proof(ordinal): φ₁ — the ε-number enumeration (BH climb rung 3, slice 2) (#238)

## Summary

New `--safe --without-K` module `Ordinal.Brouwer.VeblenPhi` (zero
postulates):
the Veblen **φ₁** level — the enumeration of the **ε-numbers** (fixed
points of
`ω^^`) — building on rung 3.1's inflationary law `α ≤′ ω^^ α`.

- `next-ε β = olim (tower-from (osuc β))` — the least ε-number
**strictly above β**.
`ω^^-next-ε-≤` / `next-ε-≤-ω^^` prove it a fixed point of `ω^^`
(bi-`≤′`); the
`≥` direction's base index uses `ω^^-infl (osuc β)`. `β<next-ε : osuc β
≤′ next-ε β`.
- `φ₁ : Ord → Ord` — `φ₁ 0 = ε₀`, `φ₁ (α+1) = next-ε (φ₁ α)`, `φ₁ (lim
f) = sup`.
- **`φ₁-ε-number`** — every value of φ₁ is an ε-number (bi-`≤′`): `oz`
reuses
`ε₀-ε-number`, `osuc` is `next-ε`'s fixed-point pair (no IH needed), the
limit
  case lifts the per-branch IH through `olim` via `f-in-lim′`.

Wired into `All.agda`; headlines pinned in `Smoke.agda`.

## Honest scope

φ₁ is the **first transfinite Veblen level** (φ₀ = `ω^^`). Γ₀ needs the
full
binary φ_α hierarchy + its diagonal fixed point; ψ₀(Ω_ω) sits far above
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`. bi-`≤′` (not `≡`) throughout, since Brouwer `olim`s
of
different ℕ-indexings of one supremum are not definitionally equal.

## Verification

- `agda proofs/agda/All.agda` exits 0 (194 modules) under `--safe
--without-K`.
- `tools/check-guardrails.sh` + `scripts/kernel-guard.sh` PASS; zero
postulates.

## Branch note

This branch also carries prior `pr-233` integration commits and an
RSR-Bronze
re-add that is content-identical to `main` (merge-neutral). The net-new
proof
deliverable here is the φ₁ module + its `All.agda` / `Smoke.agda`
wiring.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

https://claude.ai/code/session_01GJatEm2TVFSTBEkKXmserJ

---
_Generated by [Claude
Code](https://claude.ai/code/session_01GJatEm2TVFSTBEkKXmserJ)_

---------

Co-authored-by: Claude <noreply@anthropic.com>

Author: hyperpolymath

proofs/agda/All.agda

@@ -119,6 +119,7 @@ open import Ordinal.Brouwer.Arithmetic
 open import Ordinal.Brouwer.Phase13
 open import Ordinal.Brouwer.OmegaPow
 open import Ordinal.Brouwer.OrdinalExp
+open import Ordinal.Brouwer.VeblenPhi
 open import Ordinal.Brouwer.StrictLeftMonoRefuted
 open import Ordinal.Brouwer.AdditivePrincipalGenericRefuted
 open import Ordinal.Buchholz.Syntax

proofs/agda/Ordinal/Brouwer/VeblenPhi.agda

@@ -0,0 +1,95 @@
+{-# OPTIONS --safe --without-K #-}
+-- SPDX-License-Identifier: MPL-2.0
+-- SPDX-FileCopyrightText: 2025-2026 Jonathan D.A. Jewell <j.d.a.jewell@open.ac.uk>
+
+-- Veblen φ-hierarchy over Brouwer ordinals — RUNG 3 (slice 2) of the
+-- target-side climb toward ψ₀(Ω_ω) (BH order-type fidelity, open problem
+-- D-2026-06-14). 2026-06-18.
+--
+-- ## This slice: φ₁ — the enumeration of ε-numbers
+--
+-- The ε-numbers are the fixed points of ω-exponentiation (`ω^^ ε ≃ ε`).
+-- φ₁ enumerates them in increasing order:
+--
+--   * `next-ε β`   — the least ε-number STRICTLY above β: the supremum of
+--                    the ω^^-tower started at `osuc β`.  It is a fixed
+--                    point by the same shifted-tower argument as ε₀, but
+--                    its base index `osuc β` now uses the rung-3.1
+--                    inflationary law `ω^^-infl (osuc β)` (ε₀'s `oz` base
+--                    only needed `ω^^-pos`).
+--   * `φ₁`         — `φ₁ 0 = ε₀`, `φ₁ (α+1) = next-ε (φ₁ α)`,
+--                    `φ₁ (lim f) = sup`.
+--   * `φ₁-ε-number`— EVERY value of φ₁ is an ε-number (bi-`≤′`).
+--
+-- ## Honest scope (rung 3 of a long climb)
+--
+-- φ₁ is the FIRST transfinite Veblen level (φ₀ = ω^^ itself).  The
+-- Feferman–Schütte ordinal Γ₀ needs the full binary φ_α hierarchy + its
+-- diagonal fixed point; ψ₀(Ω_ω) sits far above even Γ₀ and additionally
+-- needs the ordinal-collapsing layer.  Order-type fidelity (ψ₀(Ω_ω))
+-- therefore REMAINS OPEN (D-2026-06-14); this slice neither reaches Γ₀
+-- nor plugs `Fidelity.AtHeight`.  bi-`≤′` (not `≡`) throughout, because
+-- Brouwer `olim`s of different ℕ-indexings of one supremum are not
+-- definitionally equal.
+
+module Ordinal.Brouwer.VeblenPhi where
+
+open import Data.Nat.Base using (ℕ; zero; suc)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
+
+open import Ordinal.Brouwer using (Ord; oz; osuc; olim)
+open import Ordinal.Brouwer.Phase13 using (_≤′_; f-in-lim′; ≤′-refl; ≤′-trans)
+open import Ordinal.Brouwer.OrdinalExp using (ω^^_; ε₀; ε₀-ε-number; ω^^-infl)
+
+----------------------------------------------------------------------
+-- next-ε : the least ε-number strictly above β
+----------------------------------------------------------------------
+
+-- The ω^^-tower started at an arbitrary base.
+tower-from : Ord → ℕ → Ord
+tower-from x zero    = x
+tower-from x (suc n) = ω^^ (tower-from x n)
+
+-- The supremum of the tower from `osuc β` (so it lies above β).
+next-ε : Ord → Ord
+next-ε β = olim (tower-from (osuc β))
+
+-- `next-ε β` is a fixed point of ω^^ (bi-`≤′`).  Mirrors the ε₀ proof:
+-- `ω^^ (next-ε β)` is definitionally the supremum of the tower shifted
+-- by one, so each direction is a one-step `f-in-lim′` — except the base
+-- index of the ≥ direction, which uses `ω^^-infl (osuc β)`.
+ω^^-next-ε-≤ : ∀ β → ω^^ (next-ε β) ≤′ next-ε β
+ω^^-next-ε-≤ β n = f-in-lim′ (tower-from (osuc β)) (suc n)
+
+next-ε-≤-ω^^ : ∀ β → next-ε β ≤′ ω^^ (next-ε β)
+next-ε-≤-ω^^ β zero =
+  ≤′-trans {osuc β} {ω^^ (osuc β)} {olim (λ k → ω^^ (tower-from (osuc β) k))}
+    (ω^^-infl (osuc β)) (f-in-lim′ (λ k → ω^^ (tower-from (osuc β) k)) 0)
+next-ε-≤-ω^^ β (suc m) =
+  f-in-lim′ (λ k → ω^^ (tower-from (osuc β) k)) m
+
+-- `next-ε β` lies strictly above β (witness: tower index 0 = `osuc β`).
+β<next-ε : ∀ β → osuc β ≤′ next-ε β
+β<next-ε β = 0 , ≤′-refl {β}
+
+----------------------------------------------------------------------
+-- φ₁ : the ε-number enumeration
+----------------------------------------------------------------------
+
+φ₁ : Ord → Ord
+φ₁ oz       = ε₀
+φ₁ (osuc α) = next-ε (φ₁ α)
+φ₁ (olim f) = olim (λ n → φ₁ (f n))
+
+-- Headline: every value of φ₁ is an ε-number (fixed point of ω^^), bi-`≤′`.
+-- oz reuses `ε₀-ε-number`; osuc is `next-ε`'s fixed-point pair for ANY
+-- base (no IH); the limit case lifts the per-branch IH through `olim`
+-- via `f-in-lim′` (pointwise `≤′` ⇒ `olim`-bounded).
+φ₁-ε-number : ∀ α → (ω^^ (φ₁ α) ≤′ φ₁ α) × (φ₁ α ≤′ ω^^ (φ₁ α))
+φ₁-ε-number oz       = ε₀-ε-number
+φ₁-ε-number (osuc α) = ω^^-next-ε-≤ (φ₁ α) , next-ε-≤-ω^^ (φ₁ α)
+φ₁-ε-number (olim f) =
+  (λ n → ≤′-trans {ω^^ (φ₁ (f n))} {φ₁ (f n)} {olim (λ k → φ₁ (f k))}
+           (proj₁ (φ₁-ε-number (f n))) (f-in-lim′ (λ k → φ₁ (f k)) n))
+  , (λ n → ≤′-trans {φ₁ (f n)} {ω^^ (φ₁ (f n))} {olim (λ k → ω^^ (φ₁ (f k)))}
+             (proj₂ (φ₁-ε-number (f n))) (f-in-lim′ (λ k → ω^^ (φ₁ (f k))) n))

proofs/agda/Smoke.agda

@@ -1186,6 +1186,22 @@ open import Ordinal.Brouwer.OrdinalExp using
   ; ω^^-infl
   )
 
+-- Veblen φ-hierarchy slice 2 (2026-06-18, own block per CLAUDE.md
+-- Working rules): φ₁, the enumeration of ε-numbers (fixed points of
+-- ω^^), built on the rung-3.1 inflationary law.  `next-ε β` = least
+-- ε-number above β; `φ₁` enumerates them (0↦ε₀, suc↦next-ε, limit↦sup);
+-- `φ₁-ε-number` proves every value is an ε-number (bi-≤′).  Rung 3 of the
+-- climb toward Γ₀ / ψ₀(Ω_ω) (order-type fidelity OPEN, D-2026-06-14).
+open import Ordinal.Brouwer.VeblenPhi using
+  ( tower-from
+  ; next-ε
+  ; ω^^-next-ε-≤
+  ; next-ε-≤-ω^^
+  ; β<next-ε
+  ; φ₁
+  ; φ₁-ε-number
+  )
+
 -- Recommended rank function for unbudgeted `wf-<ᵇʳᶠ_` per Echidna's
 -- design search; transport theorem deferred until Phase 1.3 lemmas land.
 open import Ordinal.Buchholz.RankBrouwer using