First-argument monotonicity; Γ₀ ≤′ φ_Γ₀(0) (BH climb rung 6)

[hyperpolymath]

What & why

Next rung toward ψ₀(Ω_ω) (BH order-type fidelity, open problem D-2026-06-14). Builds on VeblenBinary + VeblenBinaryNormal (#245/#246 — every level a normal function, the Veblen recurrence, the generic fixed-point engine).

The diagonal Γ₀ was defined in rung 4 with only ≤′-upper-bound sanity. Showing it is a fixed point of the diagonal D(α) = φ_α(0) needs monotonicity of φ in its first argument — this rung proves the two forms the diagonal needs and uses them to close one direction of the Γ₀ fixed point.

New module Ordinal/Brouwer/VeblenBinaryMono.agda

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

Theorem	Statement
commonStep-absorb	a fixed point of commonStep F is a fixed point of every family member
commonStep-cont	commonStep F is continuous when every member is
φ-mono₁-step	adjacent levels ordered: φ_α x ≤′ φ_{α+1} x
φ-mono₁-into-lim	a level below a limit is dominated: φ_{h m} x ≤′ φ_{olim h} x
Γ₀-prefixed	Γ₀ ≤′ φ_Γ₀(0) — Γ₀ is a pre-fixed point of the diagonal

Both monotonicity results rest on one mechanised idea: φ_β(x) is a fixed point of every lower level φ_α, so φ_α x ≤′ φ_α (φ_β x) = φ_β x. For successors the absorbing fixed point is the Veblen recurrence (φ-level-fixed); for limits it comes from commonStep correctness + deriv-fixed-≤.

Honest scope

This closes only the ≤′ direction Γ₀ ≤′ φ_Γ₀(0). The reverse φ_Γ₀(0) ≤′ Γ₀ (closure from above) and full "Γ₀ is the least diagonal fixed point" need general first-argument monotonicity / common-fixed-point-from-above and remain OPEN — the next slice. ψ₀(Ω_ω) sits far above Γ₀ behind the collapsing layer; order-type fidelity REMAINS OPEN (D-2026-06-14). No postulate closed.

Wiring

All.agda import; Smoke.agda pin block (5 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

🤖 Generated with Claude Code

https://claude.ai/code/session_01Rw1RpYXU5Q7rzy2bVofVeB

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Generated by Claude Code

[github-actions]

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