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docs(theory): formal GF-T theory with 30 labelled propositions + fix a stale bias comment - #378

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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

Thm 1 2·bias = offset_max — the ladder is exponent-symmetric about unity by construction
Thm 3 multiplication needs at most one renormalization step (product significand ∈ [1,4))
Thm 5 GF-T addition is commutative but not associative — 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

Honest placement in the literature (§4–5)

  • Radix economy cited correctly: 3 minimizes b/ln b over 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.
  • Setun's cost advantage is self-reported by Brusentsov, never independently benchmarked. Knuth's "prettiest number system" is an aesthetic judgment. Balanced ternary predates both (Lalanne 1840; weights problem in Bachet 1624).
  • The substantive contrast is with posits, not 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.
  • BitNet's 71.4× energy figure is labelled MODELED (arithmetic-only, excludes memory) with the published Amdahl counter-result cited.

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 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, icarus 10/10 and zig stay green.

🤖 Generated with Claude Code

…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.
@gHashTag
gHashTag merged commit 289dfcc into main Aug 9, 2026
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gHashTag deleted the docs/gft-theory branch August 9, 2026 07:12
@gHashTag
gHashTag restored the docs/gft-theory branch August 9, 2026 08:41
@gHashTag
gHashTag deleted the docs/gft-theory branch August 9, 2026 08:49
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