# Parameter-Free RELATION Report

- Grade: **RELATION** (parameter-free; measured masses + quark content only)
- PDG version: `PDG 2024 (S. Navas et al., Phys. Rev. D 110, 030001 (2024))`
- Data freeze date: `2024-05-31`
- Overall: **PASS**
- Relations: 6/6 passed (0 failed)

## Quantum-Number Gate

Result: **PASS** (42 checks, 0 failed)

| Particle | Quark content | Field | Derived | Listed | OK |
|---|---|---|---|---|---|
| proton | uud | charge | 1.0 | 1.0 | yes |
| proton | uud | strangeness | 0 | 0 | yes |
| proton | uud | baryon_number | 1.0 | 1.0 | yes |
| neutron | udd | charge | 0.0 | 0.0 | yes |
| neutron | udd | strangeness | 0 | 0 | yes |
| neutron | udd | baryon_number | 1.0 | 1.0 | yes |
| Lambda | uds | charge | 0.0 | 0.0 | yes |
| Lambda | uds | strangeness | -1 | -1 | yes |
| Lambda | uds | baryon_number | 1.0 | 1.0 | yes |
| Sigma+ | uus | charge | 1.0 | 1.0 | yes |
| Sigma+ | uus | strangeness | -1 | -1 | yes |
| Sigma+ | uus | baryon_number | 1.0 | 1.0 | yes |
| Sigma0 | uds | charge | 0.0 | 0.0 | yes |
| Sigma0 | uds | strangeness | -1 | -1 | yes |
| Sigma0 | uds | baryon_number | 1.0 | 1.0 | yes |
| Sigma- | dds | charge | -1.0 | -1.0 | yes |
| Sigma- | dds | strangeness | -1 | -1 | yes |
| Sigma- | dds | baryon_number | 1.0 | 1.0 | yes |
| Xi0 | uss | charge | 0.0 | 0.0 | yes |
| Xi0 | uss | strangeness | -2 | -2 | yes |
| Xi0 | uss | baryon_number | 1.0 | 1.0 | yes |
| Xi- | dss | charge | -1.0 | -1.0 | yes |
| Xi- | dss | strangeness | -2 | -2 | yes |
| Xi- | dss | baryon_number | 1.0 | 1.0 | yes |
| Delta | uuu | charge | 2.0 | 2.0 | yes |
| Delta | uuu | strangeness | 0 | 0 | yes |
| Delta | uuu | baryon_number | 1.0 | 1.0 | yes |
| Sigma* | uus | charge | 1.0 | 1.0 | yes |
| Sigma* | uus | strangeness | -1 | -1 | yes |
| Sigma* | uus | baryon_number | 1.0 | 1.0 | yes |
| Xi* | uss | charge | 0.0 | 0.0 | yes |
| Xi* | uss | strangeness | -2 | -2 | yes |
| Xi* | uss | baryon_number | 1.0 | 1.0 | yes |
| Omega- | sss | charge | -1.0 | -1.0 | yes |
| Omega- | sss | strangeness | -3 | -3 | yes |
| Omega- | sss | baryon_number | 1.0 | 1.0 | yes |
| nucleon:neutron | udd | charge_derivable | 0.0 | (no column) | yes |
| nucleon:proton | uud | charge_derivable | 1.0 | (no column) | yes |
| kaon:K0 | ds | charge_derivable | -0.666667 | (no column) | yes |
| kaon:K+ | us | charge_derivable | 0.333333 | (no column) | yes |
| D_meson:D0 | uc | charge_derivable | 1.333333 | (no column) | yes |
| D_meson:Dplus | dc | charge_derivable | 0.333333 | (no column) | yes |

## Relations

### R1 — Gell-Mann-Okubo baryon octet mass formula (PASS)

- Grade: RELATION
- Formula: `2(m_N + m_Xi) = 3*m_Lambda + m_Sigma`
- Predicted: `4540.202667`
- Observed: `4514.40751`
- Residual (relative): `0.00569768`
- Tolerance `gmo_octet_rel_tol` = `0.01`
  - Rationale: Gell-Mann-Okubo 2(m_N+m_Xi)=3*m_Lambda+m_Sigma holds to ~0.6% in PDG-2024; second-order SU(3) breaking is sub-percent. A >1% violation falsifies the linear-breaking assumption. Source: foundation/01_mass_method_catalog.md method 3.
- Detail: LHS=2(m_N+m_Xi)=4514.4075 MeV; RHS=3 m_Lambda+m_Sigma=4540.2027 MeV; |LHS-RHS|=25.7952 MeV; rel=0.5698% (tol 1.00%)

### R2 — Baryon decuplet equal-spacing rule (PASS)

- Grade: RELATION
- Formula: `m_Sigma* - m_Delta == m_Xi* - m_Sigma* == m_Omega - m_Xi*`
- Predicted: `147.1833`
- Observed: `[151.93, 148.97, 140.65]`
- Residual (max_relative): `0.04438908`
- Tolerance `decuplet_equal_spacing_rel_tol` = `0.1`
  - Rationale: Decuplet equal-spacing steps (Sigma*-Delta, Xi*-Sigma*, Omega-Xi*) agree to ~7% in PDG-2024; the spacing has known curvature in m_s. Steps differing by >10% beyond curvature falsify equal spacing. Source: foundation/01_mass_method_catalog.md method 4.
- Detail: steps = [151.93, 148.97, 140.65] MeV; mean=147.18 MeV; max deviation from mean = 6.53 MeV (4.44%, tol 10%)

### R3 — Isospin-splitting signs consistent with m_d>m_u + EM (PASS)

- Grade: RELATION
- Formula: `sign(m_member_heavier - m_member_lighter) == expected_sign (per multiplet)`
- Predicted: `['positive', 'positive', 'negative']`
- Observed: `['positive', 'positive', 'negative']`
- Residual (sign_match_count): `0 sign mismatch(es)`
- Tolerance `(exact sign match required)` = `0`
  - Rationale: The SIGN of each isospin splitting is the parameter-free RELATION; 0 mismatches are tolerated. Magnitudes are LATTICE-IMPORTED and not tested here.
- Detail: nucleon: neutron(939.565421) - proton(938.272089) = 1.2933 MeV => positive (expected positive) [OK]; kaon: K0(497.611) - K+(493.677) = 3.9340 MeV => positive (expected positive) [OK]; D_meson: D0(1864.84) - Dplus(1869.66) = -4.8200 MeV => negative (expected negative) [OK]

### R4:N_orbital — Regge linearity (N_orbital, M^2 vs J) (PASS)

- Grade: RELATION
- Formula: `M^2 = slope*J + intercept (least squares)`
- Predicted: `{'slope_MeV2_per_unit': 1045232.893, 'intercept_MeV2': 314682.244, 'alpha_prime_GeV-2': 0.9567}`
- Observed: `{'R_squared': 0.998656}`
- Residual (min_r_squared): `0.998656`
- Tolerance `regge_r2_floor` = `0.97`
  - Rationale: M^2 vs J (orbital) and M^2 vs n (radial) are linear to ~5-10% across PDG light-meson and N* towers. We require the least-squares coefficient of determination R^2 >= 0.97 for the linearity RELATION to PASS. Source: foundation/01_mass_method_catalog.md method 6.
- Detail: 3-point fit M^2 vs J: slope=1045232.9 MeV^2/unit (positive), R^2=0.99866 (floor 0.97) => PASS

### R4:pi_radial — Regge linearity (pi_radial, M^2 vs N) (PASS)

- Grade: RELATION
- Formula: `M^2 = slope*N + intercept (least squares)`
- Predicted: `{'slope_MeV2_per_unit': 1628310.053, 'intercept_MeV2': 33549.911, 'alpha_prime_GeV-2': None}`
- Observed: `{'R_squared': 0.999776}`
- Residual (min_r_squared): `0.999776`
- Tolerance `regge_r2_floor` = `0.97`
  - Rationale: M^2 vs J (orbital) and M^2 vs n (radial) are linear to ~5-10% across PDG light-meson and N* towers. We require the least-squares coefficient of determination R^2 >= 0.97 for the linearity RELATION to PASS. Source: foundation/01_mass_method_catalog.md method 6.
- Detail: 3-point fit M^2 vs N: slope=1628310.1 MeV^2/unit (positive), R^2=0.99978 (floor 0.97) => PASS

### R4:rho_orbital — Regge linearity (rho_orbital, M^2 vs J) (PASS)

- Grade: RELATION
- Formula: `M^2 = slope*J + intercept (least squares)`
- Predicted: `{'slope_MeV2_per_unit': 1091857.7, 'intercept_MeV2': -464690.812, 'alpha_prime_GeV-2': 0.9159}`
- Observed: `{'R_squared': 0.999353}`
- Residual (min_r_squared): `0.999353`
- Tolerance `regge_r2_floor` = `0.97`
  - Rationale: M^2 vs J (orbital) and M^2 vs n (radial) are linear to ~5-10% across PDG light-meson and N* towers. We require the least-squares coefficient of determination R^2 >= 0.97 for the linearity RELATION to PASS. Source: foundation/01_mass_method_catalog.md method 6.
- Detail: 4-point fit M^2 vs J: slope=1091857.7 MeV^2/unit (positive), R^2=0.99935 (floor 0.97) => PASS

## Disclaimer

> This suite verifies that every observed particle is quantum-number-consistent with its geometry-derived constituents and that the parameter-free QCD symmetry relations hold against PDG-2024; it does NOT compute or claim absolute hadron masses from geometry.