Gravitation as Phase Synchronisation over Finite Relational Substrate

1. Introduction

General relativity is, a century on, at once the most precisely tested account of gravitation and the least explained. Its field equations are postulated, not derived; the metric is a primitive rather than a consequence; and three things sit unresolved at its base. Galactic dynamics and the accelerating expansion require, within the theory, a dark sector (cold dark matter and a cosmological constant) that outweighs the visible universe fivefold and dominates the energy budget, yet has escaped direct detection and demands a vacuum energy fine-tuned across some 120 orders of magnitude. Gravitational collapse ends in horizons and singularities, where the theory forfeits predictivity and information. And the field resists quantisation. These are the standing reasons for a recurring conviction: that gravity is not fundamental but emergent, a coarse-grained or thermodynamic residue of microscopic degrees of freedom. The obstacle that conviction has not cleared is that those degrees of freedom are left unspecified, so its derivations consume macroscopic inputs (an entropy, a temperature, a holographic screen) rather than producing them. This paper derives gravitation, with its constant G, post-Newtonian parameters, strong-field solution, horizon thermodynamics, and galactic acceleration scale, from a microscopic substrate specified in full, with no gravitational input put in by hand.

The construction draws on several established lines. Emergent and thermodynamic gravity is the closest tradition: Sakharov’s induced gravity [1], Jacobson’s derivation of the Einstein equation as an equation of state from the Clausius relation on local horizons [2], Padmanabhan’s thermodynamic route to the field equations [3], and Verlinde’s entropic [4] and emergent-gravity [5] programmes, the latter aimed squarely at the dark sector. These lean on the holographic principle and black-hole thermodynamics: ’t Hooft and Susskind [6,7], Bekenstein and Hawking [8,9], and the Gibbons–Hawking de Sitter entropy [10] that fixes the working cardinality below. Discrete and finite substrates for quantum gravity are the second antecedent: causal sets [11], loop quantum gravity [12,13], causal dynamical triangulations [14], cellular-automaton readings of physical law [15,16], and the finitist reconstructions of mathematics and quantum theory of Lev and Zeilberger [17,18], in which the continuum is a degenerate limit and not the ground floor.

Empirically, the dark sector is the sharpest probe. The standard cosmology fits the microwave background and large-scale structure with cold dark matter and a cosmological constant [19], but no dark particle has been found [20], and at galactic scales the data obey one tight law the halo picture reproduces only by tuning: the radial acceleration relation, in which the observed acceleration tracks the baryonic one through a single curve of $\sim 0.1$ dex scatter [21,22], with a universal transition at $a_0\approx 1.2\times {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}$, numerically close to $c{H}_0/2\pi$. Modified-dynamics phenomenology posits that scale and an interpolation by hand [23,24,25,26]. In the strong field, horizon-scale imaging [27,28] and the search for horizonless ultracompact objects through their ringdown echoes [29] resolve the metric near the photon sphere. The derivation also stands on the foundations of the spin-two field: a scalar gravity deflects no light [30], the massless spin-two action is the Fierz–Pauli form [31], its nonlinear completion is the Deser bootstrap [32] with the soft-graviton uniqueness of Weinberg [33], and the post-Newtonian record is catalogued by Will [34]. The relaxation dynamics the construction uses is, mathematically, the Kuramoto equation of coupled phase oscillators [35,36].

Closest to the present approach are the emergent-gravity and discrete-substrate programmes, with which it shares the conviction that spacetime and its dynamics are derivative. It differs in three respects. The substrate is a single finite arithmetic structure (a prime field and its low extensions, the continuum a labelled degenerate idealisation) rather than a continuum lattice, a spin network, or an unspecified ensemble, and it carries an explicit observer theory [37,38,39,40], so coarse-graining is a definite projection and not a heuristic. Gravity is derived, not posited: no entropy, temperature, or screen is assumed, and the thermodynamic quantities (the horizon area law, the de Sitter entropy) emerge rather than enter. And the same substrate yields quantum mechanics [41] and the Standard-Model interactions [42], so gravitation is one sector of a single construction rather than a free-standing analogy.

1.1. Our Contribution

Against this background, the finite ring cosmology (FRC) programme reconstructs physics on the prime field ${\mathbb{F}}_{\mathsf{\Omega }}$, with $\mathsf{\Omega }\sim {10}^{122}$ fixed by the de Sitter entropy and the continuum recovered only as a degenerate idealisation [37,38,39,40]. This paper derives gravitation on that substrate: the constant G, the post-Newtonian parameters, the strong-field structure, and the galactic acceleration scale all emerge as computed numbers, with no gravitational premise.

The physical picture can be stated in one paragraph. Every massive object is an ensemble of elementary clocks, phase-locked internally (that is what makes it one object). There is no universal clock they all read: the tick belongs to the observer’s frame, and the substrate, being a field, has no proper ideals, so no embedded shell can carry the totality’s clock as its own. Every massive object is therefore a non-ideal embedded shell whose frame drifts relative to the observer’s. The two grids agree only below the shell’s horizon and diverge above it. Gravity is the relaxation of that forced drift: the mutual phase tension between two such shells lowers a free energy whose gradient along their separation is a force, and the relaxation obeys a Kuramoto-type law [35,36]. Because distance is the decoherence between two systems, reducing the mutual drift is identically motion toward smaller separation: the relaxation does not cause gravitational attraction, it is gravitational attraction.

The paper’s results divide into three layers. The Newtonian layer (Section 4): Newton’s law exactly, with each of its parts given a provenance (the $r^{-2}$ from harmonicity in three dimensions, themselves derived in the framework rather than assumed; the product $m_1{m}_2$ from the coherent additivity of locked phases; the universal sign from the single arrow of the drive; $G=\hslash c/m_P^2$ from unit channel capacity), and the equivalence principle as an identity rather than a postulate. The relativistic layer (Section 5): the source is an exact finite dust tensor on the mass shell of a quadratic extension; one further identification (time and distance are two readings of a single correlation capacity) forces the spatial metric bias to equal the temporal one, giving $\gamma =1$, the full light deflection, and, through multiplicative composition, an exponential metric with $\beta =\gamma =1$, all classical tests at their observed values, with the gauge structure constructed exactly and the nonlinear completion decided by the microscopic theory (Theorem 2): no self-sourcing, the exponential reading unique. The strong-field and observational layer (Section 6Section 7): the static problem solved exactly, a horizonless but operationally black object with parameter-free shadow and ringdown signatures a few percent from general relativity, an area-law entropy closing a consistency loop on the substrate’s own size, and a derived weak-acceleration scale $a_0=c{H}_0/2\pi$ addressing the radial acceleration relation, the one tight empirical law of galactic dynamics that current theory leaves unexplained.

Throughout, the accounting discipline of the FRC programme applies: every statement is explicitly imported, a bridge, exact, a prediction, or $\mathsf{\Omega }$-hard, with the full dependency ledger collected in Section 8. The method is Noetherian in the strict sense [43,44]: invariance is the source of physical law, and every law derived below is traced to a stated symmetry through one of Noether’s two theorems. The substrate symmetries are finite, not continuous: the drive is a cyclic shift, a gauge transformation a cell-local element of the finite frame group, the source’s translation the additive cycle. The theorems are therefore read in their exact discrete form [45]: the conserved currents and gauge identities are exact finite identities, and the continuous one-parameter statement is their degenerate idealisation. The conserved charge belongs to a generator (the order-$(p-1)$ drive), not to a reflection: the universal sign of gravity follows from the broken time-orientation $g\leftrightarrow g^{-1}$, an order-two map with no charge, which is why it is not a Noether current. By the first theorem a global symmetry yields a conserved current: cardinality and torsor equivariance conserve the source (Proposition 2), and the time-homogeneity of the drive conserves the Hamiltonian, hence the propagating wave (SubSection 5.5). By the second theorem a local (gauge) symmetry yields off-shell identities: the relational reframings give the field’s gauge structure (Proposition 7) and leave two graviton polarisations (Proposition 11), the phase-origin shift forbids self-sourcing (Theorem 2), and scale-reframing covariance fixes the form of the metric and the frame-invariance of G (SubSection 4.4). The framework’s imported structure carries the construction; the nonlinear completion is fixed by two exact arguments, a Ward identity of the relational phase and the uniqueness of the scale-covariant composition law (Theorem 2), and the remaining coefficients are traced to bounded constructions.

2. The Substrate: A Primer

This section summarises, self-containedly, the elements of the FRC framework that the derivation consumes. The framework is developed in [37,38,39,40] and applied to the spectral theory of the Riemann zeta function in [40], whose physical reading the present paper extends; no familiarity with that corpus is assumed here.

• The Carrier and the observer.

The substrate, the Carrier, is a finite prime field ${\mathbb{F}}_{\mathsf{\Omega }}$, with $\mathsf{\Omega }\sim {10}^{122}$ in Planck units, fixed by the de Sitter entropy of the observable universe. There is no completed infinity: the classical continuum is recovered only as a degenerate idealisation of the finite structure, in the finitist tradition of Lev and Zeilberger [37]. The element 0 (position), the unit 1 (scale), and a multiplicative generator (speed-like) are not absolute marks but frame data that an observer assigns (Figure 1 shows the assignment at the smallest convenient scale, the shell ${\mathbb{F}}_{13}$ framed on the generator $g_p=2$); physics is the frame-covariant content, and “no absolute frame” is the framework’s relational ontology. An observer, a Subject, is a much smaller nested shell ${\mathbb{F}}_p$, $\mathsf{\Omega }\gg p^2$; below the Subject’s horizon the Subject and Carrier agree on arithmetic, and all laboratory physics lives in that agreement window.

• Time, the drive, and the floor.

Time in the framework is scale-dilation: the multiplicative action $x\mapsto gx$ of the frame generator, advancing every substrate cell by one chronon per step (the green latitudes of Figure 1 are the orbits of this action); its idealised generator is $\widehat{H}=-i(x{\partial }_x+\frac{1}{2})$ [38,40]. The dilation is the Carrier’s own evolution, read at the largest scale as the Hubble flow; an embedded shell, not being an ideal of the field, reads only a projection of it, agreeing below its horizon and drifting above (Section 4). Two derived scales bound every observation. The coherence horizon$\mathsf{\Omega }\sim {10}^{61}$ in Planck units (the quanta of the frame that reads the Planck scale) is the Hubble radius; and the resolution floor$1/\mathsf{\Omega }$ is the finest signal any frame resolves (the depth of the totality relative to the observer, hence frame-covariant), a quantity established in [40], where it bounds the spectral resolution of arithmetic itself. The floor will appear in this paper four times: as the evaporation cutoff, the preferred-frame residual, the operational-horizon criterion, and the galactic acceleration scale.

• Chronon, cardinality, capacity.

The frame datum $(t;0,1,g)$ carries three distinct quantities ([40] §1.3). The chronon t is the observer’s present tick, the scale-dilation coordinate of the drive $x\mapsto gx$: the frame’s “now,” the one ambient drive every embedded shell reads, not a property of any shell. The cardinality ($\mathsf{\Omega }$ for the Carrier, p for a Subject) is the shell’s size and its subscript label; the Carrier, being unique, is unsubscripted. The capacity ${\kappa }_p=(p-1)/4$ is the shell’s quarter-period, the count of phase cells it resolves, derived from the cardinality and entering the framed objects as an exponent, as in the quarter-turn ${\mathrm{i}}_p=-g_p^{{\kappa }_p}$. The capacity ${\kappa }_p$ is distinct from the synchronisation stiffness $\kappa$ of Section 4: the former is the shell’s quarter-period $(p-1)/4$, the latter the per-link channel coupling normalised to unity (Lemma 1).

• Three dimensions from frame freedoms.

The observer’s chart of the Carrier is the projective line over ${\mathbb{F}}_{\mathsf{\Omega }}$, the $\mathsf{\Omega }$ field residues closed by the horizon mark, and an oriented frame is a unimodular basis of the underlying plane, so the space of frames is a torsor under ${\mathrm{SL}}_2\left({\mathbb{F}}_{\mathsf{\Omega }}\right)$ ([40] §9.6). Its three motion cycles are the three frame freedoms: translating the origin (cycle $\mathsf{\Omega }$, space), rescaling the unit (cycle $\mathsf{\Omega }-1$, scale-time), and the horizon-carrying boost (cycle $\mathsf{\Omega }+1$). The three dimensions of space are these freedoms, not three copies of the ring; the continuum shadow of the frame space is the 3-sphere, consonant with the spatial sections of de Sitter space. For the present paper the operative consequence is that the spatial chart is three-dimensional per frame: each observer carries a bounded-degree adjacency graph on the locally flat frame space, in its own quanta, with curvature radius $\mathsf{\Omega }$ far beyond any laboratory scale. The cubic lattice ${\mathbb{Z}}^3$ is used throughout as a computational representative of this chart’s quasi-isometry class, and every result below depends only on the class: the dimension enters through volume growth, the Gauss law holds on any graph, and the lattice-dependent constant of the Green’s function is absorbed into the calibration of $\kappa$. Since distance is itself the decoherence count (Definition 3), microscopic anisotropy of the adjacency cannot appear in laws expressed in r: the large-scale metric is intrinsic to the propagation. Scale is itself frame data: the substrate is scale-periodic [37], no scale is privileged, and every frame sees the full scale range around it. Adjacency in a given frame is the only relational channel that frame carries.

• Scale horizons and the rotated charts.

A frame’s inward and outward scale horizons are not boundaries of the substrate but loci where the frame’s chart has partially rotated between position and spectrum: observations there are partial fractional-Fourier readings of the space–momentum pair, and at the exact quarter-turn the reading is pure spectral [39] (on the example shell, the quarter-turn meridian, red in Figure 1, is that chart). The inward rotation is where the quantum description begins, the action quantum being the quarter-turn dual of the frame’s space quantum [40], and is developed separately in the framework’s quantum theory [41]; the outward rotation returns in SubSection 7.3, where it bears on the weak-acceleration regime.

• The quadratic extension and the boost cycle.

The Lorentzian and spinor structure lives one extension up [38]: on the quadratic extension K of the shell, with Frobenius involution $z\mapsto z^{\sigma }$, trace-zero generator $\eta$ (${\eta }^2=\nu$ a nonsquare), and norm $N\left(z\right)=z{z}^{\sigma }$. In the basis $\{1,\eta \}$ the norm is the Lorentzian form $a^2-\nu b^2$, with the nonsquare $\nu$ supplying the temporal–spatial signature. The unit-norm circle U is the boost cycle of the framework’s Dirac layer; Frobenius acts on it as inversion, supplying conjugation and time reversal.

• Mass and the Compton clock.

In the framework’s mass reading ([40] §9.5), a mass m is a Subject of m degrees of freedom (mass is cardinality), and its rest energy is the share m of the totality, with the Planck mass the locality horizon $m_P=\mathsf{\Omega }$. The object’s clock is its Compton frequency, $E=m{c}^2=hf$: frequency proportional to cardinality. This reading enters the present paper as Definitions 1 and 2, and is the hinge of the whole construction: massive objects are literally clocks, and their cardinality is literally their winding rate.

• Phase, clocks, and their comparison

The phase this paper dynamises is not an added field. Every shell contributes a phase cycle, its multiplicative group, into the phase cycle of the ambient totality. In the worked example of Figure 2, drawn from [41], the Carrier shell ${\mathcal{C}}_{157}$ carries ${\mathsf{\Phi }}^{\mathrm{C}}\simeq C_{156}$, and the nested Subject ${\mathcal{S}}_{53}$ and Object ${\mathcal{O}}_{13}$ contribute ${\mathsf{\Phi }}^{\mathrm{S}}\simeq C_{52}$ and ${\mathsf{\Phi }}^{\mathrm{O}}\simeq C_{12}$ inside it; the Object is the shell of Figure 1, whose multiplicative latitudes are the cycle embedded in ${\mathsf{\Phi }}^{\mathrm{C}}$. A cell’s phase is its position on its cycle, advanced one step per chronon by the drive: time is multiplication by the generator, winding is the generator action itself, and the Compton clock of the mass reading above is that action counted (Figure 2a). Subject and Object share a core, generically the quarter-turn subgroup $Q_4={\mathsf{\Phi }}^{\mathrm{S}}\cap {\mathsf{\Phi }}^{\mathrm{O}}$ (Figure 2b), and the comparison of phases, which is the only measurement this paper performs (rates, offsets, decoherence counts), is the Subject-relative projection of the Object carrier, read on the shared core: the Subject resolves the quotient ${\mathsf{\Phi }}^{\mathrm{O}}/Q_4\simeq C_3$ through coherent sums over its fibers (Figure 2c). The coherence counts of such projections are the correlation measures of Definition 3. The full observation formalism, with its projection operators, selection rules, and probability readout, is developed in [41]; the present paper consumes only the existence of phase cycles, generator winding, and shared-core comparison.

3. Inputs: Imported Structure and Conventions

The construction consumes three kinds of input, and none is a free physical premise. The first is the framework itself (Section 2): the substrate with its phase cycles, the global dilation as time, the cardinality reading of rest energy, the relational ontology, and the three frame freedoms; these are developed and priced in the corpus [37,38,40] and enter here as imported structure. In particular the drift is forced, not assumed: the substrate is a field, so an embedded prime shell is not an ideal of it. Its grid coincides with the Carrier’s only below the shell horizon and diverges above [40], and a frame that cannot close must drift. The relaxation of that drift is the Kuramoto-type dynamics of Section 4 [35,36]. The second is definition and unit convention. The third is the composite identification: what makes a bound cluster one object is that its cells share a single coherent phase. That coherence is not assumed here: it is derived, premise-free, in [41], where the single drive conserves the cluster’s internal phase offsets (offset superselection) and the m-body synchronisation lemma fixes the coherent cluster as one drive orbit [41]. What is left is a correspondence, stated below.

Definition 1 

(Mass is winding rate). The mass of a system is its winding rate: the number of phase quanta its state advances per chronon, with $E=m{c}^2=hf$ read as an identity. For a cluster of m phase-locked cells the rate is m quanta per chronon, the Compton clock of Section 2: cardinality is winding rate is mass.

Definition 2 

(Composite). A composite object is a cluster of cells on a single synchronised drive orbit: its cells share one coherent phase. This coherence is conserved by the common drive and superselected (not maintained by an external assumption), and the cluster couples with coefficient m, an offset-spread ensemble with $O\left(\sqrt{m}\right)$ [41]; only the coefficient m enters the present construction, through Lemma 4.

The single physical content is then a correspondence: the objects we weigh (protons, atoms, planets) are composites in this sense. Like “mass is winding rate” (Definition 1) and “distance is decoherence” (Definition 3), this is an identification of the formalism with the world, not a free parameter; it is falsifiable through the $\sqrt{m}$ coupling an incoherent source would instead exhibit (Lemma 4; the equal-force, unequal-entanglement test of [41]).

Definition 3 

(Distance is decoherence). The distance r between two systems is their decoherence, counted in relational steps of the observer’s chart. Under the relational ontology this is forced up to units: all observation is shared-core phase comparison (Section 2), so correlation is the only relational data available to define distance, and adjacency in a frame is its unit step; the decoherence metric is the graph metric of the chart by construction.

Lemma 1 

(Unit capacity is a frame normalisation). The coherence channel between adjacent cells carries at most one phase quantum per chronon, the same at every link, and the stiffness may be set to $\kappa =1$ in the frame’s units.

Proof. 

The bound: a link carrying more than one quantum per chronon would resolve structure finer than the frame’s quantum, contradicting the quantum’s definition; independently, any transfer function on a finite cycle is bounded, so saturation is forced by finiteness itself. Homogeneity: the frame torsor acts transitively on cells (no cell is distinguished, Section 2), so the capacity is the same at every link. Normalisation: if the uniform capacity differed from the frame quantum, redefining the quantum to the capacity restores $\kappa =1$; the choice is one of units, exactly as $c=1$. What the lemma fixes is the magnitude of G and the location of the saturation scale; the inverse-square form is independent of it.    □

Lemma 2 

(Channel unity). Each cell carries one correlation capacity, and the clock tick (temporal correlation with the drive) and the decoherence count (spatial correlation with neighbours) are two readings of it, so a bias of the capacity biases both.

Proof. 

A cell is one degree of freedom, and a degree of freedom is one phase on one cycle (Section 2). Temporal correlation is that phase’s coherence along the scale cycle; spatial correlation is the same phase’s coherence along the meridian: two frame freedoms of one torsor reading one coherence. Two independent capacities would require two phases per cell, contradicting the cardinality accounting in which a cell is one degree of freedom.    □

On these inputs the gravitational statement is mechanical: locked clusters are non-ideal embedded shells whose frames drift; the drift relaxes, lowering a free energy whose spatial gradient is a force; and since r is decoherence (Definition 3), reducing the mutual drift is identically motion toward smaller r (Figure 3). All inputs are scale-frame covariant, so every derivation below holds in any frame’s own units; the covariance is claimed as a result in SubSection 4.4. In summary: given the framework, the construction contains no free physical parameter and no physical premise (the coherence of composites is derived in [41]), only the correspondence that observed matter is composite; all further assumptions are those of the framework itself, stated and defended in the corpus.

4. The Newtonian Layer

4.1. The Coherence Field

Let ${\theta }_x\in \mathbb{R}$ (the cell’s phase-cycle position, linearised from the boost-cycle phase; Section 2) be the phase at cell $x\in \mathsf{\Lambda }\simeq {\mathbb{Z}}^3$ (the representative chart), with the per-chronon update

$${\theta }_x⟼{\theta }_x+\omega \tau +\kappa \tau \sum _{y\sim x}sin({\theta }_y-{\theta }_x),$$

(1)

the discrete Kuramoto dynamics [35,36] with nearest-neighbour coupling: nearest-neighbour because, by Definition 3, adjacency is the only relational channel the substrate carries. Write $u_x={\theta }_x-\omega t$ for the drift of cell x relative to the observer’s frame rate $\omega$, and linearise about alignment.

Lemma 3 

(Gradient flow). To first order in the offsets, the dynamics (1) is the gradient flow $\dot{u}=-\partial E/\partial u$ of the free energy

$$E\left[u\right]={\displaystyle \frac{\kappa }{2}}\sum _{\langle xy\rangle }{(u_x-u_y)}^2-\sum _i{m}_{i}u\left(x_i\right),$$

(2)

where the sum over i runs over the locked clusters of Definition 2, located at $x_i$ with cardinalities $m_i$.

Proof. 

Linearising $sin(u_y-u_x)$ gives the diffusive part $\kappa {(\Delta u)}_x$. The cluster enters as an intrinsic winding rate $m_x={\sum }_i{m}_i{\delta }_{x,x_i}$, the pinning frequency of its Kuramoto oscillators (mass is winding rate, Definition 1), so the forced equation of motion is ${\dot{u}}_x=\kappa {(\Delta u)}_x+m_x$. This is $-\partial E/\partial u_x$ of (2): the source appears in both the energy and the phase update. On the closed shell ${\sum }_x{(\Delta u)}_x=0$, so a static solution requires the net demand ${\sum }_i{m}_i$ to be balanced; the compensating background is the Carrier in which the shell is embedded as a non-ideal subobject, its uniform counter-winding absorbing the net (Section 2). The coupling coefficient is fixed by the next lemma.    □

Lemma 4 

(Coherent additivity). A locked cluster of cardinality m couples to the ambient field with coefficient exactly m. An unlocked ensemble of m independent phases couples with expected magnitude $O\left(\sqrt{m}\right)$.

Proof. 

Each member clock contributes a linear pull on its adjacent ambient cells toward its own phase. For a locked cluster all m pulls are toward the same phase and add as scalars: total coefficient m. For independent phases the pulls are random unit-strength terms on the phase circle; their sum is a random walk of m steps, of typical magnitude $\sqrt{m}$. Coherence converts root-mean-square addition into linear addition. The two coefficients are confirmed as exact substrate identities by the m-body synchronisation lemma and the power-additivity proposition [41].    □

Lemma 5 

(Stationary bias field). The minimiser of (2) satisfies $\kappa \Delta u=-{\sum }_i{m}_i{\delta }_{x_i}$ and equals

$$u^{*}\left(x\right)={\displaystyle \frac{1}{\kappa }}\sum _i{m}_i{G}_{\mathsf{\Lambda }}(x,x_i),G_{{\mathbb{Z}}^3}(x,y)={\displaystyle \frac{1}{4\pi |x-y|}}\left(1+O\left(\right|x-y{|}^{-2})\right),$$

the discrete harmonic Green’s function of the three-dimensional chart, whose Coulomb asymptotics is rigorous lattice potential theory [46,47].

Proof. 

Stationarity of (2) is the discrete Poisson equation; the Coulomb asymptotics holds for any graph in the quasi-isometry class of flat 3-space, with ${\mathbb{Z}}^3$ the computed representative [46,47]. The exponent is forced by the dimension: on a d-dimensional chart $G\sim r^{2-d}$, and $d=3$ is supplied by the frame-torsor derivation summarised in Section 2, not assumed here.    □

4.2. The Inverse Square Law

Theorem 1 

(Newtonian limit). For two clusters of cardinalities $m_1,m_2$ at decoherence distance r (large in the frame’s quanta), the equilibrium free energy carries the interaction term

$$U\left(r\right)=-G{\displaystyle \frac{m_1{m}_2}{r}},G={\displaystyle \frac{1}{4\pi \kappa }},$$

and the force along the decoherence coordinate is

$$F\left(r\right)=-{\displaystyle \frac{dU}{dr}}=-G{\displaystyle \frac{m_1{m}_2}{r^2}},$$

attractive, with relative corrections $O\left(r^{-2}\right)$ from the lattice Green’s function.

Proof. 

Substituting the minimiser of Lemma 5 into (2) gives

$$E\left[u^{*}\right]=-{\displaystyle \frac{1}{2}}\sum _{i,j}{\displaystyle \frac{m_i{m}_j}{\kappa }}{G}_{\mathsf{\Lambda }}(x_i,x_j)=\left(\mathrm{self}\text{-}\mathrm{energies}\right)-{\displaystyle \frac{m_1{m}_2}{4\pi \kappa r}}\left(1+O\left(r^{-2}\right)\right),$$

the self-energy terms independent of r. The cross term is $U\left(r\right)$; differentiation gives F. The sign is structural: for a field of positive stiffness with linear source coupling, the equilibrium energy decreases as like-signed sources approach. This is the scalar-mediator sign, opposite to electrostatics, where the energy is field-quadratic with no linear reward. Motion toward smaller r is available because, by Definition 3, r is the decoherence measure and the gradient flow of Lemma 3 increases mutual coherence: the relaxation is the approach.    □

Proposition 1 

(Universality of attraction). All masses are like-signed sources, hence all gravitational interaction is attractive.

Proof. 

The sign of a source in (2) is the winding direction of its clocks, and every composed object winds in the one global time direction of the dilation (the common drive, Section 2). A repulsive gravitational charge would require a time-reversed clock, which is the Frobenius branch of the boost cycle [38]: the conjugation, not an evolution available to a composed cluster riding the drive. Universal attraction is the universality of the time direction.    □

4.3. Three Corollaries

Corollary 1 

(Equivalence principle as an identity). Inertial and gravitational mass are the same cardinality by construction: gravitational mass is the coherent pull of m locked clocks (Lemma 4), inertial mass is the re-phasing cost of the same m locked clocks under a change in the state of motion. Both proportionalities are one coherent additivity applied to one cluster; there is no second parameter to tune.

Corollary 2 

(The bias field is clock rate). The stationary field $u^{*}$ is a clock-rate offset: the clock of a bound test mass is its Compton frequency, $E=hf$, so its rate at distance r from a mass m is shifted by the binding, $\Delta f/f=Gm/\left(r{c}^2\right)$. This is gravitational time dilation and redshift, derived rather than imposed. Section 5 completes this temporal reading with its spatial partner.

Corollary 3 

(Turnaround against the Hubble drive). The comoving stretch of the decoherence distance contributes an outward drift of acceleration scale $H^{2}r$; net approach requires $Gm/r^2\gtrsim H^{2}r$, i.e. $r\lesssim {(Gm/H^2)}^{1/3}$: the turnaround radius of structure formation. Bound systems are synchronised clusters decoupled from the flow, which is the observed non-expansion of bound structures.

4.4. The Magnitude of G

The form of the law used only Lemmas 3–5; the magnitude uses Lemma 1. With $\kappa =1$ (one phase quantum per cell per chronon) the constant is $G=1/4\pi$ in shell units. Restoring dimensions with the Planck mass as the locality horizon $m_P=\mathsf{\Omega }$ (Section 2) gives

$$G={\displaystyle \frac{\hslash c}{m_P^2}},$$

the definition of the Planck mass read backwards: G is not a new constant but the statement that the coherence channel has unit capacity. And since Lemma 1 is a frame normalisation, the relation holds identically in every scale frame: G is frame-covariant bookkeeping, and the construction carries a relativity principle for scale (a symmetry whose invariant is G itself) under which no observation of the gravitational sector distinguishes the observer’s scale frame. The weakness of gravity is the dilution of one clock’s bias across the substrate: the dimensionless coupling of two protons, ${(m_p/m_P)}^2\sim {10}^{-38}$, is the square of their cardinality fraction, small because $\mathsf{\Omega }$ is large.

5. The Relativistic Layer

A coherence field read as clock rate alone is the temporal half of a metric: it reproduces the Newtonian limit and the redshift but only half of the observed light deflection, and a scalar coupled to the source’s trace, as in Nordström’s theory [30], deflects light not at all [34]. The completion has two parts, and both are supplied by the framework rather than added to it: the source must be a symmetric two-index object, and the response must bias space exactly as it biases time.

5.1. The Source: The Frobenius-Symmetric Square on the Mass Shell

Proposition 2 

(Dust tensor from the quadratic extension). The state of motion of a cluster is an element $w=a+b\eta \in K$ on the Frobenius unit shell

$$N\left(w\right)=w{w}^{\sigma }=a^2-\nu b^2=1,$$

the unit mass shell of the Lorentzian form of Section 2. The source of the coherence field is the symmetric square

$$T=m(w\otimes w)=m\left(\begin{array}{cc}{a}^2& ab\\ ab& b^2\end{array}\right),$$

whose trace under the Frobenius pairing is $mN\left(w\right)=m$: the rest cardinality. Its components are the energy and momentum flux of relativistic dust; at rest ($a=1$, $b=0$) it reduces to the scalar source of Lemma 3, identified thereby as the rest-frame reading of the tensor source. The source is conserved: cardinality conservation gives the temporal component, torsor equivariance the flux components. This is the conserved current of the substrate’s translation symmetry, by Noether’s first theorem [43].

Proof. 

The mass-shell identity is the Frobenius norm in the $\{1,\eta \}$ basis; the trace identity is ${Tr}_K(w\otimes w)=N\left(w\right)$ under the pairing $\langle z,z^{\prime }\rangle =z{z}^{\prime \sigma }$. Boost covariance is the statement that the boost cycle acts on w by multiplication and on T by the symmetric square of that action, preserving N; the off-diagonal $ab$ is the momentum read by the boosted frame.    □

Proposition 3 

(From the mass shell to the $(3+1)$ tensor). The single boost plane of Proposition 2 extends to the full spacetime tensor through the frame group, with no further input. The observer’s chart is the projective line over ${\mathbb{F}}_{\mathsf{\Omega }}$, and an oriented frame is a unimodular basis of its plane, so the frame space is a torsor under ${\mathrm{SL}}_2\left({\mathbb{F}}_{\mathsf{\Omega }}\right)$ of order $(\mathsf{\Omega }-1)\mathsf{\Omega }(\mathsf{\Omega }+1)$ [40]. Its three motion cycles are the three frame freedoms: the unit rescaling (the split torus, order $\mathsf{\Omega }-1$) is the drive, hence time; the origin translation (the unipotent, order Ω) and the boost (the non-split torus, order $\mathsf{\Omega }+1$) are spatial. As the norm-one quaternion variety, four coordinates under one relation, ${\mathrm{SL}}_2\left({\mathbb{F}}_{\mathsf{\Omega }}\right)$ is three-dimensional and simply connected, carrying the spinor double cover exactly [38,41], its continuum limit the de Sitter $S^3$ [40].

Proof. 

The shell state $w=a+b\eta \in K$ is the two-component spinor of one frame direction. A spacetime vector is its Frobenius–Hermitian square $x^{\mu }\leftrightarrow w\otimes w^{\sigma }$, read over the four frame components (one temporal, from the drive; three spatial, from the freedoms), the signature fixed by the square class of the nonsquare $\nu$ [38]. The dust $T=m(w\otimes w)$ of Proposition 2 is then the rest block of the conserved momentum-flux tensor $T_{\mu \nu }=m{u}_{\mu }{u}_{\nu }$, and the three-parameter per-cell deformation of Proposition 6 is one plane of the symmetric four-frame field $h_{\mu \nu }$, ten components. The gauge of Proposition 7 is the cell-local reframing of the four-frame, ${\xi }_{\mu }$, four parameters, so the radiative count $10-4-4=2$ of Proposition 11 is this one frame read over all four directions.    □

5.2. Channel Unity: Space Bends Because Distance Is Correlation

Lemma 6 

(The isotropic pair). To first order in the bias u, a cell devoting the fraction u of its capacity to a cluster’s field presents to ambient processes the line element

$$d{s}^2=-(1-2u)c^{2}d{t}^2+(1+2u){\delta }_{ij}d{x}^{i}d{x}^j,u={\displaystyle \frac{Gm}{r{c}^2}},$$

i.e. the spatial bias equals the temporal one: the parameterised-post-Newtonian $\gamma =1$.

Proof. 

Temporal reading: the cell’s correlation with the global drive is reduced by the factor $(1-u)$, so proper time accumulates as $d\tau =(1-u)dt$. Spatial reading: the decoherence increment per lattice step is inversely proportional to the remaining capacity, $1/(1-u)\approx 1+u$, so the proper distance element is $dl=(1+u)dx$. Both factors are the one number u by Lemma 2: there is no second field to tune. The failure mode of half-deflection theories, time dilating while space staying absolute, is unavailable here, because space is itself a correlation measure and cannot stay absolute while correlation is biased.    □

Proposition 4 

(Full light deflection). Correlation propagates at one cell per chronon; in the biased region the effective index is $n=1+2u$, and Fermat’s principle gives the deflection of a ray of impact parameter b:

$$\alpha ={\int }_{-\infty }^{\infty }\left|{\nabla }_{\perp }(n-1)\right|dz={\displaystyle \frac{4Gm}{c^{2}b}},$$

the observed value: twice the clock-rate-only result, the factor restored by the spatial half of Lemma 6.

5.3. Composition: The Exponential Metric and the Classical Tests

Lemma 7 

(Multiplicative composition). Capacity renormalisations of successive cells compose multiplicatively. For an unsaturated configuration the composed factors are $exp(\mp U)$ with $U={\sum }_{i}G{m}_i/\left(r_i{c}^2\right)$ the superposed potential, giving the exponential isotropic metric

$$d{s}^2=-e^{-2U}{c}^{2}d{t}^2+e^{2U}{\delta }_{ij}d{x}^{i}d{x}^j.$$

Validity requires every per-cell factor far from saturation, which Section 6 shows holds down to a microscopic core.

Proposition 5 

(All three classical tests). The exponential isotropic metric has $g_{00}=-(1-2U+2U^2+O\left(U^3\right))$ and $g_{ij}=(1+2U+O\left(U^2\right)){\delta }_{ij}$, hence $\beta =1$, $\gamma =1$: light deflection, Shapiro delay, and perihelion precession all take exactly their observed (general-relativistic) values [34].

Beyond first post-Newtonian order the exponential metric deviates from Schwarzschild; the classical tests do not resolve the difference, and the regime where it matters is the strong-field regime of Section 6.

5.4. The Route to the Full Field Equations

The field, its gauge structure, and its coupling can be constructed exactly; the dynamics is then pinned by imported uniqueness theorems.

Proposition 6 

(The deformation space is the symmetric field). The symmetric bilinear deformations $g=g_0+h$ of the norm form $g_0=\mathrm{diag}(1,-\nu )$ form a three-parameter space per cell, $h=(h_{00},h_{01},h_{11})$. A cell-dependent assignment $x\mapsto h\left(x\right)$ is the gravitational field: the isotropic pair of Lemma 6 is the diagonal slice, and the gravitomagnetic sector is the off-diagonal $h_{01}$, conjugate to the momentum component $ab$ of the source.

Proposition 7 

(Reframings are the gauge). A reframing is a cell-dependent infinitesimal change of frame data, $A\left(x\right)=1+\epsilon \left(x\right)$, $\epsilon \in {\mathfrak{sl}}_2$, acting on forms by congruence. A constant ε moves $g_0$ inside its congruence orbit and changes no invariant; a varying $\epsilon \left(x\right)$ enters every adjacency-local functional only through link differences of $\xi =g_0\epsilon$, so the gauge orbit of the field is the discrete symmetrised gradient $h_{\mu \nu }\mapsto h_{\mu \nu }+{\nabla }_{\mu }{\xi }_{\nu }+{\nabla }_{\nu }{\xi }_{\mu }$: the linearised diffeomorphism gauge, derived from the relational ontology rather than postulated.

Proposition 8 

(Coupling, conservation, and the imported uniqueness). The unique frame-invariant linear pairing of field and source is $h_{\mu \nu }{T}^{\mu \nu }$; its gauge invariance requires ${\nabla }_{\mu }{T}^{\mu \nu }=0$, the conservation of Proposition 2. The quadratic, adjacency-local, gauge-invariant stiffness functional of a symmetric two-index field is unique up to normalisation (the Fierz–Pauli functional [31]), and its discrete form can be exhibited at once. With central differences ${\Delta }_{\mu }$ on the chart and $h={\eta }^{\mu \nu }{h}_{\mu \nu }$,

$$E_{\mathrm{FP}}\left[h\right]=\kappa \sum _x\left[\frac{1}{4}{\Delta }_{\lambda }{h}_{\mu \nu }{\Delta }^{\lambda }{h}^{\mu \nu }-\frac{1}{4}{\Delta }_{\lambda }h{\Delta }^{\lambda }h-\frac{1}{2}{\Delta }_{\mu }{h}^{\mu \nu }{\Delta }^{\lambda }{h}_{\lambda \nu }+\frac{1}{2}{\Delta }_{\mu }{h}^{\mu \nu }{\Delta }_{\nu }h\right],$$

(3)

overall normalisation and signature bookkeeping absorbed in κ. This functional is exactly invariant under the discrete gauge orbit of Proposition 7 on the periodic chart: the continuum invariance proof uses only the commutativity of derivatives and their anti-self-adjointness under integration by parts, and central differences carry both properties exactly (${\Delta }_{\mu }^{\mathsf{T}}=-{\Delta }_{\mu }$ on the periodic chart, with real Fourier symbol $sin{k}_{\mu }$ replacing $k_{\mu }$ in the continuum identity); one-sided differences, whose adjoint is not their negative, do not, and the invariance fails for them. Both statements are exact identities over the finite periodic shell, established by exhaustive evaluation in the validation suite. Uniqueness within the adjacency-local class is moreoverproven on the shell: among translation-invariant, range-one, hypercubic-invariant quadratic functionals of $h_{\mu \nu }$, the gauge-invariant space at the relevant order is one-dimensional, spanned by (3), by symbol transversality (the four two-derivative invariants reduce to the single combination $(-1,2,-2,1)$, the linearised Einstein–Hilbert functional, and the result survives the substitution $k_{\mu }\mapsto sin{k}_{\mu }$ that defines the central-difference shell operator); the Fierz–Pauli mass tuning $h_{\mu \nu }{h}^{\mu \nu }-h^2$ is recovered as the unique ghost-free completion. This is one instance of a single adjacency-local uniqueness lemma whose spin-one case is the unique-Maxwell theorem of the field paper and whose Yang–Mills case reduces to it per colour (reports/uniqueness-lemma.pdf,fierz_pauli_uniqueness.py). If the field coupled to its own energy–momentum, Deser’s first-order bootstrap [32] would complete (3) to the Einstein dynamics. Whether it does is not a choice but a question the microscopic theory answers, and Section 6 answers it: what gravitates is winding rate, the static field strain winds not at all, and the would-be self-coupling is a double count that the relational ontology forbids (Theorem 2). The functional (3) is thereby the exact linear effective theory of a field whose nonlinearity lives entirely in the capacity reading.

Proposition 9 

(No preferred-frame leakage; frame dragging at the observed value). The framework possesses a global rest frame, the drive, yet the local dynamics inherits no preferred-frame parameters at leading order: source and channel are torsor-equivariant, so boosts act covariantly on all local data, and the drive enters local physics only through the resolution floor, at $O(1/\mathsf{\Omega })$. Hence the preferred-frame parameters obey ${\alpha }_1,{\alpha }_2=O(1/\mathsf{\Omega })$, and with $\gamma =1$ the gravitomagnetic sector takes its conservative value: the Lense–Thirring precession is the observed ${\mathsf{\Omega }}_{\mathrm{LT}}=2GJ/\left(c^2{r}^3\right)$ up to $O(1/\mathsf{\Omega })$ [34]. The residual is itself a prediction: preferred-frame effects of order ${10}^{-61}$, far below the current ${10}^{-5}$ bounds, but the drive’s in-principle signature.

5.5. The Radiative Sector: The Conjugate Momentum and Propagating Gravitons

The static field of Section 4 is a gradient flow, $\dot{u}=\kappa \Delta u$, which relaxes but cannot propagate. The resolution is that the substrate carries no dissipation: the drive $x\mapsto gx$ is a permutation of the finite cycle (bijective and reversible), so it preserves the phase-space measure. The symplectic form is supplied by the quarter-turn $Q_4$: the order-four element with $Q_4^2=-1$ is the complex structure, and $\omega (a,b)=\langle a,Q_{4}b\rangle$ is the nondegenerate canonical two-form. The phase offset u is therefore one half of a conjugate pair, its partner the local winding-rate fluctuation $\pi =Q_{4}u$, the quarter-turn dual that the drive rotates into u ($u\mapsto \pi \mapsto -u$, the position–momentum rotation $Q_4$). The gradient flow is the overdamped projection in which $\pi$ is adiabatically eliminated; the reversible dynamics retains it.

Proposition 10 

(The wave equation and the speed of gravity). With the conjugate momentum restored, the Hamiltonian $H={\sum }_x{\pi }_x^2/2\chi +\frac{\kappa }{2}{\sum }_{\langle xy\rangle }{(u_x-u_y)}^2$ generates, through Hamilton’s equations $\dot{u}=\pi /\chi$, $\dot{\pi }=\kappa \Delta u$, the discrete wave equation

$$\chi {\ddot{u}}_x=\kappa {(\Delta u)}_x,$$

whose static slice ($\ddot{u}=0$) is the Newtonian Poisson equation and whose source-free modes propagate with group velocity $\sqrt{\kappa /\chi }$. By channel unity (Lemma 2) the kinetic capacity χ (temporal correlation) and the stiffness κ (spatial correlation) are one capacity read two ways, so $\kappa /\chi =c^2$: gravitational waves travel at the speed of light, with $\gamma =1$ (Lemma 6) and this result two readings of the same lemma.

Proposition 11 

(Massless spin-two, two polarisations). The same completion on the symmetric field $h_{\mu \nu }$ of Proposition 6 (the four-frame field of Proposition 3), with the Fierz–Pauli stiffness (3) as potential and kinetic term $\frac{1}{2\chi }{\pi }_{\mu \nu }{\pi }^{\mu \nu }$, gives the hyperbolic spin-two equation $\chi {\ddot{h}}_{\mu \nu }=\left(\mathrm{linearised}\mathrm{Einstein}\mathrm{operator}\right)h_{\mu \nu }$. The linearised diffeomorphism gauge of Proposition 7 and the four first-class constraints leave $10-4-4=2$ propagating degrees of freedom: the transverse-traceless modes, whose spin-two projector ${\mathsf{\Lambda }}_{ij,kl}=P_{ik}{P}_{jl}-\frac{1}{2}{P}_{ij}{P}_{kl}$ ($P=\mathbb{1}-\widehat{k}\widehat{k}$) has rank exactly 2 in every direction, and which carry helicity $\pm 2$. A rotation by ψ about the propagation axis rotates $(h_{+},h_{\times })$ by $2\psi$. The graviton is massless, spin-two, two-polarisation, propagating at c. The propagating sector uses the second-difference Laplacian (symbol $4{sin}^2(k/2)$, vanishing only at $k=0$) and so carries no doubling. On the odd prime cycle $C_p$ the first-difference symbol ${\zeta }^k-{\zeta }^{-k}$ vanishes only at $k=0$, since $2k\equiv 0\left(\mathrm{mod}p\right)$ forces $k=0$; the continuum zone-edge mode $k=\pi$ has no counterpart on the shell, so there is no spurious zero to remove.

The leading radiation then follows from the linear theory and source conservation ${\nabla }_{\mu }{T}^{\mu \nu }=0$: the monopole and dipole vanish, the mass quadrupole radiates, and the binary-pulsar damping that Remark 3 took from “the shared linear theory” is recovered, grounded in the propagating equation. The construction and the float-free checks (the dispersion and long-wave speed, the transverse-traceless rank, the helicity $\pm 2$, the absence of doubling) are in reports/radiative.pdf and validate_radiative.py. The fully nonlinear self-interaction is a bounded residue; Theorem 2 already shows the relational phase forbids tensor self-sourcing (what gravitates is winding rate), so the nonlinear completion is the exponential reading of Section 6, with corrections in field gradient over capacity rather than in graviton loops.

Remark 1 

(The radiative sector is Noetherian). Each result here is, in the method’s strict sense (Section 1), the consequence of a stated invariance. The conservative dynamics that replaces the dissipative flow is the Noether charge of time-translation: the drive acts identically at every chronon, so an autonomous reversible evolution preserves the phase-space measure, and the quarter-turn $Q_4$ ($Q_4^2=-1$) supplies the symplectic form, making the dynamics Hamiltonian. The conjugate momentum π exists because the drive is time-homogeneous (Noether’s first theorem). The collapse from ten field components to two propagating polarisations is Noether’s second theorem applied to the linearised diffeomorphism gauge of Proposition 7, the same local symmetry whose Ward identity fixes the nonlinear completion (Theorem 2). And the radiation pattern is fixed by source conservation${\nabla }_{\mu }{T}^{\mu \nu }=0$, itself the Noether current of cardinality and torsor equivariance (Proposition 2): the monopole and dipole are forbidden and the mass quadrupole radiates. Time-translation gives the wave, gauge gives the two helicities, source conservation gives the quadrupole.

6. The Strong Field

The linearisation of Lemma 3 fails where the bias approaches the channel capacity, and the failure is not a defect but the theory’s strong-field content: the capacity bound is exact, and the static problem can be solved against it in full.

Lemma 8 

(Capacity bound, exact). The interaction (1) transfers at most κ phase quanta per link per chronon ($|sin|\le 1$). No static synchronised configuration exists in any region whose maintenance demands more.

Proposition 12 

(Exact static profile). In shell units, spherically symmetric statics requires the synchronisation flux through every sphere to carry the source: $4\pi r^2\kappa sin{u}^{\prime }\left(r\right)=m$, hence

$$u^{\prime }\left(r\right)=arcsin\left({\displaystyle \frac{Gm}{r^2}}\right),u\left(r\right)={\displaystyle \frac{Gm}{r}}\left(1+\frac{1}{30}{\left(\frac{Gm}{r^2}\right)}^2+\dots \right),$$

defined exactly on $r\ge r_{*}=\sqrt{Gm}$ (in physical units $r_{*}=\sqrt{r_g{\ell }_P}$, with ${\ell }_P$ the observer frame’s quantum) and undefined inside: the static synchronised exterior exists down to $r_{*}$ and no further. Inside $r_{*}$ phase slips (the depinning of a driven oscillator past its lock-in threshold) replace synchrony: a dynamical slip core, microscopic on every astrophysical scale (for a solar mass, $r_{*}\sim {10}^{-16}$ m).

Proof. 

Gauss’s theorem for the discrete dynamics: in a static locked state the net transfer into every closed region vanishes except for the source demand, and a link carries at most $\kappa sin$ of its phase difference. Monotone radial profiles give the flux law; arcsin requires its argument at most 1, with equality at $r_{*}$; the expansion of $u={\int }_r^{\infty }arcsin(Gm/s^2)ds$ is term-by-term.    □

Remark 2 

(The slip core is the scale half-turn). The radius $r_{*}=\sqrt{r_g{\ell }_P}$ is the geometric mean of the gravitational radius and the frame quantum: the log-scale midpoint between the object’s scale and the observer’s inward horizon. In the rotated-chart reading of Section 2 that midpoint is the scale half-turn, where the frame’s chart is half-rotated between position and momentum. The failure of position-chart statics exactly there is then expected rather than accidental: the slip core marks the hand-over from the spatial to the spectral description of the same region, whose development belongs to the framework’s quantum theory [41], and its cloaking is the rotation as much as the redshift.

Proposition 13 

(Horizonless, operationally black). The clock of a bound test mass is its Compton frequency, and capacity renormalisations compose multiplicatively (Lemma 7), the composition validated down to $r_{*}$ by the exact profile: the per-cell increment, $u^{\prime }\left(r\right)$ times the frame quantum, remains far below saturation at every $r\gg r_{*}$. Hence

$$f\left(r\right)=f_{\infty }{e}^{-u\left(r\right)},$$

which vanishes at no finite r: the exact theory is horizonless. The rate reaches the resolution floor ${\mathsf{\Omega }}^{-1/2}$ at $u=\frac{1}{2}ln\mathsf{\Omega }$, i.e. at

$$r_f={\displaystyle \frac{2Gm}{c^{2}ln\mathsf{\Omega }}}\approx {\displaystyle \frac{r_s}{281}},$$

inside which no clock is resolvable in any observer frame: operational blackness. Outward correlation export across the surrounding region is redshift-suppressed by the same factor, so the surface is operationally one-way.

Theorem 2 

(The nonlinear completion: no self-sourcing, exponential reading). The nonlinear completion of the static theory is determined by the microscopic model. (i) Ward identity. Every interaction of (1) depends on link differences only (the phase origin is unphysical per cell, the cell-local shift $u_x\mapsto u_x+c_x$ being the relational ontology itself, with the global $u\mapsto u+c$ its zero mode), so for any even link potential the static equation sums over a region to an exact Gauss law: boundary flux equals enclosed winding demand, nonperturbatively, the identity that the local shift symmetry imposes on the field equations in the sense of Noether’s second theorem [43]. The potential therefore carries no self-sourcing correction: $u=Gm/\left(c^{2}r\right)·(1+O\left({(r_{*}/r)}^4\right))$, with corrections in powers of field strength over capacity, never in powers of u. (ii) Uniqueness of the reading. All nonlinearity thus lives in the capacity reading $R\left(u\right)$ of Lemma 2; series composition through nested shells requires $R(u_1+u_2)=R\left(u_1\right)R\left(u_2\right)$, since u is additive by (i), and scale-frame covariance requires the same R in every frame: by the Cauchy functional equation $R=e^{-\lambda u}$, with $\lambda =1$ fixed by the first-order redshift. The exponential metric of Lemma 7 is therefore the unique completion consistent with the microscopic theory, and the strong-field structure of this section is derived, conditional on the stated inputs alone.

Proof. (i) With $F\left[u\right]=\kappa {\sum }_{\langle xy\rangle }V(u_x-u_y)-{\sum }_i{m}_{i}u\left(x_i\right)$ and V even, the static condition is $\kappa {\sum }_{y\sim x}{V}^{\prime }(u_y-u_x)+m_x=0$; summing over any region, interior links cancel pairwise ($V^{\prime }$ odd), leaving boundary flux equal to enclosed demand exactly. A self-sourcing term in the field equation would require a free-energy term odd under the shift, which the relational ontology excludes; gradient-built corrections vary into pure divergences and renormalise the flux profile (the arcsin of Proposition 12) without creating source. (ii) Additivity of u along paths and across nested shells follows from (i); associativity and scale-frame covariance of capacity composition give the multiplicative functional equation, whose monotone solutions are exponentials.    □

Remark 3 

(Where general relativity sits, and why the bootstrap does not apply). Isotropic Schwarzschild is itself an exponential reading, $g_{00}=-e^{-2\psi }$ with $\psi =2\mathrm{artanh}(U/2)=U+U^3/12+\dots$, but of a self-sourced potential, ${\nabla }^2\psi \propto {\left(Gm\right)}^3/r^5\ne 0$. The two theories thus differ in exactly one question: which scalar is harmonic. Theorem 2(i) answers it for the present theory: the relational phase. Deser’s bootstrap [32] does not apply because its premise, that the field couples to its own energy–momentum, is false in the microscopic model: what gravitates is winding rate ($E=hf$, Definition 1), static strain winds not at all, and the self-coupling would double-count constituent phases already counted in the sources. Energy bookkeeping survives without it: the mass defect of a bound system is the redshift of its constituents’ winding demand, $m(1-u)$ in the global frame; gravitational waves gravitate because they wind; and the leading quadrupole damping, fixed by the shared linear theory and source conservation, is unchanged, keeping the binary-pulsar agreement. The first divergences from general relativity are the $U^2$ spatial and $U^3$ temporal terms, currently untested, and the strong-field signatures of Section 7.

Remark 4 

(The exponential metric and its standing objection). The isotropic solution $d{s}^2=-e^{-2u}{c}^{2}d{t}^2+e^{2u}d{\mathbf{x}}^2$ is the exponential metric of Yilmaz [48], whose geometry Boonserm, Ngampitipan, Simpson and Visser analyse [49]: horizonless, with a photon sphere at $r=2Gm/c^2$, matching Proposition 15. The standing objection to exponential-metric gravity is Misner’s [50]: taken as a field theory on a flat background with the gravitational field entering its own source, the field equation is ill posed and its many-body static limit mishandles the Newtonian term. That objection does not reach the present construction, because the exponential reading here is derived rather than posited as a self-coupled field equation. The source is the conserved cardinality dust tensor (Proposition 2); the nonlinearity lives entirely in the capacity reading (Theorem 2(ii)); and the relational shift symmetry forbids self-sourcing exactly (Theorem 2(i)), so the static potential is harmonic and the Newtonian term is the leading flux of Lemma 5, not a quantity at risk of cancellation. Where the vacuum exponential metric continues inward to a second asymptotic region, the traversable-throat reading of [49], the microscopic theory instead hands over at the slip core $r_{*}$ to the conjugate spectral chart (Remark 2): the deep interior is fixed by the substrate, and the global object is the operationally black body of Proposition 13.

Proposition 14 

(Area-law entropy, and the de Sitter closure). The externally resolvable data of the operationally black region is carried by the channels of its bounding surface. The count is the relational one: the surface is measured in decoherence area $A_f=4\pi r_f^2$, with $r_f$ the substrate (decoherence) coordinate of Definition 3, which is monotone and carries no throat; the metric proper area $4\pi e^{2u_f}{r}_f^2$ is the observer-chart reading, while the channel count is chart-invariant. The channels number $A_f/{\ell }_P^2$ up to an order-one factor, each running at capacity with one undetermined slip phase: $S=c_S^{\prime }{A}_f/{\ell }_P^2$, the area scaling derived and the coefficient $c_S^{\prime }=\frac{1}{4}$ fixed, not free. The stated ratio $S/S_{\mathrm{BH}}={(r_f/r_s)}^2={ln}^{-2}\mathsf{\Omega }\approx 1.3\times {10}^{-5}$ against the Bekenstein–Hawking value [8,9] requires $4c_S^{\prime }=1$; and the same coefficient closes on the cosmological horizon: applied to the Hubble horizon of radius $\mathsf{\Omega }$ Planck lengths the law gives $S=\pi \mathsf{\Omega }$, the de Sitter entropy that fixes the Carrier cardinality (Section 2), so the framework’s own horizon law recovers the Bekenstein–Hawking–Gibbons–Hawking $\frac{1}{4}$ rather than importing it, closing the loop between the substrate size and its horizon thermodynamics. The $\frac{1}{4}$ is determined by the substrate’s self-consistency, not counted from microstates (reports/orderone.pdf).

Remark 5 

(Phase-slip radiation). At the threshold the links hover at lock-in, where the residual floor noise drives stochastic slips with characteristic frequency set by the offset gradient, the surface gravity, giving emission at the Hawking scaling $k_{B}T\sim \hslash {\kappa }_{\mathrm{sg}}/2\pi c$ [9]. Under the exponential reading the emission is additionally redshifted by ${\mathsf{\Omega }}^{-1/2}$: luminosity below one quantum per Hubble time. Frozen objects do not evaporate on any relevant timescale.

7. Observational Consequences

The global static solution reads, from the outside in: the post-Newtonian zone $r\gg r_g$, where $\beta =\gamma =1$ and all classical tests hold; the exponential zone $r_{*}\ll r\lesssim r_g$, where the metric is $-e^{-2u}{c}^{2}d{t}^2+e^{2u}d{\mathbf{x}}^2$ with the exact potential of Proposition 12 (whose correction to $Gm/r$ at the photon sphere is zero to floor precision for any astrophysical mass); the operational horizon at $r_f=r_s/ln\mathsf{\Omega }$; and the cloaked slip core at $r_{*}$. Its observational signatures are parameter-free (Figure 4).

7.1. Strong-Field Signatures Against Current Data

Proposition 15 

(Photon sphere and shadow). Null circular orbits of the exponential zone extremise $b\left(r\right)=r{e}^{2u\left(r\right)}$: with $u=Gm/\left(c^{2}r\right)$,

$$r_{\mathrm{ph}}={\displaystyle \frac{2Gm}{c^2}}\mathrm{exactly},b_c={\displaystyle \frac{2eGm}{c^2}}\approx 5.437{\displaystyle \frac{Gm}{c^2}},$$

against the Schwarzschild $b_c=3\sqrt{3}Gm/c^2\approx 5.196Gm/c^2$: the shadow is

$$\delta ={\displaystyle \frac{2e}{3\sqrt{3}}}-1=+4.63\%$$

wider, independent of mass, distance, and environment.

Sgr A^* is the clean arena, its mass-to-distance ratio fixed independently by stellar orbits. With the GRAVITY values ($M=4.297\times {10}^6{M}_{\odot }$, $D=8.277$ kpc, ${\theta }_g=5.12\mu$as) [27], the predicted shadow diameters are $53.3\mu$as (general relativity) against $55.7\mu$as (this work): an excess of $2.5\mu$as. The Event Horizon Telescope’s fractional-deviation constraints, $\delta =-0.{04}_{-0.10}^{+0.09}$ (Keck priors) and $-0.08\pm 0.09$ (VLTI priors) [28], place the prediction about one standard deviation high (Figure 5): not excluded, not confirmed, inside the discovery window; next-generation arrays targeting percent-level shadow calibration are decisive. For M87^* the corresponding numbers are 39.7 against $41.5\mu$as, with the caveat that its mass partly derives from the ring itself.

The independent twin rides in gravitational waves: at fixed inspiral-determined mass the eikonal ringdown frequency scales as $1/b_c$, so $f/f_{\mathrm{GR}}=3\sqrt{3}/2e=0.956$: a $-4.4\%$ shift, approachable by stacked events and decisive at Einstein Telescope sensitivity. Late-time echoes, the usual signature of horizonless ultracompact objects [29], are by contrast absent: the round-trip delay to the floor surface carries the factor $e^{2u\left(r_f\right)}=\mathsf{\Omega }$, giving $\Delta t\sim (2r_g/c)\mathsf{\Omega }/{ln}^2\mathsf{\Omega }\sim {10}^{118}$ s. The object mimics a black hole in that channel, and a reported echo detection would falsify the theory. The joint signature (shadow wide by $4.6\%$, ringdown low by $4.4\%$, no echoes, no evaporation bursts) has one origin and no tuning. By Theorem 2 these signatures are conditional on the stated inputs alone: a null result at the stated precision would falsify the construction’s inputs, that is, the framework itself or the composite correspondence. The prediction is stated for slow rotation: the rotating solution is the gravitomagnetic dual of the static field (SubSection 7.2), with the spin shifting the shadow centroid at $O\left(a\right)$ and the mean diameter at $O\left(a^2\right)$, of the Kerr scale (up to $\sim 4\%$), making low-spin Sgr A^* the preferred target.

Proposition 16 

(Innermost stable orbit and accretion efficiency). The accretion flow supplies a third strong-field signature, read off the same exponential zone and independent of the photon ring. Equatorial timelike circular orbits of the metric obey the radial balance ${\dot{r}}^2=E^2-W\left(r\right)$ with $W=e^{-2u}+L^2{e}^{-4u}/r^2$ and $u=Gm/\left(c^{2}r\right)$; marginal stability, $W^{\prime }=W^{\prime \prime }=0$ at the circular value of L, places the innermost stable circular orbit at the isotropic radius

$$r_{\mathrm{ISCO}}=(3+\sqrt{5}){\displaystyle \frac{Gm}{c^2}}\left(\mathrm{exact}\right),$$

with specific energy $E_{\mathrm{ISCO}}/m{c}^2=0.94521$. The binding energy released by quasi-static inspiral to that orbit, the maximal radiative efficiency of a non-rotating accretor, is

$$\eta =1-{\displaystyle \frac{E_{\mathrm{ISCO}}}{m{c}^2}}=5.48\%,$$

against the Schwarzschild $1-\sqrt{8/9}=5.72\%$. The ISCO orbital frequency, the highest stable disc clock, is $Gm{\mathsf{\Omega }}_{\mathrm{ISCO}}/c^3=0.0633$, a fraction $0.931$ of the Schwarzschild $1/6\sqrt{6}$, a $-6.9\%$ shift at inspiral-fixed mass. With the shadow (Proposition 15) and the ringdown, these three read low by a few percent from one cause, the no-self-sourcing completion (Theorem 2), and are probed by continuum-fitting spin estimates and high-frequency quasi-periodic oscillations. Construction and float-free check:validate_strongfield.py.

7.2. The Rotating Solution: Gravitomagnetism as the Quarter-Turn Dual

A rotating cluster carries its angular momentum J on the boost cycle of the frame: its state $w=a+b\eta$ has a non-zero momentum component, and the off-diagonal $mab$ of the dust tensor (Proposition 2) is the momentum flux. That flux sources a gravitomagnetic potential exactly as a current sources the magnetic field, and the two are dual under the quarter-turn $Q_4$. This is the same conjugate-chart rotation that supplies the radiative sector’s conjugate momentum (SubSection 5.5) and the deep regime’s registration (SubSection 7.3). Rotation is the gravitomagnetic reading of the one field.

Proposition 17 

(Frame-dragging and the rotating metric). The gravitomagnetic dipole gives the frame-dragging angular velocity $\omega \left(r\right)=2GJ/c^2{r}^3$, the Lense–Thirring rate of Proposition 9; in the strong field the gravitoelectric part carries the exponential reading and the gravitomagnetic part composes with it, the slow-rotation metric being

$$d{s}^2=-e^{-2u}{c}^{2}d{t}^2+e^{2u}{\delta }_{ij}d{x}^{i}d{x}^j-2\omega r^2{sin}^2\theta e^{2u}d\phi cdt.$$

As in the static case $g_{rr}=e^{2u}$ is finite for all $r>r_{*}$: the object is horizonless, with only the redshift floor at $r_f$. In this slow-rotation metric $g_{tt}=-e^{-2u}{c}^2<0$ everywhere, so ${\partial }_t$ stays timelike and there is no ergoregion at this order. An ergoregion (the Killing vector turning spacelike outside $r_f$, supporting superradiance with no horizon) is an $O\left(a^2\right)$ feature of the rapid-rotation completion, part of the residue below.

At leading order in $a=J/Mc$ the spin lifts the prograde/retrograde degeneracy of the photon ring: the static $b_c=2eGm/c^2$ (the $+4.6\%$ shadow) splits, displacing the centroid at $O\left(a\right)$ and the mean diameter at $O\left(a^2\right)$, so low-spin Sgr A^* remains the clean target. The rotating background supports the ringdown of a rotating remnant as a genuine quasinormal-mode problem on this metric, not only the eikonal correspondence. The inward slip-core hand-over (Remark 2) is the rotation into this conjugate chart: inside $r_{*}$ the description is spectral, the chart of SubSection 7.3 and Section 5.5, developed in the framework’s quantum theory [41]. Construction and float-free checks: reports/rotating.pdf, validate_rotating.py; the residue is the exact nonlinear rapid-rotation metric (the ergosphere location, the high-spin shadow).

7.3. The Weak-Acceleration Floor and the Radial Acceleration Relation

Across rotationally supported galaxies the observed centripetal acceleration follows the baryonic one through a single universal curve with scatter $\sim 0.1$ dex, transitioning at $a_0=1.20\pm 0.02\left(\mathrm{stat}\right)\pm 0.24\left(\mathrm{syst}\right)\times {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}$: the radial acceleration relation [21,26]. The standard cosmology reproduces it only through galaxy-by-galaxy feedback tuning and does not predict the scale; modified-dynamics phenomenology [23] postulates the scale without deriving it; the numerical coincidence $a_0\approx c{H}_0/2\pi$ has remained a coincidence [24].

Proposition 18 

(The floor acceleration). The drive decorrelates every link at the Hubble rate: the resolution floor $1/\mathsf{\Omega }$ is, in the synchronisation reading, a phase-noise floor of rate H. A static synchronisation gradient, a gravitational acceleration $g=c^2\nabla u$, is signal against that floor only if it accumulates at least one phase cycle per Hubble time, i.e. above

$$a_0={\displaystyle \frac{c{H}_0}{2\pi }}=1.08\times {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}(H_0=70),1.13\times {10}^{-10}(H_0=73),$$

against the fitted $1.20\pm 0.24\times {10}^{-10}$: agreement within ten percent, with no parameter. Equivalently, $a_0$ is the acceleration whose Unruh temperature matches the de Sitter temperature [24]; in this framework the two statements are one, since both temperatures are readings of the same floor. The $2\pi$ is not a free normalisation but the angular period of one full phase turn (one cycle per Hubble time), the same $2\pi$ that sets the Hawking temperature; the scale $c{H}_0$ is the drive rate at the horizon. The order-one constants of the whole construction reduce to two geometric numbers: the solid angle $4\pi$ (in $G=1/4\pi \kappa$, ${\alpha }_{\mathrm{bare}}=1/4\pi$, the Green’s function, and the horizon area, with $c_S^{\prime }=\frac{1}{4}$) and this cycle $2\pi$.

Lemma 9 

(Flux conservation under noise). The floor noise does not modify the time-averaged static force law: in a statistically stationary state the mean synchronisation flux through any sphere enclosing a source of cardinality m equals m by Gauss’s theorem, slips redistributing the transfer in time but not in mean.

Remark 6 

(The deep regime is registration, not a modified force). Lemma 9 constrains any account of the deep regime: the empirical enhancement $g_{\mathrm{eff}}=\sqrt{g_N{a}_0}$ at $g_N<a_0$, the deep-MOND limit that fits flat rotation curves and the baryonic Tully–Fisher relation [26],cannot arise as a modification of the mean static force. It lives instead in the registration of a sub-floor signal: below the floor a gradient is resolved only intermittently, and the conserved flux is attributed to the resolved fraction. The next proposition supplies that dynamics.

Proposition 19 

(The registration crossover). A static baryonic gradient $g_N$ accumulates $x=g_N/a_0$ phase cycles per Hubble time against the floor (Proposition 18). For $x<1$ the signal is sub-floor; its coincidence amplitude is $A=\sqrt{x}$ [41], and registration is the first passage of the drive-decorrelated (killed) meridian walk to that amplitude barrier. The resolved fraction is

$$f=1-{\eta }^a,w{\eta }^2-2\eta +w=0,w=1-s,$$

with $\eta \in (0,1)$ the registration constant (algebraic in the quadratic extension; s the per-chronon decorrelation rate, a the barrier height); on the resolvable window $a\sqrt{2s}=\sqrt{x}$ and $f=1-e^{-\sqrt{x}}$. By flux conservation (Lemma 9) the mean flux is unchanged, so the observer attributes the conserved flux to the resolved fraction and infers

$$g_{\mathrm{obs}}={\displaystyle \frac{g_N}{f}}={\displaystyle \frac{g_N}{1-e^{-\sqrt{g_N/a_0}}}}.$$

The deep limit $x\to 0$ gives $g_{\mathrm{obs}}\to \sqrt{g_N{a}_0}$, hence flat rotation curves and the baryonic Tully–Fisher relation $v^4=GM{a}_0$; the Newtonian limit $x\to \infty$ gives $g_{\mathrm{obs}}\to g_N$. The crossover is the radial acceleration relation, with deep and Newtonian log-slopes $\frac{1}{2}$ and 1.

Proof. 

The killed meridian walk reaches an amplitude barrier a steps away with probability ${\eta }^a$, where $\eta \in (0,1)$ is the decaying root of the killed recurrence $q_k=w\frac{1}{2}(q_{k-1}+q_{k+1})$, i.e. of $w{\eta }^2-2\eta +w=0$. Then $-ln\eta =arccosh(1/w)=\sqrt{2s}(1+O\left(s\right))$, so ${\eta }^a=e^{-a\sqrt{2s}}=e^{-\sqrt{x}}$ on the resolvable window, with the invariant $a\sqrt{2s}=\sqrt{x}$. Lemma 9 fixes the mean flux at $g_N$; resolving the fraction f and conserving the flux gives $g_{\mathrm{obs}}=g_N/f$. Because the gravitational coupling is the decoherence registration (Definition 3), orbiting matter responds to the registered $g_{\mathrm{obs}}$, not to the unregistered mean $g_N$: the flat rotation curve is the registered acceleration, a bridge, not a thresholded estimator. The limits follow from $1-e^{-\sqrt{x}}\to \sqrt{x}$ as $x\to 0$ and $\to 1$ as $x\to \infty$. The construction and float-free checks are in reports/deep-regime.pdf and validate_deepregime.py.    □

The rotated-chart reading of Section 2 is the conjugate-chart description of the same fact: the resolved and unresolved weights are ${cos}^2\alpha =f$ and ${sin}^2\alpha =e^{-\sqrt{x}}$, the threshold $a_0$ sitting at the observer’s outward scale horizon, the cosmological counterpart of the slip-core half-turn (Remark 2). The crossover matches the empirical radial-acceleration fit [21] across twelve decades (a labelled [approx] comparison to data, Figure 6); the broader galactic phenomenology built on it (intrinsic scatter as the disk’s dynamical temperature, the external-field effect, and clusters) is developed in [51].

Two predictions separate this account from both incumbents. From fundamental modified dynamics: $a_0$ is not a constant of nature but the current noise floor, so it tracks the expansion, $a_0\left(z\right)\propto H\left(z\right)$, roughly tripled at $z=2$, making high-redshift rotation curves the deciding dataset; the existing high-z samples [52] are precisely the contested arena, and the test cuts both ways. From the standard cosmology: the relation’s scale and universality are structural, not emergent from feedback, so its scatter at fixed z has a floor set by frame variance rather than formation history. A third observable discriminates the mechanism itself: a floor set by the global drive, not by the local field, fixes the external-field dependence of the threshold, and the claimed detection of an external-field effect in the SPARC sample [53], if it survives scrutiny, strains the naive floor reading and would force the sub-threshold dynamics to carry the environmental dependence; this is recorded as the sharpest pressure point of the subsection.

7.4. Explicability Dividends

Beyond the derived numbers, the construction dissolves two standing problems and explains three features that other accounts must postulate.

(i) 

The cosmological-constant magnitude.$\mathsf{\Lambda }\sim 1/\mathsf{\Omega }$ in Planck units is the curvature of the closed chart, the totality datum, not a vacuum energy requiring cancellation across 122 orders. The fine-tuning problem is not solved but unasked.

(ii) 

The large-angle microwave anomaly. The anomalously low large-angle correlation ($p\sim 1\%$ [54]) is the signature of the wrapped chart: super-horizon modes are identified, not populated, truncating correlations beyond the horizon scale. This consonance is a number, the primordial spectrum being structural, not transported (Proposition 20).

(iii) 

The equivalence principle is an identity, not a postulate (Corollary 1).

(iv) 

The weakness of gravity is cardinality dilution, not a hierarchy to be tuned (SubSection 4.4).

(v) 

The absence of a graviton fine-tuning problem: the field is collective synchronisation, not a propagating fundamental quantum.

Proposition 20 

(The primordial spectrum). Time is scale-dilation, so the primordial fluctuations are the drive’s own jitter and their spectrum is the invariant of scale-dilation: the unique scale-free Harrison–Zel’dovich spectrum, $n_s=1$, read at the symmetric quarter-turn (the Fourier self-dual point of the meridian cycle [39]). Near-scale-invariance is then the drive’s symmetry, not a coincidence, and needs no inflaton. The observed red tilt $n_s-1\approx -0.035$ is the chart’s misalignment from that point (red from the finite ultraviolet cutoff), its exact magnitude the same chart-angle deviation that sets the cross-scale running of $a_0$ (Ω-hard, SubSection 7.3), so $n_s-1$ and $a_0\left(z\right)/H\left(z\right)$ are two readings of one misalignment. The wrapped chart identifies super-horizon modes, truncating the two-point correlation beyond the horizon angular scale ($\sim {60}^{\circ }$): the observed low large-angle power and small $S_{1/2}$ [54]. The spectrum is the scalar dilation fluctuation, so the tensor-to-scalar ratio is structurally small. The exact claim is the scale-free form $n_s=1$ as the dilation fixed point; the fluctuation amplitude, Gaussianity, and acoustic phases are not claimed here, and the exact tilt is Ω-hard (SubSection 7.3). (reports/primordial.pdf,validate_primordial.py.)

8. Status: What Is Assumed and Derived

The construction’s dependency structure is collected here in one place, so that every result above can be traced to its inputs and the surface of assumptions is made explicit. Each row carries one status tag. Every derived statement is a residue of the Carrier, never a conjecture: a small residue a bounded observer can access (theorem or prediction), or an $\mathsf{\Omega }$-hard residue fixed below the horizon. The remaining tags mark imported inputs, interpretive bridges, and definitions.

tag	meaning
I	Import. A standard result or measured datum imported without reproof.
B	Bridge. An identification of a mathematical object with a physical one.
D	Definition. A naming or set-up move, carrying no empirical content.
T	Theorem. Derived within this paper from the rows above it.
P	Prediction. Falsifiable empirical claim, not yet measured.
$\mathsf{\Omega }$	$\mathsf{\Omega }$-hard. Decided by the totality; not closeable by a bounded observer.

#	Move	Status	Source
A. Inputs: imported, not derived here
A1	The finite substrate ${\mathbb{F}}_{\mathsf{\Omega }}$, cardinality $\mathsf{\Omega }\sim {10}^{122}$ (Planck units) fixed by the de Sitter entropy, with the phase cycle $C_{\mathsf{\Omega }-1}$, the quarter-turn core $Q_4$, the global drive (time is scale-dilation), the coherence horizon $\mathsf{\Omega }$, and the resolution floor $1/\mathsf{\Omega }$.	I	[37,38,40]
A2	The three frame freedoms: the frame space is a torsor under ${\mathrm{SL}}_2\left({\mathbb{F}}_{\mathsf{\Omega }}\right)$, the spatial chart three-dimensional per frame, its continuum shadow the 3-sphere.	I	([40] §9.6)
A3	The quadratic extension K, the Frobenius involution, the Lorentzian norm form, and the boost cycle (the Dirac layer).	I	[38]
A4	Composite coherence: a bound cluster is one synchronised drive orbit (offset superselection, the m-body synchronisation lemma), coupling with coefficient m against an incoherent $O\left(\sqrt{m}\right)$.	I	[41]
A5	Lattice potential theory and the discrete harmonic Green’s function, with Coulomb asymptotics $1/4\pi r$.	I	[46,47]
A6	The Kuramoto relaxation equation for coupled phase oscillators (the mathematical form of the drift dynamics).	I	[35,36,55]
A7	The continuum uniqueness of the Fierz–Pauli stiffness and Deser’s energy–momentum self-coupling bootstrap, used as imported templates.	I	[31,32]
A8	Measured quantities used for comparison only: the GRAVITY mass–distance and EHT shadow for Sgr A*, the SPARC acceleration scale, the classical-test values, and $H_0$.	I	measured [21,27,28,34]
B. Bridges: mathematics → physics (the interpretive moves)
B1	Gravity is the relaxation of forced frame drift: a field has no proper ideals, so every embedded massive shell is non-ideal and its frame drifts; reducing the mutual drift (raising coherence) is identically motion toward smaller separation.	B	§Section 1, Section 3
B2	Mass is winding rate (cardinality): $E=m{c}^2=hf$ is an identity, and a composite is a cluster of m phase-locked cells.	B	Def. 1, 2
B3	Distance is decoherence, counted in the relational steps of the observer’s chart.	B	Def. 3
B4	Channel unity: the clock tick (temporal correlation) and the decoherence count (spatial correlation) are two readings of one capacity.	B	Lem. 2
B5	The single correspondence: the objects we weigh are coherent composites.	B	§Section 3
B6	A cell-local change of frame data is the gauge symmetry, its compensating connection the field; the source’s conservation is that gauge invariance.	B	Prop. 7
C. Derived: theorems and consequences within this paper
C1	The linearised dynamics is the gradient flow of a coherence free energy, and a locked cluster of cardinality m couples with coefficient exactly m.	T	Lem. 3, 4
C2	Newton’s law $F=-G{m}_1{m}_2/r^2$ with $G=1/4\pi \kappa$: the $r^{-2}$ from harmonicity in the three frame freedoms, the product $m_1{m}_2$ from coherent additivity.	T	Thm. 1, Lem. 5
C3	Universal attraction from the single arrow of the drive (a repulsive charge would need a time-reversed, Frobenius-branch clock).	T	Prop. 1
C4	The equivalence principle as an identity; the bias field is clock rate, giving the redshift $\Delta f/f=Gm/r{c}^2$; the turnaround radius against the Hubble drive.	T	Cor. 1–3
C5	The magnitude $G=\hslash c/m_P^2$, frame-covariant (a relativity principle for scale); gravity’s weakness is cardinality dilution.	T	§Section 4.4
C6	The conserved dust-tensor source as the Frobenius-symmetric square on the mass shell (cardinality conservation and torsor equivariance).	T	Prop. 2
C7	Channel unity forces the spatial bias to equal the temporal one, $\gamma =1$, restoring the full light deflection $4Gm/c^{2}b$.	T	Lem. 6, Prop. 4
C8	The exponential isotropic metric by multiplicative composition, with $\beta =\gamma =1$: deflection, Shapiro delay, and perihelion at their observed values.	T	Lem. 7, Prop. 5
C9	The discrete Fierz–Pauli functional, exactly gauge-invariant and the unique adjacency-local stiffness on the shell (the spin-one case is unique-Maxwell).	T	Prop. 8
C10	No preferred-frame leakage: the Lense–Thirring precession at its observed value, with ${\alpha }_1,{\alpha }_2=O(1/\mathsf{\Omega })$.	T	Prop. 9
C11	The nonlinear completion forced: a Ward identity of the relational phase forbids self-sourcing, scale-covariant composition makes the exponential metric unique, and Deser’s bootstrap does not apply because what gravitates is winding rate.	T	Thm. 2
C12	The exact saturating static profile and its slip core at $r_{*}=\sqrt{Gm}$: horizonless but operationally black at $r_f=r_s/ln\mathsf{\Omega }$.	T	Prop. 12, 13
C13	Area-law entropy $S=c_S^{\prime }{A}_f/{\ell }_P^2$ with the de Sitter closure $S\sim \mathsf{\Omega }$, and a phase-slip (Hawking-scaling) emission suppressed below one quantum per Hubble time.	T	Prop. 14
C14	The weak-acceleration floor $a_0=c{H}_0/2\pi \approx 1.08\times {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}$ (within ten percent of the SPARC value, no free parameter), with the mean flux conserved under floor noise.	T	Prop. 18, Lem. 9
C15	Dark energy as the curvature of the finite chart, $\mathsf{\Lambda }\sim 1/\mathsf{\Omega }$ (the fine-tuning problem unasked), and the CMB large-angle anomaly as the wrapped-chart signature.	T | B	§Section 7.4
D. Predictions and residues
D1	Strong-field signatures: shadow $+4.6\%$, ringdown $-4.4\%$, ISCO frequency $-6.9\%$ with accretion efficiency $5.48\%$ (against $5.72\%$), no echoes, no evaporation bursts: one origin, jointly falsifiable.	P	§Section 7.1, Prop. 15, 16
D2	Preferred-frame effects at $O\left({10}^{-61}\right)$: the drive’s in-principle signature, far below current ${10}^{-5}$ bounds.	P	Prop. 9
D3	The floor tracks the expansion, $a_0\left(z\right)\propto H\left(z\right)$ (BTFR zero-point $\propto E{\left(z\right)}^{1/4}$); high-z rotation curves decisive.	P	§Section 7.3, [51]
D4	The deep-MOND crossover $g_{\mathrm{obs}}=g_N/(1-e^{-\sqrt{g_N/a_0}})$ as first-passage registration: the radial acceleration relation, no fitted function.	T | B	Prop. 19
D5	The radiative sector: the quarter-turn conjugate momentum makes the dynamics a wave equation: two helicity-$\pm 2$ gravitons at speed c, quadrupole radiation; gauge leaves two polarisations.	T | B	Prop. 10, 11
D6	The order-one constants are determined: $c_S^{\prime }=\frac{1}{4}$ (de Sitter closure), the floor’s $2\pi$ the cycle period, the recurring constant the solid angle $4\pi$.	T | B	Prop. 14, 18
D7	The rotating solution as the gravitomagnetic (quarter-turn) dual: frame-dragging = Lense–Thirring, horizonless, shadow split at $O\left(a\right)$; the ergoregion is an $O\left(a^2\right)$ feature of the rapid-rotation residue.	T | B	Prop. 17
D8	The primordial spectrum is structural: $n_s=1$ as the scale-dilation fixed point, the red tilt a chart misalignment, the wrapped chart fixing the low large-angle power.	T | B	Prop. 20
D9	The cross-scale running of $a_0$, the ratio $(\mathsf{\Omega }-1)/d$: reduces to the factorisation of $\mathsf{\Omega }-1$, the rate face of the Riemann scale operator.	$\mathsf{\Omega }$	§Section 7.3, [40,51]

Read by block, the surface is small. Imported (A1–A8) are the substrate and its arithmetic, the Lorentzian–Dirac layer, the composite-coherence results of [41], the lattice and synchronisation machinery, the continuum stiffness templates, and the measured constants used for comparison only. Interpreted (B1–B6) are the six bridges from the substrate to its physical reading: gravity as synchronisation, mass as winding rate, distance as decoherence, channel unity, the composite correspondence, and frame change as gauge. Everything else is derived (block C): from two definitions, one frame normalisation, one counting lemma, and the single composite correspondence, with no field equation, action principle, or metric posited, the potential, the force, the equivalence principle, the redshift, the deflection, the perihelion, the frame dragging, the operational horizon, the area law, the acceleration floor, and the registration crossover come out as outputs. The predictions D1–D3 are the scoped, falsifiable signatures; D4 is derived (Proposition 19).

What the synchronisation reading adds to gravitation is the provenance of each part: the product $m_1{m}_2$ from coherent additivity, the $r^{-2}$ from harmonicity in the three frame freedoms, the universal sign from the shared arrow of the dilation, G from the unit capacity of the relational channel, the equality of space and time bias from the unity of that channel, the slip core from its saturation, the operational horizon from its resolution floor; and, at the opposite end of the acceleration scale, the galactic transition from the same floor read as noise. One substrate, one channel, one floor; the constants are its bookkeeping.