diff --git a/docs/JOURNAL.md b/docs/JOURNAL.md
index 30b18af..c91418a 100644
--- a/docs/JOURNAL.md
+++ b/docs/JOURNAL.md
@@ -13350,3 +13350,139 @@ milestone:
 - Prose round-trip for Float literals — iter 5.
 - DESIGN.md §"Float semantics" subsection (A5 determinism contract)
   and the line-2033 "supported primitive types" list update — iter 5.
+
+## 2026-05-10 — Iteration Floats.4: codegen + E2E
+
+Lowered every Float-related typecheck primitive that iter 3 installed
+into LLVM IR. The end-to-end fixture `examples/floats.ail.json` builds
+via `ail build`, runs, and produces the expected stdout
+(`4\n42\n-1.5\n`) — this is the milestone-acceptance gate, and it
+passed first try after the seven sub-tasks landed.
+
+`Float` is now a registered primitive in `synth.rs` (`llvm_type` →
+`double`, `type_descriptor` → `Fl` to avoid the single-letter `F` ADT
+prefix collision). Float literals lower as LLVM hex-float `double`
+SSA constants (`format!("0x{:016X}", bits)` — uppercase per LLVM
+convention). Both compile-completeness `unimplemented!("Floats
+milestone iter 4: codegen")` arms left by iter 1 in `lib.rs:918` /
+`:1215` are gone.
+
+Arithmetic and comparison ops dispatch on the resolved arg type. The
+old monomorphic-Int `synth::builtin_binop` was replaced by a
+3-tuple-returning `builtin_binop_typed(name, &Type) -> Option<(instr,
+operand_ll_ty, result_ll_ty)>`. Returning the result type
+explicitly eliminates the dual-meaning trap the original 2-tuple
+shape carried (caller had to know whether arm was arithmetic
+(operand-type == result-type) or comparison (result-type always
+`i1`)). The `lower_app` dispatch site was simplified accordingly —
+no more i1-result override at the caller. Float arithmetic emits
+`fadd/fsub/fmul/fdiv double`; Float comparison emits `fcmp
+olt/ole/ogt/oge/oeq double`. **`!=` Float emits `fcmp UNE double`,
+NOT `fcmp one`** — `one` is "ordered and not equal" and would
+return false for `nan != nan`, violating IEEE-`!=` and user-fixed
+constraint #2.
+
+Five new fn-builtin lowering arms in `lower_app`:
+
+- `neg` polymorphic — Int arm `sub i64 0, %x`; Float arm `fneg
+  double %x` (LLVM 8+, correct for `-0.0`). The `fsub double 0.0,
+  %x` desugar would be wrong: IEEE `0.0 - 0.0 == +0.0`, so
+  `neg(+0.0)` would erroneously produce `+0.0` instead of `-0.0`.
+- `int_to_float` → `sitofp i64 %x to double`.
+- `float_to_int_truncate` → `call i64 @llvm.fptosi.sat.i64.f64(double
+  %x)` (LLVM 12+ saturating intrinsic; clang 22 always available).
+  NaN → 0, +Inf → i64::MAX, -Inf → i64::MIN, finite-out-of-range
+  saturates, finite-in-range truncates toward zero — exactly the
+  intrinsic's documented semantics, no wrapping needed.
+- `is_nan` → `fcmp uno double %x, %x` (only NaN compares unordered
+  against itself).
+- `float_to_str` → `Err(CodegenError::Internal("...not yet
+  implemented (requires dynamic Str allocation in the runtime)"))`
+  (deferred to iter 5+; the runtime's Str path currently uses only
+  static `@.str_*` globals — no malloc-backed dynamic-Str
+  infrastructure). The deferral surfaces as a structured codegen
+  error rather than a panic, mirroring the rest of the file's
+  error discipline.
+
+Float constants `nan`/`inf`/`neg_inf` resolve as bare `Term::Var`
+references and lower to direct hex-float `double` SSA values at the
+use site — `0x7FF8000000000000` (canonical qNaN), `0x7FF0000000000000`
+(+Inf), `0xFFF0000000000000` (-Inf). Intercepted at the top of
+`lower_term`'s `Term::Var` arm, before any other resolution. **NOT**
+through the `__unreachable__` lowering pattern, which is a
+terminator instruction; constants are SSA values.
+
+`io/print_float` lowers via inline `printf("%g\n", v)` parallel to
+`io/print_int`. No new C runtime file needed — the `@printf`
+declaration already exists. Format string interned as `fmt_float`.
+
+Two plan bugs caught and corrected during the iteration:
+
+1. **Iter-4.2 plan inconsistency** — Task 2 as literally written
+   would have stranded comparison ops (Step 4 said
+   `builtin_binop_typed` returns None for them, Step 5 narrowed
+   `lower_app` matches! to `+ - * / %` only — together that
+   collapses the no-regression invariant). The implementer caught
+   this and pulled forward Task 3 Steps 3-4 as a minimal repair.
+   The orchestrator endorsed; Task 3's residual scope was narrowed
+   to Float comparison arms + `lower_eq` Float arm.
+2. **Iter-4.4 quality review** — `float_to_str` was specified as
+   `unimplemented!()` (panic), inconsistent with the rest of the
+   file's `CodegenError::Internal(...)` error discipline. A user
+   program calling the typecheck-installed `float_to_str` would
+   panic the compiler instead of producing a structured error.
+   Fixed in the 4.4 fixup commit.
+
+Two pull-forwards into iter-4.2 (`is_static_callee` extension for
+`==`) and iter-4.4 (`is_static_callee` extension for the 5 new
+fn-builtins) reflect the same pattern: any name that lowers via a
+direct `lower_app` arm must also be recognised by `is_static_callee`,
+or it falls through to the indirect-call path with `UnknownVar`.
+Future iterations adding new builtin lowering arms must remember
+this companion edit.
+
+The mono-pass and ADT slot-layout question from spec Components
+(monomorphisation produces concrete `List_Fl` slots vs. polymorphic
+erased slots needing bitcast) was NOT exercised by the E2E fixture
+— `examples/floats.ail.json` does not use Float-typed containers.
+This is not a regression: the descriptor `Fl` exists and the
+mono pass would emit `List_Fl` correctly; the bitcast question
+would only fire for an erased polymorphic container, which AILang
+doesn't have today. Queued as a follow-up if a future fixture
+exercises the path.
+
+Per-task commits:
+
+- `ac5e17e` floats iter 4.1: codegen primitive registration + Float literal hex-double lowering
+- `8044a4d` floats iter 4.2: codegen arithmetic dispatch on arg type — fadd/fsub/fmul/fdiv double
+- `2a29070` floats iter 4.2 fixup: 3-tuple return for builtin_binop_typed + classifier helper
+- `3869641` floats iter 4.3: codegen Float comparison arms (fcmp olt/ole/ogt/oge/UNE) + lower_eq Float
+- `581144a` floats iter 4.4: codegen neg/int_to_float/float_to_int_truncate/is_nan + float_to_str-deferred
+- `613aa39` floats iter 4.4 fixup: float_to_str returns CodegenError::Internal instead of panic
+- `bde5aaf` floats iter 4.5: codegen Float constants nan/inf/neg_inf as hex-double SSA values
+- `9764b61` floats iter 4.6: codegen io/print_float + examples/floats.ail.json E2E fixture
+
+Workspace at iter-4 close: 402 tests passed, 0 failed (from 395 at
+iter-3 close — added 6 codegen unit tests + 1 E2E test).
+
+Known debt deliberately deferred to iter 5:
+
+- Prose round-trip for Float literals.
+- DESIGN.md §"Float semantics" subsection (A5 determinism contract).
+- DESIGN.md line-2033 "supported primitive types" list update
+  (`Float` was not mentioned as a supported primitive there before;
+  iter 4's ship makes it true).
+- `crates/ailang-prose/src/lib.rs::Literal::Float` arm replacement
+  (currently `unimplemented!("Floats milestone iter 5: prose")`
+  from iter 1).
+
+Known debt deliberately deferred BEYOND iter 5 (this milestone):
+
+- `float_to_str` codegen lowering (requires runtime-allocated Str;
+  AILang's Str path is currently static-only). Symbol is
+  type-installed and surface-callable, but lowering errors with a
+  structured `CodegenError::Internal`. Future milestone wires the
+  runtime Str-allocator path.
+- Float-typed container slot layout (mono `List_Fl` etc.) — not
+  exercised by any current fixture; revisit if a future use case
+  surfaces.