Operational BHR/BMR Bridge: Heritage/Memory Bifurcation as Two Readouts of One Two-Qubit State, with Embedding in the JANO-1 Circuit — RCD Program, Technical Note JANO-BR-0 (v0.1)
Technical Note JANO-BR-0 of the RCD program (continues “La doble herencia de la coherencia”, DOI 10.5281/zenodo.23520439; the Adjudication Record “Relational Memory under Causal Erasure”, DOI 10.5281/zenodo.23717352; BMR-1, DOI 10.5281/zenodo.23717139; and REP-0, DOI 10.5281/zenodo.23774093). Question: can the heritage/memory bifurcation of BHR-BMR become a physical readout of one and the same circuit? Results (exact, no hardware): in BHR-3 the mixed relational-memory state ρ_mix(r) has rank 2 whenever Ω₃ > 0, so a single qubit c purifies it, |Ψ(r)⟩ = √λ₊|0⟩|ψ₊⟩ + √λ₋|1⟩|ψ₋⟩, with a three-gate {Ry, CNOT} construction and a closed map r ↦ (θ_c, η); tracing out c returns ρ_mix (R⊥), while measuring c in the HJW basis leaves the system exactly in the coherent-heritage state ρ_coh with probability p_coh = Tr D_coh/Tr D_mix and in the residual-memory state |−⟩ (independent of r) with probability p_M; the three readouts are tied by the exact identity V_mix = p_coh·V_coh − p_M (at r = 1: 145/149 = 147/149 − 2/149). Verified against the original JANO-1 code: the five-qubit JANO-1 circuit with α = α_BR(r) = arcsin √λ₋(r) is literally this purification, and its frozen observables reduce to P_rec = λ₊ and R∥ = R⊥ = √(2λ₊λ₋) = √S_lin(ρ_mix) — a spectral readout of the memory — while the HJW readout is the complementary face that separates heritage from memory. Stated limit: the construction reproduces ρ_mix(r) by design and is not a falsifiable prediction of RCD; hardware stays closed. Scripts with SHA-256 manifest. English and Spanish versions; the Spanish text prevails.
Technical Note JANO-BR-0 of the RCD program (continues “La doble herencia de la coherencia”, DOI 10.5281/zenodo.23520439; the Adjudication Record “Relational Memory under Causal Erasure”, DOI 10.5281/zenodo.23717352; BMR-1, DOI 10.5281/zenodo.23717139; and REP-0, DOI 10.5281/zenodo.23774093). Question: can the heritage/memory bifurcation of BHR-BMR become a physical readout of one and the same circuit? Results (exact, no hardware): in BHR-3 the mixed relational-memory state ρ_mix(r) has rank 2 whenever Ω₃ > 0, so a single qubit c purifies it, |Ψ(r)⟩ = √λ₊|0⟩|ψ₊⟩ + √λ₋|1⟩|ψ₋⟩, with a three-gate {Ry, CNOT} construction and a closed map r ↦ (θ_c, η); tracing out c returns ρ_mix (R⊥), while measuring c in the HJW basis leaves the system exactly in the coherent-heritage state ρ_coh with probability p_coh = Tr D_coh/Tr D_mix and in the residual-memory state |−⟩ (independent of r) with probability p_M; the three readouts are tied by the exact identity V_mix = p_coh·V_coh − p_M (at r = 1: 145/149 = 147/149 − 2/149). Verified against the original JANO-1 code: the five-qubit JANO-1 circuit with α = α_BR(r) = arcsin √λ₋(r) is literally this purification, and its frozen observables reduce to P_rec = λ₊ and R∥ = R⊥ = √(2λ₊λ₋) = √S_lin(ρ_mix) — a spectral readout of the memory — while the HJW readout is the complementary face that separates heritage from memory. Stated limit: the construction reproduces ρ_mix(r) by design and is not a falsifiable prediction of RCD; hardware stays closed. Scripts with SHA-256 manifest. English and Spanish versions; the Spanish text prevails.
We present a development of the RCD program (Radial Coherential Dynamics) in the complex scalar sector of quantum sequential growth of causal sets. The central question is not only how a causal history grows, but how much of the information about HOW IT WAS FORMED can survive when one of its elements is erased without privileging any temporal label. The central result is an exact separation between coherent inheritance and memory of the cut. For the erasure branches a_i (i = 0, …, N−1), define the branch matrix 𝒜_N = N^(−1/2) [a_0, …, a_{N−1}], the uniform register mode |u_N⟩ = N^(−1/2)(1, …, 1), and Q_N = I − |u_N⟩⟨u_N|. The memory matrix M_N = D_mix − D_coh = 𝒜_N Q_N 𝒜_N† ⪰ 0 identifies exactly the memory sector orthogonal to coherent inheritance. Its trace, Tr M_N = (1/2N²) Σ_{i,j} ‖a_i − a_j‖², is the pairwise dispersion of the causally distinct erasure alternatives: relational memory is precisely that dispersion. The Relational Inheritance Bifurcation theorem (BHR-3) shows already at N = 3 that the inherited amplitude can coincide exactly while non-zero memory of the mode of formation persists: Δ₃^coh = 0 together with Ω₃ > 0, where D_mix − D_coh = Ω₃ · [[1, −1], [−1, 1]]. At the exact point r = 1, Ω₃ = 1/294; the reading with the register traced out has spectrum {147/149, 2/149} and visibility 145/149, while the coherent reading remains pure. One and the same observable inheritance can therefore coexist with residual relational information about its formation. The second main result is an exact memory-flow law. On the star family 𝒮_N — an antichain of N−1 elements with one element above all of them — covariant coherent erasure is closed for every N. With p = ε/(t₀+ε), p′ = (N−1) p / N, P := p/N = constant, and the complete coupling trajectory is ε_n = n ε_N / (N + (N−n) ε_N). The memory has rank one, M_N = (N−1)|P|² J with J = [[1, −1], [−1, 1]], so that Ξ := M_N/(N−1) = |P|² J is an exact invariant of the flow, M_{N−1} = M_N − Ξ, and 𝓜_N − 𝓜_{N−1} = 2|P|²: each coarse-graining step transports an exactly determined amount of relational memory. The star spine 𝒮₄ also appears as an exact component of the closure variety of the 4→3 flow. The non-degenerate regular branch C₁₂ is born tangent to this spine and inherits, at leading order, its memory-contraction factor 2/3. The deposit also contains the prior results leading to this structure: absence of a non-trivial scalar fixed point under uniform incoherent erasure (𝔗_mix); absence of a non-trivial fixed point under the uniform coherent superposition (𝔗_coh), which equals the classical PF-0 condition complexified; the need to formulate erasure covariantly by classes; the rational renormalization map on the closure variety; and the minimal register compression, rank D_mix = rank M_N + 1. These results motivate the Principle of Relational Memory of the Cut, dynamical version (PMRC-D): every fundamental physical differentiation that produces causally distinguishable histories must also determine, through a covariant rule, which information of that differentiation remains recorded and which information may interfere coherently. The emerging working thesis is: "The inheritance of a history does not by itself determine the memory of how it was formed." The program remains open toward closed sectors with several independent memory modes and a covariant dynamics capable of transporting those modes under coarse-graining. Exact results are N ≤ 4 except those with closed proofs for all N (the star spine and its flow); the 5→4 step outside 𝒮_N is numerical; the physical universality of PMRC-D remains open. No claim is made about the emergence of time, Mermin's 25 % or the necessity of H > 1 as a law. Deposit contents: RCD Adjudication Record v0.2 (signed Spanish original and English version; the Spanish text prevails), diligence notes 𝔗_mix-0, 𝔗_coh-0/𝔗_flow-0 and BMR-0 (Relational Memory), reproducible code with SHA-256 manifests, and README. Panel Cerezo credits: ChatGPT (formulation of the diligences, BHR-3, PMRC-D, 𝒜QA† form, invariant Ξ, series of C₁₂) and Claude (exact computation, lemmas, curve C and map R, star spine 𝒮_N, cross-verification and drafting), under the direction and adjudication of the principal.
Technical note BMR-1 of the RCD program (Radial Coherential Dynamics), continuing "Relational Memory under Causal Erasure" (10.5281/zenodo.23630978). The previous record found an exact memory-flow law on the star family, which carries a single memory mode. This note asks for the minimal causal architecture able to carry two independent memories and survive its own coarse-graining. Three exact results. (1) Branch–rank lemma: if the N erasure positions produce r distinct branches, rank M_N ≤ r − 1, with equality under affine independence; two modes require at least three causally inequivalent erasures. (2) The P2_n sector (a chain with one pendant element at height ≤ 2) is closed under erasure for all n ≥ 4, has unbounded height and two exact memory modes: exact transfer map R_n, two flow invariants (y, J), closed-form solution, Hermitian memory M_n with rank 2 iff uv ≠ 0, and linear transport of (|u|², Re ūv, Im ūv, |v|²) with spectrum {a², ac, ac, c²}, a = (n−2)/n, c = (n−3)/n; the relative phase is carried as an independent memory component. (3) Step-closure theorem: the support of every Rideout–Sorkin (RS) amplitude is fixed by the local pair (|↓x|, #max ↓x) and always contains the antichain, so no closed family omitting it, P2_n in particular, admits an RS realization; the star is the only RS-supportable family found. An exact classification of RS closure families 5→4 on restricted supports finds only the star line, plus a certified isolated conjugate pair that closes one step and not the next. The obstruction is against the locality of the RS amplitude support, not against scalar amplitudes as such. Includes the erasure automaton of unlabeled posets up to N = 7 (1, 2, 5, 16, 63, 318, 2045 classes) and all scripts with a SHA-256 manifest. Exact results are stated as such; the enumeration of isolated points on large supports is not certified. English and Spanish versions; the Spanish text prevails. Developed at the Panel Cerezo: ChatGPT (formulation of BMR-1, branch–rank lemma, Hermitian correction), Claude (automaton, P2_n, step-closure theorem, classification, verification and drafting), under the direction and adjudication of the principal.
We propose to read spatial entanglement and interference between histories as two inheritances of a single coherence prior to the cut: a transverse residue across the partition A|B and a longitudinal residue along the causal order. The hypothesis proper to RCD is the Principle of Double Inheritance of Coherence (PDHC): one rule of differentiation determines at once what stays separated and what stays ordered. To make it an instrument we design JANO-1, a five-qubit circuit for superconducting processors in which a single cut, placed in superposition and then erased, is read twice: as entanglement between two qubits that never interacted and as an interference fringe between the histories “cut” and “no cut”. Exact simulation fixes the three predicted curves with one knob (the amplitude of the cut) and shows that a second knob (the coherence of the cut) makes the pair of residues fill a region, not a curve; the strong hypothesis is therefore stated as a precise question for the amplitude rule of PF2, and the JANO-2 phase returns to design. Nothing has run on hardware; the preregistration is conditional on a second key. Files: English version, Spanish version (the Spanish text is the reference text), and the Python code that reproduces every figure. Developed at the Panel Cerezo with cross-checking among AI tools (ChatGPT, Grok, Claude); the authorship of each contribution is stated in §10.
Can systems matched on a declared present nevertheless show different future responses because they retain different relational organisation? We test this in a six-qubit hybrid quantum-classical architecture on IBM Heron hardware. Written and sham preparations use exactly the same five coupling values and the same number of native two-qubit operations; only the edge assignment differs. Under a frozen preregistered decision rule with randomised block orders, matched fast preparations displayed different later directional responses when the prepared slow classical relational configuration differed. Because the full-sample cosine alignment and norm-based retention estimates sat outside their original percentile intervals, a post hoc audit was performed from the sealed raw-count archive. That audit is reported alongside, and does not replace, the frozen rule. Claim boundary. The result is real under its frozen analysis. It is not autonomous cross-job quantum memory, and it is not evidence of a hidden quantum degree of freedom persisting between jobs. The classical coupling configuration is part of the preparation and is named as such throughout. No claim is made about quantum coherence surviving between executions. This deposit contains the manuscript in PDF and DOCX form together with the complete evidence archive: protocol freezes, compiled QPY circuits, inspection tables, randomised block orders, backend snapshots, raw-count files for both runs, IBM job ledgers, point estimates, blockwise endpoints, the frozen decision JSON, an artefact inventory and a checksum ledger.
Nessa dissertação, abordamos o problema de dois qubits interagindo com campos externos e entre si controladamente, de acordo com um Hamiltoniano considerado realista para implementação da porta lógica quântica XOR. Introduzimos acoplamentos entre as observáveis do sistema de dois qubits e um banho de osciladores harmônicos a fim de tratarmos o problema da dissipação e da decoerência. Primeiramente nós consideramos o limite no qual a decoerência é mais rápida que qualquer processo gerado pelo Hamiltoniano do sistema. Prosseguimos então, através do método numérico conhecido como Integrador Unitário, com o estudo da matriz densidade do sistema durante a operação da porta lógica quântica sem incluir, inicialmente, o acoplamento com o banho de osciladores harmônicos. Finalmente, implementamos o método numérico conhecido como Propagador quase adiabático para estudar a decoerência e a dissipação durante a operação da porta lógica quântica XOR, a fim de analisarmos os aspectos perturbativos do sistema quântico de dois qubits.
[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] Abstract. We prove an exact algebraic theorem for the Levin-Wen string-net transfer matrix of the SU(2)_2 (Ising) modular tensor category on a cylinder of circumference L_x. The transfer matrix T commutes with the fermion (ψ) Wilson loop operator W_ψ = ⊗_x diag(1,−1,1), yielding two topological sectors T_+ and T_−. We prove: Trace Splitting Theorem: Tr(T_+) − Tr(T_−) = Tr(F^{L_x}) where F is a 3×3 matrix with exact algebraic eigenvalues {D, 2^{5/4}(2^{1/4}−1), −2(2^{1/4}−1)}, and D = 2 is the total quantum dimension of SU(2)_2. Therefore Tr(T_+) − Tr(T_−) → D^{L_x} as L_x → ∞. Proof method: The ψ-signed trace decomposes as Tr(W_ψ T) = Tr(F^{L_x}) where F = W[1,1,:,:] − W[σ,σ,:,:] + W[ψ,ψ,:,:] is the "ψ-signed diagonal channel" of the MPO column tensor. We compute F exactly: W[1,1] = W[ψ,ψ] = diag(1, √2, 1) — the vacuum/fermion channel W[σ,σ] = [[2^{1/4}, 0, 2^{1/4}], [0, 2^{5/4}, 0], [2^{1/4}, 0, 2^{1/4}]] — the σ channel F eigenvalues: 2 (= D, exact), 2^{5/4}(2^{1/4}−1) ≈ 0.450, −2(2^{1/4}−1) ≈ −0.378 All entries of the column tensor W[a,b,v',v] are powers of 2: {1, 2^{1/4}, 2^{1/2}, 2^{3/4}, 2^{5/4}}, reflecting the algebraic structure of SU(2)_2 fusion rules. Ground state splitting (numerical): We compute the exact transfer matrix eigenvalues for L_x = 2 through 9 (sector dimensions up to 9842). The signed splitting s(L_x) = λ_+(top) − λ_−(top) is decomposed via 4-term Prony analysis into: s(L_x) = A_1 ρ_1^{L_x} + Re[A_2 ρ_2^{L_x}] + A_4 ρ_4^{L_x} with four exponential rates (machine-precision fit to all 8 data points): ρ_1 = −0.5916 (real negative): dominant for L_x ≥ 8, rate α_1 = ln(C/|ρ_1|) = 2.092 ≈ ln(8) ρ_{2,3} = −0.368 ± 0.108i (complex pair): dominant for L_x ≤ 7, rate α_2 = ln(C/|ρ_2|) = 2.527 ≈ ln(12) ρ_4 = +0.130 (real positive): fastest-decaying term, α_4 = 3.607 where C = 4.79290 is the per-site growth rate. The complex pair with α_2 ≈ ln(D_total) = ln(12) dominates at moderate system sizes (explaining earlier numerical coincidences with ln(12)), while the real term with α_1 ≈ ln(D^3) = ln(8) = 3ln(2) dominates asymptotically. Key structural results: The Z_2 topological symmetry is the ψ Wilson loop: W_ψ = ⊗_x diag(1,−1,1) = (−1)^{n_σ} Both sectors share the same per-site growth rate C to 10+ significant digits The winning sector (+ or −) follows a complex pattern due to multiple interfering tunneling channels The ground state is NOT in the vacuum virtual channel — it is dominated by σ-string fluctuations The mean-field optimal state has ψ_opt ≈ (0.504, 0.702, 0.504), close to but not exactly √d_a-weighted Physical interpretation: The trace splitting D^{L_x} represents the topological amplification of sector differences — the transfer matrix enhances the trivial dimension difference (dim_+ − dim_− = 1) to a topologically protected D^{L_x} signal. The ground state splitting involves virtual σ-anyon tunneling across the cylinder, with multiple interfering channels (real and complex exponentials) reflecting the non-commutative structure of the MPO virtual space. The σ anyon tunneling amplitude is complex because the 3×3 virtual-space matrices at different sites do not commute, leading to quantum interference in the tunneling paths. Data table (L_x = 2..9): L_x λ_+ λ_− Winner δ_rel 2 22.9758 22.9680 + 3.38×10^{−4} 3 110.102 110.103 − 1.93×10^{−6} 4 527.711 527.710 + 2.05×10^{−6} 5 2529.26 2529.26 − 1.73×10^{−7} 6 12122.5 12122.5 + 1.42×10^{−8} 7 58102.0 58102.0 − 7.00×10^{−10} 8 278477 278477 − 1.58×10^{−11} 9 1334711 1334711 + 1.13×10^{−11} Relation to previous work: This paper provides the analytical foundation for the numerical hierarchy D^{L_x} observed in our earlier work (DOI: 10.5281/zenodo.19673053). The Trace Splitting Theorem proves that D appears exactly in the sector-resolved partition function, while the ground state splitting involves a richer multi-exponential structure with rates connected to D^3 and D_total = D_{SU(3)_3} × D_{SU(2)_2} = 12. Connection to AME(4,6) framework: The SU(2)_2 Ising MTC is one factor of the SU(3)_3 ⊗ SU(2)_2 product theory that emerges from the AME(4,6) quantum error-correcting code. The trace theorem generalizes: for the product theory, Tr(T_+) − Tr(T_−) = D_total^{L_x} = 12^{L_x}.