{"name":"treeOS_LOG","v":"1.1","date":"2026-11-07","scope":"OMEGA joining record; SRSD unchanged. Full numerical evidence is in the named companion package.","model":{"domain":"d in Z^120; fixed lexicographic 600-cell vertices","phi":"(1+sqrt(5))/2","Q":"339949771344778","S":"1000000000000000","D":"25000000000000000","M":"13I+A; A is 12-regular adjacency","P":"M/25","q":"Q/S","lambda":"phi/(phi+pi)","T":"floor(Q*M*d/D)","s":"d mod8 in 0..7","r":"QM*d-D*T(d), 0<=r_i<D"},"foundation":"For s_i>0, E_s(y)=(floor(y_i/s_i),y_i-s_i*floor(y_i/s_i)); D_s(d,rho)=diag(s)*d+rho. Floor inequalities give canonical residuals and both inverse identities. Retain scale and coordinate order.","proofs":[["geometry","Write doubled vertex coordinates 2C=C0+phi*C1. Exact certificates: AC0=6C1; AC1=6C0+6C1; column sums zero; C0^TC0=72I4,C0^TC1=-24I4,C1^TC1=48I4. Using phi^2=phi+1 gives AC=6phi*C and C^TC=30I4."],["embedding","B=[1 C0 C1]; L=[1^T;2C0^T+C1^T;C0^T+3C1^T]/120. Gram identities give LB=I9. Thus E_Z(t,x,z)=B(t,x,z) is injective on Z x Z^4 x Z^4. MB=BK, K(t,x,z)=(25t,13x+6z,6x+19z)."],["scalar","Omega_g(t)=1+g(t-1); F_g(v)=1+gP(v-1). M1=25*1 gives F_g(t1)=Omega_g(t)1. For a,ref>0, chi(a)=1+a/ref gives chi(lambda*a)=Omega_lambda(chi(a))."],["separation","All 120 leading principal determinants of symmetric M-9I are positive. Sylvester gives k^TMk>9 sum k_i^2 for k!=0. If ||Mk||inf<=9, k^TMk<=9 sum|k_i|<=9 sum k_i^2: contradiction. Integrality gives ||Mk||inf>=10."],["recovery","Equal outputs and mod8 witnesses give d-e=8k and ||QM(d-e)/D||inf<1. For k!=0 separation gives >=80Q/D>1. Hence d=e for every signed integer input."],["decoder","MN=NM=hI, h=4948372000; max row sum |N|=801320608. Inverse coordinate width <=D*801320608/(Q*h)<12. Fixed mod8 leaves at most two candidates per coordinate. Finite exact search recovers the unique valid predecessor; budget exhaustion is UNRESOLVED."],["records","Full decoder=N(Dy+r)/(Q*h), requiring integral result and canonical r. Full and mod8 records translate by decoding d then recomputing the other witness. Both compositions are identity on valid images."],["history","Keep terminal d_n and ordered s_0..s_(n-1). Repeated predecessor decoding recovers d0 for every finite n; n=0 is identity. Apply L to recover the nine original parameters. Witness=45 bytes/step, plus terminal state and framing."],["ideal","qPd=T(d)+r/D gives F_lambda(1+d)=1+T(d)+r/D+(lambda-q)Pd. Retain the correction. Exact enclosure at n=2S gives floor(qn)=679899542689556, floor(lambda*n)=679899542689557."]],"evidence":{"file":"OMEGA_Explicit_Joining_Theorem_2026-11-07.zip","sha256":"80d058b1807c4bb3b9ab7ba53163df3e18a5c0cb70ab2befb5cbbb2824f378e9","run":"Extract; python3 verify.py --self-test","recorded_replay":"Normal/-O matched: 32 recovery records,18 history steps,11 corruptions rejected per mode."},"gate":"Written proofs and finite certificates are distinct. Recompute before issuing INSTANCE_CHECKED; no Lean proof claimed. A hash is integrity, not mathematical proof. Without the companion evidence, numerical certificates remain unreplayed.","open":"307<=ell*<=352 unresolved; fast general decoding unproved. Flooring need not preserve the geometric family. No equivalence of all source theories or physical unification claimed."}