Every Veblen level is a normal function; φ_{α+1} enumerates fixed points of φ_α (BH climb rung 5)

[hyperpolymath]

What & why

Next rung of the target-side climb toward ψ₀(Ω_ω) (BH order-type fidelity, open problem D-2026-06-14). Builds on VeblenBinary (#245 — binary φ + the generic fixed-point engine) and VeblenPhiNormal (ω^^-monotonicity).

VeblenBinary proved continuity (φ-cont) and base-of-tower fixed-point correctness (nextFix-fixed-{≤,≥}). This slice closes the two remaining normal-function properties for every level, by induction on the level α through the generic engine.

New module Ordinal/Brouwer/VeblenBinaryNormal.agda

--safe --without-K, zero postulates, structural recursion:

Theorem	Statement
φ-mono₂	every φ_α is monotone in its argument (a ≤′ b → φ_α a ≤′ φ_α b)
φ-infl	every φ_α is inflationary (β ≤′ φ_α β)
φ-level-fixed-≤/≥	the defining Veblen recurrence: φ_{α+1}(β) is a fixed point of φ_α (φ_α (φ_{α+1} β) ≃ φ_{α+1} β, bi-≤′)

With VeblenBinary.φ-cont, this establishes that every Veblen level φ_α is a normal function (monotone + inflationary + continuous).

The generic engine lemmas are proved once for an arbitrary monotone/continuous g and instantiated at each level (the VeblenPhiNormal reuse pattern, now generic): g-tower-mono, nextFix-mono, deriv-step-≤, deriv-oz-least, deriv-mono, commonStep-mono; deriv-infl; deriv-fixed-≤ (continuous g), deriv-fixed-≥ (monotone + inflationary g); plus ≡→≤′.

Honest scope

This is the normal-function backbone for "Γ₀ is the least diagonal fixed point" (the next slice) and the eventual ordinal-collapsing layer. It does not prove Γ₀ least, does not reach the collapsing layer, and closes no postulate. Order-type fidelity (ψ₀(Ω_ω)) REMAINS OPEN (D-2026-06-14).

Wiring

All.agda import; Smoke.agda pin block (9 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; structural recursion

🤖 Generated with Claude Code

https://claude.ai/code/session_01Rw1RpYXU5Q7rzy2bVofVeB

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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 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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