Recycled bit registers on #147's reduced word: conditional κ = 4.71809569e-4

[DaysSky]

Based on main (merged #144). It adds research/bit-reuse-147/ and one workflow; no existing file changes. It builds on #147 (hpst3r): that PR's elimination plan is used unchanged, but none of its files are needed, so this branch applies to main directly.

Under #144's retained interfaces, this certifies

T(n) = O(n (log n)^(1 - κ)),   κ = 471809569/10^12 = 4.71809569e-4

That is +1.54% over #147 (4646633/10^10) and +2.36% over the merged #144 (4609169/10^10). #144 is bit-limited, so a bit-side change moves κ directly.

What changes

Only the bit word changes. Starting from #147's reduced PR97 word:

• Reads run late. The selected deferred reads keep their inherited order, but each runs as late as the word allows instead of in one block after phase 1.
• Dead registers are reused. A slot that is not an output slot is never touched after its last operation. For 3,338 pairs, such a register is moved from its last frame U_D to the entrance frame sigma_B of a later deferred slot B, where U_D is inside sigma_B, and becomes B. B's inherited read then cancels whatever the donor left. This is jamesyc's Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124 birth-read operation, applied to the bit word.

Each hand-off removes one physical role, the donor's final step and the recipient's gauge child, and adds one shorter step (993 of the hand-offs have rank zero).

	roles R	W per vertex	coarse bit saving	κ
#144 (merged)	28,866	32,408	4617656/10^10	4609169/10^10
#147	27,317	30,859	4655227/10^10	4646633/10^10
this PR	23,979	27,521	472689442/10^12	471809569/10^12

The deficit per vertex (2,024) is unchanged. The largest child falls from 66 to 60. The complex side (4856569/10^10) still has about 2.9% headroom.

The headline keeps the merged atom exponent 1/1000. At 1/2000 (#148, gupt1156) the same row gives 472026274/10^12 = 4.72026274e-4. Th0rgal's composition comment on #147 (#146 + #147 + #148) reports 4.64880716e-4, so this is +1.54% over that at the same atom exponent. #146's min-cut gauge subset is not composed here; #144's own 9,543 selected gauges are used.

Verify

python3 research/bit-reuse-147/verify.py          # stdlib only, about 2 minutes; do not use -O
python3 research/bit-reuse-147/verify.py --all    # also re-checks every inherited frame move, about 9 minutes

verify.py runs these checks:

1. It reproduces Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's coarse saving, stopped saving and κ exactly on Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's own grid, with Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's functions.
2. With no elimination and no recycling, the delayed-read word reproduces every histogram of Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's row. Delaying reads changes no charge.
3. It runs a frame ledger of the frozen plan in execution order. Every scalar gate must find its registers at one common frame, and no register may retreat. The row rebuilt from the actual moves must equal row.json.
4. It checks all 4,878 adjacent frame pairs that are not already inside one inherited certified chain for exact nesting over Q, with bases rebuilt by check_lifted.py: 3,338 hand-offs and 1,540 target-chain moves. A control must reject mismatched hand-offs.
5. It runs the literal word on the 23,979 physical registers with all 27,521 inputs as formal variables, over F2 and over Z with the exact integer coefficients, and requires the exact identity. This is complete, with no random vectors. A random replay then confirms that three tampered words break the identity: a recycled read of the register's entrance value, a recycled read taken before its donor's last write, and the inherited inverse order.
6. It prices the row with Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's assembly (47 constraints, 7 margins). The next grid points are rejected.

price147.py changes one check in #144's code, as #147 does: the bit dimension test accepts R <= 28866.

Checks run on this branch:

• verify.py passes on macOS (Python 3.12 and 3.14) and on Linux in the fork's CI (Python 3.11 and 3.14): run 37888454305.
• verify.py --all passes: all 201,824 distinct ledger moves, inherited ones included, are nested exactly over Q.
• The scalar identity was also checked on all 27,521 formal variables (every data port, target and physical register), over F2 and over Z with integer linear forms. This is complete and uses no random vectors. The second commit adds it to verify.py.
• An independently written closed-form computation of the child histogram equals row.json.
• The repository's full verification matrix on the fork: all 45 jobs pass on the first commit d0fc03e (run 37888454329): every make verify-* group on Python 3.11, 3.13 and 3.14, and the three formal jobs.
• make_plan.py regenerates plan.json byte for byte on this machine.

Not run locally: the complete make verify and the formal targets (the fork's CI runs them).

Scope and limits

PROOF.md gives the argument. Every interface #144 retains stays an assumption. Beyond that:

• The pinned PR97 frame audits are relied on for inherited chains, as in Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144 and Terminal elimination on #144's shared bit cores: conditional κ = 4.646633e-4 #147. The --all mode re-checks the nesting of every move this word actually makes, but not the other inherited audits.
• The replay runs one core on its own bank, as in Terminal elimination on #144's shared bit cores: conditional κ = 4.646633e-4 #147. That is the premise the shared-core argument uses.
• The inverse pass must undo register updates in exact reverse order (all at the full frame). The inherited order, with every V^-1 last, does not restore a recycled register; verify.py checks that it fails.
• The plan is one legal choice from a matching heuristic (make_plan.py, which needs numpy and scipy and is not used by verification). No optimality is claimed.
• This has had no review other than the checks above.

Credits

• icekylinx: Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144 and Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130 (framework, selected gauges, completed-core ledger, pricing and assembly), and Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104.
• hpst3r (William Porter): Terminal elimination on #144's shared bit cores: conditional κ = 4.646633e-4 #147 terminal elimination on this bit word; its plan is used unchanged and price147.py follows its pricing approach.
• jamesyc: Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124 birth-read reuse.
• SovereignSteak: Terminal accumulator elimination with exact majorization audit #122 terminal accumulator elimination.
• Swapnil Jain: the round-seven deferred bit word and its exact frame tools.
• Zhihao Chen: the PR97 ledger.
• an664: Completed-core sharing: conditional kappa > 2^-13 #128 completed-core sharing.
• gupt1156: Raise conditional kappa to 4.611281e-4 with tightened adapter toll #148, the 1/2000 atom exponent reported as a secondary value.

The late read timetable, the hand-off of dead bit registers on this word and their verification were prepared by DaysSky, assisted with Claude. No priority over concurrent work is claimed.