hyperpolymath · hyperpolymath · Jun 20, 2026 · Jun 20, 2026
@@ -210,7 +210,78 @@ work to `main` and refresh all documentation:
    name, the commits folded in, the remaining open pieces of the
    milestone, and the proposed smallest useful next advance.
 
-## Current rung state (2026-06-18)
+## Current rung state (2026-06-20)
+
+### Session arc 2026-06-20 — variance resolution + Veblen climb rungs 3–6 + ordinal-fidelity ladder PARKED (read this first)
+
+*Where we started:* user supplied the sharpened understanding of an
+echo-type (a tropically-graded modality of structured loss: graded-(co)monad
+machinery over the min-plus semiring, recoverable *exact on a homotopy
+fibre rather than complement-storing*, variance unsettled). Then asked to
+RESOLVE the monad/comonad/adjunction variance in `--safe` Agda, then to
+continue the ordinal track toward ψ₀(Ω_ω).
+
+*Where we ended:* SEVEN PRs merged to `main`, then the ordinal ladder
+PARKED by owner decision. All `--safe --without-K`, zero postulates, no
+`TERMINATING`, kernel-guard PASS.
+
+1. **Variance resolved (#243).** `EchoVariance.agda` — verdict: echo
+   (fibre-based loss) is a graded **MONAD of accumulation** (the `+` axis,
+   `accumulate = Echo-comp-iso-from`) with a section/retraction
+   **ADJUNCTION exact on the grade-0 fibre** (`recoverable-fibre =
+   A↔ΣEcho`); it is **NOT a graded comonad** — the comonad reading is the
+   LOSSLESS complement-storing writer (`EchoGradedComonadF1`'s `δ` is `coe`
+   along a type equality, inverted by `μ-writer`). `no-bare-recovery`
+   (via `no-section-of-collapsing-map`) is the comonad obstruction. Sharpens
+   R-2026-05-18 from "graded comonad withdrawn" to "decided against". Doc:
+   `docs/echo-types/variance-resolution.adoc`; paper Reframing-note
+   follow-up `F-2026-06-19`. DO NOT reopen: the verdict, or the
+   complement-storing-writer framing.
+
+2. **Veblen climb rungs 3–6 (#244–#247), Brouwer side.**
+   * `#244` `VeblenPhiNormal` — φ₁ a normal function; `next-ε-least`
+     (next-ε β is the LEAST ε-number above β).
+   * `#245` `VeblenBinary` — binary `φ : Ord → Ord → Ord` (the tractability
+     move: recursion on the FIRST arg returning `Ord → Ord`, second-arg
+     recursion inside the generic `deriv`; no `TERMINATING`); generic engine
+     `deriv`/`nextFix`/`commonStep`; `φ-cont`; `nextFix-fixed-{≤,≥}`; the
+     diagonal `Γ₀` defined.
+   * `#246` `VeblenBinaryNormal` — EVERY level a normal function (`φ-mono₂`,
+     `φ-infl`); the Veblen recurrence `φ-level-fixed-{≤,≥}` (φ_{α+1}(β) is a
+     fixed point of φ_α).
+   * `#247` `VeblenBinaryMono` — first-arg monotonicity (`φ-mono₁-step`,
+     `φ-mono₁-into-lim`); `commonStep` correctness; `Γ₀-prefixed`
+     (**Γ₀ ≤′ φ_Γ₀(0)**, one direction of the diagonal fixed point).
+   Reusable lesson: every `≤′-trans`/`≤′-lim`/`*-mono`/`≤′-zero` call pins
+   its implicit source explicitly — `_≤′_` is a reducing relation.
+
+3. **Ordinal-fidelity ladder PARKED (`D-2026-06-20`).** Owner decision: the
+   transfinite ladder is now CONSUMER-LESS — the Groove cleave (sole consumer
+   of ψ₀(Ω_ω) order-type fidelity) is resolved as a finite exact-round-trip
+   zipper needing well-foundedness only, and Groove RC-11 forbids ε₀+ in
+   cleave ranks. This is a PARK (resumable), NOT a kill/retraction —
+   fidelity was always flagged OPEN. **DO NOT open the Veblen mutual
+   fixed-point descent rung** (the gate to Γ₀-least). No postulate closed;
+   `D-2026-06-14` stands. Record:
+   `docs/echo-types/decisions/ordinal-fidelity-ladder-parked.adoc`.
+   * *Frontier for cold resume:* Γ₀-least OPEN; wall = reverse
+     `φ_Γ₀(0) ≤′ Γ₀` → "Γ₀ closed under every level below it" → general
+     first-arg monotonicity `φ_α x ≤′ φ_β x` → gated on the mutual
+     fixed-point descent (monotonicity + descent mutually recursive).
+     Downstream OPEN: the ψ collapsing function (produces `bh-height`), then
+     the two `Fidelity.agda` postulates `denotation` / `ordinal-upper-bound`
+     (gated on the collapse).
+   * *Disposition:* extraction to its own ordinal-notation repo (Löwe /
+     classical-order-type side, labelled OPEN) — cut FLAGGED, left to owner.
+     Firewall verified clean: `OmegaMarkers` (leaf) ← `Buchholz.Syntax` ←
+     `EchoOrdinal` STAY; everything else under `Ordinal/` MOVES. Supersedes
+     any "re-home into Groove" framing (RC-11 retired it).
+
+*Plan for the next Claude.* The ordinal ladder is PARKED — do not resume it
+unless the consumer returns (then start at the mutual-descent rung per the
+park record). The variance resolution is complete and consumed. Other
+tracks (composition, establishment Pillar E, the EchoTypes.jl mirror) are
+unaffected by the park.
 
 ### Session arc 2026-06-18 — EchoAggregation general/macro split + long-form prose relicense (read this first)
 

@@ -0,0 +1,158 @@
+// SPDX-License-Identifier: CC-BY-SA-4.0
+// SPDX-FileCopyrightText: 2025-2026 Jonathan D.A. Jewell <j.d.a.jewell@open.ac.uk>
+= D-2026-06-20 — Ordinal-fidelity ladder PARKED (consumer removed)
+:toc: macro
+:toclevels: 2
+:sectnums:
+:sectnumlevels: 2
+:icons: font
+
+[.lead]
+Records, in the same dated idiom as `D-2026-06-14` (its parent) and the
+retraction log's `R-2026-05-18`, the decision to *park* the transfinite
+ordinal-fidelity ladder. This is a *PARK* — not a kill and not a
+retraction. Nothing was over-claimed: order-type fidelity was always
+flagged OPEN (`D-2026-06-14` stands, unchanged). The ladder is correct,
+well-built, `--safe --without-K`, zero-postulate — and, as of this date,
+*unconsumed*. It is frozen at a clean rung and is fully resumable.
+
+toc::[]
+
+== Status
+
+*Ordinal-fidelity ladder: PARKED, resumable.* The Brouwer-ordinal Veblen
+climb (rungs 1–6) is frozen at its current rung. No further rung is to be
+opened (in particular *not* the Veblen mutual fixed-point descent lemma —
+see §"Frontier"). The doubled-ladder well-foundedness work and the
+`Fidelity.agda` shape remain as they were. No postulate was closed by the
+parking sweep.
+
+== Reason (the consumer was removed)
+
+The only consumer that ever required ψ₀(Ω_ω) *order-type fidelity* was the
+Groove cleave. Two independent facts one layer up retire that need:
+
+. *Groove RC-11 (ordinal economy)* forbids ε₀ and above in cleave ranks.
+  The ladder is already past ε₀ at rung 2 (ε₀ is an ε-number); Γ₀
+  (rung 4+) is far past it; ψ₀(Ω_ω) is unreachably beyond. So the ladder's
+  target is *outside the consumer's permitted range by construction*.
+. *The cleave has resolved* to a finite *exact-round-trip zipper* that
+  needs *well-foundedness only* — no transfinite order-type fidelity at
+  all.
+
+Therefore the order-type-fidelity result is now *consumer-less*. The
+project's own standing caveat — "steady honest progress up a long ladder,
+not reaching the milestone" — becomes, in this light, the argument to
+*stop*: the ladder is correct, well-built, and unconsumed.
+
+On the constructive/classical seam: every rung (the mutual-descent lemma
+included) is structure on the *near side* of the open `denotation` map —
+it builds the tower, it does not certify the tower is the claimed height.
+Even if mutual-descent closed cleanly it reaches only Feferman–Schütte
+(Γ₀), galaxies below target. The hard lemma does not reach the thing.
+
+== What is frozen (the artifact inventory)
+
+The ladder as it stands is a coherent, well-scoped artifact. Landed,
+`--safe --without-K`, zero postulates, no `TERMINATING`, kernel-guard
+PASS:
+
+[cols="1,3",options="header"]
+|===
+| Module | Content
+
+| `Ordinal.Brouwer.OrdinalExp`
+| `ω^^_` (ω to an ordinal power), `ε₀`, `ε₀-ε-number`, `ω^^-infl`.
+
+| `Ordinal.Brouwer.VeblenPhi`
+| `next-ε`, `φ₁` (ε-number enumeration), `φ₁-ε-number`.
+
+| `Ordinal.Brouwer.VeblenPhiNormal`
+| `ω^^-mono-≤′`; `next-ε-least` (next-ε β is the LEAST ε-number above β);
+  `φ₁` a normal function (`φ₁-mono` / `φ₁-strict-mono` / `φ₁-continuous`).
+
+| `Ordinal.Brouwer.VeblenBinary`
+| Binary Veblen `φ : Ord → Ord → Ord` (generic fixed-point engine
+  `deriv` / `nextFix` / `commonStep`); `φ-cont`; engine correctness
+  `nextFix-fixed-{≤,≥}`; the diagonal `Γ₀` (Feferman–Schütte) defined.
+
+| `Ordinal.Brouwer.VeblenBinaryNormal`
+| EVERY level a normal function (`φ-mono₂`, `φ-infl`); the Veblen
+  recurrence `φ-level-fixed-{≤,≥}` (φ_{α+1}(β) is a fixed point of φ_α).
+
+| `Ordinal.Brouwer.VeblenBinaryMono`
+| First-argument monotonicity (`φ-mono₁-step`, `φ-mono₁-into-lim`);
+  `commonStep` correctness; `Γ₀-prefixed` (*Γ₀ ≤′ φ_Γ₀(0)* — one
+  direction of the diagonal fixed point).
+|===
+
+The Buchholz well-foundedness apparatus (`_<ᵇ_`, the sound doubled-ladder
+`_<ᵇ²_` + `wf-<ᵇ²`, `RankBrouwer`, `BHTarget`) and the order-type-fidelity
+*shape* (`Ordinal.Buchholz.Fidelity`, two open postulates) are unchanged.
+
+== Frontier (captured for cold resumption)
+
+If the consumer ever returns, resume here:
+
+. *Γ₀-least is OPEN.* `Γ₀ ≤′ φ_Γ₀(0)` is done (`Γ₀-prefixed`). The wall is
+  the *reverse* direction `φ_Γ₀(0) ≤′ Γ₀` (closure from above), which
+  reduces to *"Γ₀ is closed under every level `φ_α` below it"*. That needs
+  *general first-argument monotonicity* `φ_α x ≤′ φ_β x` for `α ≤′ β`,
+  which is *gated on the Veblen mutual fixed-point descent lemma*: "a
+  fixed point of `φ_β` is a fixed point of `φ_α` for `α ≤′ β`", with
+  monotonicity and descent *mutually recursive* over the level. The
+  adjacent (`φ-mono₁-step`) and below-a-limit (`φ-mono₁-into-lim`) cases
+  are already proved (rung 6); the general successor/successor case is the
+  intricate mutual induction that was deliberately *not* opened.
+. *The ψ collapsing function* with fundamental sequences — the separate,
+  multi-session core that eventually produces `bh-height`. Unchanged and
+  OPEN.
+. *The two `Fidelity.agda` postulates* `denotation` (order-reflecting,
+  height-preserving `⟦·⟧`) and `ordinal-upper-bound` — *gated on the
+  collapse existing*. Unchanged and OPEN. `denotation` canNOT be `rank2`
+  (height-collapsing).
+
+*No postulate closed. Order-type fidelity remains OPEN; `D-2026-06-14`
+stands.*
+
+== Disposition / re-home (extraction FLAGGED, cut left to the owner)
+
+The transfinite ladder is now a *consumer-less, free-standing artifact*.
+The clean ending is *extraction to its own ordinal-notation repository*,
+parked on the *Löwe (classical order-type)* side of the constructive /
+classical seam and labelled OPEN. This *supersedes* any earlier "re-home
+into Groove" framing — RC-11 retired that.
+
+*The firewall is clean (verified 2026-06-20).* The three modules that
+*STAY* in echo-types as the bridge form a self-contained subgraph that
+does *not* depend on the transfinite ladder:
+
+* `Ordinal.OmegaMarkers` — a leaf (no ordinal-ladder imports);
+* `Ordinal.Buchholz.Syntax` — imports only `Ordinal.OmegaMarkers`;
+* `EchoOrdinal` — imports only `Ordinal.OmegaMarkers` +
+  `Ordinal.Buchholz.Syntax` (its *only* ordinal dependencies).
+
+So the cut is clean: *everything else* under `proofs/agda/Ordinal/`
+(the Brouwer foundations `Brouwer` / `Phase13` / `OmegaPow` / `Arithmetic`,
+the `OrdinalExp` + `Veblen*` ladder, the Buchholz `Order` / rank /
+doubled-ladder / `Fidelity` / `BHTarget` machinery, `CNF` / `Closure` /
+`PsiSimple` / `Base`) is the *consumer-less artifact* that MOVES; the three
+bridge modules above STAY.
+
+*This entry FLAGS the extraction; it does not perform it.* The actual cut
+is the owner's: it is a cross-repo operation (new repository creation +
+moving a ~70-module dependency-entangled subtree + re-rooting its
+`.agda-lib` include path) that must preserve, on *both* sides, the
+build invariant `--safe --without-K` / zero postulates / no `TERMINATING`
+/ kernel-guard PASS. The proposed cut-line above is the boundary; the
+slice is left to the owner.
+
+== See also
+
+* `docs/echo-types/decisions/ordinal-bh-order-type-fidelity-open.adoc`
+  — `D-2026-06-14`, the parent open-problem record (stands, unchanged).
+* `Fidelity-OPEN-postulates.md` — the two open `Fidelity.agda` postulates.
+* `docs/echo-types/buchholz-rank-obstruction.adoc` — carrier soundness +
+  the doubled-ladder closure.
+* `docs/bridges/buchholz-plan.adoc` — the E-stage matrix / BH staging.
+* `roadmap.adoc` §Lane 3 — the ordinal-track items (now parked).
@@ -98,6 +98,14 @@ under one of (a) discharged / (b) budgeted / (c) necessary axiom / (d) debt.
     order-type fidelity OPEN".
   - **Citation**: `D-2026-06-14`; full per-postulate spec (statement /
     what closes it / owner) in `Fidelity-OPEN-postulates.md`.
+  - **PARKED (2026-06-20, `D-2026-06-20`)**: the transfinite ladder these
+    postulates sit atop is now *consumer-less* — the Groove cleave (the
+    only consumer of ψ₀(Ω_ω) order-type fidelity) is resolved as a finite
+    exact-round-trip zipper needing well-foundedness only, and RC-11
+    forbids ε₀+ in cleave ranks. The debt is therefore *parked,
+    resumable*, not actively being closed. No postulate closed;
+    `D-2026-06-14` stands. See
+    `docs/echo-types/decisions/ordinal-fidelity-ladder-parked.adoc`.
   - **Guardrail status**: allow-listed in `tools/check-guardrails.sh`
     (`EXPLORATORY_EXEMPT`) and the inline `hypatia: allow` pragma at the
     module head. (`BHTarget` is NOT exempt — it is real kernel content

@@ -305,7 +305,20 @@ narrowed — no unique-mediator claim. Demand 4 ("parametricity") must
 be *consumer-side at the affine instance*, not full Reynolds. See
 respective §s in the synthesis doc.
 
-=== Lane 3 [PARALLEL, NOT LOAD-BEARING] — Ordinal / Buchholz
+=== Lane 3 [PARKED 2026-06-20, NOT LOAD-BEARING] — Ordinal / Buchholz
+
+*PARKED (`D-2026-06-20`).* The transfinite ordinal-fidelity ladder is
+*parked, resumable* — the only consumer of ψ₀(Ω_ω) order-type fidelity
+(the Groove cleave) is resolved as a finite exact-round-trip zipper
+needing well-foundedness only (RC-11 forbids ε₀+ in cleave ranks), so the
+ladder is consumer-less. The Veblen climb (rungs 1–6: `ω^^`/ε₀ → φ₁ normal
+→ binary φ + Γ₀ → every level normal → first-arg monotonicity +
+`Γ₀ ≤′ φ_Γ₀(0)`) is frozen at a clean rung; the next rung (Veblen mutual
+fixed-point descent → Γ₀-least) is deliberately NOT opened. No postulate
+closed; order-type fidelity REMAINS OPEN (`D-2026-06-14` stands). The
+disposition is extraction to its own ordinal-notation repo (cut FLAGGED,
+left to the owner). See
+`docs/echo-types/decisions/ordinal-fidelity-ladder-parked.adoc`.
 
 *Echo Core does NOT depend on this lane.* This is a separate
 proof-theoretic research project that happens to share the