Research: exact modular borrowed-port echoes with endpoint-sized data

[bitplane]

This supplies an explicit finite-width modular backend for PR1057's borrowed-port echo. Integer matrix factors execute modulo 2^w, restore every dirty bit pattern, and recover the exact signed output from input/output bounds. Large intermediate integer lifts need not fit the data registers. Recursive-interface construction; no new kappa.

For the 52-factor fixture, the existing plan still takes 1,376 complete leaf calls and three dirty length-two arrays, using 56-bit data. The inherited integer-prefix majorant gives 114 sufficient bits for entry values bounded by seven, or 167 for arbitrary signed 56-bit dirt. These are sufficient-bound comparisons, not integer-width minima. The retained arbitrary-dirt integer lift actually reaches 90 signed bits.

The cost is explicit: 38 scalar/access commands per two-dimensional leaf instead of 28, including ten reductions. The full kernel executes 52,292 commands, of which 13,760 are reductions. All five arrays occupy 560 model bits; the four private work registers occupy 283 bits, including a 112-bit product register. Coefficient/input conversion, final decoding, original input storage, planning, metadata, routing and physical adapters have separate paid terms. No measured speedup follows from this invoice.

Reproduce and evidence

python3 -I -B research/modular-port-borrowing/verify.py
python3 -I -B research/modular-port-borrowing/test_package.py -v

Python 3.10+ and standard library only. Expected marker: PASS_MODULAR_PORT_BORROWING.

• 5,280 exact matrix cases covering all ten input columns: all 861 ordered binary/ternary trees through seven factors at three widths and both signs, plus each parent plan from eight through 64 in both directions.
• A separate control-stack and matrix implementation checks 528,302 leaf prefixes and 667,416 recursive returns, including borrowed ancestor ports.
• 33,280 exhaustive scalar cases over the declared small quotient rings execute 2,160,640 actual scalar/access commands; another 144 dense cases vary dimension, signs, zero factors and width.
• Four primary executions compare every leaf with the integer lift, restore arbitrary dirty bits and replay the literal inverse. Closed operation and address invoices agree with literal execution.
• 19 mathematical/admission controls, 12 package tests and 25 existing selected-record tests pass. Final normal CPython 3.14.4 and read-only, relocated, isolated optimized 3.12.3 receipts are byte-identical.
• Recovery failure is explicit: residue 12 modulo 16 decodes as −4 when the true result is 12 and the bound is omitted. Saturation corrupts a helper; rounded division fails the quotient contract. The guard separately preserves input representability when a zero product makes the output small.

Scope and credit

Only research/modular-port-borrowing/ is added. Base ba82555, OS seed 52. Six exact Git-object/SHA-256 source/context pins include five unchanged PR1057 files at 78a062d. The generator/planner and additive-group identity remain credited predecessor work; the contribution is their concrete quotient-register backend, recovery guard and full cost/validation evidence.

The all-tree argument applies to integer linear factors and residue-valued recursive boundaries. Rounded, irrational or general dyadic leaves need another interface. The Python interpreter models bounded registers; its host allocation is not the stated data-space bound. A physical fixed-tape compiler, supplier/frame adapters, hosting and outer analytic transfer remain open. Leaf multiplicities and the direct all-role obstruction are unchanged. No supplier admission, full repository suite, selected cold replay, CI success or formal multiplication theorem is claimed.

The final queue was checked through #1157. Ali Romman's #1135 was read through its exact-lift proof: it studies rounded restoration and a common-denominator rational embedding with q^D initialization and numerator headroom. This integer modular backend does not remove its denominator costs or solve its rounded-leaf obstruction. #1103's rectangular batching, #1155's deferred batch restoration and #835/#1144's modular verification have different objects and are acknowledged. This backend restores every completed child boundary and does not defer those restorations. Upstream advanced to ce7ad1b, selecting conditional PR1094; the experimental context remains frozen.

Credit Gareth Davidson PR1057, shhxera PR822, androolloyd PR673, Sagar Thakrar PR730, Chafik Boukhalfa PR628 and the retained lineage. The Ben-Or–Cleve register principle and modular arithmetic are classical; no priority claim is made for them. Prepared by Gareth Davidson (@bitplane) with substantial OpenAI Codex assistance; new material Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at f38bfe5da0bed21f42ce5251d23491a56519d7c4.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Modular borrowed-port echo with endpoint-sized data, arbitrary dirt restoration and paid reductions; smaller data width is a space/work tradeoff, not a free precision theorem.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.