Aggregate operators and native transitions¶

This owner asks when several registered constituent actions are exactly one registered action on a specified aggregate. It uses the existing shared operator implementation and ordered-language descent criterion. The native collective calculus instead retains the constituent state needed to describe native execution. A sufficient coordinate description does not itself establish operator inheritance. This owner also identifies the state needed when an inherited partial transition reaches its clipping boundary. The research program owns the research queue.

The results below concern a declared arithmetic aggregation and restricted event or native-transition domains. They neither define the only possible collective NFR nor derive a universal collective law, organizational identity or physical event occurrence. A configured selector remains a supplied policy even when its decisions descend to the aggregate.

A sufficient domain for one aggregate Resonance¶

Conditional result. Two consecutive Resonance (RA) actions on a pair of nonadjacent twins can inherit exactly one registered RA on their aggregate, with unchanged factors. Internal form and capacity contrasts can both be nonzero. Paired phases, a common strict form sign and semantic kind, and duplicated consumed grammar context supply one sufficient domain. These are additional premises, not consequences of averaging or necessities of TNFR.

State, support and execution domain¶

Use the unit, undirected doubled C5 support from the native calculus: block i contains two nonadjacent nodes, each connected to both members of blocks i-1 and i+1 (indices modulo five). The comparison graph is the simple unit C5. Its node i is assigned the following aggregate channels:

x_(i,-)=M_i-d_i,       x_(i,+)=M_i+d_i,
v_(i,-)=V_i-e_i,       v_(i,+)=V_i+e_i,
theta_(i,-)=theta_(i,+)=mu_i,
Q_i=(M_i,V_i,mu_i,kind_i,context_i).

Here x=EPI is signed form and v=nu_f is capacity. The phase coordinates use an explicit common real lift of width less than both mathematical pi and the represented runtime pi. They are not form-derived angles. Each coarse edge represents one neighboring block; equal fine multiplicity, rather than summed conductance, supplies the neighbor-average identity. This statement is about RA's actual unweighted compatible-neighbor reader, not a proposed diffusion law. The exact phase argument uses the ideal circle of period 2*pi, with represented coefficients and cutoff, and compares endpoints on that common lift. Runtime modulo by represented 2*pi and trigonometric rounding are separate realization errors, not exact circle identities.

Require the following common domain at each completed-pair boundary:

Every fine form belongs to one convex interval strictly inside the shared hard-clipping rails. The interval is either wholly above the operational trigger T=1e-9 or wholly below -T. All nodes have one common nonempty epi_kind. Thus a convex blend preserves both strict sign and kind.
All primitive values and configuration consumed by the selected path pass their actual admission. In particular V_i>|e_i|, and the fine and coarse amplified capacity proposals have finite representable arithmetic. A finite continuation needs sufficient headroom at every occurrence; direct RA does not itself impose the runtime's later VF_MAX clamp.
Fine and coarse graphs share the admitted factors rho=RA_epi_diff, b=RA_vf_amplification, c=RA_phase_coupling, with 0<rho<=1, b>0, 0<c<=1, and the same admitted U3 phase limit L. Every requested block has at least one live compatible neighbor.
Both twins and their coarse node have equal, separately owned consumed grammar context: bounded trace, persistent U2 debt, prior-Coherence flag, effective local operator ordinal and retention policy. The common context must admit RA under the shared grammar policy. For example, supplied Emission (AL), Coherence (IL) history with zero debt is sufficient; it is not a necessary precursor or a derived history.

Use the strict caller enforce_canonical_grammar followed by apply_glyph, with GRAMMAR_REJECTION_MODE="raise". A supplied coarse target expands into the two fine targets consecutively. There is no interleaved action, flow, pressure preparation, phase coordinator, capacity adaptation or callback between them. Support stays fixed. Both actions retain the direct path's secondary effects; this is neither a public-class equivalence nor an atomic stage. Existing valid stored pressure can be retained, but no pressure law is inferred or refreshed by these RA requests.

Optional telemetry may be collected and compared separately. It is not included in Q: two actual fine occurrences do not become one occurrence record. Exclude custom observers that feed additional state into execution.

Same selected data, same registered action¶

On the declared common lift, the live admitted block set and its selected means are

J_i={j in {i-1,i+1}: abs(mu_j-mu_i)<=L},       J_i nonempty,
S_i=(sum_(j in J_i) M_j)/|J_i|,
eta_i=Arg_I(sum_(j in J_i) exp(i*mu_j)).

The phase lift of width less than pi makes the selected resultant nonzero and fixes its angle in the same interval. U3's wrapped differences give exactly this J_i. Both fine targets select both members of each block in J_i and no others. Consequently their selected form mean is

sum_(j in J_i) ((M_j-d_j)+(M_j+d_j))/(2*|J_i|) = S_i.

Their selected phase resultant is twice the coarse resultant, so their phase target is exactly eta_i. The two twins are nonadjacent: the first action changes none of the second's neighbors. Both consume these same selected data despite executing sequentially. The common strict form interval gives abs(S_i)>T, so every target uses the same registered capacity factor f=1+b. Common kind and convex, unclipped form mixing admit all fine identity proposals and the coarse proposal.

Applying the actual RA row to each twin and taking its mean and difference therefore gives

M_i'=(1-rho)*M_i+rho*S_i,       d_i'=(1-rho)*d_i,
V_i'=f*V_i,                    e_i'=f*e_i,
mu_i'=(1-c)*mu_i+c*eta_i,       a_i'=0.

These three mean rows and the unchanged kind are exactly one registered coarse RA with the same factors. All other blocks are unchanged. In ideal real arithmetic, writing P_i for the completed fine pair and R_i for the registered coarse action, this proves

Q(P_i(s)) = R_i(Q(s))

on the stated domain, including the consumed local grammar projection below. This is stronger than copying identical nodes: for rho<1, a nonzero internal form contrast survives, and any nonzero capacity contrast is amplified. Their event response need not be visible in the coarse primary channels for those channels to obey the same registered event rule.

Admission, context and finite composition¶

For the compared actions, sign, kind, factor and represented-arithmetic admission are premises on the whole fine domain, not deductions from a coarse pass. With these premises, paired phases identify U3 admission on both sides. An empty live J_i rejects both paths before the first fine event. Identical valid contexts identify the remaining incremental grammar decisions. Resonance is U2-neutral and has no U4 transformer/trigger requirement; it does not erase an already excessive persistent debt.

Each twin appends one RA and advances its local ordinal from k to k+1; the coarse target does the same. Persistent debt and prior-Coherence remain unchanged. The projected history is the common associated local history, not a concatenation of both traces or a sum of their ordinals. This supplies the consumed context equality needed by the partial-transition criterion.

Only completed pairs belong to this projection domain. After the first fine event, target phases and histories can differ. That intermediate state retains its fine description and pending second target; it is not the coarse endpoint. A second-target rejection outside the sufficient domain does not roll back the accepted first target.

Convex form mixing and phase motion preserve their respective common intervals, common kind and paired phases at completed-pair boundaries. Induction therefore gives the same projected successors for any finite supplied sequence of complete pairs, provided live U3 admission and output representability remain valid. Recompute J_i, S_i and eta_i after earlier completed pairs; adjacent targets consume the updated state. If the initial lift diameter is strictly less than L, phase convexity keeps every neighbor compatible throughout such a finite sequence. This stronger sufficient corollary removes the U3-renewal obligation, but not finite capacity headroom. Repeated RA fragments are not automatically closed grammatical sentences; incremental admission does not discharge unconsumed U5/U6 or complete-word closure obligations.

Exact scope and represented controls¶

The algebra above is exact-real mathematics with the specified coefficients. Binary64 reductions of two versus four selected nodes, two-stage blends, trigonometric means and capacity products need not be bit-identical. Threshold-adjacent selection/trigger decisions and overflow require their own margins. A finite coarse capacity alone cannot certify finite fine proposals. This result makes no unrestricted binary64 admission-equivalence or numerical-enclosure claim.

The shared-runtime controls freeze independent predictions before dispatch on one rational source and its all-negated form counterpart. Both have nonzero internal form and capacity contrasts, paired phases and supplied AL/IL contexts. They execute block 0 followed by adjacent block 1, using unchanged default factors. Block 0 admits two neighboring blocks; block 1 admits only the updated block 0, excluding its other neighbor by the actual U3 gate. Thus the comparison tests live partial selection rather than requiring all edges to be compatible. It verifies the mean rows, surviving contrasts, kind, local context and singleton-prefix boundary with an explicit floating residual tolerance. The controls stay in a positive non-wrapping phase chart; they do not certify runtime wrap equivalence to the ideal circle.

Separate sign, kind and persistent-debt controls keep the coarse request admitted while the second fine request rejects and the first remains committed. Their aggregate primary channels are rebuilt from the modified fine primitives. A heterogeneous kind or history has no proved aggregate map: those negative controls explicitly supply the first member's label and context. They demonstrate the insufficiency of that choice outside the theorem, not an alternative canonical projection.

Optional RA metrics expose four fine event records versus two coarse records for the two-block sequence, with twice as many selected neighbors per fine target. Stored pressure and structural time remain unchanged; these direct events create no physical temporal secants. No native selector, complete runtime call or trajectory is executed by these controls.

Why one inherited operator is not a complete collective law¶

The existing joint form/capacity calculus already separates event inheritance from the subsequent nodal row. With paired phases, write each prepared fine pressure as p_-=B+C, p_+=B-C, where C=k*d+b_v*e on the declared regular support. Its mean pressure can equal the ordinary coarse native pressure B. Nevertheless, after a common RA boost f, retaining that prepared pressure through the flow gives

fine mean form rate = f*(V*B-e*C),
ordinary coarse form rate = f*V*B.

The covariance -f*e*C generally remains nonzero in the positive event domain. This reuses the existing held-pressure result; it supplies no new pressure law. Constituent pressure peaks, normalized Si, temporal samples and native request selection have further information requirements. They do not descend merely because mean pressure or this event descends.

The admitted aggregate action consequently establishes a restricted cross-scale operator correspondence. A complete collective dynamical law, organizational identity, its maintenance or acquisition, and the occurrence of the supplied paired requests remain separate questions under the NFR ontology.

Fine Resonance does not universally inherit one aggregate Resonance¶

The complete-call map retains associated constituent state. A stronger claim would identify the admitted two-target Resonance (RA) action with one registered RA on a collective node whose capacity is the arithmetic mean V=(v_-+v_+)/2, using the same amplification factor b. This is a specified aggregation candidate, not a definition required by TNFR. Its capacity row alone has a necessary obstruction, independently of how aggregate form, phase, kind, neighbors or grammar might be completed.

Assume both constituent capacities are positive and exactly one selected signed mean activates amplification. In exact-real arithmetic with the represented factor, the admitted fine action gives

V_star=V+b*v_active/2,
0<v_active/2<V,
V < V_star < (1+b)*V,                       b>0.

These are ideal inequalities, not universal binary64 strict inequalities: an amplification can round away at a small factor or capacity. Represented separation needs its own margin, as in the default-sized control below.

One registered RA with input capacity V has only the two possible capacity outputs V and (1+b)*V, according to its own selected-mean trigger. Neither equals the fine aggregate above. Therefore no choice of aggregate trigger, retained hidden information or grammar can make these capacity rows agree while keeping this capacity map and the unchanged registered factor. This is failure of that universal operator-intertwining claim, not another claim that means omit information. It already holds for an event endpoint, before flow, coordination or capacity adaptation.

The existing composition source supplies an admitted control: v_-=7/8, v_+=9/8, V=1, and only the odd constituent activates amplification. Thus V_star=1+9*b/16, strictly between the two aggregate outputs. The control compares the existing fine response with both branches of the shared registered capacity proposal; it introduces no new source or trajectory.

When both triggers agree, this capacity obstruction disappears. Matching capacity is not sufficient: aggregate form/sign/kind, phase action, U3 and associated grammar still need their own common-domain proof under the ordered-action criterion. An effective factor b*v_active/(2*V) would reproduce this one capacity row, but would be a state-dependent collective rule, not the unchanged registered RA factor. Different aggregate capacity maps also require independent justification. The result does not exclude richer collective NFRs, formation, or all forms of operator inheritance across scales.

An actual selected native transition at uniform capacity¶

Conditional result. On a separate uniform-capacity sector, a fine doubled-C5 network and an ordinary coarse C5 have the same projected ideal native transition under the existing default selector, without supplying an operator word. Nonzero internal form contrasts are allowed. The transition retains form means, common capacity, paired primitive phases, semantic kind, associated grammar/lag context, phase gains and clock. It need not retain fine pressure normalization, Si, stability counters or complete temporal histories. This is a partial reduction of a specified runtime model, not an identification of the two complete graph states.

The proof combines the native writer domain, the conditional moment rows and the ordered-language criterion. The new obligation is descent of the actual selected transition to an ordinary coarse runtime, after discarding internal form contrast. A conditional common request or equal mean pressure alone did not prove it.

Retained state and admitted completions¶

Use the same fixed unit doubled C5 and simple C5 supports as above, with consecutive fine twins and corresponding coarse target order. Now impose one globally common positive capacity nu, interior to the runtime rails:

x_(i,-)=M_i-d_i,      x_(i,+)=M_i+d_i,       |d_i|<=D,
v_(i,-)=v_(i,+)=nu,   theta_(i,-)=theta_(i,+)=mu_i.

D>=0 is a fixed declared internal-form bound, not a fitted diagnostic. The common phase lift has width less than both mathematical and represented pi. As above, phase identities use the ideal circle with represented coefficients; runtime wrapping and reduction rounding have separate scope. Every node has the same nonempty semantic kind. At each block, retain one common, nonempty bounded grammar trace, persistent debt within its allowed capacity, prior-Coherence flag and local operator ordinal. Fine twins have separately owned copies. The shared positive retention window preserves a nonempty trace. Retain the two actual per-block lag counters a_i=since_AL and r_i=since_EN, the global phase gains, clock and runtime ordinal.

Keep the existing default mixed pressure, fresh default Si, default selector, strict grammar rejection, direct glyph path, hard clipping, default phase controller/coordinator and snapshot capacity adaptation. The form step is one held-pressure Euler interval of length delta=1/16, with the live built-in Gamma source of type none. No overridden selector, supplied word, custom pressure/integrator, optional mathematics engine or callbacks are included. Default validators still run. The fine and coarse runtime use the same declared clock, step size, policy configuration and target order.

All consumed primitive state, context, clocks and arithmetic must be admitted. Fine old rates, accelerations, physical histories and stability counts can differ; their validity is not inferred from aggregate admission. Existing temporal samples have valid times and signed forms. They are auxiliary completions of the retained state, not invented coarse observations.

For automatic REMESH, retain an admitted stable_frac history whose length is less than the configured stability window. With the specified absence of callbacks, the built-in step does not append that stream: pressure/Si, coordination, adaptation and the EPI archive have different writers. Thus the short-history guard remains invariant and the actual REMESH gate abstains. The archive itself still updates. Registering an observer that fills the stream changes this execution domain; it cannot inherit this abstention.

Let q denote the retained state above, including the short-history guard and fixed configuration. A complete coarse runtime state s_c is any admitted completion with those retained values. The claim is

q(F_f(s_f)) = q(F_c(s_c))   whenever q(s_f)=q(s_c)

on the margin domain below. F_f and F_c are the ideal real-arithmetic, common-lift maps of the declared fine and coarse native stage formulas. Actual binary64 invocations have the separate residual checks below. This does not treat q(s) as a complete checkpoint or assert that discarded outputs agree. The projected successor is independent of the admitted completion while these premises hold.

Selection and associated grammar really descend¶

Fresh Si has nonnegative normalized components. Uniform positive capacity makes its capacity term the full default normalized weight alpha, where alpha>si_hi (approximately 0.7585>0.5). Consequently every unforced base decision is Coherence (IL), even if fine and coarse pressure maxima and Si values differ. This uses the default selector, which does not apply the parametric selector's soft repetition policy.

The batch increments lags before comparison and then chooses

g_i = AL  if a_i+1 > AL_MAX_LAG,
      EN  else if r_i+1 > EN_MAX_LAG,
      IL  otherwise.

Emission (AL) has priority when both limits are exceeded. With the actual defaults the limits are five and three. These decisions depend only on retained counters and are frozen before the ordered event sweep. Duplicated nonempty grammar context admits the same choice on both twins and their coarse node. Nonempty traces avoid a coarse-zero/fine-nonzero U1a mismatch; bounded debt admits these choices without a fallback. Their direct path has no Resonance sign-identity gate. Homogeneous kind removes Reception (EN)'s otherwise constituent-dependent semantic-kind propagation.

Each constituent and its coarse representative append one selected glyph and advance the same local ordinal once, with the shared debt/prior-Coherence update. The lag successor is also the same:

AL: (a_i',r_i')=(0,min(r_i+1,EN_MAX_LAG)),
EN: (a_i',r_i')=(a_i+1,0),
IL: (a_i',r_i')=(a_i+1,r_i+1).

Neither concatenated twin histories nor a doubled local ordinal is used. The actual number of fine events remains twice the coarse count.

Mean flow and hidden contrast¶

Write N_i(z)=(z_(i-1)+z_(i+1))/2, k=w_epi and l=w_phase/pi_rep, using the resolved default pressure weights. Preparation on the common arc gives

B_i=k*(N_i(M)-M_i)+l*(N_i(mu)-mu_i),
p_(i,-)=B_i+k*d_i,       p_(i,+)=B_i-k*d_i.

Capacity contrast is zero; regular degree contrast is also zero on both supports. Thus B_i is exactly the ordinary coarse prepared pressure. It is held through this call's events and integration, not refreshed after an Emission or Reception jump.

Use the existing factors b_AL=AL_boost, m=EN_mix, h=IL_dnfr_factor. During the live ordered sweep, let Y_i be the event mean, A_i the event contrast multiplier and H_i the pressure multiplier. Their rows are

AL: Y_i=M_i+b_AL,                  A_i=1,     H_i=1,
EN: Y_i=(1-m)*M_i+m*N_i(live Y),    A_i=1-m,   H_i=1,
IL: Y_i=M_i,                       A_i=1,     H_i=h.

For Reception, earlier complete pairs have already changed the live neighbor mean. Both twins see exactly the same neighbor average as the coarse node; they are nonadjacent, so the first twin does not alter the second's inputs. Once clipping is excluded by the shared margins, these rows are precisely the ordinary coarse registered actions. At a singleton prefix the internal state/context need not belong to the paired chart.

Uniform capacity removes the covariance that obstructed the unequal-capacity flow. The shared Euler integrator gives

M_i'=Y_i+delta*nu*H_i*B_i,
d_i'=(A_i-delta*nu*H_i*k)*d_i,
mean stored form rate=nu*H_i*B_i.

The first row is the ordinary coarse held-pressure form step. The multiplier uses incoming contrast in the prepared pressure: multiplying the event contrast by a newly refreshed diffusion factor would be a different model. Old rate and physical history still determine individual accelerations and stored temporal records, but do not change this selected primary successor.

Phase gains and capacity feedback¶

Events in this sublanguage leave primitive phase and capacity unchanged. Duplicated phases give the same normalized global resultant magnitude and angle on both graphs, and the same local neighbor-resultant angle. Equal per-block glyph histories duplicate every raw count while preserving every proportion, including the disruptor fraction. The actual phase controller therefore sees the same inputs and previous gains, takes the same branch and produces the same updated gains k_G,k_L.

On the common lift its ordinary coordinator is

mu_i'=(1-k_G-k_L)*mu_i+k_G*beta+k_L*N_i(mu),
beta=Arg_I(sum_j exp(i*mu_j)).

The default nonnegative gain bounds have sum below one, so the lifted phase hull remains inside its incoming interval and fine phases stay paired. This is the actual configured relaxation step, not a new continuous phase law or an identification of capacity with angular velocity.

Capacity qualification can differ: it consumes prepared Si and held post-event pressure. The counts and write eligibility therefore need not descend. Nevertheless every immutable neighboring capacity mean equals nu, so both an eligible proposal and an ineligible no-write leave capacity equal to nu. The shared fixed-point guard preserves this equality in the represented controls too. No assumption of equal counters or equal due flags is needed for the primary capacity successor. This statement depends on global uniformity, not merely equal capacities inside each pair.

An aggregate-testable margin and its stopping boundary¶

Let (x_min,x_max) be the common form rails. Admit a step only when all incoming means, live event means and predicted flow means satisfy

x_min+D < M_i, Y_i, M_i' < x_max-D,
abs(A_i-delta*nu*H_i*k) <= 1.

All quantities in this test use retained state, configuration and the already determined lag choices; no hidden fine response is read. For the defaults 0<m<1 and 0<h<1, the common sufficient step bound delta*nu*k<=2-m implies the second inequality for all three actions. The event multipliers obey 0<=A_i<=1. Hence every fine lift with |d_i|<=D stays inside the rails at events and flow endpoints, and |d_i'|<=D. For a single Euler interval the held form interpolation also stays inside those rails. Inactive clipping justifies, rather than assumes from a single evaluated lift, the affine rows used in the proof.

The retained context, gains, common capacity and phase domain are preserved. At the successor, evaluate the next margin using the new retained state. Induction gives a unique projected transition along any finite segment for which the margins continue to hold. Margin failure stops this certificate; it does not claim that the native engine rejects or that no larger domain can work. An empty interior interval, changed callbacks/selector/support, invalid temporal evidence or nonrepresentable arithmetic cannot be repaired by a coarse pass. No all-time form-rail invariant is asserted.

Frozen complete-call controls and limits¶

The native transition controls use common capacity 1/2, nonuniform paired phases, common kind and supplied AL/IL history. Fixed lag counters select AL at block 0, EN at adjacent block 1 and IL elsewhere through the actual default machinery. Independent primitive forecasts precede every native invocation. Two fine sources have opposite nonzero contrasts and the same coarse state; one ordinary coarse source provides the comparison. Each graph executes exactly one runtime.step with delta=1/16, followed only by fresh preparation and a detached next request check. The shared existing forwarding observer captures actual stages without substituting a selector, glyph word or evolution kernel.

The chosen block 2 has B_2=k/512 and contrast k/128 on the first fine lift. Its two post-IL pressures fall on opposite sides of the default stability cutoff. The fine and coarse qualification/count/write outcomes therefore differ while all capacities stay exactly 1/2. Opposite contrasts swap constituent outcomes. This deliberately distinguishes primary closure from equality of diagnostics or complete execution state.

Mean forms, hidden contrast multipliers, held pressures, stored rates, phase gains and endpoints, context, actual lag updates, REMESH abstention and refreshed next choices have independent checks. Binary64 residuals are checked against an explicit tolerance, not promoted to bit-exact real-circle identities or a universal validated enclosure. The tests do not constitute a trajectory campaign, a source-acquisition result or a physical observation.

The theorem establishes a nontrivial lossy native transition with retained policy memory. Complete temporal histories, stability counts, fine pressure peaks, Si, sampling and telemetry remain outside q; valid completions still carry them for actual execution. Additional observables need their own descent proof. The sector contains only the default reachable IL/AL/EN actions; the separately inherited Resonance event generally leaves its uniform-capacity premise. A persistent collective NFR identity, entry into this domain and a law selecting that domain remain unproved.

Finite validity horizon under recurrent Emission¶

Conditional result. The default lag policy prevents indefinite renewal of the strict, unclipped domain of the uniform-capacity transition. Emission (AL) adds positive form recurrently. Ordered Reception (EN) and the held form flow retain a positive part of that input, while the adverse phase contribution has a finite total bound. Consequently a strict form margin must fail after finitely many calls, unless another premise fails earlier. This is a limit of that partial affine description under the specified policy, not a failure of the engine or an exclusion of collective organization in other domains.

Keep all premises of the preceding ideal transition, including the actual default selector, strict grammar, globally uniform capacity, paired phases, common kind/context, target order, no callbacks and REMESH abstention. The argument uses the ideal real-arithmetic stage formulas with resolved represented coefficients. It does not assert a validated binary64 horizon. No additional trajectory, source search or continuation law is used.

Ordered events retain positive input¶

Index calls by n=0,1,..., so M_n is the vector of incoming block means. Let N be the C5 neighbor-mean matrix, and put

z=delta*nu*k,      lambda=1-m-z,
b_AL=GLYPH_FACTORS['AL_boost']>0,
m=GLYPH_FACTORS['EN_mix'],      h=GLYPH_FACTORS['IL_dnfr_factor'].

Here k is the resolved default EPI pressure weight and delta=1/16. The proof requires lambda>0, a stronger bound than the earlier contrast nonexpansion criterion. The actual defaults satisfy it throughout the admitted positive capacity interval: using the represented VF_MAX, z_max<0.071925, 1-m>0.758546 and lambda>0.686622. These are outward rational comparisons, not identities between separately rounded constants. In particular the proof does not require h=1-m.

The complete ordered event sweep has an affine representation

Y_n=E_n*M_n+b_n.

Each Reception updates its row to a convex combination of its current value and the two live neighboring rows. Emission and Coherence (IL) have identity linear parts on form. Products of these updates give

E_n>=0,       E_n*1=1,       (E_n)_(i,i)>=1-m,
b_n>=0,       (b_n)_i=b_AL whenever block i executes AL.

Every target is updated only once. Immediately before its own Reception update its row is still the corresponding identity row, so its diagonal keeps at least 1-m; previous neighbor updates can add more. There is no factor (1-m)^5 at this stage. Similarly a block's own Emission input is not removed by later updates of other targets. Neighboring Reception may carry that positive input into additional rows.

Let H_n have entries one for AL/EN and h for IL. The independent pressure was prepared before these events, so the held Euler transition is

M_(n+1)=W_n*M_n+b_n+f_n,
W_n=E_n+z*diag(H_n)*(N-I),
f_n=delta*nu*(w_phase/pi_rep)*diag(H_n)*(N-I)*mu_n.

It is not a diffusion matrix applied to E_n*M_n+b_n. That alternative would refresh pressure from event-modified form and change the model. Since 0<h<1, the off-diagonal entries of W_n are nonnegative, its diagonal is at least lambda, and its row sums equal one. Thus W_n is nonnegative and row-stochastic. Such a matrix preserves a constant lower bound and does not increase the sup norm of an adverse forcing term. Neither the event sweep nor this argument assumes conservation of the arithmetic sum of forms; ordered Reception and target-dependent pressure retention do not supply that conservation law.

Six-call recurrence and a finite phase budget¶

The actual Emission counter increments before comparison with five and resets only on an accepted AL. With an admitted nonnegative initial lag a_i, its first AL is on one-based call max(1,6-a_i), then every sixth call. Reception never resets this counter, and AL has priority when both lags exceed their limits. Hence each block receives an AL in every six consecutive successful calls. No assumption about a periodic phase controller or an externally supplied word is needed.

Write Delta_n for the width of the inherited phase lift at call n. These three event types write no phase. The coordinator diameter bound gives

g_min=PHASE_ADAPT['kG_min']>0,       rho=1-g_min in (0,1),
Delta_(n+1)<=rho*Delta_n,
F0=delta*nu*(w_phase/pi_rep)*Delta_0,
||f_n||_infinity<=F0*rho^n,
sum_(n>=0) ||f_n||_infinity<=F=F0/g_min.

Use the stored gain floor, not its nominal pi-based expression. The shared default controller clamps the updated gain above that floor for every branch. The proof only bounds phase forcing; it does not set it to zero or identify it as an emitted form input. Gamma being zero during flow likewise does not remove the separate hybrid Emission jumps.

Within a six-call block, each coordinate has its own AL input at some position. At most five subsequent matrices follow it. Their diagonal entries retain at least

J=b_AL*lambda^5>0

of that input in its original coordinate. All other propagated emission contributions are nonnegative. Products of the stochastic matrices preserve the incoming minimum and bound the negative phase contribution in sup norm. Therefore, for any prefix that keeps the stated affine domain,

min_i M_(6K,i) >= min_i M_(0,i) + K*J
                 - F0*(1-rho^(6K))/(1-rho)
              >= min_i M_(0,i) + K*J - F.

The factor lambda^5 is a conservative self-contribution bound. It requires neither equal lag phases across blocks nor a particular Reception schedule. The phase sum may oppose the increase, but cannot offset an unlimited number of positive six-call contributions.

A strict-margin deadline and a separate closed-rail conclusion¶

Let D be the retained internal-form radius and U=x_max-D. For an initially admitted source define the exact integer bound

K_star=ceil((U-min_i M_(0,i)+F)/J),
N_star=6*K_star.

The initial strict margin ensures K_star>=1. If all N_star calls were margin-preserving, the preceding inequality would imply min_i M_(N_star,i)>=U, contradicting their strict upper margin. The first failed margin check is consequently no later than invocation N_star, unless another premise fails earlier. An event-stage or lower-rail margin may fail much earlier. Equality already exits this certificate, which is why the integer operation is a ceiling rather than a strict round-up. The expression is an upper bound, not the first-exit time or a numerical continuation through that time.

Shrinking a conservative D alone cannot sustain an indefinitely unclipped continuation. While every actual event and flow clip remains inactive, the same affine mean recurrence holds even for nonzero internal contrasts, and the means remain in the closed rails. With

K_closed=floor((x_max-min_i M_(0,i)+F)/J)+1,
N_closed=6*K_closed,

it would instead force min_i M_(N_closed,i)>x_max. The strict inequality matters for this closed-rail argument. It does not require zero fine contrast, but it also does not evaluate the runtime once clipping occurs. Clipping need not destroy every aggregate representation: the exactly paired form sector can still commute. A bounded piecewise continuation has different equations from the affine law proved here.

Exact static controls on the retained source¶

The horizon controls check the matrix mechanism with exact fractions for all 3^5 ordered AL/EN/IL assignments. They distinguish the true held-flow matrix from diffusion of the post-event state and check a negative diagonal beyond the stated monotonicity domain. Actual selector snapshots independently check lag thresholds and priority without executing event or flow sequences. Exact integer controls distinguish the strict-margin ceiling from the closed-rail strict round-up. No long native path or frozen response is replayed to obtain the bound.

For the already declared transition source, use nu=1/2, D=1/64, min M_0=3/16 and Delta_0=1/8 with unchanged defaults. Exact fractions of the resolved represented coefficients give the following displayed approximations and exact integer conclusions:

lambda ~= 0.752823447607,
J      ~= 0.019242248648,
F      ~= 0.074470176894,
K_star=46,       N_star=276,       delta*N_star=17.25,
K_closed=47,     N_closed=282,     delta*N_closed=17.625.

Thus the strict affine certificate cannot survive 276 complete ideal calls on that source. The closed-rail comparison separately excludes 282 calls with all clipping inactive and the other premises retained. These times are structural-clock durations, not laboratory seconds. The arithmetic checks do not certify accumulated floating-point or circular-lift errors; the existing one-call numerical evidence remains at its frozen horizon.

The result identifies a consequence of positive recurrent Emission in this configured native sublanguage. It establishes neither a universal TNFR lifetime nor dissolution, physical identity or failure of a clipped collective model. Valid finite transition results and their earlier admission margins remain intact.

Continuation through hard clipping with retained internal spread¶

Conditional result. The same selected uniform-capacity primary dynamics has an ideal continuation through hard clipping when each block retains its mean M_i and unsigned internal contrast s_i=|d_i|. Means alone do not determine the successor on an actual Emission (AL) clipping branch. The extended description removes independent twin labels; it retains two continuous form coordinates per block and is not another dimension-reducing collective NFR. No new operator, pressure law or executor is introduced.

Keep the support, common positive capacity, paired phases, common kind, duplicated grammar/lag context, complete-pair target order, default selector, one held Euler interval, zero Gamma and REMESH guard of the selected transition. Replace its strict unclipped margin by the exact closed-rail admission

L=x_min, U=x_max, L<U,
L<=M_i-s_i<=M_i+s_i<=U,      s_i>=0.

All other consumed state and temporal evidence must still be valid. The argument concerns ideal stage formulas with resolved represented factors; it asserts neither binary64 permutation invariance nor a validated long trajectory. The previous horizon does not locate the first clipping event or prove that any chosen boundary control is reached from the frozen source.

The exact pair projection and its branches¶

Let c(v)=max(L,min(U,v)) be the existing hard projection. For r>=0, define two observations of a projected pair:

C(a,r)=(c(a-r)+c(a+r))/2,
S(a,r)=(c(a+r)-c(a-r))/2.

They obey S>=0, L<=C-S<=C+S<=U and S<=r. Put v_-=a-r, v_+=a+r. The following branches are identities of this scalar map; overlapping equality boundaries give the same result.

Endpoint domain	`C(a,r)`	`S(a,r)`
`v_+<=L`	`L`	`0`
`v_-<=L<=v_+<=U`	`(L+v_+)/2`	`(v_+-L)/2`
`L<=v_-<=v_+<=U`	`a`	`r`
`L<=v_-<=U<=v_+`	`(v_-+U)/2`	`(U-v_-)/2`
`U<=v_-`	`U`	`0`
`v_-<=L` and `U<=v_+`	`(L+U)/2`	`(U-L)/2`

This is a derived branch description, not a second implementation of the runtime clipping contract. Its shared owner remains structural_clip. Not every algebraic branch is reachable in the selected sector: event clipping is restricted below, and the contraction bound excludes strict opposite-rail saturation by one flow step.

Ordered events and the incoming pressure association¶

At the start of a call, orient each pair for the calculation as

x_(i,-)=M_i-s_i,          x_(i,+)=M_i+s_i,
p_(i,-)=B_i+k*s_i,       p_(i,+)=B_i-k*s_i,
B_i=k*(N_i(M)-M_i)+(w_phase/pi_rep)*(N_i(mu)-mu_i).

This orientation is a representative of an unordered pair. The pressure is independently prepared by the existing map; it is not reconstructed from clipped motion. During the ordered event sweep, let Y_i,T_i be the completed event mean and nonnegative half-difference. With incoming Y_i=M_i for blocks not yet visited, the event rows are

AL: a_i=M_i+b_AL,                  A_i=1,     H_i=1,
EN: a_i=(1-m)*M_i+m*N_i(live Y),    A_i=1-m,   H_i=1,
IL: a_i=M_i,                       A_i=1,     H_i=h,
(Y_i,T_i)=(C(a_i,A_i*s_i), S(a_i,A_i*s_i)).

The two twins are nonadjacent and have identical neighboring pairs. The first twin therefore does not alter the second twin's Reception (EN) inputs. Later blocks consume the completed projected means, including earlier Emission clipping; replacing those means by unclipped event proposals would change the ordered map.

Emission has positive boost and admitted incoming forms, so only its upper clip can be active. It has three event regimes: no active clip when M_i+b_AL+s_i<=U, one-sided clipping when M_i+b_AL-s_i<U<M_i+b_AL+s_i, and full upper saturation when U<=M_i+b_AL-s_i. In contrast, Reception is a convex blend of admitted live forms because 0<m<1. It cannot actively clip outside the common rails in this ideal domain. Coherence (IL) leaves form unchanged and multiplies held pressure by h; it performs no form projection itself. The compact pair formula for IL simply returns its admitted input.

The actual held-pressure flow then gives the raw pair center and signed half-difference

u_i=Y_i+delta*nu*H_i*B_i,
v_i=T_i-z*H_i*s_i,       z=delta*nu*k,
M_i'=C(u_i,abs(v_i)),    s_i'=S(u_i,abs(v_i)).

The subtraction consumes incoming s_i, not the event spread T_i. The raw constituent values are u_i-v_i and u_i+v_i, associated with the initially lower and upper form respectively. v_i can be negative: full Emission saturation has T_i=0, yet held pressure can recreate a spread and reverse that ordering during flow. Replacing s_i by T_i would lose this response. Taking abs(v_i) in the final symmetric pair projection is legitimate; discarding the incoming association before forming v_i is not. A mid-call continuation would need to retain the incoming held pressure information explicitly.

These formulas reproduce the fine constituent map, rather than a newly refreshed pressure step. In particular the stored rates remain nu*H_i*(B_i +/- k*s_i) even when clipping changes realized motion. Neither their constituent values nor their mean generally equal the observed endpoint secants after clipping.

Complete-call closure and its limits¶

Independent twin exchanges preserve the same pairs and support. Choices remain default Coherence except for the retained lag overrides, because uniform capacity keeps fresh Si above the default upper threshold. Common nonempty grammar admits the same selected glyphs, and shared kind removes Reception's dependence on constituent kind selection. Saturated Emission still records its glyph, advances its local ordinal and resets its lag, even when it changes no form.

Neither event clipping nor integration changes primitive phases or capacity in this sublanguage. Duplicated phases and grammar proportions therefore give the same phase-controller inputs as before; common gains and the phase hull are preserved. Snapshot capacity adaptation still has the globally uniform capacity as a fixed point, independently of unequal Si, qualification and stability counters. The short-history REMESH guard also remains in force.

Let q_s retain the former primary context together with (M_i,s_i). The preceding formulas define a unique ideal primary map F_s satisfying

q_s(F_f(state))=F_s(q_s(state))

for every admitted completion in this domain. Its forms stay in the closed rails by construction. Complete histories, old rates, clipping telemetry and individual adaptation decisions remain outside this claim; their validity is still required where execution consumes them. In particular, a changed event form can restart timestamped history while a saturated unchanged twin retains its earlier samples. That difference does not affect this selected primary map, but matters for other operators. Independent twin exchange is not arbitrary relabeling, nor a quotient of every complete runtime output. The inherited execution restrictions still apply to any iterated use of the map.

The unchanged ordinary coarse runtime is the invariant slice s_i=0. It uses c(a_i) for each event and c(Y_i+delta*nu*H_i*B_i) for flow. On that slice these are exactly the extended map above. With nonzero spread, equality can hold on some branches, including the previous unclipped domain, but it does not hold universally.

A branch-specific obstruction to means alone¶

An exact family separates the descriptions under actual default factors. Use uniform incoming means M_i=U-b_AL, uniform phase, nu=1/2, delta=1/16 and the admitted common context with lag choices AL,EN,IL,IL,IL. Then every incoming B_i=0. Compare zero internal spread with s_0=epsilon, all other spreads zero, where 0<epsilon<min(b_AL,U-L-b_AL). These are conditional in-rail boundary states, not acquired sources or evaluated trajectories.

At block zero, Emission produces the pair U-epsilon,U, so Y_0=U-epsilon/2 and T_0=epsilon/2. Since the default 0<z<1/2, the held-flow endpoints are

U-(1-z)*epsilon,      U-z*epsilon.

Both are in the rails, and the successor mean is U-epsilon/2. The ordinary coarse successor is instead U. At adjacent block one, live Reception propagates the difference: its mean is below the coarse mean by m*epsilon/4; its zero incoming mean pressure does not remove it. Thus a continuously variable internal spread changes the complete-call mean with every other retained mean/context fixed. Some information about that spread is necessary on this branch. This local necessity does not assert a globally minimal coordinate system or identify an emergent NFR.

Internal spread contracts even at clipping¶

Hard projection is monotone and one-Lipschitz. The event spread satisfies 0<=T_i<=A_i*s_i, and the final projection gives

s_i'<=abs(T_i-z*H_i*s_i)
    <=max(z*H_i,abs(A_i-z*H_i))*s_i.

The default bounds established in the horizon proof give 0<z<min(1/2,1-m) throughout the positive capacity domain. For AL, the last maximum is 1-z. For EN the event never clips, so its sharper factor is 1-m-z. For IL the factor is 1-z*h. Hence, for the fixed common positive capacity of a continuation,

s_i'<=q*s_i,       q=1-z*h in (0,1),
s_(n,i)<=q^n*s_(0,i)

on every admitted ideal prefix. There is no capacity-independent strict contraction floor as nu approaches zero. The same bound and s_i<=(U-L)/2 imply abs(v_i)<(U-L)/2; strict simultaneous clipping to opposite rails cannot occur in the flow of this sector. The generic branch table above does not manufacture such an event.

This contraction concerns the two constituents within each block. It does not determine the limiting block means, establish tracking by one ordinary coarse trajectory, or prove a persistent nonuniform organization. Small local discrepancy cannot replace a cumulative comparison argument.

Static controls and evidence boundary¶

The clipping controls compare independent exact constituent formulas with the pair map on fixed partial-saturation, full-saturation and zero-spread cases, including all 32 independent twin orientations. Separate branch controls exercise the shared scalar clip contract. The family above tests a specified difference in final means and its propagation through live Reception; full saturation tests pressure-induced spread recovery and order reversal. Radius bounds are checked against constituent half-differences. Separate lower- and upper-rail flow cases check the active projection and its mean correction.

These are algebraic controls and isolated scalar/proposal checks. Native trajectory execution and frozen response replay are blocked. They do not extend the earlier one-call numerical response or certify when the original source enters clipping. The result resolves the continuation question for this configured primary model; it does not derive clipping, the selector or these constitutive choices as unique laws of TNFR.

Upper-rail limit of the selected native sector¶

Conditional theorem. Every indefinitely admitted ideal continuation of the clipped uniform-capacity sector satisfies x_(i,-) -> U, x_(i,+) -> U and therefore M_i -> U. No persistent nonzero form contrast within or between blocks survives in this sector. The conclusion follows from the unchanged positive recurrent Emission (AL), convex Reception (EN), held-pressure flow, hard rails and decaying phase forcing. It is a saturation result for this supplied policy, not an exclusion of other TNFR organizations or constitutive laws.

Keep all preceding execution and admission premises, including common positive capacity, fixed delta=1/16, paired phases in the common short lift, strict admitted grammar, complete-pair event order and REMESH abstention. The theorem is conditional on indefinite ideal admission; it does not assert indefinitely representable clocks, histories or binary64 arithmetic. No native trajectory or frozen response is evaluated here.

Fine deficits retain both pressure stages¶

Use individual constituent indices j, not only block means. Write

R=U-L>0,       y_j=U-x_j in [0,R],       D_n=max_j y_(n,j),
b=b_AL>0,     z=delta*nu*k,              lambda=1-m-z>0.

N_j denotes the mean over the four fine neighbors. Because the support has unit conductance and is regular, the EPI pressure and Reception use this same averaging map. Its phase contribution remains the admitted paired-phase term. The phase bound supplies

F_n=F0*rho^n,
F0=delta*nu*(w_phase/pi_rep)*Delta_0,       0<rho<1.

The form displacement f_(n,j) contributed by phase pressure in a held step has abs(f_(n,j))<=F_n, since its retention H_j is either one or h<1. This is a forcing bound, not an assumption that the phase term vanishes at a finite call. No internal-spread approximation is used in this proof.

The event sweep acts on deficits as follows:

AL: y_j^E=max(y_j-b,0),
EN: y_j^E=(1-m)*y_j+m*N_j(live event deficits),
IL: y_j^E=y_j.

Every target is visited once. Emission cannot increase a deficit, and Reception is convex, so every live event deficit is at most the incoming D_n. The subsequent held-pressure flow is exactly

y_j'=clip_[0,R](y_j^E-z*H_j*y_j+z*H_j*N_j(incoming y)-f_(n,j)).

The neighbor values in the event and flow terms belong to different stages. Clipping an Emission event does not refresh the prepared pressure.

One-call bounds without a monotone-map assumption¶

The complete clipped map is not generally order-preserving. For example, use the default rails [-1,1], zero internal spread, uniform phase and neighbors of block zero at form zero and the default positive boost. Compare its incoming forms M_a=U-b/2 and M_b=U-b/4 under the same AL request. Both events saturate at U, while their incoming mean pressures are -k*M_a and -k*M_b. The resulting in-rail flow forms are U-z*M_a and U-z*M_b; increasing the input decreases that output by z*b/4. Thus the unclipped stochastic matrix cannot simply be reused as a global clipped-map comparison.

Instead, bound each actual scalar row. At Emission, separate the two branches y_j>=b and y_j<b before final projection. They give

y_j'<=max(D_n-b,z*D_n)+F_n <= a*D_n+F_n,
a=max(1-b/R,z) in (0,1).

Indeed, the first branch is bounded by (1-z)*D_n+z*D_n-b, and the second by z*D_n. Since D_n<=R, D_n-b<=(1-b/R)*D_n. The maximum is nonnegative, so projection to [0,R] preserves the upper bound, including when the unprojected deficit is negative.

For Reception and Coherence respectively, the same stage distinction gives

EN: y_j'<=lambda*y_j+(1-lambda)*D_n+F_n,
IL: y_j'<=(1-z*h)*y_j+z*h*D_n+F_n.

Both coefficients retaining the target's incoming deficit are positive and at least lambda: 1-z*h>=1-m-z. The right-hand sides are nonnegative, so the final projection is again harmless for these upper bounds. In particular all three cases imply

D_(n+1)<=D_n+F_n.

This estimate allows temporary increases in the largest deficit; it does not assume that every form increases at every call.

A strict improvement in every six-call block¶

Fix a block of six calls starting at n, and define its accumulated forcing allowance

E_j=sum_(r=0)^(j-1) F_(n+r),       E_0=0.

The running maximum is bounded by D_(n+j)<=D_n+E_j. The actual lag policy gives each constituent exactly one accepted Emission in those six calls, even from an arbitrary admitted initial lag. If its Emission is at offset j, its next deficit satisfies

y_(n+j+1)<=a*(D_n+E_j)+F_(n+j)
         <=D_n+E_(j+1)-(1-a)*D_n.

There are at most five later calls in the block, and none is another Emission at that target. For either of the remaining choices, a gap g below the running allowance is retained by at least lambda*g. The next forcing contribution is included in the next allowance. Thus every constituent ends the block at most D_n+E_6-(1-a)*lambda^5*D_n, and

D_(n+6)<=Q*D_n+sum_(r=0)^5 F_(n+r),
Q=1-(1-a)*lambda^5 in (0,1).

This argument uses the true per-target lag recurrence and scalar row bounds. It does not require synchronized lag phases, conservation of total form, or monotonicity of the complete transition. In particular it remains valid when an Emission saturates and held pressure subsequently moves the same form away from the rail.

Convergence along an indefinitely admitted ideal continuation¶

Set tau=rho^6 and A=F0*sum_(r=0)^5 rho^r. Iterating the block bound gives, for K>=1,

D_(6K)<=Q^K*D_0+A*sum_(j=0)^(K-1) Q^(K-1-j)*tau^j.

For Q!=tau the sum is (Q^K-tau^K)/(Q-tau); when Q=tau it is K*Q^(K-1). Both tend to zero because 0<Q,tau<1. The K=0 sum is empty. For intervening calls 0<=r<6,

D_(6K+r)<=D_(6K)+F0*rho^(6K)*sum_(j=0)^(r-1) rho^j.

Consequently D_n -> 0 along the entire admitted continuation, proving the stated constituent and block-mean limits. The internal spread contraction is consistent with this result but was not needed to replace fine dynamics by an approximate mean law.

The phase coordinator preserves nested closed lift intervals with width at most rho^n*Delta_0. Their intersection is one angle, so all primitive phases converge to that common lifted angle. Common capacity remains fixed. The specified pressure therefore tends to zero as both form and phase differences vanish.

At exactly uniform upper-rail form and common phase, the primary nodal coordinates remain stationary under each selected AL/EN/IL event and held flow. The nominal Emission proposal still adds b; its realized clipped form increment is zero. Operator occurrences, lag counters, runtime ordinals and the structural clock can continue advancing. The theorem neither identifies the complete execution state with a fixed point nor erases its retained policy memory.

Static evidence and research consequence¶

The limit controls check the two Emission branches, the separate incoming/live neighbor bounds, the nonmonotone saturated control, six-call allowance propagation and both forms of the geometric convolution. These are exact static comparison calculations, not executed histories. The upper-rail primary state has separate stationary-coordinate checks. Runtime and integrator execution are blocked, and earlier evaluated responses remain unchanged.

This closes the limiting-organization question for this configured sector. It excludes a persistent nonuniform form or primitive-phase pattern there, but not finite transients, retained support, semantic identity or policy memory. It gives no finite saturation time or laboratory prediction and does not certify asymptotic binary64 behavior. A sustained organization outside this conclusion must fail at least one of the declared premises; changing one merely to obtain a preferred result would require a separate model and justification.

Capacity adaptation preserves saturation in a nonzero neighborhood¶

Conditional theorem. The preceding upper-rail limit is robust to a derived nonzero range of initial fine capacities, with the actual capacity adaptation enabled. All adaptation gates eventually qualify permanently; capacities converge to a common positive value, fine forms to U, and primitive phases to one common lifted angle. Gate qualification is proved before consensus is used. This excludes persistent form/phase contrast in this larger sector; it does not identify particles or a universal TNFR law.

Retain the preceding doubled unit C5, direct native default policy, delta=1/16, zero Gamma, hard rails, paired phases in the common short lift, common kind, duplicated grammar/lag contexts, complete-pair event order, strict admission and REMESH abstention. Replace uniform capacity by

0 < v_- <= nu_j(0) <= v_+ <= V < VF_MAX,
epsilon=v_+-v_-,
alpha*v_-/v_+ > si_hi.

Here V is a fixed conservative upper envelope, alpha the resolved Si capacity weight, and si_hi the common default selection/adaptation threshold. Capacities may differ within pairs. Retain all consumed fine state, including valid nonnegative stability counts. This is a fine-state theorem, not an extension of the smaller uniform-capacity quotient.

Use the resolved represented coefficients as exact constants in the ideal model, rather than replacing them by intended decimal or symbolic values:

c=w_vf>0,  k=w_epi>0,  ell=w_phase/pi_rep,
Z=delta*V*k,  lambda=1-m-Z>0,
a=max(1-b/(U-L),Z),  Q=1-(1-a)*lambda^5,
C(V)=c*(1+6*Z/(1-Q)).

The default parameters give 0<Z<min(1/2,1-m) and 0<a,Q<1 throughout this capacity range. The additional sufficient restriction is strict:

C(V)*epsilon < EPS_DNFR_STABLE.

It is not a necessary boundary of convergence. Failure of this sufficient inequality leaves the behavior undecided.

Invariant selection with arbitrary eligible adaptation subsets¶

The actual capacity adapter proposes (1-mu)*nu_j+mu*N_j(nu) from one immutable snapshot at eligible nodes and leaves the other capacities unchanged. Its admitted coefficients make every proposal convex. Thus any subset of eligible writes preserves the initial capacity hull; this does not yet prove consensus or conservation of the initial mean.

Fresh Si has nonnegative components and satisfies Si_j>=alpha*nu_j/max_l(nu_l)>=alpha*v_-/v_+>si_hi. The default selector therefore proposes Coherence (IL); the retained lags override it only by Emission (AL) or Reception (EN). These three direct events preserve capacity. This establishes an induction across complete calls: selection preserves the hull before adaptation, and adaptation preserves it afterward. No later selection or successful adaptation is assumed in advance. Unequal capacities or stability counts do not break paired choices in this high-Si sector. Phase coordination consumes no capacity and preserves the inherited paired-phase contraction.

A pressure bound before permanent qualification¶

At each call, the held flow uses its own incoming capacity, with z_(n,j)=delta*nu_(n,j)*k<=Z. The pressure capacity channel is the signed neighbor contrast c*(N_j(nu)-nu_j), not a normalized contrast. Hull preservation bounds its magnitude by c*epsilon under every eligible mask. Keep D_n=max_j(U-x_(n,j)) and the preceding phase diameter bound Delta_n<=rho^n*Delta_0. The two form forcing allowances are

F_n=delta*V*ell*Delta_0*rho^n,
G=delta*V*c*epsilon.

The pressure retained by a selected event is multiplied by H_j=1 or h<1. The fine-deficit proof therefore applies with variable z_(n,j)<=Z: Emission gives at most a*D_n+F_n+G; subsequent Reception or Coherence retains a target's gap below the running allowance by at least lambda. Every constituent still has one accepted Emission in every six-call window. Consequently, for every starting call n,

D_(n+6) <= Q*D_n + sum_(r=0)^5 F_(n+r) + 6*G.

Iterate on each of the six residue classes. The initial term and geometric phase convolution vanish; the constant forcing sum is bounded by 6*G/(1-Q). Thus

limsup_(n->infinity) D_n <= 6*G/(1-Q).

No additional allowance for intervening calls is needed because the same recurrence holds on every residue class. Neither capacity convergence nor a nonempty eligible mask has been used.

The freshly prepared pressure satisfies

max_j abs(p_(n,j)) <= k*D_n+c*epsilon+ell*Delta_n.

Indeed, all incoming forms lie in [U-D_n,U], so their difference from a neighbor mean is at most D_n, not 2*D_n. Regular support removes the topology contrast. Applying the preceding bound gives

limsup_(n->infinity) max_j abs(H_j*p_(n,j)) <= C(V)*epsilon
                                                   < EPS_DNFR_STABLE.

This is the pressure actually consumed by adaptation: prepared pressure after its glyph retention, without refresh after flow or phase coordination. The adapter also consumes the incoming prepared Si, whose strict floor already holds. The strict limiting pressure margin implies that from some finite call onward both gates qualify at every node on every call. An equality at the threshold would not suffice: the abstract sequence EPS_DNFR_STABLE+1/(n+1) approaches the threshold without ever passing. This is a counterexample to that limiting inference, not a native response.

Permanent averaging and a vanishing forcing remainder¶

Qualification increments each stability count; a successful write does not reset it. After at most VF_ADAPT_TAU further qualifying calls, all nodes therefore update on every call. With mu=VF_ADAPT_MU and the fixed fine neighbor-mean matrix N, the ideal recurrence becomes

nu_(n+1)=P*nu_n,        P=(1-mu)*I+mu*N.

On the doubled unit C5, P is doubly stochastic. The entries of P^2 in each row have four classes:

Target relative to the row node	Count	Entry
Same node	1	`(1-mu)^2+mu^2/4`
Its twin	1	`mu^2/4`
Adjacent blocks	4	`mu*(1-mu)/2`
Nonadjacent blocks	4	`mu^2/8`

For the actual default 0<mu<4/5, all entries are at least eta=mu^2/8>0. Subtracting eta from every entry leaves a nonnegative matrix with each row sum 1-10*eta. Its constant component cancels in differences, proving directly

osc(P^2*nu) <= (1-10*eta)*osc(nu),       0<1-10*eta<1.

Hence capacities converge to one positive value in the initial hull. Double stochasticity preserves the mean once full updating starts. The limit is that mean, not necessarily the initial mean: earlier partial writes need not conserve total capacity. No externally selected target capacity or physical scale follows.

Now the actual width epsilon_n=osc(nu_n) tends to zero. Replace G in the six-call bound by the actual per-call allowance G_n=delta*V*c*epsilon_n. Both phase and capacity forcing vanish. For each residue class, a recurrence with uniform Q<1 and a vanishing nonnegative remainder tends to zero: split its geometric convolution into a finite initial part and a tail bounded by the remainder supremum divided by 1-Q. Therefore D_n->0. The inherited nested phase intervals converge to one angle, and the complete prepared pressure tends to zero.

An explicit nonzero neighborhood without source search¶

Set v0=1/2 and use the conservative envelope Vbar=1. Compute Cbar from C(Vbar) above and define

r_star=min(v0/2,
           v0*(alpha-si_hi)/(2*(alpha+si_hi)),
           EPS_DNFR_STABLE/(4*Cbar)).

Every term is positive under the unchanged defaults. For any fine initial capacities in [v0-r_star,v0+r_star], the lower endpoint is positive and the upper is at most 3/4<Vbar<VF_MAX. The second term leaves a strict Si margin; the width is at most 2*r_star, so the limiting pressure bound is at most half the stability threshold. The construction admits arbitrary fine forms within the inherited rails and retains the other phase, grammar, history and execution premises. It selects no response, trajectory, natural scale or new law.

The capacity robustness controls check the derived threshold with exact represented coefficients, masked hull and mean behavior, isolated actual counter/write transitions and the exact averaging contraction. They do not execute a native trajectory or replay earlier frozen evidence. The proof is conditional on indefinitely admitted ideal execution. Finite arithmetic can reach rounding stasis; indefinite counter/clock representability, a finite consensus time, full runtime stationarity, collective formation and physical identification remain outside this result.