{ "schema": "OMEGA_JOINING_PROOF_RELATIONS/1", "package_version": "1.1.0", "date": "2026-11-07", "claims": [ { "id": "J01", "title": "Marked geometry to affine scalar dynamics", "parents": [], "from": "selected positive scale", "to": "affine scalar state", "map": "chi(a)=1+a/a_ref; chi(lambda*a)=Omega_lambda(chi(a))", "inverse_or_distinction": "chi^-1(t)=a_ref*(t-1), t>1", "domain_and_hypotheses": "a>0; a_ref>0; lambda=phi/(phi+pi); fixed selected readouts", "written_proof": "PROOF.md, section 3", "evidence_kind": "direct-algebra-proof", "local_evidence": [ "PROOF.md" ], "previous_tree_record_ids": [ "C01", "C02" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." }, { "id": "J02", "title": "Golden vertex coordinates to signal modes", "parents": [], "from": "600-cell vertex coordinates", "to": "four real network modes", "map": "2C=C0+phi*C1; A*C=6phi*C", "inverse_or_distinction": "C^T*C=30I4 gives the image inverse coefficient C^T/30", "domain_and_hypotheses": "Pinned coordinate order; phi^2=phi+1; C has the unit vertex rows", "written_proof": "PROOF.md, section 2", "evidence_kind": "finite-matrix-certificate-with-written-consequence", "local_evidence": [ "data/matrices.json#/C0", "data/matrices.json#/C1", "data/matrices.json#/Gram_blocks" ], "previous_tree_record_ids": [ "C16" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." }, { "id": "J03", "title": "Nine integer parameters to integer signals", "parents": [ "J02" ], "from": "Z x Z^4 x Z^4", "to": "Z^120", "map": "E_Z(u)=B_enc*u; M*B_enc=B_enc*K", "inverse_or_distinction": "L=L_numerator/120; L*B_enc=I9, restricted to image", "domain_and_hypotheses": "Integer parameter order t,x0..x3,z0..z3; not all integer signals are in the image", "written_proof": "PROOF.md, section 2", "evidence_kind": "finite-matrix-certificate-with-written-consequence", "local_evidence": [ "data/matrices.json#/B", "data/matrices.json#/L_numerator", "data/matrices.json#/K", "data/left_inverse_times_basis.json" ], "previous_tree_record_ids": [ "C17" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." }, { "id": "J04", "title": "Scalar and geometric real-network subdynamics", "parents": [ "J02" ], "from": "real geometric signal family", "to": "real network F_gamma", "map": "F_gamma(E(t,w))=E(Omega_gamma(t),gamma*(13+6phi)*w/25)", "inverse_or_distinction": "Image inverse t=mean(v), w=C^T*v/30; mean alone is not invertible", "domain_and_hypotheses": "Real t,w; 0= 10", "inverse_or_distinction": "Not an inverse; a separation lower bound", "domain_and_hypotheses": "k in Z^120, k nonzero; Sylvester criterion and |k_i|<=k_i^2", "written_proof": "PROOF.md, section 4", "evidence_kind": "written-proof-from-finite-matrix-certificates", "local_evidence": [ "data/matrices.json#/positive_leading_minors_M_minus_9I" ], "previous_tree_record_ids": [ "C14" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." }, { "id": "J06", "title": "Reduced output plus mod8 witness recovery", "parents": [ "J05" ], "from": "integer input d", "to": "valid record (T(d),d mod8)", "map": "d -> (floor(Q*M*d/D),d mod8)", "inverse_or_distinction": "Unique predecessor on valid images; finite search in J07", "domain_and_hypotheses": "All signed integer d; canonical residues; 80Q>D", "written_proof": "PROOF.md, section 4", "evidence_kind": "written-proof-from-finite-matrix-certificates", "local_evidence": [ "data/matrices.json#/constants", "data/recovery_examples.json#/records" ], "previous_tree_record_ids": [ "C15" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." }, { "id": "J07", "title": "Reversible full-remainder and mod8 records", "parents": [ "J06" ], "from": "valid full record (y,r)", "to": "valid mod8 record (y,s)", "map": "J(y,r)=(y,D_full(y,r) mod8); H(y,s)=(y,QM*D8(y,s)-Dy)", "inverse_or_distinction": "H o J=Id and J o H=Id on their respective valid images", "domain_and_hypotheses": "Keep y; full remainder in [0,D); integer inverse; decoder budget exhaustion is unresolved", "written_proof": "PROOF.md, section 5", "evidence_kind": "written-proof-from-finite-matrix-certificates", "local_evidence": [ "data/matrices.json#/N", "data/matrices.json#/max_abs_row_sum_N", "data/recovery_examples.json#/records" ], "previous_tree_record_ids": [ "C20" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." }, { "id": "J08", "title": "Scalar integer remainder ledger inside network", "parents": [], "from": "scalar integer dynamics", "to": "constant-vector network dynamics", "map": "T(n*1)=floor(Q*n/S)*1; r(n*1)=25rho(n)*1", "inverse_or_distinction": "Full scalar ledger reconstructs n; averaging alone is not a full-state inverse", "domain_and_hypotheses": "n in Z; M has row/column sums 25; D=25S", "written_proof": "PROOF.md, section 6", "evidence_kind": "written-proof-from-finite-matrix-certificates", "local_evidence": [ "data/matrices.json#/M", "data/recovery_examples.json#/mean_counterexample" ], "previous_tree_record_ids": [ "C13", "C19" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." }, { "id": "J09", "title": "Joined finite-history recovery", "parents": [ "J03", "J06", "J07" ], "from": "nine geometric integers", "to": "terminal state plus ordered witnesses", "map": "E_Z^-1 o DecodeHistory8 o EncodeHistory8 o E_Z=Id", "inverse_or_distinction": "Backward valid-record decoding, then L", "domain_and_hypotheses": "Every finite horizon n>=0; shared coordinate order and history framing", "written_proof": "PROOF.md, section 6", "evidence_kind": "written-induction-from-checked-parent-certificates", "local_evidence": [ "data/recovery_examples.json#/histories" ], "previous_tree_record_ids": [ "C21" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." }, { "id": "J10", "title": "Ideal and rational dynamics with exact defect", "parents": [ "J07", "J08" ], "from": "ideal lambda evolution", "to": "rational quantized trajectory and remainders", "map": "F_lambda(1+d)=1+T(d)+r(d)/D+(lambda-q)*P*d", "inverse_or_distinction": "Valid record recovers d and r, hence every correction term", "domain_and_hypotheses": "Fixed ideal lambda and rational q; finite histories use the stated defect recurrence", "written_proof": "PROOF.md, section 7", "evidence_kind": "exact-polynomial-check-and-written-induction", "local_evidence": [ "data/lambda_enclosure.json", "PROOF.md" ], "previous_tree_record_ids": [ "C22" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." }, { "id": "J11", "title": "Ideal and rational floors are not globally equal", "parents": [], "from": "floor(q*n)", "to": "floor(lambda*n)", "map": "n=2*10^15 gives 679899542689556 versus 679899542689557", "inverse_or_distinction": "Not an inverse; an explicit unequal-output counterexample", "domain_and_hypotheses": "Rational sqrt enclosure; Machin identity; alternating-series theorem", "written_proof": "PROOF.md, section 7", "evidence_kind": "exact-rational-counterexample", "local_evidence": [ "data/lambda_enclosure.json" ], "previous_tree_record_ids": [ "C23" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." }, { "id": "J12", "title": "Exact consistency cell versus enclosure", "parents": [ "J06", "J07" ], "from": "quantized output and residue", "to": "at-most-one exact lattice candidate", "map": "C_net^-1(y+[0,1)^120) intersect (s+8Z^120)", "inverse_or_distinction": "Singleton for a valid record; coordinate domains merely enclose the cell", "domain_and_hypotheses": "C_net=(Q/D)M; K_pack is a separate capacity container", "written_proof": "PROOF.md, section 5", "evidence_kind": "written-set-equivalence-and-parent-injectivity-proof", "local_evidence": [ "data/recovery_examples.json#/records", "data/matrices.json#/N" ], "previous_tree_record_ids": [ "C24" ], "universal_scope": "Only the specified maps, domain and statement; no equivalence of entire source theories is inferred." } ], "interpretation": "This is an explicit dependency graph. Parent labels are not accepted as numerical proof: verify.py recomputes the referenced finite certificates, while PROOF.md supplies the universal deductions.", "no_external_access_required": true }