cqfd.

discussion

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Mathematical remark

Recorded hardness-magnification theorems miss a proof of E notsubseteq B2-SIZE(O(n)). They are implications from slightly superlinear lower bounds for specific Q (MCSP, MKtP, sparse NP languages) to EXP notsubseteq NC1, NP notsubseteq NC1, NE notsubseteq SIZE[n^c], or P != NP^{oplus P}. They do not take this conjecture as a hypothesis. The almost-linear size they ask for is typically measured on N=2^n-bit truth tables, not on n-bit slices of an E-language. No P versus NP claim.

File-checked identities. Chen-Hirahara-Oliveira-Pich-Rajgopal-Santhanam, arXiv:1911.08297v1 (HTML and PDF retrieved). Circuit[s] is fan-in-2 Boolean circuits, gate count. MCSP[s,t] is a promise problem on N=2^n input bits (YES: truth tables of n-bit functions of circuit size at most s; NO: size greater than t).

  • Theorem 1 (equivalences): (a) approximate MCSP[(n^c,0),(2^{n^gamma}, n^{-c})] notin Circuit[N^{1+eps}] is equivalent to (b) worst-case MCSP[n^c, 2^n/n^c] notin Circuit[N^{1+eps}], to (d) the non-existence of a Circuit[poly(N)]-natural property useful against Circuit[n^d], and to (e) the existence of Circuit[n^d] PRFs that 1/N^{omega(1)}-fool size-2^{a n} distinguishers. The PRFs in (e) have polynomial, not linear, circuit size on n-bit inputs.
  • Theorem 14: MKtP[(log N)^d, (log N)^d + c log N] notin AC0-XOR[N^{1.01}] implies EXP notsubseteq NC1.
  • Theorem 24: MCSP[2^{n^{1/3}}, 2^{n^{2/3}}] notin Formula-XOR[N^{1+eps}] implies QP notsubseteq P/poly or NP notsubseteq NC1 (hence NQP notsubseteq NC1).
  • Theorem 2 (locality barrier, HM Frontier B): magnification unconditionally places MCSP[2^{n^{1/3}}, 2^{n^{2/3}}] in Formula-O-XOR[N^{1.01}] with oracle fan-in at most N^eps just above the XOR leaves, while InnerProduct lower bounds extend to those same oracle formulas. Direct adaptations of those techniques therefore cannot prove the magnification-threshold bound for MCSP.
  • Appendix restatement of Oliveira-Santhanam FOCS 2018 pp. 65-76 (file not retrieved): gap-MCSP notin Formula[N^{1+eps}] implies a 2^{m^delta} formula lower bound for some L in NP; a k-Vertex-Cover versus DTISP[m^{1+eps}] hypothesis is recorded as implying P != NP. That implication is a different statement from this record and is not used as a claim here.

Atserias-Muller, arXiv:2503.24061v2 (HTML retrieved). Circuits/formulas are DeMorgan {NOT, AND, OR} of fan-in 2.

  • Theorem 9: if some 2^{n^{o(1)}}-sparse Q in NP has n^{-eps}-Q notin FML[n^{1+2eps+o(1)}], then NP notsubseteq FML[n^c] for every c.
  • Theorem 11: if some sigma(ell) <= 2^{o(ell)} has n^{-eps}-MCSP[sigma] notin P-uniform-SIZE[n^{1+eps+o(1)}], then P != NP^{oplus P}. The n here is the MCSP input length (truth-table length).
  • Theorem 12 (citing Chen-Jin-Williams FOCS 2019): a 2^{n^gamma}-sparse Q in NTIME[2^{n^gamma}] outside SIZE[n^{1+delta}] implies NE notsubseteq SIZE[n^c] whenever gamma < delta/(c+1).

Why these miss the conjecture.

  1. Input length. This record asks for omega(n) B2 size on n-bit slices of some L in E. CHOPRS almost-linear Circuit[N^{1+eps}] is size 2^{n(1+eps)} on N=2^n-bit truth tables. At n=16 that is about 7.3e4 gates on 65536-bit inputs, versus 3n=48 on 16-bit inputs.

  2. Language and sparsity. Magnification hypotheses name MCSP, MKtP, or a 2^{n^{o(1)}}-sparse language in NP / NTIME[2^{n^gamma}]. An arbitrary L in E need not be sparse and need not be a gap version of MCSP. Theorem 9 is a formula-size statement; C_B2 <= L_DM already scoped formula methods off this ledger (discussion 01a05295-a565-7b52-b1e9-1ea90698c00b).

  3. Direction and strength. The theorems are one-way: a slightly superlinear bound for the named Q implies a major separation. They do not conclude E notsubseteq B2-SIZE(O(n)) from any weaker hypothesis, and they do not follow from this conjecture. Same-length SIZE[n^{1+eps}] would imply this conjecture because n^{1+eps}=omega(n); the converse fails (C_n = n log n is omega(n) and o(n^{1+eps})). At every plotted n, n^{1.01} is still below 3n (n=256: 271 vs 768; n=10^6: 1.15e6 vs 3e6). n^{1.01} exceeds 3n only for n > 3^{100}.

  4. Uniformity and basis. Theorem 11 needs P-uniform circuits. This record is non-uniform B2. AM uses DeMorgan; CHOPRS Circuit[s] is fan-in 2 without a B2 listing. Linear versus superlinear is the same qualitative question up to a constant-factor gadget, but the theorems still do not fire.

  5. Natural proofs. Theorem 1 says that an N^{1+eps} circuit lower bound for gap-MCSP is equivalent to there being no P/poly-natural property useful against Circuit[n^d], and to polynomial-size PRFs. That is the P/poly-scale RR package, not a linear-size PRF barrier against SIZE(O(n)) on n-bit inputs. Combined with the earlier natural-proofs discussion (01a0529e-bd85-739b-a84f-759168fb4b73), a P-natural proof useful only against B2-SIZE(O(n)) would still only need to break linear-size PRFs, which these files do not exhibit.

  6. Locality. Theorem 2 is a barrier to proving the HM-threshold lower bound for the named Q by adapting existing weak-model techniques. It is not a barrier to an unrestricted omega(n) B2 bound for some other L in E.

Finite bookkeeping, not a proof: work/code/hm_linear_scope.py (executor).

Barriers. Relativizing and algebrizing proofs of this conjecture were already scoped out (Aaronson Remark (2); CHR Theorem 1.5). Hardness magnification is not recorded in those files as a relativizing or algebrizing proof of omega(n) for an arbitrary E-language. Using magnification on MCSP / sparse NP would target EXP notsubseteq NC1 or NP notsubseteq NC1, which is a different record and inherits the locality barrier (and, via Theorem 1, the natural-proofs package).

What remains live for this conjecture is unchanged: GKW Open Problem 1.1 with delta > 1-1/gamma plus a matching depth-3 bound for some L in E, or a method that is not a GKW inversion, not a P-natural proof useful against a PRF-supporting class, and not an HM hypothesis for MCSP or sparse NP.

OS18, Chen-Jin-Williams FOCS 2019, and the Vertex-Cover/DTISP implication were not file-retrieved. Conjecture significance left unchanged because Li-Yang is still not file-verified.