A Phase-Rigidity Branch Grammar for Stable Composite Matter: Hadronic Microphysics and Weak-Field Gravitational Response on a Global Phase Field

Stable bound structures do not simply comove with the cosmological background, and ordinary visible matter derives most of its rest mass from hadronic structure rather than from bare light-quark mass parameters alone [citation]. Those two facts motivate a restricted constitutive problem. For a composite pattern stabilized against the global phase-field response, the relevant constitutive object is its effective inertial loading together with the weakest admissible weak-field response law for the same ordinary-matter quantity.

The admissible domain is restricted. Only stable composite matter is considered, with special attention to hadron-dominated ordinary matter in an adiabatic weak-field regime. Elementary Yukawa/Higgs mass parameters are outside scope. So are any identification of the strong interaction with gravity, any full cosmological synthesis, and any derivation of the Einstein field equations. The task is matter-side constitutive closure for stable composite matter. The constitutive mass law developed below is therefore not identified with any elementary non-composite mass-loading mechanism.

A stable composite object is treated as a low-response phase-condensed pattern of the global phase field. Its persistence is encoded by a binding content $\Xibind$TeX\Xibind that dominates the imported phase-field hierarchy on the declared comparison window. For hadron-dominated ordinary matter, $\Xibind$TeX\Xibind must admit a family-fixed operational bridge built from hadronic sectors that already carry mass-bearing and mechanical information; separate object-wise fits are not admitted. If positivity, continuity, and disjoint-aggregation additivity are imposed on the declared comparison family, the constitutive mass map is forced to be linear at leading order. Once an ordinary-matter rigidity density has been assigned, the remaining question is the weakest admissible unscreened local weak-field closure.

The construction contains four layers. A phase-field hierarchy tests stability. A rigidity coordinate classifies stable composite matter. A constitutive mass law reads that rigidity as effective mass. A weak-field source law then governs the long-range response of ordinary matter. The long-range step is not obtained by rebranding strong-binding microphysics as gravity. It is a separate constitutive closure after the rigidity density has already been assigned.

For hadron-dominated ordinary matter, the operational bridge is anchored in QCD energy-momentum-tensor decompositions, mechanical-stability observables, and trace-anomaly or gravitational-form-factor data. Those sectors are now constrained by lattice QCD, dispersive analyses, flavor-resolved form-factor studies, baryon-octet mechanical-structure analyses, and experimental access to gluonic gravitational form factors [citation]. Those results do not single out one unique rigidity observable. They constrain the admissible sector class within which a family-fixed rigidity coordinate must be defined for hadron-dominated ordinary matter.

Within that domain, effective mass is assigned to phase rigidity, and the ordinary weak-field response is sourced by the same rigidity content. The claim is constitutive and regime restricted. It does not replace QCD, does not replace general relativity, and does not extend beyond stable composite matter and the weak-field ordinary-matter sector.

A composite object $\mathcal B$TeX\mathcal B is evaluated on a declared coarse-grained scale window $(L,\omega)$TeX(L,\omega). Only the coarse-grained background quantities required for stability classification and rigidity assignment are used.

definition: Composite phase response on a declared coarse-grained window. Let $\mathcal B$TeX\mathcal B be a composite bound structure. Its response to a small background perturbation is written as

\delta \Hbg_{\mathcal B}(k,\omega)
=
\chiPh^{(\mathcal B)}(k,\omega)\,\delta \Hbg_{\mathrm{cos}}(k,\omega),
\qquad k\sim L^{-1},

where $\chiPh^{(\mathcal B)}$TeX\chiPh^{(\mathcal B)} is the phase-response kernel of the composite structure on the declared scale window. All downstream response, stability, and additivity statements are read only relative to this declared composite/window pairing.

definition: Rigidity functional and binding content on a declared coarse-grained window. For the same structure $\mathcal B$TeX\mathcal B, let

\Rph^{(\mathcal B)}(L,\omega)
=
\int_{\mathcal B} d^3x\, \xibind(\mathbf x;L,\omega),

where $\xibind$TeX\xibind is the local rigidity-coordinate density induced by the second variation of the coarse-grained bound-state energy under a long-wavelength phase strain. The dimensionless binding content is

\Xibind^{(\mathcal B)}(L,\omega)
\equiv
\frac{\Rph^{(\mathcal B)}(L,\omega)}{\Lambda_{\Xi}^4/M_{\Xi}^2},

with fixed background normalization scales $(M_{\Xi},\Lambda_{\Xi})$TeX(M_{\Xi},\Lambda_{\Xi}). This rigidity assignment is made only on the declared coarse-grained window with those fixed normalization scales, and all downstream constitutive claims are read only relative to that declared assignment.

proposition: Second-variation consistency of the rigidity coordinate. If the coarse-grained bound-state energy of $\mathcal B$TeX\mathcal B has a stable minimum under the declared long-wavelength phase strain on the adiabatic window, then the local rigidity density $\xibind$TeX\xibind in reference is nonnegative on that window. Accordingly,

\Xibind^{(\mathcal B)}(L,\omega)\ge 0,

and $\Xibind^{(\mathcal B)}=0$TeX\Xibind^{(\mathcal B)}=0 only when no restoring rigidity is resolved on the declared window.

proof. By definition, $\xibind$TeX\xibind is induced by the second variation of the coarse-grained bound-state energy under the declared phase strain. Stability of the reference configuration makes that quadratic form positive semidefinite on the admissible window, which yields reference.

definition: Condensation index. The bounded phase-condensation index is

\Picoh^{(\mathcal B)}(L,\omega)
=
\frac{\Xibind^{(\mathcal B)}(L,\omega)}
{\Xibind^{(\mathcal B)}(L,\omega)+\Xicos(L,\omega)}
\in (0,1).

proposition: Stable composite matter. A composite bound structure $\mathcal B$TeX\mathcal B is stable on $(L,\omega)$TeX(L,\omega) if

\Xicos(L,\omega)\ll \Xibind^{(\mathcal B)}(L,\omega)
\qquad\text{and}\qquad
\left|\chiPh^{(\mathcal B)}(L,\omega)\right|\ll 1.

In this regime $\mathcal B$TeX\mathcal B is a phase-condensed invariant pattern with low response to the global phase-field drive.

proof. The first inequality states that the coarse-grained global phase-field drive is subdominant to the internal rigidity content. The second states that the induced composite response remains small on the same operational window. Together they exclude global-field tracking behavior and identify the composite structure as a low-response pattern.

proposition: Approximate additivity under disjoint aggregation within a declared comparison family. Within one declared comparison family $\CompFam$TeX\CompFam on a common adiabatic comparison window, let $\mathcal B_1$TeX\mathcal B_1 and $\mathcal B_2$TeX\mathcal B_2 be stable composite objects separated so that cross-couplings are negligible at leading order. Then

\Xibind^{(\mathcal B_1\sqcup \mathcal B_2)}
=
\Xibind^{(\mathcal B_1)}+\Xibind^{(\mathcal B_2)}
+
O(\epssep),

where $\epssep$TeX\epssep measures residual inter-object response leakage on the chosen coarse-graining window. This additivity statement is read only on that declared comparison family and common adiabatic window, and it is not promoted here to a family-independent matter theorem.

proof. For disjoint weakly interacting composites, the coarse-grained bound-state energy is additive up to the small leakage term $O(\epssep)$TeXO(\epssep). Since $\Rph$TeX\Rph is defined by the second variation of that energy under a long-wavelength phase strain, it inherits the same additive structure. The normalization scales in reference are fixed, so the same statement holds for $\Xibind$TeX\Xibind.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The passage from stability to mass is now constrained. Once a structure is identified by reference as a preserved invariant pattern, the mass assigned to that structure should depend on the same rigidity content that makes the pattern survive. The next step is to determine how $\Xibind$TeX\Xibind can be read operationally in ordinary matter without introducing a second independent substance variable.

The rigidity content $\Xibind$TeX\Xibind is not assumed to be a primitive laboratory observable. For hadron-dominated ordinary matter it must nevertheless admit a family-fixed operational bridge that survives ordinary-matter coarse graining. That bridge has two layers. The first fixes one family-fixed hadronic bridge. The second lifts the hadronic proxy to bulk ordinary matter without allowing electronic, chemical, lattice, or environmental sectors to replace the hadronic ordering variable inside the declared family.

definition: Adiabatic comparison window. The adiabatic comparison window $\Wad$TeX\Wad consists of all scale-frequency pairs $(L_*,\omega_*)$TeX(L_*,\omega_*) satisfying

\lint^{(\mathcal B)}\ll L_* \ll \Lenv,
\qquad
\omega_* \ll \omegab^{(\mathcal B)},

where $\lint^{(\mathcal B)}$TeX\lint^{(\mathcal B)} is the internal correlation length of the composite object, $\Lenv$TeX\Lenv is the environmental variation scale, and $\omegab^{(\mathcal B)}$TeX\omegab^{(\mathcal B)} is the characteristic binding frequency.

definition: Declared comparison family. A declared comparison family $\CompFam$TeX\CompFam is a set of stable composite objects compared within the same adiabatic window $(L_*,\omega_*)\in\Wad$TeX(L_*,\omega_*)\in\Wad, the same environmental class, and the same weak-field normalization protocol. All calibration constants introduced below are fixed once for the family and are not refitted object by object. All downstream additivity, proxy, and mass-readout statements are read only within this declared comparison family and its fixed adiabatic comparison window.

definition: Hadronic rigidity tuple. For hadron-dominated ordinary matter, define the family-normalized hadronic tuple

\Qtup^{(\mathcal B)}
\equiv
\bigl(
\Eemt^{(\mathcal B)},
\Mmech^{(\mathcal B)},
\Ttr^{(\mathcal B)}
\bigr),

where $\Eemt$TeX\Eemt denotes the chosen EMT energy coordinate, $\Mmech$TeX\Mmech a mechanical-stability coordinate extracted from pressure/shear or equivalent $D$TeXD-term information, and $\Ttr$TeX\Ttr a trace-anomaly or gravitational-form-factor coordinate. Each entry is rendered dimensionless and family normalized once the bridge protocol is declared.

definition: Admissible hadronic proxy. An admissible hadronic rigidity proxy for the declared family $\CompFam$TeX\CompFam is a family-fixed map

\Ophad^{(\mathcal B)}=\BQCD^{(\CompFam)}\!\left[\Qtup^{(\mathcal B)}\right],

with the following properties:

- $\Ophad^{(\mathcal B)}>0$TeX\Ophad^{(\mathcal B)}>0 for every stable $\mathcal B\in\CompFam$TeX\mathcal B\in\CompFam; - $\BQCD^{(\CompFam)}$TeX\BQCD^{(\CompFam)} is monotone in each declared coordinate on the admissible family range; - $\Ophad$TeX\Ophad is continuous on that range; - $\Ophad$TeX\Ophad is additive under disjoint aggregation up to $O(\epssep)$TeXO(\epssep); - the map $\BQCD^{(\CompFam)}$TeX\BQCD^{(\CompFam)} is declared once and not retuned object by object.

definition: Hadron-dominated ordinary-matter family. A declared ordinary-matter family $\OrdFam\subset\CompFam$TeX\OrdFam\subset\CompFam is hadron dominated if there exists one family-fixed bulk-correction budget $\Deltabulk^{(\mathcal B)}$TeX\Deltabulk^{(\mathcal B)} such that

\Xibind^{(\mathcal B)}(L_*,\omega_*)
=
\lambda_{\OrdFam}\!\left(\Ophad^{(\mathcal B)}+\Deltabulk^{(\mathcal B)}\right)
+
O(\deltacg^{(\mathcal B)}+\deltaenv^{(\mathcal B)}+\epssep),

holds with $\lambda_{\OrdFam}>0$TeX\lambda_{\OrdFam}>0 family fixed and

\left|\Deltabulk^{(\mathcal B)}\right|\le \epshad\,\Ophad^{(\mathcal B)},
\qquad
0\le \epshad \ll 1,

for all $\mathcal B\in\OrdFam$TeX\mathcal B\in\OrdFam. The correction budget collects electronic, chemical, lattice, and environmental contributions after coarse graining to the declared ordinary-matter window. No new mass-readout coefficient is introduced on $\OrdFam$TeX\OrdFam: the same family-fixed coefficient $\mu_m^{(\CompFam)}$TeX\mu_m^{(\CompFam)} is retained after hadron-to-bulk coarse graining, while $\lambda_{\OrdFam}$TeX\lambda_{\OrdFam} records only the ordinary-matter lift of $\Xibind$TeX\Xibind on the admitted window.

remark: Representative ordinary-matter family. An admissible ordinary-matter family may be taken as a chemically matched set of neutral bulk samples, such as an isotopic series or a controlled composition class, prepared under one environmental protocol and analyzed on one adiabatic window. The hadronic bridge $\BQCD^{(\CompFam)}$TeX\BQCD^{(\CompFam)} and the bulk-correction budget $\Deltabulk^{(\mathcal B)}$TeX\Deltabulk^{(\mathcal B)} are fixed once for that family. Electronic, chemical, lattice, and environmental sectors remain inside the subleading budget of reference, and mass ordering is admitted only when that budget stays smaller than the hadronic proxy separation across the family.

*Representative admissible map and worked ordinary-matter family

A representative admissible bridge on a declared family is the positive monotone linear map

\Ophad^{(\mathcal B)}
=
a_{\CompFam}\,\Eemt^{(\mathcal B)}
+
b_{\CompFam}\,\Mmech^{(\mathcal B)}
+
c_{\CompFam}\,\Ttr^{(\mathcal B)},
\qquad
 a_{\CompFam},b_{\CompFam},c_{\CompFam}>0,

with coefficients fixed once for the family and with $\Eemt,\Mmech,\Ttr$TeX\Eemt,\Mmech,\Ttr rendered dimensionless by the same normalization convention across the full comparison class. It is a representative member of an admissible order-preserving class rather than a claim of microscopic uniqueness. An alternative family-fixed map is admissible only if, after one family calibration and under the same correction discipline, it preserves the same stable ordering across the full comparison family. All downstream proxy-dependent ordering and closure statements are read only relative to this declared family-fixed representative map or to an explicitly admitted family-fixed replacement on the same comparison family.

proposition: Representative admissible proxy map on a declared comparison family. If the coordinates in reference are dimensionless, nonnegative, and additive up to $O(\epssep)$TeXO(\epssep) on the declared comparison family, then the map in reference is an admissible hadronic proxy in the sense of reference.

proof. Positivity follows from the positivity of the coefficients and the nonnegativity of the declared coordinates. Monotonicity in each argument is immediate because all coefficients are strictly positive. Continuity holds because reference is linear, and additive coarse-graining is inherited from the declared additivity of the coordinates up to the same $O(\epssep)$TeXO(\epssep) leakage term. Since the coefficients are fixed once for $\CompFam$TeX\CompFam, no object-wise retuning is admitted.

definition: Representative isotopic ordinary-matter protocol. Let $\OrdFam^{(\mathrm{iso})}\subset\OrdFam$TeX\OrdFam^{(\mathrm{iso})}\subset\OrdFam be a chemically matched isotopic family of neutral bulk samples prepared in one environmental class, with one common lattice protocol, one adiabatic window $(L_*,\omega_*)\in\Wad$TeX(L_*,\omega_*)\in\Wad, one estimator rule $\mathsf{Est}$TeX\mathsf{Est}, and one declared bulk-correction budget $\Deltabulk^{(\mathcal B)}$TeX\Deltabulk^{(\mathcal B)}. The representative isotopic protocol keeps fixed across the family

(\BQCD^{(\CompFam)},\lambda_{\CompFam},\lambda_{\OrdFam},\mu_m^{(\CompFam)},\mathsf{Est},\Deltabulk,\Wad),

and changes only the hadronic tuple values $\Qtup^{(\mathcal B)}$TeX\Qtup^{(\mathcal B)} associated with the declared isotopic family members.

proposition: Worked ordinary-matter family closure on an isotopic series. Let $\OrdFam^{(\mathrm{iso})}$TeX\OrdFam^{(\mathrm{iso})} be the representative isotopic protocol of reference, and let the family-fixed proxy be given by reference. If

\lambda_{\OrdFam}\left|\Ophad^{(\mathcal B_1)}-\Ophad^{(\mathcal B_2)}\right|
>
\left|\Deltabulk^{(\mathcal B_1)}-\Deltabulk^{(\mathcal B_2)}\right|
+
O(\deltacg+\deltaenv+\epssep),

for every ordered pair $(\mathcal B_1,\mathcal B_2)\in\OrdFam^{(\mathrm{iso})}\times\OrdFam^{(\mathrm{iso})}$TeX(\mathcal B_1,\mathcal B_2)\in\OrdFam^{(\mathrm{iso})}\times\OrdFam^{(\mathrm{iso})}, then the same family-fixed bridge determines both

\operatorname{sign}\!\Big(\Xibind^{(\mathcal B_1)}-\Xibind^{(\mathcal B_2)}\Big)
=
\operatorname{sign}\!\Big(\Ophad^{(\mathcal B_1)}-\Ophad^{(\mathcal B_2)}\Big)

and

\operatorname{sign}\!\Big(m_{\mathrm{obs}}^{(\mathcal B_1)}-m_{\mathrm{obs}}^{(\mathcal B_2)}\Big)
=
\operatorname{sign}\!\Big(\Ophad^{(\mathcal B_1)}-\Ophad^{(\mathcal B_2)}\Big),

up to the declared estimator and coarse-graining tolerances, without object-wise retuning of either the proxy map or the weak-field source coefficient.

proof. Equation reference is the isotopic-family specialization of reference with the representative admissible map reference and one declared bulk-correction budget. Applying the leading linear mass law reference with one retained coefficient $\mu_m^{(\CompFam)}$TeX\mu_m^{(\CompFam)} then gives reference on the same family. No additional coefficient is admitted by reference, so any reversal of the ordering would force either proxy replacement or family change.

remark: Physical content of the hadronic tuple. The tuple in reference is restricted to sectors for which primary QCD or experimentally anchored information already exists. The primary anchor chain is the EMT mass decomposition [citation], lattice proton gravitational form factors [citation], lattice trace-anomaly form factors together with trace-anomaly analyses [citation], threshold $J/\psi$TeXJ/\psi access to gluonic gravitational form factors [citation], and dispersive nucleon gravitational form factors [citation]. Mechanical and force-distribution information enters through experimental and lattice determinations [citation]. Flavor-resolved proton and baryon-octet studies serve only as supplementary sector checks beyond the proton baseline [citation].

proposition: Operational realization of $\Xibind$TeX\Xibind. Within a declared hadron-dominated comparison family $\CompFam$TeX\CompFam, the rigidity content is operationally admissible only if there exists one family-fixed hadronic proxy $\Ophad$TeX\Ophad such that

\Xibind^{(\mathcal B)}(L_*,\omega_*)
=
\lambda_{\CompFam}\,\Ophad^{(\mathcal B)}
+
O(\deltacg^{(\mathcal B)}+\deltaenv^{(\mathcal B)}+\epssep),

with one family-fixed calibration factor $\lambda_{\CompFam}>0$TeX\lambda_{\CompFam}>0. On an ordinary-matter subfamily $\OrdFam\subset\CompFam$TeX\OrdFam\subset\CompFam, this condition fixes the admissible family-fixed bridge; quantitative ordinary-matter collapse is tested only through reference.

remark: Family-fixed bridge condition. Equation reference operationalizes $\Xibind$TeX\Xibind without selecting a unique microscopic formula. It excludes unrestricted proxy replacement and object-wise retuning inside the family. Once $\BQCD^{(\CompFam)}$TeX\BQCD^{(\CompFam)} and $\lambda_{\CompFam}$TeX\lambda_{\CompFam} are fixed, the same bridge must apply across the full comparison family.

proposition: Hadron-to-bulk coarse-graining for ordinary matter. Let $\mathcal B_1,\mathcal B_2\in\OrdFam$TeX\mathcal B_1,\mathcal B_2\in\OrdFam belong to a declared hadron-dominated ordinary-matter family. If their hadronic proxies obey

\lambda_{\OrdFam}\left|\Ophad^{(\mathcal B_1)}-\Ophad^{(\mathcal B_2)}\right|
>
\left|\Deltabulk^{(\mathcal B_1)}-\Deltabulk^{(\mathcal B_2)}\right|
+
O(\deltacg+\deltaenv+\epssep),

then

\operatorname{sign}\!\Big(\Xibind^{(\mathcal B_1)}-\Xibind^{(\mathcal B_2)}\Big)
=
\operatorname{sign}\!\Big(\Ophad^{(\mathcal B_1)}-\Ophad^{(\mathcal B_2)}\Big).

Hence electronic, chemical, lattice, and environmental sectors may renormalize the ordinary-matter readout, but they may not replace the hadronic ordering variable inside the declared family.

proof. Subtract reference for the two objects. If the difference of the hadronic proxies dominates the declared correction budget in the sense of reference, the sign of the total difference is fixed by the hadronic term.

remark: Coefficient hierarchy. The coefficient structure is fixed once. $\lambda_{\CompFam}$TeX\lambda_{\CompFam} calibrates the hadronic proxy-to-rigidity bridge on $\CompFam$TeX\CompFam, $\lambda_{\OrdFam}$TeX\lambda_{\OrdFam} calibrates the admitted hadron-to-bulk lift on $\OrdFam$TeX\OrdFam, and $\mu_m^{(\CompFam)}$TeX\mu_m^{(\CompFam)} is the unique retained rigidity-to-mass coefficient. Hence ordinary-matter mass collapse is tested only through

\alpha_{\OrdFam}:=\mu_m^{(\CompFam)}\lambda_{\OrdFam},

with no additional independent mass-readout coefficient.

proposition: Ranking invariance under admissible proxy replacement. Let $\mathcal O_{(1)}$TeX\mathcal O_{(1)} and $\mathcal O_{(2)}$TeX\mathcal O_{(2)} be two admissible family-fixed proxies on the same declared family $\CompFam$TeX\CompFam, each satisfying a relation of the form reference with positive family constants and the same correction discipline. Then every mass ordering that is stable against the declared correction budget is invariant under the replacement $\mathcal O_{(1)}\mapsto \mathcal O_{(2)}$TeX\mathcal O_{(1)}\mapsto \mathcal O_{(2)}.

proof. Both admissible proxies map to the same $\Xibind$TeX\Xibind up to the declared correction budget. Since the mass law below is monotone linear in $\Xibind$TeX\Xibind, any ordering that is already stable against that budget cannot change under an admissible proxy replacement.

corollary: Family-collapse consequence. If the hadronic proxy is admissible on $\CompFam$TeX\CompFam in the sense of reference, and the ordinary-matter lift reference holds on a declared hadron-dominated ordinary subfamily $\OrdFam\subset\CompFam$TeX\OrdFam\subset\CompFam, then the observed masses of objects in $\OrdFam$TeX\OrdFam must collapse onto a single family calibration line when plotted against the declared hadronic proxy after the admitted hadron-to-bulk lift, up to the explicitly declared correction terms. Object-wise retuning of the bridge is not admissible.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The admissible sector class is fixed by a primary anchor chain: EMT mass decompositions [citation], proton gravitational and trace-anomaly form factors from lattice QCD [citation], threshold $J/\psi$TeXJ/\psi access to gluonic gravitational form factors [citation], dispersive nucleon gravitational form factors [citation], and experimental or lattice stress and force distributions [citation]. Flavor-resolved proton and baryon-octet studies are used only as supplementary checks beyond that primary chain [citation].

With $\Xibind$TeX\Xibind fixed by the previous section, the mass law is constrained by positivity, continuity, and approximate additivity.

definition: Effective mass functional. Let $f_{\CompFam}:\mathbb R_{+}\to\mathbb R_{+}$TeXf_{\CompFam}:\mathbb R_{+}\to\mathbb R_{+} be the map assigning effective mass to the rigidity content of a stable composite object in the family $\CompFam$TeX\CompFam:

\mrig^{(\mathcal B)}
=
f_{\CompFam}\!\left(\Xibind^{(\mathcal B)}(L_*,\omega_*)\right).

The function $f_{\CompFam}$TeXf_{\CompFam} is required to be positive, monotone, and continuous on the admissible family interval.

proposition: Uniqueness of the leading linear map. Assume that $f_{\CompFam}$TeXf_{\CompFam} in reference satisfies:

- positivity and monotonicity on the admissible family interval; - continuity on that interval; - approximate additivity under disjoint aggregation, equation f_(x+y)=f_(x)+f_(y)+O(), equation for all admissible $x,y$TeXx,y corresponding to the aggregation regime of reference; - a single family calibration point fixing the overall mass scale.

Then the leading admissible mass law is

\mrig^{(\mathcal B)}
=
\mu_m^{(\CompFam)}\,\Xibind^{(\mathcal B)}(L_*,\omega_*)
+
O(\epssep),

with one family-fixed coefficient $\mu_m^{(\CompFam)}>0$TeX\mu_m^{(\CompFam)}>0.

proof. Ignoring the explicitly declared leakage term $O(\epssep)$TeXO(\epssep), the additivity requirement reference gives Cauchy's functional equation $f(x+y)=f(x)+f(y)$TeXf(x+y)=f(x)+f(y) on the admissible positive interval. Continuity excludes pathological solutions and yields $f(x)=\mu_m x$TeXf(x)=\mu_m x. Positivity and monotonicity require $\mu_m>0$TeX\mu_m>0. The single calibration point fixes $\mu_m^{(\CompFam)}$TeX\mu_m^{(\CompFam)}, and reintroducing the declared aggregation leakage restores the $O(\epssep)$TeXO(\epssep) remainder.

remark: Conditions for nonlinear corrections. Within the admissible family, reference is the unique leading map compatible with positivity, family ordering, continuity, and additive coarse graining for disjoint composites. A nonlinear leading law requires additional family-internal shape parameters, abandonment of approximate additivity, or exit from the declared adiabatic regime.

corollary: Rigidity ordering equals mass ordering. Within a fixed family $\CompFam$TeX\CompFam,

\mrig^{(\mathcal B_1)}>\mrig^{(\mathcal B_2)}
\qquad\Longleftrightarrow\qquad
\Xibind^{(\mathcal B_1)}(L_*,\omega_*)>\Xibind^{(\mathcal B_2)}(L_*,\omega_*),

up to the declared leakage and coarse-graining corrections.

remark: Mass readout and rigidity content. Within the declared family, the mass in reference is read from the same rigidity content that enters the stability criterion. No second constitutive scalar is introduced on top of the stabilized composite pattern. The constitutive readout in reference is restricted to stable composite matter and is not promoted here to an elementary non-composite mass-loading law.

Standard quark, gluon, binding-energy, and anomaly decompositions remain the microphysical sectors from which the family-fixed rigidity coordinate is assembled. The constitutive content of reference is limited to the composite readout layer: once a stable pattern has formed, the effective mass is assigned to the resulting rigidity content. No long-range field law follows from reference alone. The weak-field sector therefore requires a separate constitutive closure once the ordinary-matter rigidity density has been assigned.

The long-range sector requires a separate constitutive closure after the rigidity density has been assigned. The admissible closure is the weakest unscreened local weak-field law for hadron-dominated ordinary matter compatible with the declared regime.

definition: Rigidity density. For a continuum distribution of ordinary composite matter in the adiabatic window, after the hadron-to-bulk coarse-graining step of reference, define

\rhoRig(\mathbf x)
\equiv
\mu_m^{(\CompFam)}\,\xibind(\mathbf x;L_*,\omega_*),

so that the same family-fixed mass-readout coefficient is retained in the weak-field source density and, for any body $\mathcal B$TeX\mathcal B,

\mrig^{(\mathcal B)} = \int_{\mathcal B} d^3x\,\rhoRig(\mathbf x).

For a discrete collection of bodies, one may equivalently write

\rhoRig(\mathbf x)=\sum_i \mrig^{(i)}\,\delta^{(3)}(\mathbf x-\mathbf x_i).

proposition: Minimal local weak-field constitutive closure. Assume the ordinary long-range sector of a declared hadron-dominated family $\OrdFam$TeX\OrdFam is observed in a static weak-field regime with:

- linear response to $\rhoRig$TeX\rhoRig after background subtraction; - translation and rotation invariance on scales $r\gg\lint$TeXr\gg\lint; - an inverse scalar response operator that admits an even local expansion near $k=0$TeXk=0; - the empirically unscreened Newtonian limit for ordinary matter.

Then the leading local scalar closure is

\nabla^2\PhiH
=
4\pi G_{\mathrm H}\,\rhoRig
+
O(\lint^2\nabla^4\rhoRig),

with $G_{\mathrm H}\to G$TeXG_{\mathrm H}\to G in the ordinary weak-field limit.

proof. In Fourier space, linearity and translation invariance imply $\PhiH(\mathbf k)=\Gamma(k^2)\,\rhoRig(\mathbf k)$TeX\PhiH(\mathbf k)=\Gamma(k^2)\,\rhoRig(\mathbf k), where $\Gamma$TeX\Gamma is the scalar Green kernel of the weak-field response. Define the inverse local operator

\mathcal L(k^2)\equiv \Gamma(k^2)^{-1}.

Assumption (iii) gives the even local expansion

\mathcal L(k^2)=a_0+a_2 k^2+O(k^4).

Rotation invariance forbids odd powers of $k$TeXk. The empirically unscreened $1/r$TeX1/r limit of ordinary gravity excludes a nonzero mass term $a_0$TeXa_0, and background subtraction excludes source-independent offsets. Therefore the leading operator is $\mathcal L(k^2)=a_2k^2+O(k^4)$TeX\mathcal L(k^2)=a_2k^2+O(k^4). Writing $a_2=(4\pi G_{\mathrm H})^{-1}$TeXa_2=(4\pi G_{\mathrm H})^{-1} and transforming back to position space yields reference, with higher-derivative corrections suppressed by powers of the internal coarse-graining scale.

remark: Regime of reference. Equation reference is a weak-field constitutive closure for the declared ordinary-matter regime. It does not determine strong-field dynamics, it does not derive the Einstein equations, and it does not identify a microscopic gauge interaction with the macroscopic gravitational field.

remark: Failure modes of the weak-field closure. The Poisson-type form in reference is not admitted if the long-range Green kernel is screened, if the inverse response operator fails to admit the declared local analytic expansion near $k=0$TeXk=0, or if object-dependent long-range couplings are required after the family protocol has been fixed. Any such behavior lies outside the declared weak-field domain and is excluded by the protocol gates below.

corollary: Empirical universality admissibility on the declared ordinary-matter protocol. Within a declared hadron-dominated ordinary-matter protocol that passes G6 and G7,

m_{\mathrm{inert}}^{(\mathcal B)}
\doteq
m_{\mathrm{passive}}^{(\mathcal B)}
\doteq
m_{\mathrm{active}}^{(\mathcal B)}
\doteq
\mrig^{(\mathcal B)},

at the empirical precision of the admitted weak-equivalence and active/passive-mass tests.

proof. The constitutive map identifies inertial readout with $\mrig$TeX\mrig and uses the same $\mrig$TeX\mrig as source strength in reference. Gates G6 and G7 then require one family-fixed coefficient to remain compatible with observed universality of free fall and with active/passive mass equality across the admitted ordinary-matter protocol. Failure of that compatibility rejects the protocol rather than introducing independent inertial, passive, and active coefficients.

remark: Ordinary-matter regime of reference. Equation reference is an empirical constitutive closure for the admitted ordinary-matter family. It is not a microscopic derivation for arbitrary matter sectors.

remark: Microphysical bridge versus long-range response. Strong-binding microphysics acts at the composite-formation layer. The long-range weak field acts at the response layer. The construction does not identify gluon exchange with gravity. Rather, hadronic microphysics can realize the rigidity content whose coarse-grained long-range response is gravitational for ordinary matter.

definition: Declared ordinary-matter protocol. A declared ordinary-matter protocol is

\Protord
=
(\CompFam,\OrdFam,\Wad,\BQCD,\Deltabulk,\mathsf{Est},
\mathcal A_{\mathrm{disj}},
\lambda_{\CompFam},\lambda_{\OrdFam},\mu_m^{(\CompFam)},
\epshad,\deltacg,\deltaenv,\deltacoll,\deltaadd),

where $\Deltabulk$TeX\Deltabulk denotes the declared family-fixed bulk-correction budget on $\OrdFam$TeX\OrdFam, $\mathsf{Est}$TeX\mathsf{Est} denotes the common estimator and uncertainty rule, and $\mathcal A_{\mathrm{disj}}\subseteq \OrdFam\times\OrdFam$TeX\mathcal A_{\mathrm{disj}}\subseteq \OrdFam\times\OrdFam is the declared disjoint-aggregation set. Here $\lambda_{\CompFam}$TeX\lambda_{\CompFam} fixes the hadronic proxy-to-rigidity bridge on $\CompFam$TeX\CompFam, $\lambda_{\OrdFam}$TeX\lambda_{\OrdFam} fixes the admitted hadron-to-bulk lift on $\OrdFam$TeX\OrdFam, and $\mu_m^{(\CompFam)}$TeX\mu_m^{(\CompFam)} remains the unique retained rigidity-to-mass coefficient after coarse graining. No additional independent mass-readout coefficient is admitted on the declared ordinary-matter family. The induced ordinary-matter collapse coefficient is therefore $\alpha_{\OrdFam}:=\mu_m^{(\CompFam)}\lambda_{\OrdFam}$TeX\alpha_{\OrdFam}:=\mu_m^{(\CompFam)}\lambda_{\OrdFam}. All entries are fixed once for the family. Post hoc replacement of the family-fixed bridge $\BQCD^{(\CompFam)}$TeX\BQCD^{(\CompFam)}, the admitted bulk-correction budget $\Deltabulk$TeX\Deltabulk, the estimator, the normalization convention, or the aggregation set is inadmissible.

G1: Finite bulk mass-collapse gate..

Let $m_{\mathrm{obs}}^{(\mathcal B)}$TeXm_{\mathrm{obs}}^{(\mathcal B)} denote the admitted measured mass readout for $\mathcal B\in\OrdFam$TeX\mathcal B\in\OrdFam under $\mathsf{Est}$TeX\mathsf{Est}, and define

\alpha_{\OrdFam}\equiv \mu_m^{(\CompFam)}\lambda_{\OrdFam}.

The protocol $\Protord$TeX\Protord passes G1 if

\sup_{\mathcal B\in \OrdFam}
\frac{\left|m_{\mathrm{obs}}^{(\mathcal B)}-\alpha_{\OrdFam}\bigl(\Ophad^{(\mathcal B)}+\Deltabulk^{(\mathcal B)}\bigr)\right|}
     {m_{\mathrm{obs}}^{(\mathcal B)}}
\le \deltacoll,

where $\Deltabulk^{(\mathcal B)}$TeX\Deltabulk^{(\mathcal B)} is the admitted family-fixed bulk-correction budget under the declared protocol, after one family-fixed calibration and with the declared coarse-graining and environment budgets folded into $\mathsf{Est}$TeX\mathsf{Est}. This gate tests the ordinary-matter collapse after the admitted hadron-to-bulk lift; it does not introduce an additional mass-readout coefficient beyond $\alpha_{\OrdFam}$TeX\alpha_{\OrdFam}. If subfamily pruning or object-wise retuning is required, the constitutive map is rejected for that family.

G2: Finite additivity gate..

The protocol $\Protord$TeX\Protord passes G2 if

\sup_{(\mathcal B_1,\mathcal B_2)\in \mathcal A_{\mathrm{disj}}}
\frac{\left|m_{\mathrm{obs}}^{(\mathcal B_1\sqcup \mathcal B_2)}
-m_{\mathrm{obs}}^{(\mathcal B_1)}
-m_{\mathrm{obs}}^{(\mathcal B_2)}\right|}
     {m_{\mathrm{obs}}^{(\mathcal B_1)}+m_{\mathrm{obs}}^{(\mathcal B_2)}}
\le \deltaadd,

where $\deltaadd$TeX\deltaadd is declared to absorb the admitted $O(\epssep)$TeXO(\epssep) leakage on $\mathcal A_{\mathrm{disj}}$TeX\mathcal A_{\mathrm{disj}}. If systematic nonlinear residuals remain after fixing $\Protord$TeX\Protord, the linear law reference is rejected for that family.

G3: Finite hadron-to-bulk gate..

The protocol $\Protord$TeX\Protord passes G3 if

\sup_{\mathcal B\in \OrdFam}
\frac{\left|\Deltabulk^{(\mathcal B)}\right|}{\left|\Ophad^{(\mathcal B)}\right|}
\le \epshad,

on the declared comparison window, with $\Deltabulk^{(\mathcal B)}$TeX\Deltabulk^{(\mathcal B)} fixed once under the declared protocol. If electronic, chemical, lattice, or environmental sectors dominate the ordering or force replacement of the family-fixed bridge $\BQCD^{(\CompFam)}$TeX\BQCD^{(\CompFam)}, the hadron-dominated claim fails.

G4: Ranking-invariance gate..

Replacing one admissible family-fixed proxy by another may not reverse any mass ordering that is stable against the declared correction budget. If proxy replacement changes the ordering on the same family without leaving the admitted correction range, the rigidity bridge is not well defined.

G5: Second-variation gate..

The rigidity coordinate must remain compatible with the positive quadratic stability content of reference. If a candidate proxy requires negative rigidity for declared stable composites on the admissible window, the constitutive closure fails.

G6: Weak-equivalence gate..

The declared protocol passes G6 only if composition-dependent differential accelerations of admitted ordinary test bodies remain below the declared experimental tolerance and no separate passive or active source coefficient is required within the admitted ordinary-matter family. MICROSCOPE constrains composition-dependent violations of free fall for Ti/Pt test bodies at the $10^{-15}$TeX10^{-15} level [citation], and lunar laser ranging constrains equality of active and passive gravitational mass for ordinary matter [citation]. Any rigidity law that requires composition-dependent source coefficients beyond those limits is excluded.

G7: Weak-field source gate..

The closure reference must reproduce the ordinary Newtonian limit with one family-fixed coupling $G_{\mathrm H}$TeXG_{\mathrm H}. If ordinary weak-field phenomenology requires object-dependent long-range couplings once the family protocol is fixed, the response law is rejected.

G8: Domain gate..

The analysis concerns stable composite matter only. Applying reference to elementary non-composite sectors without an independent rigidity construction lies outside the domain of validity and provides neither support nor refutation of the constitutive map above. In particular, the present composite constitutive closure does not fix any elementary Yukawa/Higgs mass-loading mechanism.

Stable composite matter has been treated here as a low-response phase-condensed pattern of the global phase field. Within the declared domain, effective mass is assigned to the rigidity content that preserves that pattern against local phase-field rearrangement. For hadron-dominated ordinary matter, the rigidity coordinate is admitted only through a family-fixed bridge to EMT energy sectors, mechanical-stability observables, and trace-anomaly or gravitational-form-factor data, followed by a declared hadron-to-bulk coarse-graining step.

Under positivity, continuity, and approximate additivity, the leading constitutive mass law is

\mrig=\mu_m^{(\CompFam)}\,\Xibind.

For ordinary matter after the same coarse-graining step, the weakest admissible unscreened local weak-field closure is Poisson-type. The strong-binding sector enters only at the rigidity-carrier layer for ordinary matter; gravity is not identified with the strong interaction.

Admissibility is controlled by family-fixed proxy collapse, additive scaling under aggregation, hadron-to-bulk control, ranking invariance under admissible proxy replacement, second-variation positivity, compatibility with precision weak-equivalence tests, and the weak-field single-coupling source gate. Elementary non-composite mass-loading sectors, including Yukawa/Higgs-type elementary mass parameters, strong-field relativistic closure, and continuum moving-boundary response or wake-memory phenomena require separate analyses and are not part of the composite constitutive result stated here. No universal mass-origin claim is made.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.