docs(theory): formal GF-T theory with 30 labelled propositions + fix a stale bias comment - #378
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…a stale bias comment
Adds docs/GF_T_FORMAL_THEORY.md: the GF-T format and the compute ring
stated as theorems over the EXECUTABLE spec definitions, each discharged
by an in-spec assertion, an exhaustive sweep, or a proof given inline.
Every claim carries an epistemic label (PROVEN / MEASURED / MODELED /
FOLKLORE / OPEN), and section 6 lists the claims we deliberately refuse
to make.
The load-bearing results:
Thm 1 2*bias = offset_max, so the ladder is exponent-symmetric about
unity by construction.
Thm 3 Multiplication needs at most one renormalization step: the
product significand lies in [1,4).
Thm 5 GF-T addition is commutative but NOT associative, with a 1-ULP
counterexample confirmed on iverilog.
Cor 5.1 Therefore reduction order is PART OF THE SPECIFICATION: a
verifier folding a dot claim differently would slash an honest
executor. Systems that leave reduction order to the
implementation cannot support this dispute model at all.
Thm 7 The zero sentinel is not injective -- (0,0) collides with the
smallest normal. Recorded as a limitation, not hidden.
Thm 9 Single-op disputes need no bisection: bit-exactness makes a
fraud proof O(1), not O(log n).
Thm 12 Freshness must precede every other guard AND must not advance
the watermark on non-terminal outcomes, else one malformed
high-epoch dispute is a replay-nonce DoS.
Thm 19 The reputation cap must be computed in u64: in u32 a crafted
gain wraps the sum to 0, so honest work would ZERO a maxed
reputation.
Thm 24 (family, op) is a lossy skill proxy -- GFT8 and GFT16 multiply
share both, so a width-blind gate pays GF-T16 rates for 8-bit
work.
Section 4 places the ternary claim honestly. It cites the radix-economy
theorem correctly (3 minimizes b/ln b over the integers; 2 and 4 tie
EXACTLY, so "closest to e" is coincidence-shaped reasoning), and it
quotes the 1950 ERA source disclaiming its own result in the next
sentence. It records that Setun's cost advantage is self-reported by
Brusentsov and never independently benchmarked, that Knuth's "prettiest
number system" is an aesthetic judgment, and that balanced ternary
predates both (Lalanne 1840, weights problem in Bachet 1624).
Section 5 argues the substantive contrast is with posits, not with radix
mysticism: tapering costs scale invariance of accuracy (de Dinechin et
al., CoNGA 2019), so error analyses stop transferring. GF-T is untapered
by choice -- stated as a trade, with what it gives up.
It also states plainly that the exponent base is 2, NOT phi. The
phi^a * phi^b = phi^(a+b) comment asserts exponent additivity, true for
any base; phi's real role is generating the ladder geometry
(Et = fib(k+1)+1, M = fib(k+1)^2). Claiming a phi-radix format would be
the easiest way to lose a reviewer, and it would be false.
Spec fix found by that audit: gft_bias's comment claimed GF-T32 -> 121,
a fossil of the pre-ratification Et5 geometry. The ratified value is 364
(Et6), which the assertion three lines below already pins. Comment-only;
gen output is byte-identical and icarus (10/10) and zig stay green.
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Adds
docs/GF_T_FORMAL_THEORY.md— the GF-T format and the compute ring stated as theorems over the executable spec definitions, each discharged by an in-spec assertion, an exhaustive sweep, or a proof given inline. Every claim carries an epistemic label (PROVEN / MEASURED / MODELED / FOLKLORE / OPEN), and §6 lists the claims we deliberately refuse to make.The load-bearing results
2·bias = offset_max— the ladder is exponent-symmetric about unity by construction(0,0)collides with the smallest normal. Recorded as a limitation, not hidden.(family, op)is a lossy skill proxy — GFT8 and GFT16 multiply share both, so a width-blind gate pays GF-T16 rates for 8-bit workHonest placement in the literature (§4–5)
b/ln bover the integers, and 2 and 4 tie exactly — so "3 wins because it's closest to e" is coincidence-shaped reasoning. We quote the 1950 ERA source disclaiming its own result in the next sentence.The exponent base is 2, not φ. The
phi^a · phi^b = phi^(a+b)comment asserts exponent additivity, true for any base; φ's real role is generating the ladder geometry (Et = fib(k+1)+1,M = fib(k+1)²). Claiming a φ-radix format would be the easiest way to lose a reviewer — and it would be false.Spec fix found by the audit
gft_bias's comment claimedGF-T32 -> 121, a fossil of the pre-ratification Et5 geometry. The ratified value is 364 (Et6), which the assertion three lines below already pins. Comment-only: gen output is byte-identical, icarus 10/10 and zig stay green.🤖 Generated with Claude Code