A First Proof Sprint

TL;DR

This monograph reports a multi-agent proof sprint on ten research-level problems, combining rapid draft generation with adversarial verification, targeted repair, and explicit provenance, finding that structure-aware verification and layer-switching strategies improve reliability and calibration in compressed proof sprints.

Abstract

This monograph reports a multi-agent proof sprint on ten research-level problems, combining rapid draft generation with adversarial verification, targeted repair, and explicit provenance. The workflow uses wiring-diagram decompositions of claim dependencies to localize gaps and coordinate reviewer-driven revisions. Final outcomes are heterogeneous but explicit: the manuscript distinguishes mathematical status from QC-validation status. Mathematically, Problem~3 has a validation-complete existence path under the scoped criterion used here (uniqueness/irreducibility treated as optional), Problem 5 is solved in a scope-limited form for $F_O$-local connective spectra, Problem 10 is conditional under clearly stated assumptions (with explicit necessity counterexamples when assumptions are dropped), and Problems 4 and 6 are partial with named remaining obligations in the general case (including an unconditional $K_n$ result for Problem 6 with $c_0 = 1/3$). Problem 7 is treated as provisionally closed via the rotation-route theorem chain, pending independent ledger re-check. At the QC layer, Problems~7 and~9 have node-level validation artifacts but still contain unresolved verifier gaps. The main methodological result is that structure-aware verification and layer-switching strategies improve reliability and calibration in compressed proof sprints.

Figures (7)

• Figure 1: The Argument (Conclusion recapitulation): git hashes as raw trace data feed evidence ledgers and post-hoc pattern reconstruction; the conclusion makes this into an explicit argument (AIF node/edge typing, S-expression form, and infrastructure path via Arxana plus peripherals). The sidebar and sub-sidebar episodes are integrated as live evidence for the metacognitive interrupt claim: monitor $(\mathrm{layer},\mathrm{status})$, trigger policy switch on stationarity/falsification, and move from retrospective prose toward real-time argumentative structure.
• Figure 2: Problem 6 strategy flow (v2, 9-cycle): from leverage-split dispatch through Neumann/operator-bound attack (C4), BR1--BR4 blocking and threshold relaxation (C5), bridge B leverage-ordered greedy (C6), BMI falsification (C7), to the Sparse/Strong Dichotomy resolution (C9). Green = proved, yellow = pivot/open, dashed red = falsified. $K_n$ closed with $c_0 = 1/3$; general case conditional on the Strong Dichotomy conjecture (0/148 counterexamples).
• Figure 3: Problem 6 strategy flow (v6, Cycle 4 focus): operator-bound reduction around the Turán/greedy/pigeonhole core, with Neumann-series amplification controlled via monotonicity and an operator bound for ${\color{MPSyntaxGreekColor}\rho}_1$. Remaining open node: the trace-gap inequality (${\color{MPSyntaxGreekColor}\alpha} < 1/2$, equivalently $\mathrm{tr}(F) > 2{\color{MPSyntaxLargeOperatorColor}\sum} {\color{MPSyntaxGreekColor}\mu}_i(1-{\color{MPSyntaxGreekColor}\mu}_i)/{\color{MPSyntaxGreekColor}\mu}_{\max}$) needed for full general-graph closure.
• Figure 4: Problem 4 (v3): finite-free Stam architecture with case decomposition and proof-status stratification.
• Figure 5: Problem 8 (v3): Lagrangian polyhedral smoothing route with product-smoothing and edge generating-function steps.