Gates
Research gates, D0 → D26
Each gate is a reviewed step: exact mathematics, two independent verifiers, and a corruption suite. Status is honest, partial and blocked gates are shown as such.
SCANpass
Multi-repository research scan
Four upstream sources cloned, audited and compiled, k=9 reproduced locally in 116 s, matching upstream result_10 digit for digit.…
GATE Apass
Duplicate-coordinate traceability audit
The R2 concern rested on a false premise: the pipeline enumerates sorted multisets (a padding phase), not distinct sets, matching the tupl…
GATE Bpass
Certificate architecture
Recursive proof-trace certificate plus an independent verifier, it rejects false prunes, fabricated survivors and phantom duplicates, not …
GATE Cpass
Certified reproduction and orbit compression
C++ producer plus two verifiers plus a provenance manifest. Orbit-level certificate for the p=53 case verified with the exact completeness…
GATE C.004partial
Full k=9 campaign, estimate gate
p=19 fully certified by both verifiers. Campaign halted at preflight: p=151 exceeded the per-prime cap 9.3 times, exposing the lift-stage …
GATE C.005blocked
Orbit-compressed lift, measured infeasible
Both allowed compression families measured to fail: decision-DAG state sharing 0.999 (no sharing), orbit-stabilizer only a constant factor…
GATE D0partial
Tight-class analytic bridge
Reconstructed the polynomial-method core, isolated the load-bearing Fermat-indicator step, localized why it needs k+1 prime. The obstructi…
GATE D1partial
Composite-14 generalization, candidates refuted
Affine landing is false for k=13 because units act trivially mod 2. The CRT-idempotent indicator is exact but cannot rescue a false target…
GATE D2partial
Direct properness, the open door
Ground-truth reversal: the tight fiber is all-proper for composite k+1. The obstruction was the affine route, not properness itself. The c…
GATE D3pass
Residual closure, a working certificate
An independent bitset oracle agrees with the D2 oracle (0 disagreements). A non-unit multiplier closes the residual exactly. The k=13 resi…
GATE D4pass
Exact tight-interface closure at p=191
The extrapolation is replaced by an EXACT count: the k=13 tight residual is 16,171 orbit-classes, with an exact mass identity B0 + B1 + re…
GATE D5pass
Uniform prime transfer, all p>182
The single prime p=191 becomes a theorem over primes: the k=13 tight interface is proper for EVERY prime p>182. The witness test reduces b…
GATE D6blocked
Non-tight preflight, NO-GO-PROVISIONAL
The D5 tight theorem integrates correctly into the full reduction (rule TIGHT_UNIFORM_D5, state-equivalence proved, two verifiers, 20/20 a…
GATE D7partial
Architecture redesign, RESTART (predicate re-freeze)
Before optimizing, the find_cover predicate must reproduce frozen truth. It reproduces k=9 p=19 = 1 exactly, but NOT the upstream k=9 p=53…
GATE D7Rpass
Ground-truth harness, predicate reproduced
The pinned upstream find_cover was compiled as an oracle and reproduces the frozen truth exactly (p=19 = 1, p=53 = 4,651). A fully indepen…
GATE D8partial
Structural elimination, Fourier criterion
A strategic pivot from enumeration to STRUCTURAL elimination. A finite-Fourier identity M(u) = G0^k + relation-mass is proved: a vector wi…
GATE D9partial
Fourier tail closure & inverse structure
A rigorous barrier result: the absolute-value tail route is INSUFFICIENT. The absolute relation mass is essentially vector-independent, ab…
GATE D10partial
Moment duality & signed cancellation
Encode the signed cancellation as factorial moments of N (the number of forbidden runners): u proper iff some time has N=0. An exact ratio…
GATE D11partial
Punctured moments, inventory-wide closure
The D10 blocker was a universal atom, not a missing dual. Removing the t=0 spike EXACTLY (where all 13 runners are trivially forbidden) un…
GATE D12partial
Ratio graph: sparse closes, dense low-rank
The non-tight remainder is now a finite graph with exact rational labels, not a cloud of vectors. An exact dual-deficit identity expresses…
GATE D13partial
Connected hypergraph, tight rank-1 recovered
The higher residual left by D12 is expressed as EXACT connected correlations, and the pair graph is extended to a relation hypergraph. The…
GATE D14partial
Connected-support purity, TRUE_RELATION vs puncture
The connected correlation of each support set is separated by an EXACT identity: a primitive additive relation (the true structural signal…
GATE D15partial
Projective directions, support-3 closed over all 109 primes
The homogeneous true-relation sum is regrouped into projective relation directions (scalar orbits), with an EXACT orbit-mass identity, a c…
GATE D16partial
Modular rank repair, support-3 completeness, support-4
A load-bearing correction: relation rank must be computed over F_p, not Q, because every relation satisfies R·u ≡ 0 (mod p) with u nonzero…
GATE D17pass
Support-4 fast correlation, inventory-complete
A primitive root turns the multiplicative support-4 correlation into a cyclic 4-point correlation on Z_{p-1}, giving an exact fast engine …
GATE D18partial
Final connected order, fixed degree-5 refuted, adaptive route
D18 first tests whether a degree-4 dual alone can close the generic branch. An exact degree-4 minorant is found, positive on the independe…
GATE D19refuted
Exact cyclic cover, per-prime separation refuted
D19 returns from moment space to the exact source object. Fixing a prime and a primitive root, a speed vector becomes a multiplicity vecto…
GATE D20blocked
Simultaneous prime-product lift, blocked at scale
D20 built the full prime-product lift machinery and then hit a computational wall. The D6 finite reduction and D7R alignment were frozen a…
GATE D21partial
Signature-narrowed certified compatibility
D20 was blocked because it asked for the complete unconstrained local-cover universe before any cross-prime information had narrowed the q…
GATE D22partial
Implicit global-signature elimination
D21-S showed that writing down the complete family of starting fingerprints is astronomically large, but that only blocks the two explicit…
GATE D23pass
Early global realizability cuts
D22 was limited because its only genuine global check waited until the final reconstruction step, and until then matching fingerprints pri…
GATE D24partial
Early cuts on reachable states
D23 proved that its early cuts are safe and showed, on a bounded pilot, that they prune strongly before reconstruction. The pilot numbers,…
GATE D25partial
Adaptive rational-time separation
D24 showed that a fixed handful of clocks, the level fourteen, twenty-eight and forty-two grids, removes more than half of the reachable s…
GATE D26partial
Labeled-root CRT campaign
Every route so far has paid a heavy price at the reconstruction step, because it rebuilt the symmetric functions of the squared speeds, wh…
GATE D27blocked
Orbit-compressed unlabeled CRT: BLOCKED
D26 proved the direct coordinate CRT threshold but measured the labeling obligation as a net-negative cost. D27 keeps the lower reconstruc…
GATE D28blocked
Certified bulge bridge: BLOCKED-BEFORE-BUILD
D28 tests whether the state-count region that blocked D27 can be crossed without enumerating its orbit children, examining a bidirectional…
GATE D29partial
Pre-reconstruction cross-prime coupling: PARTIAL
D29 asks whether any label-free, sign-invariant invariant couples local cover choices across primes before full reconstruction. Phase 1 pr…
GATE D30partial
Certified bounded modular square root: PARTIAL
The Phase 3 coupling reduces its useful computation to one exact problem: given an odd squarefree modulus with known factorization, a resi…
GATE D31partial
Prime economy and the residue coset theorem: PARTIAL
D31 asks whether a prime portfolio can accumulate enough modulus bits for the coupling to become non-vacuous while the count of nonzero-re…
GATE D32partial
Implicit realizability and the symbolic CEGAR kernel: PARTIAL
D32 tests the remaining unrefuted route, representing the global realizability problem implicitly and refining it with exact predicates ra…