Paper release: r1577 Released: July 23, 2026

Claim Boundary

This paper proves a finite-channel coefficient on a declared OPH branch. It also derives continuity, dilute dust behavior, spherical deep-galaxy scaling, force, and momentum formulae conditional on the repair-charge condensate action in the dark-sector paper . The OPH derivation of that action and its dimensional source normalization are work in progress.

The exact statuses are:

Canonical Coherent-Matter Source

An OPH technology instantiation is an observer-like self-reading system: a bounded physical or software patch with local state, ports or boundaries, readback, records, feedback or repair moves, and a public evidence bundle. For a nondegenerate coherent material subfederation \(U\), define the canonical source \[ \begin{equation} S_{\mathrm{coh}}^{\mathrm{can}} =\mathbf 1_{\rm self\mbox{-}read}\,\mathsf R_U\mathsf P_U\mathsf C_U. \label{eq:source} \end{equation} \] The factors state that the patch reads itself, produces a stable record, supports a predictive boundary, and executes a registered repair or control move. If any factor vanishes, the coherent scalar source is zero.

The screen-microphysics branch supplies four structural facts :

1. \(S_{\mathrm{coh}}^{\mathrm{can}}\) descends to a protected quotient observable.

2. A nondegenerate source is nonzero.

3. Scalar edge-center exhaustion places it in the unique quotient-local scalar register.

4. The protected \(\mathbb Z_6\) reserve acts on that same register and commutes with scalar activation.

These statements identify the channel. They supply no spacetime action or dimensional gravitational charge.

Protected-Reserve Coefficient

Let \(\epsilon_{\mathbb Z_6}(y)\) be the conditional reserve-presence profile on the co-registered scalar slot, and let \(w(y)\) be the normalized scalar weight across the finite collar. The scalar opportunity survives with local factor \[ \begin{equation} \lambda_{\rm slot}(y)=1-\epsilon_{\mathbb Z_6}(y). \end{equation} \] The finite-thickness collar coefficient is \[ \begin{equation} \lambda_{\mathrm{collar}}=\int\,\mathrm dy\,w(y) \left[1-\epsilon_{\mathbb Z_6}(y)\right]. \label{eq:collar} \end{equation} \] The protected-reserve receipt gives \[ \begin{equation} \int\,\mathrm dy\,w(y)\epsilon_{\mathbb Z_6}(y)=\frac{P}{24}. \label{eq:reserve} \end{equation} \]

The theorem excludes a zero scalar-channel coefficient on the stated nondegenerate branch. It does not exclude a zero dimensional gravitational coupling \(q_\star\), a zero device repair charge, or a zero ambient repair gradient.

Engineering Coordinates

An engineering description may absorb an effective coherent capacity into the source coordinate: \[ \begin{equation} S_{\mathrm{coh}}^{\mathrm{eng}}=N_{\mathrm{coh}}S_{\mathrm{coh}}^{\mathrm{can}}. \end{equation} \] Invariance of the scalar perturbation requires \[ \begin{equation} \chi_\nu^{\mathrm{can}}S_{\mathrm{coh}}^{\mathrm{can}}=\chi_\nu^{\mathrm{eng}}S_{\mathrm{coh}}^{\mathrm{eng}}, \qquad \boxed{\chi_\nu^{\mathrm{eng}}=\frac{\chi_\nu^{\mathrm{can}}}{N_{\mathrm{coh}}}}. \label{eq:chart} \end{equation} \] Very small values of \(\chi_\nu^{\mathrm{eng}}\) can therefore represent an order-one canonical coefficient. Equation [eq:chart] is a chart identity. It does not define \(q_\star\), measure \(S_{\mathrm{coh}}^{\mathrm{can}}\), or predict force.

Repair-Charge Coupling

The dark-sector completion promotes integer scalar repair occupation and a compact repair phase to a canonical pair \((n,\theta)\). On a fixed real lift of the phase, its relevant action is \[ \begin{align} I_{\mathrm{RPC}}={}&I_{\rm Regge}+I_{\rm SM} \nonumber\\ &+\int\,\mathrm d\tau\left[ -\sum_vV_vn_vD_\tau\theta_v -\sum_vV_v\varepsilon(n_v) -\sum_eW_e\mathcal K(|(d\theta)_e|) \right.\nonumber\\ &\hspace{31mm}\left. -\sum_vV_v\theta_v \left(\mathcal Q_{b,v} +q_\star\chi_\nu^{\mathrm{can}}S_{{\rm coh},v}^{\rm can}\right) \right]. \label{eq:rpc-action} \end{align} \] The source and canonical signs are chosen so phase variation gives the positive-source balance in Eq. [eq:continuity], and translation of a positive localized source gives force \(-Q_R\nabla\theta\). A globally single-valued compact-phase coupling, the derivation of this action from finite repair dynamics, and the dimensional constants are work in progress.

The coherent repair-charge density is \[ \begin{equation} \mathcal Q_{\rm coh}=q_\star\chi_\nu^{\mathrm{can}}S_{\mathrm{coh}}^{\mathrm{can}}. \label{eq:repair-charge-density} \end{equation} \] The phase equation is \[ \begin{equation} D_\tau n+d^\dagger j_R =\mathcal Q_b+q_\star\chi_\nu^{\mathrm{can}}S_{\mathrm{coh}}^{\mathrm{can}}. \label{eq:continuity} \end{equation} \] For a finite region \(\Omega\), \[ \begin{equation} \frac{\,\mathrm d}{\,\mathrm d\tau}\sum_{v\in\Omega}V_vn_v +\sum_{e\in\partial\Omega}W_ej_{R,e} =\sum_{v\in\Omega}V_v \left(\mathcal Q_{b,v}+q_\star\chi_\nu^{\mathrm{can}} S_{{\rm coh},v}^{\rm can}\right). \label{eq:finite-balance} \end{equation} \] The boundary flux is part of the physical source ledger.

Define the integrated device repair charge \[ \begin{equation} Q_R^{\rm dev}=q_\star\chi_\nu^{\mathrm{can}} \int_\Omega S_{\mathrm{coh}}^{\mathrm{can}}(x)\,\mathrm d^3x. \label{eq:device-charge} \end{equation} \] For a prescribed external repair phase, translation of the source gives \[ \begin{equation} \boxed{ F_i^{\rm coh}=-q_\star\chi_\nu^{\mathrm{can}} \int_\Omega S_{\mathrm{coh}}^{\mathrm{can}}(x)\, \partial_i\theta_{\rm ext}(x)\,\mathrm d^3x.} \label{eq:force} \end{equation} \] If the external gradient varies slowly across the device, \[ \begin{equation} \boxed{F_i^{\rm dev}\simeq -Q_R^{\rm dev}\partial_i\theta_{\rm ext}.} \label{eq:slow-force} \end{equation} \] The sign of \(Q_R^{\rm dev}\) may select attractive or repulsive response. Positive repair energy is compatible with signed charge when \(\varepsilon(n)=\varepsilon(|n|)\).

Compact-Source Constraint

A bottom-to-top contrast \[ \begin{equation} \Delta S=S_{\rm bottom}-S_{\rm top} \end{equation} \] can shape a dipole, select the sign of injected repair charge, couple to a nonuniform external field, or drive an alternating repair current. It does not by itself define a nonzero \(Q_R^{\rm dev}\). No formula proportional only to \(g^2A\chi_\nu\Delta S/(4\pi G)\) is a force theorem for a closed compact device.

Momentum and Energy Closure

For the cubic condensed branch, the repair-field stress is \[ \begin{equation} T^R_{ij}=\kappa_R|\nabla\theta|\, \partial_i\theta\,\partial_j\theta -\delta_{ij}\frac{\kappa_R}{3}|\nabla\theta|^3+\cdots . \label{eq:stress} \end{equation} \] The exact force ledger is \[ \begin{equation} \boxed{ F_i^{\rm matter}+\frac{\,\mathrm dP_i^R}{\,\mathrm dt} +\oint_{\partial\Omega}T^R_{ij}n^j\,\mathrm dA=0.} \label{eq:momentum} \end{equation} \] A local repulsive response requires either nonzero device repair charge with a field tail or explicit momentum transfer through Eq. [eq:momentum]. A static, closed, repair-neutral device cannot lift itself.

Any switchable force also requires a toggle-energy ledger. Let \(E(q,s)\) be the energy at position \(q\) and internal state \(s\), and define \[ \begin{equation} \operatorname{Toggle}(q)=E(q,{\rm on})-E(q,{\rm off}). \end{equation} \] For an on/off spatial cycle between \(q_1\) and \(q_2\), conservation gives \[ \begin{equation} W_{\rm cycle}=\operatorname{Toggle}(q_1) -\operatorname{Toggle}(q_2). \end{equation} \] A claimed position-dependent switchable force with no corresponding toggle or field-energy difference violates the closed energy ledger.

Laboratory Evidence Conditions

A force experiment has physical standing only if it supplies:

1. an instrument-independent measurement of \(S_{\mathrm{coh}}^{\mathrm{can}}\) or its declared engineering coordinate;

2. the dimensional calibration \(q_\star\);

3. a measured ambient repair gradient or a controlled emitted repair field;

4. a nonzero integrated repair charge, or a measured outgoing repair-current and momentum flux;

5. a complete toggle, maintenance-power, and field-energy ledger;

6. detuned, dummy, reversed-sign, thermal, acoustic, electromagnetic, mechanical, and center-of-mass controls;

7. a force result that follows the repair-field geometry and sign law in Eq. [eq:force].

8. for any claimed switchable reversal, a derived signed source factor or an independently measured reversal of the external repair gradient.

Without these receipts, a null result does not measure \(\chi_\nu^{\mathrm{can}}\), and a nonzero balance signal does not establish repair charge. The lane remains outside the falsification program because the action, source calibration, and operational source observable are work in progress.

Conclusion

The canonical coherent-matter susceptibility is an order-one channel coefficient fixed by protected-reserve survival on the stated OPH branch. The repair-charge condensate completion gives that coefficient a lawful dynamical role: it weights the coherent source in the repair-current equation. The same proposed action yields exact finite continuity, a dilute dust limit, the spherical deep-galaxy scaling, repair-field stress, and a device-force law. A device force depends on the dimensional calibration, integrated repair charge, external field, and field-momentum ledger. Repulsive response is conditionally allowed. Reactionless lift from a compact neutral contrast is excluded.