Relative coupling: core-profile preparation and finite observation admission

This result admits a prospective finite reading after four actual AL → SHA preparations of the retained weighted-star contact. It supplies an error and identity certificate under one specified complete law. No reserved trajectory, finite response or physical measurement has been evaluated. The research program owns scheduling; relation foundations owns the general matched-phase remainder.

The word is admitted under the explicitly selected runtime core profile. This is not a certificate of every manuscript sentence prescription or a derivation of the uniquely canonical TNFR dynamics. The source grammar comparison states the missing development/stabilizer slots and their conditional scope. Frozen artifact labels retain their original bytes; this qualification changes neither preparation, error budget nor continued-law hypotheses.

1. Source, actual words and complete continuation

Use the complete retained central preparation in weighted_star_registration_episode/preparation.json, admitted by the full-record reader. The existing record is not regenerated. Its central successor has ten ordered nodes, nineteen undirected edges, actual represented conductances, structural time t=7 and two protected original stars (0,...,4) and (5,...,9).

On four independent copies, target donor j=5 or j=0 with AL boost a_low=1/1024 or a_high=1/512, then SHA factor 1/2. The preparation owner uses the registered emission,silence actions, validated core grammar and the shared atomic stage dispatcher. Strict live preconditions and rejection remain enabled. Each word has fixed targets, supplied lifecycle timestamp 2030-01-02T03:04:06+00:00, no pressure-refresh callback and unchanged structural time. The timestamp is a declared event input, not acquisition chronology or a physical clock. Source configuration, histories and execution seed are retained.

Both donors initially have x=5/16, capacity 4 and empty glyph histories. AL changes only its donor form by the exact represented dyadic boost, without clipping, and writes the admitted activation/lineage/history fields. SHA changes donor capacity to 2, preserves form and records latency/history. All other nodal coordinates and every edge remain unchanged. The three full prefix captures of each branch retain these primary and secondary effects, stage metadata and stored pressure. Re-admission checks the complete persistent record after primitive scalar/support admission; cached diagnostics do not replace the consumed source. Record association does not authenticate execution chronology. Only the producer performs the new words; the reader replays none.

AL has positive form-storage work on all four branches. SHA has zero immediate storage work but changes both continuing rate rows. Keep each branch's own positive capacities. In the original clock the selected unforced law is

\[ \dot x_i=\frac{\nu_i}{\pi d_i}\sum_{k\sim i}\sin(\theta_k-\theta_i), \qquad \dot\theta_i=\frac{\nu_i}{\pi d_i}(L_Wx)_i. \]

Support, conductance and capacity are held; loss is zero and exchange and beta are one. The full twenty real coordinates evolve, with declared phase lifts. Constitutive pressure is evaluated from evolving phase; absent runtime pressure-refresh callbacks do not mean held pressure. Gamma, further inputs, events and capacity evolution are absent. A local donor reset can leave the old five-block family, so that reduction supplies no continuation here.

2. Finite observation and cancellation

Fix the structural duration h=1/256. Receiver 4 has unchanged mobility nu_4/d_4=13/32. Read its phase in calibrated radian units, with initial readings at each post-word boundary and final readings at h. Branch-constant additive offsets may differ: they cancel in each increment. Define

\[ q_j=[Y_{j,\mathrm{high}}(h)-Y_{j,\mathrm{high}}(0)] -[Y_{j,\mathrm{low}}(h)-Y_{j,\mathrm{low}}(0)]. \]

Within a nominal donor pair, initial phases and post-SHA capacities coincide, and the form difference is Delta a=1/1024 at that donor only. The initial phase-rate contrast is g_j=-(13/(32*pi))*W_4j*Delta a. The two pairs can have different capacity distributions; each difference bound uses its own full law. With W_40=2 and the actual W_45=w_A, their ideal ratio is r=g_5/g_0=w_A/2, with 9/20<r<1/2.

The two nominal form velocities agree initially, so the second derivative of their phase difference vanishes. Global bounds A=16/3 and B=48/5 apply to the two coupled rows: max nu=8, max(nu_i*s_i/d_i)<72/5 and pi>3. The shared initial-state envelope rebuilds tighter source-specific coefficients before these majorants are used. The general remainder then gives

\[ D=1-ABh^2=1279/1280,\qquad |q_j^{\mathrm{nominal}}-hg_j|\le R =\frac{\Delta a AB^2h^3}{6D}=\frac1{209551360}. \]

This is a full nonlinear finite-time statement. It uses neither a frozen phase nor a tangent trajectory. It identifies a relative interaction coefficient under the supplied law, not a spatial distance or a fundamental physical force.

3. Source, input, clock and reading errors

At each post-word continuation boundary, allow independent form error e=2^-39, decomposed into source and input allowances 2^-40 each, and phase error f=2^-40 at every fine node. These boxes are supplied boundary premises: the nominal executed words do not derive uncertain event selection or transport an arbitrary pre-word error box. Held support, conductance, capacity and law remain exact. Actual membership in these boxes is a separate obligation.

Let final reading time error be c=2^-40 structural units, readout error m=2^-40 radians per reading and numerical error n=2^-40 radians per reading. The initial reading is anchored at the event boundary. The same-law envelope through h gives, per branch,

\[ T=\frac{f+Bhe}{D}=\frac{43}{43946105372672}. \]

For timing, the perturbed form span through h+c is bounded by 33/512+2e+(16/3)(h+c), because |x_dot_i|<8/3. Multiplication by B/2 bounds every phase speed strictly below M=3/4. The receiver begins within f of zero, hence |theta_4(t)|<=f+M(h+c)<1/128. Its declared local lift (-1/128,1/128) is therefore unambiguous throughout the reading window.

The complete error in each paired increment includes both endpoint source errors, two final clock errors and four readout/numerical errors:

\[ E=R+2(T+f)+2Mc+4m+4n =\frac{67283469}{14062753719255040}<4.785\,10^{-9}. \]

Using pi<22/7 gives h|g_0|>91/92274688. Therefore h|g_0|-E>151813271921/154690290911805440>0, and

\[ \left|\frac{q_5}{q_0}-r\right| <\frac{3E/2}{91/92274688-E} =\frac{2220354477}{303626543842}<\frac1{128}. \]

The prospective finite-read prediction is consequently

\[ \boxed{\frac{283}{640}<\frac{q_5}{q_0}<\frac{65}{128}}. \]

It separates this ratio from the point reference 1. Excluding a complete equal-coupling alternative would require that alternative's own preparation, continuation and finite-error admission. A common nonzero readout gain or constant clock conversion cancels in the ideal ratio; the frozen errors are already expressed in calibrated structural phase/time units. An unknown unbounded gain does not convert raw sensor noise into this error allowance.

4. Identity throughout the admitted continuations

The original stored energy obeys H0<9/20. Its form span is 1/16 and each donor strength is below 12; a boost at most 1/512 adds no more than 12*(1/16)*(1/512)+6*(1/512)^2=195/131072. SHA adds zero storage. For the post-word uncertainty box, the nineteen edges have total conductance below 39; the additional storage is at most

\[ 39[2(33/512)e+2e^2]+38f<2^{-34}. \]

Thus every admitted branch has H<29/64. The fixed protected-star barrier is 35/64, leaving a margin greater than 3/32. The retention owner checks the actual support and positive capacities; its conserved-storage argument retains both original identities for all times under each unchanged law. This is conditional maintenance after supplied events, not autonomous formation, a newly formed parent NFR or a derived scale-selection rule.

5. Frozen declaration and numerical boundary

The protocol owner re-admits preparation primitives and rebuilds all consumed static bounds. The immutable records live in docs/assets/weighted_star_relative_probe/: protocol.json, preparation.json, admission.json, source_snapshot.zip and manifest.json. The freeze records not_evaluated; it contains no response or attempt. Hashes establish byte associations, not chronology. The manifest identifies exact unchanged source bytes by Git base revision and blob; the archive contains the changed/untracked overlay and one-use freeze driver. Together they retain the package source, listed proof/contract owners and controls without copying unchanged committed files. Recovery requires those Git objects plus the overlay. Third-party dependency versions are recorded; their binaries are not archived. The evidence controls check receipts, both recovery channels and static re-admission without replay.

A future evaluation is limited to the four nominal twenty-coordinate states, sixteen equal steps 1/4096 per branch and order twelve using the shared _held_sine_neighbor_field and validated_box_taylor_step. The field uses the actual weighted neighbors and bare degrees in the original clock. The source is a point box; all endpoints are carried without discarding any coordinate, and receiver phase increments are summed from the stepper's increment fields. Independent preparation uncertainty remains in the analytic budget above. The global finite-state domain needs no reduced-family restriction.

The interval arithmetic is the shared outward dyadic128 implementation with its own rational trigonometric enclosures. Each numerical reading interval must have width at most 2n; a midpoint then meets the declared numerical error. The fixed resource policy allows 120 wall seconds per branch, 512 MiB memory and a 32 MiB response record, with no retries or changed steps/order. Failure of Picard inclusion, width or resources yields unavailable within budget, not rejection of the law. The numerical schedule is a declaration; no runtime, endpoint accuracy or measured result has been demonstrated here.

The preparation controls check actual words, independent event work and rejected record changes. The protocol controls check independent rational margins, uncertainty accounting and immutable declaration admission. Existing full-state and matched-acceleration controls retain their own scopes. This result prepares an operational comparison; it does not establish quantum statistics, a laboratory clock or physical identity.