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Mechanized parameter-frontier theorem

FPRD-T113

The retained GapCVP resource template has frontier 1/28

Exact statement

For the retained exponent inequalities of the direct deterministic 3SAT-to-Euclidean-CVP reduction, every real c with 0 <= c < 1/28 admits natural-number field, support, and moment exponents with strict slack, while no retained tuple with nonnegative field exponent exists for c >= 1/28.

StatusMechanically checked and internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The reduction spends exponent slack on field size, large-support separation, shifted valuations, and reconstruction. Keeping those resources separate exposes the exact feasibility boundary of this proof template.

Hypotheses and scope

  • The retained output-size relation is M = O(N q^2).
  • The retained valuation horizon grows as K^5, and reconstruction uses K^2 T.
  • All exponent inequalities are strict and the field exponent Q is nonnegative.

Proof or evidence

The inequalities imply Q > 2 + 7 kappa and then (1 - 28c)Q > 9 + 14c. Lean proves both real witnesses below 1/28 and impossibility at or above 1/28.

Frontier derivation →

Verification notes

The algebraic elimination, strict-inequality direction, nonnegative-Q boundary, governed claim, Lean theorem endpoints, and independent review note were reconciled against the pinned source.

Limitations

  • The value 1/28 is an unattained supremum for this retained architecture, not a universal CVP-hardness barrier.
  • The general theorem checks the resource inequalities; it is not one uniformly parameterized executable Lean reduction for every c below 1/28.
  • No novelty or external specialist review is claimed.

Open work

Compare alternative reduction architectures and seek specialist review before interpreting the retained frontier more broadly.

Notes

The fork's retained_resource_inequalities_impossible_at_or_above_frontier and frontier_real_witnesses theorems prove necessity and natural-number sufficiency. The paper uses this frontier in its mathematical reduction for every fixed 0 <= c < 1/28; the complete executable Lean reduction is concrete at c=1/30. The 1/28 value is the supremum of the retained one-hot, K^5 valuation-horizon, and K^2T reconstruction inequalities; it is not a lower bound on all direct GapCVP reductions and is not attained by this strict system.