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Experiment Rounds
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Round 10
Round 10
v1.0
2026-07-28
Sixth Non-Convex Alternating Cycle, Historical Pressure Memory, and Congruent Placement Falsification
The sixth alternating cycle C₉→γ₁₀→C₁₀ establishes a “historical pressure memory” dual ledger, expanding Round 9's transient-pressure concept into a formal classification: persistent skeleton curves (still active in the final state) and transient pressure curves (near-redundant in the final state but once under pressure) must be kept separately in the container's test set. The more critical discovery is the false-hard-case problem in the congruent placer itself — a Fourier-22 candidate curve was misjudged by a low-budget search to have exposure 0.040938380334, but a high-budget multi-seed recomputation left only 0.000225741207, an overestimate of about 181.4 times. Curve-generation error and congruent-placement error must therefore be verified separately: if the placer has not converged, an even stronger curve generator will still produce large numbers of false hard cases.
Finite curvature family, finite congruent search, and non-convex container candidate
— the package's own stated status, reproduced as-is.
Connections · Connections
Relationship to other packages, stated as far as possible in the document's own words, not my interpretation. This case's theoretical foundation is in From the Original Kakeya Needle to Moser's Worm.
Experiment round progress10 / 11
“The study of non-convex universal containers is not just a curve generation problem, but also a congruent placement global optimization problem. If the placer has not converged, even the strongest curve generator will produce a large number of false hard cases.” — quoted from this package's main report, Section 11, ‘Conclusion.’
Round 10 Core Content
The sixth alternating cycle; the attack and container response; the persistent-skeleton determination; the final-state active family; historical pressure memory; placer falsification (the 181.4× overestimate case); the Round 11 seed; the convergence assessment; honest bounds; the next natural node.
Method
The formal attack reuses the candidate saved from Round 9; the thirteen-family, two-stage container optimization pipeline; how the Fourier-22 three-parent-lineage and B-spline reserve pools are built; the follow-up falsification procedure for anomalously high-exposure candidates; verification items.
The Sixth Curve-Container Alternating Response Cycle
The attack step; direct addition; cross-basin container response; attack persistence (π₁₀=52.1623%, classified as a persistent skeleton rather than transient pressure).
Dual Ledger of Historical Pressure Memory and Final-State Activity
Formal definitions of the three different quantities (e_n, ΔA_n, ℓ_n); the classification criteria distinguishing persistent skeleton curves from transient pressure curves; the Round 10 classification results; the memory principle (container state = final-state active skeleton + historical pressure memory).
The False Hard Case Problem in Non-Convex Congruent Placers
The anomalous candidate (low-budget search: 0.0409); the high-budget recomputation result (only 0.000226 remains — a 181.4× overestimate); the conclusion (hard cases found by the generator must first survive falsification by the placer).
Fourier-22 Three-Mother Systems and Moving-Knot B-spline Audit
Robust global recomputation results (values for the four pools: mixed / distributed / localized / B-spline); this round's judgment (the distributed mother system still shows stable positive exposure).