Q.1 — Origin of the χ kinetic term

Q: The mechanism for electroweak symmetry breaking hinges on the kinetic term for the internal spinor, χ †(−Dμ Dμ)χ   Since χ is a projection-filtered composite of internal vector displacements, where does this canonical term come from?

A: The internal spinor χ parameterizes the collective orientation of the x-axis vector pair on the internal configuration manifold with coset geometry S² ≃ SU(2)/U(1). The dynamics of this geometric degree of freedom follow from a gauged nonlinear sigma model obtained by gauging the isometries of the morton configuration space.

The universal two-derivative term is 

Lgeo = (f2/4)Tr[(DμU)†(DμU)]   with  U(x) ∈ SU(2).

Expressed in a local CP¹ chart (the spinor coordinate), and after integrating out the heavy radial (“Higgs-like”) mode that enforces the CP¹ constraint, this reduces to the canonical spinor kinetic term

Lχ = (Dμ χ) †(Dμ χ) = χ†(- Dμ Dμ) χ   (up to a total derivative).

Hence Lχ is the leading term in the derivative expansion of the gauged internal geometry, and is not inserted ad hoc. This makes explicit how the master Lagrangian gives the χ kinetic term used in the electroweak construction.

References:

• Standard Model Fermion Sector — §1.3 “Nonlinear sigma model and gauging” (Eqs. (8)–(10)); §1.4 “Electroweak structure from x-vector geometry” (Eq. (11)). DOI: https://doi.org/10.5281/zenodo.16447452

• The Axis Model (foundational) — Appendix M “Master Lagrangian and field definitions” (vector basis Xμ, Zμ). DOI: https://doi.org/10.5281/zenodo.16164597

• Quantum Completion — §1.2 “Construction of the covariant derivative”; §2.2 “BRST quantization” (compatibility of the projected connection). DOI: https://doi.org/10.5281/zenodo.16489361

• Colab: see the projection/coset notebooks attached to the SM-sector and Quantum-Completion Zenodo records.

Q.2 — Universality of W/Z couplings

Q: If W and Z bosons arise from a projection mismatch in the x-axis geometry of an electron-type morton, why do they decay universally to all fermion flavors and not only to those with the same composition?

A: The W/Z are quanta of the emergent gauge connection, not of a particular morton. The local projection mismatch on the S² fiber sources the geometric Noether current of the gauged isometry; the resulting gauge field couples to the total matter current, i.e., the sum over all fermion flavors in the relevant representation:

Jmatt{aμ} =  Σf  ψ̄f γμ Ta ψf 

Thus an electron-sector disturbance can create a W or Z, but the boson itself propagates as a universal gauge excitation and can be absorbed by any current carrying the appropriate SU(2)L × U(1)Y  quantum numbers (e.g., muon or quark currents). This preserves the observed universality of the weak interactions, exactly as in the Standard Model.

References:

• Standard Model Fermion Sector — §1.3 “Nonlinear sigma model and gauging” (universal connection), §1.5–§1.6 (symmetry breaking, charges, Weinberg angle). DOI: https://doi.org/10.5281/zenodo.16447452

• Quantum Completion — §1.2 (covariant derivative from projection), §2–§3 (BRST, anomaly cancellation ensuring correct representations). DOI: https://doi.org/10.5281/zenodo.16489361

• The Axis Model (Foundational) — §4.1.1 “Master Lagrangian components” (basis fields and their projections). DOI: https://doi.org/10.5281/zenodo.16164597

• Colab: see “gauge-projection / overlap” notebooks linked from the SM-sector record.

Q.3 — Relation to the hierarchy problem

Q: The Standard Model Higgs mass suffers from quadratically divergent radiative corrections. Does the Axis Model’s projection mechanism face the same instability?

A: The framework is designed to be technically natural within its EFT window. There is no fundamental Higgs doublet here: electroweak masses arise from the stiffness of the gauged SU(2)/U(1) sigma model and from scalar-projection weights (set by coherence). Radiative corrections renormalize these geometric parameters multiplicatively (logarithmically), rather than adding large quadratic contributions to a fundamental mass term.

The theory is a predictive EFT below the scalar-coherence cutoff
ΛΦ ≈ 10⁵ GeV. Higher-dimensional operators generated by loops are suppressed by powers of 1/ΛΦ . One- and two-loop RG analysis shows stability (λ(μ) > 0) and perturbativity throughout this window, with smooth matching to SMEFT and chiral QCD in the infrared. The separation of scales v ≪ ΛΦ  is therefore technically natural within the EFT; questions of sensitivity above ΛΦ belong to the explicit UV completion, which the present papers do not assume.

References:

•  Quantum Completion — §4.1 “Characteristic energy scales and EFT cutoffs” (ΛΦ, Λq), §4.2 “One-loop corrections and coupling running,” §4.4 “Two-loop RG and positivity,” §4.5 “Precision-experiment consistency.” DOI: https://doi.org/10.5281/zenodo.16489361

•  The Axis Model (foundational)— Appendix F/X summaries (vacuum stability, SMEFT matching in the IR). DOI: https://doi.org/10.5281/zenodo.16164597

• Colab: RG flow / stability notebooks linked from the Quantum-Completion record.

Q.4 — The nature of spacetime and causality in the model

Q: The model treats the metric gµν  as an emergent, composite operator. What are the physical implications? In a scalar-incoherent domain where projection fails, does spacetime “end,” and how is causality defined if the metric itself is a statistical observable?

A: In this framework, spacetime geometry exists only where scalar coherence supports projection. The vierbein is defined by projecting internal displacement fields with the scalar-modulated map PΦμ​[⋅] and then taking expectation values in the coherent subspace; the composite metric operator is

êaμ(x) = ⟨ΨΦ | PΦμ[ v̂a(x)] | ΨΦ ⟩,

ĝμ ν(x) = êaμ êbν ηab 

and the observable geometry is ⟨gμν(x)⟩. When scalar coherence is lost, the projection factor suppresses PΦμ, so the manifold description becomes degenerate rather than terminated: curvature is suppressed and the system is better described as non-geometric, unprojected field noise (the “masz” phase for compact interiors). In such incoherent regions, gravitational signals are silent and a stable long-range causal structure does not emerge. 

Causality remains well-defined at the fundamental level: the underlying internal fields obey canonical (anti)commutators whose Pauli–Jordan functions vanish at spacelike separation, so microcausality is preserved independently of whether a smooth metric has emerged. Within coherent domains, the emergent metric reproduces the usual macroscopic causal cones and the Einstein–Hilbert dynamics derived in the effective action.

References:

• Emergent Gravity — §3 “Projection geometry and emergent vierbein/metric” (definition of PΦμ and gμν); §5 “Emergent spacetime dynamics” (EH action, graviton); §6.1 “Implications” (curvature suppression, gravitational silence/masz); §4.2–4.3 (canonical quantization, commutators). DOI: https://doi.org/10.5281/zenodo.16447059

• The Axis Model (foundational) — Appendix A/M (projection operators; field content). DOI: https://doi.org/10.5281/zenodo.16164597
  
  

Q.5 — Resolution of the cosmological-constant problem

Q: QFT predicts an enormous vacuum energy density. How does the Axis Model prevent that energy from sourcing a correspondingly huge spacetime curvature?

A: The mechanism is projection filtering: only scalar-coherent stress–energy gravitates.
The scalar alignment operator ΠΦ selects the coherent subspace HΦ. The local projection  PΦμ​[⋅] carries an explicit coherence weight that vanishes when phase gradients are large. High-frequency, stochastic vacuum fluctuations are therefore unprojectable and do not couple to the emergent geometric degrees of freedom.

At the Lagrangian level this is reflected by a projection-induced interaction that suppresses displacement dynamics when |Φ| (coherence) is small:

Lint[v, Φ] = – Σi (βi / ΛΦ²) |Φ|² (∂μ vi)(∂μ vi)

As a result, the vast bulk of QFT vacuum energy remains gravitationally inert; only the long-wavelength, coherence-aligned fraction contributes to curvature. The observed small Λ then corresponds to that tiny coherent component on cosmological scales.

References:

• Emergent Gravity — §6 “Predictive consequences” (curvature suppression and gravitational transparency), §7.4 “Falsification tests” (environment-dependent G(Φ), incoherent domains).  DOI: https://doi.org/10.5281/zenodo.16447059

• The Axis Model (foundational) — §4.1 “Lagrangian structure” and Lint term (projection-induced suppression); Appendix A (projection operator). DOI: https://doi.org/10.5281/zenodo.16164597
  
  

Q.6 — Physical intuition behind the fermion mass formula

Q: The predictive mass formula

mₙ ∝ √Nmorton · εΦ · ⟨Φₙ⟩ · ftop(n) · fcurv(ℓ,m) · fmix

is geometrically abstract. What is the physical intuition behind each factor?

A: Mass measures the scalar binding energy required to stabilize a morton composite as a coherent geometric object.

• √Nmorton · εΦ  (composite binding unit).
  The effective Hamiltonian for the mixed-axis morton yields a discrete spectrum; the ground state reproduces the electron scale and higher eigenpairs organize μ and τ. The scalar unit εΦ sets the binding energy scale, while √N_morton captures composite multiplicity and phase-locking.

• ⟨Φₙ⟩ (generation-specific VEV).
  A mode-resolved Landau functional with geometric penalties produces exactly three nonzero VEVs; coherence decay suppresses higher generations.

• ftop(n) (topological strain).
  Winding the scalar field on the internal S² stores elastic energy. Counting and stiffness give ftop(n) = {1, 4(1+αstress ), 9(1+4αstress)}. Intuitively, this is the energetic cost of “twisting” the scalar sheet.

• fcurv(ℓ,m) (curvature coupling).
  Scalar eigenmodes are spherical harmonics Yℓm on S². Their gradient energy and coupling to intrinsic curvature (parameters γᵢ) add generation-dependent contributions.

• fmix (mode overlap and mixing).
  Overlap integrals of scalar eigenmodes generate intergenerational mixing, leading directly to CKM/PMNS structure and CP phases (via Berry curvature).

These contributions combine into the complete predictive formula (with explicit lepton mass-ratio equations and uncertainty propagation) used to determine {αstress, β₂, β₃, γ₁(0), γ₃(0)}.

References:

• Standard Model Fermion Sector — §4 “VEV hierarchy from geometry” (three VEVs); §5 “Complete mass formula” (Eq. (75), ratio Eqs. (101)–(102)); §3.2 Curvature invariants on S² (γᵢ);  §6 “CKM/PMNS from eigenmode overlaps.” DOI: https://doi.org/10.5281/zenodo.16447452

• The Axis Model (foundational) — §4.2 “Morton potential and stability” (effective Hamiltonian; electron benchmark); Appendix C.4 (electron configuration solver). DOI: https://doi.org/10.5281/zenodo.16164597
  
  

Q.7 — Origin of the Master Lagrangian and its fundamental couplings

Q: The framework is built on a single Master Lagrangian with fields Φ, Xμ, Zμ, ψ and interactions (e.g., V(Φ), mass and gradient couplings, Yukawas). What deeper principle selects this Lagrangian? Why these fields and couplings, and not others?

A: Status: open (principled EFT + projection constraints; UV derivation pending).
Within the current program the Master Lagrangian is not arbitrary: it is the minimal, Lorentz-covariant, power-counting–renormalizable set of operators compatible with (i) the coherence-projection geometry, (ii) gauge/BRST consistency, and (iii) empirical constraints.

Concretely, Appendix M collects the canonical form (Eq. 202), with:

• a quartic scalar potential,

• Maxwell-type kinetic terms for the vector fields,

• dimension-4 scalar–vector mass couplings of the form gAΦ · Φ² AμAμ

• a derivative coupling gZ · (∂μΦ) Zμ 

• and a single parity-odd, gauge-invariant dimension-5 operator (gA/M) ΦA Fμν F̃μν for birefringence.
  
  

Fermions couple via standard kinetic and Yukawa terms. These are exactly the operators allowed once locality, dimensionality, and projection symmetries are enforced; adding further renormalizable terms either violates those symmetries or duplicates existing structures by field redefinitions.

Quantum consistency is handled by explicit gauge fixing, a ghost sector, and BRST nilpotency, ensuring unitary evolution in the physical cohomology. The EFT window and loop structure are summarized in the quantum-completion companion.

What remains open is a pre-geometric derivation that uniquely selects this operator basis from the scalar-coherent projection formalism. The papers emphasize that the present architecture is a minimal attractor fixed by coherence and symmetry—internally justified and empirically mirrored—but not yet derived from a deeper microtheory.

A plausible route is to derive derive 𝓛Axis as the unique local functional whose BRST cohomology matches the scalar-coherent projected Hilbert space (see Appendix A of the gravity paper) under constraints of locality, microcausality, and positivity, and whose one-loop projection-filtered effective action yields the Einstein–Hilbert term in coherent domains.

References:

• The Axis Model (foundational) — §4.1 “Foundational Lagrangians and General Field Equations”; §4.1.1 “The Axis Model Lagrangian Components” (Eqs. (55)–(61)); §4.1.4 “Modified Gravitational Field Equations”; Appendix M.7 “Canonical Lagrangian Summary” (Eq. (202)); Appendix X “Quantum Completion and BRST Consistency” (X.1–X.4). DOI: https://doi.org/10.5281/zenodo.16164597.

• Quantum Completion — §2 “Operator Formalism and Hilbert Space”; §2.2–2.3 “BRST Quantization and Gauge‑Fixed Action” (BRST invariance, ghosts); §3 “Anomaly Cancellation via Scalar Bundle Triviality”; §4.1–§4.4 “EFT Window, One‑ and Two‑Loop RG, Positivity/Dispersion constraints.” DOI: https://doi.org/10.5281/zenodo.16489361.

• Quantum Gravitational Extension — §3 “Projection Geometry and Emergent Metric” (composite metric, vierbein); §4.1 “Field Content and Action” (gravitational sector in the effective action). DOI: https://doi.org/10.5281/zenodo.16447059.
  
  

Q.8 — Ab initio calculation of geometric flavor parameters

Q: The fermion-sector predictions use geometric parameters such as αₛₜᵣₑₛₛ and βₙ, fixed today by a small calibration set {me, mµ/me, mτ/me, θC. Can these be computed ab initio from the Master Lagrangian couplings without any mass inputs?

A: Status: partially answered (EFT-level origin from RG; full UV-level ab initio open).
Within the EFT, these geometric parameters are not free: the fermion paper and quantum-completion analysis show they arise as low-energy effective quantities from RG running of the fundamental scalar–vector–fermion couplings (λ, gψΦ, gΦV, …). Explicit relations link the stabilized VEV hierarchy ⟨Φₙ⟩ to (μ², cℓ, cn, σΞ) and thereby to β₂ and β₃ (see Eq. 136). The parameter αstress and the curvature dressings γᵢ appear as subleading pieces in the same flow. This procedure reproduces the observed magnitudes and correlations once evolved from the scalar-coherence cutoff ΛΦ down to the infrared scale μIR. 

The open part is to remove the last phenomenological inputs and compute {αstress, βₙ, γᵢ} directly from UV boundary data — i.e., from a specified UV completion above ΛΦ  and Λq that fixes the bare couplings and projection weights. With those inputs, the coupled RGEs would then determine all IR flavor parameters with no lepton-mass anchors. The fermion paper flags this explicitly as an open problem. A concrete path is:

1. Specify UV initial conditions consistent with BRST symmetry and positivity.

2. Integrate the quantum-completion RGEs down to the EFT window.

3. Feed the resulting (μ², cℓ, cn, σΞ , …) into the stabilized Landau functional to produce ⟨Φₙ⟩ and then the full mass/mixing set via the derived mass formula (Eq. 75).

References:

• Standard Model Fermion Sector — §3.2 “Enhanced Scalar Field Dynamics with Curvature Invariants on S²” (definition and rationale for γᵢ, Eqs. (53)–(55)); §4 “VEV hierarchy from geometry” (exactly three nonzero VEVs; stabilization); §5 “Complete mass formula” (Eq. (75); explicit lepton ratios Eqs. (101)–(102)); §6 “CKM/PMNS from eigenmode overlaps” (rank‑1 and rank‑2 channels, Cabibbo normalization, ∆ℓ=2 uplift of |Vub|); §8 “Parameter determination and predictions” (RG‑linkage of α_stress, β₂,β₃); App. M/N/S (overlaps & Cabibbo normalization; predictions vs data; uncertainty propagation). DOI: https://doi.org/10.5281/zenodo.16447452.

• Quantum Completion — §4.1 “Characteristic energy scales and EFT cutoffs (ΛΦ, Λq)”; §4.2 “One‑loop corrections and coupling running”; §4.4 “Two‑loop RG and positivity.” (These sections provide the loop‑level origin for the low‑energy effective parameters used in the flavor fit.) DOI: https://doi.org/10.5281/zenodo.16489361.

• The Axis Model (foundational) — §4.2 “Morton potential and stability” (effective Hamiltonian & electron benchmark); App. M.7 “Canonical Lagrangian” (couplings used by the fermion sector). DOI: https://doi.org/10.5281/zenodo.16164597.

Q.9 — Dynamical mechanism for cosmological scalar coherence (and ΩΛ​)

Q: The cosmological-constant problem is addressed by projection filtering: only scalar-coherent energy gravitates, so incoherent QFT vacuum energy is inert. What sets the tiny, non-zero cosmological level of coherence (and can the model predict ΩΛ)?

A: Status: open (filtering established; sourcing/magnitude to be derived).
The program makes two firm statements:

1. Geometry and curvature exist only in scalar-coherent domains. The metric is a composite observable ⟨gμν⟩ built from projected internal fields. In decoherent (“masz”) interiors the projection vanishes, curvature is suppressed, and gravitational waves do not propagate.

2. Vacuum energy that is scalar-incoherent does not gravitate. The scalar-coherent projection operator ΠΦ dynamically filters out incoherent field fluctuations. This naturally quenches the enormous QFT vacuum contribution, leaving only the coherent component to source curvature.

These points are formalized in the emergent-gravity construction: GR is recovered in coherent domains, while departures occur only in decoherent regions.

What remains open is a cosmological evolution calculation that predicts the residual, late-time coherence fraction of the vacuum. In practice, this means deriving and solving the Φ field’s coarse-grained FRW evolution with a projection-filtered effective action, computing a large-scale coherence correlator CΦ(L), and mapping that onto an effective vacuum energy density ρΛ ∝ CΦ(L ≈ H0⁻¹).

The gravitational paper already outlines the one-loop, projection-filtered derivation of Geff and highlights that obtaining a closed one-loop expression is a priority. The same formalism would determine the magnitude of the coherent vacuum component. Until that derivation is complete, the Axis Model explains why Λ is small (filtering) but does not yet predict its exact value.

References:

• Quantum Gravitational Extension — §3 “Projection Geometry and Emergent Metric” (composite metric gμν = eaμ ebν ηab; domain of validity requires scalar coherence); §5.3 “Low‑Energy Limit and Effective Gravity” (Einstein–Hilbert term from scalar‑coherent coarse‑graining); Appendix A (A.1–A.3) “Scalar‑Coherent Projection Operator” (alignment/operator formalism that filters incoherent energy). DOI: https://doi.org/10.5281/zenodo.16447059.

• The Axis Model (foundational) — §4.1.4 “Modified Gravitational Field Equations” (role of gZ(∂μΦ)Zμ and coherence gradients); §5.* “Empirical validation roadmap” (observational signatures tied to scalar coherence). DOI: https://doi.org/10.5281/zenodo.16164597.

• Quantum Completion — §4 (EFT window and matching): coherence cutoff ΛΦ and stability needed for cosmological applications. DOI: https://doi.org/10.5281/zenodo.16489361

Q.10 — Masz domain

Q: What is a “masz domain,” and how does it replace a black‑hole singularity?

A: A masz domain is a scalar‑incoherent phase of spacetime in which the projection that makes geometry observable collapses. 

• Emergent geometry. In the Axis framework, the metric gμν is composite: gμν = eaμ ebν ηab. The vierbein eaμ  is obtained by applying a scalar‑coherent projection Pᵃμ[·;Φ] to internal displacement fields; physical geometry exists only insofar as scalar coherence supports this projection. 

• Collapse of projection. Under extreme conditions (e.g., deep inside an event horizon), scalar coherence is suppressed and the scalar‑alignment operator  ΠΦ → 0. Because eaμ = Pᵃμ[·;Φ], the composite frame loses rank and the manifold description degenerates. 

• Degenerate, not singular. The masz limit is gμν → 0 with det g → 0 (no well‑defined light cones), rather than curvature → ∞. Underlying fields and energy still exist, but become unprojectable into observable curvature (a non‑geometric, “masz” phase). 

• Topological boundary. The boundary of a compact masz region is a degeneration locus Σmasz where the rank of the vierbein drops and the geometric volume element √|det g| vanishes. Outside Σmasz, standard macroscopic causality and Einstein–Hilbert dynamics hold; inside, “gravitational silence” obtains.

• Vacuum energy implication. Projection filtering explains why most vacuum fluctuations are gravitationally inert—only the scalar‑coherent fraction projects into curvature (small effective Λ).

References:

• Quantum Gravitational Extension — §3 “Projection Geometry and Emergent Metric” (composite metric gμν = eaμ ebν ηab; domain of validity requires scalar coherence); §5.3 “Low‑Energy Limit and Effective Gravity” (Einstein–Hilbert term from scalar‑coherent coarse‑graining); §6.1 “Implications: black‑hole interiors (masz)” and gravitational‑vacuum structure. DOI: https://doi.org/10.5281/zenodo.16500059

• The Axis Model (foundational) — §3–§4 (projection operator, emergent vierbein, coherence filter; role of ΠΦ); §4.1.4 “Modified Gravitational Field Equations” (limits of applicability tied to scalar coherence). DOI: https://doi.org/10.5281/zenodo.16164597

• Quantum Completion — §4 (EFT window and matching: coherence cutoff ΛΦ and when EH dynamics are valid); Appendix/notes on projection filtering and decoupling of incoherent energy from curvature. DOI: https://doi.org/10.5281/zenodo.16489361

• Standard Model (fermion sector from internal tri‑vector geometry) — background on the internal tri‑vector structure and projection language used for matter sectors; supports consistency of the projection formalism used above. DOI: https://doi.org/10.5281/zenodo.16447452

Q.11 — Black Hole Singularities and Metric Degeneracy 

Q. General Relativity predicts a physical singularity at the center of a black hole. The Axis Model claims to resolve this with a "masz domain" where the emergent spacetime undergoes a "collapse of projection structure." What is this mechanism, and can this collapse be demonstrated concretely?  

A. The metric gμν is not fundamental but emerges from scalar-coherent projections of internal fields. A "masz domain" is a region where this scalar coherence is lost, resolving the singularity by causing the spacetime geometry itself to dissolve. The symbolic demo below constructs gμν from a coherence‑filtered vierbein. As the coherence parameter η → 0 (masz limit), every component of gμν collapses and det g → 0. 

References:

• Quantum Gravitational Extension — §3 “Projection Geometry and Emergent Metric” (composite metric gμν = eaμ ebν ηab​; scalar‑coherent projection operator ΠΦ​ and domain of validity); §5.3 “Low‑Energy Limit and Effective Gravity” (Einstein–Hilbert term from scalar‑coherent coarse‑graining); §6 “Masz interiors” (degenerate metric, det g → 0, vanishing volume element √|det g|, “gravitational silence”). DOI: https://doi.org/10.5281/zenodo.16500059

• The Axis Model (foundational) — §3–§4 (projection formalism: emergent vierbein built by scalar‑coherent projection; conditions for coherence); §4.1.4 “Modified Gravitational Field Equations” (role of gZ(∂μΦ)Zμ  and coherence gradients; limits of applicability). DOI: https://doi.org/10.5281/zenodo.16164597

• Quantum Completion — §4 (EFT window and matching: coherence cutoff ΛΦ ​; when Einstein–Hilbert dynamics are valid as a scalar‑coherent coarse‑grained limit; decoupling of incoherent energy from curvature). DOI: https://doi.org/10.5281/zenodo.16489361

• Standard Model (fermion sector from internal tri‑vector geometry) — background on the internal tri‑vector/ projection language reused across sectors, ensuring consistency of the projection operator notation referenced above. DOI: https://doi.org/10.5281/zenodo.16447452

Note on the demo: the SymPy snippet is an illustrative construction that mirrors the papers’ projection logic: a coherence weight η filters all vierbein legs; as η ⁣→ ⁣0 (masz limit), every component of gμν​ and det g collapses (√|det g| ⁣→ ⁣0). The formal ingredients and assumptions for that toy model are grounded in the sections cited above.

Q.12 — Dynamical Origin of the Three‑Generation Structure 

Q. The Fermion‑Sector paper claims that the existence of exactly three fermion generations is a dynamical consequence of minimizing a stabilized Landau functional, rather than a postulate. How robust is this result—does it require fine‑tuning? Can the “three‑generation window” be demonstrated quantitatively?

A. Within the Axis Model suite, the number of condensed internal modes is determined by the signs of the quadratic coefficients in the stabilized Landau functional (see the Standard Model/Fermion‑Sector paper for the precise definitions and conditions). With a positive quartic self‑coupling and non‑negative inter‑mode couplings, the functional is strictly convex, so the global minimum is unique and free of local‑minimum artifacts. Because the winding‑suppression term raises the quadratic coefficients monotonically with mode index, there exists a finite interval of the control parameter in which the first three coefficients are negative while all higher ones are positive. Throughout that interval, the unique minimum contains exactly three non‑zero condensates and all others vanish. This yields an open “three‑generation window,” not a single fine‑tuned point. The conditions and logic are laid out in the papers cited below.

References:

• Standard Model (Fermion Sector). Stabilized Landau functional; winding‑suppression structure; support pattern (“exactly three” when the first three quadratic terms are negative and higher ones positive); convexity/uniqueness conditions. DOI: https://doi.org/10.5281/zenodo.16447452

• Axis Model (Foundational). Projection formalism and scalar‑coherent sector that underpins the stabilized construction; cross‑reference to the fermion‑sector derivation. §3–§4; §4.1.4 for the projection/coherence framework  DOI: https://doi.org/10.5281/zenodo.16164597

• Quantum Completion. §4 for EFT window and stability context for the scalar sector; justification that the stabilized functional and parameter regime are radiatively natural in the low‑energy limit. DOI: https://doi.org/10.5281/zenodo.16489361

• Quantum Gravitational Extension. §3 for projection/coherence context emergent‑metric sector; not required for the Landau minimization proof but included for completeness. DOI: https://doi.org/10.5281/zenodo.16500059

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