[aliromman]

This scoped theorem gives the sharp worst-case adaptive tail criterion with a prescribed mean and interior mass. Two explicit endpoint laws determine the envelope, and an attaining mixture proves the strict boundary. An integer three-rank fixture passes the original tail/ordering gates where the earlier range-and-mean guarantee fails; its mirror coverage still fails. The complete kappa calculation is unchanged.

Proof and scope, exact checker, replay declaration, and source provenance are pinned to the reviewed commit. Base: canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision 7.

CPython 3.12.14 registered normal and optimized replays passed: 150 rational statistic fixtures, 1,050 envelope comparisons and attaining cases, 77 chord checks, three original gate inequalities and seven controls. Both print PASS_INTERIOR_MASS_TAIL_GATE; the launcher prints PASS_REGISTERED_RESEARCH_PACKAGE. All five package files match their committed bytes, and the base is an ancestor of the package commit.

python3 -B scripts/research_package.py --package research/interior-mass-tail-gate

Use Python 3.12 and the standard library. No external runtime inputs or third-party Python packages are required. Hosted planning and the registered package replay passed on this exact head in run 39999812959. No predecessor scientific suite, selected cold replay, native replay, compiler, Lean or Docker build was run. The optional full-history preflight is unavailable in this shallow checkout. Finite checks corroborate the written proof; they do not constitute independent human or proof-kernel acceptance.

Apply the interior-mass screen to a source-bound histogram that also passes every mirror and recurrence predicate. Source preparation, native operations, precision, movement, and supplier/outer admission remain separate obligations.

Credit @petalas's PR #5254 supplies the normalized criterion, sharp endpoint screen and explicit interior-mass question in its proof. That proof records Dillon Amburgey's PR #4532 scalar-to-gate equivalence, and the compiler lineage of zxv, Jacob Alan Sussman and Ryanson Jonathan. Those are retained premises and credits. Convex chord bounds and extremal moment methods are classical; the contribution is the sharp interior-mass adaptive boundary and the source-gate discriminator.

Prepared by @aliromman with substantial OpenAI Codex assistance; Apache-2.0.

[jwr404]

Independent review (James Reed, with Claude Code). Head ba5473db1513aa96824232dd33fcb363aca44cbd.

Verdict: the theorem and the integer discriminator are correct.

Proof check

• Envelope. On the interior mass ρ (conditional mean y ∈ [ℓ, u]) and on the remainder (outside [ℓ, u], inside [0, 1]), the mean-preserving chord replacements only raise E exp(−h(Z−x)) by convexity. The result is exactly Q_y, with the stated four masses. The constraints 0 ≤ p − ρy ≤ 1 − ρ give [L, U], and the transform is affine in y. ✓
• The "at least ρ" claim. Designate exactly ρ of the interior mass as the interior part. The rest goes to the remainder chord on [0, 1], which still works, and the endpoint laws still lie in the exact-mass class. ✓
• Sharpness. The Q* construction (mixture of the active endpoint laws whose transform derivative vanishes at h*) is correct. Its transform is convex in h, so h* is its global minimum at value E(h*). It is feasible: mean p and interior mass ρ are preserved under mixing. This avoids any minimax interchange. The x ≥ p (Jensen), x < z₀ and x = z₀ < p (strict decrease to η, not attained) cases are also right.

Independent numbers

• T(1/2) = 2 + 4 + 32·16 = 518. The transform 2y + 4 + 512/y² has its minimum 28 at y = 8, giving the threshold c > 7. ✓
• Old endpoint law: weights 14/3 and 1540/3 (total 518, same log-mean), minimum³ = (27/4)(14/3)²(1540/3) = 75460 > 32³ = 32768. ✓
• Gates:
  • u_tail = (4106/259)/16 = 2053/2072 < 1;
  • v_tail = 7³²/2⁹⁰, where 7³² = 1104427674243920646305299201 < 2⁹⁰ = 1237940039285380274899124224, about 11% margin, so it's tight but true;
  • T(1)/T(1/2) = 4106/259 > 4. ✓
• Random stress test. I generated 3,000 random classes (ℓ, u, ρ) with random finite-support feasible laws, mixing interior and exterior atoms, and evaluated each at 5 random (h, x). Across 15,000 evaluations no law exceeded the two-law envelope (maximum relative excess −1.3·10⁻⁶).

The registered replay passes locally. The scope statement is accurate: the fixture fails mirror coverage, and the scalar-to-gate equivalence is a cited premise.