|
| 1 | +// SPDX-License-Identifier: MPL-2.0 |
| 2 | +// SPDX-FileCopyrightText: 2025-2026 Jonathan D.A. Jewell <j.d.a.jewell@open.ac.uk> |
| 3 | += echo-types — Proof Status |
| 4 | +:toc: |
| 5 | +:sectnums: |
| 6 | + |
| 7 | +[abstract] |
| 8 | +Honest status of the echo-types Agda formalisation: what is *proved*, what is |
| 9 | +*open*, and the build/guardrail invariant. echo-types is a constructive Agda |
| 10 | +development (`--safe --without-K`); the main verified suite (`proofs/agda/All.agda`) |
| 11 | +is *postulate-free*. The single isolated exception (the order-type *Fidelity* |
| 12 | +scaffold) is enumerated explicitly below. |
| 13 | + |
| 14 | +== Build invariant (verified) |
| 15 | + |
| 16 | +* `--safe --without-K` throughout `proofs/agda/**`. |
| 17 | +* `agda proofs/agda/All.agda` (≈200 modules) and `agda proofs/agda/Smoke.agda` |
| 18 | + exit 0. Last verified 2026-06-18. |
| 19 | +* Guardrails are CI-enforced (`tools/check-guardrails.sh`): every file under |
| 20 | + `proofs/agda/**` must declare `{-# OPTIONS --safe --without-K #-}`, and |
| 21 | + `believe_me`, `postulate`, `TERMINATING`, `REWRITE`, `NO_POSITIVITY_CHECK`, |
| 22 | + `primTrustMe`, etc. are rejected. The main suite is postulate-free. |
| 23 | + |
| 24 | +== Proved (load-bearing, verified) |
| 25 | + |
| 26 | +=== Composition track |
| 27 | +* `Echo-comp-iso` — base accumulation isomorphism. |
| 28 | +* `cancel-iso` — per-fibre cancellation (parameterised by `s-left`/`s-right` + |
| 29 | + both triangle identities). |
| 30 | +* Pentagon coherence — `Echo-comp-pent-Σ-assoc` (full Σ-associativity iso). |
| 31 | + |
| 32 | +=== Establishment track — Pillars A–D COMPLETE |
| 33 | +* *A* `EchoFiberBridge.agda` — `echo↔fib` definitional bridge (Echo ≡ fibre). |
| 34 | +* *B* `EchoGradedComonad.agda` (graded-comonad laws) + `EchoPullback.agda` |
| 35 | + (funext-free, K-free terminal-cone universal property). |
| 36 | +* *C* `EchoSeparating.agda` — separating model; characteristic law refuted. |
| 37 | +* *D* `EchoRelModel.agda` — model-independence theorem (`GCLaws` once for any |
| 38 | + model) + `docs/echo-types/conservativity.adoc` metatheorem. |
| 39 | + |
| 40 | +=== Canonical-identity spine (Tier 1/2/3) |
| 41 | +`EchoTotalCompletion` (`A ≃ Σ B (Echo f)`), `EchoOrthogonalFactorizationSystem`, |
| 42 | +`EchoImageFactorization`, `EchoNoSectionGeneric`, `EchoLossTaxonomy`, |
| 43 | +`EchoResidueTaxonomy`, `EchoDecorationStructure`, `EchoObservationalEquivalence`, |
| 44 | +`EchoEntropy`, `EchoLLEncoding`, the funext-qualified full-OFS gate `F5` |
| 45 | +(`EchoOFSUnivF5*`), the audience moves (`EchoProvenance`, `EchoSecurity`, |
| 46 | +`EchoProbabilisticSupport`, `EchoDifferential`), `EchoCanonicalIdentitySuite`, |
| 47 | +`EchoDeniability`, and `EchoAggregation` (micro→macro aggregation as an Echo |
| 48 | +map; `aggregate-non-injective` + `no-canonical-disaggregation`). |
| 49 | + |
| 50 | +=== Ordinal / Buchholz track (sound carrier) |
| 51 | +* Doubled-ladder relation `_<ᵇ²_` over the 12 core constructors + the umbrella |
| 52 | + `rank2-mono-<ᵇ²` + `wf-<ᵇ²` (Gate 1 closed *for the sound carrier*). |
| 53 | +* `RecursiveSurfaceSound._<ᵇʳᶠ²_` + its *unbudgeted* `wf-<ᵇʳᶠ²`. |
| 54 | +* `Ordinal/Brouwer/OrdinalExp.agda` — `ω^^_` + the first ε-number `ε₀` |
| 55 | + (Bachmann–Howard climb rung 1). |
| 56 | +* `Ordinal/Buchholz/BHTarget.agda` — a real `BHNotation` order + its |
| 57 | + well-foundedness (no postulate). |
| 58 | + |
| 59 | +== Open (honest — NOT proved) |
| 60 | + |
| 61 | +* *Bachmann–Howard milestone* `ψ₀(Ω_ω)` order-type fidelity — *OPEN* |
| 62 | + (`D-2026-06-14`). ε₀ is astronomically below the target |
| 63 | + (ε₀ ≪ Γ₀ ≪ … ≪ ψ₀(Ω_ω)); the collapsing-function climb is the multi-session |
| 64 | + frontier. The headline ordinal milestone is *not* reached. |
| 65 | +* *`Ordinal/Buchholz/Fidelity.agda` — 2 isolated postulates* (`denotation`, |
| 66 | + `ordinal-upper-bound`). These are the ONLY postulates in the development, |
| 67 | + quarantined under `Ordinal/Buchholz/` and tracked in `docs/.../proof-debt.md` |
| 68 | + + `Fidelity-OPEN-postulates.md`. Dischargeable once the target collapse |
| 69 | + heights exist. |
| 70 | +* *(epi, mono) image truncation* — `∥_∥` is *constructed* only in the |
| 71 | + `--cubical` island `EchoImageFactorizationPropCubical.agda`; under |
| 72 | + `--safe --without-K` it remains a shadow (it cannot be built within |
| 73 | + `--safe --without-K` without HITs). |
| 74 | +* *Unbudgeted global `wf-<ᵇʳᶠ` over native `_<ᵇ_`* — walled: native `_<ᵇ_` is |
| 75 | + ordinally unsound (documented counterexample), so no rank embedding maps it. |
| 76 | + The achievable form landed on the sound carrier (`wf-<ᵇʳᶠ²`). |
| 77 | +* *Pillar E paper* (`docs/echo-types/paper.adoc`) — LIVING DRAFT with `[EXPAND]` |
| 78 | + tags (background, related work, evaluation, ordinal consumer-evidence |
| 79 | + appendix gated on the BH milestone). Not submitted. |
| 80 | + |
| 81 | +== Discipline |
| 82 | + |
| 83 | +* No `postulate` in load-bearing tracks (the 2 Fidelity postulates are the |
| 84 | + sole, isolated, documented exception). |
| 85 | +* Every headline theorem is pinned in `Smoke.agda`; every module is wired into |
| 86 | + `All.agda` (orphan modules that compile but are not in `All.agda` are treated |
| 87 | + as dead code). |
| 88 | +* The EI-2 integration-recipe-distinctness investigation is *terminated |
| 89 | + negatively* and must not be reopened (see `.machine_readable/6a2/STATE.a2ml`). |
0 commit comments