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error-lang: replace fabricated ABI "proofs" with machine-checked ones (#24)
The three "Safety Proofs" in src/abi/Foreign.idr (stabilityBounded, positionalDeterministic, paradoxMonotonic) were not proofs: each fabricated its evidence with cast ()/cast Refl over an IO action. Replace them with genuine, self-contained Idris2 modules, machine-checked under Idris 2 v0.8.0 (no believe_me / assert_total / cast / postulate): - Stability.idr : stability score is bounded in [0,100] (clamp model of compiler/src/Types.res calculateStability). - Positional.idr : positional-operator behaviour is deterministic over the pure model of the Zig FFI (a genuine Refl, not IO cast Refl). - Paradox.idr : the two threshold-gated factors are monotone; the blanket "paradox detection monotonic" claim is RETRACTED -- proving it honestly surfaced that scope_leakage is prime-gated and therefore non-monotone (line 7 prime, line 8 not). Foreign.idr is reduced to an honest, self-contained ABI binding layer. Add error-lang-abi.ipkg and verification/check-proofs.sh (idris2 --check all four modules). Rewrite PROOF-NEEDS.md to record what is proved, what is retracted, the toolchain, and open conformance obligations. The language's satirical "100% production-ready / formally verified" self-presentation in README/WHITEPAPER is intentional and left intact; this change only makes the underlying proof artifacts real and honest. Claude-Session: https://claude.ai/code/session_0195yA45jSSP7YDPwJSpw4bM Co-authored-by: Claude Opus 4.8 <noreply@anthropic.com>
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‎PROOF-NEEDS.md‎

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-->
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# PROOF-NEEDS.md
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## Template ABI Cleanup (2026-03-29)
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> Engineering ledger for the error-lang **formal core**. This is the honest
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> substrate beneath the language's deliberately tongue-in-cheek "100%
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> production-ready, formally verified" self-presentation: it records what is
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> *actually* machine-checked, what is not, and how to reproduce the checks.
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> The language may dissemble about itself on purpose — this file does not.
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Template ABI removed -- was creating false impression of formal verification.
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The removed files (Types.idr, Layout.idr, Foreign.idr) contained only RSR template
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scaffolding with unresolved {{PROJECT}}/{{AUTHOR}} placeholders and no domain-specific proofs.
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## Formal core (`src/abi/`)
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When this project needs formal ABI verification, create domain-specific Idris2 proofs
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following the pattern in repos like `typed-wasm`, `proven`, `echidna`, or `boj-server`.
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Three properties of the computational-haptics engine are proved in Idris2 and
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**machine-checked under Idris 2, version 0.8.0**, with **no escape hatches**
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(no `believe_me`, `assert_total`, `cast`-coerced equality, or `postulate`):
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| Property | Module | Status |
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|---|---|---|
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| Stability score ∈ [0, 100] | `src/abi/Stability.idr` | ✅ proved (`stabilityUpperBound`, `stabilityLowerBound`) |
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| Positional-operator determinism | `src/abi/Positional.idr` | ✅ proved (`positionalDeterministic`) + sanity evaluations |
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| Paradox-factor monotonicity | `src/abi/Paradox.idr` | ⚠️ partial — two factors proved, blanket claim retracted (below) |
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`src/abi/Foreign.idr` is an honest, self-contained ABI **binding-declaration**
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layer (it asserts no theorems). All four modules are listed in
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`src/abi/error-lang-abi.ipkg`.
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### Reproduce
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```sh
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# idris2 is not in apt here, and the ziglang/deno mirrors are blocked by the
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# environment network policy, so build the proof checker from source via Chez:
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sudo apt-get install -y chezscheme libgmp-dev make gcc
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git clone https://github.com/idris-lang/Idris2 && cd Idris2
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make bootstrap SCHEME=chezscheme && make install
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export PATH="$HOME/.idris2/bin:$PATH"
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# then, from the error-lang repo root:
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./verification/check-proofs.sh
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# or: cd src/abi && idris2 --typecheck error-lang-abi.ipkg
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```
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### The monotonicity retraction (an honest finding)
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The previously-advertised property *"paradox detection is monotonic with
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complexity"* is **false of the implementation** — and attempting to prove it
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honestly is what surfaced that. `error_lang_detect_paradoxes`
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(`ffi/zig/src/main.zig`) gates `scope_leakage` on `isPrime(line_count)`, which
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is not monotone: line **7** is prime → active, line **8** is composite →
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inactive, even though 8 > 7.
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`src/abi/Paradox.idr` therefore proves the part that *is* true — the two
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threshold-gated factors are monotone in their driving metric
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(`superpositionMonotone` for `var_count > 10`; `temporalMonotone` for
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`depth > 5`) — and retracts the blanket claim, recording the scope-leakage
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obstruction explicitly. Non-monotone scope leakage is intentional; it is the
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pedagogical point of the paradox. The difference now is that the proof says so
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out loud, instead of hiding it behind `cast Refl`.
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## What was removed (2026-06-23)
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`src/abi/Foreign.idr` previously carried three "Safety Proofs" —
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`stabilityBounded`, `positionalDeterministic`, `paradoxMonotonic` — that were
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**not proofs**. Each manufactured its evidence with `cast ()` / `cast Refl`
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over an `IO` action (e.g. calling an FFI function twice and coercing
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`Refl : x = x` onto the two distinct results, with a comment that it "should
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hold in practice"). An earlier note in this file claimed these files had been
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removed; in fact `Foreign.idr` was still present and still exported the fakes.
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They are now deleted and replaced by the genuine, machine-checked modules above.
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## Open obligations
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1. **CI gate.** Add an Idris2 `--typecheck error-lang-abi.ipkg` job so the core
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is checked on every push. (The dev image has no idris2 by default; it was
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built from source for this change.)
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2. **Implementation conformance.** The proofs are stated over abstract models
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that mirror `ffi/zig/src/main.zig` and `compiler/src/Types.res`. Two of those
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implementations **disagree**: positional behaviour is `column % 2` (two-way)
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in the Zig FFI but `(line*31 + column) mod 4` (four-way) in `Stability.res`.
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Reconcile them, then bind the proofs to the chosen implementation by
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extraction or conformance tests rather than parallel models.
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3. **Zig weighted-average path.** `error_lang_calculate_stability` is a convex
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combination (weights sum to 1) of per-factor scores in [0,100]; its [0,100]
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bound holds for a *different* reason than the `Stability.res` clamp proved
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here. Prove that path too.
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4. **Programs not executed in this environment.** Under the current network
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policy the Deno runtime's JSR std deps (`jsr.io`) and Zig 0.13.0
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(`ziglang.org`) are unreachable, and the ReScript compiler does not currently
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build (`return` is not valid ReScript — `VM.res:407`; `dict<string, int>`
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applies the one-argument `dict` constructor to two arguments —
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`Types.res:233`). These were **not** run or fixed as part of this change and
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are tracked as separate work — they are not claimed to pass.

‎src/abi/Foreign.idr‎

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||| All functions have type signatures and safety guarantees proven at
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||| compile-time through dependent types.
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module ErrorLang.ABI.Foreign
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import ErrorLang.ABI.Types
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import ErrorLang.ABI.Layout
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module Foreign
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%default total
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--------------------------------------------------------------------------------
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-- Minimal ABI value types (inlined so this binding module is self-contained
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-- and independently checkable: `idris2 --check Foreign.idr` from src/abi).
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-- The previous external `ErrorLang.ABI.Types` / `ErrorLang.ABI.Layout` modules
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-- were removed. The fabricated "Safety Proofs" that lived here -- which used
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-- `cast ()` / `cast Refl` over IO actions to manufacture evidence -- have been
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-- replaced by genuine, machine-checked proofs in the sibling modules
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-- Stability.idr, Positional.idr and Paradox.idr.
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--
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-- This file is a BINDING-DECLARATION layer only: it declares the C ABI of the
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-- Zig haptics library (ffi/zig). It asserts no theorems.
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--------------------------------------------------------------------------------
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||| Result codes (must match the Zig `Result` enum in ffi/zig/src/main.zig).
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public export
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data Result = Ok | Error | InvalidParam | OutOfMemory | NullPointer
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||| Opaque handle to a library instance (wraps the C pointer as Bits64).
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public export
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record Handle where
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constructor MkHandle
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handlePtr : Bits64
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||| Build a handle from a raw pointer; a null (0) pointer yields Nothing.
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export
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createHandle : Bits64 -> Maybe Handle
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createHandle 0 = Nothing
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createHandle p = Just (MkHandle p)
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--------------------------------------------------------------------------------
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-- Library Lifecycle
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--------------------------------------------------------------------------------
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pure (result /= 0)
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--------------------------------------------------------------------------------
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-- Safety Proofs
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-- Safety properties
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--------------------------------------------------------------------------------
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||| Theorem: Stability scores are always bounded [0, 100]
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|||
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||| Proof: The Zig implementation validates all score inputs in
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||| error_lang_set_stability_factor, rejecting any value < 0 or > 100.
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||| The weighted average in error_lang_calculate_stability preserves
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||| this bound through convex combination.
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export
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stabilityBounded : (h : Handle) -> (factor : Bits8) ->
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IO (Either Result (score : Double ** (0.0 <= score, score <= 100.0)))
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stabilityBounded h factor = do
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score <- getStabilityFactor h factor
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-- Runtime check (could be proven statically with refinement types)
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if score >= 0.0 && score <= 100.0
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then pure (Right (score ** (cast (), cast ())))
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else pure (Left Error)
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||| Theorem: Positional operator behavior is deterministic
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|||
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||| Proof: For any given (line, column, operatorType) triple, the Zig
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||| implementation always returns the same behavior. The calculation is
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||| purely functional (column % n) with no hidden state.
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export
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positionalDeterministic : (h : Handle) -> (line, column : Bits32) -> (op : Bits8) ->
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IO (b1 : Bits8 ** (b2 : Bits8 ** (b1 = b2)))
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positionalDeterministic h line column op = do
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b1 <- positionalOperator h line column op
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b2 <- positionalOperator h line column op
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-- In practice, should always be equal
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pure (b1 ** (b2 ** cast Refl))
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||| Theorem: Paradox detection is monotonic with respect to code complexity
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|||
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||| As lineCount, varCount, or depth increase, the set of detected paradoxes
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||| (represented as a bitmask) cannot decrease.
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export
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paradoxMonotonic : (h : Handle) ->
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(lc1, lc2, vc1, vc2, d1, d2 : Bits32) ->
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(lc1 <= lc2) -> (vc1 <= vc2) -> (d1 <= d2) ->
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IO (p1 : Bits32 ** (p2 : Bits32 ** ((p1 .&. p2) = p1)))
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paradoxMonotonic h lc1 lc2 vc1 vc2 d1 d2 _ _ _ = do
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p1 <- detectParadoxes h lc1 vc1 d1
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p2 <- detectParadoxes h lc2 vc2 d2
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-- Monotonicity should hold in practice (bitwise subset)
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pure (p1 ** (p2 ** cast Refl))
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-- The properties this ABI relies on are proved -- genuinely, with no escape
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-- hatch and machine-checked under Idris2 0.8.0 -- in the sibling modules, NOT
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-- here:
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-- * stability score in [0,100] -> Stability.idr (stabilityUpperBound)
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-- * positional operator determinism -> Positional.idr (positionalDeterministic)
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-- * paradox-factor monotonicity -> Paradox.idr (superpositionMonotone,
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-- temporalMonotone)
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--
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-- The earlier `stabilityBounded` / `positionalDeterministic` / `paradoxMonotonic`
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-- definitions here were unsound: they used `cast ()` / `cast Refl` over `IO`
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-- actions to fabricate evidence. Removed 2026-06-23. Proving the third one
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-- honestly also revealed that the *global* monotonicity claim is false of the
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-- implementation -- scope leakage is prime-gated; see Paradox.idr.

‎src/abi/Paradox.idr‎

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-- SPDX-License-Identifier: MPL-2.0
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-- Copyright (c) 2026 Jonathan D.A. Jewell (hyperpolymath) <j.d.a.jewell@open.ac.uk>
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||| Paradox-detection monotonicity (error-lang formal core, property 3 of 3).
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|||
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||| Mirrors the Zig FFI `error_lang_detect_paradoxes`
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||| (`ffi/zig/src/main.zig`, lines 280-315), which sets:
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||| * type_superposition when var_count > 10 (threshold; monotone)
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||| * scope_leakage when isPrime(line) (prime-gated; NOT monotone)
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||| * temporal_corruption when depth > 5 (threshold; monotone)
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|||
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||| HONEST FINDING. The previously-claimed blanket theorem "paradox detection
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||| is monotonic with complexity" (README, WHITEPAPER §7, COMPLETION report)
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||| is FALSE of the implementation: scope_leakage is gated on the PRIMALITY of
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||| the line number, which is not monotone (line 7 is prime -> active; line 8
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||| is composite -> inactive, although 8 > 7). The original
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||| `Foreign.idr :: paradoxMonotonic` hid this with `cast Refl`; attempting the
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||| proof honestly surfaces that the blanket claim cannot hold.
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|||
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||| What IS true, and is proved here: the two THRESHOLD-gated factors are
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||| monotone in their driving metric. The blanket claim is therefore retracted
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||| in favour of these two component lemmas (see PROOF-NEEDS.md). Non-monotone
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||| scope leakage is intentional -- it is the pedagogical point of the paradox.
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|||
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||| Self-contained; no escape hatches. Machine-check is a CI obligation.
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module Paradox
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import Data.Nat
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%default total
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||| Transitivity of <= (self-contained; the standard definition).
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lteTrans : LTE a b -> LTE b c -> LTE a c
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lteTrans LTEZero _ = LTEZero
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lteTrans (LTESucc p) (LTESucc q) = LTESucc (lteTrans p q)
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-- ───────────────────────────────────────────────────────────────────────
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-- Threshold-gated factors (monotone)
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-- ───────────────────────────────────────────────────────────────────────
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||| type_superposition fires when var_count exceeds 10 (11 <= var_count).
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public export
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SuperpositionActive : (varCount : Nat) -> Type
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SuperpositionActive varCount = LTE 11 varCount
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||| temporal_corruption fires when depth exceeds 5 (6 <= depth).
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public export
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TemporalActive : (depth : Nat) -> Type
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TemporalActive depth = LTE 6 depth
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||| THEOREM: type_superposition is monotone in var_count -- growing the
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||| variable count never deactivates it.
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public export
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superpositionMonotone : (v1, v2 : Nat) -> LTE v1 v2 ->
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SuperpositionActive v1 -> SuperpositionActive v2
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superpositionMonotone _ _ le active = lteTrans active le
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||| THEOREM: temporal_corruption is monotone in depth.
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public export
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temporalMonotone : (d1, d2 : Nat) -> LTE d1 d2 ->
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TemporalActive d1 -> TemporalActive d2
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temporalMonotone _ _ le active = lteTrans active le
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-- ───────────────────────────────────────────────────────────────────────
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-- Scope leakage is NOT monotone (the retraction, made precise)
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-- ───────────────────────────────────────────────────────────────────────
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||| scope_leakage fires on prime line numbers. The obstruction to global
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||| monotonicity, stated abstractly: for ANY predicate `p` with `p a = True`
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||| and `p b = False` at `a <= b`, the detected set decreases as the metric
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||| grows. Primality is such a `p` (witness a = 7, b = 8). Hence no global
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||| monotonicity theorem exists -- and that is by design.
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public export
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scopeLeakObstruction : (p : Nat -> Bool) -> (a, b : Nat) -> LTE a b ->
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p a = True -> p b = False ->
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(p a = True, p b = False)
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scopeLeakObstruction _ _ _ _ activeAtA inactiveAtB = (activeAtA, inactiveAtB)

‎src/abi/Positional.idr‎

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-- SPDX-License-Identifier: MPL-2.0
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-- Copyright (c) 2026 Jonathan D.A. Jewell (hyperpolymath) <j.d.a.jewell@open.ac.uk>
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||| Positional-operator determinism (error-lang formal core, property 2 of 3).
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|||
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||| Mirrors the Zig FFI `error_lang_positional_operator`
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||| (`ffi/zig/src/main.zig`, lines 222-272): for `+` the behaviour is
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||| `addition` when the column is even and `concatenation` when odd; for `*`
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||| it is `multiplication` when the column is a multiple of 3 and
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||| `exponentiation` otherwise.
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|||
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||| THEOREM. Operator behaviour is a deterministic (pure, total) function of
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||| the operator and the column: `behavior op col = behavior op col`.
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|||
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||| This is `Refl` -- *because* the model is a pure total function. That is
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||| exactly the point: the previous `Foreign.idr :: positionalDeterministic`
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||| stated the same property over an `IO Bits8` FFI action and could only
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||| "close" it with `cast Refl`, coercing `Refl : x = x` onto two distinct IO
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||| results -- a non-proof. Modelling the operation as the pure function it
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||| actually is makes determinism genuine.
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|||
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||| Self-contained; no escape hatches. Machine-check is a CI obligation.
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|||
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||| NOTE (conformance). `compiler/src/Stability.res` uses a *different* rule
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||| (`(line*31+column) mod 4`, four-way). The Zig FFI and the ReScript path
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||| therefore disagree; reconciling the two implementations is an open
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||| obligation recorded in PROOF-NEEDS.md.
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module Positional
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%default total
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||| Operators that carry positional behaviour.
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public export
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data Op = Plus | Star | OtherOp
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||| Resolved operator behaviours (mirrors Zig `OperatorBehavior`).
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public export
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data Behavior
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= Addition
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| Concatenation
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| Multiplication
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| Exponentiation
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||| Column parity (total, structural).
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public export
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isEven : Nat -> Bool
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isEven Z = True
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isEven (S Z) = False
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isEven (S (S k)) = isEven k
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||| Divisibility by three (total, structural).
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public export
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multipleOfThree : Nat -> Bool
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multipleOfThree Z = True
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multipleOfThree (S Z) = False
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multipleOfThree (S (S Z)) = False
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multipleOfThree (S (S (S k))) = multipleOfThree k
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||| The positional behaviour function (pure model of the Zig FFI).
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public export
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behavior : Op -> Nat -> Behavior
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behavior Plus col = if isEven col then Addition else Concatenation
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behavior Star col = if multipleOfThree col then Multiplication else Exponentiation
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behavior OtherOp _ = Addition
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||| THEOREM: behaviour is deterministic -- a genuine `Refl` over a pure
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||| total function (contrast the original IO-based `cast Refl`).
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public export
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positionalDeterministic : (op : Op) -> (col : Nat) -> behavior op col = behavior op col
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positionalDeterministic _ _ = Refl
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-- ───────────────────────────────────────────────────────────────────────
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-- Sanity evaluations (match the Zig integration tests + README example)
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-- ───────────────────────────────────────────────────────────────────────
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||| Column 12 (even): `+` is addition. (Zig test "positional semantics".)
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exEvenAddition : behavior Plus 12 = Addition
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exEvenAddition = Refl
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||| Column 13 (odd): `+` is concatenation.
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exOddConcatenation : behavior Plus 13 = Concatenation
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exOddConcatenation = Refl
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||| Column 9 (multiple of 3): `*` is multiplication.
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exStarMultiplication : behavior Star 9 = Multiplication
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exStarMultiplication = Refl
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||| Column 10 (not a multiple of 3): `*` is exponentiation.
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exStarExponentiation : behavior Star 10 = Exponentiation
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exStarExponentiation = Refl

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