H-007
Padding-uniform diagonalization escapes P If the universal simulator's overhead on every machine in the enumeration is bounded by a single fixed polynomial p, then the diagonal language D built stage by stage lies in NP but not in P.
∃ p ∀ i : time ( U , M i ) ≤ p ( n ) ⟹ D ∈ NP ∖ P
The first move on D-1: turn the enumeration into a diagonal language the simulator can build inside NP.
proposed by GAUSS-01
direction D-1 · Diagonalization against polynomial-time machines
recorded 2026-09-22 08:50:50 UTC
critic score 0.31 TURING family demo-critic-a 2026-09-22 09:30:50 UTC
The overhead bound is asserted for every machine in the enumeration, but p is fixed before the enumeration is taken. Machines whose own running time exceeds p appear later in the list and break the claim.
Restate with the polynomial chosen after the machine, or drop the hypothesis. As written the two quantifiers are in the wrong order.
logical completeness 0.38
novelty potential 0.22 Counterexample found
Missing assumptions Uniformity of p over the whole enumeration H-008
One diagonalization stage is sound A single stage of the construction kills exactly one polynomial-time machine, at a padding cost of a factor c on the input length, with c independent of the stage index.
n k + 1 ≤ c ⋅ n k , c independent of k
A weaker, quantifier-correct version of H-007, stated for a single stage rather than for all of them at once.
proposed by GAUSS-01
direction D-1 · Diagonalization against polynomial-time machines
recorded 2026-09-22 09:37:40 UTC
critic score 0.44 TURING family demo-critic-a 2026-09-22 10:25:20 UTC
A single stage is fine. The stage cannot be iterated: each one pads the input by a further factor, and after k stages the accumulated padding exceeds the length budget the construction has to stay inside.
Track the padding explicitly across stages before claiming the factor is stage-free.
logical completeness 0.61
novelty potential 0.28 Known barrier detected
Missing assumptions Behaviour of the padding factor under iteration H-009
Gate elimination needs only linearly many rounds The elimination procedure applied to a circuit of the restricted class on n inputs opens at most O(n) rounds before the circuit collapses to a constant.
R ( n ) = O ( n )
First attempt at counting the rounds the elimination procedure needs.
proposed by GAUSS-02
direction D-2 · Gate elimination with a controlled round count
recorded 2026-09-22 09:03:40 UTC
critic score 0.29 TURING family demo-critic-a 2026-09-22 09:44:40 UTC
The count confuses the variables fixed by a restriction with the rounds the procedure opens. The numerical check in E-03 gives roughly n·log n rounds at n = 40.
Recount. The round count is the number of rounds opened, not the number of variables fixed.
logical completeness 0.33
novelty potential 0.19 Counterexample found
H-010
Constructivity of the measure forces largeness If the elimination measure used in D-2 is computable in time 2^O(n) on a truth table of length 2^n, then it exceeds its own threshold on at least a 2^(−n) fraction of all truth tables.
f Pr [ μ ( f ) ≥ τ ] ≥ 2 − n
The natural-proofs precondition D-3 is built around: constructive plus large is exactly the pair Razborov–Rudich rules out.
proposed by GAUSS-03
direction D-3 · Natural-proofs status of the elimination measure
recorded 2026-09-22 09:09:40 UTC H-011
Every NP-complete language needs super-polynomial time For every NP-complete language L and every polynomial p, no machine deciding L runs in time p(n) for all sufficiently large n.
∀ L ∈ NPC , ∀ p : L ∈ / DTIME ( p ( n ))
The strong form of the objective, claimed directly from the diagonalization of D-1. It is the open question itself, asserted.
proposed by GAUSS-01
direction D-1 · Diagonalization against polynomial-time machines
recorded 2026-09-22 09:56:00 UTC
critic score 0.24 TURING family demo-critic-a 2026-09-22 10:07:30 UTC
Three independent problems. It restates the objective rather than establishing anything; it inherits the padding blow-up identified in H-008; and the construction relativizes, so Baker–Gill–Solovay rules it out even if the first two were repaired. Recorded so no later iteration proposes it again.
Withdraw. This is the open question restated, and the argument offered for it is the one already rejected in H-008.
logical completeness 0.21
novelty potential 0.08 Known barrier detected
Missing assumptions Padding blow-up from H-008 is not accounted for H-012
Rounds are the right measure of elimination cost The number of rounds the elimination procedure opens below n is a monotone function of n and bounds the number of gates it removes from above.
G ( n ) ≤ R ( n )
The correction TURING asked for in H-009: count rounds, not gates.
proposed by GAUSS-02
direction D-2 · Gate elimination with a controlled round count
recorded 2026-09-22 10:01:40 UTC
critic score 0.72 TURING family demo-critic-a 2026-09-22 10:48:10 UTC
The correction is right and the monotonicity claim is the only part that can be made formal without further work. Novelty is low; usefulness as a stepping stone is high.
Keep. Make the monotonicity claim precise enough to formalize.
logical completeness 0.74
novelty potential 0.36 H-013
Largeness survives averaging over the sampled family Averaging the measure of H-010 over the sampled sub-families does not destroy the largeness: the average fraction is still bounded below by a fixed constant.
A robustness check on H-010, needed before the barrier test means anything.
proposed by GAUSS-03
direction D-3 · Natural-proofs status of the elimination measure
recorded 2026-09-22 10:29:20 UTC H-014
The relativization barrier is intrinsic, not an artefact The construction of D-1 goes through verbatim relative to any oracle, so by Baker–Gill–Solovay it cannot settle the question in either direction, whatever the padding is fixed to.
Stated so the negative result is on the record rather than being rediscovered by a later iteration.
proposed by GAUSS-01
direction D-1 · Diagonalization against polynomial-time machines
recorded 2026-09-22 10:16:00 UTC
critic score 0.66 TURING family demo-critic-a 2026-09-22 10:25:40 UTC
Correctly identifies why the direction fails, and names the published barrier rather than describing it vaguely. A negative result of this shape is worth keeping: it is what stops the next iteration from repeating D-1.
Record it in research memory and close D-1. This is the most useful thing the direction produced.
logical completeness 0.70
novelty potential 0.31 Known barrier detected
H-015
The round count grows by at most one per variable fixed Fixing one further input variable opens at most one new elimination round, so the round count is non-decreasing with unit increments.
R ( n + 1 ) ≤ R ( n ) + 1
The first statement in this iteration precise enough to be worth formalizing.
proposed by GAUSS-02
direction D-2 · Gate elimination with a controlled round count
recorded 2026-09-22 10:54:40 UTC
critic score 0.78 TURING family demo-critic-a 2026-09-22 11:17:50 UTC
Two claims in one sentence. The monotonicity half is immediate from the definition and is a good formalization target; the unit-increment half needs the structure of the restriction.
Split the monotonicity from the increment bound; formalize the monotonicity first.
logical completeness 0.81
novelty potential 0.24 H-016
A round-count bound would give the size lower bound An upper bound of the form R(n) = O(log n / log log n) on the round count implies a super-polynomial size lower bound for the family on the restricted class.
R ( n ) = O ( log log n log n ) ⟹ size = n ω ( 1 )
The bridge from the round count back to the objective. Unsupported: the implication is the part that carries all the weight, and nothing in this iteration argues for it.
proposed by GAUSS-02
direction D-2 · Gate elimination with a controlled round count
recorded 2026-09-22 11:03:40 UTC
critic score 0.39 LAPLACE family demo-critic-b 2026-09-22 11:27:40 UTC
The implication is the whole content and it is asserted rather than argued. Nothing in D-2 currently connects a round-count bound to a size bound; the missing estimate is the one the objective is about.
Do not carry this implication into the proof queue. State the intermediate estimate it needs as its own hypothesis first.
logical completeness 0.36
novelty potential 0.41 Missing assumptions The passage from a round-count bound to a size bound Uniformity of the implied constant in n H-018
The elimination round counter is monotone in n For the procedure built in D-2, the number of elimination rounds recorded below n is non-decreasing: eliminationSteps n ≤ eliminationSteps (n + 1) for every natural number n.
∀ n ∈ N : R ( n ) ≤ R ( n + 1 )
The narrow, formalizable core of H-015. It says nothing about how fast the counter grows, which is what the objective actually needs, and nothing about circuits at all.
proposed by GAUSS-02
direction D-2 · Gate elimination with a controlled round count
recorded 2026-09-22 11:09:20 UTC
critic score 0.86 TURING family demo-critic-a 2026-09-22 11:13:20 UTC
Immediate from the recursive definition of the counter. Reviewed as a stepping stone, not as progress on the objective: the interesting claim is the growth rate, which this does not touch.
Send to the proof builder. It is small, it is true by construction, and it is the first thing in this iteration a formalizer can hold onto.
logical completeness 0.93
novelty potential 0.12 LAPLACE family demo-critic-b 2026-09-22 11:22:10 UTC
Reviewed without reading TURING's verdict. The recursion adds a term that is 0 or 1, so the inequality is immediate. The statement is worth formalizing precisely because it is narrow; read as progress on the separation it would be badly misleading.
Concur. Formalize as written, and record next to it that the growth rate is untouched.
logical completeness 0.95
novelty potential 0.11 SENIOR REVIEWER family demo-frontier-b 2026-09-22 11:55:00 UTC
Attacked on three fronts: the definition of the counter, the quantifier structure, and whether the statement smuggles in anything about circuit size. It does not. What is left is a monotonicity fact about a counter this iteration defined — a reasonable formalization target, and not a step towards the separation on its own.
Formalize. Freeze the statement exactly as written and do not let the formalizer widen it towards a size bound.
logical completeness 0.96
novelty potential 0.09 H-021
The measure is constructive and large on the sampled family The elimination measure is computable within the constructivity budget and exceeds its threshold on a constant fraction of random truth tables, which would place the whole of D-2 inside the natural-proofs barrier.
GAUSS-03 is testing this against the fractions measured in E-11 and E-12. If it holds, D-2 cannot give the bound it is after, whatever the round count does.
proposed by GAUSS-03
direction D-3 · Natural-proofs status of the elimination measure
recorded 2026-09-22 13:41:20 UTC