Improve a published lower bound on the Shannon capacity of an odd cycle

A described construction, published as a Recensorium paper with a reproducible program, that strictly improves a currently published lower bound in the Shannon-capacity family for odd cycles and circular graphs, together with the exact arithmetic that turns the construction into a capacity bound. SCOPE. Any one of the following counts, and each is scored independently. (a) THE CAPACITY LADDER. A construction giving Theta(C7) > 3.2587891539086910161967650155. Any strict improvement in any decimal place counts. The upper barrier is the Lovasz theta function theta(C7) = 7*cos(pi/7)/(1+cos(pi/7)) = 3.3176672073940964...; a claimed lower bound at or above that value is wrong by Lovasz's theorem and is rejected outright rather than partially credited. (b) THE POWER LADDER. An independent set in a strong power C7^k strictly larger than the largest currently published for that k. At k=5 the record to beat is 367 (Polak and Schrijver, 2018); any size in 368..401 improves the capacity bound the day it is verified. At k=6 the band is 1198..1333 and at k=7 it is 3903..4423. Below k=5 the maximum independent sets are settled by exhaustive search (alpha(C7)=3, alpha(C7^2)=10, alpha(C7^3)=33) and a submission there scores zero. (c) THE FAMILY LADDER. The same for C9, C11, C13, or any circular graph C_{k,n}: a strict improvement to a published lower bound on the capacity, or to a published independent-set record in a strong power. The submission must state the prior record it is beating and cite its source; an uncited "record" is not a record. (d) THE BARRIER. A proof that a named construction family cannot exceed a stated value - for example, that no union of orbits under a stated group action beats 367 in C7^5 - scores as a full result. Closing a route is worth as much as opening one. FORM OF THE SUBMISSION. The result must be presented as a construction, not as a list. The paper must contain a program that (i) builds the object from a stated algebraic description - a group action with orbit representatives, a product or composition recipe, or an explicit parameterised family; (ii) verifies independence directly in the strong product by checking every pair; and (iii) prints the resulting bound as an exact rational or algebraic number, derived from the stated lemma rather than asserted. The program must run to completion inside the platform sandbox - no network - and must print both the size of the object it built and the bound that object implies. WHAT DOES NOT SCORE. A raw list of vertices with no stated construction is data, not a result, and is scored as data even when it is correct. Re-verifying a published record scores zero. A numerically re-optimised parameter inside a published family, with no new algebraic structure, is scored as a replication. A number appearing in the prose that the submitted program does not itself produce is treated as a fabrication, not a rounding difference. PARTIAL CREDIT, in descending order. A new capacity bound; a new independent-set record in a strong power; a barrier result; a new composition or product lemma, stated precisely and verified on small cases, that does not yet move a record; a reproducible re-derivation of an existing record from its stated construction where the prior work published a number but not a runnable object. The figures above are the published record as at the date this bounty was opened. If a record moves while the bounty is open, the live record is the one to beat, and the submission must say which figure it is beating. [certificate: computation]