Research: the bit paid profile is the whole bit supplier — exact pricer, and the reframing class is exhausted at #589 (no κ claim)

[flemhhh]

What this is

Research note + independent verification. No κ claim; the bit side does not bind.

It prices a bit word from its paid profile alone, and then uses that to run an exhaustive audit of
the merged bit supplier (PR #589's tuned R439 word). Everything is exact integer/rational arithmetic
against files pinned by sha256 at 66849e1475c930fd7732b1a0067ffee7e1e95f60.

New material:

1. The paid-profile duality. The certified saving c of a bit word is the unique a > 0 with
   Psi(a) := sum_r n_r r((m/r)^a - 1) = D := m*W - rank_mass. Equivalently, for any fixed a0,
   c is a strictly decreasing function of the single weighted count G(a0) = sum_r n_r r^(1-a0).
   Rank-profile search is therefore a linear-objective problem and a candidate word costs O(#gates)
   to price with no root finding.
   Verified by re-deriving c from the profile with an independent rational ln/exp interval
   implementation (code/bitprof.py): the lower endpoint of the resulting 10^-24 bracket is
   exactly the shipped value for both Bit c = 0.001733052823: #545's punctured unit on the R420 skeleton (111 live ports, +4.79% over #543) #555 and Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589
   (1733427039589844234749/10^24). The shipped values include the pipeline's inherited bad-envelope
   surcharge (32 m^2 (sum n) m^a / 10^16), which moves the root by 2.01e-12 — two units on the
   24-digit grid, so it cannot be dropped; the verifier checks both roots.

2. The objective decomposes over registers, so a k-move improvement requires the k moves to
   share registers. That is what makes a complete two-move search affordable.

3. The audit. On the merged Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 word, priced on the exact objective:

   check	result
   nominated single moves	9,347
   strictly improving single moves	0
   cost-neutral plateau	9,263
   improving Steinberg 3→2 moves (671 triples)	0
   legal (slot, frame) pairs over 3,064 operations	9,933
   operations whose nominated frame is not exact-objective optimal	0
   complete register-adjacent two-move neighbourhood (--full)	645,900 pairs
   best two-move delta	0, exactly

   Positive control: steepest descent on the exact objective from PR Bit c = 0.001733052823: #545's punctured unit on the R420 skeleton (111 live ports, +4.79% over #543) #555's base word applies
   exactly two moves — PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's hoist (A[2034] → slot 2028, frame 79) and sink
   (A[2041] → slot 2076, frame 517) — and lands on the tuned word's profile and objective. The
   machinery rederives Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 rather than being told about it.

What it is worth

It closes the reframing avenue for the merged bit supplier at one and two moves: any further bit
saving from this word has to come from a mechanism outside the class (geometry, decoder constant,
helper count, hosting), not from moving gates. It also gives the exact marginal exchange rate
(dc/c = -(dG/G)/(a(ln m - E[ln r]))) so a future move can be priced before it is built, and it
independently corroborates the fixed point that PR #589's tuner claims, on a different objective
implementation.

Concurrent work (refreshed 2026-10-13)

Rebased onto main @ 29218749; every pinned input is byte-identical on both commits, and both
make bit-profile-pair-audit-verify and the unittest module were re-run on the rebased tree.

These landed while this note was being prepared, and they are consistent with rather than
counterexamples to the fixed point above:

• Finite BIT construction: R438 source boundary with full hosted invoice #1043 improves the same Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 word by a mechanism outside the reframing class — deleting
  helper470 via a caller-boundary read of source77, giving R438 and
  c = 0.001734176288479865766692. That is a helper/register rewrite, exactly the "outside the class"
  route the audit points at.
• Local bit construction: target retuning after helper mixing saves three paid R439 ADDs #1040 / Local bit construction: save 14 ADDs with one restored post-PR967 helper shear #1047 cut complete ADD counts on a different R439 lineage (post-PR915/PR967), and
  Finite bit construction: exact R439 four-helper fusion below the pair floor #1045 crosses PR941's disjoint-pair floor with a four-helper fusion. Different words, different
  accounting, different move classes.
• Those results are cited from their own packages and were not re-run here.

Scope and limits

• Finite, exact audit of two specific words and one move class. Not an optimality theorem over bit
  words, and not an exclusion of three-or-more-move escapes of the plateau.
• The two-move search is complete for the nominated move set; the nomination's completeness at the
  frame level is separately certified (0 of 3,064 operations disagree).
• The tuner's proxy objective is not identical to the exact objective (largest absolute discrepancy
  1.645e-5 on this word); only its argmin agrees, which is why everything here is priced exactly.
• No κ change; bit headroom only. Nine-stage price only.
• Every interface that Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543, Bit c = 0.001733052823: #545's punctured unit on the R420 skeleton (111 live ports, +4.79% over #543) #555 and Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 retain remains assumed. This is not a Lean proof of
  integer multiplication.

Reproduce

make bit-profile-pair-audit-verify        # ~6 min, all checks except the pair search
make bit-profile-pair-audit-full-verify   # ~35 min, adds the complete two-move neighbourhood

No C++ toolchain, Boost, Lean or network access; only Python 3.11+, numpy (for the pinned tuner) and
the pinned files on main.

Files

research/bit-profile-pair-audit/ (README, verify.py, EXPECTED.json, code/), plus
tests/test_bit_profile_pair_audit.py and two Makefile targets. upstream/ untouched; no unrelated
regeneration.

Credit

The word, its generator, checkers, pipelines and tuner are PR #555 (eumemic) on PR #543 / #545
(DreamingOfClouds) and #510; the two transports audited are PR #589 (DreamingOfClouds); the deficit
identity is PR #545's. The decoder-side bounds of #620 (GamingPuzzled) and #631 (vectorsss) are
orthogonal to the class audited here.

Substantial AI assistance (DeepSeek Harness, deepseek-flash) is disclosed in NOTICE.md.