# Frame-Aligned Records: Supplementary Information ## Calculations, study definitions and reproducibility **Supplementary Information v1.2 | 26 September 2026** **James McGaughran** **Scope.** This supplement supports the methods paper. Scientific calculations and model definitions come first; dated lineage is in the final provenance section. A reported historical run, a fresh same-code replay and a separate numerical implementation have different evidential meanings. **Version crosswalk.** Portfolio v1.8 identifies the distribution; *Frame-Aligned Records* v0.5, this supplement v1.2, the results log v1.2, protocol reader edition v1.3 and *Absolute Infinite Union* v1.4 identify its components. CAL-1.6 and similar identifiers name fixed calculation suites, not alternative current manuscripts. A historical label means a preserved prior state, even when its date is only one day earlier. The authoritative executed specification remains protocol v1.2 and its original pre-execution receipt; later reader edits do not modify that record. Finalization retains these version labels and records changes through source hashes and a dated change record. ## S1. Numerical provenance for the methods paper The source run labels S01 to S06 refer to the preserved six-script calibration suite under `historical/supporting_lineage/v1_6_code/legacy_v1_5/legacy_claude_v0_2_0/`. The original display values and sampling histories are retained, not all rerun during this editorial release. The newer calibration and pilot paths are listed explicitly below. | Statement | Value(s) | Script and output key | |---|---|---| | Lemma 1 counterexample | $L=401$ | S02 `L1_counterexample_published_expression_without_identity` | | Lemma 3 invariance; generic change | $\le6\times10^{-17}$; $0.144$ | S02 `L3_*` | | Proposition 1(c) | $1.0$ | S02 `P1_commuting_unitary_changes_pure_ground_state_by` | | Proposition 1, Remark 2 | $1.0, 0.399, 0, 0.399, 1.0$ bits | S02 `P1_stationary_average_conceals_conditional_dynamics` | | Proposition 2 | $0.5270653410$; $0.3400520132$; $-0.7615941560$; $0$; $1\to0$ bits | S01 `P2_*` | | Lemma 4 example | $0.6428571429$ (formula and enumeration) | S01 `hypergeometric_*` | | Proposition 3 checks, Table 1 | as tabulated | S03 `A_haar_fragments`, `B_dynamics` | | Table 2 | as tabulated | S04 `duplicate_inflation` | | Hoeffding half-widths | $0.200$, $0.100$, $0.050$ | S01 `hoeffding_half_width` | | Supplement Tables S4 to S6 | as tabulated | S05 `e1`, `e2`, `e3`, `e4` in the exploratory output | | Supplement Section S2.5 | as stated | S05 replication output | | Legacy S06 span checks | $0$; $7\times10^{-16}$; $0.4166$ versus $0.4170$ | S06 | | Mixed twirl, historical v1.5 | 2,000 draws; mean $0.0557464152$; SE $0.0007264476$ | Archived W03 v1.1, Section 6; not the CAL-1.6 sample | | Mixed twirl, CAL-1.6 | 1,200 draws; mean $0.05550839355$; SE $0.00093479233$; exact $0.056$ | `historical/supporting_lineage/v1_6_data/results.json`, `mixed_summary`; full draws in `mixed_draws.csv` | | Proposition 5a endpoint constant | $\sqrt{1.0025}=1.0012492197$ at $n=2,4,6$ | `historical/supporting_lineage/v1_6_data/localized_constants.csv` | | Proposition 6 spectator | error difference approximately $8.44\times10^{-15}$ against a loose bound $78.53981634$ | `historical/supporting_lineage/v1_6_data/results.json`, `spectator` | | Unequal certificate example | $0.0995$ and $0.104$, same true error $0.099$ | Section 7.3.1, direct arithmetic; not two physical observations | | Proposition 7, Figure 6 | capped bound $1$ through $n=18$; log10 bound approximately $-56.85$ at $n=19$ | `data/prop7_thresholds.csv`; direct envelope, not state simulation | | Source-off feasibility pilot | 192 independent units in two size strata; separate outcome definitions | `data/pilot_run/results.json`, `per_unit.json`; results log, not an LRA test | | Post hoc sign analysis | primary raw $p=5.4592\times10^{-5}$; four-row Holm $p=1.6377\times10^{-4}$ | `data/review_additions/sign_tests.csv`; `code/analyze_review_additions.py` | | Range-only precision budget | 32,791 units for half-width 0.015, not a power claim | `data/review_additions/review_analysis.json`, `precision` | | Initial eligibility and transient access | 5 and 10 eligible site/unit pairs in native arms briefly exceed 0.8 | `data/review_additions/initial_floor_diagnostics.csv` | | Source encoding in clean transport control | phase code $d=|f|$; vacuum/excitation code $d=|f|^2$ | `data/review_additions/transport_encodings.csv`; separate full-space checks | **Additional run crosswalk** | Target | Source and data | Boundary | | --- | --- | --- | | Factorial four/six/eight-qubit means | `data/reviewer_replay/factorial_results.json`; `factorial_n*.npz` | Same-code reproduction of an exploratory review extension | | Larger source-off sample | `data/reviewer_replay/reimpl_id1_results.json` | Fresh-stream reimplementation, not the original sample | | Third-frame means | `data/reviewer_replay/factorial_neutral.json` | Post hoc comparator, no guaranteed neutrality | | Paired range and finite-family intervals | `data/factorial_analysis/analysis.json`; `code/analyze_factorial.py` | Derived secondary sampling bounds | | Alternative numerical method | `verification/factorial_independent_checks.json` | Selected trajectories, not a second full population study | ## S2. Historical continuously driven simulation The material below retains the complete exploratory model and outcomes for reproducibility. It is not another experiment or a retrospective confirmation of the newer source-off target. References to original Sections 2 to 5 concern the methods framework; references to Section 6.x are mapped to S2.x below. Its reported causal interpretations are descriptive and family-specific, not general theorems. ### S2.0 Status **Status: EXPLORATORY.** This section reports computed results for one small model class. The configuration was changed after the first results were inspected (Section S2.2), so they are not a confirmatory test of the prospectively stated population conjectures. They can nevertheless supply counterexamples to a universal claim and guide later design; exploratory does not mean scientifically worthless. They are included because they show concretely how the design choices of Sections 2 and 5 decide outcomes. The frozen configuration was then re-run with fresh seeds (Section S2.5). The notation below uses $K_{\mathrm C}$ for the source-coupling operator, reserving C for an alternate frame label. Preserved scripts and original outputs still use their original variable C; this is a notation change only. ### S2.1 Model - **System and source.** $n=3$ system qubits and a binary source with equal priors; $H_x=H_S+(-1)^xK_{\mathrm C}$. - **Internal Hamiltonian.** $H_S$ is a random nearest-neighbour 2-local chain: Gaussian coefficients on all two-body Pauli products on adjacent pairs, plus one-body terms at half weight. It is normalised to Frobenius norm $\lambda$. - **Coupling and time window.** The source coupling is the star operator $K_{\mathrm C}=\sum_{j=1}^3X_j$, of Frobenius norm $\sqrt{24}\approx4.90$. Times are $t\in\{0.6,0.7,0.8,0.9,1.0\}$. - **Candidates.** There are $N=20$ candidates: the identity (the frame in which $H_S$ and $K_{\mathrm C}$ are local), a preparation frame $W$, and 18 Haar-random unitaries. - **Locality.** $L$ is the internal score $L^{\mathrm{int}}$ on the span of Pauli strings of weight at most 2, identity included. - **Records (simplified).** The record score is the time-averaged number of single qubits whose Helstrom error is at most $0.1$. This is a simplification of the disjoint, persistent redundancy of Section 2.5. - **Near-optimal sets.** $\mathcal N_L$ requires $L\le\min L+0.02$ and $L\le0.6$; $\mathcal N_R$ requires $R=\max R$ and $R\ge0.5$. - **Arms.** In the *shared-frame* arm the initial state is $\left| 000\right\rangle$. In the *decoupled-preparation* arm it is $W\left| 000\right\rangle$ with $W$ Haar-random, so by Section 4.3 it is Haar distributed. The coupling stays in the identity frame in both arms. ### S2.2 Development history The first configuration used a single-qubit coupling $X\otimes I\otimes I$, $\lambda=4$ and nine times in $[0.3,1.5]$. Records qualified in 1% of shared-frame models and in none of the decoupled models ($A=0.010$ and $0.000$). The coupling was then changed to the star operator, the time window was narrowed, and a grid over $\lambda$ was added. All results below use the changed configuration, which was frozen on 25 September 2026. ### S2.3 Shared versus decoupled preparation **Table S4. Association results by arm (200 models per cell; Hoeffding 95% intervals clipped at $1-1/N=0.95$).** | $\lambda$ | Arm | Records qualify | $\bar A$ | $\bar q$ | $\hat d$ | 95% interval | |---|---|---|---|---|---|---| | 0.3 | shared | 1.000 | 1.000 | 0.050 | 0.950 | [0.758, 0.950] | | 0.3 | decoupled | 0.095 | 0.050 | 0.005 | 0.045 | [-0.147, 0.237] | | 1.0 | shared | 1.000 | 1.000 | 0.050 | 0.950 | [0.758, 0.950] | | 1.0 | decoupled | 0.060 | 0.025 | 0.003 | 0.022 | [-0.170, 0.214] | | 4.0 | shared | 0.320 | 0.255 | 0.016 | 0.239 | [0.047, 0.431] | | 4.0 | decoupled | 0.030 | 0.005 | 0.002 | 0.004 | [-0.189, 0.196] | In the shared arm, the most local candidate hosts the records in every model at $\lambda\le1$, and the descriptive estimate is $0.95$ in these sampled cases. This agreement is facilitated by the shared inputs, not proved for every model by a coupling bound. At $\lambda=4$ the internal dynamics is comparable in norm to the coupling, and records qualify in only 32% of models. In the decoupled arm, records qualify rarely. In the reported cases they preferentially sit in the identity frame, where the coupling remains local. This association is consistent with the supplied interaction structure, not an isolated causal estimate. ### S2.4 Moving the preparation frame and the coupling frame **Table S5. Preparation frame rotated by $W=e^{-i\theta G}$ away from the dynamics frame** ($G$ a random Hermitian of unit spectral norm; $\lambda=0.3$; 100 models per row). The columns give the fraction of models in which records qualify anywhere, and in which the record-optimal set contains the dynamics frame, the preparation frame, or neither. | $\theta$ | Records anywhere | In dynamics frame | In preparation frame | Elsewhere | |---|---|---|---|---| | 0.0 | 1.00 | 1.00 | 1.00 | 0.00 | | 0.1 | 1.00 | 1.00 | 1.00 | 0.00 | | 0.2 | 1.00 | 1.00 | 0.97 | 0.00 | | 0.4 | 1.00 | 1.00 | 0.44 | 0.00 | | 0.8 | 0.86 | 0.82 | 0.06 | 0.00 | | 1.6 | 0.13 | 0.10 | 0.00 | 0.03 | At $\theta=0$ the "preparation frame" candidate equals the identity, a duplicated class of exactly the kind Lemma 3 requires to be merged. As the ready resource is rotated away, records do not follow it. They stay in the frame of the coupling and then disappear, in this specific exploratory family. Finite-angle exponentials of normalised random Hermitians are not Haar draws, and no Haar limit for this parameterisation has been established. **Table S6. Coupling and ready resource moved together** ($K_{\mathrm C}\mapsto U_cK_{\mathrm C}U_c^\dagger$, initial state $U_c\left| 000\right\rangle$, $U_c=e^{-i\varphi G}$; 60 models per row). The internal-locality minimum was at the identity in every model. | $\lambda$ | $\varphi$ | Records anywhere | In internal-locality frame | In coupling frame | Elsewhere | |---|---|---|---|---|---| | 0.3 | 0.4 | 1.000 | 0.583 | 1.000 | 0.000 | | 0.3 | 0.8 | 1.000 | 0.017 | 1.000 | 0.000 | | 1.0 | 0.4 | 1.000 | 0.333 | 0.917 | 0.000 | | 1.0 | 0.8 | 1.000 | 0.033 | 1.000 | 0.000 | | 4.0 | 0.4 | 0.317 | 0.133 | 0.183 | 0.067 | | 4.0 | 0.8 | 0.200 | 0.033 | 0.100 | 0.067 | The total record-generating Hamiltonian $H_x$ was more local, by $L$ averaged over $x$, in the coupling frame than in the internal-locality frame in 100%, 100% and 95% of models at $\lambda=0.3$, $1.0$ and $4.0$ ($\varphi=0.8$, 60 models each). ### S2.5 Fresh-seed repeats of the frozen exploratory configuration With the configuration frozen, the main contrasts were re-run with seeds not used during development. - **Seeds 20260926 and 20260927, $\lambda=0.3$, 200 models per arm.** The shared arm again gave $\bar A=1.000$, $\bar q=0.050$ and $\hat d=0.950$. The decoupled arm gave $\hat d=0.072$ $[-0.120,0.264]$ and $0.040$ $[-0.153,0.232]$. - **Three-frame design, $\lambda=0.3$, $\varphi=0.8$, 100 models per seed.** Records qualified in every model. They lay in the coupling frame in every model and in the internal-locality frame in 0% and 1% of models. - **Total Hamiltonian.** $H_x$ was more local in the coupling frame in 100% of models for both seeds. - **Larger model-sample decoupled run, still three qubits.** One further run was declared before execution (seed 20260928, 2,952 models, chosen for a Hoeffding half-width of $0.05$). It gave records in 10.7% of models, $\bar A=0.0515$, $\bar q=0.0055$ and $\hat d=0.046$ $[-0.004,0.096]$. ### S2.6 What the calculation shows, and what it does not 1. **Shared frames produce enrichment by construction.** The estimate $\hat d=0.95$ reflects the design, not a regularity of the model class. 2. **Records track the frame in which the ready resource and the source coupling are aligned.** A ready resource rotated away from the coupling produces few records, and they do not follow the resource (Table S5). 3. **The verdict depends on which Hamiltonian defines locality.** When the coupling and the resource move together, records move with them, away from the internal-locality frame, and into the frame in which the total Hamiltonian is most local (Table S6 and Section S2.5). Its descriptive pattern differs for internal versus branchwise total locality. Neither reading is a calibrated verdict on LRA-int or LRA-tot, because the exploratory time-averaged record outcome and label reference are not the prospective persistent-record test. 4. **The "decoupled" arm is only partly decoupled.** Its coupling remains local in the dynamics frame. Its small descriptive enrichment ($\hat d=0.046$) is compatible with influence from the shared coupling frame, but the toy does not isolate that cause; and its interval, $U=0.096$, lies just below a margin of $\Delta=0.1$. This is not evidence about any conjecture. The calculation has three qubits, 20 candidates, Haar candidates not demonstrated to form an exhaustive set of equivalence classes, a simplified record score and thresholds chosen during development. The coupling has a larger declared norm than the internal Hamiltonian at $\lambda\le1$; that norm comparison alone is not a record-persistence guarantee. Its role is to show that the questions in Section 7 need to be posed with frames declared separately. ## S3. Post hoc sign tests and window diagnostics The four-row directional analysis uses the immutable per-unit contrast data and verifies them against the full trajectories. Its null is Pr(z>0 given z is nonzero)=1/2 for independent signs; it is not a zero-mean null. No ties were observed. Exact binomial p-values and a four-row Holm adjustment are reported in the results log. The Holm family was chosen during review, not in the pre-execution plan. Dependence between the paired arms is not treated as extra sample size. The script also reports initial-eligibility and transient-access counts by frame, and compares both source encodings in an ideal exchange-chain transport control. The phase-superposition pilot code has singleton distinguishability |f|, while the vacuum-versus-excitation control has |f| squared. These are not interchangeable time-window diagnostics. The original study's thresholds, windows and data remain unchanged. ```text OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python code/analyze_review_additions.py --root . --out /path/to/fresh_review_analysis ``` The code refuses a nonempty output folder. It writes `review_analysis.json`, `sign_tests.csv`, `transport_window_checks.csv`, `transport_encodings.csv` and `initial_floor_diagnostics.csv`. The last file counts site/unit pairs, not independent observations. Mathematical derivations, same-code replay, separate-method checks and external replication remain distinct. ### S3.1 Initial eligibility and transient strong access | Qubits | Arm | Frame | Eligible / total site-unit pairs | Ever reaches d >= 0.8 | | --- | --- | --- | --- | --- | | 4 | native | L | 288 / 384 | 5 | | 4 | native | C | 192 / 384 | 0 | | 4 | rotated | L | 96 / 384 | 0 | | 4 | rotated | C | 288 / 384 | 0 | | 6 | native | L | 480 / 576 | 10 | | 6 | native | C | 384 / 576 | 0 | | 6 | rotated | L | 288 / 576 | 0 | | 6 | rotated | C | 480 / 576 | 0 | A site-unit pair is not an independent model. These are post hoc diagnostic counts from existing trajectories. A transient threshold crossing does not establish full-window persistence or redundant copying. The strict persistent-record result remains zero under the original definition. ## S4. Factorial extension and paired uncertainty ### S4.1 Design and replay The review extension crosses injection L/C with Hamiltonian orientation L/C, with $H_C=U_C H_L U_C^\dagger$. Every unit supplies all four outcomes. The time grid is 0.05 through 2.5 plus the switch-off readout; the common injection code and two-layer circuit follow the original source-off conventions. The main paper and results log define z and $e_a$. We retain supplied inputs in `feedback/current/` without editing their scripts or their predictions. `code/replay_reviewer.py` runs the supplied functions with their original seeds and sample sizes and retains per-unit observables, rather than only aggregate summaries. It replays the 41,000 factorial units, the 60,000-unit reimplementation and its smaller checks, and the 6,000-unit third-frame comparison. Each arm in a unit remains paired. The statement that two routines were written independently refers to source implementation, not statistical independence of their authors' reasoning or external peer review. ### S4.2 Range and finite-family bound At fixed injection, the initial gap cancels in the difference between dynamics arms: $$e_a=\tfrac12\bigl(g_L^a-g_C^a\bigr)\in[-1,1].$$ Let $s_a^2$ be the unbiased sample variance of e in N iid units. Theorem 4 of Maurer and Pontil (2009) applies to variables in [0,1]. Rescale by $Y=(e+1)/2$, use both tails, and allocate total error $\alpha$ over K specified targets. A simultaneous interval for each mean is its sample mean plus/minus $$r_N=\sqrt{\frac{2s_a^2\log(4K/\alpha)}N}+\frac{14\log(4K/\alpha)}{3(N-1)}.$$ **Derivation.** The theorem's one-tail bound is $\sqrt{2V_N\log(2/\delta)/N}+7\log(2/\delta)/(3(N-1))$. Set $\delta=\alpha/(2K)$, note $V_N(Y)=s_a^2/4$, multiply by two to return to e, and union over the 2K events. No independence between targets is required. The iid assumption is within each model-unit sample; paired arms are used to form one e. The variance identity $V_N=N^{-1}(N-1)^{-1}\sum_{i