boundary-theorems

name	boundary-theorems
description	Boundary Theorems BT-1 through BT-4 — formal obstruction results and how to cite them correctly
trigger	{"paths":["proofs/trinity/BoundaryTheorems.v","docs/TECH_TREE.md","paper/**/*.md"]}

Boundary Theorems Reference

What Boundary Theorems Are

Boundary Theorems (BT) are formally proven impossibility results in Coq. They prove that certain direct constructions from H4 geometry do not reproduce Standard Model parameters. They are guideposts that narrow the search space, not tombstones that end exploration.

BT-1 — φ^a π^b e^c Monomial Obstruction

Statement: No monomial of the form φ^a π^b e^c can reproduce Λ (cosmological constant) or Ω_b (baryon density).

File: proofs/trinity/BoundaryTheorems.v
Status: verified (Coq Qed.)
Meaning: Direct algebraic fits from the three fundamental constants alone are insufficient for cosmological parameters. This does not rule out more complex constructions or additional physical input.

BT-2 — NCG σ-Field Obstruction

Statement: No noncommutative geometry (NCG) σ-field can be constructed from H4 root structure alone.

File: proofs/trinity/BoundaryTheorems.v
Status: verified (Coq Qed.)
Meaning: The H4 root system does not directly yield the scalar field structure needed for spectral action models. Track B (Cl(8) / J₃(𝕆)) explores alternative algebraic structures.

BT-3 — 600-Cell Dirac Operator is Vector-Like

Statement: The 600-cell Dirac operator constructed from antipodal symmetry is vector-like (not chiral).

File: proofs/trinity/BoundaryTheorems.v
Status: verified (Coq Qed.)
Meaning: The natural Dirac operator on the 600-cell does not reproduce the chiral structure of the Standard Model fermions. The chirality problem remains open and is documented in derivations/chirality/.

BT-4 — 2I-Equivariant D_F Cannot Reproduce Lepton Mass Ratios

Statement: The 2I-equivariant Dirac operator D_F does not reproduce lepton mass ratios (e, μ, τ).

File: proofs/trinity/BoundaryTheorems.v
Status: verified (Coq Qed.)
Meaning: The simplest equivariant construction fails for lepton masses. This motivates more sophisticated spectral triple constructions or additional symmetry breaking.

Paper-Level BT-5–BT-7

These are documented in paper/CHANGELOG_v1_to_v2.md and ROADMAP_WAVE17_PLUS.md:

• BT-5: E8 plumbing eta discrepancy does not converge to -2 → high_risk_or_falsified
• BT-6: No known string / heterotic / F-theory / orbifold compactification rescues the SM hierarchy from H4 or F4 → high_risk_or_falsified
• BT-7: (Wave 17 addition) Documented obstruction in the spectral action expansion

How to Cite Boundary Theorems

Correct: "BT-2 proves that NCG σ-fields from H4 root structure alone are obstructed, directing exploration toward Cl(8) constructions (Track B)."

Incorrect: "BT-2 kills the NCG approach." — BTs only obstruct specific direct constructions, not entire research programs.

Related Files

• proofs/trinity/BoundaryTheorems.v — Formal Coq proofs
• docs/TECH_TREE.md — Layered status including BT impact
• ROADMAP_WAVE17_PLUS.md — Track A/B/C strategy post-BT
• paper/CHANGELOG_v1_to_v2.md — BT-5, BT-6, BT-7 documentation