Pillar 6

Pillar 6 — Gauge-group emergence

Tier: T4 — STRONG EVIDENCE (Math406 Pathway B SO(4) identification REFUTED at the cubic algebraic level by Math408: 6-band decomposes as A_1g ⊕ E_g ⊕ T_2g, NOT T_1u ⊕ T_1g; only one 3-dim irrep present where adj(SO(4)) requires two independent 3-dim summands. Math408 contributes (i) closure of objection O1 at T7 group-theoretic level; (ii) T3 PROOF SKETCH of Stueckelberg dynamical bridge candidate (objection O2); (iii) T2 CONJECTURE of fluctuation-induced SO(4) emergence with pre-registered falsification gates F1-F4 (objection O3 conditionally resolved via LRSM Higgs cascade). Net Pillar 6: T4 ⊕ T3 ⊕ T2 hybrid; no T5/T6 promotion claimed.)
Epistemic category: C2 — Secondary emergence (TECT-internal, standard routes EXHAUSTED 2026-05-26)
Last sync: 2026-06-05

TECT framework positioning (binding 2026-05-26): TECT is an emergent vacuum-and-gravity framework (7 C1 pillars at T6/T7) with an unresolved gauge-sector completion (Pillar 6 standard routes EXHAUSTED) and phenomenological dark-matter/quantum-origin pillars open. TOE-level claim SUSPENDED until non-standard rescue independently derived (per Math411-AddB §10).

Statement

Emergence of the SM gauge group $SU(3)\times SU(2)\times U(1)$ from BCC condensate Wilsonian RG.

Status semantics

This pillar is currently classified at tier T4 (STRONG EVIDENCE (Math406 Pathway B SO(4) identification REFUTED at the cubic algebraic level by Math408: 6-band decomposes as A_1g ⊕ E_g ⊕ T_2g, NOT T_1u ⊕ T_1g; only one 3-dim irrep present where adj(SO(4)) requires two independent 3-dim summands. Math408 contributes (i) closure of objection O1 at T7 group-theoretic level; (ii) T3 PROOF SKETCH of Stueckelberg dynamical bridge candidate (objection O2); (iii) T2 CONJECTURE of fluctuation-induced SO(4) emergence with pre-registered falsification gates F1-F4 (objection O3 conditionally resolved via LRSM Higgs cascade). Net Pillar 6: T4 ⊕ T3 ⊕ T2 hybrid; no T5/T6 promotion claimed.)) per the canonical TOE-FACT-SHEET.

Conditional on:

• Pillar-4 SO(10) emergence (gates GAP-2 + GAP-3 simultaneously)
• Reading H reinterpretation (post-Math400-AddE/AddF, 2026-05-11): in the fluctuation-stabilized disordered vacuum (Reading H), no spontaneously broken continuous symmetry — exact Goldstones replaced by quasi-Goldstones at the q_0 shell with mass sqrt(r_R) ~ 0.65. Math6 spectroscopic match programme requires reinterpretation as small-mass-mode prediction (Math403 queued).
• Math406 SO(4) Pathway B identification (post-Math404, 2026-05-12): the BCC-channel lowest-6 Hessian eigenvalue band (Math400-AddF) decomposes as adj(SO(4)) = (3,1)+(1,3) under SU(2)_L × SU(2)_R, with O_h breaking giving T_1u + T_1g triplet split (m_T1g/m_T1u = 1.18). TECT-scale gauge bosons at ~2-3 × 10^17 GeV; cascade to SM via Left-Right Symmetric Model intermediate breaking at ~10^10-10^14 GeV. T5 CLOSED@Pathway-B-structural CONDITIONAL on (i) eigenvector decomposition verification, (ii) explicit LRSM Higgs cascade, (iii) Math382-AddA multi-channel SU(3) consistency.
• ADVERSARIAL AUDIT ROLLBACK 2026-05-15 (operator hostile review of Math406): Pillar 6 T5 promotion ROLLED BACK to T4 STRONG EVIDENCE. Three concrete objections accepted: (i) Math406 used eigenvalue degeneracy pattern (1+2+2+1) as proxy for representation decomposition; T_1u + T_1g identification requires direct O_h irreducible-rep eigenvector test (NOT performed at time of Math406 promotion). (ii) Scalar Hessian normal modes → gauge-boson adjoint identification requires explicit dynamical/algebraic bridge (mode → connection → gauge generator); current Math406 inserts representation-counting argument without dynamical content. (iii) Math403 direct SM spectroscopic mismatch (TECT band factor 1.25 vs SM hierarchy factor 50-80000) is NOT resolved by Pathway B "structural" closure — only deferred to LRSM cascade with no explicit calculation. Math406 reclassified as PROMISING HYPOTHESIS (not closure). Pillar 6 RETAINS T4 STRONG EVIDENCE pending Math408 (Math406-AddA: explicit eigenvector O_h decomposition + LRSM Higgs cascade calculation).
• Math408 outcomes (2026-05-15): (i) Objection O1 CLOSED at T7 — explicit O_h character calculation gives 6-band = A_1g ⊕ E_g ⊕ T_2g (not the T_1u ⊕ T_1g implicit in Math406). The literal SO(4) adjoint identification is REFUTED at the cubic algebraic level. (ii) Objection O2 partially addressed at T3 — Stueckelberg embedding A^a_μ := ∂_μ δφ^a / m_* identified as candidate dynamical bridge; two open technical items B1 (Maxwell kinetic-term coefficient matching) and B2 (non-abelian structure-constant matching) deferred to Math410. (iii) Objection O3 conditionally resolved at T4 STRONG EVIDENCE — TECT supplies parent gauge group G_parent = SU(2)L × SU(2)R × U(1)(B-L) at M* = M_Pl; LRSM Higgs cascade with v_R ~ 10^14-10^16 GeV and v_EW = 246 GeV gives W_L, Z at EW scale as outputs. Pre-registered falsification gates: F1 (one-loop β_cubic sign in TECT regime, λ<0); F2 (residual shift symmetry of δφ^a at BCC orbit); F3 (Maxwell kinetic-term positivity); F4 (Jacobi identity for matched f^abc structure constants).
• Math410 outcomes (2026-05-26): Pillar 6 gates F1+F2 + T_{2u} shell search. (F1) 1-loop beta_cubic at TECT canonical lambda_4 < 0 (Reading H): explicit shell-mode integral gives A^* = |lambda_4| q_^2 / (48 pi r_R^{3/2}) ~ +0.021 M_Pl^2 (band [+0.007, +0.062]) — POSITIVE sign means cubic anisotropy is RELEVANT in IR (O_h symmetry retained), NOT isotropy emergent. F1 FAIL at 1-loop heuristic for Pathway B isotropy. (F2) Residual shift symmetry of T_{2g} triplet: m_{T_2g}^2 = r_R * delta_{T_2g} = 0.42 * 1.18 = 0.84 M_Pl^2 (Planck-scale mass) BREAKS shift at leading order. Stueckelberg restoration via Math408 §3 mechanism is CIRCULAR (depends on SO(4) it should enable). F2 PARTIAL. NEW NEGATIVE FINDING (not in Math408): T_{2u} parity-conjugate irrep is STRUCTURALLY ABSENT from BCC shells 1-3 (shell 1: A_1g+E_g+T_2g; shell 2: A_1g+T_1u+A_2u+T_2g; shell 3: A_1g+E_g+T_1u). T_{2u} first appears at shell 8 (12 vectors at sqrt(6) q_), beyond Brazovskii cutoff — structurally blocked. Three follow-ups: Math410-AddA (exact-RG for beta_cubic in lambda<0 regime), Math410-AddB (reformulate Pathway B without T_{2u} doubling, e.g., adopt emergent Aharony-class doubling), Math411 (independent Stueckelberg field origin decoupling from SO(4)). Pillar 6 tier UNCHANGED at T4 STRONG EVIDENCE. Verified by Math410_beta_cubic_T2g_basis.py 8/8 self-test asserts PASS.
• Math410-AddA outcomes (2026-05-26, gate F1 CONFIRMED NEGATIVE at exact-RG): Wetterich functional-RG calculation of cubic anisotropy fixed-point in λ_4 < 0 Brazovskii regime (Reading H). A*exact = +0.0177 M_Pl^2 (band [+0.015, +0.020] across Litim/sharp/exponential/polynomial regulators) — within 16% of Math410 1-loop value +0.021. Sign is POSITIVE (cubic anisotropy RELEVANT in IR, O_h locked) — opposite to standard Aharony 1977 λ_4 > 0 result (cubic IRRELEVANT, isotropy emerges). The sign flip is STRUCTURAL: β_A^{int} ∝ -λ_4 with positive-definite Brazovskii integral, universal across all regulators + at all-orders in standard truncations. NGFP scan over (λ_4 ∈ [-1.5, +1], r_R ∈ [0.1, 1.0]) finds NO alternative fixed point with A* < 0 in TECT regime (λ_4 < 0). WF fixed point IR-attractive (stability eigenvalue -1.0). Verdict: F1 CONFIRMED NEGATIVE; Pillar 6 Pathway B SO(4)-emergence via cubic-irrelevance route STRUCTURALLY REFUTED. Combined with T{2u} structural absence (Math410 §4), Pathway B is doubly blocked. Remaining Pillar 6 promotion routes: Math411 (Stueckelberg-direct, single-triplet; confidence 20-25%) or revival of Math157-class direct SO(10) construction. Pillar 6 tier UNCHANGED at T4 STRONG EVIDENCE; T5 within 12 months confidence 25-30% (predominantly via Math411). Verified by Math410_AddA_exact_RG.py 7/7 self-test asserts PASS.
• Math411 outcomes (2026-05-26, Stueckelberg-direct single-triplet route after Pathway B refutation; wording per operator audit 2026-05-26 UTC): gates B1+B2+F3+F4 evaluated. B1 (Maxwell kinetic) PARTIAL — abelian Stueckelberg F_μν vanishes for pure-gradient A_μ^a = ∂_μ δφ^a / m_*; Maxwell form contingent on B2. B2 (non-abelian f^abc) LEADING-ORDER NEGATIVE / globally PARTIAL — TECT-natural cubic anisotropy projects on A_1g (symmetric singlet of T_2g ⊗ T_2g) at leading order; SU(2) Yang-Mills requires ANTISYMMETRIC f^abc = ε^abc corresponding to T_1g projection; T_1g antisymmetric coefficient A^asym_T1g NOT yet computed (Math411-AddA queued). F3 (kinetic positivity) PASS. F4 (Jacobi) contingent on B2 closure. CANONICAL TIER VERDICT: Pillar 6 T4 STRONG EVIDENCE retained; T5 promotion BLOCKED unless Math411-AddA closes B2 via non-zero A^asym_T1g matching ε^abc structure. Combined Pillar 6 status post-Math410+AddA+Math411: Pathway B SO(4) (cubic-irrelevance) STRUCTURALLY REFUTED; Pathway B (T_2u doubling) STRUCTURALLY ABSENT; Stueckelberg-direct PARTIAL pending Math411-AddA; Pathway A (Math157 direct GUT) queued (Math411-AddB); higher-derivative gauge queued (Math411-AddC). Verified by Math411_Stueckelberg_direct.py 6/6 self-test asserts PASS. Decisive next test: Math411-AddA antisymmetric cubic coefficient. If A^asym_T1g = 0 → Stueckelberg-direct effectively REFUTED; if matches ε^abc → B2 reopens for T5 promotion.
• Math411-AddA outcomes (2026-05-26, DECISIVE TEST for Stueckelberg-direct; wording post-operator-audit 2026-05-26 EOD): Stueckelberg-direct T_2g-only gauge-emergence route REFUTED within standard local Lagrangian/Berry-phase framework. PRIMARY structural obstruction (operator-emphasised): Λ² T_2g = T_1g ≠ T_2g — the antisymmetric (Lie-bracket) product of T_2g with itself takes us OUT of T_2g into T_1g, so T_2g alone cannot close as an SU(2) Lie algebra under O_h. This is a representation-theory-level obstruction, prior to any coefficient calculation. SECONDARY obstruction (coefficient-level): antisymmetric A_2g cubic coefficient A^asym_A2g = 0 via TWO independent arguments — (a) pure-cubic ε^abc φ^a φ^b φ^c vanishes identically by bosonic commutativity (and naive derivative form ε^abc φ^a ∂φ^b ∂φ^c also vanishes by b↔c symmetry of Lorentz scalar product); (b) proper Chern-Simons-class derivative-cubic coefficient A^CS_A2g ∝ π_1(M_BCC) = {e} (Math160+Math164 anchor; R-2026-04-26-Math160-BerrySignatureTrivial). Higher-form CS rescue (Math164 §3) and instanton-CS (e^-1500 suppression) structurally available but quantitatively zero or require axiom extension. B2 FINAL VERDICT: NEGATIVE within standard Lagrangian framework. Combined Pillar 6 post-Math410+AddA+Math411+AddA: all three T_2g-based standard gauge-emergence routes EXHAUSTED. ONLY remaining standard promotion route: Math411-AddB (Pathway A Math157 direct SO(10) GUT revival via FULL internal condensate algebra, NOT T_2g-only); confidence 10-15% within 12 months. Pillar 6 tier UNCHANGED at T4 STRONG EVIDENCE. Verified by Math411_AddA_antisymmetric_cubic.py 6/6 self-test asserts PASS. CONFIRMS operator adversarial prediction.
• Math411-AddB outcomes (2026-05-26, Pathway A direct SO(10) GUT revival evaluation; LAST remaining standard Pillar 6 route): STRUCTURALLY INSUFFICIENT at standard TECT BCC channel level. (1) Dimensional gap: SO(10) adjoint = 45 dim; cumulative BCC shell modes — shells 1-3 = 20 (gap 25), shells 1-4 = 44 (1 short, but shell 4 Brazovskii-suppressed by factor 0.026), shells 1-5 = 68 (overshoots). Effective dynamically-active modes ≈ 20 at canonical TECT (r_R=0.42). (2) Lie-algebra closure failure: T_1g (needed for T_2g ⊗ T_2g bracket per Math411-AddA argument) ABSENT from shells 1-3; closure structurally fails on natural BCC channel content. Math157 group-theoretic anomaly cancellation persists independently as a "given SO(10), anomalies cancel" lemma; Pathway A revival does not rescue the dynamical realisation question. Non-standard rescues queued (Math411-AddB-AddA fermion-condensate aux channels; Math411-AddB-AddB holographic/dim-reduction) — both require substantial axiom extension. PILLAR 6 CLOSURE PROGRAMME HONESTLY COMPLETE IN NEGATIVE DIRECTION: Pathway B SO(4) (Math410-AddA) refuted, T_2u doubling (Math410 §4) absent, Stueckelberg-direct (Math411-AddA Λ² T_2g = T_1g) refuted, Pathway A direct SO(10) (this note) structurally insufficient. All four standard promotion routes EXHAUSTED. Pillar 6 effectively pinned at T4 STRONG EVIDENCE indefinitely within canonical TECT framework; T5 promotion requires axiom extension. Verified by Math411_AddB_pathway_A.py 6/6 self-test asserts PASS.
• Math411-AddB outcome (2026-05-26): STRUCTURALLY INSUFFICIENT — Pathway A direct SO(10) revival within standard BCC/Brazovskii channel framework: dimensional gap (44 vs 45 NOT a near-miss) + Lie-algebra closure failure (T_1g absent in shells 1-3)

Key Math notes

• Math157
• Math160
• Math404
• Math405
• Math406
• Math408
• Math410
• Math410-AddA
• Math411
• Math411-AddA
• Math411-AddB

Open questions

See Open-questions for active items tagged with P6. GitHub Issues with the same label provide discussion threads: issues with P6 label.

Source of truth

Canonical pillar tier: Codes/config/pillar_status.json (raw). Per-pillar narrative: Docs/status/TOE-FACT-SHEET.md Stage-1 scorecard (raw). Updates propagate through the cascade pipeline (CLAUDE.md §18) — do not edit this wiki page directly.