Conditional κ = 6.92144604250060e-4: served diagonal operations on PR200's bit word (+1.09% over #217)

[ikeboy]

Change and mathematical scope

Conditional κ = 692144604250060/10^18 = 6.92144604250060e-4, under the same retained interfaces as PR200, PR205 and PR197. No hypothesis changes.

• +1.088% over Conditional κ = 6.84696826673891e-4: paid XOR butterflies with cascade-optimized frames #217 (684696826673891/10^18), the best completed claim I know of.
• +1.330% over Pack PR200 diagonal bit residuals into completed banks (conditional κ = 0.000683061299399923) #205 (683061299399923/10^18).
• The served word alone, before packing, already gives 685800818606951/10^18, above Conditional κ = 6.84696826673891e-4: paid XOR butterflies with cascade-optimized frames #217.

Construction	Bit coarse saving	Conditional κ
#205: PR200 word, packed	6.83528e-4	0.000683061299399923
#217: butterflies and optimized frames, packed	6.85166e-4	0.000684696826673891
Served word, unpacked	6.86716e-4	0.000685800818606951
Served word, packed	6.92624e-4	0.000692144604250060

The bit branch binds. PR193's complex supplier (700918443859411/10^18) has slack.

The change: served diagonal operations. PR200's bit word mixes each partner pair on its data registers. The carrier register gets x_c + x_d, and the passive register keeps x_d at all times. The word also builds x_c + x_d again in auxiliary registers. At 373 of these operations, the control register is a pure copy of x_d: one copy operation makes it, and this operation uses it once.

This PR reads x_d straight from the passive data register at those 373 operations. It deletes the 373 copies and their copy operations. To keep the passive register's frame chain nested, the pair's partner mix moves to right after the injections. The passive chain gets one 3-dim stop: line → mix → F_i → full.

Per copy	PR200	Served
Logical auxiliary roles	18,908	18,535
Physical chains	17,114	16,741
Stock W	20,634	20,261
Deficit	1,936	1,936
Coarse bit saving	6.77774e-4	6.86716e-4 (+1.32%)

Every frame stays one of PR200's witnessed frames (24,401). PR200's 34 terminal sinks, 1,760 aliases, gauges and all other operations are unchanged. PR205's completed banks (PR197's construction) and PR193's unchanged 47-constraint assembly then apply. research/served-diagonal-bit/PROOF.md gives the argument.

This is a conditional finite construction and composition, not an unconditional theorem or a speedup claim. The all-size hypotheses (completed core, Clifford interface, uniform weighted compilation, restored rows, fixed-tape transfer) are inherited exactly as in PR200, PR205 and PR197.

Validation

Finite checks, all exact. research/served-diagonal-bit/verify.py --full:

1. runs PR193's and PR200's unchanged package verifiers at their pinned commits;
2. runs patch.py, which derives the served word again from PR200's frozen word and must match the frozen files;
3. runs bit/served_prove.py, a complete check in the style of PR200's bit/prove.py:
   • all 20,295 formal columns in F2 (identity) and over the integers (defining decoder), with no sampling;
   • exact frames and chains with PR200's PR168 v4 integer backend, including each served frame and each new passive chain;
   • PR200's 34 sinks through a forked terminal compiler, with the scalar bill retained;
   • PR200's paid-moment certification, imported unchanged;
   • seven word controls and three terminal controls, all rejected (for example, a served read of the carrier, a served chain without its stop, a served pair with a late mix);
4. applies PR205's charts, banks and colouring to the served word (W = 55,283, deficit 5,808);
5. recomputes the arithmetic, the independent audit and the bank endpoint controls; the next 10^-18 grid points are rejected.

The full run takes about five minutes on Linux, and it reproduces certificate.json exactly. The workflow .github/workflows/served-diagonal-bit.yml runs it in CI. The proof argument (why the served word computes the same columns, and why the ledger is right) is in PROOF.md; the finite checks above test it on every column.

Credit

The served operation is new. It builds on:

• Chafik Boukhalfa (A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200): the bit word, sinks, witnesses and verifier, which the served checks fork;
• Rohan Arun (Pack PR200 diagonal bit residuals into completed banks (conditional κ = 0.000683061299399923) #205) and Evan McKinney (Pack #187's bit residuals: conditional κ = 0.000676080316519385 #197): the completed banks and their code;
• Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193 for the complex supplier; icekylinx for the source-assisted method and the paired-cube framework;
• eumemic (Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168), jamesyc (Add exact dirty-target terminal compiler and 67-sink witness #166) and Finite leaf bootstrap above PR182: conditional kappa = 0.000660627334074291 #185/Globally replayed source-assisted bit word: conditional kappa = 0.000661755179959122 #187 for the backend, the terminal lemma and the leaf composition;
• hcg890 (Conditional κ = 6.84696826673891e-4: paid XOR butterflies with cascade-optimized frames #217) for the previous best claim.

I have not tried to stack the served lever with #217's butterflies and frames. #219 reports a higher number, but it says that R1 to R4 are open, so I compare with #217.

🤖 Generated with Claude Code

[ikeboy]

Correction from the author. This PR is not a new frontier, and the served operation is not new.

#210 (eumemic) already contains it. Its commit 0af9e98 (18:52 UTC, about an hour before this PR) reuses 440 original passive source registers. Those 440 include all 373 sources served here, with the same early partner mix. That head certifies κ = 6.94726e-4, and #210's current head certifies 6.94783e-4.

I compared only against new PR numbers and missed the head updates of #210. The package here is still an independent check of the 373-operation subset on PR200's word, but I do not claim a gain over #210.

[Mister-Meeseeks]

Thank you @ikeboy. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Served diagonal operations on PR200.

Review and limits: Prior exact-head changed-word geometric replay and source-bound Linux full dependency job. Retains the inherited multiplication interfaces.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 8786fec0148a41409874cad133123da227d93751. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.