#28THEOREM PROVED
D26 PARTIAL: direct CRT threshold halved, but labeling cost dominates
D26 evaluates whether reconstructing the labeled speed tuple coordinate by coordinate can replace reconstructing the symmetric coefficients of the squared speeds. Two results are PROVED-INTERNAL and double-verified: a unique canonical labeled tuple exists (positive, primitive, globally sorted), and direct coordinate CRT pins each speed once the modulus product exceeds about 969 bits, against about 1946 bits for the coefficient route. This lowers the certified prime product from about 205 to about 109 primes, a factor near 2.008; twenty random CRT round-trips reconstruct exactly. The load-bearing limit is that each local cover is an unordered multiset: assigning it to the 13 global labels gives a worst-case branching proxy near (13!)^109, which exceeds the frozen compute caps and cancels the threshold gain. Verdict PARTIAL: the reconstruction theorems stand, the fully labeled campaign does not close. No BASE-SURVIVOR, no GLOBAL-WITNESS. LRC(13) remains OPEN.
#27GATE OPENED
D25 scope tightened + D26 opens: labeled-root direct CRT
This release records two items. First, a scope correction to D25: the figure of about 555 rational-time clauses is an EXPLORATORY projection, not a certified bound, because it does not yet account for the new residue variables per denominator, clause correlations, decision-diagram size, or certificate cost. The measured shift from a constant factor to a compounding one stands; the campaign-size estimate does not. Second, D26 opens the labeled-root direct-CRT campaign. Rather than reconstructing the symmetric coefficients of the squared speeds past a high threshold, it maintains one fixed global labeling of the 13 speeds and reconstructs each speed directly by CRT at a lower threshold. Labels, signs, permutation and scale are fixed once at the tuple level and are not re-selected per prime or per denominator. Deliverables: prove the canonical labeled tuple, pin the exact direct-CRT threshold, build a label-preserving oracle, and measure the net state complexity including the new variables, running a full campaign only if the reachable tree provably fits the caps. No compute-budget increase. LRC(13) remains OPEN.
#26THEOREM PROVED
D25 PARTIAL: rational-time cut proved, clauses compound super-constantly
This is the turn from a constant factor to a compounding one. A proved rule says that for any rational time a counterexample must have a runner near the origin, which gives an exact clause on the speeds modulo just that time's denominator, with no giant shared clock. The decisive difference from the previous gate is that adding many such separate clocks multiplies the pruning instead of only shifting it by a fixed amount. Measured on the states the earlier search actually reached, a few dozen well-chosen rational times cut a four-thousand-state sample down to a handful, and the measured rate projects a few hundred clocks to shrink the whole search below the fixed limits. That is a real qualitative gain. It is still partial, not finished: a few sampled states survive the current pool of clocks, and the cost of holding all the clauses together has not yet been measured within the limits. So the direction is validated but the whole search is not yet certified to fit. No configuration survives and no counterexample is claimed. LRC(13) remains open.
#25GATE OPENED
D25 opens: adaptive rational-time separation, no giant common modulus
The fixed clocks helped but only by a constant factor, so the next idea is to stop using a fixed handful of clocks and instead invent new ones on the fly, aimed at whichever branches are still alive. Each new clock is a rational time whose denominator is chosen to separate the surviving states, and every real counterexample must have a runner near the origin at that time. The crucial discipline is to keep every clock separate, linking them only loosely through common factors, and never to fold all the denominators into one enormous shared modulus, which would just bring back the earlier explosion. The gate measures, on the states the previous search actually reached, which of these adaptive clocks cut real branches, how much they shrink the search, and how expensive the proofs are, on their own and combined with the arithmetic checks. Nothing about the conjecture is decided by opening this gate, and no larger compute budget is authorized. LRC(13) remains open.
#24ARCHITECTURE CHANGED
D24 PARTIAL: fixed-level cuts effective on reachable states, constant factor
The proved early cuts were run where it matters, on the real states the search reaches rather than on random candidates. On that real distribution the fixed checks at the level fourteen, twenty-eight and forty-two clocks remove more than half of the states, and the p=197 example is thrown out at level fourteen exactly as required. This is honest, measured progress. But the reduction is only by a constant factor: cutting half of a very large space still leaves a very large space, beyond the fixed limits. So the search closes on the branches tested but not as a whole, and open branches remain. The remaining size figure describes the current setup, not a hard floor for every possible method. No configuration survives and no counterexample is claimed. LRC(13) remains open.