G Model Minkowski Metric | Complete Current Audit

Complete Minkowski Metric audit

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.