ΔB=1,2 forbidden
Single-proton ΔB=1 width is zero while exact Z6 is preserved; ΔB=2 is also forbidden.
Technical record · Row 28 · baryon stability
Proton Decay & Stability
Detailed Row 28 technical record for the exact-Z₆ proton-stability candidate. For the concise current result, start with the Proton Stability overview.
Core result
Exact Z6 forbids isolated ΔB = 1 proton decay
Γp = 0 for an isolated proton while the exact Z6 symmetry is preserved.
Equivalent conditional statement: τp → ∞ for isolated ΔB = 1 proton decay.
Equivalent conditional statement: τp → ∞ for isolated ΔB = 1 proton decay.
The Row 28 selection rule is
ΔB = 3k
Therefore ΔB = 1 and ΔB = 2 processes are forbidden by the candidate discrete symmetry. The first allowed baryon-number violation is ΔB = 3; the gate records the lowest EFT dimension as 15 using the external theorem referenced by the executable result.
Row 28 verdict
PASS
Status: PASS_PROTON_STABILITY__UV_ALPHA_BOUNDARY_OPEN
The structural gate passes all declared anomaly, charge-lift, operator-selection, electroweak-symmetry-breaking, projector/centralizer, and order-3 split checks.
Blocking: false. The row is diagnostic and keeps its unresolved UV boundary explicit.
NP-02 · August 19 prediction
The first allowed baryon-violation signature is trinucleon-scale ΔB=3.
The exact residual Z6 rule ΔB=3k removes ordinary ΔB=1 proton decay and ΔB=2 channels from the active candidate. The first allowed local fermionic EFT order is dimension 15, with coefficient suppression Λ−11.
ΔB=3 · dimension 15
This is the lowest allowed baryon-number-violating local fermionic EFT order in the frozen candidate.
OPEN
No trinucleon lifetime is claimed because the dimension-15 Wilson coefficient / U(1)X-breaking scale remains undetermined.
Leading channel classes
| Initial class | Representative channel | Status |
|---|---|---|
| ppp | ppp → e+ π+ π+ | Allowed class |
| ppn | ppn → e+ π+ | Allowed class |
| pnn | pnn → e+ π0 | Allowed class |
| nnn | nnn → ν̄ π0 | Allowed class |
Claim boundary: this is a channel/order prediction, not a rate prediction. A finite trinucleon lifetime cannot be published until the Wilson coefficient and symmetry-breaking scale are derived independently.
Relation to the earlier v22 lifetime estimate
Why the new result is a stability statement rather than a shorter or longer conventional lifetime
The earlier v22 proton-decay treatment used conventional unified-theory decay channels and heavy-threshold scaling. Row 28 tests a different structural possibility: an exact residual Z6 selection rule that forbids the ΔB = 1 and ΔB = 2 operators themselves.
These are not two competing numerical lifetime estimates. If the exact Z6 construction is realized, the isolated ΔB = 1 width is zero, Γp = 0, rather than a finite lifetime obtained by rescaling the conventional channel.
In the protected two-loop existence diagnostic, the X,Y gauge bosons and color-triplet Higgs HC are projected out while the Σ3 + Σ8 threshold is retained. The executable therefore keeps a hard follow-up requirement: the full affected UTFANSWF gate suite must be rerun after replacing the conventional proton-decay SU(5) thresholds. Until that integration is complete, the exact-Z6 result is a passed structural gate with an explicitly open model-integration boundary.
Parent symmetry construction
Anomaly-free parent U(1)X charge lift
The candidate begins with an integer U(1)X charge assignment for three generations. Reducing these charges modulo 6 produces the residual Z6 selection rule.
| Field | Generation charges under U(1)X | Residual Z6 |
|---|---|---|
| Q | (0, 0, 0) | 0 |
| Uc | (−1, −1, −1) | 5 |
| Dc | (1, 1, 1) | 1 |
| L | (−4, 2, 2), up to generation permutation | 2 |
| Ec | (5, −1, −1) | 5 |
| Nc | (3, −3, −3) | 3 |
| H / Hu | 1 | 1 |
| Hd | Not separately recorded in the parent U(1)X block | 5 (residual Z6 block) |
Minimal lepton-doublet lift: the three equivalent permutations (−4,2,2), (2,−4,2), and (2,2,−4) have minimal norm squared 24 in the tested construction.
Anomaly checks
| Anomaly | Result |
|---|---|
| gravity² × U(1)X | 0 |
| U(1)X³ | 0 |
| U(1)X × U(1)Y² | 0 |
| U(1)X² × U(1)Y | 0 |
| SU(2)² × U(1)X | 0 |
| SU(3)² × U(1)X | 0 |
Residual Z6 operator audit
Required operators remain allowed; dangerous ΔB = 1 and ΔB = 2 operators are rejected
Allowed mod 6
QHuUc, QHdDc, LHdEc, LHuNc, NcNc, and HuHd all sum to 0 mod 6.
Weinberg operator allowed
(LHu)² also has residual charge 0 mod 6.
Dangerous terms forbidden
The tested UcDcDc, QLDc, LLEc, LHu, QQQL, and UcUcDcEc operators all carry nonzero residual charge.
| Operator | Residual charge mod 6 | Gate |
|---|---|---|
| Q Hu Uc | 0 | Allowed |
| Q Hd Dc | 0 | Allowed |
| L Hd Ec | 0 | Allowed |
| L Hu Nc | 0 | Allowed |
| NcNc | 0 | Allowed |
| HuHd | 0 | Allowed |
| (L Hu)² | 0 | Allowed |
| UcDcDc | 1 | Forbidden |
| Q L Dc | 3 | Forbidden |
| L L Ec | 3 | Forbidden |
| L Hu | 3 | Forbidden |
| Q Q Q L | 2 | Forbidden |
| UcUcDcEc | 4 | Forbidden |
EWSB safety: the representative physical charge is qphys = −B + 3L mod 6. The Higgs VEV is neutral modulo hypercharge, and charge ±6 U(1)X-breaking fields preserve the residual Z6 after acquiring VEVs.
Topology and split projector
Three generations and the 3+2 split
The topology block uses Ngen = 3 as the minimal generation count compatible with the candidate discrete anomaly condition.
| rank(P3) | 3 |
| rank(P2) | 2 |
| Tr(H) | 0 |
| Centralizer traceless dimension | 12 |
| Broken adjoint dimension | 12 |
| Tr(Y²) | 5/6 = 0.833333… |
| g1²/gY² | 5/3 = 1.666666… |
Numerical residuals
Projector diagnostics
Unitarity residual: 1.92×10−16
Order-3 residual: 1.13×10−15
|det−1|: 7.95×10−16
MΣ/Λsplit = 0.2828427125
The split is recorded as unique up to orientation within the tested construction.
Protected unification diagnostic
Two-loop crossing remains tight after the proton-protection projection
1.6614×1016 GeV
MU in the protected two-loop existence diagnostic.
4.6992×1015 GeV
MΣ for the retained Σ3 + Σ8 threshold.
0.03523%
Inverse-coupling spread after matching.
α₁⁻¹ = 25.3466335376
α₂⁻¹ = 25.3555645426
α₃⁻¹ = 25.3466334437
The diagnostic projects out the X,Y gauge bosons and the color-triplet Higgs HC, retaining the Σ3 + Σ8 threshold with weight (0,2,3). This is an existence/compatibility diagnostic using pinned v22 low-energy couplings, not a blind prediction of the electroweak inputs.
Independent UV boundary test
The primitive αU−1 = 8π boundary remains open
Row 28 deliberately preserves a separate blind primitive UV test rather than allowing the successful proton-protection structure to conceal an electroweak mismatch.
αU⁻¹ = 8π = 25.1327412287
MU = 1.0528707748×10¹⁶ GeV
MΣ = 2.9779682583×10¹⁵ GeV
predicted αem⁻¹(MZ) = 126.47475786649512
withheld v22 αem⁻¹(MZ) = 127.94049864183329
absolute percent error = 1.145642537662349%
UV test verdict
CLEAN_NEAR_FAIL_1PCT_GATE
The primitive 8π boundary misses the old 1% electroweak holdout gate by about 1.15%.
This miss is retained explicitly. It does not invalidate the internal Z6 operator-selection calculation, but it prevents treating the UV boundary as closed.
Claim boundary and falsification
What is established, what remains open, and what would kill the candidate
| Item | Status |
|---|---|
| Exact-Z6 anomaly/operator/projector gate | PASS |
| ΔB = 1 isolated proton decay | Forbidden while exact Z6 is preserved |
| ΔB = 2 baryon-number violation | Forbidden while exact Z6 is preserved |
| First allowed baryon violation | ΔB = 3; lowest EFT dimension 15 in the referenced theorem |
| Public-v22 instantiation of gauged Z6 and charge-6 Higgsing fields | Open |
| Finite 3+2 projector-Dirac law as a theorem of public v22 | Open; currently private candidate work |
| Primitive αU−1 = 8π boundary | Open; 1.14564% miss against old 1% EW holdout gate |
| Full affected UTFANSWF gate suite after replacing conventional proton-decay SU(5) thresholds | Required |
| ΔB = 3 Wilson coefficient / trinucleon lifetime | Not fixed; no lifetime prediction claimed |
Direct falsification: a confirmed ΔB = 1 or ΔB = 2 process in the regime where the candidate claims exact Z6 protection would directly defeat this exact-Z6 proton-stability construction.
Interpretation: the statement Γp = 0 is not “protons can never decay under any extension or symmetry-breaking history.” It is the narrower technical statement that the tested exact residual Z6 forbids isolated ΔB = 1 proton decay. If the residual symmetry is not realized, or is broken in a way that re-admits the forbidden operators, the zero-width statement no longer follows.
Executable references recorded by Row 28: arXiv:hep-ph/0306003, arXiv:2204.01741, and arXiv:1808.00009. The technical page reports what the Row 28 executable records; independent review of the theorem mapping and full public-v22 instantiation remains part of the open hardening work.