W963: the published TNF8 area penalty had the wrong sign — the true rung is 0.9 % cheaper, the substitute 5.0 % dearer

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#668 established that every rig here binds "tnf8" to TNFFormat(4, 3) — 11 bits, 126.91 binades — while the ladder's eighth rung is TNFFormat(3, 4) — 10 bits, 30.95 binades. The census rig carries the substitution in its own table:

("tnf8", 11, lambda: (T, T.TNFFormat(4, 3))),

The size of that error was unknown. It is now measured, in this project's original metric — an oracle-derived decode table fused with a multiply, cost as the slope of cells(N) = fixture + cost·N — each format against a float of its own physical width:

format	bits	distinct values	decoder	consumer
TNF8 = (3,4) — the ladder's rung	10	962	12.00	212.57
fp10 e5m4	10	1023	14.00	214.57
"TNF8" = (4,3) — as measured	11	2018	29.00	270.57
fp11 e6m4	11	2047	16.00	257.57

R² ≥ 0.9996 on every fit.

The penalty had the wrong sign

The ladder's true rung is 0.9 % cheaper than its width-matched float. The substitute is 5.0 % dearer. The figure this project published for TNF8 was not an overstatement — it pointed the wrong way.

The decoder is where the substitution bit: 12.00 cells against 29.00, a factor of 2.4. The substitute inherits TNF16's four-trit exponent, so its decode table spans 127 binades and resists the synthesiser's factoring; the true rung's 31 binades do not.

Two metrics, two signs, one conclusion

In the float-style MAC-lane metric (#669) the same rung is +1.1 % (380 vs 376). Here it is −0.9 %. A constant multiplicand lets the synthesiser specialise (#667), which flatters whichever format has the smaller decode table — so the two metrics tilt in opposite directions by construction.

Two metrics straddling zero is a stronger statement of parity than either number alone. A single metric landing at 1 % invites a reader to treat 1 % as a finding; two metrics with opposite signs do not.

What this does not repair

The accuracy, activation and convolution figures for "TNF8" still describe TNFFormat(4, 3). Those rigs need datasets and have not been re-run. The ladder's eighth rung now has a measured area and no measured accuracy.

python3 research/arxiv_tnf/verify_numbers.py

130 derived checks, all passing. Related: #644, #664, #668.

🤖 Generated with Claude Code