φ₁ is a normal function; next-ε β is the LEAST ε-number above β (BH climb rung 3, slice 3)

[hyperpolymath]

What & why

Target-side climb toward ψ₀(Ω_ω) (BH order-type fidelity, open problem D-2026-06-14). Slice 2 (VeblenPhi) proved every value of φ₁ is an ε-number and that next-ε β is an ε-number above β. This slice upgrades that to the two defining properties of an ε-number enumeration.

New module Ordinal/Brouwer/VeblenPhiNormal.agda

--safe --without-K, zero postulates, structural recursion (no TERMINATING):

Theorem	Statement
ω^^-mono-≤′	ω-exponentiation is monotone (α ≤′ β → ω^^ α ≤′ ω^^ β) — the one missing prerequisite; structural recursion on the _≤′_ shape
next-ε-least	the headline: next-ε β is the least ω^^-closed ordinal strictly above β (for any γ with osuc β ≤′ γ and ω^^ γ ≤′ γ, next-ε β ≤′ γ)
next-ε-mono	next-ε is monotone
φ₁-mono	φ₁ monotone
φ₁-lt-succ	φ₁ strictly increasing at successors
φ₁-strict-mono	φ₁ strictly monotone
φ₁-continuous	φ₁ continuous at limits (definitional refl)
ε₀-least	ε₀ = φ₁ oz is the least φ₁ value (the oz base case)

A normal function is precisely a strictly-monotone, continuous ordinal function — φ₁ is now mechanically one, the standard precondition for taking its fixed points (next Veblen level), the binary φ_α, and ultimately Γ₀.

The next-ε-least proof is clean because the Brouwer olim LUB is definitional: olim f ≤′ γ reduces to ∀ n → f n ≤′ γ, so it's tower-index induction using ω^^-mono-≤′ + the ω^^-closure hypothesis.

Honest scope

Does not reach Γ₀; ψ₀(Ω_ω) sits far above even Γ₀ and additionally needs the ordinal-collapsing layer. Order-type fidelity (ψ₀(Ω_ω)) REMAINS OPEN (D-2026-06-14) — this slice neither reaches Γ₀ nor plugs Fidelity.AtHeight, and closes no postulate.

Design note

Every ≤′-lim / ω^^-mono-≤′ / ·ℕ-mono-≤-left call pins its implicit source explicitly — _≤′_ is a reducing relation, so the unifier cannot infer it from the goal (the lesson the φ₁ slice already recorded).

Wiring

All.agda import; Smoke.agda pin block (8 headlines).

Verification

• agda proofs/agda/All.agda → exit 0
• agda proofs/agda/Smoke.agda → exit 0
• scripts/kernel-guard.sh → PASS
• zero postulates; --safe --without-K throughout

🤖 Generated with Claude Code

https://claude.ai/code/session_01Rw1RpYXU5Q7rzy2bVofVeB

---

Generated by Claude Code

[github-actions]

🔍 Hypatia Security Scan

Findings: 6 issues detected

Severity	Count
🔴 Critical	0
🟠 High	3
🟡 Medium	3

View findings

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    "reason": "Code scanning (Scorecard): TokenPermissionsID -- Token-Permissions -- 17 day(s) old [STALE]",
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    "reason": "Code-scanning alert TokenPermissionsID (high) at .github/workflows/scorecard.yml is 17 days old (threshold: 7 days) -- overdue for remediation",
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    "severity": "high"
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  {
    "reason": "Code-scanning alert hypatia/code_safety/agda_postulate dismissed as 'false positive' -- ensure dismissal is documented and justified",
    "type": "CSA004",
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