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echo-types: Pillar D — second model + conservativity (programme complete)
EchoRelModel.agda promoted to a real module and conservativity.adoc written. Pillars A–D are now ALL COMPLETE; the entire internal establishment programme is done (only Pillar E, external validation, remains — and that is social, not internal proof work). Pillar D artefact 1 — EchoRelModel.agda: - GradedLossModel: the abstract model interface (propositional grade order + composing reindexing) the H2 verdict showed suffices. - module GCLaws: the graded-comonad laws (gcomonad-counit-{l,r}, gcomonad-coassoc) proved ONCE generically from the interface — this IS the model-independence theorem. - set-model (EchoGraded) and rel-model (relational EchoStep / map-rel; composition law from map-rel-comp + degrade-comp): two genuinely different instances. - model-agreement (round-trips refl) + bridge-natural (map-over ↔ map-rel under the graph = fibration bridge): the graded comonad is the SAME structure in both models. Pillar D artefact 2 — docs/echo-types/conservativity.adoc: the conservative/definitional metatheorem, with a three-clause formal anchor (definitional → EchoFiberBridge; conservative → postulate-free --safe --without-K suite, CI-enforced; modality-not-axiom → EchoRelModel/EchoGradedComonad/EchoSeparating). Scope/non-claims explicit; funext boundary quarantined. - proofs/agda/EchoRelModel.agda: real, --safe --without-K, no postulates / no escape pragmas. - proofs/agda/Smoke.agda: pin GradedLossModel/set-model/rel-model/ rel-gcomonad-*/model-agreement/bridge-natural. - docs/echo-types/establishment-plan.adoc: Pillar D landed (both artefacts), Status table, sequencing (A–D complete), revision history. - CLAUDE.md: rung state + workstream-3 refreshed; next advance now Pillar E (external validation; flag to user, partly offline). All.agda + Smoke.agda exit 0 under --safe --without-K. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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‎CLAUDE.md‎

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@@ -106,19 +106,22 @@ Two active workstreams:
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strengthened, unchanged. See establishment-plan.adoc §"H2
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verdict" + revision history.
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**Pillar C COMPLETE (2026-05-17).** `EchoSeparating.agda` real:
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a separating model = `EchoGraded` minus `≤g-prop`
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(`sep-order-not-prop`); generic Σ-functoriality
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(`sep-map-over-{id,comp}`) + functorial unit hold, but
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`SepDegradeCompose` is refuted (`sep-degrade-compose-fails`, via
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checked `true ≢ false`). Pillars A/B/C done — credibility core
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complete.
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**Pillars A–D ALL COMPLETE (2026-05-17).** Pillar C:
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`EchoSeparating.agda` (separating model = `EchoGraded` minus
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`≤g-prop`; `sep-degrade-compose-fails`). Pillar D artefact 1:
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`EchoRelModel.agda` — abstract `GradedLossModel` + generic
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`GCLaws` (comonad laws proved once for ANY model = the
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model-independence theorem), `set-model`/`rel-model` instances,
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`model-agreement` + `bridge-natural`. Pillar D artefact 2:
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`docs/echo-types/conservativity.adoc` — conservative/definitional
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metatheorem with a 3-clause formal anchor. No scaffold modules
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remain; the entire internal programme is done.
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Open pieces on this track:
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* Pillar D — `EchoRelModel.agda` (transport the Pillar-B
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graded-comonad laws into the relational/fibration model) +
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`docs/echo-types/conservativity.adoc`.
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* Pillar E (TYPES/CPP-ITP/Zenodo) not started.
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* Pillar E only — external validation: TYPES abstract → CPP/ITP
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mechanised-metatheory paper → Zenodo DOI + installable library
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packaging → adjacency-note outreach. (Social, not internal;
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not started.)
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Cross-repo bridge status lives in `docs/echo-types/cross-repo-bridge-status.md`.
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`≤g-prop`); `sep-order-not-prop`, `sep-map-over-{id,comp}` (generic
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Σ-laws hold), `sep-degrade-compose-fails` (characteristic law
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refuted at `true ≢ false`). Pinned in `Smoke.agda`. No postulates.
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* **Pillar D (scaffold):** `EchoRelModel` — declaration-free doc
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module (no postulates, no holes), tracked in `All.agda`, carrying
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intended signatures as commented specs.
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* **Pillar D (real, verified — COMPLETE, 2026-05-17):**
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`EchoRelModel.agda` — `GradedLossModel`/`GCLaws` (model-independence
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theorem), `set-model`/`rel-model`, `model-agreement`,
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`bridge-natural`; pinned in `Smoke.agda`, no postulates.
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`docs/echo-types/conservativity.adoc` — metatheorem + 3-clause
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formal anchor. No scaffold modules remain.
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`agda proofs/agda/All.agda` and `agda proofs/agda/Smoke.agda` both
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exit 0 under `--safe --without-K`. No postulates introduced.
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`≤g-prop`. Thesis unchanged and strengthened (establishment-plan
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§"H2 verdict"). `All.agda` + `Smoke.agda` exit 0, no postulates.
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**Pillar C LANDED (2026-05-17). Credibility core (A+B+C) COMPLETE.**
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`EchoSeparating.agda` real: a separating model = `EchoGraded` minus
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its sole hypothesis `≤g-prop`. `sep-order-not-prop` (non-prop loss
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order), `sep-map-over-{id,comp}` + functorial unit (generic
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Σ-functoriality survives), `sep-degrade-compose-fails` (the
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characteristic composition law is refuted, checked via
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`true ≢ false`). Verdict: `degrade-compose` is carried *precisely*
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by propositionality of the loss order — genuine structure, not
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Σ-bookkeeping. `All.agda` + `Smoke.agda` exit 0, no postulates.
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Smallest useful next advance:
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1. **Pillar D — `EchoRelModel.agda`.** Second model: transport the
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Pillar-B graded-comonad laws into the relational / fibration
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model (`EchoCategorical.Fibration`, `EchoRelational`). Two
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independent models = model-independence (robustness). Then a
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`model-agreement` lemma (Set vs relational under
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`fiber-to-echo`/`echo-to-fiber`).
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2. Pillar D artefact 2 — `docs/echo-types/conservativity.adoc`:
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because `Echo = Σ`, a small graded calculus with a loss modality
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embeds conservatively + definitionally into MLTT+Σ. State as a
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metatheorem.
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3. Pillar E — TYPES abstract / CPP-ITP / Zenodo.
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Rationale: the credibility core (A pin-identity, B universal
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property + graded comonad, C separating model) is complete. What
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remains for "established" standing is model-independence (D) — the
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property reviewers actually check — then external validation (E).
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**Pillar D LANDED (2026-05-17). Pillars A–D ALL COMPLETE — the
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entire internal programme is done.** `EchoRelModel.agda` real:
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abstract `GradedLossModel` interface + generic `GCLaws` proving the
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comonad laws ONCE for any model (the model-independence theorem
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itself); `set-model` (EchoGraded) and `rel-model` (relational
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`EchoStep`/`map-rel`; composition from `map-rel-comp` +
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`degrade-comp`) instances; `model-agreement` (round-trips `refl`)
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and `bridge-natural` (`map-over` ↔ `map-rel` under the graph =
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fibration bridge). `conservativity.adoc` states the
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conservative/definitional metatheorem with a 3-clause formal anchor.
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`All.agda` + `Smoke.agda` exit 0, no postulates / no escape pragmas.
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Smallest useful next advance — **Pillar E (external validation)
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only**; the internal programme is complete:
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1. Draft the TYPES abstract: "Echo: a definitionally-grounded graded
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comonad of structured loss" (low bar, fast expert feedback).
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Material is ready: thesis + H2 verdict + separating model +
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conservativity metatheorem.
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2. Then a CPP/ITP mechanised-metatheory paper (natural venue for a
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`--safe --without-K` Agda artefact with the gate/retraction
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discipline).
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3. Zenodo DOI + installable library packaging (stable module API);
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adjacency-note outreach (HoTT fibres / QTT-modal).
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Rationale: Pillars A (pin-identity), B (universal property + graded
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comonad), C (separating model) and D (second model +
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conservativity) are all complete and verified. Authority is now
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conferred socially — Pillar E is write-up / packaging / outreach,
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not internal proof work. Note: E is partly an offline/external
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task (submissions, DOIs); flag to the user rather than auto-running.
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‎docs/echo-types/conservativity.adoc‎

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= Echo Types — Conservativity Metatheorem
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:toc: macro
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:toclevels: 3
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:sectnums:
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:sectnumlevels: 3
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:icons: font
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[.lead]
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Pillar D, artefact 2 of `establishment-plan.adoc`. A type-theoretic
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notion earns "established" standing not only by having models, but by
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being a *conservative, definitionally-grounded* extension of a
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foundational theory. This document states that metatheorem for the
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loss-graded echo modality and records its formal anchor in the
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verified Agda development.
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toc::[]
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== Statement
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[#thm-conservativity]
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.Conservativity (definitional grounding)
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****
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Let `MLTT+Σ` be Martin-Löf type theory with Σ-types and the identity
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type, as realised by Agda under `--safe --without-K` (no axiom K, no
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univalence, no postulates, no funext). Let `Echo` be the loss-graded
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echo modality: the object `Echo f y := Σ (x : A) , (f x ≡ y)`
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together with the loss-grade lattice and its reindexing action
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(`EchoGraded`), and the graded-comonad structure
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(`EchoGradedComonad`, `EchoRelModel`).
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Then the extension of `MLTT+Σ` by `Echo` and all of its laws is
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*conservative and definitional*:
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. *Definitional.* `Echo f y` is not a new type former. It is, by
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`refl`, the homotopy fibre `fib f y` (`EchoFiberBridge.echo↔fib`,
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both round-trips `refl`). No new judgement, context structure, or
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formation rule is introduced — `Echo` is a derived Σ-type.
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. *Conservative.* Every theorem of the development is a closed term
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of `MLTT+Σ`: the entire suite typechecks under `--safe --without-K`
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with *zero postulates and zero escape pragmas*. Hence any
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proposition over the base vocabulary provable using `Echo` is
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already provable in `MLTT+Σ` — `Echo` proves no new base facts.
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. *Modality, not axiom.* The graded-comonad laws
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(`gcomonad-counit-{l,r}`, `gcomonad-coassoc`) are *derived* — the
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H2 verdict (see `establishment-plan.adoc`) showed each collapses to
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`degrade-compose` + order-propositionality, with no path algebra
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and no transport. The loss modality therefore adds structure
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(a characterised graded comonad) without adding strength.
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****
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== Why this is the clean result
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The honest deflationary fact — "`Echo` is just `fib`" — is precisely
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what makes the metatheorem strong. A construction that *required* new
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axioms (K, univalence, funext, or a postulated modality) would not be
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conservative and would inherit those axioms' costs. Here the opposite
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holds: the loss-graded comonad is a *theorem schema of MLTT+Σ
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itself*, instantiable in multiple models (Pillar D, artefact 1:
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`set-model` and `rel-model`, agreeing under the graph/fibration
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bridge). "Conservative, definitionally-grounded extension" is exactly
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the result that earns trust in the coeffect / quantitative lineage.
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== Formal anchor
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The metatheorem is not asserted in prose alone; its three clauses are
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each pinned to checked Agda artefacts already in `All.agda` /
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`Smoke.agda`:
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[cols="1,3,2", options="header"]
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|===
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| Clause | Witness | Module
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| Definitional
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| `echo↔fib : Echo f y ↔ fiber f y`, round-trips `refl`
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| `EchoFiberBridge`
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| Conservative
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| Whole suite: `--safe --without-K`, no `postulate` / no escape
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pragmas; CI greps enforce it
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| `All.agda` + `Smoke.agda`
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| Modality not axiom
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| `GradedLossModel` / `GCLaws` derive every comonad law generically;
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`model-agreement` shows model-independence
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| `EchoRelModel`
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| Modality not axiom
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| `gcomonad-coassoc` etc. reduce to `degrade-compose` + `≤g-prop`
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(H2 verdict); separating model proves the law is genuine
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| `EchoGradedComonad`,
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`EchoSeparating`
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|===
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The conservativity argument is *operational*: it is discharged by the
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build itself. Any reintroduction of a postulate, a weakening of
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`--safe --without-K`, or a funext dependency would break clause 2 and
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is caught by the repository's standing guardrails
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(`establishment-plan.adoc` §"Sequencing and guardrails").
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== Scope and non-claims
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* This is conservativity over `MLTT+Σ` *as realised under
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`--safe --without-K`*. It is not a claim about extensional type
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theory, nor about theories with K or univalence.
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* It is *not* claimed that the loss modality is interderivable with
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linear or dependent type strength — that comparison is a category
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error (`establishment-plan.adoc` §"The honest type-theoretic
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verdict"). The claim is narrower and exact: the loss-graded comonad
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is a conservative, definitional *modality* in the coeffect sense.
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* The funext boundary remains quarantined: `examples/Transport.agda`
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stalls where `Field = Fin n → ℚ` forces funext. No conservativity
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claim extends across that boundary; it is isolated by design.
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== Revision policy
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Update when a formal-anchor module changes state or a guardrail is
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hit. The statement (<<thm-conservativity>>) does not change without a
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recorded reason. Append-only revision history.
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=== Revision history
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* *2026-05-17 — created.* Pillar D artefact 2. Anchored to
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`EchoFiberBridge` (definitional), the postulate-free
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`--safe --without-K` suite (conservative), and
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`EchoRelModel`/`EchoGradedComonad`/`EchoSeparating` (modality not
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axiom). Pillar D complete.

‎docs/echo-types/establishment-plan.adoc‎

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* Artefact 1: `EchoRelModel.agda` — extend
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`EchoCategorical.Fibration` / `EchoRelational`; show the
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graded-comonad laws of Pillar B transport into the relational /
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presheaf model. Two independent models = robustness.
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presheaf model. Two independent models = robustness. *Landed
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2026-05-17:* the H2 verdict is leveraged into the strongest
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model-independence form — an abstract `GradedLossModel` interface
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with the comonad laws proved *once* generically (`module GCLaws`),
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instantiated at `set-model` (EchoGraded) and `rel-model`
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(relational `EchoStep`/`map-rel`, composition from `map-rel-comp`
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+ `degrade-comp`), with `model-agreement` (round-trips `refl`) and
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`bridge-natural` (`map-over` ↔ `map-rel` under the graph bridge).
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* Artefact 2: `docs/echo-types/conservativity.adoc` (+ formal anchor)
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— because `Echo = Σ`, a small graded calculus with a loss modality
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embeds *conservatively and definitionally* into MLTT+Σ. State this
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as a metatheorem. "Conservative, definitionally-grounded extension"
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is exactly the clean result that earns trust.
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is exactly the clean result that earns trust. *Landed 2026-05-17:*
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the metatheorem is stated with a three-clause formal anchor —
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definitional (`EchoFiberBridge`), conservative (postulate-free
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`--safe --without-K` suite, operationally enforced by CI greps),
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modality-not-axiom (`EchoRelModel`/`EchoGradedComonad`/
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`EchoSeparating`). Scope and non-claims explicit; funext boundary
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quarantined.
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* Authority effect: model-independence + a conservativity theorem turn
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the deflation into a published positive result.
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*Order (revised 2026-05-17 — de-risk first):* A ✓ →
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**B(graded comonad, thin slice) ✓ → B(pullback) ✓** →
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C(separating model) ✓ → D(second model + conservativity) →
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C(separating model) ✓ → D(second model + conservativity) ✓ →
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E(write-up). The graded-comonad result was the pivotal bet, so a
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minimal slice was attempted _before_ the safe pullback work to test
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the thesis early (fail fast); it settled positively and
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over-delivered all three laws, after which the pullback and the
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separating model landed as planned. *Pillars A, B and C are now
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complete — the credibility core is done.* Next: D(second model +
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conservativity). A–C were the credibility core and needed no
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external unblocker. The ordinal / Buchholz
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over-delivered all three laws, after which the pullback, the
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separating model and the second model + conservativity landed as
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planned. *Pillars A–D are now complete — the entire internal
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programme is done; only Pillar E (external validation) remains.*
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A–C were the credibility core and needed no external unblocker. The
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ordinal / Buchholz
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track is impressive _consumer evidence_, not foundation — keep it
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firewalled from the identity claim exactly as `roadmap.md` already does.
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| B | `EchoPullback.agda` (`echo-pullback-univ` + SliceHom↔cone) | Landed
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| B | `EchoGradedComonad.agda` (`gextract`/`gduplicate` + 3 laws) | Thin slice landed (verdict <<h2-verdict>>)
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| C | `EchoSeparating.agda` (`sep-degrade-compose-fails` + Σ-laws hold) | Landed
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| D | `EchoRelModel.agda` | Scaffolded (doc-only module, in `All.agda`)
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| D | `docs/echo-types/conservativity.adoc` | Not started
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| D | `EchoRelModel.agda` (`GradedLossModel`/`GCLaws`, 2 models, `model-agreement`) | Landed
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| D | `docs/echo-types/conservativity.adoc` (metatheorem + 3-clause anchor) | Landed
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| E | TYPES abstract / CPP-ITP paper / Zenodo / library packaging | Not started
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|===
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Pillars A, B and C are fully landed (`EchoFiberBridge`,
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`EchoPullback`, `EchoGradedComonad`, `EchoSeparating` — all real,
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verified, no postulates). The remaining Pillar D entry
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`EchoRelModel` is still a documentation-only Agda module (no
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declarations, hence no postulates or holes — it typechecks under
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`--safe --without-K` and is tracked in `All.agda` per the
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rung-consolidation policy), carrying the intended theorem signatures
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as commented specifications so a fresh session can start filling it
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without re-deriving the design. `conservativity.adoc` (Pillar D
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artefact 2) and Pillar E are not started.
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Pillars A, B, C and D are fully landed (`EchoFiberBridge`,
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`EchoPullback`, `EchoGradedComonad`, `EchoSeparating`,
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`EchoRelModel` — all real, verified, no postulates;
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`conservativity.adoc` stated with formal anchor). There are no
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remaining scaffold modules. Only Pillar E (external validation)
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is not started.
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== Revision policy
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precisely by propositionality of the loss order. *Pillars A, B, C
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complete* — the credibility core is done. Next per sequencing:
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Pillar D — `EchoRelModel.agda` (second model) + `conservativity.adoc`.
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* *2026-05-17 — trigger: Pillar D artefact state change (both
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artefacts).* Pillar D landed. Artefact 1: `EchoRelModel.agda`
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is real — abstract `GradedLossModel` + generic `GCLaws` (the
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model-independence theorem: comonad laws proved once for any
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model), instances `set-model`/`rel-model`, `model-agreement` and
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`bridge-natural`. Artefact 2: `conservativity.adoc` — the
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conservative/definitional metatheorem with a three-clause formal
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anchor to checked modules. *Pillars A–D all complete; no scaffold
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modules remain.* Only Pillar E (external validation) is left.

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