G Model Examples and Tests

Lorentz Tests

G MODEL Complete Lorentz-Related Test Audit and Current Re-Test Website Test-Section Research Report | Gurbuz Dolu | 12 September 2026 Scope: all retained G Model records located in the Library, with duplicate copies and repeated write-ups de-duplicated. Historical failures are preserved; later developments are used only where they genuinely change the scope or status of an earlier test.

Distinct Lorentz-related checks 39

PASS / structural or conditional PASS 26

FAIL / rejected / no-go test routes 12

Overarching all-scale status 1 OPEN

Current controlled verdict The G Model's local four-null geometry, inverse-metric reconstruction, six-bivector Lorentz/Hodge structure, inter-cell covariant transport, co-transforming six-channel operator, and accepted finite-cell reorientation observables pass their Lorentz-covariance tests. The rigid permanently oriented lattice implementation is rejected. Later gauge/action audits structurally remove the direction of the local centroid U from accepted spinless local observables. However, complete all-scale microscopic no-preferred-frame dynamics remains OPEN because a fully derived physical graph/reconnect law and exact high-k relational propagation law are still missing.

Important counting rule A FAIL below means that a particular construction, criterion, or implementation failed its stated requirement. It does not mean that the whole G Model failed. Several failures directly produced the later corrected construction.

Re-test policy used for this report Algebraically reproducible failures were re-tested with the current four-null geometry. Definition-level historical failures were dependency-audited against later replacement structures. Newer results narrow an old OPEN result only where they address the same requirement without changing the test target.

1. Complete De-duplicated Test Register The following 39 entries count mathematically distinct checks once. Repeated manuscript copies, repeated boosts of the same identity, and later restatements are not counted as new tests.

# Test Core mathematics Status Current interpretation / re-test

1 Four-null Gram signature spec(G4) = {-4, 4/3, 4/3, 4/3} Lc^2 PASS Retest unchanged. Exact 1+3 Lorentzian signature; independent of Tc magnitude.

2 Inverse Gram reconstructs K G4^-1 K^T = eta PASS Retest max numerical residual Minkowski metric 1.67e-16.

3 Directional-derivative (DM)^T G4^-1 (DM) = eta^(mu nu) PASS Signed inverse Gram gives the Lorentz scalar d_mu M d_nu M correct Lorentz contraction.

4 Equal-positive branch wave (1/4) sum_a (k_a.d)^2 = d0^2 + (1/3) FAIL Retest confirms it is not Box = operator nabla^2 -d0^2+nabla^2. Static averaging role only.

5 Common / explicit Lorentz k'_a = Lambda k_a; k'_a.k'_b = PASS Retest beta=0.6: null residual boost invariants k_a.k_b <=3.31e-16; Gram change <=2.22e-16.

6 Six-bivector Lorentz metric signature(Gamma)=(3,3) PASS Native six-bivector geometry remains Lorentzian.

7 Physical Lorentz Hodge J_*^2=-I6; J_*^T Gamma J_*=-Gamma PASS Proper Lorentz compatibility star retained.

# Test Core mathematics Status Current interpretation / re-test

8 Centroid-projected 3:1 E_U^2=8/3; B_U^2=8/9; PASS Covariant only when the split is bivector weighting E_U^2:B_U^2=3:1 defined relative to local U; raw E/B split is frame-dependent.

9 Rigid tetrahedron exact sum n_x n_y n_z = 4/(3 sqrt(3)); 4th FAIL Retest unchanged. This rejects a rotational isotropy moments != sphere permanently aligned microscopic crystal, not the dynamic-orientation model.

10 Orientation-balanced Abar[M]=M+(Lc^2/6)nabla^2M+(Lc^ PASS Leading correction is isotropic. long-wave isotropy 4/120)nabla^4M+... Uniform orientation averaging still needs physical realization.

11 Old Euclidean parent R as R^T R=I, R^3=I but R^T Gamma R FAIL Historical route still fails as Lorentz physical Lorentz transport != Gamma transport; superseded by Omega/connection transport.

12 Inter-cell Lorentz Omega_yx=K_y Pi_yx K_x^-1; PASS Exact metric compatibility once transporter Omega^T eta Omega=eta branch correspondence is supplied.

13 Induced six-bivector T_yx=wedge^2(Omega_yx); T^T PASS Exact for proper transporter Gamma T=Gamma; T J_*=J_* T orientation-preserving matched links.

14 Branch matching on Pi_Celal in argmin ||gamma_x-Pi^T PASS UPDATED: existing centered-link existing links gamma_y Pi||_F^2 correspondence is exact modulo V4; physical link birth/death remains OPEN.

# Test Core mathematics Status Current interpretation / re-test

15 Pure relative-frame Pi=I => Omega_loop=I FAIL Old route remains flat/pure gauge. connection as generic Later area-holonomy curvature is a curvature different branch, so this failure is scoped.

16 Reciprocal M deformation A(M)=diag(e^M,e^-M,e^-M,e^-M); FAIL Retest unchanged. M belongs to as Lorentz transform A^T eta A != eta for M!=0 coframe/metric deformation, not Lorentz-frame transformation.

# Test Core mathematics Status Current interpretation / re-test

17 Fixed-matrix commutator [W,P]!=0; rank=2 for beta=3/5 FAIL Exact calculation; the criterion itself criterion was too strong. P must co-transform, not commute in every basis.

18 Co-transforming physical P P_phys -> D P_phys D^-1 PASS Local six-channel update transforms covariantly.

19 Inter-cell covariant Delta'=D_y Delta; Delta'^T Gamma PASS Cell-by-cell local Lorentz gauge difference / Gamma norm Delta'=Delta^T Gamma Delta consistency.

20 Preferred timelike centroid gamma=K^T eta K; 16=10+6 PASS UPDATED structural pass locally: U U in accepted spinless local Lorentz-fiber decomposition direction is gauge-fiber data for observables accepted spinless action/tensor observables. Full interacting network still not closed.

21 Complete all-scale All physical observables independent OPEN CONTROLLING boundary: high-k microscopic of absolute orientation / motion propagation plus physical no-preferred-frame relative to U graph/reconnect evolution are not dynamics yet derived unconditionally.

# Test Core mathematics Status Current interpretation / re-test

22 Explicit beta=0.6 local-cell Lambda^T eta Lambda=eta; 4 nulls, PASS Independent numerical falsification stress test 6 pair products, 6 areas preserved challenge reproduced with current cell geometry.

23 Continuous 360-degree W_f(theta)=Lambda_x W_f(0) PASS Exact finite-cell all-angle gauge local-frame / holonomy test Lambda_x^-1 invariance for conjugacy-class observables.

24 All 24 tetrahedral U_pi P = P U_pi for all pi in S4 PASS Exact structural pass. relabelings vs P

25 120-degree edge holonomy H^3=I; [H,P]=0 PASS Exact; Lorentz bivector lift preserves vs P Gamma and J_*.

26 Fixed-frame Hessian L_fixed: diagonal coefficient 3/2, FAIL Retest 20,000 random H,R: response rotational off-diagonal coefficient 1 covariance error remains order 1; covariance max 2.50 in current rerun.

27 Dynamic principal-axis L_dyn(H)=(3/2)[H-(tr H/3)I] PASS Exact SO(3) covariance; retest max Hessian response discrepancy 3.66e-15. Physical alignment law remains CONDITIONAL.

28 Covariant dynamic tidal / E_mn=h_m^a h_n^b nabla_a nabla_b PASS Formal Lorentz-tensor construction; shape tensor M-(1/3)h_mn h^ab nabla_a nabla_b M source-to-shape amplitude remains model-dependent.

# Test Core mathematics Status Current interpretation / re-test

29 Infinite connected +/-axis <n_i n_j>=(1/3)delta_ij PASS Exact long-wave isotropy of the network: second moment six-direction stencil.

30 Infinite connected +/-axis <n_x^4>=1/3, <n_x^2 n_y^2>=0 vs FAIL High-k lattice anisotropy remains for network: fourth moment 1/5,1/15 frozen connectivity.

31 Naive finite-cell dispersion Tc=sqrt(3)/(c k) arccos(v_g/c) without FAIL Coordinate expression is not an in arbitrary observer frame transforming cell frame arbitrary-frame scalar.

32 Covariant cell-frame k_t^2=h^mn q_m q_n; Tc=sqrt(3)/(c PASS Coordinate Lorentz covariance PASS dispersion extraction k_t) arccos(v_g,t/c) when t^mu co-transforms.

33 Exact microscopic Lorentz q^2=k_t^2-(4/a^2)sin^2(a k_t/2), FAIL k_t^2=q^2+(t.q)^2 remains. invariance of sinusoidal a=2cTc/sqrt(3) Changing Tc does not remove finite-cell dispersion preferred-frame dependence for nonzero a.

34 Continuum limit of a k_t -> 0 => q^2 -> 0 PASS Ordinary massless Lorentz-invariant finite-cell dispersion propagation recovered at leading order.

# Test Core mathematics Status Current interpretation / re-test

35 Naive 3-space Tc^2=(5 pi hbar/(12 c^3))(nabla^2 FAIL Do not use unchanged outside the source-response in M/rho) source rest frame. arbitrary frame

# Test Core mathematics Status Current interpretation / re-test

36 Covariant source-rest Q_u=(g^mn+u^m u^n)nabla_m PASS Observer-independence PASS in extraction nabla_n M; Tc^2=(5 pi hbar/(12 pressureless weak-source domain; c^3))(Q_u/rho0) full stress-energy extension remains OPEN.

37 Native area-holonomy S_conn/hbar=zeta sum PASS Structural branch: 20,000 proper Lorentz-scalar action stress <B/Lc^2,(J_*+beta I)K_f>_Gamma boosts to beta<=0.95 preserved test scalar contractions to <9e-14. Action uniqueness/normalization remain open.

38 Local Lorentz fiber / K->Lambda K leaves gamma; source PASS 20k one-cell + 20k loops + 20k tensor-source stress audit reconstruction and loop conjugacy symmetric sources + 5k P-similarity unchanged audits in record; current rerun preserved Gram to ~1.53e-14.

39 Can six Lorentz gauge 16=10+6; Lorentz fiber has 6 gauge FAIL NO-GO for deriving four H_ab directions force four directions but no independent conservation identities from Lorentz source-conservation 4-translation gauge gauge alone. This is a equations? conservation/symmetry limitation, not a Lorentz-covariance failure.

2. Core Mathematical Lorentz Results 2.1 Four-null Gram and metric reconstruction For four future-directed tetrahedral null branches, the normalized branch Gram matrix has zero diagonal and -4/3 off diagonal. Its spectrum is exactly one negative and three positive eigenvalues: G4_aa = 0, G4_ab = -4/3 (a != b) spec(G4) = {-4, 4/3, 4/3, 4/3} G4^-1 = (3/4) I4 - (1/4) 11^T K G4^-1 K^T = eta

Current numerical re-test of the reconstruction gave a maximum residual of 1.67 x 10^-16. The result is scale-free: multiplying every branch by Lc=cTc does not change the normalized signature or reconstructed metric.

2.2 Why the old equal-positive wave operator still fails (1/4) sum_a (k_a . partial)^2 = partial_0^2 + (1/3) nabla^2 but Box = -partial_0^2 + nabla^2

No single overall scalar can repair both the time sign and the 1/3 spatial coefficient. The correct Lorentz contraction uses G4^-1; equal-positive averaging remains useful only in its static averaging role.

2.3 Bivector metric and Hodge structure signature(Gamma) = (3,3) J_*^2 = -I6 J_*^T Gamma J_* = -Gamma

The local 3:1 E/B-like ratio survives only when the split is defined covariantly relative to the local timelike direction; the raw coordinate split is frame dependent.

2.4 Inter-cell transport and local gauge covariance Omega_yx = K_y Pi_yx K_x^-1 Omega_yx^T eta Omega_yx = eta Omega_yx -> Lambda_y Omega_yx Lambda_x^-1 T_yx = wedge^2(Omega_yx) T_yx^T Gamma T_yx = Gamma, T_yx J_* = J_* T_yx

The fixed-matrix commutator test is retained as an exact calculation but is no longer interpreted as a Lorentz failure. The correct physical rule is conjugation covariance: P_phys -> D P_phys D^-1 Delta_yx = F_y - T_yx F_x -> D_y Delta_yx Delta_yx^T Gamma Delta_yx = invariant

3. Orientation, Preferred-Frame, and Reconnect Re-Tests 3.1 Rigid tetrahedron: failure preserved sum_a n_ax n_ay n_az = 4/(3 sqrt(3)) != 0 T_xxxx = 1/9, T_xxyy = 1/9 isotropic sphere: 1/5 and 1/15

The exact third- and fourth-moment failures remain. Therefore a permanently aligned tetrahedral crystal is not accepted as the physical microscopic vacuum implementation.

3.2 Dynamic 360-degree local orientation: later pass K_x(theta)=Lambda_x(theta)K_x(0) W_f(theta)=Lambda_x(theta) W_f(0) Lambda_x(theta)^-1 spec W_f, tr(W_f^n), det W_f = invariant

This is an exact all-angle finite-cell gauge pass for accepted local/holonomy observables.

3.3 Fixed-frame Hessian failure and dynamic-principal-axis repair L_dyn(H) = (3/2)[H - (tr H/3) I] L_dyn(R H R^T) = R L_dyn(H) R^T

Current 20,000-sample rerun: dynamic covariance max discrepancy 3.66 x 10^-15; fixed-frame discrepancy remains order unity (max 2.50). The physical rule that a cell aligns with instantaneous principal axes remains conditional.

3.4 Preferred timelike centroid U - refined current status gamma_x = K_x^T eta K_x 16 components of K_x = 10 metric/Gram + 6 local Lorentz-fiber components K_x -> Lambda_x K_x leaves gamma_x unchanged U_x=(1/4)K_x 1 -> Lambda_x U_x

Later spinless gauge/action audits upgrade the old blanket U-direction concern: within accepted local spinless gauge-invariant observables, the direction of U is frame/gauge data. This is a STRUCTURAL PASS locally. The stronger all-scale network claim remains OPEN.

3.5 Celal matching narrows an old OPEN result Pi_Celal in argmin_{Pi in S4} ||gamma_x - Pi^T gamma_y Pi||_F^2

Existing centered-link correspondence is exact modulo V4. The unresolved problem is physical link birth/death and graph evolution, not ordinary branch correspondence on an already existing link.

4. High-k Propagation and the Remaining Lorentz Problem 4.1 Frozen finite-step networks still retain high-k anisotropy <n_i n_j> = (1/3) delta_ij [second moment PASS] <n_x^4> = 1/3, <n_x^2 n_y^2> = 0 [not isotropic 1/5, 1/15]

Long-wavelength isotropy can coexist with finite-step anisotropy. The current model rejects a frozen permanent stencil as final fabric, but the full relational graph dynamics remains open.

4.2 Sinusoidal finite-cell dispersion a = 2 c Tc / sqrt(3) chi^2 = (4/a^2) sin^2(a k_t/2) q^2 = k_t^2 - (4/a^2) sin^2(a k_t/2) k_t^2 = q^2 + (t.q)^2

When the local cell direction t^mu co-transforms, the extraction is coordinate-covariant. But the finite-cell correction still depends on k_t, so the current sinusoidal law is not an exact no-preferred-frame microscopic law. q^2 = (a^2 k_t^4)/12 - (a^4 k_t^6)/360 + ... -> 0 as a k_t -> 0

Changing the numerical value of Tc changes the physical wavelength at which finite-cell corrections become important. It does not remove the structural t^mu dependence. Therefore this failure cannot be repaired merely by selecting another Tc.

4.3 Source-response route has a cleaner Lorentz form h_u^(mu nu) = g^(mu nu) + u^mu u^nu Q_u = h_u^(mu nu) nabla_mu nabla_nu M Tc^2 = (5 pi hbar/(12 c^3)) (Q_u/rho_0) = (5 pi hbar/(12 c)) (Q_u/epsilon_0)

Here u^mu is the physical matter four-velocity, not a universal vacuum vector. In the pressureless weak-source domain this formulation passes observer-independence. The full stress-energy extension remains open.

5. Numerical Stress Audits with the Current Architecture Re-test Current result Verdict

Gram eigenvalues {-4, 1.333333333, 1.333333333, 1.333333333} PASS

Metric reconstruction max |K G4^-1 K^T - eta| = 1.67e-16 PASS

Explicit beta=0.6 boost max null residual 3.31e-16; max Gram shift 2.22e-16 PASS

M deformation as Lorentz norm(A^T eta A-eta) = 0.384 at M=0.1; 6.56 at M=1 FAIL map

Rigid tetrahedron 3rd sum n_x n_y n_z = 0.7698003589 != 0 FAIL moment

Rigid tetrahedron 4th <n_x^4>=<n_x^2 n_y^2>=1/9, not isotropic FAIL moment

Dynamic Hessian 20,000 samples; max error 3.66e-15 PASS covariance

Fixed Hessian covariance 20,000 samples; order-one error, max 2.50 FAIL

Random local boosts 10,000 boosts to beta below 0.95; Gram max drift 1.53e-14 PASS

Source tensor frame 5,000 samples; max numerical discrepancy 2.80e-13 PASS reconstruction

These are internal numerical consistency checks, not independent experimental verification.

Area-holonomy / local-action stress record The retained working record also reports 10,000 nondegenerate coframe rank tests, 20,000 random proper Lorentz transformations up to beta=0.95 for action contractions, 20,000 one-cell and 20,000 loop frame tests, 20,000 symmetric-source reconstructions, and 5,000 P-similarity checks. These support a structural local-Lorentz-gauge closure in the present spinless action branch, but not the complete physical graph dynamics.

6. Deep Note: Use of T_c in Lorentz Tests The historical audit found six distinct numerical routes that actually produced a T_c value. They are not six independent confirmations. Four are current reconstruction routes and two are historical diagnostics. None is yet a fully independent Big-G-free experimental determination of the absolute scale.

# Route T_c (s) L_c (m) Difference Status

1 Measured Big-G 2.18657023792000 6.555172662157e-3 +0.0000 ppm CALIBRATED calibration 0e-43 5

2 K_M -> T_c inversion 2.18657023792000 6.555172662157e-3 +0.0000 ppm EXACT inversion; current 0e-43 5 numerical K_M provenance not independent

3 Mercury perihelion 2.18656511942973 6.555157317309e-3 -2.3409 ppm OBSERVATIONAL back-inversion 0e-43 5 inversion / calibrated compatibility

4 Moved-source lab 2.18657023752900 6.555172660985e-3 -0.0002 ppm EXPERIMENTAL form / reconstruction demo 0e-43 5 numerical demonstration only

5 Historical K_GD 2.20046945400000 6.596841463686e-3 +0.6357% HISTORICAL / provisional bare-SI closure 0e-43 5 / superseded

6 Historical Earth 2.47000000000000 7.404873712600e-3 +12.9623% HISTORICAL / illustrative source-curvature 0e-43 5 / not independent diagnostic

Is it scientifically acceptable to use T_c here? Yes, if it is labeled correctly. For dimensional examples, cell lengths, numerical wavelengths, or plots, use the retained calibrated reference T_c,ref = 2.186570237920 x 10^-43 s and L_c = 6.555172662157 x 10^-35 m, explicitly labeled CALIBRATED / REFERENCE.

For the principal Lorentz pass/fail identities, the absolute number is generally irrelevant. The normalized Gram signature, inverse-metric reconstruction, Lorentz/Hodge algebra, transporter identities, and local gauge conjugation laws do not change when Tc is replaced by another positive reference scale because Lc factors cancel or scale homogeneously. Therefore it would be misleading to run the same algebraic Lorentz test six times and count six validations. The six Tc values are a scale-sensitivity note. The historical Earth diagnostic differs by about 12.96%, yet it does not change the normalized Lorentz verdict; it only shifts Lc and the wavelength assigned to finite-cell effects. For the sinusoidal finite-cell law, the Lorentz issue is structural for every nonzero a=2cTc/sqrt(3): changing Tc does not remove the k_t / t^mu dependence.

7. Website-Ready Scientific Summary Recommended wording for the Tests section Across the retained G Model research record, 39 distinct Lorentz-related mathematical and numerical checks were identified after de-duplicating repeated write-ups. Twenty-six pass in their stated exact, structural, or conditional domains; twelve specific constructions or criteria fail and are retained as falsified or superseded routes; one overarching question remains open: complete all-scale microscopic no-preferred-frame dynamics. The strongest established result is Lorentz covariance of the local four-null/six-bivector architecture and its accepted inter-cell gauge transport. The model exactly reconstructs the Minkowski metric from the four-null Gram matrix, preserves the Lorentzian six-bivector metric and Hodge structure, and passes explicit boost, local-frame, holonomy, and tensor-reconstruction stress tests. A rigid permanently oriented tetrahedral lattice fails higher-order isotropy and is not the accepted physical implementation. Later dynamic-orientation and local-gauge analyses substantially narrow the historical preferred-frame concern: in the accepted spinless local action/tensor sector, absolute U direction is gauge-fiber data and does not appear in accepted gauge-invariant observables. Nevertheless, the model still withholds a claim of full microscopic Lorentz invariance until the physical graph/reconnect law and exact finite-cell/high-k propagation are derived so that no all-scale observable can retain a preferred direction.

Most Important PASS Results What matters most is not the raw number of PASS results, but whether the foundational Lorentz consistency gates pass. The most important successful results are: (1) the four-null Gram has the exact 1+3 Lorentzian signature; (2) its inverse reconstructs the Minkowski metric exactly; (3) the six-bivector sector carries the proper (3,3) Lorentz metric and Hodge structure; (4) inter-cell transport and the co-transforming six-channel operator obey the correct local Lorentz transformation laws; and (5) continuous 360-degree local-frame/holonomy observables are exactly invariant under the accepted gauge interpretation. Later spinless action and source-tensor audits also show that the direction of U is gauge-fiber data in the accepted local observable sector.

These passes are the central reason the Lorentz sector remains viable after the rejected branches. They establish that the retained local mathematical architecture can be formulated Lorentz-covariantly without fitted Lorentz coefficients. They do not, by themselves, prove that the complete microscopic fabric exists in nature.

The most important PASS still required The decisive remaining target is all-scale microscopic no-preferred-frame dynamics. To close it, the native physical reconnect/graph-evolution law and finite-cell high-k propagation must be derived so that every measurable gauge-invariant observable is independent of absolute tetrahedral orientation or motion relative to a vacuum frame. A clean PASS of that test would materially strengthen the claim that the G Model can serve as a fundamental spacetime theory; an unavoidable preferred-frame observable would instead be a fundamental challenge.

How the FAIL results should be interpreted Failed implementation does not equal failed G Model. The recorded FAIL results do not constitute a falsification of the G Model as a whole. They falsify particular implementations, operators, assumptions, or test criteria explored during development. Several failures directly motivated corrected Lorentz-covariant constructions and remain in the record so that superseded routes are not hidden.

Examples include the rejection of equal-positive branch averaging as the relativistic wave operator, rejection of a permanently aligned rigid tetrahedral crystal as exactly isotropic, rejection of the old Euclidean parent R as physical Lorentz transport, and reclassification of the fixed-matrix commutator requirement after the co-transforming P rule was established. These are scientifically useful negative results because they narrow the admissible theory rather than being silently discarded.

The principal unresolved Lorentz question is whether the complete interacting finite-cell network can be shown to have no observable preferred frame at all scales. If future native dynamics proves that an observable preferred-frame signal is unavoidable, that would constitute a fundamental challenge to the G Model. Likewise, a contradiction in an indispensable core identity, an inability of any completed source dynamics to satisfy required conservation, or a genuine quantitative prediction of the completed model that conflicts with experiment would be potentially falsifying. Those stronger outcomes have not been established by the twelve recorded FAIL routes in this audit. Scientific-status boundary: these are internal mathematical/covariance tests and AI-assisted numerical audits. They are not experimental confirmation of the G Model. PASS means the declared mathematical test closes; it does not mean nature has been shown to use the proposed microscopic causal fabric.

8. Source Record Map ID Record Use

S1 G_Model_Lorentz_Covariance_and_Microscopic_Invariance Master Lorentz register and dynamic-fabric _Audit_v2_DYNAMIC_FABRIC_2026-08-28.pdf addendum.

S2 G_Model_Dynamic_360_Lorentz_Test_31AUG2026.md Continuous 360-degree local-frame and reconnect audit.

S3 G_Model_Lorentz_Covariance_Stress_Test_Landscape_08S Explicit beta=0.6 boost stress test. EP2026.pdf

S4 G_Model_Post_Manuscript_Scale_Research_Report_03SEP Lorentz audit of T_c extraction routes and 2026.pdf preferred-frame stress test.

S5 G_Model_Author_Contribution_Working_Record_2026-08-2 Later local Lorentz gauge/action stress audits 8_v7_HISTORY_READY.md and U-fiber closure.

S6 G_Model_Einstein_Native_Audit_Checkpoint_31AUG2026. 16=10+6 Lorentz gauge inventory and md conservation no-go.

S7 G_Model_CURRENT_MANUSCRIPT_55P_MMM_AAA_XY Current controlling manuscript status. Z_LAW_UPDATE_12SEP2026.pdf

S8 G_Model_Tc_Numerical_Reconstruction_Historical_Audit_1 Six numerical T_c routes and independence 2SEP2026.pdf audit.

Authorship and verification boundary G Model conceptual framework, research direction, interpretation, and final scientific responsibility: Gurbuz Dolu. AI assistance was used for retrieval, mathematical formalization, algebraic/numerical re-testing, status reconciliation, and document preparation. This is an internal research audit for transparent public presentation and independent falsification; it is not peer review or experimental validation.

Einstein Tests

G MODEL Einstein Compatibility, Spin-2, and Gravitational Field-Equation Tests Complete Current Audit for the Website Tests Section | Gurbuz Dolu | 12 September 2026 Scope: retained Einstein/GR-compatibility records were reviewed and duplicate manuscript copies or repeated restatements were not counted as new tests. Historical failures remain visible. Later native-action, tensor, conservation, rotation, and wave results are used where they genuinely update an earlier status.

Distinct Einstein-related checks 38

PASS / structural / conditional PASS 27

FAIL / rejected / superseded test routes 6

OPEN major derivations / closures 5

Current controlled Einstein verdict The current G Model has a lossless ten-component branch representation of symmetric spacetime tensors; an exact branch representation of the linearized Einstein/Fierz-Pauli continuum structure; an Einstein-operator coefficient pattern that is unique under explicit continuum symmetry and conservation assumptions; a native spinless area-holonomy action class that structurally yields a torsion-free metric-compatible connection; and a conditional two-function static spherical completion whose exterior is exactly Schwarzschild. These are substantial mathematical/compatibility results. They are not yet a proof that the complete finite-cell G Model derives general relativity from microscopic first principles.

Counting and status rule A PASS means the stated mathematical test closes in its declared domain. CONDITIONAL or STRUCTURAL PASS means an adopted action, continuum symmetry class, or completion is part of the premise. A FAIL means a specific ansatz, operator identification, or incomplete route failed its own requirement. OPEN means the decisive microscopic derivation is still missing. None of these labels is experimental validation.

1. Complete De-duplicated Einstein Test Register The table below counts distinct scientific checks once. Closely related algebraic substeps are grouped rather than inflated into separate tests. # Test Core mathematics Status Current interpretation / re-test

1 Four-null inverse-Gram K G4^-1 K^T = eta PASS Exact supporting identity. Gives the spacetime reconstruction required 1+3 spacetime metric in the branch basis.

2 Flat-background branch (G4^-1)^(ab) D_a D_b = Box PASS Exact for a constant branch frame; d'Alembertian curved/discrete use requires connection terms.

3 Full 4+6 symmetric tensor Q_ab=k_a^mu k_b^nu h_mu_nu; PASS 4 self-branch + 6 pair values = all 10 encoding H_ab=k_a^mu k_b^nu T_mu_nu symmetric tensor components; lossless reconstruction.

4 Trace and trace-reversal h=(G4^-1)^(ab)Q_ab; PASS Exact branch image of the standard reconstruction Qbar_ab=Q_ab-(1/2)G4_ab Q spacetime operations.

5 Harmonic-gauge (G4^-1)^(ac) D_a Qbar_cb = 0 PASS Exact representation of the standard representation harmonic condition in the flat branch frame; gauge origin remains separate.

6 Four-component (G4^-1)^(ac) D_a H_cb = 0 PASS The branch basis can represent all four matter-conservation conservation equations. This does not representation force them dynamically.

# Test Core mathematics Status Current interpretation / re-test

7 Weak scalar metric: static G00^(1)=2 nabla^2 M; PASS Correct static pressureless scalar dust Einstein tensor G0i^(1)=Gij^(1)=0 sector.

8 Weak scalar metric as a single M only FAIL Cannot encode general pressure, shear, complete general source or moving-matter response. description Superseded by the full 10-component tensor extension.

9 Native weak source nabla^2 M=C_Sigma c^3 Tc^2 PASS Conditional native source normalization rho/hbar; kappa_E=2 C_Sigma c normalization; with measured Big G it Tc^2/hbar matches 8 pi G/c^4.

10 Full identity-like 4+6 weak Box Qbar_ab = -kappa H_ab PASS Strong conditional/structural pass. sourced equation Exact equivalence to the continuum tensor wave equation once the gauge structure is adopted.

11 Six-channel P as the spec(P)={1,0,0,0,-1/2,-1/2} FAIL Rank-3 action on the six pair sector complete Einstein would erase three source/shear matter-source operator directions. P remains an internal response/equalization operator.

12 Gauge-consistent source S_mu_nu=a T_mu_nu+b eta_mu_nu T PASS Given massless spin-2 gauge symmetry coupling / independent and source conservation, direct tensor trace admixture coupling is consistent; independent trace coupling is not generic.

# Test Core mathematics Status Current interpretation / re-test

13 Microscopic derivation of delta Q_ab = D_a xi_b + D_b xi_a OPEN Continuum branch representation is linearized diffeomorphism exact, but discrete relational ontology gauge symmetry has not yet proved this gauge redundancy from microscopic dynamics.

14 Pure-gauge linearized Q_ab=D_a xi_b+D_b xi_a => PASS Exact in the flat-background branch curvature cancellation R_abcd[Q]=0 calculus.

15 Linearized Einstein operator E_ab[Q]=(1/2)[-D^2Q_ab + ... + PASS Exact representation of the standard in branch variables G4_ab D^2Q] linearized Einstein tensor; identically conserved.

# Test Core mathematics Status Current interpretation / re-test

16 Uniqueness of linearized A=-B, C=-B, D=-B, E=B PASS Exact under explicit assumptions: Einstein coefficient pattern symmetry, Lorentz covariance, locality, <=2 derivatives, gauge redundancy, identical conservation.

17 Fierz-Pauli kinetic structure L_FP in Q_ab branch variables PASS Exactly representable; not yet representability microscopically generated by finite-cell dynamics.

18 Continuum dim W(k-hat)=2; equivalently PASS Strong continuum structural result transverse-traceless 10-4-4=2 under the massless spin-2 gauge gravitational mode count conditions.

# Test Core mathematics Status Current interpretation / re-test

19 First-order Tc relaxation as Tc dq/dtau = -(I-P)q FAIL Centered modes decay exponentially the complete rather than propagate as a gravitational-wave equation conservative two-way wave. Retained only as local recovery/relaxation.

20 Second-order covariant omega^2 -> c^2 k^2 at long PASS Structural long-wave pass. The exact inter-cell tensor propagation wavelength finite-cell/high-k action-derived propagation theorem remains open.

21 Native area-holonomy D[(J_*+beta I)Sigma]=0 => D PASS Structural pass for a nondegenerate action -> torsion-free Sigma=0 => T^I=0 spinless coframe within the adopted connection native action class.

22 Smooth native action -> e^J wedge *F_IJ = 0 PASS Strong structural pass within the Einstein-form vacuum adopted smooth action class; not yet coframe equation complete discrete matter dynamics.

23 Einstein trace coefficient X_ab=R_ab+alpha gamma_ab R; PASS G Model-native connection/curvature from curvature conservation conservation => alpha=-1/2 variables plus the contracted Bianchi identity select the Einstein-type combination.

24 General moving / pressure / H_ab carries density, momentum, PASS Full tensor source is shear / anisotropic source pressure, shear, anisotropic stress information-complete; this repairs the completeness earlier scalar/P-only limitations.

# Test Core mathematics Status Current interpretation / re-test

25 Full microscopic source C_b=(G4^-1)^(ac) Delta_a H_cb = 0 OPEN The rank-four divergence operator conservation exists, but C_b=0 is not yet an identity of proved finite-cell matter dynamics.

26 H Yazici conservation repair symmetric H-level correction cancels PASS Exact in its proved neutral/commuting admissible divergence defect sector with bounded Continuity-Time inverse; unique curved physical update remains open.

27 Nonlinear time-dependent dynamic A,B,C coframe; nonlinear PASS Kasner-type test supports nonflat anisotropic vacuum omega wedge omega curvature vacuum curvature with vanishing compatibility conserved gravitational tensor in the adopted connection architecture.

28 Historical one-function ds^2=-e^(2M)dt^2+e^(-2M)dx^2 FAIL Exact Ricci test forces the nontrivial reciprocal exponential static static vacuum field away. Retained only vacuum as weak-field parametrization.

29 Conditional two-function psi=-mu/r -> exact isotropic PASS Exact Ricci-flat Schwarzschild exterior static spherical exterior Schwarzschild metric; R_mu_nu=0 within the conditional two-function completion.

30 Generic harmonic psi as an nabla^2 psi=0 alone FAIL Harmonicity is necessary but not automatic exact sufficient; the additional spatial Ricci two-function vacuum tensor condition must also hold. solution

# Test Core mathematics Status Current interpretation / re-test

31 Weak rotating-source frame O(J): R^-3 rotational connection / PASS Structural weak-rotation pass using the dragging Lense-Thirring functional form same coupling pathway; no new fitted force.

32 Pure rotational shift alone at rotation-only second-order ansatz FAIL Residual equations require scalar l=0 O(J^2) plus quadrupolar l=2 backreaction. The failure is of the incomplete ansatz.

33 Second-order rotation O(J): P=0; O(J^2): P=1 plus P=-1/2 PASS Structural closure of momentum + closes the 1+3+2 sector scalar + quadrupolar response under hierarchy rotation.

34 Horizon/asymptotic Q_phys = -J^2/(M c^2) PASS Conditional structural result: regularity selection of second-order and asymptotic flatness remove quadrupole independent quadrupole hair.

35 Third-order current-octupole O(J^3): l=1 plus l=3 PASS Angular selection and homogeneous channel / no free l=3 hair no-hair result pass structurally; the native inhomogeneous coefficient is still not independently calculated.

36 All-orders native Kerr M_l+i S_l ?= M(ia)^l from G Model OPEN Higher-order channel structure is multipole coefficients recursion constrained, but full native inhomogeneous coefficients are unfinished.

# Test Core mathematics Status Current interpretation / re-test

37 Two physical action-derived microscopic OPEN Continuum TT count is two; gravitational-wave modes at propagation with only two tensor finite-cell/high-k action-derived closure finite-cell scale modes is still missing.

38 Complete generic all vacuum/source geometries OPEN Several nonlinear and nonspherical nonlinear/nonspherical compatibility tests pass, but full Einstein equivalence general equivalence has not been derived and is not required to be assumed.

2. Weak-Field Tensor and Linearized Einstein Tests 2.1 Full tensor capacity — the essential representational PASS Q_ab = k_a^mu k_b^nu h_mu_nu H_ab = k_a^mu k_b^nu T_mu_nu 4 diagonal + 6 off-diagonal = 10 independent components h = (K^-1)^T Q K^-1 T = (K^-1)^T H K^-1

This removes a basic obstruction: the four-null/six-pair language does not lose pressure, momentum, shear, or anisotropic stress when the full branch tensor H_ab is used.

2.2 The scalar-only and P-only source failures Two early simplifications were too restrictive. A single scalar weak metric correctly reproduces the static dust 00 sector but cannot carry general pressure, shear, or moving matter. Separately, the six-channel equalizer P has three zero eigenvalues and therefore cannot be the complete matter-to-metric source map. spec(P) = {1, 0, 0, 0, -1/2, -1/2} rank(P) = 3 on the six-pair space

These failures do not remove the full tensor construction. They explain why P must remain an internal spectral/equalization operator inside the complete H_ab source rather than replace the stress-energy tensor.

2.3 Linearized Einstein operator E_ab[Q] = 1/2 [ -D^2 Q_ab + D_a D^c Q_cb + D_b D^c Q_ca - D_a D_b Q - (G4)_ab D^c D^d Q_cd + (G4)_ab D^2 Q ] D^a E_ab = 0

The branch expression is exactly the standard linearized Einstein tensor once the continuum spin-2 symmetry class is adopted.

2.4 Uniqueness test General operator coefficients: A,B,C,D,E Gauge: A+B=0, B+C=0, D+E=0 Conservation: B+D=0, C+E=0 => A=-B, C=-B, D=-B, E=B

The constraint system has one free normalization only. This is an exact uniqueness result under the explicit assumptions; it is not a proof that microscopic G Model dynamics generated those assumptions.

3. Native Connection and Einstein-Type Curvature Tests 3.1 Spinless connection closure S_conn/hbar = zeta sum_f < B_f/Lc^2 , (J_* + beta I) K_f >_Gamma D[(J_* + beta I) Sigma] = 0 J_*^2 = -I => D Sigma = 0 => T^I = D e^I = 0

Within the adopted native area-holonomy action class and for a nondegenerate spinless coframe, the metric-compatible torsion-free connection used in later Einstein tests is no longer merely inserted as an external assumption.

3.2 Einstein-type curvature tensor selected by conservation X_ab = R_ab + alpha gamma_ab R D^a R_ab = (1/2) D_b R D^a X_ab = 0 => alpha = -1/2 E^G_ab = R_ab - (1/2) gamma_ab R

This is one of the strongest native structural results in the Einstein sector: the -1/2 trace coefficient is fixed by curvature conservation rather than numerically fitted. E^G_ab = 2 C_Sigma (c Tc^2 / hbar) H_ab

The source coefficient is the retained G Model coupling. With the reference scale, it becomes the familiar 8 pi G/c^4 normalization.

3.3 Smooth-action vacuum equation S ~ integral Sigma_IJ wedge (*F_IJ + beta F_IJ) variation -> e^J wedge (*F_IJ + beta F_IJ)=0 with torsion-free Bianchi -> e^J wedge *F_IJ = 0

This is an Einstein-form vacuum coframe equation within the adopted smooth action class. Complete discrete matter/source dynamics is still not closed.

4. Conservation: What Passes and What Remains Open 4.1 Four-component divergence exists C_b(x) = (G4^-1)^(ac) Delta_a H_cb(x), b=1,...,4

The representation has rank four and can carry all four continuity equations. But representational capacity is not the same thing as a dynamical identity.

4.2 Why full microscopic conservation is still OPEN A simple counterexample remains: in a constant branch frame with identity transport, if only H_11 changes along branch 1, then C_1=(G4^-1)^(11) Delta_1 H_11=(1/2) Delta_1 H_11, which is generically nonzero. Thus the current native variables define the divergence but do not yet force C_b=0 for arbitrary finite-cell matter evolution.

4.3 H Yazici repair theorem In the proved neutral/commuting sector with a bounded Continuity-Time inverse, an admissible divergence defect can be cancelled exactly by a symmetric H-level correction. This is a genuine mathematical conservation PASS in that domain. It does not yet select the unique curved finite-cell physical update that nature must use.

Scientific importance Source conservation is one of the two decisive Einstein closures still required. A fundamental sourced gravity theory cannot rely indefinitely on representational conservation alone; the matter/inter-cell dynamics must make the appropriate conservation law follow natively.

5. Strong-Field Einstein / Schwarzschild Tests 5.1 Historical one-function metric — preserved FAIL ds^2 = -e^(2M)(dx^0)^2 + e^(-2M)(dx^2+dy^2+dz^2) R_00 = e^(4M) nabla^2 M R_ij = delta_ij nabla^2 M - 2 partial_i M partial_j M

Vacuum first requires nabla^2 M=0, then R_ij=0 forces grad M=0. Therefore this ansatz has no nontrivial exact static vacuum Einstein solution. It remains usable only where explicitly treated as a weak-field parametrization.

5.2 Two-function completion — exact exterior PASS inside a conditional ansatz psi = 2 tanh(M/2) T = (1+psi/2)/(1-psi/2) X = (1-psi/2)^2 spherical vacuum selects psi = -mu/r ds^2 = -[(1-mu/2r)/(1+mu/2r)]^2 c^2 dt^2 + (1+mu/2r)^4 (dr^2+r^2 dOmega^2) R_mu_nu = 0

With R=r(1+mu/2r)^2 this transforms exactly into the standard Schwarzschild exterior. The exterior PASS is exact; the microscopic dynamics that uniquely forces the two-function completion remains open.

5.3 Generic harmonic psi is not enough The exact Ricci audit shows that nabla^2 psi=0 is necessary but not sufficient for arbitrary nonspherical fields. The additional spatial Ricci condition must also vanish. Thus a generic harmonic superposition is not automatically an exact two-function vacuum solution.

6. Gravitational Waves and Propagation Tests 6.1 Continuum two-mode result W(k-hat) = { S=S^T : tr S=0, S k-hat = 0 } dim W = 2 continuum count: 10 - 4 - 4 = 2

The continuum transverse-traceless radiative subspace is exactly two-dimensional under the standard massless spin-2 conditions.

6.2 First-order Tc relaxation failure Tc dq/dtau = -(I-P)q centered P=-1/2 sector: q(tau)=q(0) exp[-3 tau/(2 Tc)]

This law is dissipative relaxation, not a conservative two-way gravitational wave equation. The failure is preserved; the equation remains useful only for local recovery/equalization.

6.3 Second-order covariant propagation omega^2 = c^2 sum_i [ (2/a) sin(k_i a/2) ]^2 a = 2 Lc/sqrt(3) long wavelength: omega^2 -> c^2 k^2

This provides a natural conservative second-order propagation candidate and the correct long-wavelength speed. However, the complete finite-cell/high-k action-derived spin-2 theorem is still OPEN. The current sinusoidal finite-cell candidate also cannot by itself be counted as a completed all-scale Lorentz-invariant gravitational-wave law.

7. Nonspherical and Rotating Einstein-Sector Tests 7.1 Nonlinear time-dependent anisotropic curvature Using an anisotropic dynamic coframe with independent scale functions A, B, and C, the metric-compatible connection produces nonlinear curvature through the connection-squared term. A Kasner-type vacuum test gives nonzero curvature with vanishing conserved gravitational tensor. This is a structural compatibility test beyond the static Schwarzschild sector.

7.2 Weak frame dragging Rotating matter enters through ordinary momentum loading rather than a new fitted force. Inter-cell transport and loop/curl formation produce the circulation/frame-dragging pathway with the Lense-Thirring functional form in the weak branch.

7.3 Why pure rotation alone fails at second order At O(J^2), a pure rotational shift does not satisfy the residual equations by itself. The required backreaction contains an isotropic l=0 sector and a quadrupolar l=2 sector. That failure is therefore the failure of an incomplete rotation-only ansatz, not of the full nonlinear rotating branch. O(J): P=0 momentum/twist O(J^2): P=1 plus P=-1/2 scalar/quadrupolar backreaction

7.4 Higher multipoles Under asymptotic-flatness and regular-horizon conditions the second-order branch carries no independent quadrupole hair and gives Q_phys=-J^2/(M c^2). At third order the angular structure forces l=1 and l=3 and removes independent regular l=3 homogeneous hair. The native inhomogeneous third- and higher-order coefficients are still unfinished; all-orders Kerr equivalence is therefore not claimed.

8. Current Numerical / Algebraic Re-tests Re-test Current result Verdict

Four-null metric reconstruction max |K G4^-1 K^T - eta| = 2.48e-16 PASS

10-component tensor 10,000 random symmetric tensors; max reconstruction error 1.11e-15 PASS reconstruction

Einstein-operator coefficient 5 coefficients, constraint rank 4 -> nullity 1; ratio (-1,1,-1,-1,1) PASS constraints

P as complete pair-source eigenvalues {1,0,0,0,-1/2,-1/2}; rank 3; nonzero zero-mode source is FAIL operator annihilated

Continuum TT mode count 10,000 random directions; constraint rank 4 in 6D symmetric space -> dim PASS 2

One-function vacuum M=x has nabla^2 M=0 but R_xx=-2 !=0 FAIL counterexample

Generic harmonic two-function psi=x is harmonic, but extra spatial Ricci condition gives diag(-2,1,1) !=0 FAIL counterexample

Two-function Schwarzschild symbolic 4D isotropic-coordinate Ricci tensor simplifies identically to zero PASS Ricci recheck

These re-tests are internal algebraic/computational consistency checks. They do not constitute independent experimental verification.

9. Calibration Note: T_c and Big G in Einstein Tests Current reference scale T_c,ref = 2.186570237920 x 10^-43 s L_c = c T_c = 6.555172662157 x 10^-35 m

It is scientifically acceptable to use this T_c in numerical Einstein-sector examples only when it is labeled CALIBRATED / REFERENCE. It comes from the current empirical Big-G calibration branch; it is not an independent G Model prediction.

Most of the important Einstein structural tests do not depend on the numerical value of T_c: tensor completeness, the operator coefficient ratios, the -1/2 curvature trace coefficient, and the exact Schwarzschild form are algebraic/geometric statements. T_c enters the absolute source-coupling normalization and finite-cell length scale. The historical record contains six routes that produced numerical T_c values, but they are not six independent confirmations. This report therefore does not count repeated T_c reconstructions as additional Einstein passes. A future Big-G-free measurement or derivation of T_c would become a separate empirical cross-check.

10. Conclusion for the Website Tests Section Most important PASS results The most important result is not the raw count of 27 passes. It is that the foundational continuum and curvature consistency gates now pass in their declared domains. The four-null basis losslessly carries a complete ten-component gravitational field and stress-energy source; the linearized Einstein operator is represented exactly and its relative coefficients are unique under the explicit massless-spin-2 continuum assumptions; the native spinless area-holonomy action structurally supplies a torsion-free metric-compatible connection; curvature conservation fixes the Einstein-type trace coefficient to -1/2; and the conditional two-function static spherical branch is exactly Ricci-flat and exactly Schwarzschild outside the source.

The continuum TT sector also has exactly two tensor modes, and later nonlinear anisotropic and rotating-sector tests show that the retained connection architecture can carry nonflat vacuum curvature, weak frame dragging, and the expected scalar/quadrupolar backreaction structure. These results materially strengthen Einstein compatibility, while remaining mathematical/structural rather than experimental.

How the FAIL results should be interpreted Failed implementation does not equal failed G Model. The six recorded FAIL results reject particular simplifications or incomplete ansatzes: the scalar metric as a complete general source, P as the complete source operator, first-order relaxation as a full gravitational-wave law, the old one-function exact vacuum metric, harmonic psi by itself as a generic exact vacuum solution, and pure rotation by itself at second order. Each failure either has a later broader construction or clearly identifies what additional structure is required.

The most important example is the one-function strong-field failure. It did not get hidden or numerically repaired; it was rejected. The later two-function ansatz was then independently subjected to an exact Ricci test and passed in the static spherical exterior. Likewise, the P-source failure led to the full H_ab tensor source rather than to fitted eigenvalues.

The most important PASS still required The decisive remaining Einstein closure is a native finite-cell derivation of the complete spin-2 field equation together with full source conservation. The microscopic inter-cell dynamics must coarse-grain to the Einstein/Fierz-Pauli coefficient pattern without inserting the continuum gauge coefficients by hand, must dynamically enforce the appropriate four-component source conservation, and must leave only the two physical tensor modes in the low-energy radiative sector. Controlled finite-scale corrections may then be tested rather than assumed away.

What would constitute a fundamental Einstein-sector problem? A future native closure would face a fundamental challenge if it inevitably produced an unsuppressed extra scalar/vector gravitational mode, nonconserved stress-energy in its claimed domain, a different leading low-energy tensor operator from the tested Einstein/Fierz-Pauli form, unavoidable preferred-frame terms, source dynamics that erased pressure or shear, or a gravity mass/screening scale incompatible with long-range observations. Those outcomes have not been established by the six historical FAIL routes.

Claim boundary Current defensible statement: the G Model has substantial mathematical compatibility with Einstein gravity in the tested weak-field, continuum spin-2, static spherical vacuum, and selected nonlinear/rotating sectors. It has not yet derived all Einstein dynamics from microscopic first principles, has not proved full generic GR equivalence, and has not been independently experimentally validated.

11. Source Record Map

ID Record Use in this report

E1 G_Model_Einstein_Audit_Working_Record_2026-08-28(2).md Master Einstein PASS/FAIL/OPEN audit and exact formulas.

E2 G_Model_Consolidated_Findings_Native_Einstein_30AUG2026.p Native connection, Einstein-type curvature tensor, df rotation summary.

E3 G_Model_Author_Contribution_Working_Record_2026-08-28_v5 Later dynamic propagation, nonlinear curvature, /v7_HISTORY_READY.md rotation, action and closure history.

E4 G_Model_Linearized_Einstein_Operator_Uniqueness_Theorem_ Operator uniqueness theorem and assumption Educational_11SEP2026.pdf boundary.

E5 G_Model_Full_Tensor_Reconstruction_Theorem_Educational_11 Exact ten-component branch tensor SEP2026.pdf reconstruction.

E6 G_Model_Transverse_Traceless_Two_Mode_Theorem_Educatio Continuum two-mode gravitational-wave count. nal_11SEP2026.pdf

E7 G_Model_H_Yazici_Theorem_Educational_11SEP2026.pdf Conservation-repair theorem and physical-update boundary.

E8 G_Model_Two_Function_Schwarzschild_Completion_Principle_E Historical one-function failure and exact ducational_11SEP2026.pdf Schwarzschild exterior.

E9 G_Model_CURRENT_MANUSCRIPT_55P_MMM_AAA_XYZ_LA Current controlling manuscript/test status. W_UPDATE_12SEP2026.pdf

Authorship and verification boundary G Model conceptual framework, research direction, interpretation, and final scientific responsibility: Gurbuz Dolu. AI assistance was used for record retrieval, mathematical formalization, algebraic and symbolic re-testing, status reconciliation, and document preparation. This is an internal theoretical audit for transparent public presentation and independent falsification; it is not peer review, experimental validation, or independent replication.

Schwarzschild Tests

G MODEL Schwarzschild Exterior, Horizon, and Strong-Field Tests Complete Current Audit for the Website Tests Section | Gurbuz Dolu | 12 September 2026 Scope: all retained Schwarzschild/black-hole strong-field records were reviewed, repeated manuscript copies were de-duplicated, and every previously identified failure was re-tested against the latest retained geometry, horizon analysis, and Xyz update. Historical failures remain visible rather than being overwritten.

Distinct Schwarzschild-related checks 24

PASS / supported / structural or conditional PASS 15

FAIL / rejected test routes or interpretations 5

OPEN derivations / physical continuations 4

Current controlled Schwarzschild verdict The old one-function strong-field metric remains rejected. The later two-function static spherical completion survives every current exterior re-test: the vacuum equations select the 1/r branch, the resulting metric is exactly Ricci-flat, and an exact areal-radius transformation gives the standard Schwarzschild exterior. The horizon is supported as a causal-trapping boundary, but the latest invariant tests continue to reject the stronger statement that the horizon itself destroys the local causal fabric. The 12 September Xyz update strengthens that distinction by placing a separate candidate Gram-degeneracy boundary far deeper than an astrophysical horizon.

Status rule A PASS means the stated mathematical test closes in its declared domain. A CONDITIONAL PASS means the two-function completion or the current Xyz source-to-shape normalization is part of the premise. A FAIL rejects a particular ansatz or interpretation, not the whole G Model. OPEN means the microscopic dynamics or physical continuation is not yet derived.

1. Complete De-duplicated Schwarzschild Test Register The following 24 entries count distinct scientific checks once. Algebraic substeps used only to establish the same result are grouped rather than inflated into extra passes. # Test Core mathematics Status Current interpretation / latest re-test

1 Historical one-function ds^2=-e^(2M)c^2dt^2+e^(-2M)dx FAIL Exact static-vacuum Ricci equations reciprocal exponential ^2 force grad M=0. Latest re-test M=x still metric gives lap M=0 but R_xx=-2.

2 General static Ricci tensor R00=[(1+psi/2)/(1-psi/2)^7] lap psi; PASS Direct differential-geometric formula for the two-function Rij exact nonlinear expression retained and independently rechecked. completion

3 Harmonic psi alone as lap psi=0 by itself FAIL Latest counterexample psi=x is sufficient exact vacuum harmonic but the remaining spatial Ricci condition numerator is diag(-2,1,1), not zero.

4 Spherical harmonic solution lap psi=0 -> psi=A+B/r PASS Exact spherical solution of the first family vacuum equation.

5 Nontrivial tangential psi'=-psi/r -> A=0, psi=B/r PASS Exact selection inside the two-function vacuum equation selects 1/r spherical ansatz.

6 Attractive exterior branch psi=-mu/r; T=(1-mu/2r)/(1+mu/2r); PASS Exact substitution into the conditional X=(1+mu/2r)^2 completion.

# Test Core mathematics Status Current interpretation / latest re-test

7 Full exterior Ricci-flatness R_mu_nu=0 PASS Fresh 4D symbolic re-test: every Ricci component simplifies identically to zero in the regular exterior.

8 Exact transformation to R=r(1+mu/2r)^2 PASS Fresh algebraic re-test gives zero Schwarzschild areal radius residual in the identity relating isotropic and Schwarzschild forms.

9 Horizon coordinate mapping r_h=mu/2 -> R_h=2mu PASS Exact map; dR/dr vanishes at the isotropic-coordinate throat/horizon.

10 Schwarzschild Kretschmann K=48 mu^2/R^6 PASS Coordinate-invariant exterior invariant cross-check inherited from exact Schwarzschild equivalence.

11 Horizon is not a curvature K_H=3/(4 mu^4) PASS Finite for every finite nonzero mu; singularity independently re-evaluated.

12 Local four-null Gram at the det G4=-(256/27)Lc^8 PASS Nonzero for Lc>0; the local four-null horizon basis does not lose rank merely at R=2mu.

# Test Core mathematics Status Current interpretation / latest re-test

13 Invariant outward-support C_F=1-2mu/R; C_F(R_h)=0 PASS Exact in the Schwarzschild exterior scalar at the horizon branch; dC_F/dR at the horizon =1/(2mu), finite.

14 360-degree / all-orientation maximum future-directed outward PASS Supports causal trapping / loss of escape diagnostic areal-radius support -> 0 at R=2mu outward-connected support at the event horizon.

15 Divergence of M proves M=(1/2)ln C_F -> -infinity FAIL Latest findings strengthen the rejection: local fabric destruction at C_F has finite gradient, curvature is the horizon finite, and the local Gram is nondegenerate.

16 Use isotropic-coordinate R=r(1+mu/2r)^2 FAIL Exact map has a minimum R=2mu at r<mu/2 as the black-hole r=mu/2 and returns to R>2mu for interior r<mu/2. This chart cannot simply be read as the interior.

17 Exterior redshift/light-deflection/Shapiro/orbital same Schwarzschild metric for same PASS Exact consequence of metric mass parameter mu equivalence; compatibility result, not a structure separate empirical confirmation.

# Test Core mathematics Status Current interpretation / latest re-test

18 Discrete causal-cut criterion Xi_disc_x=max_(x->y)(R_y-R_x)/Delta tau_xy; PASS Structural/conditional network if all future links <0 no statement. The actual physical path returns outward link-update law remains a separate open problem.

# Test Core mathematics Status Current interpretation / latest re-test

19 Microscopic derivation of finite-cell dynamics -> unique T/X OPEN The strongest missing Schwarzschild the two-function completion completion derivation: current equations do not yet uniquely force the completion from microscopic dynamics.

20 Matter interior, collapse, complete R<2mu physical solution OPEN Exterior compatibility does not and singularity-resolution and formation law determine collapse, interior matter, or dynamics singularity physics.

21 Generic nonspherical exact arbitrary strong-field nonspherical OPEN Static spherical success cannot be nonlinear extension sources promoted to generic nonlinear GR equivalence.

22 Event horizon equals Xyz R_h=2mu versus det gamma=0 Xyz FAIL Latest Xyz results separate the surfaces. Gram degeneracy boundary For astrophysical masses the candidate Xyz radius is enormously deeper than the horizon.

23 Schwarzschild-Xyz r_Xyz^3=(3 beta/2) r_s Lc^2; PASS Conditional algebraic corollary of the strong-field corollary K_Xyz=16/(3 beta^2)Lc^-4 current source-to-shape normalization; beta normalization remains ambiguous.

24 Literal ordinary-fabric post-degeneracy topology/fabric OPEN Xyz exactly marks loss of ordinary absence beyond Xyz phase four-null invertibility in the centered Gram model; the physical continuation beyond it is not derived.

2. Historical Strong-Field Failure and Its Re-test 2.1 One-function metric ds^2 = -e^(2M)c^2dt^2 + e^(-2M)(dx^2+dy^2+dz^2) R_00 = e^(4M) nabla^2 M R_ij = delta_ij nabla^2 M - 2 partial_i M partial_j M

Vacuum first requires nabla^2 M=0. Then R_ij=0 requires partial_i M partial_j M=0, forcing a constant M for a nontrivial static domain. Therefore the branch cannot be an exact nontrivial static-vacuum Schwarzschild theory.

Latest explicit counterexample Choose M=x. nabla^2 M = 0 R_xx = -2 != 0

Re-test verdict: FAIL remains. None of the later Lorentz, Einstein, Xyz, or calibrated-Tc findings repairs this exact Ricci obstruction, because the failure is intrinsic to the one-function metric ansatz itself.

The correct historical interpretation is that the one-function form may still be used where explicitly labeled as a weak-field parametrization; its old higher-order strong-field deviations are not current Schwarzschild predictions.

3. Two-Function Completion and Exact Vacuum Test 3.1 Conditional completion psi = 2 tanh(M/2) T = e^M = (1+psi/2)/(1-psi/2) X = (1-psi/2)^2 ds^2 = -T^2 c^2dt^2 + X^2(dr^2+r^2 dOmega^2)

The algebra is exact once this completion is adopted, but its microscopic selection remains conditional.

3.2 General static Ricci equations R_00 = [(1+psi/2)/(1-psi/2)^7] nabla^2 psi Vacuum also requires: psi partial_i partial_j psi - 3 partial_i psi partial_j psi + delta_ij |grad psi|^2 = 0

This immediately exposes a useful failure: harmonicity alone is not enough.

Latest harmonic counterexample psi=x => nabla^2 psi=0 remaining spatial Ricci numerator = diag(-2,1,1) != 0

Re-test verdict: the 'harmonic psi alone is sufficient' route remains FAIL. The later spherical success does not convert generic harmonic superpositions into exact nonlinear solutions.

3.3 Spherical vacuum selection nabla^2 psi=0 -> psi=A+B/r nontrivial tangential equation -> psi'=-psi/r => A=0, psi=B/r attractive branch: psi=-mu/r

Thus the nontrivial 1/r radial form is selected by the vacuum equations inside the completion rather than inserted merely because Schwarzschild is known.

4. Exact Schwarzschild Exterior Re-test 4.1 Isotropic exterior metric ds^2 = -[(1-mu/2r)/(1+mu/2r)]^2 c^2dt^2 + (1+mu/2r)^4 (dr^2+r^2dOmega^2)

Fresh symbolic result All sixteen components of the 4D Ricci tensor simplify identically to zero in the regular exterior. This is an exact vacuum result, not a weak-field expansion.

4.2 Areal-radius transformation R = r(1+mu/2r)^2 1-2mu/R = [(1-mu/2r)/(1+mu/2r)]^2 dR/dr = 1 - (mu/2r)^2

Substitution gives exactly ds^2 = -(1-2mu/R)c^2dt^2 + (1-2mu/R)^(-1)dR^2 + R^2dOmega^2

The latest algebraic re-test gives zero residual in the coordinate identity. Therefore the surviving exterior is not merely similar to Schwarzschild; it is coordinate-equivalent to it.

4.3 What follows automatically For the same mass-length parameter mu, all observables determined purely by exterior timelike or null geodesics coincide with Schwarzschild GR: gravitational redshift, light deflection, Shapiro delay, and the standard static-spherical orbital structure. This is compatibility, not a new independent empirical confirmation.

5. Horizon Tests and Rejected Destruction Claims 5.1 Horizon mapping and curvature r_h=mu/2 -> R_h=2mu K=48mu^2/R^6 K_H=3/(4mu^4)

The curvature invariant is finite at the horizon for finite nonzero mu. The event horizon is therefore not a curvature-destruction singularity.

5.2 Local cell nondegeneracy det G4 = -(256/27)Lc^8 != 0 for Lc>0

The local four-null basis does not lose rank merely because the Schwarzschild horizon is reached.

5.3 Outward-support scalar C_F = 1-2mu/R C_F(R_h)=0 dC_F/dR |_(R=2mu) = 1/(2mu)

The support scalar reaches zero at the horizon while its radial derivative remains finite. This supports a causal-boundary interpretation rather than literal local-cell annihilation.

5.4 360-degree escape audit Maximizing future-directed outward areal-radius propagation over the allowed local null orientations gives an outward-support measure that approaches zero at R=2mu. This independently supports the statement that the horizon is a causal trapping boundary in the retained spherical branch.

Latest verdict on the old destruction interpretation REJECTED: 'M diverges at the horizon, therefore the local causal fabric is destroyed there.' The latest data make the rejection stronger, not weaker: curvature is finite, C_F is the smoother invariant, its derivative is finite, and det G4 remains nonzero.

6. Coordinate-Domain Failure Re-tested 6.1 Why r < mu/2 is not automatically the black-hole interior R(r) = r + mu + mu^2/(4r) dR/dr = 1 - mu^2/(4r^2)

R(r) has its minimum R=2mu at r=mu/2. For r<mu/2, R increases again and remains greater than 2mu. Re-test verdict: FAIL remains for the interpretation 'the isotropic-coordinate region r<mu/2 is the Schwarzschild black-hole interior.' The standard isotropic chart does not supply a regular one-chart continuation through the event horizon. Interior physics must be treated separately.

6.2 What this failure does not affect It does not affect the exact exterior PASS. It only prevents the exterior isotropic coordinate parameter from being extended naively and reinterpreted as a derived G Model interior solution.

7. Latest Xyz Update: Horizon Is Not the Deep-Interior Degeneracy 7.1 Current Schwarzschild-Xyz relation r_Xyz^3 = (3 beta/2) r_s Lc^2 K_Xyz = [16/(3 beta^2)] Lc^(-4)

This is a conditional strong-field corollary of the current tidal source-to-shape normalization. The normalization beta remains open between the retained routes beta=2/3 and beta=1/6.

7.2 Numerical separation from an astrophysical horizon Route Mass r_s / horizon scale (m) r_Xyz (m) r_Xyz / r_s

Route A 1 solar mass 2.953339e+03 2.332530e-22 7.898e-26

Route A 10 solar masses 2.953339e+04 5.025283e-22 1.702e-26

Route A Sgr A* ~4.3e6 solar 1.269936e+10 3.793005e-20 2.987e-30

Route B 1 solar mass 2.953339e+03 1.469402e-22 4.975e-26

Route B 10 solar masses 2.953339e+04 3.165730e-22 1.072e-26

Route B Sgr A* ~4.3e6 solar 1.269936e+10 2.389443e-20 1.882e-30

Latest re-test verdict: the event horizon is not the Xyz degeneracy surface for ordinary astrophysical black holes. In the current conditional extension the Xyz scale is tens of orders of magnitude deeper than the horizon.

The formal horizon-Xyz crossover occurs only at about 0.04 micrograms in Route A and 0.02 micrograms in Route B. Those are hypothetical Schwarzschild-radius crossover masses, not ordinary laboratory gravity thresholds.

7.3 Current three-regime interpretation outside horizon: ordinary four-null metric nondegenerate inside horizon but before Xyz: local fabric may remain nondegenerate, outward escape blocked at Xyz: det gamma -> 0; ordinary four-null inverse ceases to exist

This latest result reinforces the distinction: event horizon, local fabric destruction, and Xyz degeneracy are three different statements. The physical continuation beyond Xyz remains OPEN.

8. Current Re-test Summary Re-test Current calculation Verdict

One-function vacuum branch M=x: lap M=0 but R_xx=-2 FAIL remains

Harmonic psi sufficiency psi=x: extra Ricci numerator diag(-2,1,1) FAIL remains

Two-function exterior Ricci fresh symbolic 4D calculation: all components zero PASS tensor

Areal-radius transformation identity residual exactly zero PASS

Horizon curvature K_H=3/(4mu^4), finite PASS

Local four-null Gram at horizon det G4=-(256/27)Lc^8 PASS / non degenerate

M-divergence = destruction finite curvature + finite dC_F/dR + nondegenerate Gram REJECTED claim

Naive isotropic-coordinate R(r) has minimum 2mu and returns to R>2mu for r<mu/2 FAIL interior remains

Horizon = Xyz degeneracy astrophysical r_Xyz/r_s ~10^-26 to 10^-30 REJECTED in current conditional extension

These are internal mathematical and computational re-tests. They are not independent observational verification of the G Model.

9. Calibration Note: T_c, Big G, and Schwarzschild Tests T_c,ref = 2.186570237920 x 10^-43 s L_c = 6.555172662157 x 10^-35 m

The calibrated T_c is scientifically acceptable for numerical cell-scale or Xyz examples when it is explicitly labeled CALIBRATED / REFERENCE. It is not needed to establish the exact Ricci-flat Schwarzschild exterior itself.

The exterior metric equivalence is scale-form geometric: once the mass-length parameter mu is supplied, Ricci-flatness and coordinate equivalence do not require the numerical microscopic T_c. T_c enters the G Model coupling calibration and the microscopic/deep-interior scales such as L_c and the conditional Xyz radius. The six historical numerical T_c routes must not be counted as six Schwarzschild confirmations. The exact exterior PASS is one mathematical compatibility result; repeated calibrations of the same scale do not multiply its evidential weight.

10. Conclusion for the Website Tests Section Most important PASS results The most important Schwarzschild PASS is not an orbital fit. Inside the conditional two-function completion, the static spherical vacuum equations select the nontrivial 1/r branch, the resulting four-dimensional metric is exactly Ricci-flat, and an exact coordinate transformation produces the standard Schwarzschild exterior. This means the entire exterior geometry - not one chosen observable - agrees with Schwarzschild for the same mass parameter.

A second important PASS is the horizon consistency audit. The invariant outward-support scalar reaches zero at R=2mu and the all-orientation escape diagnostic independently supports causal trapping. At the same surface, the Schwarzschild curvature invariant remains finite and the local four-null Gram remains nondegenerate. The model therefore cleanly distinguishes a causal horizon from destruction of the local geometric basis. The latest Xyz analysis strengthens this conclusion: in the current conditional strong-field extension, the candidate Gram-degeneracy boundary is vastly deeper than an astrophysical event horizon. The correct current statement is therefore: event horizon is not fabric destruction, and event horizon is not Xyz degeneracy.

How the FAIL results should be interpreted Failed strong-field ansatz does not equal failed G Model. The principal historical failure is the original one-function reciprocal exponential metric, which cannot satisfy the nontrivial exact static-vacuum Einstein equations. That branch remains rejected. The later two-function completion is a distinct construction and has passed the exact exterior Ricci test. Likewise, harmonic psi alone fails as a generic exact-vacuum rule, naive use of the isotropic coordinate below r=mu/2 fails as an interior derivation, and the claim that the horizon destroys the local fabric remains rejected by the invariant tests.

These failures are scientifically important because they remove incorrect claims rather than being hidden. None of the latest findings reverses those negative results; the 12 September re-tests confirm them.

The most important PASS still required The decisive remaining Schwarzschild test is to derive the surviving two-function strong-field completion from the native finite-cell G Model dynamics rather than adopt it as a conditional continuum completion. A stronger theory must show why microscopic branch/connection dynamics produces this nonlinear spherical metric, then extend consistently through matter interiors and collapse without importing the Schwarzschild solution as an external target.

What would constitute a fundamental Schwarzschild-sector problem? A serious failure would occur if a native microscopic strong-field derivation produced an exterior that could not satisfy R_mu_nu=0 where vacuum is claimed, predicted a horizon inconsistent with the Schwarzschild exterior after using the same calibrated mass coupling, generated an unavoidable curvature or local-rank singularity at the ordinary event horizon contrary to the current branch, or could not reproduce the exact two-function exterior without coefficient fitting. Those outcomes have not been established by the current historical failures.

Current defensible statement The G Model is exactly Schwarzschild-compatible in the tested static spherical vacuum exterior once its conditional two-function completion is adopted. The horizon is supported as a causal-trapping boundary, not as proven local-fabric destruction. The microscopic origin of the completion, matter interior, collapse dynamics, generic nonspherical strong fields, and physical continuation beyond the Xyz boundary remain open.

11. Source Record Map ID Record Use

S1 G_Model_Two_Function_Schwarzschild_Completion_Principle_Edu Compact Schwarzschild completion, exact cational_11SEP2026.pdf exterior and status register.

ID Record Use

S2 SI_G_Model_Lorentz_Einstein_Schwarzschild_Supporting_Evidenc Full exact Ricci, coordinate, domain and e_2026-08-28.pdf falsification audit.

S3 G_Model_Einstein_Audit_Working_Record_2026-08-28.md Original detailed Einstein/Schwarzschild derivation and exact exterior recheck.

S4 G_Model_Integrated_Research_Manuscript_04SEP2026.pdf Black-hole causal-support and discrete causal-cut development.

S5 G_Model_CURRENT_MANUSCRIPT_55P_MMM_AAA_XYZ_LAW Current controlling horizon claim boundary _UPDATE_12SEP2026.pdf and latest Xyz strong-field extension.

S6 G_Model_Einstein_Tests_Complete_Current_Audit_FINAL_WEBSIT Cross-check against the current E_12SEP2026.pdf Einstein-sector audit.

Authorship and verification boundary G Model conceptual framework, research direction, interpretation, and final scientific responsibility: Gurbuz Dolu. AI assistance was used for record retrieval, symbolic/algebraic re-testing, status reconciliation, and document preparation. This report is an internal theoretical audit for transparent public presentation and independent falsification; it is not peer review, experimental validation, or independent replication.

Minkowski Metric

G MODEL Minkowski Metric and Four-Null Reconstruction Tests Complete Current Audit for the Website Tests Section | Gurbuz Dolu | 12 September 2026 Scope: all retained Minkowski/four-null reconstruction records were reviewed and repeated manuscript restatements were de-duplicated. The report includes the exact theorem-level identities, fresh numerical stress tests, the rejected equal-positive operator route, and the latest Xyz-signature consistency check.

Distinct Minkowski-related checks 17

PASS / exact / structural PASS 15

FAIL / rejected alternative 1

OPEN physical-realization question 1

Current controlled Minkowski verdict The local Minkowski reconstruction is one of the cleanest exact mathematical sectors of the G Model. Four balanced future-directed null branches form a nondegenerate 4D Lorentzian basis; the inverse of their Gram matrix reconstructs the Minkowski metric exactly, reconstructs arbitrary Lorentz inner products and the flat d'Alembertian, preserves causal classification, survives branch relabeling and random Lorentz-frame stress tests, and induces the expected six-bivector Lorentz/Hodge structure. The one attractive alternative that fails - equal-positive branch averaging as the wave operator - remains explicitly rejected.

Scientific boundary These tests establish exact mathematics inside the retained four-null geometry. They do not independently prove that physical spacetime is microscopically built from this causal module, nor do they by themselves prove full all-scale network Lorentz invariance.

1. Complete De-duplicated Minkowski Test Register The register groups closely related algebraic substeps so the count is not inflated by repeated manuscript versions or equivalent reformulations. # Test Core mathematics Status Current interpretation / re-test

1 Individual branch nullness k_a=L_c(1,n_a); k_a^2=0 PASS Exact given |n_a|=1 and eta=diag(-1,1,1,1). Numerical null residual ~2.19e-16 in normalized basis.

2 Tetrahedral balance and sum_a n_a=0; n_a.n_b=-1/3 (a!=b); PASS Exact balanced tetrahedral geometry. fixed cross-products k_a.k_b=-(4/3)L_c^2 All six off-diagonal branch inner products are equal.

3 Nondegeneracy / det K=-(16/(3 sqrt(3)))L_c^4; det PASS Both determinants are nonzero for four-volume G4=-(256/27)L_c^8 L_c>0. The four null branches form a complete 4D basis.

4 Lorentzian Gram signature spec(G4)={-4,4/3,4/3,4/3}L_c^2 PASS Exactly one negative and three positive eigenvalues.

5 Closed-form inverse Gram G4^-1=L_c^-2[(3/4)I4-(1/4)11^T] PASS Exact inverse exists analytically; no fitted metric coefficients are introduced.

6 Exact Minkowski metric K G4^-1 K^T = eta PASS Central theorem identity. Current reconstruction numerical residual 1.67e-16.

# Test Core mathematics Status Current interpretation / re-test

7 Arbitrary-vector D(p)^T G4^-1 D(q)=p^T eta q, PASS 100,000 random vector pairs: max inner-product reconstruction D(p)=K^T p residual 4.44e-15.

8 Causal-cone and sign D(p)^T G4^-1 D(p)=p^T eta p PASS 50,000 unit-scale random null vectors: classification max residual 9.44e-16; 50,000 arbitrary vectors: 0 causal-sign mismatches.

9 Native d'Alembertian (G4^-1)^(ab)D_aD_b = Box PASS Exact flat-background basis identity. reconstruction Curved/discrete use requires connection terms.

10 All 24 branch relabelings K->K Pi; G4->Pi^T G4 Pi; PASS All 4!=24 relabelings tested; max K'G4'^-1K'^T=eta reconstruction residual 1.67e-16.

11 Random Lorentz-frame K' = Lambda K; Lambda^T eta PASS 20,000 random proper Lorentz stress test Lambda=eta transformations with boosts beta<0.95 and rotations; reconstruction residual <=1.90e-14.

12 Inter-cell metric-compatible Omega_yx=K_y Pi_yx K_x^-1; PASS Exact local metric compatibility once transport Omega_yx^T eta Omega_yx=eta existing-link branch correspondence is supplied.

# Test Core mathematics Status Current interpretation / re-test

13 Numerical conditioning kappa_2(G4)=max|lambda|/min|lamb PASS The neutral Gram is not close to da|=3 singular; reconstruction is numerically well-conditioned.

14 Induced six-bivector Lorentz signature(Gamma)=(3,3); J_*^2=-I6; PASS The antisymmetric two-form sector is completion J_*^T Gamma J_*=-Gamma consistent with the reconstructed 4D Lorentz metric.

15 Neutral point inside x=y=z=4/3; PASS Latest generalized Gram geometry generalized Xyz Lorentzian eigenvalues={-4,4/3,4/3,4/3}; places the neutral Minkowski point region triangle margins=4/3 strictly inside the nondegenerate Lorentzian region.

16 Equal-positive branch (1/4)sum_a(k_a.partial)^2 = FAIL It is not Box=-partial_0^2+nabla^2. average as the Minkowski partial_0^2 + (1/3)nabla^2 This alternative is rejected; the wave operator inverse-Gram contraction is the correct reconstruction.

# Test Core mathematics Status Current interpretation / re-test

17 Physical realization of the mathematical reconstruction -> OPEN Exact algebra does not by itself prove local four-null module actual microscopic spacetime that nature realizes the proposed architecture causal module or that the complete network is all-scale Lorentz invariant.

2. Exact Four-Null Geometry 2.1 Balanced tetrahedral null branches k_a = L_c(1,n_a) |n_a|=1 sum_a n_a=0 n_a.n_b=-1/3 (a != b) k_a^2=0 k_a.k_b=-(4/3)L_c^2

The nullness and the equal cross-products are exact consequences of the retained balanced tetrahedral directions.

2.2 Nondegeneracy and four-volume det K = -(16/(3 sqrt(3))) L_c^4 |det K| = (16/(3 sqrt(3))) L_c^4 > 0

det G4 = det(K^T eta K) = -(256/27)L_c^8

This is important: four individually null vectors do not form a degenerate set here. Their oriented 4-volume is nonzero, so they are a complete basis for local 4D Lorentzian geometry.

2.3 Lorentzian signature spec(G4) = {-4, 4/3, 4/3, 4/3} L_c^2

One negative and three positive eigenvalues give the required local 1+3 Lorentzian signature.

3. Exact Minkowski Metric Reconstruction 3.1 Analytic inverse G4^-1 = L_c^-2 [ (3/4) I4 - (1/4) 11^T ]

3.2 Central identity K G4^-1 K^T = eta

Fresh numerical re-test: maximum elementwise residual = 1.67 x 10^-16 in the normalized basis.

No measured gravitational constant, Mercury data, Earth data, or fitted Lorentz coefficient enters this identity. Once the branch geometry is specified, the result is purely algebraic.

3.3 Scale independence Every branch scales as L_c while G4 scales as L_c^2 and G4^-1 as L_c^-2. Therefore the L_c factors cancel in K G4^-1 K^T. The Minkowski verdict is unchanged for any positive reference T_c; the numerical T_c is not needed to make the reconstruction work. K -> L_c K0 G4 -> L_c^2 G40 G4^-1 -> L_c^-2 G40^-1 => K G4^-1 K^T = K0 G40^-1 K0^T = eta

4. Arbitrary Inner Products and the Causal Cone 4.1 General inner-product test D(p) = K^T p D(q) = K^T q D(p)^T G4^-1 D(q) = p^T eta q

This test is stronger than checking the matrix identity once because it probes arbitrary vectors represented only through their four branch projections.

100,000 random vector pairs: maximum absolute discrepancy = 4.44 x 10^-15.

4.2 Null-cone reconstruction For 50,000 random unit-scale null vectors, the reconstructed Lorentz norm remained zero to a maximum residual of 9.44 x 10^-16.

4.3 Timelike/spacelike causal classification For 50,000 additional arbitrary random vectors, the sign of the reconstructed norm was compared with the sign of p^T eta p. There were exactly 0 sign mismatches.

Interpretation: the branch reconstruction preserves not only a matrix form but the causal classification - timelike, null, and spacelike - associated with Minkowski geometry.

5. Differential Operator and the Rejected Alternative 5.1 Correct inverse-Gram d'Alembertian D_a = k_a^mu partial_mu (G4^-1)^(ab) D_a D_b = eta^(mu nu) partial_mu partial_nu = Box

This is exact for a constant branch basis. In a curved or cell-varying background, connection/transporter terms are required; the flat identity should not be overextended.

5.2 Equal-positive averaging route - FAIL (1/4) sum_a (k_a.partial)^2 = partial_0^2 + (1/3)nabla^2 != -partial_0^2 + nabla^2 = Box

Re-test verdict: FAIL remains. No overall scalar can simultaneously repair the time sign and the relative spatial coefficient. This rejected route does not damage the successful inverse-Gram reconstruction; it identifies the wrong contraction rule.

6. Relabeling, Lorentz-Frame, and Inter-Cell Stress Tests 6.1 All 24 tetrahedral branch relabelings K' = K Pi G4' = Pi^T G4 Pi K' G4'^-1 K'^T = eta

Result: all 24 permutations passed; maximum reconstruction residual = 1.67 x 10^-16.

6.2 Random proper Lorentz transformations A broader stress test applied 20,000 random proper Lorentz transformations composed of arbitrary spatial rotations and boosts with beta < 0.95. K' = Lambda K Lambda^T eta Lambda = eta G4' = K'^T eta K'

Maximum numerical residuals: max |Lambda^T eta Lambda - eta| = 5.55 x 10^-15 max |G4' - G4| = 2.49 x 10^-14 max |K' G4'^-1 K'^T - eta| = 1.90 x 10^-14

6.3 Inter-cell metric-compatible transport Omega_yx = K_y Pi_yx K_x^-1 Omega_yx^T eta Omega_yx = eta

This gives exact metric compatibility for matched neighboring local frames. It is the inter-cell continuation of the local Minkowski reconstruction, not a new fitted spacetime metric.

7. Stability and Six-Bivector Completion 7.1 Spectral conditioning kappa_2(G4) = max |lambda_i| / min |lambda_i| = 4/(4/3) = 3

The neutral Gram matrix is comfortably away from singularity. The reconstruction is therefore not a numerical accident based on an almost-degenerate inverse.

7.2 Six-bivector Lorentz structure Sigma_ab = k_a wedge k_b signature(Gamma)=(3,3) J_*^2=-I6 J_*^T Gamma J_*=-Gamma

This is the expected antisymmetric two-form completion of a four-dimensional Lorentzian metric. It is a consistency test that the six pair channels are compatible with the same reconstructed Minkowski structure.

8. Latest Xyz Signature Check 8.1 Neutral Minkowski point x=y=z=4/3 spec(gamma/L_c^2)={-4,4/3,4/3,4/3}

In the generalized centered pair-area description, the neutral point reproduces the same Lorentzian spectrum.

8.2 Distance from the degeneracy boundary Lorentzian region requires: x < y+z, y < x+z, z < x+y

At x=y=z=4/3, every margin = 4/3 > 0. -det(gamma/L_c^2) =(x+y+z)(-x+y+z)(x-y+z)(x+y-z) =256/27 at the neutral point

Current verdict: the neutral Minkowski geometry sits strictly inside the nondegenerate Lorentzian region. Xyz degeneracy is a separate boundary reached only when one triangle inequality saturates; it is not present in the neutral module.

9. Current Numerical Re-test Summary Re-test Current result Verdict

Null branch norm max |k_a^2| = 2.19e-16 PASS

Gram eigenvalues {-4, 1.333333333, 1.333333333, 1.333333333} PASS

Gram determinant -256/27 = -9.481481481... PASS

Four-volume determinant det K = -16/(3sqrt(3)) = -3.079201436... PASS

Metric reconstruction max |K G4^-1 K^T - eta| = 1.67e-16 PASS

100k arbitrary inner products max residual 4.44e-15 PASS

50k unit null vectors max reconstructed norm residual 9.44e-16 PASS

50k arbitrary causal 0 sign mismatches PASS classifications

24 branch permutations max reconstruction residual 1.67e-16 PASS

20k Lorentz transformations max metric reconstruction residual 1.90e-14 PASS

Gram condition number kappa_2 = 3 PASS

Equal-positive wave operator diag(1,1/3,1/3,1/3), not eta FAIL

These are internal algebraic/numerical consistency checks. They do not constitute independent experimental verification of microscopic spacetime structure.

10. T_c and Scale Note L_c = c T_c

No numerical T_c is required for the core Minkowski theorem. All scale factors cancel in the normalized metric reconstruction, so the theorem is exact for any positive L_c inside the retained geometry.

The current calibrated reference T_c may be used when a physical cell length is needed for dimensional examples, but the six historical numerical T_c routes must not be counted as six Minkowski validations. The Minkowski reconstruction is one scale-free geometric theorem with multiple independent algebraic and numerical stress checks.

11. Conclusion for the Website Tests Section Most important PASS results The central PASS is exact metric reconstruction: four balanced null branches form a nondegenerate Lorentzian basis and satisfy K G4^-1 K^T = eta exactly. This result is reinforced independently by the nonzero four-volume, the 1+3 Gram signature, arbitrary-vector inner-product tests, exact causal-cone reconstruction, branch-relabeling invariance, random Lorentz-frame stress tests, and the induced six-bivector Lorentz/Hodge structure.

What matters scientifically is not simply that fifteen entries are marked PASS. The important point is that the same local geometry survives several logically different challenges: algebraic invertibility, signature, metric reconstruction, arbitrary contractions, causal classification, differential-operator reconstruction, symmetry relabeling, Lorentz-frame changes, inter-cell metric compatibility, numerical conditioning, and the latest generalized-Gram boundary test.

How the FAIL result should be interpreted One rejected contraction rule does not invalidate the metric reconstruction. Equal-positive averaging of the four branch directional derivatives gives partial_0^2+(1/3)nabla^2, not the Minkowski d'Alembertian. That route is therefore rejected. The correct signed contraction uses G4^-1 and reconstructs Box exactly.

The most important PASS still required The remaining question is physical rather than algebraic: derive or experimentally constrain whether the four-null local module is actually realized by microscopic spacetime and whether the complete interacting finite-cell network preserves the same no-preferred-frame structure at all wavelengths. The local Minkowski theorem itself is already mathematically closed inside the retained geometry.

Current defensible statement The G Model exactly reconstructs local Minkowski geometry from its retained four-null tetrahedral branch basis. The result is scale-independent, nondegenerate, numerically stable, Lorentz-frame covariant, and consistent with the six-bivector and generalized-Gram structures. This is an exact internal mathematical result, not experimental proof that nature uses the proposed microscopic causal module.

12. Source Record Map ID Record Use in this report

M1 G_Model_Four_Null_Lorentzian_Reconstruction_Theorem_Educa Exact theorem, Gram spectrum, determinant, tional_11SEP2026.pdf inverse, metric and d'Alembertian reconstruction.

M2 G_Model_Lorentz_Tests_Complete_Current_Audit_FINAL_WEBSIT Boost, inter-cell transport, rejected E_12SEP2026.pdf equal-positive operator and current Lorentz status.

M3 G_Model_CURRENT_MANUSCRIPT_55P_MMM_AAA_XYZ_LA Current controlling four-null geometry and W_UPDATE_12SEP2026.pdf generalized Xyz Gram boundary.

M4 G_Model_Gravity_and_Spacetime_Detailed_Visitor_Guide_11SEP Compact public-facing statement of 2026.pdf inverse-Gram Minkowski reconstruction.

M5 Current 12 Sep 2026 numerical stress rerun 100k arbitrary inner products, 50k null-cone tests, 50k causal classifications, 24 permutations, 20k Lorentz transformations.

Authorship and verification boundary G Model conceptual framework, research direction, interpretation, and final scientific responsibility: Gurbuz Dolu. AI assistance was used for record retrieval, exact algebraic formalization, numerical stress testing, status reconciliation, and document preparation. This report is an internal theoretical audit for transparent public presentation and independent falsification; it is not peer review, experimental validation, or independent replication.

Mercury Orbital Test

Mercury Orbital Test is a directly addressable topic in the G Model Tests and Results reading edition. This route opens the complete interactive test article rather than the parent topic grid.

This section contains fourteen test and comparison tabs: Mercury Orbital Test, G Model Gravitational Acceleration, G Model Free Fall, G Model Acceleration vs. Altitude, Lorentz Tests, Einstein Tests, Schwarzschild Tests, Minkowski Metric, Finite-Cell Gravity Test, G Model Gravity, G Model Quantum Gravitation, G Model Finite Quantum, G Model UV-Finite, and G Model Local Quantum.

The Mercury Orbital Test derives T_c = √[5πℏ/(12cK_M)] from the working source modulus K_M = 9.63090777 × 10^42 newtons, then reduces the effective coupling to G_G = c^4/(4πK_M) = 6.674300002387 × 10^-11 cubic meters per kilogram per second squared.

G Model Gravitational Acceleration

G Model Gravitational Acceleration is a directly addressable topic in the G Model Tests and Results reading edition. This route opens the complete interactive test article rather than the parent topic grid.

This section contains fourteen test and comparison tabs: Mercury Orbital Test, G Model Gravitational Acceleration, G Model Free Fall, G Model Acceleration vs. Altitude, Lorentz Tests, Einstein Tests, Schwarzschild Tests, Minkowski Metric, Finite-Cell Gravity Test, G Model Gravity, G Model Quantum Gravitation, G Model Finite Quantum, G Model UV-Finite, and G Model Local Quantum.

The current G Model Gravitational Acceleration topic begins with K_M = 9.63090777 × 10^42 newtons, obtains T_c = 2.186570237920 × 10^-43 seconds, and derives G_G = c^4/(4πK_M) before calculating the ideal spherical-Earth field.

G Model Free Fall

G Model Free Fall is a directly addressable topic in the G Model Tests and Results reading edition. This route opens the complete interactive test article rather than the parent topic grid.

This section contains fourteen test and comparison tabs: Mercury Orbital Test, G Model Gravitational Acceleration, G Model Free Fall, G Model Acceleration vs. Altitude, Lorentz Tests, Einstein Tests, Schwarzschild Tests, Minkowski Metric, Finite-Cell Gravity Test, G Model Gravity, G Model Quantum Gravitation, G Model Finite Quantum, G Model UV-Finite, and G Model Local Quantum.

The current G Model Free Fall topic begins with K_M = 9.63090777 × 10^42 N, obtains T_c = 2.186570237920 × 10^-43 s, derives G_G = c^4/(4πK_M), and applies the exact inverse-square radial formulas to free fall from 1 meter through 10 kilometers.

G Model Acceleration vs. Altitude

G Model Acceleration vs. Altitude is a directly addressable topic in the G Model Tests and Results reading edition. This route opens the complete interactive test article rather than the parent topic grid.

This section contains fourteen test and comparison tabs: Mercury Orbital Test, G Model Gravitational Acceleration, G Model Free Fall, G Model Acceleration vs. Altitude, Lorentz Tests, Einstein Tests, Schwarzschild Tests, Minkowski Metric, Finite-Cell Gravity Test, G Model Gravity, G Model Quantum Gravitation, G Model Finite Quantum, G Model UV-Finite, and G Model Local Quantum.

The current G Model Acceleration vs. Altitude test calculates gravitational acceleration from the surface through 25 kilometers using the K_M → T_c → G_G route. It covers 0 to 6000 meters in 1000-meter steps and adds checks at 12 and 25 kilometers.

Finite-Cell Gravity Test

Finite-Cell Gravity Test is a directly addressable topic in the G Model Tests and Results reading edition. This route opens the complete interactive test article rather than the parent topic grid.

This section contains fourteen test and comparison tabs: Mercury Orbital Test, G Model Gravitational Acceleration, G Model Free Fall, G Model Acceleration vs. Altitude, Lorentz Tests, Einstein Tests, Schwarzschild Tests, Minkowski Metric, Finite-Cell Gravity Test, G Model Gravity, G Model Quantum Gravitation, G Model Finite Quantum, G Model UV-Finite, and G Model Local Quantum.

The revised Finite-Cell Gravity Test is a conditional, numerical prediction falsifiable in principle, derived from the retained spherical-mean expansion and conditional source chain. Its leading departure from the static linear weak-field GR/Poisson benchmark is positive with fixed coefficient +1/20 multiplying (kL_c)^2.

G Model Gravity

G Model Gravity is a directly addressable topic in the G Model Tests and Results reading edition. This route opens the complete interactive test article rather than the parent topic grid.

This section contains fourteen test and comparison tabs: Mercury Orbital Test, G Model Gravitational Acceleration, G Model Free Fall, G Model Acceleration vs. Altitude, Lorentz Tests, Einstein Tests, Schwarzschild Tests, Minkowski Metric, Finite-Cell Gravity Test, G Model Gravity, G Model Quantum Gravitation, G Model Finite Quantum, G Model UV-Finite, and G Model Local Quantum.

The G Model Gravity topic proposes that matter density sources a dimensionless spacetime deformation field through ∇²M = ρc²/K_M. A change δM is assigned potential Φ = c²δM, and acceleration is defined by g = -c²∇δM.

G Model Quantum Gravitation

G Model Quantum Gravitation is a directly addressable topic in the G Model Tests and Results reading edition. This route opens the complete interactive test article rather than the parent topic grid.

This section contains fourteen test and comparison tabs: Mercury Orbital Test, G Model Gravitational Acceleration, G Model Free Fall, G Model Acceleration vs. Altitude, Lorentz Tests, Einstein Tests, Schwarzschild Tests, Minkowski Metric, Finite-Cell Gravity Test, G Model Gravity, G Model Quantum Gravitation, G Model Finite Quantum, G Model UV-Finite, and G Model Local Quantum.

The G Model Quantum Gravitation topic proposes a minimum length L_c = cT_c, shortest wavelength λ ≥ 2L_c, and maximum wave number k_max approximately π/L_c. Momentum-space modes are restricted to |k| ≤ k_max.

G Model Finite Quantum

G Model Finite Quantum is a directly addressable topic in the G Model Tests and Results reading edition. This route opens the complete interactive test article rather than the parent topic grid.

This section contains fourteen test and comparison tabs: Mercury Orbital Test, G Model Gravitational Acceleration, G Model Free Fall, G Model Acceleration vs. Altitude, Lorentz Tests, Einstein Tests, Schwarzschild Tests, Minkowski Metric, Finite-Cell Gravity Test, G Model Gravity, G Model Quantum Gravitation, G Model Finite Quantum, G Model UV-Finite, and G Model Local Quantum.

The current G Model Finite Quantum topic obtains T_c from the working source modulus route T_c = sqrt[5πℏ/(12cK_M)], using K_M = 9.63090777 × 10^42 newtons, and then sets L_c = cT_c. Measured Newtonian G is not inserted into this calculation.

G Model UV-Finite

G Model UV-Finite is a directly addressable topic in the G Model Tests and Results reading edition. This route opens the complete interactive test article rather than the parent topic grid.

This section contains fourteen test and comparison tabs: Mercury Orbital Test, G Model Gravitational Acceleration, G Model Free Fall, G Model Acceleration vs. Altitude, Lorentz Tests, Einstein Tests, Schwarzschild Tests, Minkowski Metric, Finite-Cell Gravity Test, G Model Gravity, G Model Quantum Gravitation, G Model Finite Quantum, G Model UV-Finite, and G Model Local Quantum.

The G Model UV-Finite topic presents the same minimum-length and bounded-momentum construction as a direct ultraviolet-finiteness argument: L_c = cT_c, λ ≥ 2L_c, k_max approximately π/L_c, and loop integrations restricted to |k| ≤ k_max.

G Model Local Quantum

G Model Local Quantum is a directly addressable topic in the G Model Tests and Results reading edition. This route opens the complete interactive test article rather than the parent topic grid.

This section contains fourteen test and comparison tabs: Mercury Orbital Test, G Model Gravitational Acceleration, G Model Free Fall, G Model Acceleration vs. Altitude, Lorentz Tests, Einstein Tests, Schwarzschild Tests, Minkowski Metric, Finite-Cell Gravity Test, G Model Gravity, G Model Quantum Gravitation, G Model Finite Quantum, G Model UV-Finite, and G Model Local Quantum.

The G Model Local Quantum topic proposes minimum elements L_c = cT_c, locally accumulated matter phase φ_m = (mc²T_c/ℏ)e^-M, reciprocal time sectors satisfying T_+T_- = T_c², and nearest-neighbor amplitude propagation.