Abstract
In v2.2.1 of the Unified Topological Mass Framework (UTMF), a minimal integer-tuple catalog is presented that reproduces the Standard Model hypercharge assignments, preserves SU(2) doublet structure, and yields anomaly-free generations under a lattice family ladder. While the construction is internally consistent, the minimality criterion stated therein does not, by itself, uniquely determine the published catalog. In this addendum we identify the precise source of this non-uniqueness and introduce a single additional axiom—the Canonical Doublet Exclusion Axiom—which renders the catalog unique up to ladder symmetry. No physical predictions, anomaly cancellations, or downstream results are altered. This note completes the determinism of the v2.2.1 catalog construction.
1. Scope and Purpose
This addendum is not a revision of the UTMF charge rules, anomaly analysis, or family-ladder structure. Its sole purpose is to close a logical gap in the derivation of the v2.2.1 minimal one-family catalog by making explicit a constraint that was previously implicit.
All results in v2.2.1 and UTMF-II remain valid as written. The additional axiom introduced here upgrades the catalog construction from a valid minimal choice to a uniquely determined solution.
2. Review of the v2.2.1 Charge Layer
States are labeled by integer tuples , with distinguishing lepton () and quark () sectors.
The hypercharge functional is:
Define:
so that hypercharge depends only on and .
The SU(2)-covariant triality is defined as:
SU(2) doublets are generated by the ladder:
which preserves and hence preserves hypercharge across a doublet.
3. Minimality Criterion and Its Limitation
The v2.2.1 construction specifies a minimality functional:
For a given particle type (fixed and ), the hypercharge equation fixes but leaves infinitely many integer solutions along the line .
Minimization of on this line generically admits multiple degenerate minima. Consequently, the stated minimality criterion does not uniquely select a tuple, nor a unique SU(2) doublet pair.
This degeneracy is especially visible in the quark doublet sector (, ), where the tuple strictly minimizes while satisfying all published constraints, yet does not appear in the catalog.
This demonstrates that an additional exclusion principle is required to recover the published catalog uniquely.
4. Canonical Doublet Exclusion Axiom
We now state the missing axiom explicitly.
Canonical Doublet Exclusion Axiom
For any SU(2) doublet state with fixed hypercharge and sector label , the tuple is forbidden.
Equivalently: SU(2) doublet assignments may not occupy the origin of the -plane.
5. Consequences of the Axiom
Lemma 5.1 (Removal of Dominant Degenerate Minimizer)
Consider an SU(2) doublet sector with fixed hypercharge and sector label . Suppose the hypercharge constraint fixes and the triality constraint requires . Under the minimality functional
the tuple is the unique global minimizer prior to additional constraints.
The Canonical Doublet Exclusion Axiom forbids for SU(2) doublets, thereby removing this dominant minimizer from the admissible set.
As a result, the minimal admissible solutions satisfy and lie on the nearest nontrivial SU(2) ladder orbit.
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Theorem 5.2 (Uniqueness of the One-Family Catalog)
Assume the following data:
- The v2.2.1 hypercharge rule ,
- The SU(2)-covariant triality ,
- SU(2) ladder invariance under ,
- The minimality functional ,
- The Canonical Doublet Exclusion Axiom.
Then, for each Standard Model fermion representation in one generation, the set of integer tuples minimizing subject to the above constraints is unique up to the SU(2) ladder symmetry.
In particular, the v2.2.1 published one-family catalog is the unique minimal solution consistent with these assumptions.
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Corollary 5.3 (Stability of Physical Results)
The Canonical Doublet Exclusion Axiom does not modify:
- hypercharge values,
- representation content,
- anomaly cancellation,
- or the family ladder structure.
All physical and mathematical results derived in v2.2.1 and UTMF-II remain unchanged.
6. Relation to UTMF-II
The construction analyzed here operates entirely within the v2.2.1 charge-layer formalism, which is naturally defined on an extended lattice labeled by . The Canonical Doublet Exclusion Axiom constrains only the admissible SU(2) doublet representatives within this layer.
UTMF-II, by contrast, presents an affine reduction derived from anchor constraints, kernel invariants, and a family-translation generator . Its hypercharge functional, projection maps, and anomaly arguments do not depend on the v2.2.1 minimality construction.
Accordingly:
- The Canonical Doublet Exclusion Axiom does not alter the UTMF-II charge map.
- No modification of the UTMF-II family ladder or anomaly proofs is required.
- The two formalisms remain compatible but distinct in scope, with v2.2.1 supplying a refined catalog selection and UTMF-II supplying an affine reduction.
7. Conclusion
This addendum completes the logical structure of the v2.2.1 catalog construction by making explicit a single, physically natural exclusion principle. With this axiom stated, the catalog becomes uniquely derivable rather than heuristically chosen.
No previously published results are invalidated, and no physical predictions are altered. The Unified Topological Mass Framework remains internally consistent, with the v2.2.1 layer now fully deterministic.
Appendix A. Explicit Excluded Minimizer in the Quark Doublet Sector
We exhibit the precise configuration excluded by the Canonical Doublet Exclusion Axiom.
For the quark doublet (, ), the hypercharge constraint implies:
The triality condition reduces to:
Under the minimality functional , the unique global minimizer is:
for which .
This tuple satisfies the hypercharge rule, triality condition, and minimality requirement, yet does not appear in the v2.2.1 catalog. Its exclusion is therefore logically necessary to recover the published result.
By forbidding for SU(2) doublets, the Canonical Doublet Exclusion Axiom removes this configuration and yields the published quark doublet ladder as the unique minimal admissible solution.
Appendix B. Singlet Stability Under the Exclusion Axiom
The Canonical Doublet Exclusion Axiom applies exclusively to SU(2) doublet assignments. Right-handed singlets are unaffected, since their identification does not rely on the SU(2) ladder structure and does not invoke the exclusion of the origin.
All singlet assignments in the v2.2.1 catalog remain minimal and unchanged under the refined admissibility criteria.
References
- Unified Topological Mass Framework (UTMF), v2.2.1: Erratum and Consolidated Charge Construction.
- Unified Topological Mass Framework (UTMF-II): Complete Proofs and Affine Charge Reduction.
- Standard Model anomaly cancellation and SU(2) ladder structure (canonical references).
Notation and Conventions
All integer tuples are elements of unless otherwise stated. Absolute value bars denote the standard absolute value on . Congruences are taken in . SU(2) doublets are understood as ordered pairs related by the ladder . Right-handed fields are treated as singlets under SU(2).
Hypercharge and triality are defined exactly as stated in Section 2. Electric charge is given by with for doublet components and for singlets.
Acknowledgments
The author acknowledges helpful internal consistency checks and discussions that clarified the determinism requirements of the v2.2.1 catalog construction.
Data Availability
No external datasets were used in this work. All results follow from the stated algebraic definitions and integer-lattice constructions.
Author Contributions
The author carried out the analysis, identified the missing axiom, proved the uniqueness results, and prepared the manuscript.
Competing Interests
The author declares no competing interests.
License
This work is released under the same license as the Unified Topological Mass Framework v2.2.1 publications.