C2 algebra and gravitational response: Algebraic structure, continuum obstructions, and observable tests

1 The problem and the status of the argument

A discrete model does not become a theory of gravity merely because it has a geometrical interpretation, a large symmetry group, or a positive quadratic response. Gravity concerns how physical clocks, rods and matter respond to a dynamical geometry. A derivation must therefore connect the microscopic variables to fields, identify their equations and redundancies, and compute the response of physical sources and probes. This paper organizes those requirements for C2, the internal name of an algebraic construction in the Holographic Circlette research programme.

The manuscript is intended for readers familiar with linear algebra, Fourier analysis, classical field theory and introductory general relativity. No familiarity with the project’s file formats or review history is assumed. Its starting C2 premises are stated below, and every continuum calculation specifies the extra assumptions it uses. The scope is classical and predominantly linear: the nonlinear completion, quantum constraint algebra and controlled continuum limit remain separate problems.

Four kinds of statement will occur: inherited C2 premises, used at their stated algebraic scope; certificate-backed native results, evaluated on the specified project records and separately reviewed; proved conditional statements, derived from named hypotheses; and open matching tasks, whose physical or computational inputs are not yet supplied. The native projected-module result is presented in Section 2.3. A continuum calculation performed after inserting the Einstein operator is conditional; it is not evidence that C2 has generated that operator.

The central formula motivating the investigation is

Here $J$ would map physical spatial strain into an active module, and $\mathsf H$ would be a physically justified bilinear form on that module. We use $\mathsf H$ for this form to avoid confusing it with the Einstein tensor $G_{\mu\nu}$ or a Newton constant. Formula (1.1) specifies how a form is transported; it does not construct $J$, select $\mathsf H$, or put time derivatives into the theory. The native free-theory proposal records these as separate obligations [8].

The paper’s mathematical contribution is the organization and explicit evaluation of the links around this formula: active-sector invariance, symmetry transfer, compatibility and curvature, constraints and physical response, and source-sensitive observational diagnostics. The incompatibility identity, linearized Einstein equations, stress-tensor improvements, soft-coupling argument and scalar-tensor tests all have antecedents in the literature. We derive the needed statements in one convention and use counterexamples to prevent unjustified identifications. No priority claim for the standard results, or exhaustive novelty assessment of the complete synthesis, is made.

1.1 Conventions

Spatial indices run from 1 to 3 and spacetime indices from 0 to 3. The metric signature is $(-+++)$, $x^0=\tau=ct$, and $\Box=-\partial_\tau^2+\Delta$. Spatial tensor products use the Euclidean metric and Frobenius inner product $A:B=\mathop{\mathrm{tr}}(A^\mathsf TB)$. We write

The perturbation convention for spatial strain is $h_{ij}=-2\epsilon_{ij}$. This follows to first order if the proposed inverse spatial metric is $g^{ij}=[(I+\epsilon)^2]^{ij}$: inversion gives $g_{ij}=\delta_{ij}-2\epsilon_{ij}+O(\epsilon^2)$. It is a specified continuum field assignment, not an established native map. In the GR benchmark $G^{(1)}=\kappa T$ with $\kappa=8\pi G_N/c^4$. In the scalar-tensor benchmark we instead use $\kappa=8\pi G_*/c^4$ and distinguish the bare Einstein-frame constant $G_*$ from the measured weak-body constant $G_N$.

2 What the C2 algebra supplies

2.1 Substrate geometry and physical ownership

The project’s fixed polyhedral embedding is a bond-centred tiling by oblate square bipyramids. Each bipyramid has apex-to-apex distance equal to an equatorial edge, not to the equatorial diagonal. There are three face-normal orientations. The single-cell metric point group is $D_{4h}$, while the colour-permuting embedding has the cubic point group $O_h$. The physical matter registers are separately owned: coincident polyhedral coordinates do not identify their degrees of freedom. Inter-cell couplings require independently specified bridge variables. These distinctions are part of the C2 source convention, not consequences of continuum elasticity [7].

In particular, the existence of a geometric $O_h$ action does not show that a physical response is $O_h$ invariant. A law may distinguish orientations, and a state can break a symmetry of the law. Nor does an abstract group isomorphism identify the geometric action with an action on coefficient records. The relevant representation and its intertwining map have to be written. These points will matter in Section 3.

2.2 A surviving sector and the active target

We formulate the algebra over $\mathbb Q$ for the rational vector-space problems used here. A finite list of generators need not make the full raw-action carrier finite-dimensional. A Laurent-module version needs its own specialization or extension-of-scalars justification. Suppose the reported C2 sector supplies

where $P_0$ is the projection onto $H_0$, $R_H$ is the relation subspace, and the restricted action $T_g=Q_g|_{H_0}$ satisfies its strict group law. Write $S_g=Q_g|_{H_1}$, without assuming the raw complement action is strict. Equation (2.1) implies $R_H\subset H_1$ and that the quotient map $q:H\to H/R_H$ is injective on $H_0$.

The reported finite presentation of the reached sector has 1,872 labels, indexed by 48 actions on 39 projected units. The label count is not its rank. The retained account bounds its rational dimension between 48 and 1,872; the separate original-active lower bound is 44. The two lower bounds concern different objects and are not added. We use this reported state as context, not as a new native computation [7].

Let the finite-dimensional original active target be $\mathcal A=j(A_0)\subset H$, and set

Projection is onto $U$ by definition, but may lose $K_{\mathcal A}$. The distinction between the whole reached sector $H_0$, its active image $U$, and the original active space $\mathcal A$ cannot be resolved by matching dimensions.

Thus an injective active projection establishes survival. A noninjective projection does not establish destruction: the unresolved intersection with $R_H$ is the deciding object.

Choose a linear section of the projection, written $u\mapsto u+f(u)$ with $f:U\to H_1$. Every active vector is uniquely $u+f(u)+k$ for $k\in K_{\mathcal A}$. Define the complement defect, once $T_gU\subset U$, by

For a small nondegenerate illustration take $H_0=\mathbb R^2$, $H_1=\mathbb R$, $R_H=0$, $T=I_2$, $S=-1$, and $f(u_1,u_2)=u_1$. This is an order-two action. The graph $\mathcal A=\{(u_1,u_2,u_1)\}$ projects isomorphically onto the invariant $H_0$, yet $(1,0,1)$ maps to $(1,0,-1)$ outside the graph. Survival and projected invariance therefore do not establish active invariance. This artificial vector-space example preserves no native geometry; it isolates exactly the missing implication.

If $\mathcal A$ has column basis $M_A$ and $P_0M_A$ has independent column basis $M_U$, invariance of $U$ can be tested by

That is a test of the image only; the remaining two conditions in (2.5) cannot be dropped. This gives an explicit algebraic stopping point before assigning physical field content.

2.3 A certified native projected module

The projected problem now has a native result, rather than only the criterion above. Two completed computations and their separate result reviews establish the following statement over $\mathbb Q$: the specified projection of the 768 ordered original active generators has dimension 24 and is invariant under all 48 supplied actions [3, 5]. “Native” here means evaluation on the frozen project records, with their source-coordinate and action correspondence checked at the stated scope. It does not identify those records with physical matter or prove correspondence to every component of the full original-active subject. We report these reviewed results; this manuscript revision performs no new native computation.

Let $E\in\mathbb Q^{39\times768}$ contain the selected seed coefficients of those generators. The retained rational matrices have shapes

and satisfy Here $\mathsf J$ is an ordered selector: each column is a distinct coordinate unit of $\mathbb Q^{768}$, in increasing original-column order. This matrix is unrelated to the proposed physical strain map $J$ in (1.1). The first identity bounds $\mathop{\mathrm{rank}}E$ above by 24; the second and third bound it below by 24. The selector condition, which the four equations alone do not imply, retains the selected generators’ identities.

The source review reconstructs all 78 nonzero contributions. The resulting matrix has 78 nonzero entries, all $\pm1$, in 48 nonzero columns, and 720 zero columns. As a second rank proof, the 48 nonzero columns form 24 opposite pairs. Representatives have pairwise disjoint nonempty row supports, covering all 39 rows. Selecting one row from each support gives the reviewed determinant-one $24\times24$ minor. Thus the rank is exact, rather than a numerical tolerance or an inference from the number of labels. A zero column of $E$ means zero projection, not a zero original generator.

To check invariance without discarding escaping components, let $F=\mathbb Q^{104}$ be the support envelope containing the 39 seed coordinates and all their supplied first-step images. Let $S_0:\mathbb Q^{39}\to F$ be coordinate inclusion and $R_0:F\to\mathbb Q^{39}$ the corresponding restriction, so $R_0S_0=I_{39}$. For each action, $B_g:\mathbb Q^{39}\to F$ contains its complete seed-image columns. The evaluated residuals are

The last equality follows from the first two and was also checked directly. Since $(WR_0)S_0V=I_{24}$, the image of $S_0V$ has dimension 24. Under the inherited action correspondence, (2.11) proves that this image is preserved. Both tests are needed: $Y_g$ detects departure from the seed-coordinate space, while $Z_g$ detects departure from the 24-dimensional image within that space. The 104 coordinates are an envelope for these calculations, not an asserted invariant carrier or the whole reached sector.

All 48 induced matrices $H_g$ are distinct signed permutation matrices of size 24. The reviewed calculation verifies all $48^2=2,304$ identities

against the supplied multiplication table, with its identity represented by $I_{24}$. This is a strict representation on the projected module. It neither computes a physical isotypic decomposition nor turns 24 algebraic dimensions into a count of propagating fields. In particular, it is not a two-polarization graviton result.

2.4 The full lift and what the projection leaves undetermined

The rank-24 result determines the next calculation much more sharply than a general request to reduce the 1,872-label reached sector. Let $\iota:\mathbb Q^{39}\to H_0$ be the full seed inclusion, and write $\mathsf U=\iota V$. The correspondence to the complete original-active source requires a map

The columns must be the complete original generators under the specified injection, including both rational coefficient components and their coordinate identities. The last equality alone does not certify that correspondence: arbitrary complementary additions would leave it unchanged.

Define the source-selected lift and discarded-component map by

The following reduction has been proved and checked at construction scope; its native $\mathsf D$ and $N$ remain unevaluated [4].

The 744-dimensional formal kernel of $E$ is therefore not the discarded active dimension. Because $N\mathsf J=0$, the 744 nonselected labelled columns of $N$ generate $K_{\mathcal A}$; they need not be nonzero or independent. Of the original 768 labels, 720 have zero projected column and retain $Ne_a=\mathsf D e_a$; 24 have nonzero projection but are not selected, giving $Ne_a=\mathsf D e_a-LAe_a$; and the 24 selected columns of $N$ vanish. None of the 720 may be discarded merely because its projection vanishes. If the inherited lower bound $\dim\mathcal A\ge44$ applies to the complete source in (2.13), then $\dim K_{\mathcal A}\ge20$. This is a conditional lower bound, not a new native kernel measurement or a survival statement.

An exact basis of $K_{\mathcal A}$ can be certified by retaining matrices

with $\mathsf J_K$ an ordered selector and $r=\mathop{\mathrm{rank}}N$. The first equality gives spanning, the third independence, and the second preserves source-column identities. A full-source adapter implementing (2.14) and (2.17) has passed construction review on artificial inputs [2]. That is an executable certificate method, not a certificate already obtained for the native discarded space.

2.5 Remaining quotient tests and the scope of averaging

Put $F_g=Q_gL-LH_g$. Its image lies in $H_1$, by projected invariance. For the complete, raw-action-stable relation space $R_H\subset H_1$, the exact remaining invariance criterion is

Indeed, $Q_g(Lu+k)=LH_gu+F_gu+Q_gk$, proving sufficiency. Conversely, $(\mathcal A+R_H)\cap H_1=K_{\mathcal A}+R_H$, by the injective projection of $L$. Apply quotient invariance first to $k\in K_{\mathcal A}$ and then to $Lu$. This proves necessity and explains why the two tests examine different possible failures. Full survival separately requires $K_{\mathcal A}\cap R_H=0$. From (2.3), Thus the premises already guarantee a 24-dimensional projected quotient of the active image. They do not say that all original-active vectors survive.

A useful formal retraction is available for the pure projected module. Let $r_s:H_0\to\mathbb Q^{39}$ extract the seed coefficients on the full finite-support carrier, and set $\ell_0=Wr_sP_0$. Then

For the second equality, use $Q_g\mathsf U=\mathsf U H_g$ and $\ell_0\mathsf U=I$. For the third, $\ell_0$ kills $H_1$, so strictness of the $H_0$ action gives $\ell_0Q_gQ_h=\ell_0Q_{gh}$ even if the raw complement action is not strict. Reindexing $gh$ proves equivariance. Consequently $\mathsf U\Pi$ is an equivariant idempotent on the full carrier under these premises. Its range is $U\subset H_0$, which need not lie in $\mathcal A$; it is not the required retraction onto the full active image. The finite-envelope map $WR_0$ alone was not asserted equivariant.

If (2.18) and strictness of the full quotient have been established, ordinary averaging can split the active extension. Writing $\rho_g$ for its action and $\bar L=qL$, the section

satisfies $\pi\sigma=I$ and $\rho_h\sigma=\sigma H_h$, where $\pi$ reads projected coordinates. Reindexing $hg$ proves the latter identity. This existence argument begins after the missing quotient and invariance hypotheses; it cannot replace their native construction.

2.6 Complete relations and why a finite quotient is not a stopping rule

The inherited carrier is generated by finite words in the raw actions from the full old source space $W_0=j(V_0)$. Therefore its complementary carrier is generated from $W_1=(I-P_0)W_0$ by the $S_g=Q_g|_{H_1}$. This domain is not merely $K_{\mathcal A}$, the 39 projected seeds or the 104-coordinate envelope. For $\delta_{g,h}=S_gS_h-S_{gh}$ define

Then $R_*=R_H$: it is exactly the smallest stable space containing every identity and composition defect on the carrier [4].

To prove completeness, the full relation space contains $R_*$ by stability. Conversely, in $H_1/R_*$ the seed identities allow reduction of a word on $w\in W_1$ to $S_{g_1\cdots g_n}w$, by successively reducing its rightmost pair. The quotient is therefore spanned by $[S_gw]$. Composition and identity defects vanish on every such class, hence on the whole quotient. The explicit transport identity is

It follows by expansion and associativity of the group table, without a raw action law. This proves coverage, but it does not license stopping at the initial seeded relations.

Even the consequence $\dim(H_1/R_H)\le48\dim W_1$ for finite-dimensional $W_1$ does not prove termination of successive relation spans. For the two-element group on $H_1=\mathbb Q[x]$, take $S_e=I$, $S_af=xf$ and $W_1=\langle1\rangle$. The partial relation space

has dimension $n+1$ for every $n$, whereas the complete quotient by $(x^2-1)$ has dimension two. Distinct leading degrees prove strict growth; polynomial division proves the quotient statement. A native computation needs a justified completion or finite presentation, not a stopping rule inferred from the dimension of the quotient. Similarly, an annihilator of an incomplete relation list cannot certify nonmembership in $K_{\mathcal A}+R_H$.

2.7 Computational feasibility and the scope of this advance

The latest construction provides full-source reconstruction, both rational components, source-selected lifts and the four kernel equations. Its artificial supervised integration runs the entire computation and its source rebuild in one child. All seven scientific fixtures reproduce, including a 768-column fixture of kernel rank 744; limit and incomplete-capture controls produce no mathematical outcome [6]. These tests validate an implementation path on artificial inputs. The resource/supervision integration is a new candidate awaiting its own external review; predecessor reviews are not transferred to it.

At the largest allowed dimensions and $r=744$, the conservative ceiling for both passes is

(Appendix E). It includes the shared projection additions that the sparse meter itself does not count. It is not a CPU-time or memory bound. The sparse one-bit rank-744 fixture used about $0.62$ child CPU seconds per attempt on the tested host, including the rebuild; that measurement does not test native fill-in or coefficient growth. A large upper bound and an easy artificial example establish neither practical native feasibility nor infeasibility. They also do not diminish the already completed projected test.

The resulting status can be stated without conflating its subjects:

The positive result is a concrete algebraic subsystem on which further matching questions can be posed. Its relevance to gravity remains conditional on the physical identifications developed in the following sections.

3 From an invariant module to a strain response

3.1 What symmetry transfer requires

Suppose a strain space $\mathop{\mathrm{Sym}}^2(\mathbb R^3)$ carries $\rho(R)\epsilon=R\epsilon R^\mathsf T$, a physical active module $V$ carries $\mathcal T_R$, and

Then $\mathcal K(\epsilon)=\left\langle J\epsilon,\mathsf HJ\epsilon\right\rangle$ is invariant under $\rho(R)$. This follows by substitution. Positivity of $\mathsf H$ implies nonnegativity of $\mathcal K$, with null directions determined by $J$ and the nullspace of $\mathsf H$. Neither equation in (3.1) follows from the cardinality of the group.

A useful formulation includes internal relaxation. Let $W(\epsilon,z)$ be the microscopic or effective energy, and assume $z\mapsto\Theta_Rz$ is a bijection between the admissible internal configurations at $\epsilon$ and at $\rho(R)\epsilon$, with

Relabelling the infimum gives $W_{\rm eff}(\rho(R)\epsilon)=W_{\rm eff}(\epsilon)$, where $W_{\rm eff}(\epsilon)=\inf_zW(\epsilon,z)$. If this function is twice differentiable at an invariant reference state, its Hessian is invariant. Unique minimizers are unnecessary. A selected branch or a nondifferentiable minimum can invalidate the Hessian conclusion. This is the hypothesis behind the use of property-tensor symmetry in Neumann’s principle [16]; it cannot be replaced by the symmetry of an unweighted embedding.

3.2 The three cubic channels

For the signed-permutation action of $O_h$, define

These are mutually orthogonal projectors with ranks $1,2,3$. Sign changes force cross terms between distinct off-diagonal entries, and between diagonal and off-diagonal entries, to vanish. Permutations equate the three off-diagonal coefficients and leave a scalar quadratic form on the diagonal traceless plane. Thus every invariant quadratic form has the form Its conventional representation labels are $A_{1g}\oplus E_g\oplus T_{2g}$. These labels describe spatial tensor transformation properties, not spacetime particles. To connect them to a C2 module, the actual multiplicity spaces and the map $J$ still have to be constructed.

Under the conventional elastic normalization $W=\frac12C_{ijkl}\epsilon_{ij}\epsilon_{kl}$,

This is a normalization map, obtained by expanding both forms, not a claim that C2 has measured elastic moduli. Rotational isotropy of (3.4) is equivalent to $k_E=k_T$. Necessity follows by rotating a diagonal traceless shear into an off-diagonal one; sufficiency follows because the traceless sector then carries its full Frobenius norm.

For example, rotate $\epsilon_0=\mathop{\mathrm{diag}}(a,-a,0)$ by angle $\theta$ around the third axis. Direct multiplication gives

This is a fourfold law for the displayed family, not a universal directional diagnostic for arbitrary strain. A uniaxial strain rotated about its own axis remains unchanged.

3.3 Exact directional splitting of tensor costs

For unit $n$, set $P=I-nn^\mathsf T$ and define

Choose a basis $E_1,E_2$ with $E_a:E_b=2\delta_{ab}$. With $\delta_s=k_E-k_T$, the cost matrix is $B(n)=2k_TI_2+\delta_sM(n)$, where $M_{ab}=\sum_i(E_a)_{ii}(E_b)_{ii}$.

For $k_E\ne k_T$, the splitting vanishes exactly at the eight oriented body diagonals, since $\sigma_4\ge1/3$, $\pi_6\le1/27$, with simultaneous equality only there. It is maximal, $2|k_E-k_T|$, on the six coordinate-axis directions. The representative values are

The common body-diagonal cost still depends on the moduli. It is the splitting that is blind there. These are static costs, not dispersion eigenvalues.

The symmetry premise is essential. Consider $\mathcal K_w(\epsilon)=\kappa_0\left\lVert\epsilon\right\rVert_F^2+\lambda\epsilon_{33}^2$, with $\kappa_0>0$ and $\lambda>0$. It is positive but distinguishes the third axis. For a $[111]$ transverse frame its cost matrix is $\mathop{\mathrm{diag}}(2\kappa_0+4\lambda/9,2\kappa_0)$. At $(\kappa_0,\lambda)=(1,9)$ the two costs are 6 and 2. The unweighted tiling can retain cubic geometry while this response does not. Hence body-diagonal degeneracy is a theorem about an invariant form, conditional in its application to C2.

4 Compatibility, curvature and the tensor obstruction

4.1 An explicit Fourier decomposition

Let $k\in\mathbb R^3\setminus\{0\}$, $s=k\cdot k$, $Q=kk^\mathsf T/s$ and $P=I-Q$. We use real cosine amplitudes and absorb the sine phase of a displacement into its coefficient. Define

Every symmetric tensor has the unique decomposition Indeed $D_ku\,k=(su+k(k\cdot u))/2$, so the displayed $u$ exactly reproduces $\epsilon k$. The remainder is transverse and its two-dimensional trace fixes $\beta$. The three mutually orthogonal spaces have dimensions $3,2,1$. This is a direct sum of tensor spaces at fixed nonzero $k$, not yet a count of physical modes.

For $k$ along the third axis, compatible strain can have entries in the third row or column but none in the transverse two-by-two block. A plus-polarized tensor $\mathop{\mathrm{diag}}(1,-1,0)$ cannot be produced this way. A displacement-only realization would therefore miss the proposed radiative tensor field. This is a conditional obstruction for that field dictionary, not a theorem excluding every C2 dictionary.

4.2 Incompatibility and its inverse on the transverse block

The Saint-Venant incompatibility operator is

Smooth compatible strains have zero incompatibility by commutation of derivatives. Global reconstruction from zero incompatibility requires topological and regularity hypotheses; a simply connected setting with the appropriate function spaces is treated by [18]. We require only the exact nonzero-mode algebra below and do not import an unrestricted global converse.

Let $C(k)v=k\times v$. The Fourier symbol is $\mathop{\mathrm{inc}}_k\epsilon=-C(k)\epsilon C(k)^\mathsf T$. On the transverse plane, $C(k)/\sqrt s$ is a quarter-turn. For a symmetric two-by-two matrix $B$, this turn conjugates $B$ to $(\mathop{\mathrm{tr}}B)I-B$. It follows that

Thus The kernel equality follows because the inverse reconstructs the entire transverse block; no additional transverse null vector exists. Incompatibility retains three components, of which two are TT. For example $\epsilon=P\cos(k\cdot x)$ is incompatible, with $\mathop{\mathrm{inc}}\epsilon=-sP\cos(k\cdot x)$, but has zero TT projection. Finding incompatibility alone therefore does not identify a tensor wave.

The differential operator has an established role in defect theory, but an identification with a native defect density would require further variables and constitutive relations. It is not supplied by the operator’s name. Here the useful conclusion is quantitative: any native field mapped to a TT strain must map under incompatibility to $sE$ at this mode.

4.3 The exact spatial Einstein identity

For a spatial metric $\gamma_{ij}=\delta_{ij}+h_{ij}$, the three-dimensional version of (1.2) gives

Expanding the product of the two Levi-Civita symbols in (4.3) yields precisely the parenthesis. Consequently This known identity is also displayed by [10]; its significance here is the fixed sign and normalization for the proposed strain dictionary. It relates differential operators, not equations of motion or sources. Nothing in it alone requires $G_{\mu\nu}=\kappa T_{\mu\nu}$.

At $k=(1,2,2)$, choose $v=(2,-1,0)$ and $w=(2,4,-5)$, both transverse and mutually orthogonal. The tensor

has $Ek=0$, $\mathop{\mathrm{tr}}E=0$ and $\left\lVert E\right\rVert_F^2=2$. Equation (4.4) gives $\mathop{\mathrm{inc}}_kE=9E$ and (4.8) gives ${}^{(3)}G^{(1)}[-2E]=-9E$. This non-axis example evaluates the map and makes its lost compatible sector explicit.

4.4 Why local cubic channel occupancy is not a gauge invariant

The continuum spatial gauge transformation changes strain by a compatible term. For a static cosine mode, the family

is obtained from $\epsilon_0=\psi I\cos(k\cdot x)$ under $\delta\epsilon=-\operatorname{sym}\nabla\xi$. All members have the same transverse trace $\beta=\psi$ and the same incompatibility. Yet $\mathop{\mathrm{tr}}\epsilon_\lambda=(3-\lambda)\psi\cos(k\cdot x)$ can be zero.

At a unit-amplitude crest, take $\lambda=3$. The squared $(A,E,T)$ channel norms are $(0,6,0)$ for $k\parallel[001]$, $(0,0,6)$ for $k\parallel[111]$, and $(0,2/3,16/3)$ for $k\parallel[122]$. The isotropic representative had $(3,0,0)$. All these figures follow directly from the projectors in Section 3; the curvature is unchanged by the corresponding gauge transformation. Ordinary strain trace therefore cannot serve as a gauge-invariant source-sector label.

Even minimizing the local cost over compatible shifts does not automatically give a GR kinetic operator. For positive $k_A,k_E,k_T$, on $[001]$ and $[111]$ one obtains respectively

The longitudinal parameter $\lambda$ in (4.10) gives the quadratic $\psi^2[k_A(3-\lambda)^2/3+2\gamma\lambda^2/3]$; its minimum is (4.11). Transverse displacement terms add nonnegative independent contributions on these axes, so this is the full minimum. It is a quotient cost with residual material parameters, not a source law or a proof of diffeomorphism invariance.

5 Dynamics and the gravitational constraints

5.1 A spatial cost does not determine a wave equation

For a quadratic spacetime action $S_2=\frac12\left\langle h,\mathscr Kh\right\rangle$ and a linear gauge generator $\delta h=\mathscr R\xi$, gauge invariance under the chosen domain and boundary conditions requires $\mathscr K\mathscr R=0$ for a self-adjoint $\mathscr K$. Suppose its Fourier symbol is local and analytic at zero momentum, with derivative-free term $\mathsf M$. For linearized diffeomorphisms $\mathscr R(q)a=q\otimes a+a\otimes q$. The term linear in $q$ in the Ward identity is

Those tensors span all symmetric matrices: set $q=a$ to basis vectors for the diagonal and take distinct basis vectors for the off-diagonal entries. Hence $\mathsf M=0$. A nonzero local strain cost cannot simply be assigned as a derivative-free mass matrix for a freely gauge-redundant metric about flat stationary vacuum. Extra fields, changed gauge structure, nonlocal terms, or a nonstationary background change the premises. A cosmological term around flat space does not evade the stated result: with nonzero cosmological constant that background is not a vacuum stationary point.

The same static matrix can enter different dynamics. For two tensor amplitudes $q$,

gives $\det(\omega^2M-B-C^{ij}k_ik_j)=0$. A matrix inserted in $B$ can produce a gap, whereas one inserted in $C^{ij}$ can change speeds. The time kinetic form $M$, its sign, the constraints and physical units are indispensable. They are absent from a spatial pullback alone.

5.2 The Einstein benchmark in synchronous strain variables

We now assume linearized Einstein dynamics in order to derive a benchmark. Let $h_{00}=h_{0i}=0$ and $h_{ij}=-2\epsilon_{ij}$ in a local synchronous gauge. With primes denoting $\partial_\tau$, direct substitution into (1.2) gives

The first four equations are the Hamiltonian and momentum constraints. The last term in the spatial equations cannot be dropped before imposing them.

For the decomposition (4.2), the constraint amplitudes are

Vacuum therefore gives $\beta=\beta'=0$ and $Pu'=0$. The Bianchi identity propagates initially satisfied constraints when the spatial equations hold. Taking the trace of the spatial equations then gives $\mathop{\mathrm{tr}}\epsilon''=0$, and the evolution reduces to The residual synchronous gauge transformations are Taking $g=-u_0\sin(k\cdot x)$ and $f=-\lambda\cos(k\cdot x)$ generates exactly the compatible part of (5.7). Thus the physical nonzero-mode initial data are the two TT amplitudes and their two velocities. This is an explicit quotient construction, not merely the subtraction of gauge and constraint counts.

In contrast, extending $\epsilon''+\mathop{\mathrm{inc}}\epsilon=0$ to all six components without the constraints gives a transverse scalar equation $\beta''-s\beta=0$. At arbitrarily large $|k|$ its exponential growth destroys uniform Sobolev estimates. A finite Fourier cutoff gives an unstable but finite-dimensional well-posed system. The compatible equation $u''=0$ has only linear drift, with a solution bound uniform in $k$ in $H^r\times H^r$; drift is not itself ill-posedness. This six-component extension is not GR, because it has discarded part of (5.5).

5.3 A gauge-invariant source dictionary

To describe sources and lapse, use amplitudes with phase $\vartheta=\omega\tau+k\cdot x$:

For this subsection use an isotropic representative instead of $\beta P$: The definitions of $\beta$ and $E$ are unchanged. Put $v=b+\omega u$, $\ell=k\cdot v/s$, and For $\xi_\mu=a_\mu\sin\vartheta$, the transformation law is $\delta\phi=-\omega a_0$, $\delta b=\omega a+ka_0$, $\delta u=-a$, $\delta\ell=a_0$. Hence $\Phi_g,\beta,V,E$ are invariant. Choosing $a=u$, $a_0=-\ell$ produces The four-dimensional gauge map is injective for $k\ne0$, and these six invariant amplitudes parametrize its quotient before field equations. Relative to the variables of [12], the map is $\Theta=-2\beta$, $\Phi=\Phi_g$, $\Xi=V$, $h^{\mathrm{TT}}=-2E$. It is established by comparing metric components, not by their names.

Contracting the Ricci tensor of (5.12) gives

Its Bianchi contraction vanishes identically. For example $-\omega G_{00}+k_iG_{i0}=0$; the spatial contraction cancels the $\beta$ and $V$ terms separately. Thus a source must satisfy Write $j_T=Pj$, $S_{\mathrm{TT}}=PSP-\mathop{\mathrm{tr}}(PSP)P/2$, and $\pi_s=(3Q:S-\mathop{\mathrm{tr}}S)/2$. Projection of $G=\kappa T$ then gives Conservation implies $Q:S=\omega^2e/s$, and hence This is a response to a conserved source, not permission to specify density, momentum and stress independently. On $\omega^2=s$ the tensor inverse is singular; homogeneous waves and resonant sources require a Green function or initial data. Off shell all six source components determine a response modulo gauge.

In vacuum at nonzero spatial $k$, (5.17) sets the scalar and vector invariants to zero. Two tensor polarizations remain on the null cone; none remains off it. This is a full-space Fourier statement. It does not remove a Coulomb field from an exterior vacuum region with boundary data. The decomposition is spatially nonlocal, and its elliptic constraints cannot by themselves establish instantaneous signalling [12].

6 Newtonian response, tides and isolated stresses

6.1 Lapse and the pressureless limit

For a stationary weak source, (5.18) is

When $e=\rho c^2$ and stress is negligible, $\Phi_N=c^2\Phi_g$ satisfies $\Delta\Phi_N=4\pi G_N\rho$. The Green solution tending to zero at infinity is This normalization uses $\Delta(-1/r)=4\pi\delta^{(3)}(x)$. It recovers Newton’s law after inserting Einstein’s source coefficient; it does not derive $G_N$ from C2.

For a static isotropic metric $h_{00}=-2\phi$, $h_{ij}=-2\psi\delta_{ij}$, the spatial equations with negligible stress imply $\delta_{ij}\Delta(\phi-\psi)-\partial_i\partial_j(\phi-\psi)=0$. The trace gives $\Delta(\phi-\psi)=0$; then its Hessian vanishes. The difference is affine and vanishes under the stated decay condition. Thus equality of the two potentials follows from equations and boundaries, not a notation choice. Omitting the lapse would generally violate the spatial equations.

Stress matters beyond the pressureless limit. At $k=(0,0,1)$ and $\omega=0$, the conserved source $e=0$, $j=0$, $S=\mathop{\mathrm{diag}}(1,1,0)\cos z$, in units $\kappa=2$, gives $\beta=0$ but $\Phi_g=-2\cos z$. It produces a force through the lapse although its density vanishes. This plane-wave control is not an isolated finite body or an energy-condition-approved matter model.

6.2 Tidal curvature and spatial curvature

The physical geodesic-deviation tensor is $\mathcal E_{ij}=R^{(1)}_{0i0j}$. For (5.12) its amplitude is

For an arbitrary linear metric, without imposing this gauge, a useful identity is To derive it, split the four-dimensional Ricci contraction into temporal and spatial parts: $R^{(4)}_{ij}=R^{(3)}_{ij}-\mathcal E_{ij}$, $R^{(4)}_{00}=\mathop{\mathrm{tr}}\mathcal E$, and $R^{(4)}=R^{(3)}-2\mathop{\mathrm{tr}}\mathcal E$. Substituting in the Einstein tensors gives (6.4). In linearized vacuum, In a TT wave this is also $\epsilon''=-\mathop{\mathrm{inc}}\epsilon$. The equality is therefore more general than the plane-wave example that suggests it.

For a point source of mass $M$, outside the origin let $\psi=-\mu/r$ with $\mu=G_NM/c^2$. Then

The radial and two tangential eigenvalues are $-2\mu/r^3$, $\mu/r^3$, $\mu/r^3$. At $x=(1,2,2)$ and $\mu=1$, Its trace vanishes and the complete exterior Einstein tensor vanishes, while the tidal curvature does not. Vanishing Ricci response and vanishing curvature are different tests.

6.3 Laue’s theorem and the finite-source distinction

For a stationary conserved stress with sufficient decay or vanishing boundary flux,

The second equality follows by the divergence theorem. These are the hypotheses of the classical Laue statement, carefully discussed by [15]. Therefore the coefficient of the far-field monopole in (6.1) depends on $E_{\rm tot}=\int e\,\mathrm d^3x$ alone: Stress still affects local curvature and higher multipoles. For constant $S=pI$ restricted to a ball, the omitted boundary integral equals the nonzero volume integral; such a source without its containing stresses does not satisfy the isolated-source hypothesis.

Conservation is also weaker than an energy condition. If a stationary source has $e=j=0$ and obeys the weak energy condition in all frames, then $v^\mathsf TSv\ge0$ for every $v$ by using timelike vectors proportional to $(1,v)$. Thus $S$ is positive semidefinite. With the decay hypothesis and (6.8), its integrated nonnegative trace is zero, forcing $S=0$ for a continuous source. Compact conserved stresses without energy density can therefore be useful controls while failing this additional condition. Pointwise energy conditions are stated premises for the selected matter model, not universal laws for all effective fields.

7 Matter actions and what a source map must select

7.1 A conserved source derived from an action

An auxiliary real scalar illustrates the issue without identifying a native C2 variable. In this section set $c=1$ and take

Metric variation gives The scalar field equation therefore supplies conservation. In an arbitrary local orthonormal frame, put $a=\partial_0\sigma$ and $b=\nabla\sigma$. Then and when $V\ge0$. Since the argument applies in every orthonormal frame, the flux is future causal for every observer: the dominant energy condition holds. This is an argument quantified over frames, not an inference from a finite boost sample.

This scalar does not automatically provide an isolated static body. For $V=m^2\sigma^2/2$, a time-independent decaying solution obeys $\Delta\sigma-m^2\sigma=0$. Multiplication by $\sigma$ and integration by parts gives $\int(|\nabla\sigma|^2+m^2\sigma^2)=0$. With the stated decay there is no nontrivial localized static solution, including at $m=0$. A body model needs additional matter or interactions.

7.2 Flat-space dynamics does not select a unique curved source

Add $-\frac12\xi\int\sqrt{-g}R\sigma^2\mathrm d^4x$. On flat space this does not change the free scalar equation, but variation through curved metrics changes the flat stress tensor:

The improvement is identically conserved for arbitrary smooth $F$. On a time slice $I_{00}=-\Delta F$ and $I_{0i}=-\partial_0\partial_iF$, so its translation charges vanish under the relevant boundary conditions. Neither the flat matter equation nor those global charges fix $\xi$.

The source change can have curvature. A particular solution of the linearized Einstein equation for $\xi I[F]$ is

because $G^{(1)}[\eta F]=I[F]$. It is generally not pure gauge. For a static $F$, its linked density $\delta e=-\xi\Delta F$ and stress $\delta S=\xi(I\Delta F-\operatorname{Hess}F)$ give $\delta\Phi_g=\kappa\xi F/2$. Retaining that stress and discarding the density instead doubles the lapse response. Those are different sources, even though the stress components agree.

Pointwise energy conditions can distinguish covariant completions, but their use is an additional selection premise. Classical nonminimal scalar models need not satisfy them [11]. For a massless on-shell field,

Requiring the trace to vanish for every such configuration selects $\xi=1/6$ in this one-parameter family. It is not enough to test one null travelling wave, for which $(\partial\sigma)^2=0$ regardless of $\xi$. The exact solution $\sigma=\tau x$ has $\Box\sigma=0$ and $(\partial\sigma)^2=3$ at $(\tau,x)=(2,1)$, so the test is nonvacuous. Conversely, requiring pointwise weak energy for every smooth flat massless solution selects $\xi=0$ in this family: at a point where $\sigma\ne0$ and all first derivatives vanish, the free wave equation permits second derivatives of either sign and arbitrarily large magnitude, and $T_{00}^{(\xi)}=-2\xi\sigma\Delta\sigma$ can be negative for any $\xi\ne0$. Minimal coupling satisfies the condition by the preceding proof. These two selection premises conflict; covariance alone selects neither.

8 Universal long-range exchange and local freedom

8.1 The conserved-source pole

Let $q_\mu$ be a Fourier covector, $q^2=\eta^{\mu\nu}q_\mu q_\nu$, and let symmetric source amplitudes $T,S$ satisfy $q^\mu T_{\mu\nu}=q^\mu S_{\mu\nu}=0$. Set $t=\mathop{\mathrm{tr}}_\eta T$, $s_T=\mathop{\mathrm{tr}}_\eta S$. In harmonic gauge, $G^{(1)}_{\mu\nu}=q^2\bar h_{\mu\nu}/2$. Away from the pole,

This defines a bilinear response, not a signed potential energy or a self-action with its own extra symmetry factor. An actual propagator also requires a boundary or causal prescription. The numerator and denominator are imported Einstein benchmarks here; their native derivation is the principal open matching problem.

In momentum space the improvement is $I_{\mu\nu}[F]=(q_\mu q_\nu-\eta_{\mu\nu}q^2)F$, with trace $-3q^2F$. Conservation gives $I:S=-q^2Fs_T$. Therefore

If both sources change, the difference is The last coefficient follows from $I(F):I(H)=3(q^2)^2FH$ and half the product of traces $=9(q^2)^2FH/2$.

When the form factors are regular at the pole, these terms add no residue of the exchanged $1/q^2$ propagator. In a local-operator setting they are contact terms and their derivatives. This is not a claim that an infrared-singular form factor or a complete diagram containing other propagators cannot have long-range effects. Without conservation an extra $Fq_\mu q_\nu S^{\mu\nu}$ remains and the cancellation would be false. On shell a transverse-traceless polarization satisfies $e:I(F)=0$ directly, without dividing by $q^2$.

The harmonic-gauge metric change is

Its second term is pure gauge off shell, so its curvature agrees with (7.6). Nonzero local curvature and unchanged long-range pole residue are consistent. An unfixed local improvement coefficient therefore does not by itself invalidate the universal-pole target.

8.2 What the soft Ward condition proves

Assume a Lorentz-covariant asymptotic scattering description, a massless spin-2 external state, leading soft factorization, and decoupling of longitudinal polarizations. The classic argument of [19] uses these hypotheses; a finite cubic symmetry does not supply them. For future-directed on-shell hard momenta $p_i$ and signs $\eta_i=-1$ incoming, $+1$ outgoing, write the leading soft tensor as

Its longitudinal contraction is $q_\mu\mathcal B^{\mu\nu}=\sum_i\eta_i g_ip_i^\nu$. For a nonzero momentum exchange between two species, $\Delta p_A=-\Delta p_B\ne0$, this requires Equality propagates through a connected graph of nontrivial processes. Forward scattering with $\Delta p_A=0$ imposes no such constraint, and disconnected species retain independent couplings in this test. Terms regular in the soft limit cannot cancel the leading gauge variation under the stated factorization assumptions.

This result fixes equality, not the common magnitude, a nonzero interaction, or the sign of a physical kinetic term. It neither derives $G_N$ nor selects $\xi$. Extending a free massless spin-2 theory to nonlinear self-coupling requires its own consistency analysis; Deser’s construction is a standard benchmark [9]. For C2, using that benchmark before deriving the native free operator would merely insert the desired theory at the starting point.

9 Extra scalar poles: force, lensing and empirical scope

9.1 A fully specified auxiliary alternative

To see what a tensor-only test can miss, add a canonical positive-kinetic massless scalar $\vartheta$ and let matter couple to $\widetilde g_{\mu\nu}=e^{2\lambda\vartheta}g^E_{\mu\nu}$. In this subsection use $c=1$ and $\kappa=8\pi G_*>0$. At linear order, solving the Einstein and scalar equations gives the conserved-source response

with residue The extra term vanishes when a source is traceless, so a TT-only test cannot exclude it. On a null covector, choose the transverse symmetric source block $\left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ after imposing conservation. Its self-pairing is The positive source-coupling rank is two at $\alpha=0$ and three for any $\alpha>0$, however small. This rank counts the directions coupled by this residue under its assumptions, not every mode of an unspecified native theory.

For several massless scalars with positive-definite kinetic matrix $K$ and matter coupling vector $\ell$, the same argument gives

Under $\theta=L\theta'$, $K'=L^\mathsf TKL$ and $\ell'=L^\mathsf T\ell$, so the expression is basis invariant. It vanishes only when $\ell=0$ for finite positive-definite $K$. Opposite signs in the components of one universal coupling cannot cancel this positive self-residue. Massive fields, singular kinetic operators or nonuniversal source couplings require separate analysis.

9.2 Two potentials and a normalization-sensitive lensing test

For a weak static point mass,

where $\widetilde g_{00}=-(1+2\Phi_N/c^2)$ and $\widetilde g_{ij}=(1-2\Psi_N/c^2)\delta_{ij}$. Thus For an unperturbed unit ray direction $n$, the transverse coordinate acceleration at leading order is $-P_n\nabla(\Phi_N+\Psi_N)/c^2$. The complete coordinate-time equation also contains a longitudinal term proportional to $n(n\cdot\nabla\Phi_N)$; projection alone cannot test that term. The impact-parameter integral uses and gives bending The scalar conformal factor cancels from the fixed-$G_*$ light response, but the comparison at measured $G_N$ still changes. At $\alpha=1/6$ the relative bending is $3/4$ and $\gamma=1/2$. These are the familiar linear response factors appearing in the van Dam–Veltman–Zakharov discussion; the matching response is not an identification of the complete auxiliary model with massive gravity [17].

9.3 Use the measured variable, not an extrapolated uncertainty

The historical Cassini result of [1], as reported in the inspected account by [20], is

We do not refit that experiment or claim it is the strongest current constraint in every scalar theory. Its use here is an explicit empirical calibration of (9.6). The exact map and inverse are With $\mu=21/10^6$ and $w=23/10^6$, the inverse reverses interval endpoints. Intersecting the transformed one- and two-width intervals with $\alpha\ge0$ gives respectively. They are transformations of quoted-width intervals, not a new posterior or confidence construction. Will’s separate summary $\omega_0>40000$, under $\omega_0=1/(4\alpha)-3/2$, gives $\alpha<1/160006$. The close endpoints are not exactly equal.

The direct displacement of $\alpha=1/6$ is $|d(1/6)-\mu|/w=500021/23$, in quoted widths. No Gaussian tail claim is warranted at such an enormous displacement. Conversely, $\alpha=10^{-7}$ lies within one displayed width, and $10^{-6}$ within two. Finite accuracy constrains a scalar residue without proving that it is exactly zero. Small coupling is a logically distinct possibility from a mass gap or screening, although these examples do not establish global viability against all observations.

10 Binary radiation requires body-dependent charges

10.1 The derivative that weak-field measurements do not supply

Define $\phi_s=\sqrt{\kappa/2}\,\vartheta$ in $c=1$ units, and $a_0=\partial\ln A/\partial\phi_s$. Then $\alpha=a_0^2/2$. A compact body’s effective action is

The derivative follows an equilibrium branch at fixed conserved matter content. Rotating or multicomponent bodies require the remaining conserved quantities to be specified as well. Weak self-gravity gives $a_A\simeq a_0$; strong binding can change it. This effective-body distinction is explicit in [13].

For background $A=1$, the conservative scalar potential is $-q_Aq_B/(4\pi r)$, so

The compact-body force constant and weak Cavendish constant need not coincide. We assume $G_{AB}>0$ for the bound orbit below. Scalar kinetic positivity fixes the flux sign but not the sign of a bilinear force between arbitrary unlike charges.

10.2 Derivation of the dipole flux and period decay

For fixed leading-order masses and charges, $\Box\vartheta=\sum_Aq_A\delta^3(x-x_A)$ has far-zone expansion

where $D=\sum_Aq_Ax_A$ and $Q_{ij}=\sum_Aq_Ax_A^ix_A^j$. Integrating the outgoing flux $R^2\dot\vartheta^2$ using $\int n_in_j\mathrm d\Omega=4\pi\delta_{ij}/3$ gives $\mathcal P_D=|\ddot D|^2/(12\pi)$. In the inertial centre of mass, For a circular orbit, restoring units, The leading dipole measures a charge-to-mass-ratio difference. Unequal masses alone do not produce it.

With $E_b=-G_{AB}\mu M/(2r)$, $\Omega^2=G_{AB}M/r^3$ and $P_b=2\pi/\Omega$, energy balance gives $\dot P_b=3P_b\mathcal P_D/(2E_b)$. For eccentricity $e$, the required average is

To see the time weight, use $r=a(1-e^2)/(1+e\cos f)$ and $\mathrm dt=r^2\mathrm df/h$, where $h=\Omega a^2\sqrt{1-e^2}$. Integrating $(1+e\cos f)^2$ supplies $1+e^2/2$. Thus This agrees with [14] under $n_b=2\pi/P_b$, mass ratio $q=m_A/m_B$, and $m_Bq/(q+1)=\mu$. It is one contribution to the observed period derivative. Kinematic corrections, tensor radiation, other scalar multipoles and the inference of masses belong to an actual timing analysis.

10.3 A permitting control and what survives it

The tensor quadrupole power on the same circular orbit is $\mathcal P_T=32G_*\mu^2r^4\Omega^6/(5c^5)$, so

Take masses 3 and 1 with $a_A=a_B=10^{-3}$. The leading dipole vanishes, but the scalar force does not. Nor does scalar radiation vanish: angular integration of the quadrupole term gives, in $c=1$ units, For equal charges per mass on a circular orbit the trace is constant, $Q_{ij}=\sqrt{\kappa/2}a_0\mu r_ir_j$, and $I'''_{ij}I'''_{ij}=32\mu^2r^4\Omega^6$. Consequently This is a leading multipole statement for a constant-charge circular source, not an all-orders cancellation or a complete post-Newtonian prediction.

In constant-coupling Brans–Dicke theory introduce the fixed-content sensitivity

Since $m_A^E=A m_A^J$ and $G_J\propto A^2$ in this restricted case, the chain rule yields For prescribed $s_A=1/5$, $s_B=0$, $a_0=10^{-3}$ and $v/c=10^{-3}$, the dipole-to-tensor ratio is $1/120$. Setting both sensitivities to zero at the same $a_0$ gives zero dipole. These are algebraic examples, not computed neutron-star models.

The observational comparison is accordingly conditional. The J1738+0333 analysis is less restrictive than Cassini along its constant-coupling Brans–Dicke line, while other nonlinear-coupling regions can be more strongly constrained [14, 20]. A proposed native $\alpha_{\rm eff}$ does not by itself choose the relevant pulsar bound. The coupling function, background, equation of state and equilibrium charge map must accompany it. In several canonical scalar fields, the static correction uses $\bm a_A\cdot\bm a_B$, whereas dipole emission uses $\left\lVert\bm a_A-\bm a_B\right\rVert^2$. This does not contradict the positive self-residue (9.4): the two source trajectories enter a difference, not a self-contraction.

11 The remaining C2 matching problem

The completed native projected-module checkpoint changes the starting point: $U$ is now a specified, certified 24-dimensional representation, rather than an unevaluated projected candidate. Equations (2.14)–(2.16) identify the discarded data whose survival and invariance remain unresolved. Neither those omissions nor an unestablished full-run budget erase the projected result. The full original-active quotient and its physical interpretation remain separate tasks. The first physical task is a field dictionary

where $\mathcal V_{\rm native}$ is a specified perturbation space and $\mathcal N_{\rm native}$ its specified redundancy subspace at this linear stage. Its domain, assignment, locality, units, kernel and image must be given. The map from spatial strain has only six components and cannot supply the lapse and shift merely by renaming variables. A nonzero-mode test should exhibit a mapped tensor configuration and a mapped source, and state which information is discarded. The invariant module and multiplicities used by $J$ must be those of the actual active target, with (2.5) established where needed.

Second, the native law must select an action or an equivalent dynamical prescription. For a quadratic limit $S_2=\frac12\left\langle\psi,\mathscr K_0\psi\right\rangle$, one needs a physical measure, temporal and spatial operators, boundary conditions and a symmetry generator $\mathscr R_0$. The tests are then $\mathscr K_0\mathscr R_0=0$, the constraint reduction, physical kinetic signs and dispersion. The three static coefficients in (3.4) cannot substitute for these data. A proposed approximation, such as a local two-derivative truncation about a specified background, must be justified by an error estimate or explicitly retained as a hypothesis.

Third, matter variation must give the source and identify the physical probe metric. A successful leading gravitational comparison would compute a conserved-source response with

in a declared scaling regime. Here the Lorentzian variable $q$, its massless dispersion surface and the physical conserved-source space are themselves derived objects. An extra scalar pole belongs in $\mathcal X_{\rm extra}$; a regular local improvement may belong in $\mathcal X_{\rm regular}$. Their separation requires analyticity and locality assumptions to be checked, not inferred from a fit at finitely many momenta.

Equation (11.2) gives a useful first target without demanding that every local coefficient be fixed before any gravity evidence is possible. At the same time, matching its leading tensor structure would not prove nonlinear Einstein dynamics, universal finite-body motion, quantum consistency or the absence of additional modes. Those are subsequent tests. A compact-body calculation would further construct $m_A(\phi_{s,\infty})$ and its derivative before claiming a pulsar constraint on a native coupling.

11.1 What the bounded result establishes

The most informative conclusion combines a native algebraic result with conditional constructions and obstructions. The specified projection is an invariant rank-24 module, with all induced group products checked. The lift theorem reduces the remaining full-active calculation to a precise kernel and quotient problem. A symmetry-preserving pullback does produce the three-channel cost under its hypotheses. Incompatibility does recover the spatial linearized Einstein tensor in the stated convention. The Einstein benchmark does yield two physical tensor polarizations together with constrained scalar and vector source responses. Local improvement freedom can leave the long-range tensor residue unchanged. Scalar-force and binary tests can distinguish couplings invisible to a tensor-only calculation.

The obstruction is equally concrete. A displacement-only strain dictionary cannot supply a nonzero TT mode. A static positive cost does not establish the kinetic operator. Geometric cubic symmetry alone does not establish physical response symmetry. A gauge-dependent local channel decomposition cannot identify a physical source sector. A value of a weak-field scalar coupling does not determine compact-body dipole radiation.

Together these results define a conditional route from C2 to a gravitational interpretation. They do not establish that gravity has already been derived from C2. The next direct algebraic checkpoint is the complete source lift and kernel certificate, followed by the complete correction quotient and the two invariance tests. A physical advance would additionally construct a native field/action/source map and test (11.2), or prove an obstruction for a delimited class of such maps. Further agreement obtained by inserting the continuum benchmark again would not close that gap.

11.2 Reproducibility and review status

The accompanying materials retain the source notes, their arithmetic programs and selected review documents with exact file bindings. The replayed scripts use artificial continuum or finite algebraic inputs and exact rational arithmetic; they do not construct or evaluate a native C2 worker. The new native claims are supported by the retained rank and invariance result reviews, whose completed computations are not repeated by this paper update. The source notes distinguish proofs from finite checks, and retain the original failures and corrections where applicable. The fifteen continuum companions and one artificial lift-reduction program are replayed for this revision, with per-program results recorded rather than summed into a purported number of independent discoveries.

The earlier notes have undergone the reviews identified in the source index. This revised synthesis remains to be independently reviewed as a manuscript. The rank and projected-invariance reviews use separate arithmetic implementations, but producer and reviewer are Codex; no independent-author or third-party audit is claimed. A review of predecessor bytes is not a review of a rewritten proof, a new combination of premises, or an added conclusion. In particular, the present paper’s selection, notation changes and exposition require their own scholarly assessment.

This manuscript was prepared with Codex at the owner’s request. This release reports existing computations and conditional arguments; it performs no new native execution or model adoption. The internal ledger remains 261 / 33,930. That bookkeeping is included for continuity only and is not mathematical evidence. The physics claims stand on the displayed assumptions, proofs, source records and executable checks.

A Split identities and an invariant-subspace trap

One recurrent algebraic issue deserves a compact derivation. Over a commutative unital ring, let $W$ be any $q\times p$ matrix and define maps in free-then-pivot coordinate order by

Block multiplication gives The two completeness summands have opposite lower-left blocks $W$ and $-W$. This cancellation, not a shape check, is the substantive equality.

For an arbitrary ambient matrix $A$, put $\rho_C=P_CAJ_C$. Then

for every $A$. Projecting this residual cannot test whether the chosen complement is invariant. The full residual is $AJ_C-J_C\rho_C=J_FP_FAJ_C$, so complement invariance requires $P_FAJ_C=0$. Similarly, flat-subspace invariance requires $P_CAJ_F=0$. In split coordinates these are the two different off-diagonal blocks of the ambient action. Neither is supplied by the inverse-split identities alone.

For $p=2$, $q=1$, $W=(1\;2)$, choose split-coordinate $A'=\left(\begin{smallmatrix}1&0&1\\0&1&0\\0&0&1\end{smallmatrix}\right)$. Its complement projects correctly onto itself, yet its image has a nonzero flat component. This three-coordinate example is a counterexample to the projected test, not a native value certificate. The block identities apply to an individual linear operator; semilinear composition and a group law require their separately specified coefficient action.

B The physical curvature convention and checks

To reconstruct (1.2) directly, use

Contracting all four-dimensional components and applying the reduced formulas in Section 5 are the two arithmetic routes used in the source companions. Gauge controls verify that a pure $\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$ has zero linearized Riemann tensor. Sourced controls are separately constructed to obey conservation; constructing a source by applying $G$ to a metric would not independently test the conservation boundary.

For the scalar lensing model, the complete coordinate-time ray acceleration can be written

for a static diagonal metric at leading order, with the transverse part used in (9.8). Equivalently it is $-P_n\nabla(\Phi_N+\Psi_N)/c^2+n[n\cdot\nabla(\Phi_N+\Psi_N)]/c^2$. This displayed full expression fixes what the projected bending calculation would otherwise leave untested. It includes the coordinate-time non-affine contribution; the vector $n$ is the unperturbed unit coordinate direction. In the optical derivation one may instead parameterize the spatial path by Euclidean arclength; the transverse deflection is the same but the longitudinal equation changes with parameterization.

C Evidence map and reproducibility instructions

The original sources are indexed in SOURCE_INDEX.json; the added C2 records are indexed in C2_UPDATE_SOURCES.json. An exact manifest binds the revision files, including REPLAY_REPORT.json. The index identifies the original path, byte count and SHA-256 digest of each retained note, program, result and review. It records the corrected versions of the tensor-field, Ward, Newtonian and scalar-pole reviews rather than silently treating superseded wording as current. The original packages are not edited by making this manuscript.

The source-note families cover active-sector algebra, shear response, tensor kinematics, incompatibility and direction dependence, response symmetry, Ward dynamics, constraints and tides, Newtonian sources, scalar gauge channels, the invariant source decomposition, isolated stress, matter actions, universal poles, scalar lensing, the empirical amendment and binary charges. The index identifies the retained notes and result files; the original result schemas and per-script counts remain unchanged in those files. Repeated source-pair or angular-grid assertions are implementation checks, not independent physical discoveries.

The reproduction driver runs retained continuum scripts in separate scratch directories with Python’s isolated, no-bytecode, no-site flags. It compares their stdout with the retained result bytes. It does not run the independent reviewer’s programs, a native extractor, a production launcher or a native arithmetic evaluator. Historical mutation tests are retained as historical records unless the replay report explicitly says they were rerun. This distinction keeps a successful reproduction from being presented as an independent audit.

The following commands build the document and replay the packaged artificial checks:

latexmk -pdf -interaction=nonstopmode -halt-on-error paper.tex
python3 -I -B -S replay_update.py

The Python command uses only the standard library. Its scope is fifteen retained artificial continuum programs and one artificial lift-reduction program, not validation of the entire native provenance chain. The bibliography supplies public source identifiers; the local source archive is not an archive of all cited literature or of the native runtime.

For the C2 update, the new source index retains the exact rank and invariance review documents, the lift theorem and its review, the adapter review and its meter correction, and the resource candidate’s note and result records. Native matrix and action payloads, private execution credentials and the complete runtime archives are not bundled with this revision. The native certificate claim is reported at the source/action scope of those reviews; the displayed certificate implications are proved in the text. The artificial lift replay corroborates the general reduction but is not a re-evaluation of native data.

The prior PDF-only release is version 1.0, DOI 10.5281/zenodo.22699108. This version 1.1 is a PDF-only update, DOI 10.5281/zenodo.22714538. Supporting records discussed here are retained separately by the author; neither a bibliography entry nor a source digest makes an unpublished record publicly available.

D Reference editions and attribution

Equation numbering in different editions of the same source need not agree. For Will’s 2014 review, the publisher text uses equations (61)–(62) for deflection and its grazing specialization, whereas arXiv:1403.7377v1 uses (59)–(60); the publisher Shapiro-delay equation is (64), versus arXiv (62). The present manuscript uses section-level citations for the empirical discussion and identifies the publisher edition in the bibliography.

For Hinterbichler, the stated comparison uses arXiv:1105.3735v2: equation (3.26) gives the two potentials and (3.27) the fixed-$G$ bending; the subsequent Newtonian rescaling is in unnumbered prose. Freire et al.’s binary formula is equation (21) of arXiv:1205.1450v1. The reported Cassini numerical result is taken from Will’s inspected account of the original experiment; the original Nature citation is provided, but a fresh reanalysis or direct reading of its inaccessible text is not claimed.

Fecko is cited as a 2021 lecture, not a refereed journal article. Maggiani, Scala and Van Goethem supply the function-space and topology setting for compatibility. The local Fourier inverse used here is derived directly and does not inherit a global converse without those hypotheses. The research-programme sources are explicitly unpublished technical records; their role differs from that of the external primary and review literature.

E A conditional cost bound for the full-kernel construction

This bound concerns the accepted-input algorithm implementing (2.14) and (2.17), not a new native evaluation [6]. Let the full matrix have shape $m\times p$, the selected projection have $d$ rows and rank $k$, and the original source have $n_{\rm src}$ coordinate-row occurrences, each with two rational components. Let $r=\mathop{\mathrm{rank}}N\le\min(m,p-k)$. Count one rational addition, subtraction, multiplication or division as one operation; gcd, hashing, parsing, comparison, indexing, allocation and serialization are excluded.

A sparse vector update against at most $m$ nonzero terms costs at most $2m$ operations. Allowing full fill-in yields the following bounds for one successful computation:

The minor inverse has $r$ pivot stages on at most $2r$ augmented entries per row: at most $2r^2$ divisions and $4r^2(r-1)$ update operations. The bounds for products with $\mathsf Z$ use its support on $r$ selected ambient rows. Selectors and row restrictions introduce no rational binary operations.

Let $T$ be the sum of this table. The unchanged projection extractor adds at most $n_{\rm src}$ further rational additions, outside the sparse meter. The formatter adds no rational binary operations, although it performs integer and byte work. Validation repeats the complete calculation with a fresh meter. Therefore the two-pass ceiling is

At the inherited input counts and projected rank, enumeration of $0\le r\le744$ gives its maximum $1{,}235{,}628{,}732{,}192$ at $r=744$. Here $m=2\times132078$ is the coordinate count of a representation retaining both coefficient axes, including zero axes; it is not a measured full rank. The table gives a conservative algorithmic upper bound, not a lower bound on required work.

The cost of an individual rational operation also depends on coefficient size. For accepted operands whose numerators and denominators have at most $b$ bits, the unreduced products have at most $2b$ bits, and an addition’s unreduced numerator has at most $2b+1$ bits. This observation does not bound the whole heap, and the shared projection arithmetic is outside that meter. Output-size checks run after record construction and serialization. Only limit-triggered meter refusals are sticky; arithmetic-domain exceptions such as division by zero are not. These mechanisms are protective checks, not a proof that native computation will complete within a budget.

The artificial integration encloses both full passes in its supervised child. Two suites of sixteen attempts retain all outcomes; seven scientific fixtures per suite complete with certificates, five complete with refusals, and four have incomplete/failing capture statuses with no mathematical outcome. Twelve stage records and the summary replay byte-identically. The rank-744 fixture is sparse with one-bit coefficients. Neither its approximately $0.62$ CPU seconds per attempt nor (E.1) determines the cost of the actual native full-kernel problem. A resource limit, should it stop that problem, would establish failure of that attempt to complete under its limits, not a failure of a C2 identity.

References

1. Bruno Bertotti, Luciano Iess, and Paolo Tortora. A test of general relativity using radio links with the Cassini spacecraft. Nature, 425: 374–376, 2003. doi:10.1038/nature01997. URL https://www.nature.com/articles/nature01997. Original experiment. Numerical input here is attributed to Will’s inspected 2014 account; no new fit or direct inspection of the original full text is claimed.
2. C2 research programme. Full-generator kernel adapter: construction review and documentation correction. Unpublished construction review, 11 September, 2026. Artificial-input reconstruction and certificate review; only limit-triggered meter refusals are sticky. Retained as adapter-review.md and meter-correction.md in the C2 update companion.
3. C2 research programme. Native projected-invariance result: mathematical and execution review. Unpublished project result review, 11 September, 2026. Both residual families for 48 supplied actions and all 2,304 induced multiplication identities. Retained as invariance-review.md in the C2 update companion; full source and byte bindings in C2_UPDATE_SOURCES.json.
4. C2 research programme. Lifting the rank-24 module to the complete original-active subject. Unpublished construction note and review, 11 September, 2026. Full-lift decomposition, quotient criterion and stable-relation closure under the inherited premises. Retained as lift-note.md and lift-review.md, with artificial checker and result, in the C2 update companion.
5. C2 research programme. Native active-projection result review. Unpublished project result review, 11 September, 2026. Exact rank of the frozen 39-by-768 matrix, four basis equations and determinant-one minor. Retained as rank-review.md in the C2 update companion; byte bindings in C2_UPDATE_SOURCES.json. Separate arithmetic review by Codex, not a third-party audit.
6. C2 research programme. Full-generator resource analysis and artificial supervision. Unpublished construction and resource candidate, 11 September, 2026. Two suites of 16 artificial attempts, conservative rank-dependent operation ceilings and retained outcomes. Candidate awaiting its own review; no native full-kernel evaluation. Retained as resource-note.md and resource-results.md with rank-bound and retention records in the C2 update companion.
7. C2 research programme. C2 physics technical notes and review records, 2026. Unpublished records, 10–11 September 2026. Exact source snapshots, current review versions and SHA-256 bindings accompany this manuscript in supporting/ and SOURCE_INDEX.json. These are project evidence, not external journal publications.
8. Codex. From native strain to a nonlinear-consistency evaluator, 2026. Source-backed proposal prepared for the C2 programme, 8 September 2026, Sections 3–6. Retained in supporting/native-free-theory/PROPOSAL.md; native field map, dynamics and matter coupling explicitly open.
9. Stanley Deser. Self-interaction and gauge invariance. General Relativity and Gravitation, 1: 9–18, 1970. doi:10.1007/BF00759198. URL https://arxiv.org/abs/gr-qc/0411023v3. Later arXiv posting of the original article with additional references.
10. Marián Fecko. Saint-venant’s compatibility condition and Einstein tensor. Lecture, Student Colloquium and School on Mathematical Physics, Stará Lesná, Slovakia, 22–28 August, 2021. URL https://davinci.fmph.uniba.sk/~fecko1/referaty/stara_lesna_2021.pdf. Equation (45); lecture slides, not a journal article.
11. Christopher J. Fewster and Lutz W. Osterbrink. Averaged energy inequalities for the non-minimally coupled classical scalar field. Physical Review D, 74: 044021, 2006. doi:10.1103/PhysRevD.74.044021. URL https://arxiv.org/abs/gr-qc/0606009v2.
12. Éanna É. Flanagan and Scott A. Hughes. The basics of gravitational wave theory. New Journal of Physics, 7: 204, 2005. doi:10.1088/1367-2630/7/1/204. URL https://arxiv.org/abs/gr-qc/0501041. Gauge-invariant decomposition and tidal response: Sections 2.2–2.3, equations (2.45)–(2.70) of the arXiv text.
13. Paulo C. C. Freire and Norbert Wex. Gravity experiments with radio pulsars. Living Reviews in Relativity, 27: 5, 2024. doi:10.1007/s41114-024-00051-y. URL https://link.springer.com/article/10.1007/s41114-024-00051-y. Section 5.5.1, equations (70)–(73), for the field normalization and effective body charges.
14. Paulo C. C. Freire, Norbert Wex, Gilles Esposito-Farèse, Joris P. W. Verbiest, Matthew Bailes, Bryan A. Jacoby, Michael Kramer, Ingrid H. Stairs, John Antoniadis, and Gemma H. Janssen. The relativistic pulsar–white dwarf binary PSR J1738+0333 – II. the most stringent test of scalar-tensor gravity. Monthly Notices of the Royal Astronomical Society, 423: 3328–3343, 2012. doi:10.1111/j.1365-2966.2012.21253.x. URL https://arxiv.org/abs/1205.1450v1. Section 5, in particular equation (21); historical constraints within the paper’s theory and stellar assumptions.
15. Domenico Giulini. Laue’s theorem revisited: Energy-momentum tensors, symmetries, and the habitat of globally conserved quantities. International Journal of Geometric Methods in Modern Physics, 15: 1850182, 2018. doi:10.1142/S0219887818501827. URL https://arxiv.org/abs/1808.09320. Section 1.5 and Theorem 1.
16. Ervin Hartmann. An introduction to crystal physics: Description of the physical properties of crystals. International Union of Crystallography, Teaching Pamphlet 18, n.d. URL https://www.iucr.org/what-we-do/education/pamphlets/introduction-crystal-physics. Section 4, Neumann’s principle. Online edition inspected 11 September 2026; no publication year inferred from the web page.
17. Kurt Hinterbichler. Theoretical aspects of massive gravity. Reviews of Modern Physics, 84: 671–710, 2012. doi:10.1103/RevModPhys.84.671. URL https://arxiv.org/abs/1105.3735v2. The version-specific comparison uses the expanded arXiv v2, Section 3.3, equations (3.26)–(3.27) and subsequent unnumbered rescaling discussion.
18. Giovanni Battista Maggiani, Riccardo Scala, and Nicolas Van Goethem. A compatible-incompatible decomposition of symmetric tensors in $L^p$ with application to elasticity. Mathematical Methods in the Applied Sciences, 38 (18): 5217–5230, 2015. doi:10.1002/mma.3450. URL https://webpages.ciencias.ulisboa.pt/~vangoeth/M2AS_2015.pdf.
19. Steven Weinberg. Photons and gravitons in $S$-matrix theory: Derivation of charge conservation and equality of gravitational and inertial mass. Physical Review, 135: B1049–B1056, 1964. doi:10.1103/PhysRev.135.B1049. URL https://journals.aps.org/pr/abstract/10.1103/PhysRev.135.B1049. Cited for the soft-factorization and Lorentz-covariance hypotheses; the leading Ward step is derived in the present text.
20. Clifford M. Will. The confrontation between general relativity and experiment. Living Reviews in Relativity, 17: 4, 2014. doi:10.12942/lrr-2014-4. URL https://link.springer.com/article/10.12942/lrr-2014-4. Publisher text, Sections 4.1.1–4.1.2 and 6.4. Deflection equations (61)–(62) differ in numbering from arXiv:1403.7377v1.