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Jusang Lee edited this page Jun 5, 2026 · 12 revisions

Pillar 5 — Topological charge quantisation

Tier: T6 — PROVED CONDITIONAL
Epistemic category: C1 — Core emergence (TECT-internal, rigorously derived)
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

BCC-channel topological defect (Reading H: topological defects of the dominant BCC fluctuation channel) winding number quantisation; topological charge as integer-valued homotopy invariant.

Status semantics

This pillar is currently classified at tier T6 (PROVED CONDITIONAL) per the canonical TOE-FACT-SHEET.

Conditional on:

  • H_Math10-master-theorem: DISCHARGED at publication-grade level per Math421-AddA. Math10 (Paper-05) contains 47+ sections including TECT Master Theorem for Emergent Dirac Fermions (§47) + Generic Nonvanishing of Shell-Protected Dirac Couplings + Single-Channel Genericity + Simultaneous Two-Channel Genericity + Symmetry-Forbidden vs Generic Failure + Dirac Closure Consequence + Spectral Isolation. Publication-grade verification; theorem-grade per-section verification queued as paper-grade refinement.
  • H_Math157-anomaly-cancellation: DISCHARGED rigorously per Math157 (replaces RETRACTED Math148). 6 SM perturbative anomaly coefficients = 0 identically via SO(10) > SU(5) x U(1)_chi > G_SM representation-theoretic traces on spinor multiplet 16. GUT-normalised Y_GUT = sqrt(3/5) Y_SM. Numerical verification via Codes/supplementary/Math157_anomaly_trace_verification.py.
  • H_Reading-H-topological-invariance: DISCHARGED unconditionally per Math421-AddA §3 (inherited from Math420-AddD Atiyah-Singer pattern). Topological structures (Chern classes, AS index, BCC bundle classification) are gauge-bundle invariants; Reading H reframes scalar sector only; topological-charge content preserved.
  • H_AS-index-c_2-determination: DISCHARGED CONDITIONAL on standard QFT bundle-topology axiom per Math421-AddB. c_2(E_16) = 2 * p_1(P_SO(10)) per Slansky 1981 representation theory; with p_1 = 0 (standard QFT axiom: no instanton background on chiral fermion sector; Peskin-Schroeder Ch.19 + Weinberg vol II Ch.22), c_2(E_16) = 0 and ind(D_E^c) = 16 as required for SO(10) spinor matching.
  • H_Math106-intra-sector-stability: DISCHARGED CONDITIONAL on Math400-AddF BCC TRUE LOCAL MIN inheritance per Math106-AddA. 4-sector Z^2 Chern classification: Sector (1,1) (BCC + LRSM-gauge-coupling) is physically realised; sectors (0,0)/(1,0)/(2,1) excluded by canonical TECT parameter analysis. Inheritance combines Math400-AddF (BCC stability at μ^2=+5e-3) + Math408 LRSM coupling (gauge-bundle defect-mediated c_2=1 contribution).

Key Math notes

Open questions

See Open-questions for active items tagged with P5. GitHub Issues with the same label provide discussion threads: issues with P5 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.

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