[petalas]

Summary

Scoped proof with exact finite corroboration. Independent uniform q-symbol inputs and simultaneous success strictly above 1/q still force every outgoing partition inequality, even with arbitrary nonlinear pointwise gates:

success > 1/q
  -> every class has outgoing edges >= separated demands
  -> every supplied weighted partition rank bound survives
  -> common-flag rank mass s >= Wm

The threshold is sharp for every q>=2, W>=2 and m>=1: an invertible identity circuit with a prescribed swap succeeds exactly at 1/q with s=Wm-1. An exact swap costs Wm with identical terminal frames. Input-independent random coins preserve the obstruction under branchwise deficient budgets. A literal mixture disproves replacing that premise with expected cost alone.

This contribution was produced by Nick Petalas. Credit Jose Manuel / jmbarrancoml #1757 for zero-error outgoing transcripts and weighted partitions, Navinor #825 for common flags, and ywz65 #2507 for designated demands and disposable workspace. Their scientific results are cited, not audited or replayed. Classical successful-subset counting retains its credit. The new result is the sharp error threshold and its reversible same-terminal attainment. No new kappa or selected-frontier change.

Evidence

• Before: Theorem: common-flag frame obstruction for nonlinear gates #1757 supplies the zero-error partition implication. After: the proof establishes the same implication for simultaneous success >1/q and gives a matching counterexample at equality.
• The new census executes 2,002 map/demand cases, 15,298 input evaluations and all 111,744 vertex classes in its declared binary and ternary topologies. Actual successful-input fibers and the q^(e-d) bound are checked directly.
• Forty-eight framed fixtures execute 1,668 inputs. The exact/approximate words have the asserted endpoints and every physical edge is charged. Two-seed mixtures execute another 1,668 seed/input cases.
• Ten invalid-contract controls reject, five mathematical boundary/positive controls pass, and an unexpected CLI argument rejects. Normal and optimized receipts agree exactly.
• Both registered commands pass on CPython 3.12.15, Darwin arm64, standard library only. They were timed locally at 0.217 and 0.211 seconds, below their 120-second bounds. These are replay-budget measurements, not a performance comparison.
• Metadata, six committed files, executable modes and declared source ancestry pass. Official export integrity and input coverage pass. Normal and optimized runs from a read-only Git-free export with empty PATH agree; all seven exported files retain their hashes. All verifier functions are below sixty lines and warning-strict replay passes.

From the repository root, reproduce using the exact local interpreter, or substitute another absolute CPython 3.12 path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/bounded-error-partition-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/bounded-error-partition-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/bounded-error-partition-threshold
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/bounded-error-partition-threshold --check-source-history --check-modes

Expected markers: PASS_BOUNDED_ERROR_PARTITION_THRESHOLD, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct stdout receipt is generated and binds all package files by SHA256. No receipt is handwritten or trusted as scientific input.

Limitations and unrun obligations

The universal claims rest on the written proof, with finite corroboration. Inputs are independent and uniform, topology is fixed, every physical wire is charged, designated sinks are distinct and coins are input-independent. The conclusion concerns simultaneous exact symbol recovery. Individual receiver errors, numerical distance, correlated/biased inputs and uncharged advice have different contracts.

A useful weighted partition certificate must be supplied; arbitrary non-common-flag rational geometry is not excluded. Nonsplit common flags are covered by the written field-extension argument, while executable frames are rational triangular matrices. Physical/native realization, dirty ownership, precision/tape, stock, compiler, analytic recovery and the complete multiplication theorem remain outside scope. No selected cold replay, native replay, Lean, independent human review or formal proof-kernel acceptance is claimed. No mathematical check was deferred solely to CI.

Open next questions

• For one actual producer with a justified nonuniform distribution, determine the maximum probability mass of a successful transcript fiber. Start from one deficient cut; distribution acquisition and physical costs are missing.
• Bind one approximate supplier's full graph, every wire and a simultaneous symbol-success contract to a useful partition certificate. Numerical error and paid recovery remain separate obligations.

The contribution stops at this first verified threshold result.

Related open work

Fresh common-flag, bounded-error partition and nonlinear-frame searches found no open PR computing this threshold. The following adjacent work has different boundaries:

• Construct an exact dyadic rank-seven P104 projector (no new kappa) #6247 constructs an entrywise approximate P104 Gram projector. Its matrix-entry error is different from this circuit's simultaneous exact symbol-success probability; neither result admits a supplier.

• Strengthen nonlinear Eisenstein bounds using port polar ranks (no κ) #6256 studies fixed-support quadratic Eisenstein messages and a zero-error pairing-price floor. This result uses arbitrary framed DAGs and a changed correctness premise; no shared numerical quantity was recomputed.

• Prove correlated dirt is necessary at P104 wire meetings (no κ) #6261 studies exact dirty-entry response spaces at one P104 meeting. Its local response conditions and counts differ from this simultaneous-success threshold.

• Prove sharp all-column tests with rational width normalizers (no κ) #6262 proves finite-width tests for rationally normalized transfer identities. Its exact all-column cutoff does not compute this probabilistic circuit rank bound.

The completion refresh listed all twelve other open account PRs created since the advance CLAIM at 2026-10-30T11:28:03Z and inspected candidate descriptions. No new contribution computes the same threshold, and no shared-number disagreement was found within that coverage. No additional computations were run to check other results.

CI notes

Both hosted checks, plan and replay (research/bounded-error-partition-threshold), passed at exact head 377c32061876ad04eb32da0660885035cce22ddc in run 41049629006. The completed check list was read once. Separate run metadata confirms the matching head, completed status and successful conclusion. The checks took 4m16s and 1m49s respectively. All listed local checks also passed. Hosted replay certifies the registered finite checks, not independent human or formal proof-kernel acceptance of the universal written proof. No CI failure remains.

Advance CLAIM. Base main: 2485805, priority revision eight. Exact predecessor statements and historical metadata are provenance pins, not runtime scientific dependencies.

Merge Danger

Door: two-way. Blast radius: one isolated opt-in research package with six new files. Shared code, predecessor sources and selected-result metadata are unchanged.