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| 1 | +{-# OPTIONS --safe --without-K #-} |
| 2 | +-- SPDX-License-Identifier: MPL-2.0 |
| 3 | +-- SPDX-FileCopyrightText: 2025-2026 Jonathan D.A. Jewell <j.d.a.jewell@open.ac.uk> |
| 4 | + |
| 5 | +-- EchoAggregation: micro→macro economic aggregation as structured loss. |
| 6 | +-- |
| 7 | +-- This module mechanises the keystone claim of the oikos/betlang |
| 8 | +-- "aggregate library" design note (oikos |
| 9 | +-- `docs/alib-aggregate-bridge.adoc` §2): economic *aggregation* — |
| 10 | +-- rolling a micro ledger up into a macro observable — is literally an |
| 11 | +-- `Echo` map, and the *non-identifiability* of the micro state from |
| 12 | +-- the macro observable ("you cannot disaggregate") is literally the |
| 13 | +-- repo's `no-section` theorem. |
| 14 | +-- |
| 15 | +-- The honest minimal instance. The alib's `MacroState` is a rich |
| 16 | +-- record (population, elites, capital stock, …). Each of its fields |
| 17 | +-- is an aggregation of the same shape: a sum (a Godley column) of |
| 18 | +-- micro entries. The load-bearing structural fact is visible already |
| 19 | +-- at the smallest faithful case — a two-account ledger collapsing to |
| 20 | +-- a total: |
| 21 | +-- |
| 22 | +-- * `MicroLedger = ℕ × ℕ` two sector balances (e.g. household, |
| 23 | +-- firm) — the micro state; |
| 24 | +-- * `MacroTotal = ℕ` the aggregate money stock — one Godley |
| 25 | +-- column sum, the macro observable; |
| 26 | +-- * `aggregate (a , b) = a + b` the rollup. |
| 27 | +-- |
| 28 | +-- The full `MacroState` is then a product of such projections; the |
| 29 | +-- structural story (many-to-one ⇒ no canonical disaggregation) is |
| 30 | +-- identical field-by-field, so the single-column instance is the |
| 31 | +-- right place to pin it. |
| 32 | +-- |
| 33 | +-- What is proved. |
| 34 | +-- |
| 35 | +-- * `ConsistentLedgers m = Echo aggregate m` — the fibre: ALL micro |
| 36 | +-- ledgers consistent with the macro total `m`. This IS the |
| 37 | +-- economist's "aggregation is many-to-one", as a type. |
| 38 | +-- * `aggregate-non-injective` — two distinct micro ledgers, |
| 39 | +-- `(0,1)` and `(1,0)`, are distinct echoes at the SAME macro |
| 40 | +-- total `1`. The fibre is genuinely non-trivial. |
| 41 | +-- * `no-canonical-disaggregation` (keystone) — `aggregate` admits |
| 42 | +-- NO section: there is no `raise : MacroTotal → MicroLedger` |
| 43 | +-- recovering the micro split from the macro total for every |
| 44 | +-- input. This is the aggregation / non-identifiability problem, |
| 45 | +-- as a theorem, obtained by instantiating the generic |
| 46 | +-- `EchoNoSectionGeneric.no-section-of-collapsing-map`. |
| 47 | +-- |
| 48 | +-- This is the SAME `no-section` machinery that underwrites the |
| 49 | +-- affine⊑linear story in the wasm proof layer (`EchoLinear.weaken`, |
| 50 | +-- machine-checked equal to AffineScript subtyping in |
| 51 | +-- `nextgen-typing`'s `EchoTyping.agda`). One type language serves |
| 52 | +-- micro→macro aggregation, cross-language ABI, and uncertainty. |
| 53 | +-- |
| 54 | +-- Headlines (pinned in Smoke.agda): |
| 55 | +-- |
| 56 | +-- * aggregate -- the rollup map |
| 57 | +-- * ConsistentLedgers -- its fibre, as an Echo |
| 58 | +-- * aggregate-non-injective -- the fibre is non-trivial |
| 59 | +-- * no-canonical-disaggregation -- the keystone: no section |
| 60 | +-- |
| 61 | +-- Scope guardrail. `aggregate` here is a concrete finite ℕ-valued |
| 62 | +-- map; the theorem is about THIS map's non-injectivity. It does NOT |
| 63 | +-- claim a quantitative bound on the size of fibres, nor anything |
| 64 | +-- about the rich `MacroState` record's joint identifiability — those |
| 65 | +-- are downstream, and named in the alib note's open questions. The |
| 66 | +-- minimal claim is exactly the load-bearing one: aggregation is an |
| 67 | +-- Echo, and the macro observable cannot in general be disaggregated. |
| 68 | + |
| 69 | +module EchoAggregation where |
| 70 | + |
| 71 | +open import Echo using (Echo; echo-intro) |
| 72 | +open import EchoNoSectionGeneric using (no-section-of-collapsing-map) |
| 73 | + |
| 74 | +open import Data.Nat.Base using (ℕ; _+_) |
| 75 | +open import Data.Product.Base using (Σ; _×_; _,_; proj₁) |
| 76 | +open import Relation.Binary.PropositionalEquality |
| 77 | + using (_≡_; _≢_; refl; cong) |
| 78 | +open import Relation.Nullary using (¬_) |
| 79 | + |
| 80 | +---------------------------------------------------------------------- |
| 81 | +-- The micro / macro types and the aggregation map. |
| 82 | +---------------------------------------------------------------------- |
| 83 | + |
| 84 | +-- A micro ledger: two sector balances (e.g. household, firm). |
| 85 | +MicroLedger : Set |
| 86 | +MicroLedger = ℕ × ℕ |
| 87 | + |
| 88 | +-- The macro observable: one aggregate total (a Godley column sum). |
| 89 | +MacroTotal : Set |
| 90 | +MacroTotal = ℕ |
| 91 | + |
| 92 | +-- Aggregation: roll the micro ledger up into the macro total. |
| 93 | +aggregate : MicroLedger → MacroTotal |
| 94 | +aggregate (a , b) = a + b |
| 95 | + |
| 96 | +---------------------------------------------------------------------- |
| 97 | +-- The fibre, as an Echo. |
| 98 | +-- |
| 99 | +-- `ConsistentLedgers m` is the type of ALL micro ledgers whose rollup |
| 100 | +-- is exactly the macro total `m`. Definitionally it is |
| 101 | +-- `Σ MicroLedger (λ l → aggregate l ≡ m)` — the fibre of `aggregate` |
| 102 | +-- over `m`. "Aggregation is many-to-one" becomes "this type can have |
| 103 | +-- more than one inhabitant" (witnessed below). |
| 104 | +---------------------------------------------------------------------- |
| 105 | + |
| 106 | +ConsistentLedgers : MacroTotal → Set |
| 107 | +ConsistentLedgers m = Echo aggregate m |
| 108 | + |
| 109 | +---------------------------------------------------------------------- |
| 110 | +-- The fibre over macro total 1 is non-trivial: two distinct micro |
| 111 | +-- ledgers, (0,1) and (1,0), both aggregate to 1. |
| 112 | +---------------------------------------------------------------------- |
| 113 | + |
| 114 | +ledger₁ : MicroLedger |
| 115 | +ledger₁ = 0 , 1 |
| 116 | + |
| 117 | +ledger₂ : MicroLedger |
| 118 | +ledger₂ = 1 , 0 |
| 119 | + |
| 120 | +-- The two micro ledgers are distinct: their household balances differ |
| 121 | +-- (0 vs 1). Refuted at the first projection by constructor clash. |
| 122 | +ledger₁≢ledger₂ : ledger₁ ≢ ledger₂ |
| 123 | +ledger₁≢ledger₂ eq with cong proj₁ eq |
| 124 | +... | () |
| 125 | + |
| 126 | +-- … yet they collapse to the same macro total (both 1). |
| 127 | +aggregate-collapses : aggregate ledger₁ ≡ aggregate ledger₂ |
| 128 | +aggregate-collapses = refl |
| 129 | + |
| 130 | +-- As echoes at the same macro total. |
| 131 | +echo-ledger₁ : ConsistentLedgers 1 |
| 132 | +echo-ledger₁ = echo-intro aggregate ledger₁ |
| 133 | + |
| 134 | +echo-ledger₂ : ConsistentLedgers 1 |
| 135 | +echo-ledger₂ = echo-intro aggregate ledger₂ |
| 136 | + |
| 137 | +-- The fibre is genuinely non-trivial: two distinct inhabitants at the |
| 138 | +-- same macro observable. This is "aggregation is many-to-one", as a |
| 139 | +-- checked theorem. |
| 140 | +aggregate-non-injective : echo-ledger₁ ≢ echo-ledger₂ |
| 141 | +aggregate-non-injective eq = ledger₁≢ledger₂ (cong proj₁ eq) |
| 142 | + |
| 143 | +---------------------------------------------------------------------- |
| 144 | +-- The keystone: no canonical disaggregation. |
| 145 | +-- |
| 146 | +-- There is no section `raise : MacroTotal → MicroLedger` recovering |
| 147 | +-- the micro split from the macro total for every input — i.e. no |
| 148 | +-- function with `raise (aggregate l) ≡ l` for all micro ledgers `l`. |
| 149 | +-- This is the aggregation / non-identifiability problem of |
| 150 | +-- macroeconomics, obtained as a one-instance application of the |
| 151 | +-- generic no-section theorem. |
| 152 | +---------------------------------------------------------------------- |
| 153 | + |
| 154 | +no-canonical-disaggregation : |
| 155 | + ¬ Σ (MacroTotal → MicroLedger) |
| 156 | + (λ raise → ∀ l → raise (aggregate l) ≡ l) |
| 157 | +no-canonical-disaggregation = |
| 158 | + no-section-of-collapsing-map |
| 159 | + aggregate |
| 160 | + ledger₁ ledger₂ |
| 161 | + ledger₁≢ledger₂ |
| 162 | + aggregate-collapses |
| 163 | + |
| 164 | +---------------------------------------------------------------------- |
| 165 | +-- Companion remark. |
| 166 | +-- |
| 167 | +-- Why this is the right level of generality: |
| 168 | +-- |
| 169 | +-- * The fibre `ConsistentLedgers m` is `Echo aggregate m` ON THE |
| 170 | +-- NOSE (definitional), so every downstream `Echo`/`EchoResidue` |
| 171 | +-- result applies to aggregation without restatement. In |
| 172 | +-- particular the residue machinery names what the macro layer is |
| 173 | +-- entitled to observe after the loss. |
| 174 | +-- |
| 175 | +-- * `no-canonical-disaggregation` refutes a LEFT inverse (a |
| 176 | +-- section of `aggregate`). It does NOT refute the existence of |
| 177 | +-- SOME right inverse / choice of representative — economists pick |
| 178 | +-- representatives all the time (a "typical household"). The |
| 179 | +-- content is precisely that no such choice is CANONICAL: it |
| 180 | +-- cannot satisfy `raise ∘ aggregate ≡ id`, so it always discards |
| 181 | +-- information about ledgers it did not pick. |
| 182 | +-- |
| 183 | +-- * Promoting this to the rich `MacroState` record is mechanical: |
| 184 | +-- each field is an aggregation of this shape, and a section of |
| 185 | +-- the product would restrict to a section of each projection, |
| 186 | +-- which this theorem already refutes. No new proof idea is |
| 187 | +-- needed; see the alib note §3–§4. |
| 188 | +---------------------------------------------------------------------- |
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