Native collective calculus

This owner combines conditional moment identities, finite admission margins and bounded collective observations for the general native operator runtime. The source and branch studies own the underlying preparations and counterexamples. These are detached mathematical descriptions of the declared execution path, not a new runtime or a derived occurrence law. The aggregate transition owner separately proves a restricted lossy reduction under the actual uniform-capacity native policy, retaining its admission margins and discarded-output boundary. The research program remains the sole research queue.

Conditional collective calculus for five native operators

Emission (AL), Reception (EN), Coherence (IL), Resonance (RA) and Transition (NAV) share an exact conditional event-plus-held-flow description on doubled C5. Unequal constituent forms are allowed. The event mean and internal contrast have distinct roles; the complete native state also retains constituent-associated memory. This synthesis reuses the block descent, four-operator extension, Transition map and correlated Reception results. It does not assert that the default selector realizes every assignment of these operators.

Domain and request origin. Keep the block-descent execution premises: unit undirected doubled C5, consecutive nonadjacent twins, whole-pair order, default mixed pressure/Si and direct glyph execution, strict live grammar, hard clipping, live built-in zero Gamma and one Euler substep delta=1/16. No external nodes, custom callbacks, optional mathematics engine or optional Resonance tracking are included. Automatic REMESH must independently abstain; insufficient retained stable_frac history is one sufficient reason. It is not removed from the law. Each node receives exactly one effective request.

Forms are finite admitted signed scalars (including admitted uniform-real BEPI), with common nonempty semantic kind. Capacities are positive and paired; primitive phases are paired and admit an inherited common lift of width less than pi. Every proposed event and flow form lies strictly inside its rails. Post-Resonance capacities also lie inside their rails: integration uses the amplified value before the later clamp. Consumed histories, counters, rates, clocks, normalized diagnostics and produced arithmetic retain their admission. The equations below are ideal-real identities over represented coefficients, not universal binary64 enclosures.

Both effective requests in each pair must agree and pass their own live grammar, identity and operator gates. Retain their actual origin, including selection preparation and the increment-before-comparison AL/EN lag priority. A supplied paired transcript can test the algebra conditionally; it is neither a default-selector witness nor a derived occurrence law. Prepared requests can differ within a pair because its pressures and Si differ. Rejection is sequential: a successful first twin or earlier pair is not rolled back by a later rejection. The theorem concerns successful complete calls.

Prepared pressure. Write each incoming pair as

x_(2i)=M_i-d_i,                  x_(2i+1)=M_i+d_i,
N_i(z)=(z_(i-1)+z_(i+1))/2,
B_i=w_phase*(N_i(theta)-theta_i)/pi_rep
    +k*(N_i(M)-M_i)+b*(N_i(V)-V_i),
p_(2i)=B_i+k*d_i,                p_(2i+1)=B_i-k*d_i.

Here k=w_epi>0, b=w_vf, and the common arc justifies the neighbor-resultant midpoint. Regular support makes degree contrast zero. Preparation fixes this independent pressure before any event. Its actual global normalization peak is P=max_i(abs(B_i)+k*abs(d_i)), with divisor one only when P=0. Si also consumes individual absolute pressure, paired capacity divided by its maximum, and circular phase offset. The diagnostics are generally non-affine; means alone do not determine selection or stability qualification.

One ordered event table. Initialize live means, phases and capacities with M,theta,V, then process blocks 0..4. Each block is still at its incoming state when its pair begins; earlier pairs have updated its neighbors. Denote its event mean by Y_i, event contrast by A_i*d_i, event capacity by U_i, and event phase by phi_i. Let H_i be its pressure multiplier and g_i,zeta_i the extra pressure mean/contrast. The following are moment identities, not a parallel operator registry:

Registered action	Y_i	A_i	H_i	g_i	zeta_i
Emission (AL)	M_i+e	1	1	0	0
Reception (EN)	(1-m)*M_i+m*N_i(live Y)	1-m	1	0	0
Coherence (IL)	M_i	1	h	0	0
Resonance (RA)	(1-r)*M_i+r*S_i	1-r	1	0	0
Transition (NAV)	M_i	1	1-eta	eta*sigma_i*V_i+xi_bar_i	xi_delta_i

Use the actual default factors e=AL_boost, m=EN_mix, h=IL_dnfr_factor, r=RA_epi_diff, eta=NAV_eta. Except for Resonance, U_i=V_i and phi_i=theta_i. Live neighbor averages include earlier event changes but never refresh B_i or its incoming contrast. Nonadjacent twins share the same live neighbor input, so their two fine proposals yield these moments. Equality holds after each complete pair, not at its singleton prefix.

For Resonance, with the actual U3 limit L and current paired phases, define

J_i={j in {i-1,i+1}: abs(wrap(phi_j-phi_i))<=L},
S_i=mean_(j in J_i)(live Y_j),
alpha_i=Arg(sum_(j in J_i) exp(i*phi_j)),
phi_i=theta_i+c*wrap(alpha_i-theta_i),
U_i=(1+a*I(abs(S_i)>T))*V_i.

Here c=RA_phase_coupling, a=RA_vf_amplification, and T is the represented runtime amplification trigger. Require nonempty J_i; the common arc then makes the selected resultant defined. Recompute this subset at the live prefix. Pairwise phase equality selects whole adjacent blocks, but it does not justify selecting every neighbor or freezing the preparation-time subset. Both fine sign identities must pass, not merely the mean sign identity. Zero is neutral under the actual sign contract; the common kind remains explicit. Amplification reads the selected signed mean, not mean absolute constituent magnitude. Outside this domain an empty subset rejects, and an undefined resultant requires the engine's separate phase policy; neither is an arbitrary assigned angle.

For Transition use the default nonstrict random path, with common strict prepared pressure sign sigma_i:

sigma_i*B_i>k*abs(d_i),
xi_bar_i=(xi_(2i)+xi_(2i+1))/2,
xi_delta_i=(xi_(2i+1)-xi_(2i))/2,
abs(xi_bar_i)+abs(xi_delta_i)<=NAV_jitter.

Each innovation is its actual target-associated stream input. Both fine nonidentity gates require eta*(sigma_i*V_i-p_n)+xi_n!=0. Common sign alone does not prove them. The native zero-pressure branch takes positive sign; it is outside this strict-sign sector, not an engine defect. Opposite signs require the two sign contributions separately. Averaging innovations gives the displayed observation, not a replacement RNG law or permission to resample one collective draw.

Held flow and sufficient means. Every row has

q_bar_i=H_i*B_i+g_i,
q_delta_i=-H_i*k*d_i+zeta_i,
M_i_next=Y_i+delta*U_i*q_bar_i,
d_i_next=(A_i-delta*U_i*H_i*k)*d_i+delta*U_i*zeta_i,
rate_bar_i_next=U_i*q_bar_i,
rate_delta_i_next=U_i*q_delta_i.

Subtract the associated prior stored rate and divide by delta to obtain each acceleration moment. The held pressure contains incoming d_i, not event contrast A_i*d_i; replacing the contrast factor by A_i*(1-delta*U_i*H_i*k) changes the model. Post-event capacity U_i is used before later adaptation. Integration does not recompute pressure or Si.

Conditionally on the same admitted paired requests, common Transition signs, innovation means and retained phase/capacity inputs, form/rate means factor through incoming means. Live Resonance subsets, amplification and phase rows then agree by ordered induction. This is useful observation sufficiency; internal contrasts still determine fine reconstruction and joint admission. It does not eliminate histories, old rates, counters, stream association or the actual grammar context. A finite composition uses these rows only while each successor is re-admitted. It is not a complete bare-node stochastic realization or an invariant five-operator sector.

Temporal records and phase policy. Each same-time event form jump is

jump_(2i)=Y_i-M_i-(A_i-1)*d_i,
jump_(2i+1)=Y_i-M_i+(A_i-1)*d_i.

History restarts precisely for the constituent with a realized nonzero jump; its flow endpoint is then appended with the existing four-entry cap. For an EN pair with live neighbor mean n and M_i=n+d_i, the even form already equals n and is unchanged, while the odd form changes by -2*m*d_i. The analogous RA witness has m replaced by r, with its separate admission. Thus a common glyph need not give common temporal-reset status. Do not replace the two admitted histories by one block history. Prior rate association also remains necessary, as the Reception same-form/different-rate witness shows.

The post-event phase coordinator uses the actual full fine glyph histories to compute disruptor load, and the retained global gains and phase resultant. Unequal traces may not be replaced by one representative trace. Given this actual load, paired phases have identical local/global updates. Default gain rails keep the common lifted arc convex and phases paired. This retains primitive circular phase and its inherited lift, not a phase derived from form. Different gain/load branches are allowed; their inputs must remain available.

Capacity exit and hidden counters. Let s_n be the incoming individual stability count and use fresh prepared Si with held post-event pressure:

chi_n=I(Si_pre_n>=si_hi and abs(q_n)<=EPS_DNFR_STABLE),
s_n_next=s_n+1 if chi_n else 0,
due_n=I(s_n_next>=tau),
W_i=N_i(U),
v_n_next=U_i+mu*due_n*(W_i-U_i),          n in {2i,2i+1}.

W comes from the immutable post-event capacity snapshot; fixed-point and hull guards remain in force. In this unclipped ideal-real domain the successor capacity half-contrast is exactly

dV_i_next=mu*(due_(2i+1)-due_(2i))*(W_i-U_i)/2.

Consequently capacity pairing survives exactly when this product vanishes. Equal due flags suffice, but are not necessary at a neighbor fixed point. Zero mixing also annihilates the product algebraically; default mixing is positive. Equal no-write flags do not imply equal qualification or counts: from zero counts, one qualifying constituent can reach one while the other stays zero, with neither write due. A later unequal due decision with nonzero mixing and W_i!=U_i leaves the paired-capacity domain. This is a declared exit, not a contradiction of the form map already evaluated with U_i. Retain both counters even when the capacity values happen to remain paired.

The analytical controls compare independently written fine-node sweeps and collective rows, including live neighbor inputs and held-pressure timing. Detached boundary controls retain Resonance subset/identity/amplification, Transition sign/nonidentity, clipping, split temporal resets and split qualification/write decisions. Production runtime, pressure, selection, operator and integrator evaluation are blocked. The result supplies a reusable conditional calculus without a new trajectory or engine API. It does not prove autonomous request selection, repeated-call retention, formation or physical identification.

Finite internal-contrast retention with admission margins

The five-operator calculus gives a finite retention criterion once its actual request and admission boundaries are included. A contracting form contrast alone is insufficient: normalized selection, sign identity, temporal resets and adaptation have independent boundaries. This result concerns transverse form differences at fixed block means, primitive phases and capacities, under that calculus's complete law. It supplies neither an invariant domain for all time nor autonomous formation.

Reference and comparison. Fix a finite independently admitted reference segment, its complete state, structural clock, whole-pair order, grammar, request origin and post-event policies. Its physical form is pair constant, with means M_i; retain every constituent's own history, rate and counter. The comparison has the same means, phases and capacities but forms M_i-d_i,M_i+d_i, with abs(d_i)<=D_i. Its actual temporal histories must end at these forms and pass their own admission. Corresponding glyph traces, debt, prior-Coherence flags, operator ordinals, AL/EN lags, stability counts and global phase gains agree with the reference; they need not agree between the two constituents. Initial rate differences remain associated and are used to reconstruct acceleration. All other calculus premises still apply.

For Transition (NAV), keep each declared innovation mean xi_bar_i and bound the associated half-difference by Z_i, with abs(xi_bar_i)+Z_i<=NAV_jitter. The reference's duplicated mean input is an algebraic comparison within that amplitude domain. It is not automatically a seed-realizable native path. An actual stochastic claim must independently retain and admit each target's stream and draws; do not replace them by a new collective RNG. The concrete reference below uses only Resonance (RA) and Coherence (IL), so it consumes no innovations. Other conditional words retain their supplied origin unless native occurrence is separately proved.

Prepared selection margins. Let B_i be the reference's independently computed pressure, P0=max_i(abs(B_i)) and Dmax=max_i D_i. For P0>0, the exact correlation between pair pressures gives

p_(2i)=B_i+k*d_i,                p_(2i+1)=B_i-k*d_i,
P=max_i(abs(B_i)+k*abs(d_i)),
P0<=P<=P0+k*Dmax,
rho0_i=abs(B_i)/P0,
abs(abs(p_n)/P-rho0_i)<=E_i=k*(D_i+Dmax)/P0.

To prove the last bound, separate the numerator change, at most k*D_i, from the denominator change, at most k*Dmax; use P>=P0 and abs(B_i)<=P0. This does not assume a fixed maximizing node. Intersect [rho0_i-E_i,rho0_i+E_i] with [0,1] to obtain rho_lo,rho_hi. All nonpressure Si terms are identical, so abs(Si_n-Si0_i)<=gamma*E_i. Equivalently, the sharper endpoints are Si0_i-gamma*(rho_hi-rho0_i) and Si0_i-gamma*(rho_lo-rho0_i); call them S_lo,S_hi. These are sufficient enclosures, not independent freedom for Si and pressure.

Retained lag counts first determine the actual higher-priority requests: increment both counts, then force Emission (AL) when its strict lag limit is exceeded, otherwise Reception (EN) when its strict limit is exceeded. Matching counts proves the same override; it does not waive prepared-state admission. Without an override, sufficient conditions for the default selector are:

Reference request	Sufficient comparison inequalities
Coherence (IL)	S_lo>=si_hi
Resonance (RA)	S_lo>si_lo, S_hi<si_hi, rho_hi<=dnfr_hi
Transition (NAV)	S_lo>si_lo, S_hi<si_hi, rho_lo>dnfr_hi

These reproduce the actual precedence and strictness; in the intermediate Si band, equality at the pressure threshold selects Resonance. The default path does not apply the parametric selector's score, acceleration decision, hysteresis or soft filters. It still admits and normalizes old acceleration. A possible low-Si request, Mutation abstention or unresolved interval exits this sufficient five-request certificate. It is not silently substituted by Coherence. Original grammar records remain admitted, and live grammar still checks each effective request. No threshold is retuned to obtain agreement.

Live event margins. At block i, previous complete pairs already have their reference means. Consequently all live EN neighbor means, RA selected means and paired phase inputs agree exactly. In particular, a zero selected mean can stay exactly zero under these correlated form perturbations; an arbitrary independent perturbation estimate would discard that information. Live U3 subsets, resultant angles and capacity amplification branches agree with the reference, including an inactive amplification boundary. They must be recomputed at each reference prefix, rather than frozen at preparation.

Use Y_i,A_i,U_i,H_i from the shared event rows. The following sufficient conditions admit both fine proposals:

• Resonance: abs(M_i)>D_i, M_i*Y_i>0 and abs(Y_i)>abs(A_i)*D_i, with the common retained kind and admitted U3. These nonzero sign margins are sufficient, not necessary. A sign-neutral zero reference form or proposal does not itself give a positive transverse admission radius.
• Transition: abs(B_i)>k*D_i retains the common strict sign sigma_i. With G_i=eta*(sigma_i*V_i-B_i)+xi_bar_i, require abs(G_i)>eta*k*D_i+Z_i. This keeps both fine pressure changes nonzero; common pressure sign alone does not do so.
• Form rails: the incoming interval M_i+[-D_i,D_i], event interval Y_i+[-abs(A_i)*D_i,abs(A_i)*D_i] and the flow interval below must all lie strictly inside the admitted rails. Reference post-event capacities and the later capacity proposals must also remain in their declared domain.

No pair transaction is inferred. The margins admit each singleton prefix; failure of a later check does not imply native rollback of an earlier action.

Contrast propagation. Set Z_i=0 outside Transition. Prepared pressure is held through the event and form flow, so

lambda_i=A_i-delta*U_i*H_i*k,
d_i_next=lambda_i*d_i+delta*U_i*xi_delta_i,
D_i_next=abs(lambda_i)*D_i+delta*U_i*Z_i.

The input half-difference is zero for the other four actions. For a finite sequence of admitted calls, write a_(t,i)=abs(lambda_(t,i)) and b_(t,i)=delta*U_(t,i)*Z_(t,i). Induction gives

D_(N,i) = (product_(t=0..N-1) a_(t,i))*D_(0,i)
          +sum_(s=0..N-1) b_(s,i)*product_(t=s+1..N-1) a_(t,i).

An empty product is one. If throughout this finite admitted segment a_(t,i)<=a<1 and b_(t,i)<=b, then D_(N,i)<=a^N*D_(0,i)+b*(1-a^N)/(1-a). This is a conditional finite bound, not a proof that the required branches persist indefinitely. Transition differences supply forcing; even with contracting multipliers, a nonzero forcing budget need not yield convergence to exact constituent equality. Positive capacity alone does not give a horizon-independent rate: some multipliers approach one as capacity approaches zero.

Memory and capacity margins. Let J_i=Y_i-M_i. If J_i!=0, the inequality abs(J_i)>abs(A_i-1)*D_i makes both event jumps nonzero, preserving the reference's reset status. If J_i=0 and A_i=1, both jumps vanish identically. If J_i=0 and A_i!=1, an unknown nonzero contrast can change the reset mask; the matching-history certificate abstains unless this is separately resolved. It does not discard or replace the fine histories. Admitted histories then restart/append by the actual rules, with the existing retention cap.

After the event, write Q_i=H_i*k*D_i+Z_i. The held fine pressures lie in q_bar_i+[-Q_i,Q_i]. The adaptation qualification is certified true when S_lo>=si_hi and abs(q_bar_i)+Q_i<=EPS_DNFR_STABLE; it is certified false when S_hi<si_hi or abs(q_bar_i)-Q_i>EPS_DNFR_STABLE. Otherwise this bound leaves qualification unresolved. The original pressure/Si correlation may support a sharper independent proof; an unresolved enclosure is not an observed execution failure.

Certified matching qualification and the retained individual counts give the same next counts and due decisions as the reference. The immutable capacity snapshot and neighbor means therefore give its same capacity proposals. Reference pairing must itself satisfy the previously derived mu*(due_odd-due_even)*(W-U)=0 condition. Equal capacities alone never replace the counts. Identical glyph traces and global gains give the same phase controller load, branch and updates, preserving the paired phases. REMESH abstention and all requested output/history records remain separately checked.

Finite induction and meaning of failure. At each call, certify prepared selection first, then every live event, flow rails, temporal update and adaptation. Update the reference and contrast bound only after these checks. The shared means, phases, capacities and retained policy context then supply the next induction premise. Store the earliest unresolved boundary if any; do not extrapolate the formula beyond it. Such an outcome rejects this certificate, not every possible collective description. Actual rejection, changed selection and a conservative inconclusive bound remain distinct.

An admitted box with unequal constituents. Reuse the existing live Resonance prefix source with q=15/128, means (q,q,-q,-1/2,-q), capacities (1/4,1/4,1,1,1), zero phases and its original supplied AL/IL/EN context, lags, rates and zero stability counts. Vary only five independent signed half-contrasts in [-1/4096,1/4096]. This is a different domain from unrestricted independent form perturbations: every neighboring complete pair retains its mean.

Let epsilon=k/4096. Exact represented-coefficient bounds give P0>0.0701, epsilon<0.000045 and ratio error below 0.0013 on the whole box. For this source, direct ratio endpoints

max(0,abs(B_i)-epsilon)/(P0+epsilon)
    <=rho_n<=min(1,(abs(B_i)+epsilon)/P0)

give RA ratios below 0.007, RA Si strictly between 0.4307 and 0.4309, and IL Si above 0.94. The original lags force neither AL nor EN. The actual default requests are therefore RA/RA/IL/IL/IL uniformly, before any response is evaluated. Fine corner/axis controls check the algebra independently; these uniform rational inequalities, not corner sampling of non-affine Si, admit the complete box.

The first live RA neighbor mean remains exactly zero. The second becomes -r*q/2, so the first amplification stays inactive and the second active throughout the box. Both fine proposals exceed 0.09, retain their original positive sign/kind, and have event jumps below -0.018. Coherence has no form jump. All event/flow forms have absolute value below 0.6, and all post-event capacities remain interior. Adaptation qualification is false at the RA targets because Si is too low, and false at IL targets because held pressure magnitude exceeds 0.026; thus all counts remain zero. The matched traces preserve phase load/gains and zero phases. No innovation is consumed, and the separately insufficient REMESH history still abstains.

The five contrast factors are positive and strictly below 0.992 (approximately 0.8379833,0.8378694,0.9913168,0.9913168,0.9913168). The comparison consequently retains exactly the reference's terminal block means, phases and capacities, with a strictly smaller bound on internal form differences. Fine pressures, histories, rates and accelerations keep their constituent association. This is a one-call ideal-real box certificate using an existing source; it makes no claim that its successors retain those requests indefinitely or that every box member is a newly acquired state.

The zero-pressure boundary. The positive-peak hypothesis cannot be discarded from the general ratio estimate. Set phases to zero, V=(1/4,1/4,1,1,1) and M_i=-(b/k)*V_i. All forms are negative and interior, and the independent law gives B_i=0 exactly. At zero contrast, each low-capacity pair has rho=0 and Si=alpha/4+beta+gamma, approximately 0.43109; without a lag override, its default request is Resonance. With any sufficiently small nonzero d_0 alone, the peak is k*abs(d_0), both block-zero ratios are one, and Si=alpha/4+beta, approximately 0.37279. Both requests at block zero become Transition; other pairs retain their previous requests. The means, phases, capacities and supplied context have not changed. This proves ideal-real selection discontinuity, not an arbitrarily small binary64 witness or an executed alternative response.

Zero peak does not make every request unstable. A fixed lag override retains its request when the other gates still admit. Likewise, if the nonpressure Si contribution already satisfies alpha*V_i/max(V)+beta*phase_score>=si_hi, the default request remains Coherence for every pressure ratio in [0,1]. Uniform positive capacities and zero phases give this robust case under the defaults. Request robustness alone still does not certify temporal or adaptation branches. These two controls separate a source-specific decision margin from an assumed universal continuity of the normalization policy.

The detached retention controls retain production-execution guards and unchanged earlier source producers. They check uniform enclosures, fine ordered prefixes, associated memory, finite forced-contrast products and explicit boundary failures. No new native trajectory, observation model, occurrence law or engine API is introduced.

Joint form and capacity calculus across the adaptation exit

The five-operator description extends through unequal-capacity adaptation if it retains the signed form and capacity contrasts with their constituent association. The new term is the covariance of event capacity and held pressure. It affects the collective mean, including its response to Transition (NAV) innovations. This extends the earlier capacity-feedback identities and joint admission to the shared event calculus; it does not turn an invertible coordinate change into a reduced model.

Domain and sufficient coordinates. Keep the shared calculus's execution premises, including doubled C5, unit conductances, nonadjacent twins, whole-pair order, one direct request per constituent, default preparation/policies, held event pressure, live built-in zero Gamma and one Euler substep delta=1/16. Keep paired primitive phases on their common arc, the common semantic kind, actual request origin, grammar and all associated histories, rates, counters and streams. Automatic REMESH must independently abstain. Supplied conditional transcripts are not default-selection or fixed-seed occurrence certificates.

Replace paired capacities by

x_(2i)=M_i-d_i,                  x_(2i+1)=M_i+d_i,
v_(2i)=V_i-e_i,                  v_(2i+1)=V_i+e_i.

Both capacities must be strictly positive and interior to their rails; in particular V_i>abs(e_i). Zero capacity is admitted by the engine under its own contracts but is outside this theorem. All individual event and flow forms, amplified capacities and adapted capacities remain unclipped and admitted. Exact formulas below are ideal real over represented coefficients; concrete floating execution also needs representable arithmetic. These four coordinates reconstruct the pair's four scalar values exactly. No physical degree of freedom has been eliminated by naming them collective moments.

Fresh pressure and selection. With N_i(z) the adjacent-block mean, the independently supplied mixed law becomes

B_i=w_phase*(N_i(theta)-theta_i)/pi_rep
    +k*(N_i(M)-M_i)+b*(N_i(V)-V_i),
C_i=k*d_i+b*e_i,
p_(2i)=B_i+C_i,                  p_(2i+1)=B_i-C_i,
P=max_i(abs(B_i)+abs(C_i)),       Vmax=max_i(V_i+abs(e_i)).

Neighbor contrasts cancel because each adjacent block contributes both constituents with equal multiplicity. Own capacity contrast still enters pressure. For the corresponding even/odd signs, prepared Si is

phase_term_i=beta*(1-abs(N_i(theta)-theta_i)/pi_rep),
Si_(2i)=alpha*(V_i-e_i)/Vmax+phase_term_i
        +gamma*(1-abs(B_i+C_i)/(P or 1)),
Si_(2i+1)=alpha*(V_i+e_i)/Vmax+phase_term_i
          +gamma*(1-abs(B_i-C_i)/(P or 1)).

Both normalizers therefore require the joint fine association. Rebuild these fields from admitted primitive state before selecting, retain the actual lag priority and validate each live request. Equal means do not imply equal constituent requests. The earlier form-only robustness bounds cannot simply be reused with nonzero e: their capacity and pressure premises have changed.

Shared event rows. Reuse Y_i,A_i,H_i from the five-operator table: event form is Y_i-/+A_i*d_i, and H_i multiplies prepared pressure. Emission (AL), Reception (EN) and Coherence (IL) retain their original rows. Reception and Resonance (RA) still use the ordered live neighbor form means. Let

f_i=1+a*I(abs(S_i)>T) if the request is RA, otherwise 1,
U_i=f_i*V_i,                    E_i=f_i*e_i,
q_bar_i=H_i*B_i+g_i,
q_delta_i=-H_i*C_i+zeta_i.

U_i,E_i are the event capacity mean and half-contrast. Outside Transition, g_i=zeta_i=0. Resonance retains the same live whole-block U3 subset, selected form mean S_i and phase proposal. Its two capacities acquire the same factor, rather than becoming equal. Admit both sign identities (M_i-d_i)*(Y_i-A_i*d_i)>=0 and (M_i+d_i)*(Y_i+A_i*d_i)>=0, the common kind, nonempty subset and every individual post-amplification capacity. Mean identity alone remains insufficient. The first twin does not change the second's live neighbors.

For default nonstrict random Transition with common strict pressure sign,

sigma_i*B_i>abs(C_i),
g_i=eta*sigma_i*V_i+xi_bar_i,
zeta_i=eta*sigma_i*e_i+xi_delta_i,
abs(xi_bar_i)+abs(xi_delta_i)<=NAV_jitter.

The extra eta*sigma_i*e_i is required because each target uses its own capacity. Define G_i=eta*(sigma_i*V_i-B_i)+xi_bar_i and J_i=eta*(C_i+sigma_i*e_i)+xi_delta_i; the two pressure changes are G_i-J_i,G_i+J_i. Both must be nonzero. Opposite or zero prepared signs need their separate native branches, not this common-strict-sign formula. Innovations remain the actual associated inputs; a mean draw cannot replace their target-dependent contribution.

Held flow and the missing mean term. After every complete event pair, the independent fine products give

rate_bar_i=U_i*q_bar_i+E_i*q_delta_i,
rate_delta_i=U_i*q_delta_i+E_i*q_bar_i,
M_i_next=Y_i+delta*rate_bar_i,
d_i_next=A_i*d_i+delta*rate_delta_i.

Subtract associated old rate moments and divide by delta for acceleration. The pressure contrast contains incoming C_i, not the post-event form or capacity contrast. Current flow uses U_i-/+E_i; later adaptation must not be inserted early. Stored Si remains the preparation value even if Resonance has changed the live maximum capacity. A cache update is not a Si refresh.

For Emission, Reception, Coherence and Resonance, the mean rate is f_i*H_i*(V_i*B_i-k*e_i*d_i-b*e_i^2). This recovers the old Coherence covariance and variance terms with the correct event-specific multiplier. For Transition, the expanded mean rate is

(1-eta)*(V_i*B_i-k*e_i*d_i-b*e_i^2)
+eta*sigma_i*(V_i^2+e_i^2)+V_i*xi_bar_i+e_i*xi_delta_i.

Thus capacity variance has a different coefficient in this action. In particular, swapping the two innovations at fixed admitted state and fixed innovation mean reverses xi_delta_i; the corresponding mean-form difference is 2*delta*e_i*xi_delta_i. Innovation means alone were sufficient only in the paired-capacity sector. This is a conditional input-association result, not a new stochastic law or a claim that both assignments occur at one seed.

The exact error of discarding the covariance term in one admitted flow is delta*E_i*q_delta_i. If only abs(d_i)<=D_x, abs(e_i)<=D_v are retained, the default nonnegative factors imply

abs(mean_error)<=delta*f_i*D_v*(H_i*(k*D_x+b*D_v)+Zeta),
Zeta=0 outside NAV; Zeta=eta*D_v+Z_input for NAV,
abs(xi_delta_i)<=Z_input.

This is a useful conditional one-call bound that need not reconstruct each constituent. It does not propagate the next mean domain, admit the next request or assert that independent marginal extremes are jointly realizable.

Phase, memory and adaptation. The native coordinator reads paired phases, actual full glyph histories and global gains, without a capacity-driven free angular advance. The five admitted events and this coordinator therefore preserve phase pairing despite unequal capacities. This is a property of the declared native law; the separate capacity-driven phase model has different premises. Common-arc/resultant admission still applies.

Temporal event jumps remain Y_i-M_i-/+(A_i-1)*d_i; retain each actual reset, sample value, clock and old rate. Compute qualification from individual fresh Si and held post-event pressure, update each count, then obtain due flags a_minus,a_plus. Set a_bar=(a_minus+a_plus)/2, a_delta=(a_plus-a_minus)/2 and W_i=N_i(U) from the immutable event-capacity snapshot. The final writer is exactly

V_i_next=U_i+mu*(a_bar*(W_i-U_i)-a_delta*E_i),
e_i_next=E_i+mu*(a_delta*(W_i-U_i)-a_bar*E_i).

Fixed-point and hull guards remain in force. Unequal due flags can create capacity contrast from zero. When both are due the contrast becomes (1-mu)*E_i; when neither is due it remains E_i. Equal flags alone do not erase it. Counts do not reset after a successful write. The next preparation uses these new capacities and flow forms, not the old stored pressure or a capacity substituted into the preceding flow rate.

Admitted entry and limits. The existing positive joint-admission source supplies a conditional entry through this boundary: its first call's node-two/node-three due split creates e_1=mu*epsilon/4, with epsilon=1/1024. Fresh Si/pressure and retained lag and grammar state admit its second all-Coherence call. Applying the joint rows to that already-declared source recovers its independent fine forecast, including active capacity writes, associated rates/history and the nonzero mean gap between opposite internal form assignments. No response is rerun to manufacture a new preparation or occurrence claim.

The joint-calculus controls compare independent fine sweeps with all five joint moment rows, live phase and neighbor inputs, held pressure and post-flow adaptation. They retain the actual-source continuation separately from supplied mixed transcripts, and test the capacity-innovation association and covariance-error bound. Production runtime, pressure, selector, operator and integrator evaluation are blocked. The result supplies sufficient exact coordinates across the former capacity exit. It neither proves mean-only closure, a smaller complete NFR state, unlimited retention, autonomous formation nor physical identification.

Bounded prediction from a lossy collective observation

The joint moment identities permit a genuinely lossy one-call observation: retain intervals for the two means and upper bounds for the two internal contrasts, without retaining their signed values. The prediction is a set of possible successors. Its width and unresolved decisions matter separately; an enclosing interval is not a unique response or a smaller complete NFR.

Preparation and information boundary. Keep the complete law, support, whole-pair order and successful-call premises of the joint calculus. For each pair retain

M_i in X_i,             V_i in N_i,
abs(d_i)<=D_xi,         abs(e_i)<=D_vi.

Here N_i is a capacity interval; write neighbor_i(z) for the adjacent-block mean. Retain exact paired primitive phases and their inherited common arc, actual supplied grammar/lag context, node-associated sets of possible counts, global gains and target-associated innovation constraints. Old rates and temporal sample values may be represented by associated envelopes. Any nonempty admitted history ends at its fine lift's actual current form; empty histories remain explicit and supply no measured trigger evidence. This is a premise on the family, not permission to invent independent samples. The predictor consumes neither exact signed contrasts nor exact history endpoints from which to recover them. Fine lifts are evaluated only after freezing the observation, protocol and output budget.

The proposed requests retain their origin. For actual default occurrence, certify the fresh selection and lag precedence uniformly. A supplied paired transcript instead asserts the requests and still has to pass every live grammar, identity, nonidentity, support and state gate. Passing such a transcript does not prove that the default selector would choose it. Supplied selector requests remain subject to the batch's Emission/Reception lag overrides; the retained effective transcript must agree with those overrides.

Fresh normalization from retained data. Form the interval pressure mean B_i from X,N and the exact phase row. Let minabs(J)=min_(z in J)|z| and maxabs(J)=max_(z in J)|z|. For the nonnegative default pressure weights,

R_i=k*D_xi+b*D_vi,
p_n in B_i+[-R_i,R_i],                   n in {2i,2i+1},
P_lo=max_i minabs(B_i),
P_hi=max_i(maxabs(B_i)+R_i),
Vmax_lo=max_i N_i.lo,
Vmax_hi=max_i(N_i.hi+D_vi).

These bounds follow from the exact correlated peaks P=max_i(|B_actual_i|+|k*d_i+b*e_i|) and Vmax=max_i(V_i+|e_i|), where the actual scalar pressure mean B_actual_i belongs to B_i. A contrast radius supplies no positive lower bound for its realized value. When P_lo>0, divide the pressure magnitude enclosure by the peak bounds and intersect the ratio with [0,1]. If zero peak remains possible, use [0,1], except that P_hi=0 gives the exact ratio zero. Bound each capacity ratio from N_i+[-D_vi,D_vi] and the positive capacity-maximum interval. Insert these bounds and the exact phase score into the fresh Si formula. This admits requests with the actual strict/inclusive thresholds; it does not freeze a maximizing node or assume affine normalized diagnostics.

Ordered event admission. Use interval arithmetic on the shared event rows, updating live form-mean intervals in order. Reception (EN) consumes the live adjacent means. Resonance (RA) recomputes its whole-block U3 subset from the live paired phases and its selected form-mean interval S_i. Amplification is inactive if maxabs(S_i)<=T, active if minabs(S_i)>T. Otherwise this fixed-branch certificate is unavailable; it must not choose one branch from a fine witness. The same requirement applies to unresolved request, sign and identity gates. An unresolved temporal reset instead needs associated alternative history records or an unavailable history certificate. A more elaborate event union is possible but is not supplied by a silently selected branch.

For Transition (NAV), admit a common strict sign with min(sigma_i*B_i)>R_i. With retained innovation mean interval Ibar_i and half-difference bound Z_i, require maxabs(Ibar_i)+Z_i<=NAV_jitter. Its nonidentity changes lie in

G_i+[-Jmax_i,Jmax_i],
G_i=eta*(sigma_i*N_i-B_i)+Ibar_i,
Jmax_i=eta*(k*D_xi+(b+1)*D_vi)+Z_i.

Excluding zero admits both changes. This sufficient symmetric bound is not an assertion that every mean/difference combination is realizable by a fixed seed. Keep the original target association and declared input domain. Both Resonance sign identities and every individual incoming, event, flow and capacity rail require uniform admission. A failed enclosure can be conservative; it is not itself a native rejection witness.

Mean and contrast bounds. Let Y_i,A_i,H_i,f_i be the admitted interval event mean and exact branch factors. Write n_i=1 for Transition and zero otherwise, with Ibar_i=0,Z_i=0 outside Transition. Define

Qbar_i=H_i*B_i+n_i*eta*sigma_i*N_i+Ibar_i,
Cov_i=f_i*[-H_i*k*D_xi*D_vi,H_i*k*D_xi*D_vi]
      +f_i*(n_i*eta*sigma_i-H_i*b)*[0,D_vi^2]
      +f_i*[-D_vi*Z_i,D_vi*Z_i],
X_i_next=Y_i+delta*(f_i*N_i*Qbar_i+Cov_i),
D_xi_next=maxabs(A_i-delta*f_i*H_i*k*N_i)*D_xi
    +delta*f_i*maxabs(H_i*(B_i-b*N_i)
                     +2*n_i*eta*sigma_i*N_i+Ibar_i)*D_vi
    +delta*f_i*maxabs(N_i)*Z_i.

Terms multiplied by n_i=0 vanish without requiring a pressure sign for the other actions. The covariance includes both the signed capacity-variance coefficient and capacity-input uncertainty. These equations enclose the exact fine products algebraically. Replacing them by a product of mean capacity and mean pressure would discard a consumed term. Independent interval extrema provide an enclosure, not simultaneous attainable values.

Associated policies and temporal evidence. Use the prepared Si enclosure and held post-event pressure to classify each stability qualification as true, false or unresolved. Update each node's count set accordingly, then derive its possible due flags using the original threshold. With U_i=f_i*N_i, D_Ei=f_i*D_vi, W_i=neighbor_i(U), each possible due pair gives a_bar=(a_minus+a_plus)/2, a_delta=(a_plus-a_minus)/2 and

N_i_candidate=U_i+mu*a_bar*(W_i-U_i)
              +[-mu*abs(a_delta)*D_Ei,mu*abs(a_delta)*D_Ei],
D_vi_candidate=abs(1-mu*a_bar)*D_Ei
               +mu*abs(a_delta)*maxabs(W_i-U_i).

Take the capacity hull and largest contrast radius over those pairs. This reuses the joint-admission owner and preserves unresolved count outcomes instead of replacing them by false. The Cartesian enumeration is a safe outer bound; shared pressures may forbid some enumerated combinations. Recheck positive capacity, rails and the fixed-point/hull writer premises. Adaptation follows the current form flow.

The admitted event jumps determine each temporal reset. A reset sample lies in the individual event-form envelope; otherwise preserve the associated prior sample envelopes. Append the individual flow endpoint envelope with the original timestamps and retention cap. Old-rate envelopes and the new rate enclosure bound acceleration. They do not prove arbitrary combinations of samples and rates consistent, or admit a future Mutation trigger. Paired phase continuation uses its own retained glyph-load/gain evidence, without inserting capacity as a free phase advance. Automatic REMESH still requires its independently declared abstention. This output is a scoped observation certificate, not a complete restart record.

Frozen one-call classes and budget. The detached enclosure controls freeze three observations before evaluating any fine lift. Keep the shared default represented coefficients, source/support order 0..9, clock t=1, one call with delta=1/16, zero built-in Gamma and the original no-REMESH premise. The source convention is common kind seed, empty incoming temporal histories, zero old acceleration, the admitted EN trace with prior Coherence, and associated rates, debt, ordinals, lags and count sets. Old rates remain declared state; empty history supplies no measured trigger evidence.

1. Supplied five-action transcript. Use Emission (AL), Reception (EN), Coherence (IL), Transition (NAV), Resonance (RA) in block order. Retain mean intervals of radius 2^-16 around the joint-calculus transcript's form and capacity means, its absolute form/capacity contrast bounds and paired phases (0,1/8,1/4,1/8,0). Innovation constraints are abs(xi_bar)<=NAV_jitter/4, abs(xi_delta)<=NAV_jitter/2, a rectangle inside the admitted amplitude diamond. This is a supplied input domain, not a fixed-seed realization. All five event maps are covered conditionally; default occurrence is not claimed for this transcript.
2. Actual default selection. Reuse the live-prefix source means (q,q,-q,-1/2,-q), q=15/128, capacity means (1/4,1/4,1,1,1), zero phases and associated context/rates. Keep these means exact but discard all ten signed internal form/capacity contrasts, retaining only D_xi=D_vi=1/4096. Fresh interval Si/ratios admit RA/RA/IL/IL/IL uniformly. The first live RA selected mean stays exactly zero and does not amplify; the second is -r*q/2 and amplifies. Exact means here are substantive: an independent mean envelope around the first zero signal could leave its amplification branch unresolved. No adaptation qualifies in this class.
3. Actual ambiguous adaptation. Reuse the joint-admission observation with all form means 1/8, capacity means (1+epsilon,1,1,1,1), epsilon=1/1024, D_x0=59/8192, other form radii zero, and zero capacity radii. Node zero's incoming count is zero; all other counts are four. Default requests are all Coherence, while node one's due decision remains unresolved. The enclosure must preserve both outcomes.

All three must have maximum successor form-mean interval width below 1/200, form half-contrast radius below 1/20, capacity-mean interval width below 1/100, and strict individual source/event/output rails. These budgets are fixed before fine controls, not selected from their errors. The shared 128-bit outward dyadic interval kernel encloses ideal-real arithmetic over represented coefficients. Exact fine containment adds no tolerance; this is not a universal binary64 execution certificate.

The outward calculations give the following displayed approximations. The exact interval endpoints in the controls, not rounded table entries, supply the certificates.

Frozen class	Largest form-mean width	Largest form radius	Largest capacity-mean width	Largest capacity radius
Supplied five-action transcript	0.000352828	0.0311475	0.0000317319	0.00812335
Actual RA/RA/IL/IL/IL	1.64472e-9	0.000243106	0	0.000253855
Actual ambiguous all-IL	5.87748e-39	0.00713955	0.000048828125	0.000048828125

The exact uniform bound on the actual RA/IL class's deviation from the conditional product-of-means form prediction is below 1e-9 (approximately 9.35272e-10). Its mean enclosure is narrow despite ten discarded signed coordinates. Internal capacity differences can grow under RA amplification; form-radius contraction alone is not complete-state contraction.

For the ambiguous class, two admissible lifts with opposite d_0 have the same retained observation but distinct node-one adaptation decisions. Their terminal first-block capacity means differ by mu*epsilon/2. Any single common capacity-mean prediction must err by at least mu*epsilon/4 on one of them. This is an information obstruction, not merely interval overestimate; the enclosing union remains useful without selecting a hidden branch.

The controls evaluate a fixed 32/4/2 list of exact fine lifts after the three coarse certificates. Their finite inclusion checks are separate from the uniform interval proof. Boundary controls preserve a possible zero pressure peak, unresolved Transition sign and actual lag-override priority. A single shared phase-origin interval retains phase correlations after coordination; individual temporal envelopes retain right=left+delta*rate at times 1,17/16. Arbitrary Cartesian combinations are not declared valid histories. These results establish bounded one-call prediction and its limits. Reusing the successor description for another call still needs its own admission; complete collective NFR identity, formation and physical identification remain separate questions.

Finite composition with the propagated collective memory

A useful one-call enclosure need not supply an admissible next input. The second call consumes the evolved physical observation, histories and policy state. Restoring the initial context or reconstructing a representative fine state would answer a different question. This result studies two calls of the existing actual Resonance (RA)/Coherence (IL) class using only its first abstract successor.

Composition criterion. Let A contain the admitted mean intervals, contrast bounds and associated context. Write Lift(A) for the fine states that satisfy these bounds and their retained relations. In particular, an individual history endpoint and the live form are the same variable, not independent values drawn from overlapping intervals. The latest flow relation's velocity is also that constituent's retained rate coordinate; equal marginal rate intervals alone do not establish this association. Configuration, support, node order, clock and the actual complete law remain fixed. If the detached observer F proves

every s in Lift(A) admits the declared complete call T,
T(Lift(A)) is a subset of Lift(F(A)),

and proves the same statements again on F(A), then T(T(Lift(A))) is contained in Lift(F(F(A))). This follows by set inclusion and induction. The second observer may enclose more states than are reachable; all consumed decisions must still be admitted on its declared family. Neither one representative lift nor agreement of marginal means proves this inclusion. The exact ordered-action criterion has stronger equality obligations; a bounded successor set does not establish an exact smaller node or a uniquely determined collective response.

Source and numerical protocol. Retain the previous actual RA/RA/IL/IL/IL observation: means (q,q,-q,-1/2,-q), q=15/128, capacity means (1/4,1/4,1,1,1), zero paired phases and form/capacity contrast radii 1/4096. All ten signed contrasts remain discarded. Use its unchanged associated rates, grammar, counts, kinds and empty initial histories, default represented coefficients and no additional input. The fixed horizon is two complete calls from t=1 with delta=1/16, ending at t=9/8.

Before evaluating fine controls, require every final form-mean and capacity-mean interval width below 1e-7, every form contrast radius below 1/4096, every capacity contrast radius below 1/3500, and strict individual rails throughout. These are cumulative endpoint budgets, not an extra local tolerance or a budget reset after the first call. The shared outward dyadic 128-bit kernel encloses ideal-real arithmetic over represented coefficients. Fine inclusion is exact, without an added tolerance. Keep the already frozen one-call producer unchanged.

What the successor must retain. The second preparation reads the first output's mean/radius bounds and rebuilds pressure, both normalization peaks and Si. Stored first-call pressure or Si cannot stand in for this preparation. The retained old-rate interval gives the next acceleration through (new_rate-old_rate)/delta. The old-acceleration envelope also remains admitted although the default selector does not use acceleration to choose its mark. An unknown diagnostic is not replaced by zero.

Carry each constituent's trace, persistent U2 debt, prior Coherence flag, operator ordinal, kind, lag counts and stability-count set. The actual batch increments lags before applying its Emission/Reception priority. Traces from the first call already contain the new event. For this sector, Resonance leaves debt unchanged and Coherence reduces it by one down to zero. Replacing Coherence's decrement by an unconditional reset would be incorrect outside the source's zero/one debt stratum. Runtime invocation ordinals have their own provenance and are not physical time or constituent operator counts. With the source's initially absent runtime marker, the two calls use ordinals zero and one and leave the next ordinal two; constituent operator ordinals instead advance 3 -> 4 -> 5.

Retain the first coordinator's gains and common phase-origin observation, with its relative phases. In this source every phase stays exactly zero, so the shared origin is exactly zero as well. The gain recurrence nevertheless continues from the first output rather than restarting at configured initial gains. The complete updated traces contain no disruptors; the stable branch has phase resultant one and zero disruptor load. This scoped composition does not establish a general phase-rotation quotient for dispersed phases.

The second event is applied to a state whose recorded endpoint already matches its live form. A uniformly nonzero Resonance form jump at 17/16 restarts that constituent's temporal record at the new event form; the subsequent flow adds its endpoint at 9/8. Coherence leaves form unchanged at the event and preserves its first two samples, then appends the new one. Thus the final RA records have times (17/16,9/8) and the IL records have times (1,17/16,9/8), within the four-sample limit. Each retained flow segment keeps its actual sample/rate association. These predicted envelopes are not measured evidence for a future Mutation trigger.

The abstract successor retains envelopes for the successive whole-support snapshots written by native REMESH-history sampling. The admitted runtime has no callback adding stability samples; its initially empty stable_frac remains empty, so the automatic REMESH policy abstains on both calls before reading delayed form values. Declaring that abstention does not remove REMESH from the law or promise it indefinitely.

Re-admitting the second call. Use the same interval pressure/Si, live-neighbor, covariance and adaptation identities as the one-call owner. The second call's input is its propagated observation, never an initial-state adapter with substituted traces, time, acceleration or histories. Both fresh requests and live operator gates must pass over the whole abstract family. The actual second requests remain RA/RA/IL/IL/IL. The first live selected-mean interval is now strictly negative, separated from the amplification trigger; the first RA therefore changes from inactive amplification to active amplification. The second RA also amplifies. Their live inputs are recomputed in order; freezing the first-call amplification pattern would omit a real event-capacity factor from the second form flow.

Both RA targets remain below the Si stability threshold. The IL targets' held pressures stay outside the adaptation band. Hence every qualification is false and each stability count remains zero. This is a re-admitted decision, not an assumption that unchanged counts waive capacity adaptation. The amplified capacities are used by the current held-pressure flow before the later writer. Covariance continues to affect means despite the small enclosure widths.

Concrete bounds and controls. The second preparation has pressure peak between 0.0691967 and 0.0692554. RA pressure ratios are below 0.032; their Si intervals remain strictly inside the default middle band, while the other blocks have Si above 0.941. The two ordered RA selected-mean intervals are contained respectively in (-0.010904808,-0.010904804) and (-0.018872395,-0.018872390), strictly beyond the magnitude trigger. These whole-family bounds precede the fine controls.

At t=9/8, the largest form-mean interval width is below 3.094e-9, the largest form contrast radius below 0.000242093, and the largest capacity contrast radius below 0.000263956. Capacity means remain point intervals, approximately (0.259947184,0.270290154,1,1,1). The exact outward endpoints, not these rounded displays, verify every predeclared budget. Increasing capacity contrast under amplification remains compatible with decreasing form contrast; no complete-state contraction is claimed.

The composition controls freeze the source, complete law, both abstract outputs and budgets before evaluating four independent exact paths. Their initial contrasts are d_i=s_d*(-1)^i/4096, e_i=s_e*(-1)^i/4096, with independent signs s_d,s_e in {-1,1}. Each second fine call continues its own first endpoint, rates and history. Their inclusion checks validate the implementation of the interval argument; four paths alone do not prove the continuous domain. Runtime, production pressure, selector, operator and integrator evaluations remain disabled in this new owner. Earlier producers remain unchanged.

The observer binds each current-form and rate coordinate to its corresponding history relation. Missing, stale or reassigned records reject this detached admission; it never reconstructs hidden values or restarts with initial defaults. An unavailable certificate is not an executed failed runtime call and consumes no native invocation ordinal. Reusing initial phase gains could leave the zero-phase endpoint unchanged while giving the wrong policy state; the controls therefore check the actual gain recurrence separately. Likewise, restoring initial rates changes the acceleration calculation even when mean form remains enclosed. These are consumed-state distinctions, not extra physical laws.

The result establishes reusable bounded collective information across this two-call source class, including a changed live event branch. It retains constituent-associated policy and temporal state; it does not establish indefinite retention, a minimal autonomous NFR, newly formed organization, general dispersed-phase closure or physical identification.

Relative phases for the complete native call

The common phase origin can be removed from the admitted five-operator map without removing its relative phases or constituent-associated context. This is an exact symmetry reduction of the declared ideal circular model. It is not a proof that an arbitrary rotated binary64 graph executes identically, or that the remaining state constitutes a smaller autonomous NFR.

Model and equivalence. Retain the joint-calculus domain: doubled C5, consecutive nonadjacent twins, paired primitive phases, positive individual capacities, admitted signed forms and common kind, fresh default pressure/Si, direct ordered glyphs, zero live Gamma and held Euler flow. Keep all rails, grammar, clock, history and independently inactive REMESH premises. Every effective pair request must agree. Its origin is either the fresh admitted default selector or an explicitly supplied transcript; these are different inputs. Exclude anchored callbacks, additional phase writers and optional tracking that consumes absolute phases.

For this theorem the circular operations use mathematical radians and their true period 2*pi. Configured finite coefficients, including the pressure normalization pi_rep, retain their represented values as exact real parameters. On an inherited common lift of width less than pi, all selected phasor targets use the representative in that lift. Default Resonance phase coupling lies in [0,1]; coordinator gains are nonnegative with sum at most one. Their convex proposals stay in the same lifted hull. This separates the ideal circular law from modulo normalization with a represented period and from materialized trigonometric arithmetic.

Write theta_i=beta+r_i, with beta=theta_0 and r_0=0. Let H retain the consumed non-phase state: individual rates and acceleration, kinds, associated temporal samples, grammar traces/debt/prior-Coherence flags, operator and runtime ordinals, lags, stability counts, phase gains and controller history, random inputs/stream progress and REMESH history/abstention evidence. The complete law, support, node order and clock are also retained. Define

Q(s)=(M,d,V,e,r,H),
rho_a(s): theta_i -> theta_i+a for every constituent,
          all retained non-phase state and associated inputs unchanged.

The equivalence is one shared rotation of every participating primitive phase, not independently rotating blocks or deleting their relative state. For the admitted successful call T, its consumed semantic-state map obeys T(rho_a(s))=rho_a(T(s)). Consequently Q(T(s)) depends only on Q(s) and the retained input, defining T_relative(Q(s))=Q(T(s)). A declared coherent lift is a chart for this statement; raw centered extrema need not describe that chart after rotation. This quotient removes one common angle from the five paired phases. It retains the joint form/capacity moments and all their constituent associations. Arbitrary graph dictionaries, absolute-angle debug logs and untransformed trigonometric caches are not this state equivalence.

Fresh preparation and live events. The pressure phase term is w_phase*(N_i(r)-r_i)/pi_rep. The other channels, both normalization peaks, individual absolute pressures and capacity ratios are unchanged. The Si phase term also uses relative offsets. Thus fresh default selection, its incremented lag priority and all grammar decisions have the same admitted inputs. Stored old acceleration still requires admission even when the base selector does not use it to choose a mark.

Emission (AL), Reception (EN), Coherence (IL) and Transition (NAV) on this direct native path have no phase write. Their form/pressure actions depend only on retained invariant quantities. Reception still reads live ordered neighbor forms; Coherence still leaves primitive phase unchanged. Transition retains each constituent's own capacity and innovation, including their association and stream progress.

For Resonance (RA), initialize live offsets y=r. At every ordered prefix,

J_i={j in {i-1,i+1}: abs(y_j-y_i)<=L},
S_i=mean_(j in J_i)(live Y_j),
a_i=mean_(j in J_i)(y_j),
y_i <- (1-c)*y_i+c*a_i.

These expressions use the admitted common lift. The selected adjacent blocks have equal multiplicity, so their circular mean is exactly the lifted midpoint (or the one selected phase). Nonempty selection and the open-semicircle domain exclude an undefined resultant. Recompute J_i at each live prefix: it may gain a neighbor during the event batch. The scoped compatible-edge retention result excludes losing an already compatible edge under these default paired events; it does not replace the live subset with its initial value. A common rotation preserves every U3 inequality, both fine sign/kind identity gates, the selected signed form mean and the capacity-amplification decision. The phase proposal gains precisely the same common rotation. Induction over the ordered pairs proves event equivariance without freezing their subsets. This proof does not admit arbitrary unequal requests inside a pair or supply rollback for a rejected sequential prefix.

Coordinator and its retained policy. After the events, write l_i(y)=N_i(y) and choose the global target representative b(y) in the same common interval. The global resultant magnitude is

R(y)=abs(sum_i exp(1j*y_i))/5.

The twin multiplicity cancels in this expression. Rotating all phases multiplies the phasor sum by one unit complex factor, so R is invariant. Disruptor load depends on the actual updated traces and is also invariant. The configured controller therefore selects the same branch, updates the same incoming gains and appends the same state/R/load/gain records. Its global angle need not be calculated to advance the relative coordinates:

r_i_next=(1-kG-kL)*(y_i-y_0)+kL*(l_i(y)-l_0(y)),
beta_next=beta+(1-kG-kL)*y_0+kG*b(y)+kL*l_0(y).

The first row is closed on retained relative phases and policy state; r_0_next=0. The second row reconstructs an absolute lift only if its initial origin is supplied. Global coupling still changes relative dynamics through its coefficient; cancellation of the common target does not remove global coupling. Nor does it remove R or the gain history needed by the next invocation. The policy acts per call, not per unit of physical time.

Conditional phase-resource consequence. Let D_j be the lifted phase diameter before call j. Every admitted Resonance update is a convex move inside the current hull, including a move with a newly compatible neighbor. The other four events, held form flow and capacity adaptation leave phase unchanged. The coordinator then gives

D_(j+1) <= (1-kG_j)*D_j,
D_N <= D_0*product_(j=0..N-1)(1-kG_j) <= q^N*D_0,
q=1-m < 1,  m=the represented default adaptive global-gain floor.

The last bound assumes the default enabled adaptive policy. The hull step and the coordinator diameter theorem prove the result by induction for any successful segment remaining in the five-action domain. Unequal evolving capacities do not defeat it because they supply no free phase advance on this native path. The phase chart is preserved; admission of later identities, rails, selections and histories is a separate obligation. Infinite successful invocation count is not proved.

This extends the earlier uniform-capacity reachable IL/AL/EN restriction to the admitted live five-action class. Along an infinite admitted ideal sequence with the gain floor, the diameter tends to zero. This excludes a uniform positive lower bound, not positive diameter at every finite invocation: D_N=q^N*D_0 is positive for every finite N when D_0>0 and 0<q<1. It does not exclude persistent form/capacity organization or finite identity. Binary64 defects retain their separate ledger. An identity requiring a fixed positive phase margin eventually leaves that identity domain if the admitted sequence continues. Maintaining such a margin would require a justified departure from these contraction premises, not necessarily strict diameter increase after a complete call. The observable admission separates that obligation from Mutation admission and phase recharge.

Flow, adaptation and memory. Every event form, capacity and held pressure is unchanged by the rotation. The nodal products, covariance contribution, form endpoints and accelerations are therefore unchanged. Event jumps reset exactly the same individual temporal records, and each history endpoint and flow-rate relation retains its association with the live constituent. The clock is unchanged. This is not a phase-derived clock or a time change.

Adaptation reads the unchanged prepared Si and post-event held pressure, updates the same stability counts and due flags, and uses the same immutable capacity snapshot. Thus both capacity moments, including any contrast created by unequal due flags, are invariant. Grammar records, lags, ordinals, whole-support EPI snapshots and the independently empty stability series have the same updates. These component identities complete the whole-call proof; the earlier pressure or coordinator symmetry alone did not establish it.

Caches containing absolute trigonometric components are not invariant physical records. They must be transformed consistently or rebuilt from the admitted primitive phase. The quotient does not authorize reusing a stale cache, replacing retained memory with initialization defaults, or treating an absolute phase log as an invariant observation.

What the quotient cannot predict. Signed forms, capacities, pressures, Si, event decisions, relative phases, R and the retained non-phase histories factor through Q. An absolute phase reading or the signed real/imaginary components of a phasor sum generally do not. A callback selecting a mark from an absolute phase threshold introduces an external reference; it needs that reference as retained state/input and its own covariance argument. The present exclusion is substantive, not a declaration that every callback is unphysical.

Relative phase cannot be discarded with the origin. Keeping the same forms, capacities and context but changing a nonconstant r changes fresh phase pressure; it can also change Si, live U3 subsets and the controller. Likewise, restarting incoming phase gains can preserve a synchronized endpoint while giving the wrong next controller state. Origin reduction does not resolve these distinct state-sufficiency obligations. A future physical sensor may require an external phase reference even when the internal model does not.

Numerical boundary. A binary64 implementation is not an exact action of the continuous rotation group. Forming float(theta+a) can change a gap before any operator runs. Subsequent trig reduction, branch representatives, represented-period normalization and summation introduce separate defects. The runtime's signed circular difference and centered phase storage also use different endpoint conventions. Even exact arithmetic with a represented 2*pi_rep modulo cannot be identified with true-radian periodicity across all branch crossings. The regular lifted identities above do not waive any of these implementation boundaries.

For a claimed numerical comparison, retain the chosen lifts, backend and every realization defect, with independent margins for U3, resultants, selection, identity, amplification and controller thresholds. At a decision boundary, a small angular discrepancy can change a discrete branch; an endpoint tolerance alone does not establish equivalent calls. This section supplies neither a uniform roundoff bound nor permission to replace the native executor by an origin-centered implementation.

Frozen controls. The detached owner prepares two origin-free forecasts before six independent fine lifted controls: one complete call for each source at origins -7/4,0,5/2, with exact comparisons and no tolerance. Both sources have nonzero relative phase. The source, law, context, input, horizon and output protocol digest is bad6429813ae9fdcd07b8cd904862429a26bbd2f518818e37585ff6b6540e1bb.

The joint five-action transcript retains its supplied request origin: fresh default selection on that source would instead give Coherence everywhere. Its updated trace load is 1/10; exact cosine-tail bounds certify the stable controller branch. Its phase diameter falls from 1/4 to approximately 0.212219. The separate unchanged neighbor-gain source has actual fresh RA/RA/IL/IL/IL requests. Its live RA subsets are first (4,), then (0,2) after the earlier phase change. Exact resultant-square bounds inside (0.2547626,0.2547695) and zero disruptor load certify the dissonant controller branch. The event diameter 2.998046875 contracts to approximately 2.165845 or less. Rounded displays are not the exact comparisons.

Both complete-call sources have no capacity qualification; a separate admitted static adaptation control checks dependence on the retained input and unequal due flags. Prefix hulls, fresh/default requests, identities, amplification, held products, reset/debt masks and associated temporal/rate records have their own checks. Absolute readings, omitted relative phases and binary64 materialization supply distinct counterexamples to stronger claims. No new native scientific execution occurs and earlier producers stay unchanged. The proof supplies the all-origin ideal symmetry; finite controls check its algebra and retained-state implementation. No physical identification, formation or indefinitely admitted execution follows.

Mutation admission and the complete phase-renewal obligation

Mutation (ZHIR) supplies a phase-changing event that is absent from the five-action hull argument. Its request, temporal eligibility, grammar, physical proposal and the remaining live call have independent obligations. The following bounded analytical study separates them. It proves an admitted Mutation prefix followed by a later Resonance identity obstruction, and a distinct conditional renewal map whose simple history acquisition is excluded. These results cannot be combined into a successfully executed renewing organization.

Actual native boundary. Reuse the same ideal circular doubled-C5 law, joint coordinates, default represented coefficients, strict direct-glyph grammar, zero Gamma and held Euler flow with delta=1/16. The added pressure event is the default non-noise Dissonance (OZ) primitive: it multiplies stored pressure by its configured factor when abs(p)>1e-9; its separate near-zero branch assigns 0.1. Do not replace that piecewise law by a universal multiplier. The declared source below stays strictly on the multiplying branch. It changes no form, capacity or phase at the event.

Mutation requires active capacity, a live signed two-sample EPI secant strictly above the default represented threshold, and target-local grammar. Both the retained recent handler for U4a and the recent destabilizer/prior Coherence requirements for U4b matter. A lifetime prior-Coherence flag is not a recent handler. The five earlier actions cannot create a canonical destabilizer: Transition (NAV) contributes to the phase controller's configured disruptor load but is not a U2/U4b destabilizer.

Fresh default selection requests Mutation only if the high-Si branch has not already applied, Si<=si_lo, and abs(p)/P<=dnfr_hi. The divisor is one when the pressure peak is zero. With positive capacities, the necessary low-pressure-branch bound is

Si >= alpha*(nu/vmax)+beta*(1-abs(phase_offset)/pi_rep)
      +gamma*(1-dnfr_hi).

This excludes some capacity/phase domains but not all heterogeneous states. The default selector admits old acceleration independently even though its base choice does not use it. AL/EN lag priority can still replace that base request. Missing, stale or subthreshold temporal evidence instead causes the native Mutation abstention and a Coherence request before grammar. Valid evidence with invalid grammar reaches strict rejection; these are not the same outcome.

The signed direct Mutation proposal retains form, capacity and stored pressure and changes only primitive phase and its own telemetry. Write its ideal signed increment as s*copysign(1,p_stored), with s=ZHIR_theta_shift_factor*pi_rep/4. This uses exact real arithmetic over represented factors. The default binary64 multiplication yields 1/4; that rounded scalar and the ideal product need not be identical. Signed zero has the primitive's explicit copysign convention. Neither direction nor magnitude is inferred from the positive historical secant.

General renewal test. Let y be the final admitted event-phase lift and let the controller recompute its branch and gains from y and the actual updated glyph load. On the paired common-arc domain, put

A(g)y=(1-kG-kL)*y+kL*N(y),
osc(z)=max_i(z_i)-min_i(z_i),
D_out=osc(A(g)y).

The omitted global target is a common additive term. For the declared diameter observation, strict one-call renewal is exactly D_out>D_in. This criterion needs every event, flow and policy boundary to be admitted before it describes a completed call. A proposed phase pulse or a favorable event-only diameter is insufficient. A common phase increment alone is not renewal. Positive diameter is also not necessary for every possible form/capacity organization, nor does its renewal establish sustained form work, formation or a physical identity.

On doubled C5 this bound can be sharpened without another trajectory. Let a_g=1-kG-kL. The default gain rails give a_g>=kL/2. Rows of A have mass 1-kG, diagonal entry a_g and two neighbor entries kL/2. Adjacent rows overlap by kL, while nonadjacent rows overlap by kL/2. The exact oscillation contraction coefficient is therefore

tau=1-kG-kL/2,
osc(A(g)y)<=tau*osc(y).

The bound is sharp over real phase vectors, including vectors inside the common-arc domain after a sufficiently small rescaling. For one whole-pair Mutation theta -> theta+s*e_k, with all other event-phase rows inert, osc(A(g)e_k)=a_g and hence

D_out <= tau*D_in+a_g*abs(s).

Strict renewal consequently requires a_g*abs(s)>(kG+kL/2)*D_in. This necessary resource condition is not sufficient: the geometry, sign, controller and full execution must still agree. Use the actual post-event gains. If another event changes phase, rebuild its live proposal instead of treating this single-shift formula as the entire call.

Frozen source and preparation. The analytical source family is

M=(a,0,-9/10,9/10,0),   abs(a-19/20)<=1/4096,
V=(27/50,27/50,6,6,27/50),   d=e=0,
r=(0,3/2,0,3,3/2).

Keep all ten node kinds seed, supplied initial traces (AL,IL,EN), prior Coherence true, zero debt, constituent ordinal three and zero lags/stability counts. Initial temporal histories are empty at t=1; old rates and accelerations are zero. These are explicit preparation data, not a claim that the engine generated this initial organization. The retained IL handler is essential: importing the earlier EN-only trace would reject the first OZ despite its prior-Coherence flag. The earlier source is not rewritten.

Fix support/order 0..9, default controller gains, empty REMESH stability history and the existing no-callback/no-math-engine premises. Allow the two first-call Transition inputs independently in the full configured jitter amplitude interval. Their target association is retained; this input family does not claim that one seed realizes every combination. The declared horizon is at most two calls, stopping at the first live rejection. No request, threshold, initial form or clock is adjusted after evaluating the control.

First complete call: acquisition of conditional evidence. Fresh preparation uniformly selects OZ/OZ/IL/IL/NAV by block. All target-local grammar checks pass; lags do not override. The first two pressure multipliers and the two Transition inputs then act on the held prepared pressure. No event changes form. The flow therefore predicts two associated samples for each constituent, (1,x_initial) and (17/16,x_endpoint), without a jump being mistaken for continuous growth.

For the first pair, the exact domain certificate places the sampled secant inside (0.1339,0.1341), strictly above the default 0.1 threshold. The first call also appends its own OZ, creating the missing recent destabilizer and debt one. This is a conditional forecast of the recorder's output from the same admitted flow, not measured native evidence inserted into a graph. An actual later execution must consume its actual records. The general Mutation-history contract does not permit replacing those records by the predicted nodal product.

All first endpoints stay inside the form rails. The held pressures remain outside the capacity-adaptation band, so qualification is false, counts remain zero and capacities do not change. Exact cosine-tail bounds give R^2 inside (0.23555,0.23557), certifying the dissonant phase-controller branch with the actual load 3/20. Its relative map supplies the next phases without evaluating an absolute global angle. All consumed memory, gains, rates, input progress and REMESH sampling/abstention are retained.

Second preparation and accepted prefix. Refresh pressure and Si from the first endpoint. The prospective batch is ZHIR/RA/IL/IL/NAV. For the first pair, the certificate has fresh pressure inside (0.0842,0.0846), Si inside (0.2403,0.2404) and normalized pressure inside (0.1156,0.1162). Its live positive history, active capacity, recent IL/OZ context and debt allowance admit Mutation on both constituents. Lags advance to two without forcing another mark. Each accepted Mutation changes phase by the positive s, appends its mark, raises debt to two and advances its operator ordinal. Form, capacity, pressure, rate and the associated EPI history stay unchanged.

The next block has Si inside (0.2540,0.2541) and normalized pressure inside (0.0496,0.0498), so its fresh default request is Resonance (RA). Its first constituent, fine node two, still has a strictly negative form below -0.008828. Both neighboring blocks remain U3-compatible after Mutation; their live selected form mean lies inside (0.1188,0.1192). The actual Resonance blend proposes a strictly positive form inside (0.01149,0.01154). Thus the sign-preservation clause rejects the proposal throughout the declared source/input domain. Kind preservation and phase compatibility cannot waive the independent sign clause.

This is a live identity obstruction, not a grammar rejection, undefined phase, missing Mutation evidence or numerical solver failure. The same compatible subset and sign failure occur if the preceding phase reset is omitted in a detached counterfactual. Mutation is therefore not established as the cause of this failure. The fresh selector proposes an action whose live identity gate does not admit it; a base request is not an admission certificate.

Partial-call accounting. The native sequential path preserves the two accepted Mutation events. The failing Resonance commits no form, phase, capacity or glyph history. Later targets are not executed. There is no second flow, coordinator, adaptation or terminal REMESH snapshot, and no second-call Transition input is consumed. Physical time remains 17/16; all first-call EPI records, old rates and incoming phase gains remain. Second preparation has already refreshed pressure/Si and incremented every node's lag counters. The runtime invocation ordinal would still record the admitted failed invocation; a detached certificate consumes no runtime ordinal. This chronology is not all-target rollback.

The accepted phase prefix itself narrows the diameter, but no complete second-call diameter or renewed organization is available. Continuing its algebra past the rejected event would silently replace the specified law. The eight fixed fine controls use the endpoints of the initial-form interval and both independent first-call innovation intervals. They check the whole-domain forecast; their number does not prove its interval inequalities. The detached owner blocks production execution and preserves the earlier frozen producers.

Separate conditional renewal and its history obstruction. A second static family illustrates why the preceding rejection is not a universal geometric impossibility. On the same support let

M=(1/8,0,3/4,-3/4,0),   V=(1/64,1,1,1,1),   d=e=0,
r=(1/10,0,0,0,0).

Its fresh default base choices are ZHIR/IL/IL/IL/IL, with positive pressure at the first pair. Without qualifying history, the actual selector abstains from Mutation and requests Coherence. Conditional on separately admitted Mutation evidence and grammar, the increment increases the sole phase maximum to 1/10+s. The default gain rails give 1-kG-kL>0.6816 and 1-kG-kL>kL/2, so the resulting diameter is (1-kG-kL)*(1/10+s)>0.2385>1/10 for every admitted default controller branch. No extra low-disruptor-load premise is needed. This is a conditional action/geometry calculation; no qualifying temporal record is supplied and no Mutation is executed on that source.

Indeed, let P_bound=w_phase+2*w_epi+w_vf*VF_MAX. Under the same admitted form/capacity rails and a common lift narrower than pi_rep, fresh pressure satisfies abs(p)<=P_bound. For a target held at capacity 1/64, with no form jumps between samples and at most the declared multiplying OZ pressure event before its held flow,

abs(secant) <= (1/64)*OZ_dnfr_factor*P_bound < 0.031 < xi.

The separate near-zero OZ branch also lies below this bound. The simple history-producing route therefore cannot supply the required trigger on this conditional renewing source. Different past capacities, inputs, laws or sampling histories need their own provenance and rate argument. The prospective phase benefit cannot manufacture missing evidence.

Finally, Mutation's primitive always retains branch-specific telemetry. The signed mode includes absolute before/after angles and quadrant labels; these do not descend through the common-origin quotient even though the physical circular shift does. Retain their reference or report them as reference-dependent readouts. Public/staged Mutation has additional lifecycle effects and is not the direct route used here. The two complementary obstructions leave successful complete renewal open, with its missing joint conditions identified rather than hidden in a favorable operator name.

Joint history and renewal: a reflection-sector obstruction

The previous study leaves a joint question: can a predecessor supply the history that admits Mutation (ZHIR), while the next complete call renews the relative-phase diameter? A declared family now has a negative answer. This is an analytical exclusion of a continuum of preparations, not a failed sampled trajectory or a universal obstruction to TNFR organizations.

Complete question and frozen scope. Keep the preceding native direct law, doubled-C5 support, whole-pair order, represented defaults, zero Gamma, held Euler delta=1/16, initial gains and supplied (AL,IL,EN) context. Initial debt, lags, stability counts, rates and accelerations are zero; temporal records are empty at t=1. Prior Coherence is true, constituent ordinals are three, and callbacks, math engine and initial REMESH stability history are absent. The preparation's origin remains supplied.

One target pair has capacity u; all four other pairs have the same capacity W, with 0<u<W<=VF_MAX. All capacities remain fixed over the two-call horizon: from zero counts, even qualifying nodes cannot reach the default adaptation threshold of five. Their counts need not remain zero. Because each other pair has the network maximum capacity, its Si is at least alpha, above the default high cut, so its fresh base choice is Coherence (IL). Lags one and two do not override. This removes the earlier Resonance obstruction through an explicit source restriction, without changing the selector or weakening an identity contract.

The target must first receive freshly selected Dissonance (OZ), then freshly selected and temporally admitted Mutation. No word is imposed to obtain those actions. All forms and both unclipped flow endpoints must lie within the default rails. Renewal means D2>D1, comparing the second completed call with its own incoming diameter, not with the initial preparation. The initial, first coordinated and post-Mutation phase states consumed by the second coordinator must all admit the common-lift calculation.

General necessary inequalities with arbitrary phase orientation did not decide this question. Before any source evaluation, the decision domain was restricted explicitly to reflection-symmetric block coordinates

M=(x,y,z,z,y),             (x,y,z) in [-1,1]^3,
theta=(D,a,0,0,a),        0<=a<=D<=3<pi,
nu=(u,W,W,W,W),           0<u<W<=VF_MAX,
d=e=0.

A common absolute phase origin may be retained separately. The target is initially at a phase maximum. Freeze eighteen algebraic outer regions: two directions of the second Mutation, three first controller branches, and three second branches. Each second gain update starts from the first updated gains. No sampled sources, candidate scans, native responses or additional horizons belong to this protocol. An outer region may include unrealizable controller branches; excluding it is still sufficient. This avoids needing trigonometric branch selection to establish the obstruction.

Shared gap and pressure identities. Write pressure weights as (k,b,w), Si weights as (alpha,beta,gamma), d_p=w/pi_rep, OZ multiplier as f and IL retention as ell. For controller j write (G_j,L_j) for its updated global/local gains and A_j=1-G_j-L_j. These are configured quantities, not additional TNFR laws. The common global target cancels from phase gaps.

Let H=D-a. After the first coordinator define h as target-minus-neighbor phase and t as neighbor-minus-far phase. The exact paired map gives

h=(A1-L1/2)*H+(L1/2)*a,
t=(L1/2)*H+A1*a,
D1=h+t.

The gain rails preserve h,t>=0. With det=A1*(A1-L1/2)-L1^2/4>0, set

c_a=(A1-L1/2)/det,
c_t=L1/(2*det),
c_h=A1/det-1,
a=c_a*t-c_t*h,
H-h=c_h*h-c_t*t.

All three coefficients are positive for the admitted default gain updates. Reflection and the regular support also give p0+2*p1+2*p2=0. On the nonzero multiplying OZ branch, putting lambda=delta*k*W*ell and mu=delta*k*u*f yields the exact two-boundary row

p0_next=(1-mu-lambda/2)*p0-lambda*p2+d_p*(H-h).

Here p0_next is freshly computed pressure at the next boundary, not a pressure reconstructed from the response. The two associated flow samples would have secant u*f*p0; its required strict lower bound is xi. This identity applies only when the first OZ uses its multiplying branch and no form clipping changes the endpoint. Its tiny-pressure branch is handled separately below.

Let c be the default normalized-pressure cut. The second Mutation base choice requires rho<=c and low Si. Consequently

u <= U(h)=VF_MAX*(r0+r1*h),
r0=(si_lo-beta-gamma*(1-c))/alpha,
r1=beta/(alpha*pi_rep).

Both coefficients are positive here. Keep pressure and Si weights separate in this derivation, even when their configured values agree. Missing actual history would still cause selector abstention; these necessary inequalities do not certify execution by themselves.

Positive increment. Suppose the second pressure selects the positive increment s. After the event the phase gaps are (h+s,t), so the completed coordinator has diameter

D2=A2*(h+s)+(A2+L2/2)*t.

Strict renewal therefore requires

(G2+L2)*h+(G2+L2/2)*t < A2*s,
h < h_max=A2*s/(G2+L2),
t < T-J*h,
T=A2*s/(G2+L2/2),   J=(G2+L2)/(G2+L2/2).

The first flow endpoint's form rails give -k+d_p*t/2<=p2_next<=k+d_p*t/2. Combined with pressure zero-sum and the normalized-pressure cut, nonnegative p0_next satisfies p0_next<=kappa*(k+d_p*t/2), where kappa=c/(1-c/2). This also covers a zero pressure peak: then every fresh pressure is zero and the divisor-one selector convention does not invalidate the inequality. The first rails similarly give p2<=k+d_p*a/2.

Use lambda_max=delta*k*VF_MAX*ell, and define

n0=k*(kappa+lambda_max),
nt=d_p*(kappa/2+lambda_max*c_a/2+c_t),
nh=d_p*(lambda_max*c_t/2+c_h),
N0=n0+nt*T,              N1=nh+nt*J,
den_min=1-lambda_max/2-delta*k*f*U(h_max).

The exact certificates have den_min>0. A positive acquired secant then obeys

u*f*p0 <= f*VF_MAX*(r0+r1*h)*(N0-N1*h)/den_min,
0<=h<=h_max.

If the numerator is nonpositive, positive acquisition is already impossible. Otherwise its concave quadratic has an exact maximum at an endpoint or its rational vertex. Across all nine gain pairs that maximum gives u*f*p0<0.037<xi. No source choice or unresolved phase-controller branch inside the declared sector can supply the required trigger on this branch. An independent certificate using the weaker oscillation condition h+t<T also leaves a strictly positive quadratic gap to xi in every cell.

Negative increment. If D1>=s, lowering the target by s keeps all event phases in the incoming hull. The coordinator's strict oscillation contraction then excludes renewal. Thus renewal with a negative increment requires h+t<s, and in particular u<=U(s).

For p0_next<=0, the pressure identity, inverse gap map and first form rails give the necessary bound

u*f*p0 <= f*U(s)*[lambda_max*k
                  +d_p*(lambda_max*c_a/2+c_t)*s]
             /[1-lambda_max/2-delta*k*f*U(s)]
          < 0.008 < xi.

All three possible first gain updates satisfy it; the second gain branch is irrelevant to the hull argument. Include zero pressure with either declared signed-zero direction in the corresponding case. Neither zero pressure nor a zero network peak leaves a missing boundary case.

The actual small-pressure OZ branch. When abs(p0)<=1e-9, direct OZ assigns its represented literal 0.1; it does not multiply p0. The same selection and renewal constraints, independently of the transport identity, give u<=U(h_max) in the positive-increment cells and u<=U(s) in the negative-increment cells. These imply secants below 0.076 and 0.026, respectively, still below xi. The equality boundary at the small-pressure cut belongs to this branch. Thus the exclusion applies to the complete piecewise OZ law, not just its multiplying part.

Result and retained limits. In this reflection sector, no two-call preparation can simultaneously acquire the required Mutation history, select/admit the target event and complete a strict diameter renewal under the declared law. Even relaxing the first selector and grammar inequalities does not remove the contradiction; restoring them cannot create a witness. The family cannot repair its missing trigger by extending a predicted history, suppressing a gate or continuing an unavailable execution.

The exact certificate owner checks the eighteen outer regions, gain carry, positive denominators, coefficient identities, quadratic extrema, signed-zero/peak boundaries and the separate OZ fallback. Production execution is guarded. Algebraic basis controls verify identities; they are not sampled physical preparations. No fine response, new temporal record or completed call is reported. All inequalities concern the ideal real model over represented constants; they are not an unproved bit-for-bit statement about floating-point paths.

The obstruction does not cover asymmetric block states, targets away from the initial phase maximum, different capacity layouts, clipped histories, other words or longer horizons. Nor does it exclude organizations sustained without this diameter-renewal criterion. It identifies incompatible joint requirements of this declared preparation and policy, not a new universal law or a physical identification.

Reflection matters to the pressure-peak bound. Without it, the target transport row still uses the mean far pressure (p2+p3)/2, by the same regular-support zero-sum identity. However, opposite pressure contrasts can increase the actual network maximum without changing that mean. Its bound cannot be transferred from a reflected mean to the full pressure peak. Independent phase contrasts also affect extrema and the resultant consumed by the gain controller. Retaining these quantities is necessary when using the calculation outside the reflection sector.

Arbitrary paired geometry with a uniform-capacity environment

Removing reflection does not remove the joint obstruction under the same two-call protocol. The following conditional theorem allows all five paired forms and phases to vary independently. It concerns acquisition followed by one strict phase-diameter renewal, not equilibrium, all NFR maintenance or physical geometry.

Declared domain and decision budget. Keep the preceding support, law, context, default coefficients, initial gains, clock and horizon. Forms range over [-1,1]^5; the initial phase lift has oscillation at most three radians, and the target need not be an extremum. Capacities are still (u,W,W,W,W), with 0<u<W<=VF_MAX, and within-pair contrasts remain zero. Require both unclipped flow endpoints in the form rails and the common phase chart at every consumed boundary, including after Mutation (ZHIR). The requested sector remains freshly selected OZ/IL/IL/IL/IL followed by ZHIR/IL/IL/IL/IL, with an acquired positive secant above xi. Preparation is supplied; predicted records cannot replace an actual later recorder. The source-width bound and default gains also imply 3*(1-G1)+s<pi_rep: the first coordinator and subsequent single-pair shift stay within a common arc. No unexamined circular branch is introduced.

Before calculation, freeze eighteen direction/gain outer cells. A reserved partition allowed at most 7,200 regions: those eighteen cells, twenty ordered renewal witnesses, ten signed pressure-peak owners and two local-gap signs. No source evaluation, parameter scan or optimizer was authorized by that partition alone. The coarse inequalities below close all eighteen cells, so the larger partition and any optimizer are unused. No native research response or independent fine response is evaluated.

What asymmetry changes. Write the paired pressure as

p=(p0, p_near+d_near, p_far+d_far, p_far-d_far, p_near-d_near).
P=max(abs(p0), abs(p_near)+abs(d_near), abs(p_far)+abs(d_far)).

The peak depends on the retained contrasts. Replacing it by a peak of means would change selection. The first and second controller resultants also consume the full phase state. The outer cells allow all three target branches at each update, with the actual preceding gains carried forward; excluding even this relaxation does not require evaluating a resultant.

Nevertheless the target transport row still closes in paired means. Let y be the first coordinated phase lift, and define

h=y0-(y1+y4)/2,
t=(y1+y4-y2-y3)/2,
D1=osc(y).

Use H and a for the same two signed mean gaps before the first coordinator. Unlike the reflected case, h, t, H and a can be negative, and h+t is not the diameter. The same two-by-two restriction and its inverse remain exact:

H-h=c_h*h-c_t*t,
a=c_a*t-c_t*h,
p0_next=(1-mu-lambda/2)*p0-lambda*p_far+d_p*(H-h).

The last identity uses p0+2*p_near+2*p_far=0, fixed capacity and the multiplying OZ branch. The meanings of lambda, mu, d_p, c_a, c_t and c_h are those derived above. Arbitrary antisymmetric components cancel from this mean identity, but not from the peak or controller.

The affine phase pressure is valid on the admitted common arc: each fine node has two equally repeated neighboring block phases whose circular resultant has their lift midpoint as its angle. This is ideal-circle algebra; no exact binary64 trigonometric covariance is inferred.

General geometric restrictions. The exact C5 oscillation bound gives

D2 <= (1-G2-L2/2)*D1+A2*s,
D2>D1  =>  D1<T=A2*s/(G2+L2/2).

For a positive shift, the column carrying the impulse is (A2,L2/2,0,0,L2/2). Only eight ordered output pairs can have a favorable impulse difference. If the renewed maximum is on an adjacent row, renewal forces D1<(L2*s/2)/(G2+L2/2)<s. If it is on the target, translate the incoming lift to minimum zero. For h>=0, its coordinated row is at most (1-G2)*D1-L2*h, hence renewal requires A2*s>G2*D1+L2*h>=(G2+L2)*h. For h<0, the target obeys y0<=D1+h, giving abs(h)<s instead. Reversing the lift proves the corresponding negative-shift statements.

Thus an outward shift, whose direction agrees with the sign of h, requires abs(h)<H_max=A2*s/(G2+L2); an inward shift requires abs(h)<s. All admitted gain pairs have H_max>s. The Si argument consequently gives u<=U(abs(h)) using the same r0 and r1, now without reflection. In particular u<0.752 throughout the renewing domain. The literal small-pressure OZ branch therefore has secant below 0.076<xi, including its closed threshold boundary, independently of any multiplied-pressure transport identity.

These restrictions concern the full incoming diameter. Comparing an output pair only with its own incoming gap would not prove renewal. Also, an event remaining inside the old phase hull cannot renew the diameter after the strictly contracting coordinator. A phase pulse alone is not the observation.

Bounds retaining the complete pressure peak. At any non-target block, the endpoint rails and common lift give

abs(p_i_next) <= B(D1)=2*k+b*VF_MAX/2+d_p*D1.

Only the two adjacent blocks have a capacity-contrast contribution; the bound covers them and the far blocks. If the positive target pressure is not zero, its low normalized-pressure ratio prevents it from owning the peak. Hence p0_next<=c*B(D1) for the positive-shift case. If the whole peak is zero, the same inequality holds with the actual divisor-one convention. This bound replaces the reflection-specific peak estimate; it never drops the antisymmetric pressure contribution.

The first rails still imply p_far<=k+d_p*a/2. Independently of reflection, t<=D1 and h+t<=D1. These inequalities allow positive upper bounds before replacing the actual lambda by its maximum. Set

lambda_max=delta*k*VF_MAX*ell,
K=lambda_max*c_a/2+c_t,
S=lambda_max*c_t/2+c_h,
C0=c*(2*k+b*VF_MAX/2)+lambda_max*k,
den(v)=1-lambda_max/2-delta*k*f*v.

All used denominators are strictly positive. The four orientation cases can now be excluded by scalar bounds over the declared gain cells:

Mutation direction and local gap	Necessary acquired secant bound
Positive, h>=0	<0.091
Positive, h<0	<0.089
Negative, h>=0	<0.030
Negative, h<0	<0.078

For the first row, use t<=D1-h, D1<T and 0<=h<=H_max. A positive history-producing pressure satisfies

u*f*p0 <= f*U(h)*[C0+d_p*(c+K)*T-d_p*(K+c_h)*h]/den(U(H_max)).

Its numerator is a concave quadratic in h, so endpoint and rational-vertex evaluation certifies the bound for the whole interval. A nonpositive pressure numerator already excludes the positive trigger.

For the second row put z=-h<s. Using t<=T in the mean transport gives the sufficient upper bound

u*f*p0 <= f*U(s)*[C0+d_p*((c+K)*T+S*s)]/den(U(s)) < 0.089.

For the third row 0<=h<s and p0_next<=0; discard the latter nonpositive term, use the positive far-pressure upper bound and obtain

u*f*p0 <= f*U(s)*[lambda_max*k+d_p*K*T]/den(U(s)) < 0.030.

The final row needs the capacity/phase relation instead of maximizing them independently. With z=-h>=0, nonpositive fresh target pressure and form rails require b*(W-u)+d_p*z<=2*k. Put R(z)=r0+r1*z; low Si gives u/W<=R(z)<1. Consequently

u <= R(z)*(2*k-d_p*z)/(b*(1-R(z))),
0<=z<=H_max.

Bound u by the exact maximum of the concave numerator on this interval, divided by b*(1-R(H_max)). Call this bound u_star; every gain cell has u_star<0.292. The independent transport bound then gives

u*f*p0 <= f*u_star*[lambda_max*k+d_p*(K*T+S*H_max)]/den(u_star)
          < 0.078.

This proof retains the correlation between capacity and the signed phase pressure. The simpler independent bound u<=U(H_max) is too loose for this case. Both possible signed-zero directions are included by the non-strict pressure inequalities; zero local gap is also covered.

Conclusion and execution boundary. Every multiplying-branch bound is below 0.091<xi, and the fallback bound is below 0.076<xi. Therefore no preparation in this entire arbitrary-paired-geometry domain can both acquire the required Mutation history and complete strict second-call phase-diameter renewal. This conclusion already holds after relaxing some first-selection and grammar constraints, so restoring them cannot create a witness.

The general certificate owner freezes the constant certificates before independent coefficient and optimization checks. It verifies general five-coordinate pressure transport, retained peaks, signed-gap geometry, gain carry, positive denominators and the separate OZ fallback. Production execution is guarded; no preparation, temporal record, completed call or optimizer result is reported. The earlier reflection and admission producers remain unchanged. All arithmetic belongs to the ideal real model over represented coefficients.

The result removes the freedom of paired forms and phases as an escape within this protocol. It still assumes a uniform-capacity environment, the declared pressure and selection laws, one OZ acquisition interval, unclipped forms, the common chart and this diameter observation. It does not exclude other operator histories, longer organizations or maintenance that does not require this particular renewal. No material geometry or fundamental physical property is identified by the exclusion.

Heterogeneous capacities: retained association and a necessary spread

Unequal surrounding capacities change the target transport even when the support, pressure law and operators are unchanged. The following calculation retains those terms and excludes a finite neighborhood of the equal-capacity environment. It does not exhibit an acquired organization or identify a physical geometry.

Domain and prospective budget. Retain the preceding arbitrary paired forms and phases, doubled C5, represented defaults, strict grammar, supplied context, empty initial temporal history and two-call horizon. Replace the capacity vector by (u,v1,v2,v3,v4), with 0<u<=VF_MAX and 0<=vi<=VF_MAX. The actual maximum is V=max(u,v1,v2,v3,v4)>0. Both unclipped endpoints, every consumed common phase chart and all live-event obligations remain required. The requested sectors are freshly selected Dissonance (OZ) and then acquired Mutation (ZHIR) at the target, with high-Si Coherence (IL) on every surrounding pair at both boundaries. These are selection conditions, not an imposed operator word. Capacity adaptation remains inactive over this horizon from the declared initial counts.

Before coefficient evaluation, freeze the eighteen carried-gain/direction cells and a single neighborhood width eta0=1/16. No other width, tuning, source search, optimizer, native trajectory or finer partition belongs to this decision budget. Algebraic coordinate and tensor-basis controls check identities rather than physical preparations.

What must be retained. Let n and F be the near and far capacity means, and C the covariance of source capacity and freshly computed source pressure over the two neighboring pairs:

n=(v1+v4)/2,              F=(v2+v3)/2,
C=(v1*p1+v4*p4)/2-n*(p1+p4)/2
 =(v1-v4)*(p1-p4)/4,
lambda_n=delta*k*ell*n,   mu=delta*k*f*u.

The pressure in C precedes the first IL multiplication and flow. Neither the held post-event pressure nor the refreshed next-boundary pressure can replace it. The regular-support identity p0+2*p_near+2*p_far=0 still holds. The exact target row, on the multiplying OZ branch, is now

p0_next=(1-mu-lambda_n/2)*p0-lambda_n*p_far
        +delta*k*ell*C+d_p*(H-h).

With source form means M_near and M_far, the far pressure is

p_far=k*(M_near-M_far)/2+b*(n-F)/2+d_p*a/2.

Thus both capacity contrast and its pressure association survive in the target row. In particular, equal neighboring capacities can make C vanish while n-F remains nonzero. Also, the maximum V used by Si is a different observation from either mean. Retaining only capacity and pressure means does not determine this transport term.

Define the signed capacity contribution J and the positive denominator A:

J=delta*k*ell*[b*n*(n-F)/2-C],
A=1-mu-lambda_n/2,
A*p0=p0_next+lambda_n*[k*(M_near-M_far)/2+d_p*a/2]
     +J-d_p*(H-h).

Here A is positive throughout the default capacity rails. A negative C assists the removal of initial target pressure, holding the other terms fixed; positive C has the opposite sign. This is a pressure-transport identity, not an energy reservoir or a new microscopic law.

The required positive secant u*f*p0>xi, together with the source form rails, implies the necessary joint resource inequality

J > A*xi/(u*f)-p0_next-lambda_n*[k+d_p*a/2]+d_p*(H-h).

For the positive Mutation direction use p0_next<=c*P_next; for the negative direction use p0_next<=0. The complete pressure peak P_next must be retained. The right-hand side can be negative: this argument does not assert that J or its covariance component must independently be positive. Satisfying the inequality does not supply grammar, temporal evidence, selection or renewal.

Actual Coherence selection. At each of the two fresh boundaries every surrounding pair must satisfy

alpha*vi/V + beta*(1-abs((L*theta)_i)/pi_rep)
           + gamma*(1-rho_i) >= si_hi.

The full peak defines rho, including its zero-peak convention. The other two terms are nonnegative on the admitted chart and are bounded above by beta and gamma. Therefore vi/V>=(si_hi-beta-gamma)/alpha>0 is necessary; a zero-capacity surrounding pair cannot belong to this high-Si sector. This coarse floor is not equivalent to the full mask. Conversely alpha*vi/V>=si_hi is sufficient for the base IL choice, with the declared lags and history still retained.

A finite excluded neighborhood. Define

eta=(V-min(v1,v2,v3,v4))/V,
0<=eta<=eta0=1/16,
epsilon0=eta0*VF_MAX,
P_bar=2*k+b*VF_MAX+3*d_p.

All source pressures are bounded by P_bar using the source form rails, capacity rails and phase width three. The entire neighborhood satisfies

abs(C)<=epsilon0*P_bar/2,
n-F<=epsilon0,
abs(p_i_next)<=2*k+b*VF_MAX/2+b*epsilon0/2+d_p*D1,  i!=0.

For the last bound, each far capacity row is at most eta*V in magnitude; each adjacent row is at most (1+eta)*V/2. The bound thus retains every non-target pressure contrast. Since alpha*(15/16)>si_hi, all four surrounding high-Si base choices actually follow at both boundaries within this neighborhood. No missing selection mask is silently replaced by IL.

The coordinator does not consume capacity. Hence the previously derived geometric restrictions remain valid: D1<T, outward abs(h)<H_max and inward abs(h)<s, with actual gains carried between the two updates. Target low Si gives u<=U(abs(h)); the OZ literal fallback again has secant below 0.076<xi, independently of the multiplying-branch identity.

The additional pressure numerator can be bounded before dividing:

R_minus=delta*k*ell*epsilon0*(P_bar+b*VF_MAX)/2,
R_plus=R_minus+c*b*epsilon0/2.

The two terms in R_minus cover adverse covariance and far capacity contrast; the extra term in R_plus covers the enlarged non-target pressure peak. Positive-shift bounds require R_plus, while negative-shift bounds use R_minus. These are rederived joint bounds, not a subtraction of the old numerical gap from a covariance term alone.

The negative outward case additionally changes its capacity bound. With z=-h>=0, nonpositive fresh target pressure requires b*(n-u)+d_p*z<=2*k. Since n>=(1-eta0)*V and u/V<=R(z), it follows that

u <= R(z)*(2*k-d_p*z)/(b*(1-eta0-R(z))).

The denominator is positive on 0<=z<=H_max; a negative numerator would already contradict u>0. Let u_eta be the maximum of the concave numerator on that interval divided by b*(1-eta0-R(H_max)). Exact endpoint/vertex evaluation gives u_eta<0.315. The old uniform capacity bound cannot be reused without this correction.

Use the previously defined K, S, C0, T and den(v), recomputed from the same admitted coefficients. Let B_out be the exact positive-outward scalar quadratic bound in the preceding proof. The following bounds cover all four orientations. For the positive-outward row, a nonpositive old numerator is discarded upward before bounding the added resource; it no longer alone excludes acquisition. A positive old numerator uses B_out, which is positive.

Direction and local gap	Necessary acquired secant upper bound
Positive, h>=0	B_out+f*U(H_max)*R_plus/den(U(H_max)) < 0.096
Positive, h<0	f*U(s)*[C0+d_p*((c+K)*T+S*s)+R_plus]/den(U(s)) < 0.087
Negative, h>=0	f*U(s)*[lambda_max*k+d_p*K*T+R_minus]/den(U(s)) < 0.031
Negative, h<0	f*u_eta*[lambda_max*k+d_p*(K*T+S*H_max)+R_minus]/den(u_eta) < 0.085

All denominators remain positive. These are ideal-real inequalities over represented coefficients, not an assertion of bitwise trigonometric equivariance. Both shift directions, zero local gap, zero pressure peak and the literal OZ threshold boundary are included.

Result. Every bound is strictly below xi=0.1. Consequently any successful acquired renewal in the declared heterogeneous sector must have eta>1/16: at least one surrounding capacity must be strictly below 15*V/16. This is a necessary spread, not an optimal threshold, a universal constant or a sufficient condition for renewal. The general high-Si floor further requires eta<=1-(si_hi-beta-gamma)/alpha. The resulting interval is only a necessary domain; it does not establish that any source in it is jointly admissible.

The capacity-budget certificate checks the generalized transport and covariance on coefficient bases, the uniform limit, pressure envelopes, actual Coherence floor and all eighteen carried-gain cells. It preserves the earlier producers and guards native execution. No source, new history or completed trajectory is claimed. The retained association is a dynamical dependency; it does not by itself establish a smaller autonomous NFR, formation or sustained organization.

Joint-source constraints select the remaining renewal orientation

The heterogeneous resource budget does not permit its pressure, capacity and phase extrema to be realized independently. Enforcing their common source excludes every positive Mutation (ZHIR) increment, every inward increment and every renewal whose new extreme is owned by an adjacent pair within the declared two-call protocol. The remaining negative outward target-extreme case is unresolved; this result is a necessary orientation, not a jointly admitted source or a full-domain impossibility theorem.

Unchanged domain and finite calculation. Keep the complete heterogeneous domain above, including arbitrary five paired forms/phases, capacity rails, both actual high-Si Coherence (IL) masks, the default selector and gains, strict grammar, empty initial temporal history and the supplied context. All pressure fields must arise from the same source and its declared flow. Retain the unclipped source/endpoints and common phase charts. Successful acquisition on the multiplying Dissonance (OZ) branch requires p0>0 and an independently acquired positive secant above xi. The separate literal fallback remains below xi throughout the renewing domain. These are ideal common-lift certificates with represented coefficients; they do not assert exact binary64 wrap equivariance.

The prospective budget contains eighteen carried-gain/direction cells, exact source identities, signed-row support bounds and piecewise polynomial certificates. It contains no source search, optimizer, linear-programming run or native response. A declared extension used degree-at-most-three Bernstein bounds only at natural geometric/capacity breakpoints for the negative outward case. Its failure to exclude that case is retained below; no adaptive subdivisions or posterior threshold changes followed.

The source fixes the covariance. Put m=(si_hi-beta-gamma)/alpha; the admitted default coefficients give m>1/3. Both surrounding high-Si masks imply vi>=m*V. Write near and far capacity contrasts as e and e_far, and use r and s for the corresponding contrasts in another coordinate. The freshly computed source pressure contrast is

e_p=k*(s_x/2-r_x)+b*(e_far/2-e)+d_p*(s_theta/2-r_theta),
C=e*e_p.

It cannot be chosen independently after the forms and phases have been fixed. With zeta=delta*k*ell, the recovered target numerator contains zeta*(n*p_far-C). Its source-form coefficients, in order (x1,x4,x2,x3), are

(k/4)*(n+2*e, n-2*e, -n-e, -n+e).

Since 2*abs(e)<=n, their exact absolute row sum is k*n. The phase and capacity association therefore does not supply an additional freely chosen form contribution. The capacity contribution is b times

Q_capacity=n*(n-F)/2+e^2-e*e_far/2.

Both far-capacity coefficients are nonpositive. Its upper bound places the far capacities at m*V, then keeps the near capacities jointly: the three endpoint types, up to reflection, are (m*V,m*V), (V,m*V) and (V,V). Convexity gives this endpoint support; it is not a scan of preparations. The actual IL masks can make those extrema unattainable, which only makes the necessary relaxation more permissive.

Joint renewal geometry. Let y denote the first coordinated lift, h=y0-(y1+y4)/2 and D=osc(y). For a positive increment whose renewed maximum is on the target, set delta_top=max(y)-y0. A necessary inequality retains the distance from the target to the old maximum:

G2*D+(1-G2)*delta_top+L2*h < A2*s.

For h>=0, write a for the near mean above the incoming minimum and b_phase=D-a=delta_top+h. Then

a>=0, b_phase>=h, a+b_phase<T,
b_phase+G2*a<A2*(s+h).

When the relevant linear support is nondecreasing in a at fixed D, taking delta_top=0 enlarges its upper bound and relaxes the last constraint. This is a support maximization, not an assumption that the original target was an extremum. It leaves

a<=a_max(h)=min(T-h, [A2*s-(G2+L2)*h]/G2),
0<=h<=H_max.

The two affine branches meet at h=T/2. Also min(a_max(h),h) equals h on the first branch and a_max on the second. This joint polygon is strictly stronger than maximizing D and h separately. An adjacent renewed extreme instead requires D<s; that separate case does not use the target-extreme constraint. Negative directions follow by reflecting the phase lift, with the corresponding pressure sign retained.

Keep the signed pressure peak in the same phase calculation. Define

K=zeta*n*c_a/2+c_t,       S=zeta*n*c_t/2+c_h,
E_theta=c_a+c_t/2,       F_theta=c_t+(1+c_h)/2.

The inverse first coordinator gives the recovered phase numerator

d_p*[K*t-S*h+zeta*e*(E_theta*r_y-F_theta*s_y)].

For the next pressure peak, retain each of the eight signed non-target rows rather than separately assigning its phase contribution D:

(L*y)1=(h-t)/2-r_y+s_y/2,
(L*y)4=(h-t)/2+r_y-s_y/2,
(L*y)2=t/2+r_y/2-3*s_y/2,
(L*y)3=t/2-r_y/2+3*s_y/2.

Adding the selected signed row multiplied by c to the recovered phase expression gives A*t+B*h+C_r*r_y+C_s*s_y, with A nonnegative in all declared cells. If the far mean is f_phase, then t=a-f_phase and abs(s_y)<=min(f_phase,D-f_phase). Exact far support is

sup[-A*f_phase+C_s*s_y]=D*max(abs(C_s)-A,0)/2.

Let R_peak=max(abs(C_s)-A,0)/2 and raise A to A_prime=max(A,abs(C_r)). Since a is nonnegative, this is an upper bound. It makes near support monotone in a at fixed D, as required above. The remaining phase bound is

(A_prime+R_peak)*a_max(h)+(B+R_peak)*h
    +abs(C_r)*min(a_max(h),h).

The endpoint form contribution to a non-target peak is at most 2*k; its capacity contribution is bounded using the heterogeneous rails, not the old equal-capacity value. In particular (1-m/2)*V bounds every non-target capacity contrast. All source, capacity and phase support terms retain their coefficients before taking the maximum over endpoint types.

For outward positive h, discarding the favorable term -zeta*n*c_t*h/2 permits S=c_h in an upper bound. This alone does not make every absolute-value expression monotone in V. The full numerator support is convex in V; endpoint support at zero and VF_MAX must be checked before replacing the actual maximum. Zero here is an algebraic bounding endpoint, not an active-capacity preparation. The required rate is then bounded with the admitted U(h) and a positive denominator. Each natural phase branch gives an exact scalar quadratic certificate.

Excluded orientations. Recomputed coefficient certificates give:

Mutation increment and renewed extreme	Necessary acquired secant bound
Positive, outward, target maximum	<0.095
Positive, inward, target maximum	<0.094
Negative, inward, target minimum	<0.032
Either direction, adjacent extreme	<0.053
OZ literal fallback, any orientation	<0.076

Every bound is below xi=0.1. Inward cases retain abs(h)<s; adjacent cases retain D<s; both keep the same source pressure and capacity constraints. Zero local gap is covered by the inward bounds. A zero next pressure peak does not erase Si admission: its actual zero-ratio convention is retained, while both possible signed-zero increment directions are included in the respective necessary inequalities.

The remaining question is open. Any successful route in this protocol must use a negative Mutation increment, have h<0, and place the renewed minimum on the target. Its first acquired positive secant requires p0>0; the second selected pressure must satisfy p0_next<=0. This need not be a strict sign reversal because negative signed zero is included. Neither the history requirement nor the current pressure can be substituted for the other.

The same coefficients give R(H_max)<m, so every surrounding capacity exceeds the target capacity u. With z=-h>0, the fresh target pressure p0_next<=0 requires the post-flow form contrast

x0_next-(x1_next+x4_next)/2 >= [b*(n-u)+d_p*z]/k > 0.

Thus a still-possible target is below its neighbors' mean in phase and above their mean in form, while retaining lower capacity. This is a necessary compatibility condition, not a constructed state or a new constitutive law.

For this outward negative case, writing z=-h gives the exact restriction

n<=n_bar(z)=min(VF_MAX, U(z)+(2*k-d_p*z)/b).

The finite cubic attempt joined this with source capacity support and the renewal polygon. Its rational Bernstein upper envelope exceeds 0.157 in the worst cell and therefore does not exclude the required rate 0.1. That is an inconclusive upper bound, not an attained rate, an admissible source or evidence of renewal. No finer partition or response was evaluated.

The remaining actual IL masks still couple near capacity to geometry and pressure. For example, averaging the second near masks gives n/V>=m+beta*abs(h-t)/(2*alpha*pi_rep), even after discarding the nonnegative pressure penalty. The owner of the actual maximum V and the common source/endpoint form rails also remain unresolved joint constraints. They cannot be replaced by a favorable covariance or independent extrema.

The joint-orientation certificate retains the closed orientations and the inconclusive negative outward attempt separately. Earlier producers remain unchanged. This is progress on the declared acquisition/renewal mechanism; formation, maintenance, equilibrium and physical identification remain separate questions.

Same-source feasibility with retained pressure peaks

The remaining negative-outward renewal question is still open. Two finite algebraic attempts below retain more of the shared source, but neither certifies exclusion or supplies a feasible source. An exact pressure-coordinate reduction now exposes the remaining compatibility problem directly, including all three form boundaries. It preserves the complete two-call domain above; no threshold, support, history or horizon changes, source search or native response belongs to this calculation.

Actual capacity ownership. Because u/V<R(H_max)<m, the target cannot own the maximum capacity. A maximum must occur at a near or far pair; ties can belong to both cases. This is an exhaustive decomposition of the same domain, not a selection of prepared capacity profiles. If a near pair owns V, then n>=(1+m)*V/2, and the negative fresh target pressure gives

u <= min(VF_MAX*R(z), R(z)*(2*k-d_p*z)/(b*((1+m)/2-R(z)))).

All denominators are positive. In either ownership case, n>u also requires z<2*k/d_p. If a far pair owns V, the same-source capacity resource obeys Q_capacity<=(1-m)*V^2/4. To see this, reflect the cycle so v2=V and maximize over v3 at m*V. The four near-capacity corner values, divided by V^2, are

-m*(1-m)/4, (1-m)^2/8, 3*(1-m)^2/8, (1-m)/4.

Coordinatewise convexity and m>1/3 give the stated bound. The stronger fixed-mean expression keeps

Q_capacity <= n^2/2+e^2-V*((1+m)*n-(1-m)*abs(e))/4,
abs(e) <= min(n-m*V,V-n).

The frozen first attempt combined the coarse ownership bounds with the renewal polygon over nine carried gain pairs. Its worst acquired-rate upper bounds are approximately 0.10994 for a near maximum and 0.17459 for a far maximum. Both exceed xi; neither is an attained rate. The exact failed calculations remain with the certificate rather than being replaced by a favorable interpretation.

The first pressure peak remains associated with the acquired history. On the multiplying branch, successful history requires p0>0. The actual default Dissonance (OZ) selection also requires p0/P0>c, where c=dnfr_hi and P0 is the first full pressure peak. Hence P0<p0/c. This comparison is strict; hysteresis and score overrides belong to the separate parametric selector, and strict grammar cannot substitute OZ. The declared lag priorities cannot create an OZ request either.

Write q=p_next. Before zero-sum elimination, the same-source target row is

q0=(1-delta*k*f*u)*p0
   +zeta*(v1*p1+v4*p4)/2-d_p*(c_h*z+c_t*t).

Because n>0, its near contribution is strictly greater than -zeta*n*p0/c. This couples the allowed pressure transport to the very pressure that must generate the positive secant; the two cannot be maximized independently.

Put B(z)=(2*k-d_p*z)/b, R(z)=r0+r1*z, and J(z,t)=m+r1*abs(z+t)/2-R(z). The second near Coherence (IL) masks and fresh target pressure imply

n<=u+B(z),        u<=R(z)*B(z)/J(z,t).

These use the actual t of the source. A phase value chosen solely to maximize another bound cannot be charged to the IL mask. If t<-z, replacing t by -z raises the phase numerator and removes a nonnegative IL penalty, giving a valid upward relaxation. The renewal geometry also gives t<=A2*(s+z). With t>=-z, the resulting acquired-rate envelope is

f*d_p*R*B*(c_h*z+c_t*t)
---------------------------------------------------------------
[m+r1*(z+t)/2-R]*(1-zeta*B/c)-(delta*k*f+zeta/c)*R*B.

The frozen certificate proves the substituted denominator positive. At fixed z the numerator and denominator are affine in t, so their ratio is bounded by its two endpoint values. These have cubic numerators and quadratic denominators in z. Exact rational Bernstein bounds on the declared natural intervals give a positive denominator margin above 0.17749, but the threshold-gap coefficients do not certify exclusion. The resulting conservative rate envelope remains above xi. This second attempt is also inconclusive. No subdivision, alternative envelope, source evaluation or threshold adjustment followed it.

Exact source coordinates and all fifteen form rails. Let L=N-I on the five-cycle, with indices modulo five. On zero-sum pressure, the inverse on the mean-zero subspace is

(Q*p)i=-4*p_i/5+2*(p_(i-2)+p_(i+2))/5,
L*Q=Q*L=I-1*1^T/5.

For fixed capacities v and a source phase lift theta, the declared pressure law p=L*(k*x+b*v+d_p*theta) is equivalent to

sum(p)=0,
a0=(Q*p-b*v-d_p*theta)/k,
x0=a0+eta*1.

This inverts the supplied constitutive pressure law before evolution; it does not reconstruct pressure from an evaluated form derivative. The common form offset eta is a free source coordinate subject to the form rails. With F1=diag(f,ell,ell,ell,ell) and the first relative coordinator y,

a1=a0+delta*diag(v)*F1*p,
q=p+delta*k*L*diag(v)*F1*p+d_p*L*(y-theta),
a2=a1+delta*diag(v)*F2*q,
F2=diag(1,ell,ell,ell,ell).

Mutation changes phase without a form jump. Its terminal flow still uses the selected, held q, not a pressure refreshed after the phase event. Thus the three form boundaries are xj=aj+eta*1. For the fifteen coordinates of (a0,a1,a2), a single offset exists exactly when

max(a0,a1,a2)-min(a0,a1,a2)<=2,
eta in [-1-min(a0,a1,a2), 1-max(a0,a1,a2)].

Separate offset choices at each boundary would admit inconsistent histories. The associated first-flow secant remains u*f*p0; the second endpoint is retained for complete-call rails, not a second Mutation trigger. Since the admitted capacities are positive, the exact weighted balances are

sum_i (x1_i-x0_i)/(v_i*F1_ii)=delta*sum_i p_i=0,
sum_i (x2_i-x1_i)/(v_i*F2_ii)=delta*sum_i q_i=0.

These identities constrain one source and its evolution. They neither admit an event nor establish conservation of unweighted form.

Conditional linear feasibility; the parameter domain remains open. Fix capacities and primitive relative phases, admit their common charts, and determine the actual two controller branches and complete phase renewal. Then four independent source pressures and eta suffice for the remaining source/flow constraints. The three forms and q above are affine in these five unknowns. Retaining eta uses thirty linear rail inequalities.

For a positive pressure peak P with fixed owner j and sign sigma, impose P=sigma*p_j>0 and -P<=p_i<=P. Define, at each selection boundary, S_i=alpha*v_i/V+beta*(1-abs((L*theta)_i)/pi_rep)+gamma. The surrounding high-Si masks are the two linear inequalities gamma*p_i<=(S_i-si_hi)*P and -gamma*p_i<=(S_i-si_hi)*P. Source target selection, its positive secant, and successor target selection similarly retain

(S0_initial-si_lo)*P0-gamma*p0<=0,
p0>c*P0,             u*f*p0>xi,
p0>OZ_TINY,
(S0_next-si_lo)*P1+gamma*q0<=0,
q0<=0,               -q0<=c*P1.

There are nine possible signed source-peak owners: positive target or either sign at a non-target pair. When P1>0, c<1 excludes the target as its owner, leaving eight signed non-target choices. Thus each fixed admitted parameter configuration has at most 72 linear nonzero-peak systems, plus nine separate q=0 outer systems. Ties may overlap. At q=0, retain the actual zero-ratio Si tests; multiplying by a zero peak would erase them.

Here OZ_TINY is the actual multiplying-branch cutoff. Strict inequalities require exact positive slack or a verified witness, not a solver tolerance. A common slack must use fixed declared scales for pressure and rate units; separate exact margins avoid that extra convention. Rational capacities and relative phases, with admitted controller branches and represented law coefficients, give rational systems; arbitrary fixed real parameters need not. A constructive negative action requires q0<0 or a separate proof that the fresh represented producer yields negative zero. An assigned algebraic zero cannot supply that sign bit. Both directions may still be retained in an exclusion domain.

This is a reduction of the conditional pressure/form problem. The capacities, phases and their admissible controller branches remain joint parameters; the full problem has not become a solved linear program. No linear solver or source was evaluated. The source-constraint certificate retains the failed envelopes and checks the inverse, transport, shared offset and weighted balances independently. These are conditional compatibility restrictions of the declared model. Autonomous occurrence, formation and selection of a fundamental law remain separate questions.

Exact conditional pressure-system compilation

The private research compiler implements the preceding reduction for the unchanged paired unit five-cycle, two-call Dissonance (OZ)/Coherence (IL) then Mutation (ZHIR)/Coherence (IL) protocol. It is a detached algebraic instrument. It selects no source, installs no execution law and evaluates neither a native trajectory nor a linear program.

NativeMutationSlice supplies five capacities, five primitive phase lifts and two declared controller branches. compile_native_mutation_slice re-admits those primitives and rebuilds the law from fresh owned defaults. Rational inputs remain exact; other admitted reals use their represented values. Raw represented default weights are normalized in rational arithmetic, as in the preceding ideal certificates. This is not bitwise equivalence to floating-point normalization, trigonometry or wrapped native execution. Removing a common phase origin is justified only by the declared ideal relative-phase quotient.

The result contains the three affine form boundaries, fresh successor pressure, associated ideal temporal samples and all 81 canonical cells in the fixed coordinate order (p0,p1,p2,p3,eta). Each cell retains the thirty common-offset rails and labelled weak or strict inequalities. Signed peak ties can overlap; the nine zero-successor-pressure cells retain constant zero-ratio Si masks instead of multiplying them by zero. A zero target pressure in any cell still does not prove the represented negative-zero sign required for that action.

Parameter and execution obligations remain explicit. Algebraic checks retain the common charts, negative outward displacement, complete diameter renewal and target-minimum geometry. A failing check marks the parameter slice incompatible; passing them yields only conditional compilation. Actual controller branches remain unevaluated obligations. For the paired source,

R^2 = 1/5 + (2/25)*sum_(i<j) cos(theta_i-theta_j).

The first predicate uses source phases; the second uses the first coordinated phases after the negative Mutation jump. Glyph histories give disruptor loads 1/20 and 2/25; both lie below the owned low-load threshold. The ordered native policy therefore requires R>=R_hi for the stable branch, R<=R_lo for the dissonant branch, and R_lo<R<R_hi for transition. The obligations store these equivalent squared bounds and carry the first gain update into the second. Compilation does not declare a chosen branch realized. Actual associated measured history, strict grammar context and represented execution retain their separate admission duties; ideal affine samples are not observed native samples.

One-way exact contradiction checking. Write each compiled row as a_i dot u <= b_i, or < b_i for a strict row. For nonnegative exact weights lambda_i, the conditions

sum_i lambda_i*a_i = 0,
sum_i lambda_i*b_i < 0

contradict the weighted sum of the rows. The same follows with zero summed right-hand side if at least one strict row has positive weight: it would require 0<0. A zero-weight strict row contributes no strictness. This proves the checker directly; no solver completeness or numerical tolerance is assumed. Weights apply to the declared row units, with reciprocal-unit factors when combining pressure, form, rate and dimensionless inequalities. No shared unscaled slack variable is introduced.

check_cell_contradiction receives primitive parameters, a canonical CellKey and labelled multiplier pairs. It rebuilds the consumed cell rather than trusting a compiled report. Unknown or repeated labels, invalid or negative scalars and any nonzero coefficient residual reject the certificate. Exact cancellation without a contradictory bound returns an inconclusive result, never feasibility. The result retains normalized parameters, the rebuilt law, execution premises and unevaluated controller obligations alongside its weights and exact residuals. A successful check excludes only its conditional fixed-parameter cell. It does not cover the other cells, the unfixed capacity/phase domain or a different law.

The independent controls exercise coefficient identities, carried gains, row equivalence, strict and zero-peak boundaries and certificate admission. Their algebraic frames are not admitted sources. No source search or controller-branch response is evaluated. The earlier frozen envelopes and producers remain unchanged; heterogeneous acquired renewal existence, autonomous occurrence and formation remain open.

Joint phase and pressure-mask parameter restrictions

The same two-call negative-outward target-minimum problem has two useful parameter projections. One constrains the actual second phase controller from renewal geometry; the other eliminates pressure ratios from the selector masks and zero-sum law. Neither supplies a source or changes the constitutive law. Both preserve the preceding source, history, support, clock and form rails.

Renewal constrains the actual second controller. Write the carried second gains as (g,l), A=1-g-l, the Mutation shift magnitude as s, and the first coordinated phase diameter as D. For the five-cycle and A>=l/2, the largest half-l1 difference of two coordinator rows is A+l/2. A negative target jump adds at most A*s to a row difference. Complete renewal therefore requires

D < T = A*s/(g+l/2).

If the jumped target remained inside the old phase interval, the positive coordinator, whose row sum is 1-g<1, could not renew its diameter. Gauge the old minimum to zero. The jumped target must be -a, with 0<a<=s, while the four other phases lie in [0,D]. These are necessary consequences of renewal, not a prepared phase profile.

Suppose T+s<pi. For fixed other phasors, their resultant is nonzero and its angle lies in the common arc. The squared resultant as a function of one environmental phase has no interior minimum on that arc: its possible interior stationary point is a maximum. Moving each of the four phases to an endpoint therefore gives a lower bound. Increasing a to s and D to T decreases every nonconstant cosine term. With h of the four phases at the upper endpoint,

R_event^2 >= min_(h=0,...,4) [1+(4-h)^2+h^2
    +2*(4-h)*cos(s)+2*h*cos(T+s)+2*h*(4-h)*cos(T)]/25.

The paired ten-node resultant equals this five-block resultant. The bound retains the single shifted target; replacing all five phases by an arbitrary interval loses that information. The endpoint patterns belong to an enlarged algebraic domain and need not satisfy first-phase descent, pressure or form rails.

The controller restriction owner rebuilds the default law and both carried gains. It uses the shared rational cosine enclosure with eight terms (degree fourteen lower polynomial) and 128-bit outward dyadic rounding. Its finite budget is the nine branch pairs and five endpoint multiplicities per pair, with no refinement or source search. The common-arc and row-sign prerequisites are checked separately. The actual second disruptor load is 2/25, below the low-load threshold; the native ordered predicates, including their strict boundaries, determine which branch claims contradict the resulting lower bound.

This phase-box check excludes (transition,dissonant) and (dissonant,dissonant): their universal squared-resultant lower bounds exceed 0.49153 and 0.52391, respectively, whereas R_lo^2<0.40529. Thus seven of the nine carried branch pairs are not excluded by this necessary test. The pair (stable,dissonant) is not excluded by this box: its lower bound is about 0.404720, below the actual cutoff. The other six pairs also remain conditional. Its retained relaxed-minimum upper bound is also below the cutoff, by more than 1/2000. Greater cosine precision cannot close this relaxation; the missing correlations with the actual first-phase image and source remain essential. Here dissonant names the phase coordinator's configured branch; it is distinct from Dissonance (OZ), which remains the required first target operator. The actual first-phase image supplies the missing exclusion separately. Excluding controller pairs does not prove that a surviving pair is realized.

Exact single-boundary pressure-mask projection. At either selection boundary set

S_i = alpha*v_i/V + beta*(1-abs((L*theta)_i)/pi_rep) + gamma,
B_i = min(1,(S_i-si_hi)/gamma), i=1,...,4,
H = max(0,(S_0-si_lo)/gamma), c=dnfr_hi.

All four B_i must be nonnegative. When the pressure peak P is positive, Coherence (IL) is equivalent to abs(p_i/P)<=B_i at the surrounding blocks. A signed non-target peak at j additionally requires B_j=1. Let M_j=sum_(i!=0,j) B_i. The remaining three normalized pressures fill the entire interval [-M_j,M_j]: sums of closed intervals have no gaps. This proves the following exact existence conditions for the mask/peak/zero-sum subsystem.

Boundary and signed peak	Additional necessary and sufficient conditions
Source target positive peak	H<=1, sum_(i=1,...,4) B_i>=1
Source non-target negative peak j	B_j=1, H<=1
Source non-target positive peak j	B_j=1, H<=1, M_j>1+c, M_j>=1+H
Successor non-target positive peak j	B_j=1, H<=c, M_j>=1-c
Successor non-target negative peak j	B_j=1, H<=c, M_j>=1+H
Successor zero pressure	S_0<=si_lo, with all four surrounding masks passing

For the source, p0/P=t obeys t>c and t>=H; a positive non-target peak requires the others to sum to -1-t. Thus M_j>1+c is strict, whereas equality in M_j>=1+H is allowed when H exceeds c. For a negative non-target peak, the permitted target tie t=1 balances it. At the successor, q0/P=-t obeys H<=t<=c: a positive peak requires the others to sum to t-1, and a negative peak requires 1+t. Choosing t=c or t=H, respectively, proves sufficiency for these projected conditions. Peak ties remain admitted.

In particular, a lone pressure-capable surrounding block cannot balance a nonzero successor peak when the other three B_i vanish: the target's allowed ratio is below one. More generally the other blocks must supply at least 1-c of normalized pressure for a positive peak, or 1+H for a negative one. If every B_i is below one, every nonzero successor peak owner is excluded. Zero pressure retains its separate constant masks and no acquired negative-zero direction is inferred.

project_native_mutation_masks in the private compiler re-admits primitive parameters and computes these two boundary projections. The three-form shared-offset rows and fresh transport still couple the source and successor pressures; passing both projections does not solve that coupling. The source's strict OZ cutoff and acquired-rate threshold also constrain its absolute pressure scale. No normalized-pressure witness here is an admitted source, measured history or native Mutation event. The original 81 cells remain available with all those obligations intact.

The controller controls and mask projection controls check the finite coefficient bounds, interval-sum derivation, strict endpoints, zero-peak masks and agreement with the original affine rows. They preserve all earlier frozen producers and responses. Joint source existence and actual formation remain open until the full associated conditions are discharged.

Actual first-phase image and joint capacity boundaries

The first phase coordinator and both Coherence (IL) masks act on one source. Retaining that association gives two additional conditional obstructions in the same ideal paired five-cycle model. No prepared source, solver, trajectory, threshold adjustment or refinement of the preceding cosine bounds is used.

High-resultant phase averaging. For five real phase lifts in a common interval of width less than pi, set S=sum(exp(i*theta_j)) and R=abs(S)/5. If R>=3/5, every doubly stochastic averaging of these lifts preserves or increases R. This is a conditional phase-resultant theorem, not a universal monotonicity statement about the diagnostic C or the operator runtime.

First consider a pair with midpoint m and half-gap d, where abs(d)<pi/2. Its phasor sum is a*exp(i*m), with a=2*cos(d)>0. Write psi=arg(S). Projecting the other three unit phasors onto psi gives

a*cos(m-psi) = abs(S) - sum_(other three j) cos(theta_j-psi)
             >= abs(S)-3 >= 0.

A pair average reduces the absolute gap and adds b*exp(i*m) to S for some b>=0. Therefore

abs(S+b*exp(i*m))^2-abs(S)^2
    = 2*b*abs(S)*cos(m-psi)+b^2 >= 0.

The new phases remain in the old interval and retain R>=3/5. A pair average past the midpoint differs only by a permutation, which leaves R unchanged. These facts allow the pair operation to be repeated.

For completeness, a finite construction extends the pair result to any doubly stochastic matrix B. Sort x and y=B*x decreasingly. The sum of the largest k coordinates of y is sum(c_j*x_j) with 0<=c_j<=1 and sum(c_j)=k; it is at most the sum of the largest k coordinates of x. The total sums agree. If y1=x1, keep that coordinate and continue on the remainder. Otherwise choose adjacent sorted coordinates with x_k>=y1>=x_(k+1) and average them to (y1, x_k+x_(k+1)-y1). The residual lies between the original pair. Move y1 to the front. For a prefix of length r in the remaining sorted list, if r<k, every chosen coordinate is at least y1 and hence at least each remaining target coordinate. If r>=k, the prefix sum is the old largest r+1 sum minus y1, which dominates the corresponding target sum. Induction constructs y with at most four pair averages and permutations. All intermediate vectors stay in the same phase interval, so the preceding high-resultant argument applies at every step.

The high-resultant premise matters. As a formal lemma counterexample, take two unit phasors (24+7i)/25, (24-7i)/25 and three copies of (-4+3i)/5. They occupy a common arc shorter than pi. Replacing the first pair by two copies of 1 averages their phase lifts but decreases R squared by exactly 44/15625. Its initial R is below 3/5. This is neither a native source nor a counterexample to the particular five-cycle coordinator; it prevents an unsupported general averaging claim.

The stable first branch cannot lead to the dissonant second branch. Write its actual first gains as (g,l) and let N be the two-neighbor mean. Up to a common angle, which does not change R, the first phase image is

y=C1*theta, C1=(1-g-l)*I+l*N=(1-g)*B,
B=C1/(1-g).

For 0<g<1, 0<l<=1-g, B is nonnegative and doubly stochastic. After averaging by B, multiplication by 1-g also cannot lower R: every phase difference has absolute value below pi, and contracting it increases its cosine in R^2=(5+2*sum_(i<j) cos(theta_i-theta_j))/25. The first controller's stable branch requires R_source>=R_hi. Provided R_hi>=3/5, its actual coordinated phase thus has R_y>=R_hi.

Mutation (ZHIR) then changes one block phase by the negative shift s. The paired ten-node resultant equals the five-block resultant, so the triangle inequality gives

R_before_second >= R_y-abs(exp(-i*s)-1)/5 >= R_hi-s/5.

The source-image instrument rebuilds the owned law, first stable gains and actual disruptor loads, and checks these exact coefficient prerequisites. It applies no trigonometric approximation and consumes no prepared source or cached verdict. Under the current owned law R_hi-s/5>R_lo: the successor lower bound is about 0.708547, versus R_lo about 0.636620, with a positive exact gap greater than 0.071927. The actual second load 2/25 is below disr_lo, so the dissonant branch would require R_before_second<=R_lo. This contradiction excludes (stable,dissonant) through the actual first image. It leaves the old, inconclusive phase-box calculation unchanged.

Together with the preceding two renewal obstructions, this excludes every second-dissonant branch in the declared problem. Six carried branch pairs remain conditional: each of the three first branches followed by stable or transition. This does not establish that any of those pairs has a compatible source, acquired Mutation history or complete renewal. The new stable-first obstruction itself needs no complete-renewal inequality; the other two do.

Two saturated IL capacity floors are incompatible with outward geometry. Let h=L*theta, L=N-I, and m=(si_hi-beta-gamma)/alpha. At a surrounding block i, the actual IL mask at either boundary is equivalent to

alpha*(v_i/V-m) >= beta*abs(h_i)/pi_rep + gamma*rho_i,

where rho is the normalized absolute pressure, or zero when the full peak is zero. Both right-hand terms are nonnegative. Since capacities are held, a block at v_i/V=m that passes both masks must satisfy

p_i=q_i=0, h_i=(C1*h)_i=0.

Here L*C1=C1*L and the common phase angle is annihilated by L. For every admitted first branch, C1=(1-g)*I+l*L with l>0; hence also (L*h)_i=0. Moreover sum(h)=0. Two distinct such blocks force h to vanish on C5: up to rotation/reflection, an adjacent pair has h0=h1=0, giving h4=h2=0 through the two Laplacian rows and then h3=0 by the sum. A distance-two pair has h0=h2=0; its rows give h4=h3=-h1, and the zero sum forces h1=0. Since the cycle is connected, L*theta=0 makes theta constant. The first image is constant too, contradicting the strictly negative outward target gap. Consequently at most one of the four surrounding capacities can equal the common IL floor in a valid source. Capacities below the floor already fail a single mask. This excludes exact boundary configurations, not a neighborhood of them and not every heterogeneous configuration.

There is also a same-source pressure constraint at any saturated block. On the admitted multiplying OZ branch p0>OZ_TINY, set D=diag(f*u,ell*v1,...,ell*v4). The exact pressure transport is q=p+delta*k*L*D*p+d_p*(C1-I)*h. At a saturated block all terms except delta*k*(L*D*p)_i vanish, so (L*D*p)_i=0. If that block neighbors the target, its other neighbor j must obey

ell*v_j*p_j=-f*u*p0 < 0.

With r_j=v_j/V, r0=u/V, the source IL mask at j and strict source ratio p0/P0>c further require

r_j*[alpha*(r_j-m)-beta*abs(h_j)/pi_rep]
    > gamma*(f/ell)*r0*c.

The joint-boundary instrument retains the fresh law and exact two-distance-class reconstruction identities. Its controls check formal coordinate bases and associated pressure cancellation. The source-image controls check the averaging algebra, actual matrix, threshold premises and low-resultant counterexample. These are conditional proof and coefficient controls, not source evaluations. Full pressure transport, both masks, common form rails, history and actual branch admission still have to hold together. None of these results derives autonomous occurrence, formation or physical identification.

Exact elimination when the entire successor pressure vanishes

The zero-successor-peak subcase retains the complete equation q=0, with all five components zero. This is stronger than the target condition q0=0. It determines the source pressure from the same capacities and primitive phases, rather than allowing separately compatible pressure masks to choose different fields. The result is an exact conditional reduction of the nine zero-peak cells. It neither excludes the whole parameter domain nor supplies an admitted source.

Use the unchanged multiplying Dissonance (OZ) source branch, held capacities, Euler interval delta, and pressure coefficients (k,b,w) from the owned law. The phase coefficient in the common-lift pressure is d_p=w/pi_rep, not w. Write

L=N-I, h=L*theta, y=C1*theta,
D=diag(f*v0,ell*v1,...,ell*v4), tau=delta*k,
A=I+tau*L*D,
q=A*p+d_p*(C1-I)*h.

The common global phase angle is annihilated by L, and the first coordinator commutes with L. The matrix D belongs to the first call's nodal form rate; using D to express transport does not make it the second call's rate law.

The pressure transport is invertible throughout the admitted capacity rails. Set D_max=max_i D_ii. Every column of A sums to one. Its diagonal is 1-tau*D_ii and its two nonzero off-diagonal entries are tau*D_ii/2. Thus A is nonnegative when tau*D_max<1. The stronger uniform condition

2*delta*k*VF_MAX*max(f,ell) < 1

implies strict column diagonal dominance for every admitted capacity tuple. Here is an elementary invertibility proof. If A^T*z=0, choose i with maximal nonzero abs(z_i). Its row would require (1-tau*D_ii)*abs(z_i)<=tau*D_ii*abs(z_i), contradicting 2*tau*D_ii<1. Hence the kernel of the square matrix A^T is trivial, so A is invertible. Nonnegativity of A does not imply nonnegativity of its inverse. This proof establishes invertibility; evaluating one owned coefficient bound only verifies that the present model lies in its domain. The frozen owned-coefficient check gives tau*D_max<=0.094819 and a uniform strict-dominance margin 1-2*tau*D_max>0.81036. These are bounds over the capacity rails, not measurements of a selected source.

Consequently q=0 has the unique pressure

p_star=-d_p*A^(-1)*(C1-I)*h.

Since the forcing has zero sum and 1^T*A=1^T, sum(p_star)=0. There is no independent pressure amplitude or peak owner left to tune. Once the primitive parameters are fixed, the actual peak is max(abs(p_star)); ties may admit more than one canonical owner, but they describe the same pressure. The strict positive source peak, source OZ ratio, multiplying cutoff and acquired-rate threshold still have to hold for that pressure. If it vanishes, the strict source requirements reject it.

One shared form offset remains. At the first endpoint, q=0 is equivalent to L*(k*x1+b*v+d_p*y)=0. The connected cycle therefore gives

r1=-(b*v+d_p*y)/k,
r0=r1-delta*D*p_star,
x0=r0+eta1*1, x1=r1+eta1*1, x2=x1.

The last equality follows because the second form row uses held q, including after Mutation (ZHIR). The phase event and subsequent coordination can still change phase. Zero pressure at this boundary is therefore not an equilibrium of the complete state, and a later refreshed pressure need not remain zero.

For the ten coordinates of (r0,r1), the exact rail condition is

eta1 in [-1-min(r0,r1), 1-max(r0,r1)],
max(r0,r1)-min(r0,r1)<=2.

A single-point interval is allowed because the rails are closed. Different offsets for the two endpoints are not allowed. The original compiler uses eta relative to its mean-zero pressure-potential inverse. That eta differs from eta1 by one determined common constant. The implementation retains the original gauge and all thirty labelled rail rows for comparison, even though only the ten source/first-endpoint coordinate slots determine their interval.

Exact substitution, with source admission still conditional. The private zero-pressure reducer re-admits NativeMutationSlice, rebuilds the complete law and original rows, checks the uniform invertibility premise, and uses the shared exact rational five-by-five matrix inverse. It substitutes p_star into all nine original zero-peak cells and eliminates their one common form offset. The source's strict peak, OZ and history inequalities retain exact strictness; the successor uses its actual zero-ratio Si masks. No division by a zero peak or zero-scaled mask is used to certify those masks.

The resulting algebraic compatibility is necessary and sufficient for the pressure/form rows of each fixed-parameter zero-peak cell. Actual controller predicates, common-lift and renewal geometry, supplied execution context and native realization retain their separate obligations. In particular, an exact algebraic zero does not establish a represented negative-zero Mutation action. The six surviving controller pairs and global source existence remain open.

The independent controls use the previously declared capacity frame, formal phase coordinate bases and the previous algebraic phase frame. They check exact inverse residuals, ten-constituent pressure/flow identities, common-offset equivalence, original cell rows and rejected domains. These are coefficient and contract controls, not admitted preparations or a parameter search. The preceding controller exclusions, failed envelopes and frozen source evidence remain unchanged.

Joint zero-pressure restrictions and an inconclusive rate test

The complete zero-successor-pressure equation also couples the endpoint form rails to the actual zero-ratio selector masks. Keep the preceding law, notation, both masks, coherent phase lifts and complete negative-outward target-minimum renewal. The argument concerns multiplying Dissonance (OZ); the separate source fallback result remains unchanged.

Write the stability weights as (alpha,beta,gamma), target capacity as u and actual maximum capacity as V. At zero pressure, the difference between each surrounding high-Si mask and the target low-Si mask gives

alpha*(v_j-u)/V >= si_hi-si_lo-beta*(abs((L*y)_0)-abs((L*y)_j))/pi_rep
                > si_hi-si_lo-beta.

The strict inequality uses the common arc, which gives abs((L*y)_0)<pi_rep. The owned coefficient check has kappa=(si_hi-si_lo-beta)/alpha>0, so v_j-u>kappa*V>0 for every surrounding block. Constant endpoint potential and the common form rails therefore imply

d_p*(y_j-y_0) = k*(x1_0-x1_j)-b*(v_j-u) < 2*k-b*kappa*V.

A renewing negative Mutation (ZHIR) must put the target below the previous phase hull: otherwise positive phase averaging contracts that hull. Thus the postevent diameter satisfies D_event<s+(2*k-b*kappa*V)/d_p<B, where B=s+2*k/d_p. If D1=osc(y) and D2 is the complete second-call diameter, renewal and the second phase coordinator give D1<D2<=(1-G2)*D_event<(1-G2)*B. This is an associated phase/capacity restriction; it is not an independently selectable phase preparation.

Let z=(y1+y4)/2-y0>0, and let t be the near-block mean minus the far-block mean. The preceding target-minimum renewal geometry gives z<A2*s/(G2+K2) and t<A2*(s+z), with A2=1-G2-K2. The endpoint potential also gives d_p*z<2*k. Define

H=min(A2*s/(G2+K2), 2*k/d_p),
r(z)=r0+r1*z,
r0=(si_lo-beta-gamma)/alpha, r1=beta/(alpha*pi_rep),
m=(si_hi-beta-gamma)/alpha.

The zero-pressure masks require u<=V*r(z) and v_j>=m*V for surrounding blocks. Their near mean n obeys b*(n-u)+d_p*z<=2*k. When m>r(H), these joint relations license the uniform capacity ceiling

U=min(VF_MAX*r(H),
      max_{0<=z<=H} r(z)*(2*k-d_p*z) / (b*(m-r(H)))).

The numerator is a concave quadratic; its maximum is obtained from the two endpoints and clipped vertex. Zero is a closure endpoint of this calculation, not an admitted active-capacity source or outward gap.

Full transport bounds the pressure from phase forcing. The exact identity A*p=d_p*(C1^(-1)-I)*L*y follows from the preceding zero-pressure reduction. The matrix J=(C1^(-1)-I)*L is circulant with zero row sums. If K is the sum of its positive coefficients in one row, then abs((J*y)_i)<=K*osc(y) by subtracting the common minimum of y. At a row where abs(p_i)=P, the diagonal term of A is at least 1-tau*D_max, while the sum of absolute off-diagonal terms is at most tau*D_max. Consequently

P <= d_p*K*D1 / (1-2*tau*D_max).

This uses row bounds as well as the complete q=0 equation, and does not assert positivity or contractivity of A^(-1). The zero-pressure target mask also yields u/V<=(si_lo-gamma)/alpha. The checked owned inequality f*(si_lo-gamma)/alpha<=ell permits D_max<=ell*VF_MAX here.

For first gains (G1,K1), put A1=1-G1-K1 and det=A1*(A1-K1/2)-K1^2/4. The grouped phase inverse gives c_h=A1/det-1, c_t=K1/(2*det) and the exact target row

(1-tau*f*u)*p0 + tau*ell*(v1*p1+v4*p4)/2 = d_p*(c_h*z+c_t*t).

Check positive determinant, nonnegative coefficients and all denominators before taking upper bounds. The frozen calculation uses

Dbar=min(A2*s/(G2+K2/2), (1-G2)*B),
Pbar=d_p*K*Dbar/(1-2*tau*D_max),
Fbar=d_p*(c_h*H+c_t*A2*(s+H)),
u*f*p0 <= f*U*(Fbar+tau*ell*VF_MAX*Pbar)/(1-tau*f*U).

The fixed six-pair pass is inconclusive. In order (stable,stable), (stable,transition), (transition,stable), (transition,transition), (dissonant,stable), (dissonant,transition), the rate ceilings are approximately 0.143303, 0.114018, 0.132666, 0.105984, 0.128148, and 0.101957. Every ceiling exceeds the strict acquired-rate threshold xi=0.1; none excludes a remaining pair. These are upper bounds, not attainable rates, source witnesses or probabilities. The separate maxima discard correlations even though each bound follows from joint premises. No precision refinement, subdivision, parameter search or native response was performed after this outcome.

The same private owner provides derive_native_mutation_zero_obstruction(), a no-argument fresh-law coefficient report with intermediate bounds and explicit inconclusive status. The original reducer remains available. The independent controls check the exact phase/forcing identities, support norm, quadratic extremum, coefficient admission and absence of source or cached-report consumption. Global feasibility still requires all pressure, form, phase, controller and history conditions at the same primitive parameters. A source search failure would not prove exclusion, and algebraic zero would still not realize a native negative-zero Mutation. Formation and physical identification remain separate.

Bounded joint source proposal: no admitted candidate

The frozen protocol tests the constructive question with all quantities associated to one source. It targets strictly negative successor target pressure, q0<0, and a positive successor peak. The all-zero-pressure boundary remains a separate open subcase; a small numerical residual is never substituted for algebraic zero or its sign.

The complete two-call law, support, clock, supplied grammar/history, both actual high-Si Coherence (IL) masks, multiplying Dissonance (OZ), acquired Mutation (ZHIR), form rails and negative-outward target-minimum renewal remain unchanged. The search uses thirteen coordinates: five capacities, four source phase differences with theta0=0, and four independent source pressures with p4=-sum(p0,p1,p2,p3). The complete pressure-potential inverse then determines all fifteen source/endpoint form offsets. Their one common interval eliminates the remaining form origin; there is no independent endpoint adjustment.

The finite search restricts capacities to [1/64,VF_MAX] and source phase differences to [-3,3], with the original common width at most three. The pressure coordinates lie within the necessary source bound Pbar=2*k+b*VF_MAX+3*w/pi_rep; the fifth pressure must obey the same bound. The positive lower capacity cutoff belongs to this search policy, not to the complete model or a new physical hypothesis.

Proposal budget and evidence. The six surviving declared controller pairs each use the same four Sobol initial points, with seed 20261018. Each SLSQP run has a limit of 400 iterations and 5,000 objective evaluations, including finite-difference probes. The fixed objective penalizes violated normalized constraints, with a desired margin 1e-6 for strict predicates. That margin does not replace any exact admission threshold. Maxima, pressure ratios, both Si fields, carried gains, both controller resultants and all geometry are recomputed from the same proposed state. Binary64 calculations only rank proposals; a solver termination flag certifies no source.

The recorded summary contains 71,204 objective evaluations across 24 runs; two runs exhaust their evaluation budget. One lowest-penalty evaluated point per run is retained, with the first point chosen in a tie. The remaining 22 optimizer success flags describe numerical termination, not scientific admission. No additional restart, objective change, threshold change or parameter search followed. The artifact manifest associates the protocol, environment and selected proposals, the losslessly compressed full evaluation trace, exact results and source archive. Byte hashes establish consistency, not authenticated chronology or scientific truth.

Exact primitive readmission. Each selected physical coordinate is converted once to the nearest dyadic with denominator 2^40, with ties to even. The private point verifier rebuilds the complete law and compiler expressions from those primitives. It determines the actual pressure peaks and tied owners, intersects all fifteen form-offset slots, and uses their single midpoint. An empty interval is a rejection, not a correction. Wrong target signs or unavailable canonical peak owners leave the cell explicitly unavailable. Otherwise all original strict/weak cell rows are checked at that point, including all thirty form rails and both Si masks.

The first secant is calculated from the associated ideal form samples at t=1 and t=17/16. No supplied temporal rate can replace those samples. This remains a calculated ideal history, not measured native evidence. Both controller predicates are admitted independently from their consumed phase differences with the shared rational cosine enclosure, fixed at twelve terms and 128 outward dyadic bits. A straddling interval is unresolved, not passing; precision is not increased after seeing a result. The second Si mask uses the pre-Mutation phases, while the second controller uses the post-Mutation phases and carried gains.

Finite result. All 24 retained points are exactly rejected. Every retained point fails both strictly negative successor target pressure and complete phase-diameter renewal; some also fail other conditions. These statements concern the 24 retained, rationalized points, not an exact assessment of all 71,204 optimizer evaluations. The rejected sources supply neither a joint witness nor an all-domain contradiction. In particular, they do not prove that negative pressure and renewal are universally incompatible. Both the remaining nonzero-pressure domain and the separate zero-pressure domain stay open.

The point controls independently reconstruct ten-constituent fields, flows, temporal samples and controller enclosures on earlier formal coefficient frames. Explicit mocked wiring checks cell and interval boundaries without adding searched sources. The read-only evidence controls check archive association, the finite budget and selected evaluations, then rebuild the consumed primitive fields and phase diameters that reject the recorded candidates. They do not rerun the proposer or the point evaluator.

Scope of the research observable. Increasing complete-call phase diameter is the renewal observable chosen for this study. It is not itself a grammar requirement, a Mutation admission condition or a general criterion for collective organization. This finite failure therefore cannot obstruct every canonical operator investigation. Any change of observable requires its own scientific justification and prospective declaration; it cannot turn the frozen failed attempt into a positive result. Native execution, maintained collective identity, formation and physical identification retain their independent obligations.

Canonical action, identity and the phase-diameter observable

The scientific target is a declared NFR identity and its admitted structural response under a complete law. A successful registered action, a maintained identity and a useful diagnostic are different claims. The ontology and ordered-action criterion already supply their obligations; no new universal scalar criterion is added.

What the renewal problem assumes. The frozen two-call problem retains paired doubled-C5 support, represented mixed pressure, held capacity, a declared clock and held Euler step, zero Gamma and the native selector/controller. Its target Dissonance (OZ) followed by Mutation (ZHIR), surrounding high-Si Coherence (IL), strict complete-call diameter increase and remaining negative-outward orientation are restrictions of that question. The finite proposal adds its own capacity cutoff and numerical budget. None is a general definition of an NFR or a canonical Mutation requirement. The conditional exclusions and all 24 rejected retained points remain valid with their original definitions; this audit does not reclassify them.

Mutation admission instead consumes an actual target with active capacity, live associated signed EPI samples giving a secant above the configured threshold, and independent grammar context. Current pressure determines its signed phase action, not the historical growth sign. Invocation still requires a supplied request or a declared selector. These contracts are implemented choices whose structural necessity remains a separate foundational question.

The existing acquired-history calculation already admits Mutation on both target twins while its accepted phase prefix narrows the diameter. The later Resonance (RA) sign gate rejects independently; there is no complete second call or maintenance conclusion. Conversely, that owner's conditional expanding phase map lacks qualifying history on its specified acquisition route. Expansion therefore neither supplies admission nor is required by it. No new trajectory is needed to establish this distinction.

A phase margin is not finite extinction or strict renewal. Fix a declared identity requirement D>=epsilon>0. In the five-action contraction domain, define N_epsilon=min{n>=0: q^n*D_0<epsilon}, using exact comparisons. Any successful prefix reaching that index has left this margin, or has already left the theorem's premises earlier. Equality of the bound with epsilon does not exclude the closed condition. The index counts calls, not physical time, and proves neither successful continuation nor finite exact consensus.

The phase-writer ledger separately records intervening diameter increase r and realization defect e: D_next<=q*(D+r)+e. At the level of this inequality, e=0, r=(1/q-1)*D_star and D_next=D=D_star>0 are compatible. Thus maintenance with a phase margin need not mean D_next>D at every call, or even at one call of a constant-diameter sequence. This is a counterexample to that logical implication, not a native source, selected event or realized writer. Such equality is impossible for positive D under the unchanged five-action law with r=e=0. The ledger supplies no phase recharge mechanism or input.

Diameter and resultant do not retain relational response. On the same unit paired C5, take homogeneous form and positive capacity, fix target block zero and let 0<a<pi. The phase lifts

r=(0,a,0,0,0),       r_tilde=(0,0,a,0,0)

have the same phase multiset, diameter a and resultant magnitude abs(4+exp(1j*a))/5. Nevertheless the freshly prepared target pressure is w_phase*a/(2*pi_rep) in the first state and zero in the second: block one is a neighbor of zero and block two is not. The form, capacity and degree contrast channels vanish in both. No relabeling may discard the fixed target and its support association. Even the combined observation (D,R) therefore fails to determine this consumed pressure or its unforced form rate. This does not claim that the two states have different default operator marks. The static controls use a=1/4, independently compare the prepared pressure and check the exact margin/ledger distinctions. They install no history, event or selection law.

Current nodal channels alone also fail to determine Mutation admission when associated histories differ. The existing context controls hold the physical state fixed and distinguish fresh positive growth from a flat record, or equal recent traces with different prior Coherence/debt. These are supplied-context counterexamples, not acquired histories. Together the controls identify distinct missing information: relational geometry and target-associated execution context. Resultant magnitude does not solve these omissions; any proposed coherence score needs its own state-sufficiency argument.

Canonical continuation contract. Use the existing sufficient candidate Q=(M,d,V,e,r,H) with its complete law, support, clock and retained inputs. For a declared action, equal Q must give equal admission decisions and equal projected successors; a retained occurrence mark needs its own equality as well. Rejection, abstention and committed prefixes retain their actual state. If an identity predicate I(Q) is claimed, declare it and its admissible symmetries independently, then check it on the relevant prefixes and flow intervals. Admission does not establish I, and a prospective predicate must not be chosen merely because the evaluated source satisfies it. Formation additionally requires initially absent identity and subsequent acquisition.

The common-origin quotient does not retain absolute Mutation telemetry such as before/after angles and quadrant labels. Preserve their reference or mark them as reference-dependent outputs; semantic-state descent is not whole-graph equivariance. Retaining sufficient state likewise proves neither a smaller autonomous macro-node nor an autonomous occurrence law. This admission result returns the question to source-bound identity and complete action, while the execution plan alone chooses the next proof obligation.

Mutation preserves paired phases only for matched realized shifts

The joint native state retains unequal constituent forms and capacities but one primitive phase per pair. Its five-action theorem does not establish preservation of that chart by Mutation (ZHIR). The general equal-shift criterion has an existing event owner; the result here specializes it to the native mixed-pressure law, actual constituent admission and sequential execution. No source search, acquired-history claim or diameter-renewal condition is used.

Domain, admission and origin. Retain fixed unit undirected doubled C5, consecutive nonadjacent twins and declared node order. Use signed forms, positive individual capacities, associated temporal records, grammar and the direct native glyph map. This is a supplied-target action question. A default selector's request still requires its own live Si, pressure and lag admission. Direct apply_glyph validates Mutation's temporal/capacity gate and grammar state representation; it does not itself enforce incremental grammar. The strict native selector or an explicitly declared grammar-enforcement caller must admit each request. Setting GRAMMAR_REJECTION_MODE="raise" alone does not add a grammar call to direct dispatch.

For the exact equations use a common real phase lift and mathematical circle period, with represented coefficients treated as exact parameters. Runtime endpoints retain their binary64 proposal/wrap arithmetic separately. Supplied temporal samples end at the live form and declared clock, and their positive secants must independently exceed the configured threshold. Their presence does not prove that the current law acquired them. Mutation changes no form, capacity or stored pressure; it retains its phase telemetry and glyph context.

Exact chart boundary. On the incoming paired-phase chart, the existing fresh mixed-pressure rows give p_-=B+C, p_+=B-C, with C=k*d+b*e. These equal the consumed stored pressures only if preparation and intervening events justify that association. For two actually admitted unwrapped shifts delta_- and delta_+, write the extended phase coordinates as

theta_-=mu-a,                  theta_+=mu+a,
mu_next=mu+(delta_-+delta_+)/2,
a_next=a+(delta_+-delta_-)/2.

Starting at a=0, circular pairing survives if and only if delta_+-delta_- is an integer multiple of 2*pi. This follows by subtracting the two phase endpoints; the mean/contrast rows are their sum and difference. An inherited coherent lift with equal small shifts gives a_next=0. For the default nonzero signed shift s, with 2*s not a full turn and both pressures strictly nonzero, the two signs agree exactly when

(B+C)*(B-C)>0  <=>  B^2>C^2  <=>  abs(B)>abs(C).

If abs(B)<abs(C), the signs are opposite: mu_next=mu and a_next=-s*sign(p_-). Identical operator names and positive prior EPI growth therefore do not preserve paired phases. This is exit from the old coordinate family, not proof that an independently specified organizational identity dies.

At abs(B)=abs(C)>0, one pressure is zero; its stored sign bit must agree with the nonzero pressure's sign for the default shifts to match. If both pressures are zero, both actual sign bits matter. Exact real cancellation does not determine binary64 signed zero. A shared fixed shift preserves pairing when both events execute, independently of pressure. Zero shifts, full-turn differences and other configured factors require the general modular criterion rather than the default nonzero-sign shortcut.

Sequential prefixes and abstention. After the first even-twin shift s*sigma_-, but before the second action, the pair already has mu_next=mu+s*sigma_-/2, a_next=-s*sigma_-/2. A second equal shift restores pairing at the completed-pair boundary. It cannot make the earlier singleton prefix paired. Each target still needs its own live history and grammar.

On direct dispatch, invalid second-target temporal evidence raises before that target's mutation. On the native selector path, missing/stale/subthreshold evidence normally substitutes Coherence (IL) before grammar and records the abstention after successful dispatch. The resulting ZHIR/IL pair retains the half-shift split; IL also changes its own held pressure. Strict grammar does not disable this temporal substitution. If the second target has valid temporal evidence but fails strict grammar, the earlier Mutation remains committed and the batch stops. The native batch has already incremented its lag counters. These outcomes differ from all-target atomic rejection.

The action controls use fixed analytic preparations with M=1/4,d=1/16,V=1,e=1/8, initially common phase one, and adjacent-block mean-form shifts 0,+1/2,-1/2. They independently reconstruct B and C before actual direct actions and distinguish opposite-sign splitting from either common sign. Their matching timestamped samples and AL/IL/OZ context are supplied. Separate controls retain signed-zero/fixed branches, direct rejection and the actual native batch's IL abstention versus strict grammar failure. The batch uses a supplied callable; none of these cases is a default-selector occurrence or a full trajectory.

The split contains consumed relational information. For all five blocks, retain theta_(2i)=mu_i-a_i, theta_(2i+1)=mu_i+a_i alongside M,d,V,e. Require one common lift of width less than min(pi,pi_rep); width at most three is sufficient here. Then cos(a_i)>0, and each twin pair contributes

exp(1j*(mu_i-a_i))+exp(1j*(mu_i+a_i))=2*cos(a_i)*exp(1j*mu_i).
L_i=Arg_I(cos(a_(i-1))*exp(1j*mu_(i-1))
          +cos(a_(i+1))*exp(1j*mu_(i+1))).

The open-semicircle cone makes this resultant nonzero and selects its representative inside the common lift I. Both constituents of block i see the same four neighbors, hence the same L_i. With t=w_phase/pi_rep, the next fresh preparation is exactly

B_i=t*(L_i-mu_i)+k*(N_i(M)-M_i)+b*(N_i(V)-V_i),
C_i=k*d_i+b*e_i+t*a_i,
p_(2i)=B_i+C_i,                  p_(2i+1)=B_i-C_i,
P=max_i(abs(B_i)+abs(C_i)),       Vmax=max_i(V_i+abs(e_i)),
Si_(2i)=alpha*(V_i-e_i)/Vmax+beta*(1-abs(L_i-mu_i+a_i)/pi_rep)
         +gamma*(1-abs(B_i+C_i)/(P or 1)),
Si_(2i+1)=alpha*(V_i+e_i)/Vmax+beta*(1-abs(L_i-mu_i-a_i)/pi_rep)
           +gamma*(1-abs(B_i-C_i)/(P or 1)).

Here P or 1 denotes the exact zero-peak convention. Regular support removes degree contrast. Form/capacity neighbor contrasts cancel; the own phase contrast does not. The width and positive-capacity premises keep these Si rows in their unclamped domain. Other domains retain actual wrapping, resultant availability and the outer clamp. Never replace held pressure or prepared Si inside the current event batch with these refreshed values.

Two effects require a: its signed own contribution t*a_i and its neighboring resultant weight cos(a_i). To isolate the latter, keep phase means (0,1/4,1/2,-1/2,-1/4) plus common origin one, and split only block zero by a_0=s, 0<s<pi/2. Block one's neighbor angle, relative to that origin, changes from 1/4 to

Arg(cos(s)+exp(1j/2))=1/4+atan(tan(s/2)^2*tan(1/4)) > 1/4.

Multiplying by exp(-1j/4) gives real part (1+cos(s))*cos(1/4) and imaginary part (1-cos(s))*sin(1/4), proving the identity and strict sign. Unchanged mean phases therefore do not determine next pressure. Reversing a leaves the cosine weights unchanged but changes the own signed pressure contrast, so magnitudes alone do not retain the required association either. The static readout controls compare independent block formulas with shared fine pressure and Si, including nonzero form/capacity contrasts. They do not execute a trajectory or create history.

A coordinator contracts the split without restoring exact pairing. Let g,l be the actual shared coordinator gains, with their retained controller/load state. Its global target b and resultant magnitude are obtained from sum_i cos(a_i)*exp(1j*mu_i), with R=abs(sum)/5. The fine affine proposals give

mu_i_next=(1-g-l)*mu_i+g*b+l*L_i,
a_i_next=(1-g-l)*a_i.

Default gain rails have 0<1-g-l<1. A nonzero ideal split remains nonzero after this coordinator. Thus, conditional on a completed call with no other phase writer, held form flow, phase-inactive capacity adaptation and the existing REMESH-abstention premises, the old paired-phase chart is not restored. This is an ideal conditional consequence, not a guarantee that the native batch completes, an exact binary64 inequality near zero, or maintenance of a newly formed NFR.

The extended coordinates Q_plus=(M,d,V,e,r,a,H), r_i=mu_i-mu_0, retain nine relative fine-phase coordinates instead of four. They reconstruct the fine phases up to a common origin in this chart. Jointly permuting twins also transports d,e,a and their histories/ports; independent inventories lose that association. Absolute Mutation telemetry still needs its reference. This adds information already present in the fine engine, not a new force, operator or constitutive law. The pressure/readout and coordinator rows are closed on this retained state in their domains; a complete future operator word and any organizational identity require their own admission and proof.

Resonance on the constituent-specific split-phase chart

The Mutation boundary retains an internal phase contrast because admitted constituents need not keep a common phase. Resonance (RA) consumes that contrast through its live U3 neighborhoods. The earlier partial-subset result already permits split neighbors of a paired target. The additional boundary here is that the two target constituents select different subsets of the same neighbor snapshot. Their phase targets, signed form means and capacity triggers can consequently differ.

Domain and execution. Keep the fixed unit undirected doubled C5, ordered nonadjacent twins, positive capacities and admitted signed forms. Use the extended chart Q_plus above, a common phase lift I of width less than min(pi,pi_rep), and associated constituent kinds and grammar context. These are supplied states and requests, not histories acquired by Mutation or requests produced by the default selector. Each direct action below uses an explicitly declared strict grammar-enforcement caller before shared glyph dispatch. The native selector path retains its own preparation, lags and sequential admission. No continuous trajectory, threshold search, new operator implementation or collective executor is introduced.

The equations describe the ideal action with represented coefficients and an admitted subset. Actual binary64 angle comparisons, resultant arithmetic, two-stage form blending, wrapping and boundaries remain separate numerical operations. The fixed controls below keep a strict U3 margin; a tolerance comparison does not certify a boundary decision for arbitrary inputs.

Live constituent selection. Write the current associated coordinates as

x_(i,sigma)=M_i+sigma*d_i,    v_(i,sigma)=V_i+sigma*e_i,
theta_(i,sigma)=mu_i+sigma*a_i,                    sigma in {-1,+1}.
J_(i,sigma)={(j,tau): j in {i-1,i+1}, tau in {-1,+1},
             abs(wrap(theta_(j,tau)-theta_(i,sigma)))<=L}.

Indices are modulo five; the even constituent has sign -1. The actual configured U3 limit satisfies 0<=L<=pi_rep/2. Each admitted subset must be nonempty. It counts selected fine neighbors once, without conductance weighting. A selected neighbor does not imply admission of its twin. All neighbor phases are admitted before selecting; only selected forms enter the propagation mean. Recompute these sets after any earlier adjacent action.

For each target and adjacent block j, let n be the number of selected members and m their signed count (selected odd minus selected even). Then

N_sigma=sum_j n_(j,sigma),
S_sigma=sum_j (n_(j,sigma)*M_j+m_(j,sigma)*d_j)/N_sigma,
Z_sigma=sum_j exp(1j*mu_j)*(n_(j,sigma)*cos(a_j)
                          +1j*m_(j,sigma)*sin(a_j)),
eta_sigma=Arg_I(Z_sigma).

The selected phasors lie in the same open semicircle, so their projection onto its midpoint direction is positive and Z is nonzero. The formula for Z follows by adding the selected positive/negative phase exponentials. Whole-block selection has m=0; selecting one member retains both its signed form contrast and its signed phase contribution. The cosine weights used by the fresh four-neighbor pressure are therefore insufficient for this action. Neither unordered contrasts nor independent form and phase inventories recover S and eta. Preserve their common constituent/port association.

Admission and pair action. Denote the actual factors by rho=RA_epi_diff, b=RA_vf_amplification, c=RA_phase_coupling, with 0<rho<=1, b>0, 0<c<=1. Let T be the shared represented amplification trigger, and let clip denote the admitted operator boundary policy. Each target proposes

F_sigma=clip((1-rho)*(M_i+sigma*d_i)+rho*S_sigma),
u_sigma=I(abs(S_sigma)>T),
v_sigma_star=(1+b*u_sigma)*(V_i+sigma*e_i),
theta_sigma_star=(1-c)*(mu_i+sigma*a_i)+c*eta_sigma.

The common lift justifies the last row; outside it retain the actual circular policy and resultant availability. A strict nonzero form-sign inversion rejects. Exact zero remains neutral, and an established kind must independently survive the selected-neighbor kind rule. A finite nondecreasing capacity proposal, finite arithmetic, U3 and grammar are additional gates. The trigger uses the selected signed mean, not selected magnitude or the block mean. An admitted zero selected mean suppresses amplification without suppressing form or phase propagation. Kind/sign rejection precedes target-state, glyph history and RA-telemetry writes; it does not promise rollback of graph caches or an earlier accepted target.

For consecutive actions on the two nonadjacent twins, the first changes none of the second's neighbors. If both requests pass, their completed-pair moments are consequently

M_i_star=(F_-+F_+)/2,                  d_i_star=(F_+-F_-)/2,
V_i_star=V_i+b*(u_-*(V_i-e_i)+u_+*(V_i+e_i))/2,
e_i_star=e_i+b*(u_+*(V_i+e_i)-u_-*(V_i-e_i))/2,
mu_i_star=(1-c)*mu_i+c*(eta_-+eta_+)/2,
a_i_star=(1-c)*a_i+c*(eta_+-eta_-)/2.

With inactive clipping, the first two rows become M_i_star=(1-rho)*M_i+rho*(S_-+S_+)/2 and d_i_star=(1-rho)*d_i+rho*(S_+-S_-)/2. Thus unequal subsets introduce a contrast source and can introduce unequal amplification even from equal capacities. The older common-subset multiplier is a special case, not a coefficient transferable to every split target. If only one action executes, average its accepted endpoint with the untouched twin instead. Rejection does not complete that pair.

Order preservation on the same neighbor snapshot. Suppose a_i>0 and both target subsets are nonempty. Their selected mean angles satisfy eta_-<=eta_+. To prove this, move the closed interval [theta-L,theta+L] right from the even to the odd target. If the selected sets overlap, every removed phase is below every common phase and every newly selected phase is above every common phase. Within the open-semicircle cone, adding lower phasors cannot raise the resultant angle, and adding higher phasors cannot lower it. Thus the mean of the common set lies between the two means. If the sets are disjoint, all phases of the first set are below those of the second, giving the same ordering. Boundary ties retain weak inequalities. Consequently,

a_i_star >= (1-c)*a_i > 0,                         for 0<c<1.

For negative a, exchange the constituent labels to obtain the corresponding absolute-value bound; their associations must move with the labels. At a=0 the two sets and mean angles agree. The default factor is strictly below one, so this ideal admitted pair action cannot exactly remove an existing split. The bound is sharp when both targets select the same mean; c=1 can merge them in that case. It is not a bound for different neighbor snapshots, undefined resultants, failed actions or binary64 differences near zero.

Internal separation need not contract. A static default-limit control sets target block zero to mu_0=2,a_0=1/8, and both neighboring blocks to mu_j=2,a_j=3/2. Their phases occupy the common interval [1/2,7/2] of width three. For either target, the same-sign neighbor gap is 11/8<pi_rep/2 and the opposite-sign gap is 13/8>pi_rep/2. Hence the even target selects only phases 1/2, and the odd target only phases 7/2. Both selected resultants are defined. Positive interior forms with a common kind independently admit the identity gates. For the default positive phase factor,

eta_-=1/2,    eta_+=7/2,
a_0_star=1/8+(11/8)*c > 1/8.

Each target moves toward its own compatible neighbors, while the unchanged neighbor extrema keep the global phase hull fixed. This is compatible with the earlier global-hull retention argument. It refutes universal contraction of an internal twin contrast under RA; it does not establish global diameter growth, a completed native-call increase, sustained separation or formation. The later coordinator still has its own contraction row and actual gains.

Checks and composition boundary. The static action controls also use target phases 1,3/2, neighboring block-one phases 5/2,11/4, and block-four phases 1/4,1/4. The default subsets are respectively {2,8,9} and {2,3,8,9}. Reversing block one's form contrast at fixed means changes only the first target's selected mean. Independent predictions check each form, circular phase target, capacity and reconstructed pair contrast. Later-target controls distinguish live adjacent propagation from a frozen snapshot and preserve an accepted first action when a second fails its independent sign or kind identity gate. These controls are supplied action requests with explicit grammar, not a default-selector occurrence witness.

RA changes no stored pressure. A later adjacent RA consumes the updated neighbor form/phase/kind and its own capacity, without refreshing the batch's prepared pressure or Si. Neighbor capacity changes enter later preparation and adaptation, not that local RA mean. Histories and bookkeeping follow the actual execution path. Atomic stages instead propose from one immutable snapshot; they cannot replace this sequential map. Q_plus retains enough associated coordinates to reconstruct these local decisions, but that invertible chart is neither a smaller autonomous NFR nor a new identity predicate. A complete call must separately join the admitted event prefix to held-pressure flow, individual temporal/rate records, coordination and capacity adaptation.

Conditional complete-call composition with split phases

The constituent-specific Resonance action, joint held-flow calculus and split readout/coordinator compose on the same retained state. The essential distinction is between post-event capacity, held pressure and terminal adapted capacity. Their association determines the form response and the temporal evidence supplied to a later request. This is a conditional complete-law description, not a new execution API.

Scope and order. Fix unit undirected doubled C5, consecutive nonadjacent twins, node order 0..9, signed forms, positive capacities and one common phase lift of width less than min(pi,pi_rep). Retain Q_plus, associated context, incoming rates, the declared clock and the actual request origin. Consider one admitted complete batch with either Resonance (RA) on both members of a pair or Coherence (IL) on both. Each member has one request and its own strict live grammar and physical gates. Incoming phase splits may be supplied or separately acquired; Mutation is not inserted as another request on those targets during this call.

Use default mixed pressure and Si preparation, the live built-in zero Gamma source, a single Euler substep delta=1/16, and default phase/capacity policies. Require every event, flow and adapted form/capacity to remain strictly inside its rails; canonical post-flow clamps then leave those channels unchanged. The exact equations use ideal arithmetic with represented parameters and a declared lift. Represented phase storage, rounding, finite arithmetic and decision boundaries remain separate execution obligations. No external callbacks, optional mathematics engine or support-changing event is included; validators must pass and automatic REMESH must independently abstain.

The actual runtime order is: left temporal boundary, fresh pressure/Si, prepared requests with live sequential admission, post-event temporal boundary, held integration, canonical clamps, phase coordination, capacity adaptation, final temporal boundary, whole-support EPI archive and REMESH admission. Failure stops at its actual prefix. Rejection does not supply the missing flow, coordinator or terminal state. A supplied request schedule does not establish that the default selector chooses that schedule.

Event moments and held flow. Reconstruct incoming pressures B_i+/-C_i and prepared Si from the split-phase readout above. After the sequential batch, denote its actual admitted endpoints by

x_(i,sigma)^E=Y_i+sigma*D_i,     v_(i,sigma)^E=U_i+sigma*E_i,
p_(i,sigma)^E=qbar_i+sigma*qdelta_i,
theta_(i,sigma)^E=psi_i+sigma*alpha_i.

On an RA pair, use the two distinct selected means, phase targets and amplification indicators from the preceding action theorem; qbar=B and qdelta=-C. On an IL pair, form, capacity and phase retain their incoming values, while (qbar,qdelta)=IL_dnfr_factor*(B,-C). Earlier adjacent events enter each RA's live subset and propagation inputs. They never replace the prepared pressure or Si by a refreshed value within this batch.

Expanding the two fine nodal products gives

fbar_i=U_i*qbar_i+E_i*qdelta_i,
fdelta_i=U_i*qdelta_i+E_i*qbar_i,
M_i^F=Y_i+delta*fbar_i,          d_i^F=D_i+delta*fdelta_i,
f_(i,sigma)=fbar_i+sigma*fdelta_i.

The stored rate is f_(i,sigma). Its stored acceleration is (f_(i,sigma)-old_rate_(i,sigma))/delta, retaining each old rate's association. For RA, the mean is U_i*B_i-E_i*C_i, where C includes the incoming phase contrast. Substituting the new phase contrast into C would silently change the held-pressure law. Replacing the mean product by U_i*qbar_i loses the exact term E_i*qdelta_i; its one-step form error is delta*E_i*qdelta_i. Unequal RA triggers can create that term even when incoming capacities are equal. All these are identities of the admitted held model, not licenses to reconstruct independent pressure from response.

Individual temporal evidence. Let x_n be the live form at the initial boundary t and x_n^E its event endpoint. If they differ, the recorder resets that node's same-time record to [(t,x_n^E)]. If they agree, it retains its admitted earlier history. The final recorder appends the flow endpoint at t+delta, with the existing bounded history length. Thus the last two samples give the ideal secant

(x_n^F-x_n^E)/delta=f_n.

The event jump divided by delta is excluded. RA and IL constituents may therefore keep different history lengths despite sharing the same clock. Represented samples give their actual represented secant; exact equality to the separately rounded stored rate is not assumed. Capacity adaptation later changes neither these form endpoints nor the already stored rate. Their values remain measured-flow evidence, not a prediction using terminal capacity. The next Mutation request still needs its active-capacity, threshold and independent grammar gates; a valid record alone does not select or admit it.

Coordinator and relative state. Rebuild the post-event phase targets from all four support neighbors, not the RA-compatible subset:

L_i^E=Arg_I(sum_(j=i-1,i+1) cos(alpha_j)*exp(1j*psi_j)),
b^E=Arg_I(sum_j cos(alpha_j)*exp(1j*psi_j)),
R^E=abs(sum_j cos(alpha_j)*exp(1j*psi_j))/5.

All these resultants are nonzero on the common open semicircle. The controller consumes this R, the actual updated glyph-history load and retained incoming gains. Let g,l be its resulting gains, not restarted defaults. The shared coordinator gives

mu_i^C=(1-g-l)*psi_i+g*b^E+l*L_i^E,
a_i^C=(1-g-l)*alpha_i,
r_i^C=(1-g-l)*(psi_i-psi_0)+l*(L_i^E-L_0^E).

Retain the gains and their history for the next invocation. A common rotation cancels from the relative row, R and local decisions, extending the existing ideal origin quotient to the split chart. Absolute telemetry still requires a reference, and bit-exact rotation symmetry of the implementation is not asserted.

For a complete RA pair with incoming a_i!=0, the same-neighbor-snapshot order theorem and default 0<c<1, 0<1-g-l<1 imply

abs(a_i^C)>=(1-g-l)*(1-c)*abs(a_i)>0.

For IL the corresponding equality is abs(a_i^C)=(1-g-l)*abs(a_i). Meanwhile every RA event stays in the current common phase hull and the coordinator gives diameter_next<=(1-g)*diameter_in. Nonzero internal separation at every admitted finite call is therefore compatible with global contraction. This neither supplies an infinite admitted sequence nor a fixed positive identity margin. Unequal requests inside a pair, Mutation, other phase writers or failed admission require their own phase-order argument.

Capacity adaptation and next preparation. Qualification uses the stored prepared Si and post-event held pressure. Apply its actual threshold/count policy per constituent and let h_-,h_+ be the resulting due flags. With hbar=(h_-+h_+)/2, hdelta=(h_+-h_-)/2, m=VF_ADAPT_MU and the immutable event-capacity neighbor mean W_i=(U_(i-1)+U_(i+1))/2,

V_i^A=U_i+m*(hbar*(W_i-U_i)-hdelta*E_i),
e_i^A=E_i+m*(hdelta*(W_i-U_i)-hbar*E_i).

The existing fixed-point/hull guards and admitted capacity rails remain in force. Counters count qualifying calls and do not reset after a due write. If v_n^A!=v_n^E, replacing the consumed capacity in this call's rate would add the spurious term (v_n^A-v_n^E)*p_n^E. The new capacity belongs to subsequent laws and observations. Refreshing Si before this writer would also change the selected adaptation policy.

The physical successor is (M^F,d^F,V^A,e^A,r^C,a^C), together with the actual updated H: rates, samples, kinds, grammar/debt, counters, lags, ordinals, controller records and REMESH evidence. Its old stored pressure/Si remain distinguishable from a next preparation. To evaluate that preparation, rebuild the split pressure/Si equations using these terminal coordinates, including their new pressure peak and maximum individual capacity. Re-admit every next request and live prefix. Current-call success alone proves no next-call request pattern, identity preservation or future successful execution.

Verification boundary. The composition controls freeze the existing asymmetric cancellation geometry, supplied RA requests on nodes 0/1 and IL on nodes 2..9, distinct incoming rates and associated temporal samples. The selected signed means at the two RA targets are zero and 5/32, so their capacity multipliers differ. Independent predictions precede shared stage execution in runtime order. They test the covariance, individual resets, controller/coordinator, actual adaptation eligibility and terminal refreshed readouts. Separate algebraic adaptation controls retain their own supplied flags rather than inventing eligibility for this source. The checks execute neither runtime.step nor a trajectory campaign; existing runtime wiring controls supply the stage-order integration boundary. This closes the declared conditional composition, not a smaller autonomous NFR, acquired source, default-selection witness or maintained collective identity.

The aggregate Resonance capacity obstruction and its separate sufficient inheritance domain now belong to the aggregate operator owner. This anchor preserves published links.