Stojnic uses parametric fl-RDT on the symmetric binary perceptron to obtain a satisfiability threshold of 1.8159 and an algorithmic threshold near 1.6021, pointing to a computational gap of about 0.21

The author proposes using a change in the ordering structure of the c-sequence within fl-RDT to distinguish satisfiability from algorithmic thresholds: the first two lifting levels keep the natural decreasing order and give the satisfiability threshold, while higher levels break the ordering and give algorithmic-threshold candidates. Previously fl-RDT mainly characterized random dual structures; this work directly links the c-sequence ordering change to the SBP transition from αc to αa and provides a numerically testable sequence. Based on the fl-RDT theorem and numerical solution of Corollary 2; at κ=1 the second level αc≈1.8159 agrees with the established satisfiability threshold, and the seventh level gives αa≈1.6021 with a predicted convergence interval of about 1.59–1.60.

In the small-κ regime α→0, the third lifting level gives κ≈1.2385√(α/−log α), qualitatively matching OGP predictions and identically matching local-entropy 1RSB predictions. This result aligns the parametric fl-RDT approach with two independent routes, OGP and local entropy, in the low-α limit, further supporting the OGP–local-entropy correspondence. The third-level derivation is exact as α,κ→0 and numerically yields the coefficient 1.2385; the author notes that the fourth level produces redundant stationary-point equations at leading order, suggesting the third level may suffice.

The author designs a CLuP-SBP algorithm whose simulated performance approaches the fourth lifting level estimate αc(4)(κ) over a wide κ range. CLuP had been used for ABP and negative Hopfield models; this work extends it to the SBP feasibility problem and provides parameter-tuning experience. Simulations are run at n=50, 300, and 1000 with at most 2000 restarts; the author notes limited practicality at high κ, but the results closely follow the theoretical curve.

The author formulates two conjectures: the SBP algorithmic threshold conjecture and the parametric fl-RDT algorithmic conjecture, suggesting the c-sequence ordering change may be a more general fl-RDT property. It unifies similar phenomena observed in SBP, ABP, negative Hopfield, and SK models under one parametric framework and proposes a general mechanism. Based on numerical and phenomenological consistency with this work and [98,100]; it is a conjecture rather than a rigorous proof, and the author explicitly states that rigorous confirmation remains open.

The results apply to Gaussian SBP in the large-dimensional proportional regime; κ=1 gives concrete numbers, and other κ values are covered by numerical curves. The third- and fourth-level results in the low α, κ regime are exact as α,κ→0 and approximate for small but nonzero values. The CLuP-SBP algorithm is simulated at moderate dimensions over a wide κ range, with limited practicality at high κ. The framework provides a starting point for future rigorous proofs, algorithm design, and numerical-method research.

The author explicitly frames the core propositions as conjectures, and rigorous proof remains open; numerical evaluation beyond the seventh level is currently infeasible, and the convergence interval 1.59–1.60 is a prediction; Table 3 parameters may not be fully accurate, though the author expects little effect on the αc(r)(1) estimates; CLuP-SBP has limited practicality at high κ, and success emulation for feasibility problems is difficult.