storage and signed boundary work remain observable; the target does not require the stronger low-storage protection condition above.

The producer freezes the protocol, source archive and runtime before invoking the shared Picard/Taylor forecast. The regional observer distinguishes certified finite retention, exclusion on a fully covered horizon and unresolved or partial evidence. No amplitude, horizon or solver budget is adjusted in response to the reserved evolution. A numerical enclosure failure cannot become a mathematical impossibility result.

The frozen response certifies all 256 whole-time tubes, covering the entire declared horizon. Its outcome is acute_winding_excluded_on_horizon, with no evaluation error or unresolved numerical step. Every receiver edge remains acute and all principal-branch offsets remain zero, so receiver winding is exactly zero throughout. The minimum certified acute margin exceeds 0.348 radian. Each tube also independently excludes acute unit winding through a nonnegative two-hop phase cosine. These are full-interval exclusions, not an inference from endpoint sampling.

The exchange nevertheless changes the receiver. Its endpoint internal storage is approximately 5.19076, consisting of form storage 4.17198 and phase storage 1.01878. Every tube bounds phase storage below 1.263, whereas acute unit winding on this C5 requires at least V_5=5*(1-cos(2*pi/5)), approximately 3.45492. The signed regional power is strictly positive on 224 tubes and strictly negative on 23; its interval contains zero on the remaining 9. Thus incoming and returned work both occur, but total storage exceeding V_5 does not supply the required phase geometry. The endpoint's nonzero relative phase rates also preclude interpreting it as phase locking or equilibrium. The exact interval record, not these rounded display values, owns the evidence.

This closes the declared H=3 control negatively without retuning preparation, horizon or target. It does not exclude later winding, other preparations or other definitions of an organized pattern. In particular, nonuniform form and locally acute phases can be organized without nonzero winding; winding is the selected identity discriminator here, not a universal NFR definition. This response does not establish how boundary exchange and reversible internal form/phase conversion supply directed phase organization. The total-work ledger alone cannot decide it. The separate same-orbit construction proves a conditional existence result with a different preparation and support; the failed response here remains retained without replay.

Controls, implementation and limits¶

Resetting the two nonzero environmental forms to zero, while retaining the same support, law, receiver state and phases, gives an exact stationary control. This is a preparation control, not an equal-energy comparison. A distinct support control removes 1-6 before initialization and keeps the supplied forms. Then the receiver reflection (6 9)(7 8) fixes the entire preparation and law. By the existing single-port symmetry obstruction, receiver winding stays zero whenever no receiver edge is antipodal. Neither control introduces a support event into the evaluated trajectory.

certify_sine_conservative_winding_entry reuses admitted complete-law state and reports the analytic phase enclosure, declared cycle offsets and whole-interval branch margins. The proof supplies its numerical budget before evaluating any response; a trajectory solver is unnecessary for this witness. Regional work observations retain their own exact edge ledger and full-support rates rather than replacing them with the phase enclosure.

comparison.regional_storage_balance(region=...) in the shared sine owner also supports the existing general sine coefficients and nonnegative capacities. In its original structural clock t, the same calculation gives dE_R/dt=-e*sum_R(nu_i/d_i)*q_i^2+P_R, where phase storage and phase boundary work carry the declared beta. The complement has its own full-node loss; the cross-edge derivative remains -P_R-P_Q. This snapshot identity does not integrate boundary work or substitute for the finite-window theorem. No inverse capacity is needed at zero capacity.

The analytic declaration precedes the retained certificate and controls. The producer reuses the write-once evidence transport and retains its source archive. There is no reserved trajectory: the global inequality certifies every time in the declared window. Independent tests check the complete nodal jet, telescoping, storage partitions and both controls without fitting a response.

The analytic winding theorem establishes one conditional mechanism for acquiring and retaining a regional winding through conservative environmental exchange. The large, asymmetric environmental preparation and both ports are supplied premises. Their origin, a mechanism selecting this preparation, a low-budget formation law and physical identification remain open. This proof neither selects the sine constitutive law nor derives a connection-occurrence law.

36. Regional channel balance and phase accessibility¶

Separate internal conversion from boundary input¶

Retain the complete smooth sine law on fixed simple connected unit support, held nonnegative capacities and no forcing or events. Set K=diag(nu_i/d_i) using full-network degrees, a=w/pi and storage scale beta>0. For a region R, let q_R=L_R*x_R and S_R be its internal form gradient and internal sine current. External contributions are

\[ f_i=\sum_{j\notin R,\,j\sim i}(x_j-x_i),\qquad s_i=\sum_{j\notin R,\,j\sim i}\sin(\theta_j-\theta_i). \]

Thus the restriction of the full form gradient is q_R-f, and the full sine current is S_R+s. The actual regional rows in original structural time are

\[ \dot x_R=K_R[-e(q_R-f)+a(S_R+s)],\qquad \dot\theta_R=\frac a\beta K_R(q_R-f). \]

Write internal storage as E_R=F_R+beta*V_R, with F_R=x_R^T L_R x_R/2 and V_R=sum_internal(1-cos(delta)). Its gradients are q_R and -S_R. Define

\[ J_R=a q_R^\mathsf T K_R S_R,\quad B_F=a q_R^\mathsf T K_R s,\quad B_V=a S_R^\mathsf T K_R f,\quad D_F=e q_R^\mathsf T K_R(q_R-f). \]

Direct differentiation yields the exact identities

\[ \boxed{\dot F_R=J_R+B_F-D_F,\qquad \frac{d(\beta V_R)}{dt}=-J_R+B_V.} \]

Positive J_R converts phase storage into form storage, while its negative converts form into phase. B_F is the input to internal form storage from external phase currents; B_V is the input to internal phase storage from external form contrasts. They can have either sign. D_F is signed and is not the nonnegative full-node loss in the existing regional ledger.

Indeed that ledger has P_R=f^T*xdot_R+beta*s^T*thetadot_R and D_R=e*(q_R-f)^T K_R(q_R-f). Expanding both gives P_R-D_R=B_F+B_V-D_F. The cross term f^T K_R s cancels, while internal conversion cancels between the two storage rows. The old form and phase conjugate boundary-work fields are therefore not the new inputs to the individual stores and must not be renamed as such. These identities remain valid at zero capacity and require no inverse capacity.

For e=0, the cumulative phase-directed work satisfies

\[ \beta[V_R(t)-V_R(0)]=\int_0^t[-J_R(u)+B_V(u)]\,du. \]

This is a derived balance under the declared complete law, not an extra pressure, damping term or controller. Instantaneous rates and their running integrals are different observations. Retained environmental state supplies both boundary inputs; eliminating it requires the separate memory contract.

Equal storage does not determine the initial phase response¶

If all receiver phases are initially equal, the internal phase gradient vanishes. The second derivative of its weighted phase storage is then

\[ \boxed{\frac{d^2(\beta V_R)}{dt^2}(0) =\beta\sum_{\{i,j\}\in E_R} [\dot\theta_j(0)-\dot\theta_i(0)]^2.} \]

The terms involving phase acceleration vanish because every internal sine is zero. The rates on the right are those of the full support, including external form contrasts and full degrees. A common initial phase rotation does not grow phase storage. Relative velocities, rather than total storage alone, determine this initial curvature. This is a local-in-time identity; it cannot certify later branch passage or retention.

An exact control reuses the two-C5/intermediary support and return edge 1-6. Take e=0,w=beta=1, unit capacity, all phases zero and all forms zero except x_10=H and x_1=+H or -H. Both preparations have exactly the same storage partitions: receiver 0, complement 3H^2/2, cross-boundary H^2. In tau=t/pi, receiver phase rates and initial phase-storage curvature are

Environmental form	Receiver phase rates in cycle order	`d^2 V_R/dtau^2(0)`
`x_10=H, x_1=-H`	`(-H/3,+H/3,0,0,0)`	`2H^2/3`
`x_10=H, x_1=+H`	`(-H/3,-H/3,0,0,0)`	`2H^2/9`

For H!=0, orientation changes initial phase growth by a factor of three at identical storage, including each regional partition. Original-time curvature divides these entries by pi^2. The raw conserved form means differ and are retained; if a common mean leaf is required, independent uniform form shifts set them equal without changing either field or any storage. No response is simulated, and neither preparation is thereby proved to form an acute twist.

A running phase barrier for any acute unit-winding entry¶

The pure-phase C5 minimax is 7/2: every continuous phase path from consensus to a uniform unit twist reaches V_R>=7/2. This is a geometric result on the phase torus, independent of the positive-loss assumptions of other results using it.

It also applies when the endpoint is any fully acute state of winding +1 or -1. At that endpoint keep its integer edge offsets and linearly interpolate its principal gaps to the uniform gaps +2*pi/5 or -2*pi/5. Their sum remains +2*pi or -2*pi, so the change lifts to a continuous nodal phase path. All gaps remain in (-pi/2,pi/2); convexity of 1-cos there bounds this appended path's storage by its initial endpoint storage. If the original path never reached 7/2, the concatenation would contradict the minimax. This establishes a necessary running, not final, condition:

\[ \boxed{\text{flat receiver reaches an acute unit-winding state by }T \quad\Longrightarrow\quad \max_{0\le t\le T}\int_0^t[-J_R(u)+B_V(u)]\,du \ge \frac72\beta.} \]

Exceeding this bound does not supply oriented passage or a particular target; the two signs of winding have identical storage. Remaining below it excludes acute entry, not arbitrary nonacute winding. Nor does the barrier exclude an already prepared acute state. The zero initial potential in this formula uses flat phase; the larger sector argument retains the initial potential and gives a broader exclusion without it. A signed branch crossing still depends on the actual relative phase velocity, and protected retention still requires its own full-state and work bounds.

Shared implementation and evidence boundary¶

regional_storage_balance in the comparison owner exposes these channels for both region and complement, preserving the original conjugate work ledger. The same kernel consumes admitted whole-time boxes in assess_sine_regional_channels. It integrates interval rates over each complete step, retains endpoint balance residuals and distinguishes complete horizons from validated prefixes. The 7/2 exclusion is conditional on certified flat initial receiver phase and applies only to the validated prefix. Taylor endpoint enclosures remain numerical premises; neither export nor re-reading authenticates their origin. The API contract owns admission and projection details. These observers add no evolution law.

Retrospective channel diagnosis of the frozen control¶

The separate analysis record binds the original response, protocol and source archive by hash and retains its own analysis source archive. No solver is replayed and no preparation, horizon or original verdict changes. All 256 original tubes pass primitive/chain admission and recomputed Picard inclusion. Integrating their signed channel intervals on t in [0,12] gives the following outward-rounded bounds; the record retains exact fractions.

Cumulative quantity	Certified enclosing interval	Meaning
`integral J_R dt`	`[2.982,3.205]`	Positive net internal conversion from phase to form
`integral B_F dt`	`[0.952,1.203]`	Positive net boundary input to internal form storage
`integral B_V dt`	`[4.036,4.187]`	Positive net boundary input to internal phase storage

Their balances contain the independently rebuilt endpoint stores F_R(12) approximately 4.17197693 and V_R(12) approximately 1.01878426. They do not imply that each channel stays positive at every instant. The running phase-work upper bound is less than 1.377; direct primitive phase-storage enclosures give the tighter bound V_R<1.263. Both stay below the necessary 7/2 barrier over the entire validated horizon. In particular, the boundary phase input alone exceeds the barrier, but the net retained phase-directed work does not. Counting gross input as available phase storage would give the wrong accessibility diagnosis.

This identifies a realized conversion mechanism under the declared law. It does not establish that another orientation suppresses conversion long enough to create the target, nor that phase-to-form conversion prevents every other kind of pattern. The equal-budget curvature control supplies only an initial discriminator. The distributed boundary theorem below supplies one sufficient finite-passage class; it does not reopen this frozen control.

37. Distributed source geometry can produce finite acute winding¶

The conservative full law admits a sufficient preparation-to-passage result when the boundary supplies enough distinct phase directions. This changes the supplied support and preparation, not the nodal law. The result is finite acute winding, not attraction, an indefinitely maintained pattern or an autonomous choice of its initial source.

The environmental source image uses full degrees¶

Keep Section 35's finite connected simple unit support, unit held capacities, e=0, w=beta=1, no forcing or events, and tau=t/pi. Let the receiver R have uniform initial form c, and let the whole network start with one common phase. Put Q=V\setminus R and y=x_Q(0)-c. If A_RQ is the receiver-to-environment adjacency block and D_R retains the full receiver degrees, then the complete phase row gives

\[ \boxed{v_R(0)=T y,\qquad T=-D_R^{-1}A_{RQ}.} \]

The internal form gradient is initially zero; the external contrasts give the displayed row. Columns for environmental nodes without receiver contact are zero and remain part of the retained state. They can affect later evolution and are not declared dispensable.

Subtracting one receiver row of T from the others removes common phase velocity. Full rank |R|-1 of this relative map is sufficient to prescribe arbitrary initial relative phase velocities. It is not necessary to reach one particular acute phase cell. The relevant geometric admission is whether the source image intersects that cell, with declared integer edge offsets. Rank deficiency alone does not settle this intersection or nonlinear later accessibility.

For example, the earlier adjacent two-port C5 has initial receiver velocity of the form (u,v,0,0,0). Its three non-port phases coincide in every affine initial-velocity profile. An acute unit-winding C5 instead has five distinct circular phases: its five principal gaps share a sign and their proper partial sums have magnitude strictly between zero and 2pi. Consequently that two-port image cannot contain an acute unit-winding profile. This is an obstruction to this direct short-time construction, not a proof that subsequent nonlinear redistribution is impossible.

A source image plus a global remainder gives actual finite passage¶

Let R induce the declared C5 cycle. Supply a source shape y and scale H>0, so the full initial forms are c on R and c+H*y on Q. Write u=T*y. Select 0<s_0<s_1, integer cycle offsets h_ij, and a strict margin m>0 such that for every s in [s_0,s_1],

\[ -\frac\pi2+m \le s(u_j-u_i)-2\pi h_{ij} \le\frac\pi2-m,\qquad -\sum_{C_5}h_{ij}\in\{-1,1\}. \]

For fixed offsets these are strict-interior linear phase-cell conditions on the source profile. At one admitted interior point, continuity supplies a nonempty surrounding s interval. Their feasibility is a geometric property of T; it does not assert that a preparation is selected by the dynamics.

The global full-law enclosure, which uses bounded sine currents rather than a linearized solution, gives

\[ \left|\theta_i(\tau)-\theta_i(0)-\tau v_i(0)\right|\le\tau^2. \]

At tau=s/H, every cycle gap therefore differs from the admitted affine profile by at most 2s_1^2/H^2. Hence

\[ \boxed{\frac{2s_1^2}{H^2}<m} \]

is sufficient for the actual complete nonlinear solution to have the declared acute winding throughout [s_0/H,s_1/H], with margin at least m-2s_1^2/H^2. Its initial winding is zero, so an earlier branch passage has occurred. No prescribed boundary trajectory, extra pressure, damping or support event has been inserted. All environmental coordinates continue to evolve under the same nodal rows.

The full regional state remains bounded as well: |x_i(tau)-c|<=tau, and for its five cycle edges F_R(tau)<=10tau^2. Actual phase rates satisfy |theta_i'(tau)-v_i(0)|<=2tau; each relative edge rate has error at most 4tau. These are finite-time statements about the consumed form and phase rows, not just a phase snapshot. The lifetime in the original clock is pi*(s_1-s_0)/H: increasing source scale improves this enclosure while shortening its certified passage window. It does not establish persistence on a fixed time scale or approach to equilibrium.

One supplied distributed preparation¶

Take receiver cycle (0,1,2,3,4) and five private environmental leaves (5,6,7,8,9), with leaf 5+i joined only to receiver node i. Every receiver degree is three, each leaf degree is one, and T=-I/3. Keep the same unit-capacity conservative law and zero common phase, and supply

\[ x_i(0)=0,\qquad x_{5+i}(0)=-30(i-2),\qquad i=0,\ldots,4. \]

Thus v_R(0)=(-20,-10,0,10,20). Freeze

\[ \tau\in[1/8,2/15],\qquad t\in[\pi/8,2\pi/15]. \]

The four forward cycle edges have affine gap 10tau; the closing edge 4->0 has affine gap -40tau. With the common global edge error r=2(2/15)^2=8/225, the complete solution obeys

\[ \begin{aligned} \delta_{i,i+1}&\in[5/4-r,\;4/3+r],\quad i=0,1,2,3,\\ \delta_{40}+2\pi&\in[2\pi-16/3-r,\;2\pi-5+r]. \end{aligned} \]

All these principal gaps are positive and smaller than pi/2-1/5. For the first four the upper endpoint is 308/225; for the closing edge the margin is 5-3pi/2-8/225. The elementary bounds 3.14<pi<22/7 already prove both strict margins greater than 1/5. The offsets are (0,0,0,0,-1), so actual winding is exactly +1 throughout the window. Its original-clock duration is pi/120. The earlier two-port source's time restriction does not apply: the non-port velocity degeneracy has been removed by the separately supplied distributed contacts.

The same enclosure proves that this witness is a transit: at tau=1/6, the four forward principal gaps belong to [29/18,31/18], strictly inside (pi/2,pi). Thus it has exited the acute sector by t=pi/6. This does not say that its winding has disappeared. It separates the certified finite acute episode from captured or maintained organization without a new trajectory calculation.

Here F_R<=8/45, and the full relative phase rates retain the bounds just proved. The initial global storage is exactly 4500, entirely on the five cross-boundary edges: receiver and environmental internal stores are zero. The degree-weighted form mean and initial phase mean are both zero. Global storage is conserved, while regional work obeys the existing channel and cross-edge ledgers. This high-budget preparation encodes a desired relative phase arrangement in environmental form; the theorem does not derive that information or identify its storage with physical energy.

The same storage and mean do not select the acquired winding¶

On exactly the same graph, keep all phases zero and instead supply

\[ x_i(0)=9\quad(i\in R),\qquad (x_5,x_6,x_7,x_8,x_9)=(9,-21,-51,-51,-21). \]

Equivalently, begin with receiver form zero and leaf contrasts -30*(0,1,2,2,1), then shift every form by 9. This common shift changes neither field nor storage and sets the degree-weighted form mean to zero. Both sources thus have the same support, coefficients, capacities, initial phase, total storage 4500, all three regional storage partitions and conserved means. Their receivers satisfy the same uniform-form preparation condition; their environmental arrangements are different.

The control is fixed by the full-graph reflection i -> -i mod 5 on receiver nodes and its corresponding leaf permutation. Equivariance of the complete sine rows and uniqueness preserve that reflection for all time. It reverses cycle orientation, so principal increments cancel in reflected pairs whenever no cycle edge is antipodal. In particular the receiver can never have acute winding +1 or -1. This is an all-time conditional symmetry obstruction, independent of the positive source's finite window. It neither claims consensus nor excludes other form or phase organization.

Shared admission and scope¶

analyze_sine_conservative_source_geometry rebuilds the environmental map from admitted full-support state, retaining all columns, relative rank, actual initial velocities and identical-row groups. The same owner supplies certify_sine_conservative_winding_entry and its global enclosure. Source image geometry, a verified finite phase window, reflection invariance and conditional regional storage retention are separate contracts; one must not substitute a rank, a tangent derivative or an approximate midpoint for their full hypotheses.

The declared matched control, analytic certificate and source archive retain this positive preparation and its equal-budget reflection control. The shared analytic instrument records all eight gates as satisfied, with source/declaration hashes and exact interval projections. This is a reproducible evaluation of analytic bounds, not a sampled or reserved numerical trajectory. Its duration pi/120 is a separately declared target; it does not satisfy or rewrite the earlier control's quarter-unit duration criterion.

This supplies a genuine sufficient mechanism under the declared model: initially phase-flat receivers can acquire finite acute winding through their evolving environment, and equal scalar storage does not determine which source does so. The support and the structured initial form are supplied. Pattern selection, a primitive microscopic law, connection birth, indefinite maintenance and physical identification remain unresolved.

38. Low regional storage does not close the retention handoff¶

The regional work theorem requires a bound on future accumulated work, not just small entry storage. The following class demonstrates why that premise cannot be removed, even for an identity actually acquired from flat phase under the unchanged complete law. It is a source-class obstruction to a proposed implication, not a search for an optimal amplitude or support.

Admit the actual full-support velocity ramp¶

Retain Section 37's finite connected simple unit graph, unit capacities, e=0, w=beta=1, no forcing or events, and tau=t/pi. Let the receiver induce the oriented C5 (r_0,r_1,r_2,r_3,r_4). Its initial forms are one constant c, and every initial network phase is common. Environmental forms are supplied and subsequently evolve under the same full law.

Suppose the actual initial row v=KLx(0), with full graph degrees, satisfies

\[ v_{r_i}=v_c+\sigma(i-2)\omega,\qquad \sigma\in\{-1,1\},\quad\omega>0. \]

Thus the first four oriented edge velocities equal sigma*omega, while the closing velocity is -4sigma*omega. This hypothesis is a test of admitted state and contact geometry, not a prescribed phase drive. The private-leaf construction realizes it with leaf forms -3omega(i-2) and receiver form zero, but that particular support is not required.

Put

\[ \alpha=\frac{2\pi}{5},\qquad \tau_* = \frac{\alpha}{\omega},\qquad \tau_{\mathrm{out}}=\frac{5}{3\omega}. \]

The following explicit sufficient conditions define the admitted class:

\[ \begin{aligned} 2\tau_*^2&<\frac\pi{10},\\ 10\tau_*^2+10\tau_*^4&<B-V_5,\\ 2\tau_{\mathrm{out}}^2 &<\min\left\{\frac53-\frac\pi2,\;\pi-\frac53\right\}, \end{aligned} \]

where V_5=5(1-cos(alpha)) and B=5-4cos(3pi/8) are the existing regional minimum and acute-face barrier. They hold, for example, for every omega>=64. Indeed tau_*<1/50,

\[ B-V_5 =\frac{5(\sqrt5-1)}4-2\sqrt{2-\sqrt2} >\frac{13}{1000}, \]

and 2tau_out^2<=50/(9*64^2)<2/21<5/3-pi/2. The other exit inequality is weaker. The value 64 is a convenient sufficient rate, not a dynamical threshold or an optimized preparation.

Acquired entry near the storage minimum¶

At tau_*, the affine receiver profile is the uniform twist. Use edge offsets zero on the first four edges and -sigma on the closing edge. The actual principal gaps have the form

\[ \delta_e(\tau_*)=\sigma\alpha+\epsilon_e,\qquad |\epsilon_e|\le2\tau_*^2,\qquad \sum_{e\in C_5}\epsilon_e=0. \]

The global nonlinear enclosure supplies the individual bounds. The last identity is exact telescoping on the cycle, not an assumption of independent errors. The first admission inequality keeps every gap acute, so the actual winding is sigma; initially it was zero.

For U(delta)=1-cos(delta), Taylor's inequality U''<=1 and the zero error sum cancel the linear phase-storage term. Consequently

\[ V_5\le V_R(\tau_*)\le V_5+10\tau_*^4,\qquad F_R(\tau_*)\le10\tau_*^2, \]

and hence

\[ \boxed{E_R(\tau_*)\le V_5+10\tau_*^2+10\tau_*^4<B.} \]

The form bound uses |x_i(tau)-c|<=tau on all five receiver nodes. Thus these acquired entry states approach the minimum regional storage as omega increases. This controls internal form and phase, but not the environmental form contrasts consumed by KLx.

Strict finite exit and a necessary positive work transfer¶

At tau_out, each of the first four oriented principal gaps satisfies

\[ \sigma\delta_e(\tau_{\mathrm{out}}) \in\left[\frac53-2\tau_{\mathrm{out}}^2, \frac53+2\tau_{\mathrm{out}}^2\right] \subset\left(\frac\pi2,\pi\right). \]

Thus a first acute-face exit tau_exit after entry exists, with

\[ 0<\tau_{\mathrm{exit}}-\tau_* <\frac{5/3-2\pi/5}{\omega}. \]

The first exit is strictly before the displayed nonacute endpoint by continuity. Multiply this duration by pi in the original clock. No conclusion about disappearance of winding follows from an acute-face exit.

At the first exit, all receiver gaps still belong to the closed acute chart with the same winding. The existing face theorem gives E_R(t_exit)>=B. The exact regional balance in the original clock therefore forces

\[ \boxed{\int_{t_*}^{t_{\mathrm{exit}}}P_R(t)\,dt \ge B-E_R(t_*) \ge B-V_5-10\tau_*^2-10\tau_*^4>0.} \]

For omega>=64, the displayed coarse bounds already make this integral greater than 1/125. Actual environmental work has crossed the missing retention allowance; this is not merely a failed upper-bound calculation. The full law conserves global storage, and internal form/phase conversion remains part of the regional channel balance.

This class has arbitrarily small regional excess storage and arbitrarily short acute lifetime. Accordingly, acute membership and E_R<B alone cannot supply a uniform positive retention duration across admitted environments. The class does not keep global storage fixed as omega increases. It neither excludes finite maintenance for other sources nor establishes or excludes another definition of regional identity.

Zero instantaneous work does not remove the missing information¶

There is a separate instantaneous counterexample. At an exact receiver twist delta_e=sigma*alpha with uniform receiver form, the internal quantities satisfy q_R=S_R=0. Every regional channel in Section 36 and P_R is therefore zero at that instant, regardless of the retained environment. Nevertheless differentiation of the internal edge storage under the complete law gives the exact identity

\[ \boxed{\frac{d^2E_R}{d\tau^2} =\sum_{e\in C_5}(\Delta_e x')^2 +\cos\alpha\sum_{e\in C_5}(\Delta_e\theta')^2.} \]

The phase-acceleration term cancels because the oriented edge accelerations telescope. Both sums retain actual full-support nodal rates. On the private-leaf support, match each leaf phase to its receiver phase, keep receiver form zero and supply leaf forms -3omega(i-2). Then the first sum is zero and the second gives E_R''=20cos(alpha)*omega^2, although E_R=V_5 and P_R=0. This constructed snapshot is not claimed to be an exact state attained by the phase-flat acquisition family.

Likewise, in that acquisition family the actual first-four phase rates at entry satisfy |Delta_e theta'(tau_*)-sigma*omega|<=4tau_*. They can grow while regional excess storage tends to zero. Regional storage and instantaneous power discard precisely this environmental rate information. A sufficient prediction must retain the source state or its justified causal memory; it cannot replace them by a small local storage observation.

Relative rhythm means compatible phase rates, not identical phases¶

The exact kinematic row on every retained edge is

\[ \delta_{ij}'=(KLx)_j-(KLx)_i,\qquad \delta_{ij}(T+s)=\delta_{ij}(T) +\int_T^{T+s}\bigl[(KLx)_j-(KLx)_i\bigr]\,d\tau. \]

The integral is read on the same continuous phase chart. A fixed relative shape requires equal phase rates on its connected receiver, and that equality must be preserved by the complete field. Equality at one instant does not establish its continuation. An evolving identity instead permits relative oscillations provided their accumulated differences stay within the declared sector throughout the interval. Equal phases or constant relative geometry are not necessary for such finite retention. Capacity is still the held nodal mobility; it is not identified with these measured phase rates or a physical oscillation frequency.

For this unforced fixed-support law, sum_i d_i theta_i'=sum_i(Lx)_i=0. Consequently a common nonzero phase velocity across the whole connected network is impossible under these premises. A regional common velocity can be compensated by the retained environment. Neither this balance nor an admitted finite relative rhythm establishes a universal permanent pulse.

Shared admission and boundary of the result¶

assess_sine_conservative_handoff and SineConservativeHandoff share the admitted full-state source and global enclosure with the entry owner. The theorem concerns the declared conservative ramp class on unchanged support. Its sufficient inequalities may fail without deciding another preparation. The existing future-work retention theorem remains valid; this result identifies a class whose actual work violates its premise after a genuinely acquired acute entry. No damping, contact removal, prescribed feedback, autonomous preparation, indefinite-stability claim or physical identification is added.

42. A larger cycle sector gives a global conservative barrier¶

The flat-receiver phase-path bound in Section 36 has a stronger geometric form. It applies to any source outside a specified unit-winding sector, including nonuniform initial phases with zero winding. This is a direct boundary argument for the existing phase potential; it introduces neither a changed law nor a new initial preparation.

The boundary minimum is exactly `7/2`¶

On an oriented C5 let delta_i be its principal edge gaps and W=(sum_i delta_i)/(2pi). For sigma=+1 or -1, define

\[ \Omega_\sigma= \{W=\sigma,\quad |\delta_i|<2\pi/3\text{ for every edge}\}, \qquad V_R=\sum_i U(\delta_i),\quad U(s)=1-\cos s. \]

Every fully acute state of winding sigma lies in Omega_sigma. The sector contains no principal-branch seam: a path leaving it first reaches a gap of magnitude 2pi/3. Reflection reduces the proof to sigma=+1, with sum delta_i=2pi.

The following tangent inequality holds on the whole required interval:

\[ U(s)\ge U(\pi/3)+\sin(\pi/3)(s-\pi/3), \qquad -2\pi/3\le s\le2\pi/3. \]

Indeed sin(s)-sin(pi/3) is nonpositive to the left of pi/3 and nonnegative from there to 2pi/3. Integrating this derivative difference on either side proves the inequality without assuming that U is convex on the entire interval.

If one boundary gap is +2pi/3, the other four sum to 4pi/3. Summing their tangent bounds cancels the linear terms and gives

\[ V_R\ge U(2\pi/3)+4U(\pi/3) =\frac32+4\cdot\frac12=\frac72. \]

Equality occurs at the five permutations of (2pi/3,pi/3,pi/3,pi/3,pi/3). All five edge sine currents agree there, so these are actual critical cycle configurations, rather than a bound from independently incompatible gaps. If a boundary gap is -2pi/3, the other four must all equal +2pi/3, giving V_R=15/2 instead. Consequently the minimum over the complete sector boundary is exactly

\[ \boxed{\min_{\partial\Omega_\sigma}V_R=\frac72.} \]

This does not replace the smaller acute-face minimum B=5-4cos(3pi/8), or the minimum principal-branch seam cost 6-2sqrt(2). They concern different boundaries. In particular, Omega_sigma contains nonacute states; its protection does not prove that every regional edge remains acute.

A conserved full budget separates the sector in both directions¶

Retain a finite connected unit-support graph containing the receiver C5, unit held capacities, e=0,w=beta=1, no inputs or events, and tau=t/pi. The existing full law conserves

\[ H=\frac12\sum_{\{i,j\}}(x_i-x_j)^2+ \sum_{\{i,j\}}[1-\cos(\theta_j-\theta_i)]\ge V_R. \]

The smooth finite-dimensional flow exists in both time directions: bounded form rates preclude finite-time form blowup, and the linear phase row then has bounded rates on every finite interval. If H<7/2, continuity and the boundary minimum give two conclusions for all real time:

A receiver initially in Omega_sigma remains there with winding sigma.
A receiver initially outside Omega_sigma cannot enter it, and therefore cannot become an acute unit-winding state of that sign.

The same finite-time conclusions hold at H=7/2. At a sector boundary, equality in the phase minimum requires one principal gap sigma*2pi/3 and the other four sigma*pi/3. Indeed the tangent inequality above is strict away from pi/3 for each of those four remaining gaps. Equality in H>=V_R also forces every full-edge form difference to zero and every non-cycle edge phase difference to zero modulo 2pi. On connected support, the form is therefore a common constant, which need not be zero. The equal oriented cycle sine currents cancel at every receiver, and all other sine currents vanish. Both full rows vanish: such a state is an equilibrium. If additional edges make these simultaneous equalities impossible, there is no boundary state at this budget. Otherwise uniqueness of the smooth flow prevents a distinct orbit from reaching that equilibrium at a finite time. A boundary equilibrium itself remains there. Thus H<=7/2 excludes finite crossing in either direction; an asymptotic approach is a different question.

An initial W=0 receiver is outside both sectors. Thus a conservative source with that winding and H<=7/2 cannot acquire either acute orientation, even if its initial phases are not equal. More generally any continuous phase path from outside to an acute endpoint must have max V_R>=7/2. For an actual finite-time orbit of this complete law, the necessary full budget is strictly H>7/2, not equality. The geometric phase-path bound alone does not imply that stronger dynamical conclusion. Using the original-time channel rates of Section 36, the corresponding running phase-input condition over original time [0,t_*] is

\[ V_R(0)+\max_{0\le s\le t_*}\int_0^s[-J_R(v)+B_V(v)]\,dv \ge\frac72 \]

when the initial receiver is outside the target sector. Section 36's flat-phase statement is its V_R(0)=0 specialization. Full uncertainty boxes must certify the storage bound and initial sector membership or exclusion, rather than substituting a midpoint or cached winding label.

Negative collective feedback can still lie below this barrier¶

Take Section 41's two matched contact-lag preparations with u=1, gamma=1/4 and rho=1/10. Both start with uniform receiver form and phase, so W_R=0, and their exact full storage is

\[ H=\frac52+5-(1+4\cos(1/10))\cos(1/4). \]

The alternating Taylor bounds C_6(z)=1-z^2/2+z^4/24-z^6/720 <= cos(z) <= C_4(z) at these two positive arguments give the rational enclosure 2674/1000 < H < 2675/1000 < 7/2. The barrier therefore excludes acute winding +1 and -1 for both cyclic arrangements at all times. The reflected source retains its independent symmetry obstruction; the asymmetric ordering removes that obstruction but not this conserved-budget one. Their negative collective transfer and positive initial internal derivatives remain valid local results.

For the same uniform-form family and general declared u,gamma,rho, a necessary budget condition for acute entry is

\[ \boxed{u^2>\frac25 \left[\frac72-5+(1+4\cos\rho)\cos\gamma\right]} \]

when the right side is positive. Equality does not permit finite-time entry. This neither identifies a successful amplitude nor guarantees oriented passage or maintenance after the budget obstruction disappears.

Compatible conserved quantities still do not determine reachability¶

Section 40's reference with h=1/50 has H_*=V_5+1/10>7/2; this barrier does not exclude every zero-winding source with that total storage. There is even a continuous geometric path preserving those scalar invariants. Take receiver phases theta_i=i*s for i=0,...,4 and 0<=s<=2*pi/5. Their potential is V_R(s)=4(1-cos(s))+1-cos(4s); its derivative 8*sin(5s/2)*cos(3s/2) changes sign at s=pi/3, where V_R=7/2. Set each private leaf phase equal to its receiver phase, and along that path choose

\[ u(s)=\sqrt{\frac25[H_*-V_R(s)]},\qquad x_R(s)=\frac{u(s)}4\mathbf1,\qquad x_Q(s)=-\frac{3u(s)}4\mathbf1. \]

Then contact phase storage vanishes, H=V_R+5u^2/2=H_*, and the degree-weighted form mean is zero. A common continuous phase shift fixes the lifted weighted phase mean as well. The endpoint is the compatible unit twist with u=1/5. This path is a geometric compatibility witness, not a solution of the nodal equations. Its uniform, phase-consensus starting state in fact has a rotationally invariant zero-winding evolution; scalar compatibility alone cannot prescribe a different trajectory.

Nor can an outside trajectory enter the exact invariant reference family at a finite time. A positive-width retention neighborhood has a separate entry question. Its own correlated storage bounds may exclude a named source even above 7/2, as Section 41's H=14201/4000 control and Section 40's fixed small-error box demonstrate. When that energy test passes, the two-sided finite retention estimate still places the earliest first appearance before its certified backward interval. None of these constraints supplies the missing directed autonomous handoff.

assess_sine_cycle_barrier re-admits the full source and computes the strict storage flag and the closed H<=7/2 finite-time gate separately from each sector's membership. Other nodes, contacts and any receiver chords remain in the law and full storage. An undecided principal branch or energy bound can leave a conclusion unavailable; no report chooses a missing preparation or executes a trajectory.

43. Fast contact motion can exclude finite receiver organization¶

The full storage barrier supplies a necessary lower resource bound, not a monotone relation between supplied form contrast and acquisition. On the same C5/private-leaf support, the actual contact phases can turn rapidly enough that their accumulated sine currents stay small over a fixed horizon. The following estimate retains the nonlinear receiver and every leaf; it does not prescribe a periodic input or replace the environment by a fitted average.

The full rows bound the accumulated contact currents¶

Retain unit capacities, e=0,w=beta=1, no events or inputs, and tau=t/pi. Initially let receiver forms be the constant a_0, leaf forms the constant b_0, and receiver phases one common value theta_0. The five initial contact phases may be arbitrary. Write u=a_0-b_0, fix a horizon T>0, and define continuous contact lifts and their integrated currents by

\[ \phi_i=\theta_{l_i}-\theta_{r_i},\qquad I_i(\tau)=\int_0^\tau\sin\phi_i(s)\,ds. \]

\[ M=\frac43|u|-4T>0, \]

each contact rate keeps its sign and satisfies |phi_i'|>=M throughout [0,T]. Integration by parts using sin(phi)=-(cos(phi))'/phi' then bounds every prefix:

\[ |I_i(\tau)|\le\frac2M+\frac{4\tau}{M^2} \le\epsilon_T:=\frac2M+\frac{4T}{M^2}, \qquad 0\le\tau\le T. \]

This bound concerns the actual moving contacts. Their rates need not be constant and their phases need not be acute; the smooth sine law supplies the derivative bound everywhere.

An exact change of variables retains the nonlinear receiver¶

Let L_R be the combinatorial Laplacian of C5 and Id the identity matrix. Since x_Q=b_0*1-I exactly, use the corrected receiver coordinates

\[ z=x_R-a_0\mathbf1-I/3,\qquad q=\theta_R-\theta_0\mathbf1-u\tau\mathbf1/3. \]

Here I is the five-vector of integrated contact currents, not a matrix. Put V(q)=sum_i[1-cos(q_(i+1)-q_i)] and g=grad V; the receiver internal sine-current vector is -g. Direct substitution into the full degree-three receiver and degree-one leaf rows gives

\[ z'=-\frac13g(q),\qquad q'=\frac13(L_R+\mathrm{Id})z+ \frac19(L_R+4\mathrm{Id})I. \]

No current has been discarded. Both z(0) and q(0) are zero. For the nonnegative auxiliary function

\[ \mathcal E=\frac12z^\mathsf T(L_R+\mathrm{Id})z+V(q), \]

the two autonomous terms cancel exactly:

\[ \mathcal E'=\frac19g^\mathsf T(L_R+4\mathrm{Id})I. \]

This is a proof function, not the total storage or a new physical reservoir. The cycle incidence matrix and sin^2(delta)<=2[1-cos(delta)] imply ||g||^2<=8V; the same graph has ||L_R+4Id||_2<=8. With ||I||_2<=sqrt(5)*epsilon_T it follows that

\[ |\mathcal E'|\le\frac{16\sqrt{10}}9 \epsilon_T\sqrt{\mathcal E},\qquad \boxed{V_R(\tau)\le\mathcal E(\tau) \le\frac{640}{81}\epsilon_T^2\tau^2.} \]

The square-root integration starts from E(0)=0; applying it first to sqrt(E+eta) and then taking eta down to zero justifies the bound at zero without dividing by a vanishing quantity.

A prospective finite exclusion despite a large total budget¶

The receiver starts flat. If

\[ \frac{640}{81}\epsilon_T^2T^2<\frac72, \]

its entire phase path stays below the acute-entry barrier. It cannot acquire either acute unit-winding orientation anywhere on [0,T], whatever the five initial contact phases or their cyclic ordering. The original structural-time horizon is pi*T.

For the exact supplied values u=20, T=1, one has M=68/3, epsilon_T=111/1156 and (640/81)*(111/1156)^2<73/1000<7/2. Yet its full initial storage is at least 5u^2/2=1000, because all additional contact phase costs are nonnegative. This is an analytic family control, without a numerical trajectory or a search over contact orderings.

The result is finite-horizon and preparation-specific. It establishes neither damping nor absence of later acquisition, and it does not exclude nonacute winding. If M<=0 or the upper bound is too large, this sufficient test is unavailable rather than evidence of formation. The nonuniform environmental form ramps in Section 37 do not have these uniform block forms and are outside the theorem. Combining this estimate with Section 42 removes some low-budget and rapid-contact preparations; it selects neither a successful intermediate source nor its occurrence.

assess_sine_contact_averaging rebuilds the admitted C5/private-leaf source, verifies its uniform block forms and flat receiver phase, and evaluates these finite-horizon bounds. The detached reader preserves the actual source and full storage; it does not integrate the source or change its form contrast to make the test pass.

44. A distributed preparation has a robust full-state passage¶

The finite acute-entry construction of Section 37 can be joined to an explicit nonzero-width full-state retention set. This answers an existence question for the same distributed-form preparation class: initially absent winding can be acquired and then retained over a declared short interval, with independent perturbations of every form and phase coordinate. It does not require the particular moving-reference neighborhood of Section 40. Nor does it establish prolonged maintenance or solve Section 41's distinct uniform-form, ordered-contact-phase preparation question.

Direct full-row bounds compose into an independent target box¶

Retain the conservative unit law and clock tau=t/pi. On the complete fixed support write A=D^(-1)L, so the actual rows are x'=D^(-1)S(theta) and theta'=Ax. For every state, the normalized rows give

\[ |x_i'|\le1,\qquad |\theta_i''|=|(Ax')_i|\le2, \qquad \|A\|_\infty=2. \]

Supply nominal initial data (x^0,theta^0) and independent coordinate errors bounded by epsilon>=0 in both form and continuously lifted phase. These bounds hold at every node, including the environment. All members share the stated fixed capacity, support and complete law. For their actual solutions,

\[ \boxed{ |x_i(\tau)-x_i^0|\le\epsilon+\tau,\qquad |\theta_i(\tau)-\theta_i^0-\tau(Ax^0)_i| \le\epsilon(1+2\tau)+\tau^2, \quad \tau\ge0.} \]

The second inequality uses an initial phase-rate error at most 2epsilon and the global acceleration bound. It is not a linearized trajectory or a truncated local jet; its remainder bounds the complete nonlinear rows for the stated interval. No small-time or exponential estimate is needed to derive it.

Choose a checkpoint a>0 and a later endpoint b>a, with h=b-a. Every actual checkpoint belongs to the explicitly supplied box centered at

\[ x^*=x^0,\qquad \theta^*=\theta^0+aAx^0, \]

with per-node form and phase radii

\[ r_x=a+\epsilon,\qquad r_\theta=a^2+\epsilon(1+2a). \]

Consider now any independent member of that target box, not only those already shown to be reachable from the source. Evolving the same full law for 0<=s<=h gives

\[ |\theta_i(a+s)-\theta_i^0-(a+s)(Ax^0)_i| \le r_\theta+2sr_x+s^2, \qquad |x_i(a+s)-x_i^0|\le r_x+s. \]

This follows by admitting its initial form and phase deviations at the checkpoint and applying the same acceleration bound afresh. In particular, the whole-interval edge-gap error is bounded by

\[ \boxed{q=2(r_\theta+2hr_x+h^2) =2[b^2+\epsilon(1+2b)].} \]

Thus the direct source enclosure and the independent target continuation compose exactly at this level of bounds. If the nominal affine edge gaps on [a,b], enlarged by [-q,q], lie strictly within one declared acute winding sector, all source members reach that sector and all target members retain it over the subsequent interval. The target is a sufficient full-state set; it does not assert that every one of its points is reached from the smaller source set.

An exact distributed source and a fixed short window¶

Use the unchanged C5/private-leaf graph, with receiver nodes i=0,...,4 and corresponding leaves 5+i. The nominal source is Section 37's ramp

\[ x_i^0=0,\qquad x_{5+i}^0=-3\omega(i-2),\qquad \theta_i^0=0\text{ at all ten nodes},\qquad \omega=64. \]

Supply independent source radii and times

\[ \epsilon=\frac1{4096},\qquad a=\frac{11}{512},\qquad b=\frac3{128},\qquad h=\frac1{512}. \]

Every source member initially has receiver winding zero: each initial raw receiver gap has magnitude at most 2epsilon=1/2048<pi, and their sum telescopes to zero. The nominal full-support phase rates are

\[ (Ax^0)_i=\omega(i-2),\qquad (Ax^0)_{5+i}=-3\omega(i-2). \]

The target center therefore retains ten explicit rational form values and ten rational phase values; no response supplies that center. Its independent coordinate radii are

\[ r_x=\frac{89}{4096},\qquad r_\theta=\frac{751}{1048576},\qquad q=\frac{211}{131072}<\frac1{500}. \]

On [a,b] the four forward affine receiver gaps range from 11/8 to 3/2; the closing raw gap ranges from -6 to -11/2. After the single closing-edge turn correction, the whole target-box continuation satisfies

\[ \begin{aligned} \delta_0,\ldots,\delta_3 &\in[11/8-q,\ 3/2+q],\\ \delta_4&\in[2\pi-6-q,\ 2\pi-11/2+q]. \end{aligned} \]

All five intervals lie strictly in (0,pi/2). For example the elementary bounds 157/50<pi<22/7 give an acute margin greater than 1/16; the limiting forward-gap margin is already greater than 7/100-q=224101/3276800. The raw gaps telescope, so their principal sum is exactly 2pi, and the receiver has winding +1 throughout. This is a whole-interval conclusion for every independent target member, not a collection of sampled windings.

The target also bounds the receiver's internal form. Each cycle form contrast has magnitude at most 2(r_x+h), hence

\[ F_R\le\frac52[2(r_x+h)]^2 =\frac{47045}{8388608}<\frac6{1000}. \]

The original structural-time checkpoint is pi*a and the guaranteed subsequent duration is pi/512. Full environmental form, phase, storage and weighted means remain those of each actual trajectory. No controller, support event, damping or removed environmental coordinate performs the handoff.

What this passage establishes and what it does not¶

This proves robust finite acquisition followed by a separately admitted full-state retention window. The source and target both have positive width in all twenty form/phase coordinates. It makes the finite existence claim quantitative rather than relying on unspecified continuous dependence.

The interval is deliberately short and the distributed environment is strongly prepared. Its nominal full storage is 45omega^2=184320, and the source box retains the corresponding perturbed budgets. The regional phase storage need not lie below the acute-face barrier B; this is not Section 35's low-regional-storage work certificate. The rapid relative phase rates have not disappeared, and no extended lifetime, attraction, autonomous preparation or physical identification is established.

Section 40's small moving-family neighborhood is an optional sufficient maintenance target, not a definition of every coherent identity. Requiring entry into that particular neighborhood would be an additional research question. Conversely this finite passage does not settle a prescribed longer duration, a lifetime that does not shrink with the preparation's rapid transport scale, or the separate ordered-contact-phase source class. Those stronger questions need their own target and acceptance conditions; they do not invalidate the positive bounded result here.

The existing certify_sine_conservative_winding_entry owns this source-error and checkpoint-box extension. Its optional source_error_bound retains a nominal admitted source with full-coordinate uncertainty and exposes the composed phase enclosure and form bound. The analytic radii above describe exact real boxes. If interval construction rounds their endpoints outward, the implementation measures the resulting effective radii about the exact centers and uses those in its separate entry-box continuation bounds. It does not promote a rounded display into an exact-radius proof. The original source-trajectory bounds retain their own tighter calculation; the explicit dyadic witness needs no such enlargement. It remains an analytic certificate, not a numerical trajectory or evidence that a source is selected autonomously.

The full rows identify the information needed for a stronger retention question. Let P subtract the five-node receiver mean, q_R=P x_R, and r=x_R-x_Q in matched receiver/leaf order. The actual relative receiver phase rate is

\[ P=I_5-\frac15\mathbf1\mathbf1^\mathsf T,\qquad \boxed{v=P\theta_R'=\frac13\bigl(L_Rq_R+Pr\bigr).} \]

Small internal form or phase storage does not control the environmental contrast Pr. The ramp has q_R=0 initially but supplies the large relative rates v_i=omega(i-2). Cancellation between the two terms can reduce relative motion, but an instantaneous cancellation does not prove its continuation under the full law. Longer retention therefore requires bounds on their actual accumulated phase transport and regional work over the declared interval. A moving compatible family is one sufficient tool for those bounds; it is not a necessary definition of identity. Any longer prospective duration is an evaluation requirement, not a new physical constant or a change to the finite claim proved above.