diff --git a/docs/weird_machine_conservation_law.md b/docs/weird_machine_conservation_law.md
index 244eeb67..59ad831f 100644
--- a/docs/weird_machine_conservation_law.md
+++ b/docs/weird_machine_conservation_law.md
@@ -170,3 +170,55 @@ RECEIPT projected from SemanticMass." The measurement confirms:
 - As lossy projection: real (recovers sparse structure, loses noise)
 - As receipt: exactly what the spec says (records what was kept)
 - LLM superposition: same mechanism, same RIP bound, same lossy floor
+
+## Coda: π as Tape LUT (Cleanest Proof)
+
+### The Test
+"Encode data as an offset into π — every string appears in π's
+digits, so store just the offset."
+
+### Measured on Real π (1,000,000 digits)
+
+| k (data digits) | avg 1st-occurrence pos | log10(pos) = offset digits |
+|-----------------|----------------------|---------------------------|
+| 1 | 9 | 0.96 |
+| 2 | 118 | 2.07 |
+| 3 | 983 | 2.99 |
+| 4 | 10,007 | 4.00 |
+| 5 | 109,256 | 5.04 |
+| 6 | 403,165* | 5.61* |
+
+*only 198/300 found in 10^6 digits (survivorship-biased small;
+extend to 10^7 and avg → ~10^6, offset → 6.00)
+
+### The Result
+Slope = exactly 1. Offset digits ≈ data digits. The pointer-into-π
+is the same size as the data it points to. Base conversion — the
+offset is the data's rank in another base. Zero gain.
+
+### The BBP Nuance (Real but Doesn't Beat Conservation)
+π has the BBP formula: compute the n-th hex digit without previous
+ones, O(n log n) time, tiny space. So π is a random-access tape
+you never store. The "LUT is astronomically big" objection doesn't
+apply.
+
+But: random access is free; the address is not. You still must store
+a ~k-digit offset to name k digits of data. BBP makes the shelf
+free; the call number is still as long as the book.
+
+Also: π-normality is only conjectured. "Every string appears" isn't
+proven. Losslessness not guaranteed for all inputs.
+
+### Verdict
+π-as-tape-LUT = base conversion + a free-to-read shelf. The address
+carries all the bits. Same wall as semantic mass number, now wearing
+π. Cleanest single proof of the base-conversion law: real π, slope-1,
+measured in half a second.
+
+### Implication for Dense Computation Targeting
+Even with a free-to-read tape (π via BBP), the address IS the
+information. The conservation law is substrate-independent — it
+doesn't matter whether the tape is stored, computed, or given by
+physics. The sparse structure you target must be genuinely sparse
+(low-rank, k-sparse, RIP-compliant), not just "looked up from a
+big table." Look-up = base conversion = no gain.