Collective Transition with retained lifecycle context¶
On the successor of the admitted EN/IL/SHA word, NAV selects its latent regime because SHA sets a lifecycle flag, even though capacity exceeds the numerical latent cutoff. Its phase and capacity reset survives the declared sine-pressure refresh; its raw-pressure jitter does not survive as an independent continuous coordinate. The same contextual chart admits this conditional event. Preservation of the old acute phase sector needs an additional margin: a counterexample in the full Reception source box fails it, while an explicit smaller source domain passes.
This owner closes a bounded analytical question, without a new retained-source word, trajectory or physical observation. The research program owns subsequent work. Regime labels, event timing and jitter configuration remain supplied policies, not autonomous dynamical causes.
1. The actual successor and the context NAV consumes¶
Keep the Reception model: twenty associated nodes on the fixed 45-edge weighted contact, with two singleton ports 0 and 1 and nine exact physical twin pairs. Its conservative continuous rows and storage are
The old conductances are one, the five added conductances lie in (3/4,1],
and bare degrees retain all fine neighbors. There is no forcing or loss.
Capacities are held between events; time is the same structural clock.
Geometric lengths equal conductances only for the finite U6 comparison.
The prior source domain and word
give port form in (1/4,3/8) and post-SHA capacity in (3/16,5/16).
SHA preserves form and phase and writes latent=True, a supplied
latency_start_time, the current preserved_epi, zero silence_duration
and its initial-form classification. The complete word leaves persistent
U2 debt zero and the associated history ending in SHA. Retain the freshly
evaluated pressure after that word. No intervening flow or form event is
inserted before the NAV event considered here.
The Transition owner selects the latent regime whenever the flag is true or capacity is below 0.05. Thus the actual successor uses the latent branch, although its capacity is strictly greater than 3/16. Deleting the flag from an otherwise identical physical state selects the active branch: its default phase shift and capacity multiplier differ. The resonant branch is excluded here because form remains below 3/8 and capacity below 5/16. Other branches are not inferred to be reachable from this successor.
Consequently the predictive chart must retain lifecycle fields in addition to form, phase, capacity and weighted support. It also retains target association, ordered histories, debt, all consumed configuration, a timezone-aware stage timestamp and any consumed per-node random stream identity and progress. Phase/capacity equivalence alone does not select NAV's map. The lossless eleven-cell physical chart still lifts these records to the twenty associated members before executing the shared atomic stage. It does not replace them with cell averages.
The supplied NAV instant must not precede the SHA timestamp. Their elapsed wall-clock duration is latency telemetry, not the structural flow clock or a finite interval spent evolving (1). With no intervening form change, the preserved-form drift is zero. An explicit maximum silence duration, when configured, still has its own admission and warning behavior. NAV clears its consumed latency flag/start/preserved-form fields on successful commit and records the duration and regime origin; it does not erase every historical lifecycle record.
2. Selected primary reset and raw pressure¶
Let T be the two ports and let chi_T be its indicator. The ideal primary
map for the latent preset is
where A has entry a at each port and one elsewhere. Use the represented preset parameters as exact inputs to this ideal-real map:
The implemented phase write normalizes modulo one turn; equation (3) uses the corresponding continuous lift. Actual multiplication and modulo arithmetic have separately retained endpoint defects. A requested phase increment does not prove that arbitrary large represented phase origins realize it exactly.
Select the admitted non-strict handler with NAV_eta=1/2. If p denotes the
pre-event stored pressure, the target handler and final raw pressure are
where s_i=+1 for p_i>=0 and s_i=-1 otherwise. The selected jitter
amplitude satisfies 0<=j<=1/24, including the configured default
1/(8*pi). In random mode, the supplied per-node draw has |zeta_i|<=j.
In deterministic mode, the handler uses zeta_i=j*sign_+(b_i); setting
NAV_RANDOM=False does not by itself remove jitter. Setting j to zero
is a separate configuration. Both flags use their actual Boolean admission.
The low-level direct glyph has its own pressure-only contract. Equation (3), regime retention, lifecycle updates and the following chart concern the public/atomic staged path, not that direct glyph. The strict handler uses a different base and is outside the bounded preset in (5).
The full previous source domain already supplies a margin for a nontrivial
handler. The prior IL phase bound gives port error E=343/2000 and
untargeted error less than 1/8 about the reference. The port has six external
neighbors and the other port; since the reference sine sum vanishes,
Together with nu_i>3/16, this bounds every admitted jitter draw away from
the handler's no-change rejection:
The default live checks nu>=0.01, |p|<=1 and the latent form advisory
threshold 0.05 also have strict margins here. All-target atomic admission
still validates finite values, timestamps, flags, sinks, phase, requested
parameters and actual proposals before committing. Equation (7) is an
ideal-domain bound, not permission to bypass those represented checks.
3. Event work, changed means and refreshed rows¶
For an actual admitted phase/capacity endpoint with form and support held, use a declared lift delta of its actual phase change. Its exact structural work is
The phase potential uses bare edges. Conductance weights do not multiply these terms, and capacities add no separate term to (2). Equal shifts on both ports leave their internal-edge phase difference fixed; only eight old edges and four cross edges change. Adding two isolated-port work calculations would incorrectly include their internal-edge contributions. The sign of (8) is not determined by the regime name.
At the reference phase, EN and IL leave phase at that reference and SHA also preserves it. The latent NAV reset there has
This is a source in the analytical successor class, with a supplied event and positive work, not a new evaluation of the historical preparation. The pressure attenuation in (5) does not make this phase event passive.
Put rho_i=d_i/nu_i, rho_i^+=rho_i/a_i, and R^+=sum_i rho_i^+.
The post-event normalized means, using the same lifts, satisfy
Form itself is unchanged, but its capacity-weighted mean can change. The continuing conservative flow preserves the new mean leaves. A circular representative difference is not automatically the lift used in (11).
Constitutive refresh computes p^+=S(theta+delta)/(pi*d) independently of
(5). With Delta S=S(theta+delta)-S(theta), the complete fresh rows give
The phase jump changes fresh pressure and can change form rates even at untargeted neighbors. Their phase rates are unchanged at this immediate boundary because their capacity and q were unchanged. The two rows do not generally share one global clock factor. Work and changed responses must be evaluated on this actual reset rather than inferred from an activation label.
Under immediate pressure-only refresh with no intervening flow, all
accepted draws in (5) have the same refreshed continuous coordinates
(x,theta,nu,U,W): the raw pressure is overwritten, while the phase and
capacity reset remains. This is a conditional projection equivalence.
Histories, transition telemetry and random progress can still differ, so
the complete execution states are not identified. Consuming raw pressure
over a held interval, allowing another event to inspect it, or declaring
an independent pressure-memory law would be a different model.
4. The previous phase box does not guarantee acute preservation¶
An acute EN/IL/SHA successor is not enough to infer that the fixed positive NAV phase shift preserves every incident acute gap. The following exact counterexample lies strictly inside the earlier source box.
Write alpha=2*pi/5 and epsilon=3/25<1/8. Give ports 0 and 1, forward
neighbors 2 and 3, and cross neighbors 10 and 11 phase error +epsilon;
give trailing neighbors 8 and 9 error -epsilon. Set the other errors to
zero and retain all physical twins, admitted forms, weights and capacities.
EN does not change phase. The target neighbor resultant before IL is
The denominator is positive. Also cos(alpha)<31/100,
sin(epsilon)>epsilon-epsilon^3/6, and monotonicity of cosine on this
interval give tan(mu)>2/25. Therefore
mu>arctan(2/25)>1/16. After IL with any 0<=lambda<=3/10 and then
latent NAV, the trailing port-edge gap is
The final inequality follows from pi<22/7. The counterexample is still
well below the principal cut pi: it proves loss of the acute sector, not
a change of either constituent winding or destruction of organization.
It also does not prove that a subsequent trajectory is unstable.
5. Explicit acute successor domain and whole-word grammar¶
For an actual pre-NAV lifted state, the direct admission criterion is
with the original marked reference integers kept. Endpoints and their straight phase interpolation then stay in the same convex acute sector. This criterion retains actual phase geometry instead of replacing it by a winding label.
A positive-width sufficient family is obtained by explicitly restricting
the earlier EN source to max_i|eta_i|<1/16. This is a new admission
premise for NAV, not a revision of the earlier EN domain or evidence.
All other form, capacity, weight, history and configuration premises stay
as previously declared. The reference target resultant exceeds four, and
its perturbation is less than 7/16, so its mean angle satisfies
After the latent shift, every port-to-old-neighbor edge has error below
Its old reference magnitude is alpha, leaving the full gap strictly below pi/2. Untargeted old edges have error below 1/8; the internal port edge cancels the common shift and has error below 49/304. Cross-edge reference gaps are zero and obey the bound in (17). Thus the ideal reset preserves the marked acute sector and its two original windings on this positive-width source stratum. It supplies no future lifetime bound.
The selected complete successor word is a singleton NAV on both ports,
using ValidatedSequence with the retained nonzero-form context, strict
rejection and live preconditions. NAV already has both generator and
closure roles and introduces no U2 debt; the successor debt remains zero.
The actual history ends in SHA, matching its retained predecessor context.
No additional IL or SHA is implied by this analytical reset. Applicable
stage validation and the finite U6 observations below remain distinct
from the geometric criterion (15).
6. Finite U6 and implementation controls¶
For the declared geometry, the prior structural-potential estimate is
Only two stored pressures change in (5). Using (4), |p_i|<=1/pi,
nu_i<5/16 and j<=1/24, their changes are bounded by
For equal port shifts sigma, the internal-edge contribution cancels. The sine Lipschitz estimate with the actual degrees gives
Combining (18)--(20) and pi>3 bounds the largest selected before/raw,
before/refreshed or raw/refreshed comparison by
This holds on the full prior EN successor, including the nonacute counterexample in section 4, for accepted draws under the selected non-strict handler. Thus passing this finite U6 policy does not prove acute-sector preservation. Other strict-mode pressure bases, jitter budgets, geometric lengths or baselines need their own assessment.
7. A conditional continuous rate budget at the ports¶
In the retained marked acute sector, every edge of an oriented original C5 cycle has a strictly positive principal gap. Indeed, its five gaps sum to 2pi by winding +1; if one were nonpositive, the other four strict acute bounds would make that sum less than 2pi. Every original fine edge belongs to such a constituent cycle, including either marked port.
At a port, the four original neighbors consequently supply two positive and two negative sine terms, whose total has magnitude less than two. The other port and two cross neighbors add at most three in magnitude. Together with post-NAV capacity below 3/8 and degree seven, this gives
A true continuous-time EPI secant over a nonzero interval wholly governed by (1), with these held capacities and the same retained phase sector, obeys the same bound by integration. Two acute endpoints alone do not establish the intermediate-sector premise. An interval containing an operator form jump, a different clock or a capacity reset also has different assumptions.
This is a conditional budget relevant to a later temporal-trigger admission: it cannot exceed the configured default mutation threshold 0.1 on that clock and domain. It is not measured trigger evidence, a statement about operator-step histories, or an exclusion of every ZHIR configuration or source. The execution plan alone selects any later investigation.
8. Representation and verification boundaries¶
Ideal input-parameter bounds, represented endpoint checks and finite software observations retain separate scopes. Synthetic controls reuse the existing EN/IL/SHA preparation and NAV atomic proposal, including live grammar, latency, raw/refresh boundaries and random progress. They do not replay a frozen preparation, install a new pressure law or establish autonomous occurrence or physical identification.
The collective Transition controls exercise the existing public stage on independent preparations, including the latent/active context discriminator, actual phase work and fresh rows, changed mean leaves, accepted random progress, invalid-lifecycle rollback and the nonacute counterexample despite passing grammar and finite U6. The verification guide owns the maintained selection.