Research: all-word dirty-connector bounds and signed triangle (no new κ)

[conorholds]

Change and mathematical scope

Scoped query-model proofs and constructions; no new κ. Adds research/bidirectional-connector/, centered on two results:

• All-word lower bounds with matching asymptotics. For (M=2^n-1) outputs, the optimal one-root and two-root query tariffs approach (4M) and (3M), respectively, over F2 and characteristic zero when (a=1), (b\sim\sqrt{2M}), port prices agree, and transfer prices remain fixed and finite. The lower bounds include source-locality corrections and allow arbitrary source-call counts and transfer depth. Matching constructions establish a 25% limiting query-tariff reduction under these assumptions.
• Signed zero-sum triangle. A characteristic-zero construction uses two arbitrary dirty roots, with every child call at physical z0. For three source blocks with (q_i=2^{|P_i|}), its complete operator/transfer invoice is:

  B = sum(q_i)
  A = 2M + product(q_i - 1)
  T = sum(q_i) + 4
  

  At (n=15), (b=285), (a=\tau=1), its tariff is 122,785, versus a proved lower bound of 138,778 for every one-root word: at least 11.52% less in this model.

Sources remain immutable, outputs are unobserved until completion, maps may be singular and noncommuting, and incoming root values are arbitrary and restored. Signed inverse calls subtract map applications; they do not require inverse matrices.

Supporting material includes a 73-call exclusion, a minimum-read walk compiler, minimum-source bounds, and signed dirty-cache constructions. The compiler does not minimize transfers across all minimum-read decompositions. Cache constructions explicitly require additional dirty rows.

Limits: extra root reservation, routing, preparation, precision, tape/head access and other costs outside the fixed call tariffs remain unresolved. No qualifying recursive caller, physical supplier adoption, integer-multiplication runtime improvement or exponent improvement is established. RAM benchmarks exercise deliberately expensive B maps and cheap A maps; they are checks of that tariff regime.

Task and overlap

• Original task: connector-alphabet-change.
• Scope status: changed, from a tiny alphabet-extension experiment to scalable query-model bounds and constructions.
• Scientific baseline: 0b1cccf86e2f5132ed286a26ef0a681a495740ea.
• Current-main/guidance checkpoint: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4.

The connection to the bottleneck is conditional: an actual caller must support the required map tariffs, offline targets and paid root operations before these query savings can affect a recursive budget.

Nearest overlaps:

• #2065, head 2721b1e31d718a6e70266a99521f629ce2e1d259, establishes the six-transfer five-source endpoint. That endpoint is credited supporting evidence here.
• Our #2182, head dc252de1cc594829c3031abdb4de104713bc474a, addresses fixed-port optimality and the two-root boundary with exactly two calls per operator. Those results are not claimed again.
• #1915 and #1945 supply related framed one-connector bounds.
• Our #2227 concerns a separate scalar compensation network; no combined saving is claimed.

The last complete overlap snapshot, 2026-10-14 23:53 UTC, covered 445 open PRs, 896 discussion/comment/reply records and 1,538 local records. This metadata search is not an exhaustive novelty certificate. Exact source bindings and historical provenance are retained in SOURCES.json and TASK.json.

Validation

Recorded local environment: macOS arm64, CPython 3.12.11 for registered replay.

python3.12 -B scripts/research_package.py \
  --package research/bidirectional-connector --check-only

python3.12 -B scripts/research_package.py \
  --package research/bidirectional-connector

All eight registered commands and 27 regression groups passed, completing in 32.944 seconds. Coverage includes symbolic endpoints and restoration, separate dense forward/inverse replays, omission/sign/port failure controls, all 23 benchmark function hashes, and rejection of altered measured source.

A separate fresh native census reproduced 71,856,432 assignments in 11.585 seconds, using:

python3 -B research/bidirectional-connector/certify_cost.py --full \
  --sdk /Library/Developer/CommandLineTools/SDKs/MacOSX15.2.sdk

The default registered census command checks the source-bound native receipt; it does not perform that fresh native enumeration. Recorded verification is reused for byte-identical scientific files and fixtures.

Written arguments support the all-size claims; finite replays check implementations and failure modes. Not checked: hosted CI, proof-assistant formalization, a second native implementation of the 73-call exclusion, selected global cold replay, or physical caller/stock/precision/tape admission.

Credit

Credit Ross Richardson’s #934/#943 for the generic connector and Gray-block lineage; Douglas Colkitt’s #1185 for the six-source boundary; woklerx/#1438, Gareth Davidson/#1498, Hanson Sun/#1615 and Cinna Davis/#1887 for preceding root contracts and constructions; @nedveder/#1812 for the bidirectional-transfer question; and Adam Gall/#1915/#2065 and Arham Garg/#1945 for related endpoint and frame results.

Math PR1/PR2 remain the canonical earlier accounts of our six-transfer result and two-call obstruction. Remaining signed and dirty-commutator predecessor credits and source notices are retained in the package.

Open next questions

1. Bind one actual recursive caller. Price its complete stock, routing, preparation, restoration, precision and head-return costs against the saved query invoice; no qualifying caller or completed physical bill is established.
2. Optimize transfers across read decompositions. Start with a complete small-instance state model and the retained counterexample before extending synthesis; current placement optimality is limited to the compiler’s chosen decomposition.
3. Bound cache costs at positive transfer prices. Fix the available dirty stock and transfer tariff, then seek a matching construction or lower bound; unrestricted transfer optimality remains open.