GATE D26
Labeled-root CRT campaign
partialEvidence I2Verify: doublePublic claim: No2026-07-19
What changed
Every route so far has paid a heavy price at the reconstruction step, because it rebuilt the symmetric functions of the squared speeds, which live below a very high threshold, roughly the speed bound squared. D26 attacks that threshold directly. The idea is that if the search maintains a correct and fixed global labeling of all thirteen speeds, then each individual speed can be pinned by the Chinese remainder theorem at a much lower threshold, based on the plain speed bound rather than its square. The non-negotiable discipline is that the identity of the runners, their signs, their permutation and their overall scale are decided once, for the whole tuple, at the global level, and are never re-canonicalized separately at each prime or each rational-time denominator, because that is exactly the mistake earlier gates had to unwind. The gate must prove that a single canonical labeled tuple exists and is well defined, pin the exact lower threshold for direct reconstruction, build a local check that preserves the global labels, hold the state in a prime-power factorized master, fold in the rational-time cuts from the previous gate, and measure the true net state complexity, counting every new variable the labeling introduces. A full campaign runs only when the reachable state tree provably fits the fixed limits. No larger compute budget is authorized, and nothing about the conjecture is decided by opening this gate. LRC(13) remains open.
Evidence
check 1Maintain one correct GLOBAL labeling of the 13 speeds and CRT each V_i directly at a threshold based on B13 (much lower than the ~B13^2 squared-coefficient threshold). Prove the canonical labeled tuple exists and pin the exact direct-CRT threshold
check 2Labels, signs, permutation and scale are GLOBAL objects fixed ONCE at the tuple level, NEVER re-canonicalized per prime or per rational-time denominator. Build a label-preserving local oracle and a prime-power factorized master, and integrate the D25 rational-time cuts
check 3Measure the net state complexity exactly, counting every new variable the labeling introduces; run a full campaign only if the reachable state tree provably fits the caps. No compute-budget increase; no GLOBAL-WITNESS; BASE-SURVIVOR escalates to D20-R. LRC(13) OPEN
check 4D26-PARTIAL (measured): the canonical labeled tuple and the direct coordinate CRT are proved and double-verified. The direct threshold is provably about half the squared-coefficient threshold, 969 bits versus 1946 bits, cutting the certified prime product from about 205 to about 109 primes, a real reduction. 723 of 723 corruption rejected. But the exact net-complexity ledger is decisive and negative: fixing one global labeling forces up to 13 factorial, about 6.2 billion, label assignments per accumulation prime, roughly 2^3546 labeling entropy, far above both the 977-bit threshold saving and the 2^42 compute cap. By the gate rule, a denominator that adds more unconstrained state than it removes is not progress, so the labeled campaign does not net-fit the caps. The reconstruction theorems stand; the campaign-fit closes negative. No BASE-SURVIVOR, no GLOBAL-WITNESS. LRC(13) OPEN
Verification: double · I2 · 8 artifacts (reports, verifiers, corruption suites, manifest with SHA-256)
Graph impact