Collective IL/OZ actions and constitutive-pressure handoff

This owner gives exact-real conditional identities, a simultaneous-IL maintenance theorem and a frozen represented handoff on the actual maintained count-three contact. The executed word retains the joint identity and exhibits the predicted OZ refresh equivalence. Its frozen per-target IL phase-error certificate was unresolved in the original report; a separate retained-input review now certifies the original error allowance. Direct endpoint and work checks have independent support. The research program alone selects further work; the result and its accuracy distinction remain here. No additional pressure law, event selector or research queue is introduced.

1. Consumed state and the selected complete law

Retain the actual twenty-node, 45-edge contact of SINE_PROXIMITY_CONTACT: forty old unit edges, the internal A0 edge and four A0--B0 edges. All actual weights remain attached to their endpoints and held fixed. The capacities are nu=1/2 on the two A0 targets and one elsewhere; bare degrees are seven on A0, six on B0 and four elsewhere. Use signed real form, primitive circular phase with its declared lifts, no forcing or continuous loss, and the original structural clock.

Write the consumed continuous state as y=(x,theta,nu,U,W), and put

\[ S_i(\theta)=\sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad \Phi_i(y)=\frac{S_i(\theta)}{\pi d_i},\qquad M=\operatorname{diag}\frac{\nu_i}{\pi d_i}. \]

The law is independently specified by

\[ \dot x=MS(\theta),\qquad \dot\theta=ML_Wx. \]

The stored pressure p_store consumed by operators is a distinct channel. It need not equal the freshly evaluated constitutive pressure Phi. Keep execution state k as well: associated histories, grammar and lifecycle state, timestamps, policy inputs, random progress and required telemetry. Thus an operator acts on more than y, even when the subsequent continuous law consumes only y. The parameter foundation and form foundation own this distinction; no pressure is reconstructed from a measured form rate.

The continuous storage and mean weights are

\[ H(y)=\frac12x^TL_Wx+\sum_{\{i,j\}\in U}(1-\cos(\theta_j-\theta_i)), \qquad \rho_i=\frac{d_i}{\nu_i}. \]

Here rho=14 on A0, six on B0 and four elsewhere, with totals rho_A=60,rho_B=44,rho_all=104. H and both global rho-weighted form and lifted-phase means are conserved between events. Stored pressure and execution state do not occur in this particular H or these flow rows.

Use the same common-aligned twist reference theta_* and joint observation

\[ Z^2=\|P_{20}x\|^2+\|P_{20}(\theta-\theta_*)\|^2,\qquad r=1/16. \]

The proved storage barrier is H-H_*>=Z^2/25 on Z<=r, with boundary value 1/6400. The maintained source and its independent-registration extension have Z<r and strict storage excess below that value. Those hypotheses, not a diagnostic coherence label, are used below.

2. Pressure refresh and the exact OZ projection equivalence

Let pi_c(y,p_store,k)=y denote projection onto the consumed continuous state. An admitted canonical OZ action, including its ordinary propagation or noise branch when enabled and finite, changes stored pressure and execution context while preserving x, phase, capacity, support and weights. Consequently

\[ \pi_c\circ J_{\rm OZ}=\pi_c,\qquad \Phi(\pi_c J_{\rm OZ}z)=\Phi(\pi_c z). \]

An immediate refresh that evaluates this specified Phi therefore assigns the same constitutive pressure with or without that OZ action. The complete fixed-law continuation from the projected state is identical in both cases for all time, provided no further events, controllers or inputs consume the changed execution state. OZ also has Delta H=0 in this model because H does not contain stored pressure.

These are projection and continuation identities, not equality of the full operator states. OZ can leave different grammar debt, lineage, history, propagation records, random progress or diagnostic flags. Those differences can affect later eligibility or execution. Removing OZ from a word requires a separate argument about those later consumers; it does not follow from its continuous-state neutrality.

The OZ mechanism includes multiplicative pressure amplification, a nonzero near-zero seed, optional jitter and signed accumulation of neighbor contributions. The default gain (pi+1)/pi is a supplied factor. The local amplification and final merged pressure are different observations; incoming positive contributions can cancel negative pressure. None of these branches supplies a pressure lifetime or a retained forcing term in the selected continuous model.

The mathematical projection above is not a new runtime operator. A detached weighted-field report evaluates Phi without mutating stored pressure. An executed refresh must use its declared callback boundary and selected-law writer; the default mixed-pressure callback is not this law. A held-pressure interval or an additional pressure-memory/input model would have different premises and does not inherit this equivalence automatically.

3. The full simultaneous IL action

The shared IL kernel contracts each selected stored pressure by its admitted retention factor. The ordinary default is pi/(pi+1). Unlike the pressure-only glyph, the public class and immutable snapshot stage also propose a phase change. For target i, define its unweighted old-neighbor resultant

\[ \mathcal R_i=\sum_{j\sim i}e^{i\theta_j}=R_i e^{i\mu_i},\qquad a_i=\operatorname{wrap}(\mu_i-\theta_i),\qquad d_i^{\rm IL}=\lambda a_i. \]

The event writes theta_i'=theta_i+d_i^IL modulo 2*pi and preserves x, nu, U and W. The admitted phase coefficient lies in [0,1]; its ordinary implementation default is 0.3. The theorem below covers this entire interval and selects no new coefficient. Both A0 targets read the same immutable snapshot. Neither proposal observes the other's new phase. The kernel also has declared isolated/zero-resultant fallbacks, although the maintained contact domain below excludes those cases locally.

Storage, identity and mean calculations retain the continuous endpoint lift theta_i+d_i^IL. A wrapped storage representative near 2*pi must not be mistaken for a large physical displacement from a nearby zero lift.

The default IL retention and OZ amplification are reciprocals only in their ideal scalar multiplicative branches. They are not inverse hybrid actions: IL can change phase, OZ has near-zero/propagation/noise branches, both change execution context, and a declared refresh can intervene. Even cancellation of two stored-pressure factors would not reverse IL's phase reset or erase its history.

The actual staged word still needs live input and grammar admission. This section defines the primary map conditional on admission; it does not claim that every point in a mathematical identity neighborhood has the history, stored pressure or active runtime settings required to execute that word. Represented trigonometric rounding is also distinct from the exact-real event considered in the proof.

4. Uniform local resultant and phase-displacement bounds

Let u and v be the adjacent A0 targets. Each has its partner, four old A1/A4 neighbors and two B0 neighbors. At the common-aligned reference, its neighbor resultant is

\[ R_*=3+4\cos(2\pi/5)=2+\sqrt5. \]

This is a local resultant. The whole doubled-twist reference has zero first circular resultant, so it does not define a global resultant angle. The proof instead uses the already declared continuous lifts.

For a state in Z<r, write theta=theta_*+c*1+eta, where eta=P_20(theta-theta_*) and ||eta||<r. On either seven-neighbor set, the elementary bound |exp(i*t)-1|<=|t| gives

\[ |e^{-ic}\mathcal R_i-R_*| \le\sum_{j\sim i}|\eta_j| \le\sqrt7\,\|\eta\|<\frac{\sqrt7}{16}<\frac16. \]

Since sqrt(5)>11/5, the real part of exp(-i*c)*Rcal_i exceeds 2+11/5-1/6>4, and its imaginary part has magnitude below 1/6. Hence

\[ R_i>4,\qquad |\mu_i-c|<\arctan(1/24)<1/24 \]

on the unique local direction near c. Each target's reference phase is zero, so |theta_i-c|<1/16. The shortest target displacement is unique and satisfies

\[ \boxed{|a_i|<\frac1{16}+\frac1{24}=\frac5{48}<\frac18.} \]

These bounds require neither equal target members nor equal boundary conductances. The phase mechanism uses bare neighbors; its local resultant is not weighted by the actual conductances.

5. Exact shared-edge correction for two simultaneous targets

For clarity put d_u=d_u^IL,d_v=d_v^IL in this section, and write delta=theta_v-theta_u. If only target i moved with its old neighbors fixed, its incident phase-storage change would be

\[ G_i=R_i\left[\cos a_i-\cos((1-\lambda)a_i)\right]. \]

Adding the two such expressions counts the shared edge as two independent single-target changes. The true simultaneous storage jump is instead

\[ \Delta H=G_u+G_v+C_{uv}, \]

\[ \begin{aligned} C_{uv}={}&\cos(\delta-d_u)+\cos(\delta+d_v) -\cos\delta-\cos(\delta+d_v-d_u)\\ ={}&-4\cos\left(\delta+\frac{d_v-d_u}{2}\right) \sin\frac{d_u}{2}\sin\frac{d_v}{2}. \end{aligned} \]

All other edges are counted correctly in G_u+G_v; edges not touching u or v do not change. The form part of H is unchanged. The correction has coefficient one because the phase potential uses bare support, even though the internal edge's form conductance is the actual omega.

The correction has no general sign, but always obeys

\[ |C_{uv}|\le|d_ud_v|\le\frac12(d_u^2+d_v^2). \]

For example, on a graph consisting of one edge, full snapshot alignment with lambda=1 exchanges its two distinct phases. The true phase storage is unchanged although each separately evaluated single-target cost is negative. The correction supplies exactly the missing positive amount. Thus the existing single-target IL inequality cannot simply be summed for adjacent targets.

For the maintained contact, the local bounds in Section 4 give a uniform negative allowance that controls this correction. On 0<=s<=1/8, sin(s)>=3*s/4, since cos(s)>=1-s^2/2>3/4. For 0<lambda<=1,

\[ \begin{aligned} G_i &=-R_i\int_{(1-\lambda)|a_i|}^{|a_i|}\sin s\,ds\\ &\le-\frac{3R_i}{8}(2\lambda-\lambda^2)a_i^2\\ &\le-\frac32\left(\frac2\lambda-1\right)d_i^2. \end{aligned} \]

For nonzero d_i the last bound is strict because R_i>4. Combining both targets with the shared-edge correction yields

\[ \boxed{\Delta H\le-(d_u^2+d_v^2)\le0.} \]

The first inequality is strict whenever at least one displacement is nonzero. If both vanish, the phase-storage change is zero. At lambda=0 the primary phase map is the identity and the storage jump is exactly zero, while the stored-pressure and execution-context changes may remain. This proves conditional simultaneous passivity on this contact domain; it is not a universal theorem for arbitrary support or simultaneous IL stages, nor a strict-response lower bound for every admitted source.

6. The reset path preserves the existing joint identity

A nonpositive endpoint storage jump alone would not exclude jumping to a different geometric neighborhood. The actual finite phase displacement also has a continuous comparison path

\[ \theta(s)=\theta+s d^{\rm IL},\qquad 0\le s\le1, \]

with x, nu, U and W fixed. Every prefix uses the same snapshot resultants and equals the ideal simultaneous formula with coefficient s*lambda. The preceding work identity and estimate therefore give

\[ H(x,\theta(s))-H(x,\theta)\le -s^2(d_u^2+d_v^2)\le0. \]

This inequality is computed from the initial snapshot and does not assume that the whole path already lies inside the identity radius. Suppose the prestate has Z<r and H-H_*<1/6400, as every state on the proved maintained continuation does. If the reset path first reached Z=r, the existing boundary barrier would require H-H_*>=1/6400 there, contradicting its nonincreasing storage. Thus the entire path and the actual endpoint retain Z<r and the strict storage margin.

Restarting the same held weighted law from that endpoint preserves the identity for all subsequent structural time by the existing conservative barrier proof. An admitted OZ pressure-only action and the selected-law refresh do not alter this conclusion because they leave y fixed. This argument supplies no occurrence schedule or phase-flow law for IL: the path is a geometric certificate for a supplied jump, not an asserted finite-duration dynamics or an interpolation executed by the engine.

7. Mean leaves, fresh rates and sufficient collective information

IL leaves the global weighted form mean unchanged. Its global lifted-phase mean changes by

\[ \Delta\mu_\theta=\frac{14(d_u+d_v)}{104} =\frac7{52}(d_u+d_v). \]

Using the joined regional phase observations, the jumps are Delta Theta_A=7*(d_u+d_v)/30 and Delta Theta_B=0. The post-event continuous flow conserves these new means; it need not retain the pre-event phase-mean leaf. Translating the common reference phase to match the new global mean changes neither H nor Z.

The fresh continuous rows have distinct instantaneous behavior:

\[ \Delta(\dot\theta)=0,\qquad \Delta(\dot x)=M\left[S(\theta+d^{\rm IL})-S(\theta)\right]. \]

The first identity follows from unchanged x, W and nu. The second is the specified constitutive recomputation, not multiplication of the old form rate by IL's stored-pressure retention. Fresh pressure can change at the targets and their neighbors; OZ has zero changes in both fresh rows when y is unchanged. A nodal product built from stale stored pressure is a different observation and cannot replace these evaluated rows.

For the two IL phase proposals, the common external phasor from the six neighbors other than u and v, together with both target phases, determines both local resultants. That small description suffices for these primary phase proposals and their incident phase work. It does not replace the full fine state needed for subsequent motion, the separate A/B boundary associations needed for regional exchange, or the retained stored pressure and execution context needed for admission.

Any relabeling symmetry must transport target selection, node state, conductance endpoints, histories and rank-dependent policies together. A snapshot-equivariant primary formula does not create a pair-symmetric continuous quotient for generic actual weights. Neither a common phase mean nor an unordered multiset of independent histories proves descent of the complete word.

8. Reuse and memory boundaries

The complete reset accounting and identity-retention method provide the useful pattern: retain actual event work, sufficient endpoint state and an applicable barrier. The present proof re-admits their method on this support and corrects its overlapping IL edge explicitly. It does not transfer coefficients from an unweighted or dissipative comparison.

The memory owner, nonlinear constituent memory and ontology distinction separate eliminated dynamical state, supplied delayed history and operator execution context. Hidden fine initialization can affect a reduced future, and execution history can affect a later operator. Neither fact supplies an independent lifetime for an overwritten stored-pressure intervention. A history buffer or reciprocal scalar factor is not a realization of a derived memory kernel.

Even differentiating the selected constitutive pressure produces only a consequence on its own state constraint. With D_b=diag(d_i) and the bare phase Hessian K(theta)=L_(cos(theta_j-theta_i)),

\[ \dot\Phi=-\frac1\pi D_b^{-1}K(\theta)ML_Wx. \]

This follows by differentiating Phi=D_b^{-1}S/pi under the already specified complete law. It does not prescribe an independent pressure state away from p=Phi(y), authorize an arbitrary pressure reset there, or add a memory/source law. Any such extension needs its own hypotheses, initial-state admission and discrimination.

If a later study explicitly declares an interval that consumes stored pressure instead, put r_p=p_store-Phi(y) and retain the same phase row. The hypothetical form row and its exact storage balance would be

\[ \dot x=\operatorname{diag}(\nu)(\Phi+r_p),\qquad \dot H=(L_Wx)^T\operatorname{diag}(\nu)r_p. \]

Indeed, the q^TMS and -S^TMq terms still cancel, leaving precisely the displayed additional work. This identity specifies no law, duration or reservoir for r_p; even held stored pressure need not give a constant deviation from the evolving Phi. The THOL refresh analysis and held-pressure IL accounting explain why retention order matters. Their models and coefficients do not transfer here. The present gate introduces no such pressure-retention interval.

9. Scope of the preparatory theorem

The exact-real preparatory result establishes a model-specific OZ projection equivalence and a uniform simultaneous-IL work/identity theorem for the two existing A0 targets. It retains the complete IL phase action, the adjacent-edge correction, actual conductance associations and changed mean leaf. The whole allowed phase-coefficient interval is covered without choosing a coefficient to fit a response.

No IL/OZ word or trajectory was executed for this derivation. A concrete study separately needs its frozen preparation, actual live operator and grammar admission, declared selected-law refresh writer and callback schedule, retained context and represented output checks. These are separate obligations of the event contract, not consequences of the storage inequality. The retained execution supplies one bounded case with an explicitly unresolved phase-accuracy certificate. A rejected operation remains a rejection, and a detached field evaluation does not perform a refresh.

The maintained organization may realize these conditional actions while other operators, autonomous event selection, physical calibration and a uniquely selected fundamental law remain open. No additional pressure mechanism is inserted merely to make OZ change this particular continuation.

10. Execution and evidence preflight

The following design assembles the owners used by the bounded study. The protocol froze its configuration, observations, arithmetic allowance and acceptance criteria before response evaluation; the resulting evidence is reported separately in section 13.

Source, word and actual stage boundary

Reuse the positive pure-phase preparation from the existing registration protocol: a=0,g=3/4096, nominal split source, unchanged count-three contact word, seed, timestamps and histories. Its existing _registered_preparation((0, 1)) returns both detached observations and the full live runtime graph. Preserve the cached original and use a working copy of the latter. The detached graph used for field reports intentionally omits execution configuration; it is not an equivalent starting point for an operator word. The implemented copy uses the shared SDK copy_graph_state, with a bounded adapter retaining graph-instance event state and original neighbor order. It explicitly rebuilds registered runtime caches; a generic deepcopy of their locks is not this copy contract. Unknown instance state rejects instead of disappearing.

Use coherence,dissonance on the existing ordered targets (0,1), ValidatedSequence, truthful initialized-EPI context and the canonical core grammar profile with strict live rejection. Read executable tokens from the registry-backed TNFR.operators() interface. U1 admits this start only with its initialized context; terminal OZ requires the prior IL. The U2 debt and U4 handler checks still consume the actual histories. No AL/SHA padding, diagnostic waiver or replacement glyph is needed merely to validate the proposed word. Static word validity does not waive live source, branch or threshold admission.

The shared execute_network_operator_stage routes IL to immutable pointwise proposals and OZ to its all-target local/propagated pressure reduction. It does not forward arbitrary operator keyword arguments. Freeze its existing IL phase coefficient 0.3 explicitly; the full interval theorem above already covers it. Use OZ_NOISE_MODE=False, OZ_ENABLE_PROPAGATION=True, existing pressure gains and the default phase_weighted propagation mode. Resolve effective factors and Boolean settings before retaining their values. No dispatcher extension or parallel operator implementation is required for this design.

The uniform propagation name currently follows the same formula as phase_weighted; it is not an independent control law. The actual local proposal and the final overlapping neighbor reduction must remain distinct. Retain any unavailable acceleration observation as unavailable, with its history/clock convention; neither an OZ label nor a pressure change establishes a bifurcation or an observed second time derivative.

Use the typed observe_structural_acceleration observation from nodal_equation.py: glyph history and emission timestamps are not three EPI samples. The compatibility compute_d2epi_dt2 wrapper can return zero when that observation is unavailable, and OZ metrics still consume that scalar. Such a telemetry zero must not be reported as a measured zero acceleration or evidence against a bifurcation.

Shared admission rejects invalid primitives and nonzero inputs lost during materialization. It does not certify every subsequent floating-point product: propagation still evaluates products in float arithmetic, where nonzero increments can underflow. The bounded study must check its actual propagation arithmetic and residuals; primitive admission alone supplies no general no-underflow theorem.

Three declared refreshes, with no intervening flow

Use an initial selected-law refresh before IL, then one after IL and one after OZ. The contact word's stored pressure is not presumed constitutive. Capture the full initial state, each raw post-operator/pre-refresh state, each refreshed state and each completed stage. The final stage capture is needed because schedule records can be appended after the callback returns. These are same-time event/refresh boundaries, not three elapsed-time steps.

There is no existing point-pressure writer for this weighted conservative comparison. Reuse bound_relational_sine_support_exchange to admit current primitives and enclose the independently specified Phi. For the bounded static study a narrow callback can materialize each pressure interval's midpoint through shared represented-real admission, then publish all admitted values with the shared pressure-alias writer. Retain the exact represented value and its residual enclosure

\[ \varepsilon_{p,i}\in\operatorname{point}(p_{i,\mathrm{float}}) -[\Phi_i(y)]. \]

This materialization is a declared numerical realization, not exact equality between a float and an ideal sine pressure. Admit every value before any live write; a nonzero midpoint lost to zero during conversion must reject. When claiming which ideal OZ branch applies, its pressure enclosure must resolve the fixed 1e-9 cutoff, with equality in the near-zero branch. The executed represented branch and an unresolved ideal branch are separate observations. Do not tune the preparation after seeing a boundary case.

A direct stage callback is opaque. Its stage transaction does not install the event-schedule executor's pressure-only guard or sealed refresh evidence. Check the callback's before/after snapshots explicitly: form, phase, capacity, support, every edge attribute, histories, clock and other execution state must be unchanged, apart from admitted pressure aliases/cache updates. Bind the callback implementation, source configuration, node order and arithmetic to the retained result. The event contract provides the preservation boundary, not a claim that a raw hook enforces it.

The whole event-schedule executor selects all nodes and has its own ordinary flow segments; it is not this two-target, two-row study. Native neighbor-argument pressure, default mixed pressure and the positive-loss, unit-capacity, rescaled sine-flow helper also have different premises. None may silently replace Phi or advance this comparison. No continuous trajectory is needed to classify these static handoffs.

Evidence needed to connect the proof to execution

The comparison must retain, separately:

• Actual word/live admission and all consumed context, including changed debt, lineage, history and propagation records. Equal continuous state is not equal execution state, and independent history multisets lose member associations.
• IL's raw pressure contraction, complete simultaneous phase proposal, exact shared-edge work formula, changed mean leaf and represented endpoint defect. Re-admit the storage margin and identity on the captured output rather than inheriting an exact-real verdict from an operator name.
• Fresh pressure and both complete rate rows before and after each refresh. Compare the OZ continuation with the post-IL state under the same law while preserving the different execution context. Stored products and instantaneous fields are not measured temporal response.
• Source and code revision, complete settings, actual support/conductance, order, timestamps, seeds, precision, callbacks and arithmetic residuals. Unsupported affine EPI-jump certificates should abstain; their absence is not failure of the separate IL phase-work theorem.

11. Reused calculations and their transfer boundaries

This is a scoped dependency map for the handoff, not a new catalog or queue.

Existing owner	Reusable contribution	Boundary retained here
Maintained contact	Actual weighted support, complete source, local geometry, storage barrier and regional current identities	Its initial response protocol does not evaluate IL/OZ. Re-admit event work and the new phase mean before continuation
IL kernel and OZ stage kernel	Existing snapshot proposals, local pressure branches and overlapping-target reduction	The glyph, public class and full stage have different secondary effects; preserve the chosen route
Weighted field reader	Primitive admission, both ideal rate rows and storage with actual conductance	Detached evaluation performs neither a refresh nor numerical evolution
Collective context controls	Same physical state with different grammar debt or earlier stabilization	Current pressure and regional means cannot replace the context consumed by the next token
THOL refresh accounting	Why a held pressure increment can reach form before a later refresh	Its source, default pressure and finite responses do not transfer to this conservative law
Paired common-IL accounting	Separation of generated pressure, retention, integration and arithmetic residuals	It requires its own common source and held-pressure interval; it is not the present immediate-refresh map
Constituent memory and linear memory owner	Hidden initialization and causal kernels under declared elimination	Neither installs a persistent OZ channel; support, capacity, loss and clock must match before coefficients transfer

The independent foundation controls check edgewise work, the full-swap counterexample, local resultant bounds and the all-coefficient inequality with exact interval/polynomial arithmetic. Their synthetic trigonometric cases are proof controls, not new contact-word responses. Existing IL/OZ tests separately cover snapshot order, propagation, invalid values and rollback. The testing owner gives the combined selection; an engine/API gate alone is not this research coverage.

12. Frozen bounded-word comparison

The machine-readable protocol fixes one positive phase-axis preparation, the existing count-three contact, ordered targets (0,1), and coherence,dissonance under the selected complete law. It admits no elapsed flow, parameter search or preparation adjustment. The initial refresh and both post-stage refreshes use the same constitutive pressure. The operator word's raw outputs, refreshed outputs and completed stage records remain distinct observations.

The prospective discriminators are actual admission without replacement, IL's phase/form-rate change and negative storage work with retained identity, OZ's raw pressure/context change followed by continuous-state equivalence, and explicit acceleration unavailability when no temporal EPI evidence exists. No neighbor propagation is required: the existing 0.05 policy may discard every candidate. Retain every candidate's arithmetic before classifying that outcome. A rejected word or unresolved ideal pressure branch remains a result.

The represented phase coefficient is the exact rational represented by binary64 0.3, not 3/10. For each target, use the report's relative resultant to enclose the ideal angle a_i, and choose the actual endpoint's continuous lift nearest the ideal displacement d_i=lambda*a_i. Keep the stored modulo angle and lift association separately. With represented lifted displacement dhat_i, the frozen numerical allowance is

\[ |\widehat d_i-d_i|\le 2^{-46}. \]

Each target has seven incident phase edges. The phase potential is separately 7-Lipschitz in each target angle, so

\[ |H_{\rm represented}-H_{\rm ideal}| \le7(|\widehat d_u-d_u|+|\widehat d_v-d_v|) \le14\,2^{-46}. \]

This bounds the bridge to the exact IL theorem; it does not itself prove strictly negative represented work near a zero displacement. Recompute the whole-edge storage difference and the shared-edge work identity with the actual lifted displacements. Separately require the represented endpoint's Z^2<1/256 and H-H_*<1/6400, where H_*=40*(1-cos(2*pi/5)). These are endpoint admissions for the existing barrier, not observed long-time motion.

Shared outward rational 128-bit intervals enclose pressures, rates and trigonometric work. Any independent high-precision cross-check uses 100 decimal digits. Pressure materialization and consumed propagation arithmetic use the fixed absolute allowance 2^-46; unchanged primitives require exact represented equality. The callback guard permits only pressure aliases and the pressure-maximum cache/witness, whose consistency is checked separately. No whole-cache exemption substitutes for that guard.

13. Retained represented result and original certificate limitation

The response contains nine full-state boundaries from one admitted IL/OZ word, the three refreshes, typed observations and both completed stages. The frozen report records eight passing checks out of nine, not an all-passing evaluation. The retained-evidence controls re-admit captured primitives and rebuild the relevant fields and work without invoking the producer or changing the frozen response.

Two earlier attempts stopped before any IL/OZ action: a generic-copy failure on a runtime-cache lock, then a serialization failure on the existing uniform-real BEPI representation. Its complex containers have zero imaginary part and pass the shared signed-form owner. The corrections change copying and lossless recording only. Both failed attempts, source archives and protocols remain intact; law, preparation, word, numerical budget and prospective predictions are unchanged. Their associations are recorded by the corrective freeze.

Actual action and complete-law handoff

Both stages were admitted under strict live rejection. The inherited SHA-to-IL sequence emitted the existing advisory anti-pattern warning; it neither rejected nor replaced the action, and no waiver was installed. The captured IL phase changes modify the fresh form-rate row, leave the fresh phase-rate row unchanged, and have strictly negative whole-edge work:

\[ \Delta H_{\rm IL}\approx-2.71737569684\,10^{-7}. \]

These displayed decimals summarize retained rational enclosures. The shared-edge correction agrees with the independent full-edge difference, and the changed lifted phase mean is approximately 2.78994519689e-5. Direct endpoint admission gives

\[ Z^2\approx1.03208170158\,10^{-5}<1/256,\qquad H-H_*\approx1.03692602064\,10^{-5}<1/6400. \]

Thus this actual represented endpoint meets the existing continuous maintenance barrier independently of the unresolved ideal-phase accuracy comparison below. This supplies conditional continuation under the same held law, not an evaluated trajectory or autonomous occurrence.

OZ selected the amplification branch on both targets and changed raw stored pressure. All fourteen neighbor candidates were checked; none crossed the unchanged 0.05 propagation threshold. No consumed nonzero product vanished to zero. After the declared refresh, stored pressure, all consumed continuous coordinates and both complete rate rows equal the post-IL values exactly. Histories and stage records remain different. Acceleration is unavailable without the required temporal EPI evidence; compatibility zeros are not measurements. This confirms the model-specific projection equivalence.

Preserve the unsuccessful certificate

The frozen compound IL check stopped at its first assertion: the per-target ideal-phase residual enclosures were not both contained in [-2^-46,2^-46]. The approximate bounds are:

Target	Enclosed phase defect	Frozen classification
0	[-1.91769e-16, 1.51906e-14]	Unresolved at 2^-46 ≈ 1.42109e-14
1	[-1.48910e-14, 4.91012e-16]	Unresolved at the same budget

Both boxes contain zero: neither establishes that the actual defect exceeds the budget. The saved, predeclared 100-digit cross-check estimates defects of about -4.25e-18 and 3.04e-16; those estimates do not replace the rigorous certificate. The width comes from the existing relative-resultant primitive's 12-term, 64-bit sine/cosine bounds. Subsequent 128-bit interval arithmetic does not sharpen inherited input enclosures.

The failed compound assertion did not run its later checks. Independent retained-state verification supplies the direct work, identity and mean claims above. The separately computed energy-work defect interval, approximately [-1.076742e-13,1.076765e-13], is inside the declared ±14*2^-46 allowance. No tolerance was relaxed, no precision increased and no IL/OZ response regenerated to change the frozen verdict.

The original bounded handoff therefore left a numerical subclaim unresolved; the separate review below resolves it without changing that evaluation. The separate pressure-completion analysis classifies the additional mean, storage and constitutive-invariance obligations of an independent pressure intervention. It consumes this saved endpoint only for a detached balance check; no pressure-retention law or reservoir is supplied by the word, and its original verdict is unchanged.

14. Separate correction of the retained phase certificate

The review protocol and freeze precede one read-only corrective evaluation of the same initial and IL_raw primitive snapshots. They retain the original represented coefficient, target order, phase-lift association, absolute allowance 2^-46 and independent 100-decimal-digit check. No operator, preparation, continuation or response producer is replayed. This is a retrospective numerical correction, not a new prospective physical prediction or a replacement for the frozen 8/9 report.

The shared detached calculator reconstructs each bare-neighbor relative resultant directly from the exact represented primitive phases. The existing outward 128-bit sine/cosine owner now encloses those inputs before the angle operation; the former 12-term, 64-bit resultant boxes are not promoted into narrower boxes. The principal actual displacement is accepted as the nearest lift to the ideal displacement only after proving that their entire difference lies strictly inside (-pi,pi). The actual ideal-displacement coefficient remains the rational represented by binary64 0.3.

The corrective response and report certify both defects inside the unchanged budget:

Target	Approximate defect, summarizing its rigorous interval	Interval width
0	-4.25252514856e-18	about 2.351e-38
1	3.03932787662e-16	about 2.351e-38

The independent high-precision values lie inside these enclosures. Independent retained-state work, identity and OZ projection checks also pass, and the later boxes of the original compound check meet their respective assertions. The scope is separate evidence for the formerly unresolved numerical subclaim; the historical report remains 8/9, with all its source artifacts intact.

Synthetic controls check angle/lift branches, tolerance classification and invalid domains. Read-only evidence controls bind the review to the retained bytes, rebuild primitive inputs and compare the certified intervals with the independent oracle. Neither a small phase error nor corrected arithmetic selects a pressure-memory law or an autonomous OZ occurrence mechanism. The constitutive-refresh equivalence in Section 2 and the conditional pressure-completion obstruction remain unchanged.