[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"yT80Iw27qw":3},"# P vs NP: Lean Formalization (Honest Status)\n\nStatus date: 2026-05-30.\n\n`STATUS.md` is the authoritative, most-recently-updated snapshot; this README\nis a stable high-level overview and may lag it by a few days.\n\nCanonical checklist for unconditional readiness:\n`CHECKLIST_UNCONDITIONAL_P_NE_NP.md`.\nCurrent release posture:\n`RELEASE_RC.md`.\n\n## What This Project Is\n\nThis repository contains a Lean 4 formalization framework for a magnification\nroute around Partial MCSP:\n\n`SAL -> Covering/Lower Bounds -> anti-checker -> magnification -> final wrappers`.\n\nThe active code lives in `pnp3/`. Historical material under `archive/` is kept\nfor provenance only and must not be treated as the status source for the\ncurrent branch.\n\n## Variant Boundary\n\nActive `pnp3/` development uses **Partial MCSP** (`GapPartialMCSP*` names).\n\n- Working model: `pnp3/Models/Model_PartialMCSP.lean`.\n- Active language/promise names: `gapPartialMCSP_Language`,\n  `GapPartialMCSPPromise`.\n- Legacy total-table / older MCSP variants are historical unless explicitly\n  linked from active status docs.\n\n## Current Verified State\n\n- `pnp3/` builds and `./scripts/check.sh` passes on the current tree.\n- Active project-local `axiom` declarations in `pnp3/`: `0`.\n- Active `sorry/admit` in `pnp3/`: `0`.\n- Inclusion is internalized:\n  `proved_P_subset_PpolyDAG_internal : P_subset_PpolyDAG`.\n- DAG endpoint plumbing is substantial, including the fixed-slice\n  `PpolyDAG -> PpolyFormula` bridge and final wrappers.\n\n## Current Audit Result\n\nThere is still **no unconditional in-repo theorem** `P != NP`.\n\nThe earlier support-bounds route was not merely incomplete.  It was formally\nrefuted:\n\n- `FormulaSupportRestrictionBoundsPartial -> False`\n- `FormulaSupportBoundsFromMultiSwitchingContract -> False`\n- `MagnificationAssumptions -> False`\n- `FormulaSupportBoundsPartial_fromPipeline -> False`\n\nThe root cause is fixed-slice truth-table hardwiring.  The old predicates were\nstrong enough to apply to arbitrary polynomial-size formula witnesses, including\nhardwired formulas whose support is all variables.\n\nBeyond the formula-side refutation above, the **canonical asymptotic track**\n(the iso-strong and promise-YES route family) is now also formally closed at\nthe conclusion level.  The kernel-checked, in-build witness is\n`isoStrong_conclusion_negative_general` in\n`pnp3/Tests/GeneralIsoStrongNoGoProbe.lean`; see `STATUS.md` for the full\naudit chain.  This does not prove `P != NP`; it rules out that route family.\n\n## Fixed-Params Candidate\n\nThe current nontrivial candidate shape is:\n\n```text\nFormulaSupportBoundsPartial_fromPipeline_fixedParams ac0 sb\n```\n\nIt fixes AC0 parameters externally, so the known singleton-provider attack does\nnot directly port.  But it is not a proved lower-bound theorem.  Also,\n`fixedParams + uniformProvenance` reconstructs the old false support-bounds\npredicate and is therefore inconsistent as currently stated.\n\nThe theorem\n`NP_not_subset_PpolyDAG_final_under_fixedParams_and_uniformProvenance` exposes\nthis research gap.  It does not close it.\n\nThe one-file closure boundary is\n`pnp3/Magnification/UnconditionalResearchGap.lean`.  It defines\n`ResearchGapWitness` and proves the conditional bridge from that witness to\n`P != NP`.\n\n## What This Means\n\nThe repository is useful as a formal framework and audit harness for future\nmagnification attacks.  It does not currently prove `P != NP`, and the\nremaining gap is mathematical: a non-vacuous fixed-params support/locality\nsource theorem that cannot be satisfied by truth-table hardwiring or singleton\nprovenance.\n\n## Verification\n\n```bash\n./scripts/check.sh\nfor f in pnp3/Tests/AxiomsAudit.lean \\\n         pnp3/Tests/BarrierAudit.lean \\\n         pnp3/Tests/BarrierBypassAudit.lean \\\n         pnp3/Tests/BridgeLocalityRegression.lean \\\n         pnp3/Tests/WeakRouteSurfaceTests.lean \\\n         pnp3/Tests/FormulaSupportBoundsFalsifiabilityProbe.lean; do\n  lake env lean \"$f\"\ndone\n```\n\n## Primary Documents\n\n- `STATUS.md` - authoritative current snapshot.\n- `TODO.md` - remaining execution order.\n- `CHECKLIST_UNCONDITIONAL_P_NE_NP.md` - exact closure checklist.\n- `pnp3/Magnification/UnconditionalResearchGap.lean` - single-file remaining\n  research gap boundary.\n- `RELEASE_RC.md` - release posture and wording guardrail.\n- `AXIOMS_FINAL_LIST.md` - axiom/sorry hygiene only.\n\n## Wording Policy\n\nUntil the checklist is fully closed, any statement of `P != NP` in this\nrepository must explicitly say that the current final theorem surface remains\nconditional and that the support-bounds source theorem is still open.\n",1780846776883]