Research/partial complex stopped

[seanabreau]

This builds on #104 (948ce1510df750f4c18b96bdaef436a86f8bf834) and the exact #113 cyclic-strip producer (0eee4d507092a96bde88de703f07574ba17401a8).

Under those interfaces,

$$ T(n)=O!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{942802139}{10^{13}}=0.0000942802139. $$

The saving is 54.4687% above $2^{-14}$.

Construction

Bit network (23,23). Apply #104's common-basis, opposite-bank stopped compiler to every proper projector of Swapnil Jain's frozen round-seven word, pinned at 741e7aa078392553815df7926ee17ac5e25a8c38. Every nonzero projector residual is one reversed child of its full rank. The strict coarse saving is 15513/125000000; stopping at atom exponent 1/1000, with ordinary leaf saving 384599/10^10, gives ordinary bit saving 1240183559/10^13.

Complex network (24,24). Keep #113's producer and legal matching: 68,188 additions, 8,120 roots, 37,802 matched links and 38,506 physical auxiliary roles. Reconstruct the invertible matched scalar word L, signed rational scatter J, and exact complete adjoint C=JL.

A 32,488-event closure computes the 24 retained centers first, preserving every role's full access order, including read/read dependencies. Checks cover 209,340 accesses, retained-center boundaries and 8,096 ordinary root reads. Selected scratch is untouched by the closure and injection.

Select 9,356 nondegenerate partial subframes (3,309 distinct frames), each contained in its first owner and every target hyperplane receiving its adjoint coefficient. Target readouts follow actual nested chains. All residuals are nondegenerate; alternating cases use the retained inverse-Gram/Gauss interface.

For dimension d inside an owner of rank b, the first child changes from b to b-d, and the exterior from m-h to m-h+d. Source and sink phases change together: in stage one use +q_sigma and q_F+q_sigma; in stage two use -q_sigma and q_F-q_sigma. The completed operator remains q_F. Target-front chain replacements preserve rank mass. The strict complex saving is 5893637/62500000000 = 0.000094298192.

Complete paid inventory	Bit	Complex
m / auxiliary roles	529 / 28,866	576 / 38,506
W	108,516,254	164,065,440
Recursive rank sum	57,403,754,177	94,499,831,360
Positive bins / maximum child	46 / 528	40 / 572
Rank-one copied-output children	3,136,441	4,096,576

The complete adjoint has 11,678,073 nonzero coefficients per invocation, with numerator magnitude at most 42 over denominator 42. Charge 16 scalar operations per term and retain the predecessor allowance:

G = 2019773888 + 2*2024*11678073*16 = 758385205952.

Recompute the denominator-21 exact grid, literal precision guard, completed-child height induction and C1=1 contract. Halving factors 367, 100 and 9 give row coefficient 12,961; p^27000 supplies degree gap 13989/25 > 0. All copies, scalar work and wrappers are paid. With beta=10^-6 and eta=10^-8, all 47 assembly constraints and seven margins are strict.

Contribution sources and prior PRs

Source	Contributor and retained role
Original manuscript/framework	OpenAI, Douglas Colkitt, David Harvey and Joris van der Hoeven: multiplication reduction and analytic context.
Round-seven witness	Swapnil Jain: frozen bit word, lifted frames and deferred readouts.
#104	icekylinx: stopped compiler, rational centers, odd grid, precision/row bridge and semantic assembly, retaining the #10/#18/#24/#32/#36 lineage.
#108/#111/#113	Rohan Arun: dual-suffix pair stars and cyclic-strip composition; exact producer bytes and Anthropic Claude disclosure preserved.
#62/#110	Avi Eisenberg: cyclic intervals and independently developed deferred-readout pipeline.
#55	Rohan Gupta / gupt1156: dual-suffix identity credited by the producer.
Retained #104 notices	Zhihao Chen, Aurel Prosz, RaD/hipotures, Dominik Scholz, Bortlesboat, dleen, eumemic and other predecessors; Baker-Harman-Pintz prime packing.

Proof and attribution

PROOF.md gives the physical schedule, paired endpoint gauges, stopped bit recurrence, exact grid and all-size transfer dependencies. MANIFEST.json and SWAPNIL_SOURCE.json bind sources and inputs; NOTICE and imported headers preserve Apache-2.0 and source-specific attribution.

New work used OpenAI Codex and twenty GPT-6-sol agents at medium reasoning effort, with independent audits. EXPERIMENTS.md records more than twelve tested directions and rejection of a scalar-valid schedule with unpaid frame descents.

This is a conditional research bound. Analytic, prime, ordered-affine streaming, restored-row, exact-recovery and fixed-tape interfaces, constructive setup and eventual thresholds remain premises. Bit stage-two reversal and complex complement/time-reversal remain written dependencies. Finite Lean arithmetic does not formally verify the complete multiplication theorem.

Validation

• Rebuilt the pinned complex producer, C++ matcher and physical role word, and all 102 pinned bit-source files.
• Checked every selected owner/residual/target chain and every physical access/root boundary; independently audited alternating frames and phase identities.
• Exact signed-rational dirty-scratch replay restores scratch and gives the required side map. Selected roles' full adjoint supports and frame ranks are cross-bound to the assembly.
• Recomputed complete paid histograms and strict rational moment gaps above 1.1417e-10 (bit) and 1.5791e-13 (complex), G, precision, rows, all 47 constraints and seven margins. Controls reject omitted copies, wrong sinks, degenerate frames and illegal selections.
• Lean 4.34.1 checks both finite moments and signed assembly. bit_grid, cx_grid and signed_assembly print empty axiom lists; analytic enclosure interpretations and all-size transfers remain external.
• Focused package/source checks, staged diff checks and the complete inherited make verify pass, covering all eleven groups, isolated regressions and historical patch checks.

Reproduction

make -C research/partial-complex-stopped verify
make -C research/partial-complex-stopped verify-sources \
  SWAPNIL_SOURCE=/path/to/pinned/round7 PR104_SOURCE=/path/to/pinned/pr104
cd research/partial-complex-stopped/kernel
python3 generate.py
lean KernelNew.lean

The README supplies pinned checkouts; CI repeats source and kernel checks.