From 2de557c1dc7dca7344d35b0d7e0b2c2452de49ca Mon Sep 17 00:00:00 2001
From: hyperpolymath <6759885+hyperpolymath@users.noreply.github.com>
Date: Thu, 28 May 2026 10:06:53 +0100
Subject: [PATCH] =?UTF-8?q?ordinal(buchholz):=20Slice=203=20lex-rank=20com?=
 =?UTF-8?q?panion=20=E2=80=94=20bpsi-source-at-equality=20=CF=88-rank=20di?=
 =?UTF-8?q?scharge?=
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New module Ordinal.Buchholz.RankLexSlice3 closes the ψ-rank-level
discharge of the bpsi-source-at-equality sub-case of <ᵇ-+1 joint-bplus.

Composes with the existing Slice 3 head-Ω headline
(rank-mono-<ᵇ-+1-via-head-Ω in RankPowSlice3Headline) which covers
the strict-head premise case; this slice adds the equality-marker
case where the strict jump must come from the ψ-argument α's rank
rather than the leading Ω-marker.

Headlines pinned in Ordinal/Buchholz/Smoke.agda under their own
using block per CLAUDE.md Working rules; module wired into
proofs/agda/All.agda adjacent to RankLex:

  * rank-adm-bpsi-strict-at-equality — wrapper around RankAdm's
    rank-mono-<ᵇ⁺-ψα-from-pow named for the joint-bplus consumer;
    rank-pow α <′ rank-pow β → rank-adm (bpsi ν α) <′ rank-adm (bpsi ν β).
  * rank-lex-bpsi-strict-at-equality — same content lifted to the
    existing RankLex's <lex order via <lex-second.
  * rank-adm-bplus-decompose-left — definitional refl on the bplus
    decomposition of rank-adm for downstream chain composition.

Honest scope (documented inline + in the module preamble):

The task's intended bplus-chain-level conclusion `rank-pow (bplus
x₁ x₂) <′ rank-pow (bplus y₁ y₂)` is structurally NOT derivable at
the bpsi-source-at-equality sub-case — rank-pow collapses both
ψ-sources at the same ν to the SAME ordinal (ω-rank-pow ν), so no
strict step exists at the rank-pow level. Promoting to rank-adm
gives a clean strict step on the left summand
(rank-adm-bpsi-strict-at-equality), but lifting through the bplus
sum requires either additive-principal closure on `rank-adm y₁`
(NOT available — `ω-rank-pow ν ⊕ rank-pow β` is a generic sum, not
an ω-power) or strict-left-mono of `_⊕_` (a non-theorem; documented
counterexample in Phase13).

The clean follow-on slice would introduce a parallel rank
`rank-lex-jb : BT → RankLex` whose bplus case carries the leftmost-
ψ-α info in the second component; that requires reshaping the
existing rank-lex (its `<ᵇ-ψΩ≤` boundary discharge depends on the
oz-second bplus case) and is deferred as a non-local design slice.

Build invariants held: --safe --without-K throughout; zero
postulates / no funext / no TERMINATING pragma; proofs/agda/All.agda
+ proofs/agda/Smoke.agda + proofs/agda/Ordinal/Buchholz/Smoke.agda
all exit 0 under the current library configuration.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
---
 proofs/agda/All.agda                          |   1 +
 .../agda/Ordinal/Buchholz/RankLexSlice3.agda  | 233 ++++++++++++++++++
 proofs/agda/Ordinal/Buchholz/Smoke.agda       |  20 ++
 3 files changed, 254 insertions(+)
 create mode 100644 proofs/agda/Ordinal/Buchholz/RankLexSlice3.agda

diff --git a/proofs/agda/All.agda b/proofs/agda/All.agda
index 190b75f..1adbc63 100644
--- a/proofs/agda/All.agda
+++ b/proofs/agda/All.agda
@@ -134,6 +134,7 @@ open import Ordinal.Buchholz.RankPartial
 open import Ordinal.Buchholz.RankPow
 open import Ordinal.Buchholz.RankAdm
 open import Ordinal.Buchholz.RankLex
+open import Ordinal.Buchholz.RankLexSlice3
 open import Ordinal.Buchholz.HeadOmega
 open import Ordinal.Buchholz.HeadOmegaInversion
 open import Ordinal.Buchholz.RankPowDomination
diff --git a/proofs/agda/Ordinal/Buchholz/RankLexSlice3.agda b/proofs/agda/Ordinal/Buchholz/RankLexSlice3.agda
new file mode 100644
index 0000000..a097769
--- /dev/null
+++ b/proofs/agda/Ordinal/Buchholz/RankLexSlice3.agda
@@ -0,0 +1,233 @@
+{-# OPTIONS --safe --without-K #-}
+
+-- Lex-rank companion to the Slice 3 head-Ω discharge, covering the
+-- bpsi-source-at-equality sub-case of `<ᵇ-+1` joint-bplus, 2026-05-28.
+--
+-- ## Where this sits
+--
+-- The Slice 3 headline `Ordinal.Buchholz.RankPowSlice3Headline.
+-- rank-mono-<ᵇ-+1-via-head-Ω` discharges `bplus x₁ x₂ <ᵇ bplus y₁ y₂`
+-- (from `x₁ <ᵇ y₁`) UNDER A STRICT-HEAD PREMISE `head-Ω x₁ <Ω head-Ω y₁`.
+-- The premise is supplied externally by the umbrella's case-split:
+--
+--   * x₁ = bOmega μ  : `head-Ω-inv-bOmega` gives `μ <Ω head-Ω y₁`
+--                       directly (STRICT for all three Ω-source
+--                       constructors).  Slice 3 closes.
+--   * x₁ = bpsi ν α  : `head-Ω-inv-bpsi` gives `ν ≤Ω head-Ω y₁`
+--                       (NON-STRICT — `<ᵇ-ψΩ≤` permits equal markers).
+--                       Strict sub-case (`ν <Ω head-Ω y₁`) closes
+--                       via Slice 3.  EQUALITY sub-case
+--                       (`ν ≡ head-Ω y₁`) is THIS MODULE'S TERRITORY.
+--   * x₁ = bplus a b : recurse on the leftmost-leaf shape; reduces to
+--                       one of the above two atomic cases.
+--
+-- This module handles the bpsi-source-at-equality remainder via the
+-- existing lex-pair rank `rank-lex` from `Ordinal.Buchholz.RankLex`
+-- (whose second component carries `rank-pow α` for the ψ case) plus
+-- the existing admissibility-aware rank `rank-adm` from
+-- `Ordinal.Buchholz.RankAdm`.
+--
+-- ## What this slice closes (the lex headline)
+--
+-- The headline:
+--
+--   rank-mono-<ᵇ-+1-via-rank-lex : ∀ {ν α β x₂ y₂}
+--     → rank-pow α <′ rank-pow β       -- second-component lex witness
+--     → rank-lex (bplus (bpsi ν α) x₂) <lex rank-lex (bplus (bpsi ν β) y₂)
+--
+-- ...is NOT directly stateable: `rank-lex (bplus x y) = mkLex
+-- (rank-pow (bplus x y)) oz` in the existing `RankLex` module, so the
+-- second component is *both* `oz` and the lex-second-strict
+-- constructor `<lex-second : b₁ <′ b₂` cannot fire at `oz <′ oz`.
+-- The existing `rank-lex` was deliberately shipped that way to scope
+-- the slice to the `<ᵇ-ψΩ≤` ν=μ boundary headline alone (see the
+-- companion-remark in `RankLex.agda` lines 142-148).
+--
+-- The HONEST closure THIS module ships:
+--
+--   rank-adm-bpsi-strict-at-equality : ∀ {ν α β}
+--     → rank-pow α <′ rank-pow β
+--     → rank-adm (bpsi ν α) <′ rank-adm (bpsi ν β)
+--
+-- This is a thin convenience re-export of `RankAdm.rank-mono-<ᵇ⁺-ψα-
+-- from-pow` named for the joint-bplus consumer.  It is the LOAD-
+-- BEARING primitive for the bpsi-source-at-equality sub-case at the
+-- ψ-rank level.  The bplus-extension under `rank-adm` (= `rank-adm
+-- (bplus x₁ x₂) <′ rank-adm (bplus y₁ y₂)`) is documented in the
+-- design note below as the open structural gap.
+--
+-- ## What this slice deliberately DOES NOT close (honest scope)
+--
+-- The task's intended conclusion `rank-pow (bplus x₁ x₂) <′ rank-pow
+-- (bplus y₁ y₂)` is *structurally not derivable* from `α <ᵇ β` alone
+-- at the bpsi-source-at-equality sub-case:
+--
+--   * `rank-pow (bpsi ν α) = ω-rank-pow ν = rank-pow (bpsi ν β)` —
+--     the provisional `rank-pow` shape collapses both ψ-sources at
+--     the same ν to the SAME ordinal, so no strict-on-left-summand
+--     witness exists at the `rank-pow` level.
+--   * Promoting the conclusion to `rank-adm` produces `rank-adm
+--     (bpsi ν α) <′ rank-adm (bpsi ν β)` cleanly (this module's
+--     `rank-adm-bpsi-strict-at-equality`), but the bplus
+--     extension `rank-adm (bplus x₁ x₂) <′ rank-adm (bplus y₁ y₂)`
+--     requires either (a) additive-principal closure on `rank-adm
+--     y₁` (NOT generally available — `rank-adm (bpsi ν β) =
+--     ω-rank-pow ν ⊕ rank-pow β` is generically a sum, not an
+--     ω-power), or (b) strict-left-monotonicity of `_⊕_` (a NON-
+--     theorem; the `α ⊕ ω ≡ β ⊕ ω` counterexample documented in
+--     `Phase13.agda`).
+--   * Promoting to a redesigned lex rank `rank-lex-jb` whose bplus
+--     case carries the leftmost-ψ-α info in the second component
+--     would close the headline at the LEX-rank level (lex-first
+--     comparing rank-pow's then lex-second comparing α-ranks);
+--     however, this requires reshaping the existing `rank-lex` (or
+--     introducing a parallel rank), breaking the existing `<ᵇ-ψΩ≤`
+--     boundary discharge.  Deferred as a follow-on design slice.
+--
+-- The bpsi-source-at-equality sub-case is therefore CLOSED AT THE
+-- PSI-RANK LEVEL (via `rank-adm-bpsi-strict-at-equality`) and OPEN
+-- AT THE BPLUS-CHAIN LEVEL pending one of (a)/(b)/(c) above.
+--
+-- ## Headlines (pinned in `Ordinal/Buchholz/Smoke.agda`)
+--
+--   * `rank-adm-bpsi-strict-at-equality` — ψ-rank strict step from
+--     the α<β witness, joint-bplus-consumer-facing wrapper.
+--   * `rank-lex-bpsi-strict-at-equality` — same content lifted to
+--     `rank-lex`'s second component via `<lex-second`.
+--   * `rank-adm-bplus-decompose-left`   — `rank-adm` decomposes
+--     definitionally on bplus into the left and right summands (used
+--     by the joint-bplus consumer to chain `rank-adm-bpsi-strict-
+--     at-equality` against the source's left summand).
+
+module Ordinal.Buchholz.RankLexSlice3 where
+
+open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+
+open import Ordinal.Brouwer            using (Ord)
+open import Ordinal.Brouwer.Arithmetic using (_⊕_)
+open import Ordinal.Brouwer.Phase13    using (_<′_)
+open import Ordinal.Buchholz.Syntax    using
+  ( BT
+  ; bzero
+  ; bOmega
+  ; bplus
+  ; bpsi
+  )
+open import Ordinal.Buchholz.RankPow   using (rank-pow; ω-rank-pow)
+open import Ordinal.Buchholz.RankAdm   using
+  ( rank-adm
+  ; rank-mono-<ᵇ⁺-ψα-from-pow
+  )
+open import Ordinal.Buchholz.RankLex   using
+  ( RankLex
+  ; mkLex
+  ; _<lex_
+  ; <lex-second
+  ; rank-lex
+  )
+
+----------------------------------------------------------------------
+-- ψ-rank strict step at the equality sub-case (rank-adm side)
+----------------------------------------------------------------------
+
+-- Re-export of `RankAdm.rank-mono-<ᵇ⁺-ψα-from-pow` named for the
+-- joint-bplus consumer.  Given a `rank-pow α <′ rank-pow β` witness
+-- (produced by `Ordinal.Buchholz.RankMonoUmbrella.rank-pow-mono-<ᵇ⁰`
+-- for any `α <ᵇ⁰ β`), conclude `rank-adm (bpsi ν α) <′ rank-adm
+-- (bpsi ν β)`.
+--
+-- This is the LOAD-BEARING primitive of the bpsi-source-at-equality
+-- sub-case at the ψ-rank level.  The joint-bplus headline at the
+-- bplus-chain level is structurally blocked as documented in this
+-- module's preamble; this primitive closes the part that DOES close.
+
+rank-adm-bpsi-strict-at-equality : ∀ {ν α β}
+  → rank-pow α <′ rank-pow β
+  → rank-adm (bpsi ν α) <′ rank-adm (bpsi ν β)
+rank-adm-bpsi-strict-at-equality {ν} {α} {β} p =
+  rank-mono-<ᵇ⁺-ψα-from-pow {ν} {α} {β} p
+
+----------------------------------------------------------------------
+-- ψ-rank strict step at the equality sub-case (rank-lex side)
+----------------------------------------------------------------------
+
+-- Lift the strict α-rank step into the `rank-lex` lex order.  Since
+-- `rank-lex (bpsi ν α) = mkLex (ω-rank-pow ν) (rank-pow α)` and
+-- `rank-lex (bpsi ν β) = mkLex (ω-rank-pow ν) (rank-pow β)`, the
+-- first components are syntactically equal (no funext, no `with`
+-- elimination needed) and the `<lex-second` constructor fires
+-- against the second-component witness.
+--
+-- This is the LEX-rank-level mirror of `rank-adm-bpsi-strict-at-
+-- equality`; consumers that work at the lex-rank level use this one
+-- rather than the rank-adm form.  Both convey the same structural
+-- content (the α-rank discriminates), differing only in which rank
+-- the consumer is composing against.
+
+rank-lex-bpsi-strict-at-equality : ∀ {ν α β}
+  → rank-pow α <′ rank-pow β
+  → rank-lex (bpsi ν α) <lex rank-lex (bpsi ν β)
+rank-lex-bpsi-strict-at-equality {ν} {α} {β} p =
+  <lex-second {ω-rank-pow ν} {rank-pow α} {rank-pow β} p
+
+----------------------------------------------------------------------
+-- Definitional decomposition of `rank-adm` on `bplus`
+----------------------------------------------------------------------
+
+-- Pinned as `refl` for the joint-bplus consumer that needs to chain
+-- a strict step on the left summand against a tail bound on the
+-- right summand.  Documents the additive structure of `rank-adm` on
+-- `bplus` for callers that want to reason about it without re-opening
+-- `RankAdm`.
+
+rank-adm-bplus-decompose-left : ∀ x y →
+                                rank-adm (bplus x y) ≡ rank-adm x ⊕ rank-adm y
+rank-adm-bplus-decompose-left _ _ = refl
+
+----------------------------------------------------------------------
+-- Design note: why the headline `rank-pow (bplus x₁ x₂) <′
+-- rank-pow (bplus y₁ y₂)` does NOT close from this slice's primitives
+----------------------------------------------------------------------
+
+-- The task's stated headline conclusion
+--   rank-pow (bplus x₁ x₂) <′ rank-pow (bplus y₁ y₂)
+-- at the bpsi-source-at-equality sub-case
+--   (x₁ = bpsi ν α, y₁ = bpsi ν β, x₁ <ᵇ y₁ via the α<β witness)
+-- requires a strict step on the bplus left summand at the rank-pow
+-- level.  But
+--   rank-pow (bpsi ν α) = ω-rank-pow ν = rank-pow (bpsi ν β)
+-- by definition; the strict step exists ONLY at the rank-adm level
+-- (`rank-adm (bpsi ν α) <′ rank-adm (bpsi ν β)`, which this module
+-- closes via `rank-adm-bpsi-strict-at-equality`).
+--
+-- Lifting the strict rank-adm step to a strict rank-adm sum
+--   rank-adm x₁ ⊕ rank-adm x₂ <′ rank-adm y₁ ⊕ rank-adm y₂
+-- requires either
+--   (a) additive-principal closure on `rank-adm y₁`, which holds
+--       only when `rank-adm y₁ = ω-rank-pow _` exactly — i.e., when
+--       y₁ is bOmega μ or bzero, NOT when y₁ = bpsi ν β where
+--       `rank-adm y₁ = ω-rank-pow ν ⊕ rank-pow β` is a generic sum.
+--   (b) strict left-monotonicity of `_⊕_`, which is a NON-theorem
+--       under Brouwer arithmetic — the `α ⊕ ω ≡ β ⊕ ω`
+--       counterexample documented in `Ordinal.Brouwer.Phase13` (see
+--       the block comment near `⊕-mono-≤-left`'s declaration).
+--
+-- The clean resolution is a parallel rank with a richer bplus second
+-- component (`rank-lex-jb : BT → RankLex` whose `bplus`-case carries
+-- the leftmost-ψ-α info), but introducing it parallel to `rank-lex`
+-- requires a follow-on design slice — the existing `rank-lex`'s
+-- `<ᵇ-ψΩ≤`-boundary discharge depends on `rank-lex`'s bplus-case
+-- being `oz`-second, so the redesign is non-local.
+--
+-- Therefore the joint-bplus bpsi-source-at-equality sub-case is
+-- CLOSED at the ψ-rank level (one source-level discriminator
+-- between source ψ-α and target ψ-β is now formal) and OPEN at the
+-- bplus-chain level (transporting that discriminator through bplus
+-- requires additional ordinal-arithmetic infrastructure that this
+-- session does not land).
+--
+-- The honest contribution of this slice: pin the ψ-rank discharge as
+-- a NAMED THEOREM accessible from the joint-bplus consumer, plus an
+-- explicit declaration of the bplus-chain obstacle so the next
+-- session can see exactly which lemmas would unblock it without
+-- re-deriving the analysis.
diff --git a/proofs/agda/Ordinal/Buchholz/Smoke.agda b/proofs/agda/Ordinal/Buchholz/Smoke.agda
index 13eaa65..1731305 100644
--- a/proofs/agda/Ordinal/Buchholz/Smoke.agda
+++ b/proofs/agda/Ordinal/Buchholz/Smoke.agda
@@ -497,3 +497,23 @@ open import Ordinal.Buchholz.RankMonoUmbrellaSlice3 using
   ; <ᵇ¹-+1-+
   ; rank-pow-mono-<ᵇ¹
   )
+
+-- Slice 3 lex-rank companion 2026-05-28 (own block per CLAUDE.md
+-- Working rules): the bpsi-source-at-equality sub-case of
+-- `<ᵇ-+1` joint-bplus.  Closes the ψ-rank-level discharge (via
+-- `rank-adm-bpsi-strict-at-equality` / `rank-lex-bpsi-strict-at-
+-- equality`) under the α/β strict-rank witness from the umbrella;
+-- the bplus-chain-level extension to a strict step on the full
+-- bplus sum is structurally blocked (additive-principal closure on
+-- a generic `ω-rank-pow ν ⊕ rank-pow β` sum doesn't hold, and
+-- strict-left-mono of `_⊕_` is a non-theorem).  Honest scope: pins
+-- the ψ-rank discharge as a named theorem accessible from the
+-- joint-bplus umbrella consumer + documents the bplus-chain
+-- obstacle inline so the next session sees exactly which lemmas
+-- would unblock it.  Complements `RankMonoUmbrellaSlice3`'s
+-- gated documentation of the same sub-case from the umbrella side.
+open import Ordinal.Buchholz.RankLexSlice3 using
+  ( rank-adm-bpsi-strict-at-equality
+  ; rank-lex-bpsi-strict-at-equality
+  ; rank-adm-bplus-decompose-left
+  )