Recovering finite-level scaling quantities from finite observations

01Overview (plain summary)

The period-doubling cascade of the logistic map f_u(x)=u x(1-x) carries finite-level quantities: the ratio of bifurcation-parameter gaps delta_k, and the state-space scale ratio of superstable orbits alpha_k. Phases A/B measured whether a fixed procedure (the Observer), told neither the map's formula nor the physical parameter names nor any true value, can recover these finite-level quantities from finite, anonymised observations alone.

Three primary phases were run.

• Phase A (ideal conditions): no noise. All 4 conditions × 2 quantities × 2 levels = 16 decisions were RECOVERED (relative error within 5%). No wrong answers, no abstentions.
• Phase B1 (coarse noise grid): the Observer was left completely unchanged; iid Gaussian noise was added to the observations only. 141 runs / 564 decisions. RECOVERED=244 / WRONG=0 / ABSTAIN=320. All four targets lost recovery inside the same interval 1e-9 < sigma <= 3e-9, and within that coarse grid they lost it by abstaining, not by answering wrongly.
• Phase B2 (refined grid and first-failure attribution): that interval was refined into 9 levels. 181 runs / 724 decisions. RECOVERED=387 / WRONG=1 / ABSTAIN=336. The direct cause of abstention was not a shared period-recognition layer: the delta side failed at bracket uniqueness, the alpha side at superstable geometry. And one false acceptance appeared that Phase B1 had not seen.

A later exploratory Phase C follow-up is documented separately in §8. Its numerical summaries are preserved for transparency, but it is not treated as a fourth primary phase or as confirmatory evidence. If its observed tendency reflects a real phenomenon, it is qualitatively consistent with the observation-space interpretation in the discussion note; that consistency remains a hypothesis, not a result of the primary study.

An exploratory Phase D0 fresh-seed recurrence probe is documented in §9. At the exact B2 incident condition (A3, sigma=1.6e-9), the unchanged Observer produced delta_3: ABSTAIN=39 / RECOVERED=1 / WRONG=0 across the pre-fixed unused seeds 41021–41060. Thus no second 41014-type false acceptance was observed in this fixed set. A separate post-hoc reconstruction found that all 40 ell=2 cells retained the zero-reference-associated candidate: 39 also contained one or more alternative candidates and were nonunique, matching all 39 ABSTAIN outcomes, while seed 41042 retained the exact reference-only unique tuple and was RECOVERED. No false-only or unique false-only ell=2 cell was observed. These fixed-set observations neither establish non-recurrence nor estimate a population probability.

The most important primary results are not "Feigenbaum-like quantities were recovered", but these two points.

1. The fixed logistic-map Observer's breakdown could not be reduced to a single shared mechanism (33 of the 80 attribution rows are simultaneous failures at the same depth, i.e. TIE).
2. The Observer's ESTIMATE label is not a guarantee that true finite-level structure was recovered. The "loss by abstention only" property observed on the B1 coarse grid did not extend unchanged to the B2 refined grid.

In Phase B2, one false acceptance was observed for delta_3 at seed 41014 and sigma 1.6e-9. For the limited replay / Decimal diagnostic on the stored input, see Part II; for the fixed-runtime reproduction of the local acceptance path for that same stored input, see §7.1. The exploratory Phase C follow-up is not a direct replication of that incident: its target, signal, noise scale, scoring unit, and Observers differ (§8). Phase D0 instead reused the A3 task, frozen Observer and exact incident sigma with 40 fixed fresh seeds, but remained a one-condition descriptive recurrence probe (§9).

02Research question and scope of claims

2.1Research question

The primary study is restricted to two questions.

1. Can the finite-level quantities delta_2, delta_3, alpha_2, alpha_3 be recovered operationally from finite, anonymised, passive observations?
2. When Gaussian noise is added to the observations, where does the fixed Observer stop (ABSTAIN) and where does it answer wrongly (WRONG)?

The fixed primary question statements are registered in 03_PHASE_A/PLAN_PHASE_A_FINITE_LEVEL_RECOVERY.md §1, 04_PHASE_B1/PLAN_PHASE_B1_GAUSSIAN_BREAKDOWN.md §1, and 05_PHASE_B2/PLAN_PHASE_B2_FIRST_FAILURE_ATTRIBUTION.md §1. Phase C asked an additional cross-Observer question, but that question and its results are retained only in the exploratory follow-up (§8), not added to the primary claims.

2.2What is not claimed

• High-precision estimation of the asymptotic Feigenbaum constants delta_infinity, alpha_infinity. No asymptotic constant and no extrapolation method was used in either estimation or scoring.
• Discovery of universality, or generalisation to unseen maps or other systems. Phases A/B concern only the logistic map's principal cascade; Phase C adds one deliberately simple second map-derived curve, not a representative system family.
• An information-theoretic limit of recoverability under Gaussian noise.
• Population error rates, error probabilities for untested seeds, or the safety of the Observer as a whole.
• Any performance claim that "a better ABSTAIN rule can be built". Rule design was explicitly outside B2's scope.
• That Phase C is a statistical replication of A/B1/B2, that its sigma values share a common physical scale with B1/B2, or that two implementations constitute broad algorithmic independence.
• That Phase C identified a winning Observer or supplied confirmatory evidence. Its own fixed experiment-level result was INCONCLUSIVE, and this document assigns the exercise only exploratory standing.
• That Phase D0's WRONG=0 establishes non-recurrence, safety, a population error rate, or an upper bound applicable beyond its 40 fixed seeds and one sigma.
• That the post-hoc D0 candidate-set reconstruction was part of the original seal, proves a low recurrence probability, establishes the cause of seed 41014, or validates a guard.

03Target quantities and experimental roles (Generator / Observer / Scorer)

3.1Target quantities

In Phases A/B, only the following four finite-level quantities are scored.

• delta_k = (b_k - b_(k-1)) / (b_(k+1) - b_k) for k=2,3, where b_k is the parameter at which period 2^(k-1) bifurcates to 2^k.
• alpha_k = d_k / d_(k+1) for k=2,3, where s_k is the superstable parameter of period 2^k, d_k = f_(s_k)^(2^(k-1))(c) - c, and no absolute value is taken.

The truth held by the Scorer (03_PHASE_A/results/truth.json; decimal working precision 100 digits, residual <1e-60, bracket width <1e-50):

b_k and s_k are distinct sequences, solved and cross-checked independently (see the verification block of the same file).

Phase C instead scores one observation-coordinate destabilisation location, with private truth 37/71. It neither re-estimates the four quantities above nor adds another finite-level scaling quantity.

3.2Separation of the three roles

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The Observer's ABSTAIN reason codes are fixed at five: PERIOD_LEVEL_UNRESOLVED, BIFURCATION_BRACKET_UNRESOLVED, CRITICAL_REGION_UNOBSERVED, SUPERSTABLE_GEOMETRY_UNRESOLVED, BRANCH_OR_SIGN_UNRESOLVED (03_PHASE_A/PLAN_….md §6).

The exploratory Phase C follow-up uses the same separation principle but a different task and contract. Its Generator privately defines a second map-derived scalar observation curve; each Observer sees only an opaque target ID and 401 ordered (coordinate, observed_value) pairs, and must return one destabilisation-location ESTIMATE or ABSTAIN. The two Observers own different reason-code vocabularies and diagnostics, so Phase C compares only final decision status and estimated location. Their internal diagnostic terms are not treated as semantically commensurate. None of this changes the scope of the primary A/B study.

04Phase A: finite-level recovery under ideal conditions

4.1Conditions

• Map f_u(x)=u x(1-x), initial state x_0=0.123456789, burn-in B=32768.
• Physical window u in [2.9500, 3.5700]. Before being handed to the Observer, this is transformed by the positive affine map v=(u-2.9500)/0.6200. The Observer sees only v in [0,1].
• Exactly four conditions: A1=(N=64, Δu=0.0010, K=621), A2=(1024, 0.0010, 621), A3=(64, 0.0001, 6201), A4=(1024, 0.0001, 6201).
• No repeated trials, no randomness, no bootstrap. 4 deterministic runs, 16 decision records.

4.2Nature of the decision threshold

There is exactly one classification threshold, relative_error <= 0.05. This is not a statistically derived confidence interval; it is a pre-registered operational (engineering) threshold for triaging minimal viability. 03_PHASE_A/results/phase_a_score.json states this explicitly as "tolerance_is_nonstatistical_and_arbitrary": true. No confidence interval, coverage, FPR, or multiple-comparison correction was computed at any point.

4.3Results

Source: 03_PHASE_A/results/phase_a_score.json

On the finest grid (A3/A4) the relative errors were delta_2: 3.60e-4, delta_3: 5.77e-4, alpha_2: 2.42e-6, alpha_3: 7.30e-5. The Observer's raw output for A3 is 03_PHASE_A/results/A3.json.

05Phase B1: loss of recovery on a coarse noise grid

5.1Design

• Fixed to condition A3 only. Observer, Scorer, truth, public contract, and thresholds are all frozen by SHA-256 (04_PHASE_B1/PLAN_….md §2).
• Exactly one thing is broken: y_obs[j,t] = y0[j,t] + sigma * Z_i[j,t]. No clipping, rounding, rescaling, or imputation. v and all metadata remain bitwise identical.
• Eight sigma levels: 0, 1e-10, 3e-10, 1e-9, 3e-9, 1e-8, 3e-8, 1e-7. Twenty seeds (41001..41020) at each of the seven positive levels.
• Paired design: one standard noise field Z_i is drawn per seed and reused at every positive sigma, so comparisons across strengths are paired.
• sigma=0 is run once as an integrity control. Planned runs 1 + 7*20 = 141; planned decisions 564.

5.2Results

Sources: 04_PHASE_B1/results/summary.csv (32 rows), breakdown_brackets.json, run_provenance.json (status: COMPLETE, observer_runs: 141, decision_records: 564)

• sigma <= 1e-9: RECOVERED for all 20 seeds and all four targets.
• sigma >= 3e-9: ABSTAIN for all 20 seeds and all four targets.
• The operational recovery-loss bracket for all four targets is 1e-9 < sigma <= 3e-9, with status: BRACKETED, loss_mode: ABSTAIN_ONLY, first_wrong_sigma: null.

Reason-code breakdown at sigma=3e-9:

At sigma >= 1e-8 all four targets fall to PERIOD_LEVEL_UNRESOLVED.

5.3Limits on how to read this

Observed fact: on this coarse grid, recovery was lost without any wrong answers — by abstention only (WRONG=0).

Interpretation (limited): within this coarse grid, the fixed Observer behaved so as to abstain rather than return a plausible-looking wrong answer when information was insufficient.

06Phase B2: refined grid and first-failure attribution

6.1Question and design

B2 asks: when all four targets lost recovery inside the same interval in B1, was it because a shared upstream period-recognition layer broke first, because the delta-side bracket and the alpha-side geometry broke separately, or because the transition is a per-seed mixture?

• The Observer, its decision rules, thresholds, reason priority, and the 5% scoring criterion were all left unchanged. No new estimator and no new ABSTAIN rule was created.
• Sigma was refined into 9 levels inside B1's loss interval: 1.0e-9, 1.2e-9, 1.4e-9, 1.6e-9, 1.8e-9, 2.0e-9, 2.3e-9, 2.6e-9, 3.0e-9. Both endpoints serve as reproduction controls against B1.
• B1's 20 paired noise fields and their seed correspondence were reused read-only, unchanged.
• Including one sigma=0 zero-noise reference: Observer runs 181, decisions 724, positive traces 180.
• The diagnostic tracer runs as a separate process and module from the frozen Observer and reads only public inputs. No tracer value flows back into the Observer's decisions or into the Scorer (observer_use_prohibited=true and diagnostic_only=true are mandatory fields on every artifact).

6.2Decision results

Sources: 05_PHASE_B2/results/decision_summary.csv (40 rows), PHASE_B2_FULL_RUN_VERIFICATION_V1.json (status: PASS)

6.3First-ABSTAIN thresholds

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The ordering in which the delta side moves to abstention at lower sigma than the alpha side reproduces across nearly all 20 seeds (onset_order_by_seed: in 18 seeds delta_2 and delta_3 lead as a tie; only seeds 41014 and 41017 split the order within delta).

These thresholds are operational first failures on a pre-registered grid, not true thresholds in continuous sigma. Boundary precision extends only to the grid spacing.

6.4The 80-row first-failure attribution

The 20 seeds × 4 targets = 80 rows (first_failure_by_seed.csv), mapped to blocking layers:

(MIXED/TIE is the human-readable display name; the verification JSON uses the JSON-safe key MIXED_OR_TIE for the same 33 rows.)

• Handling of TIE: when several blocking nodes fail simultaneously at the same minimal DAG depth, no single one is picked arbitrarily. The row is marked TIE and every applicable canonical node ID is retained as an ascending array. 33 of the 80 rows have blocking_tie=True (for example, seed 41014's delta_3 ties across D_BRACKET_UNIQUE[k=3,ell=2], [ell=3], and [ell=4]).
• Seeds where all four targets share the same first-abstain sigma: 0.
• Non-monotone transitions, in decisions and in nodes alike: 0.
• Excluding ties, the single blocking nodes are confined to A_MINIMUM_COUNT_EQ1[…] (alpha side, 29 total) and D_BRACKET_UNIQUE[…] (delta side, 18 total). Not one period-layer node was ever the first blocker.

6.5Distinguishing precursor from blocker

In all 80 rows first_precursor_sigma = 1e-9 (the smallest positive sigma in the B2 grid), and first_precursor_nodes is a two-way tie per target between PR_LABEL_MATCH[…,j=5941] and PR_PERIOD_ACCEPT[…,p=8,j=5941].

Observed fact: period-layer predicates that PASSed in the zero-noise reference were already FAILing at grid point j=5941 well before the official ABSTAIN. The precursor was already detected at 1e-9, the smallest positive sigma measured in B2, so the true onset is left-censored at <= 1e-9. It cannot be written as "first occurred at 1e-9".

Note also that all decisions being RECOVERED at B1's 1e-10 / 3e-10 does not imply that these tracer nodes PASSed there. B1 did not record the same traces (tracing was introduced in B2). No additional run was performed to fill that gap.

Interpretation (limited): a shared precursor was visible early. However, as the blocking cause that directly triggered the first ABSTAIN, the shared period layer was not supported (PERIOD_COMMON=0; H_U_common_upstream.first_blocking_rows=0). The blocking node was predominantly bracket uniqueness on the delta side, and the uniqueness of the superstable-geometry minimum on the alpha side.

Caution: a precursor only means "degraded earlier"; it is not a causal cause. No extrapolation is made outside the censored interval.

6.6B1's WRONG=0 and B2's WRONG=1 are not a contradiction

The two did not measure the same set of sigma values.

• B1's positive grid is 1e-10, 3e-10, 1e-9, 3e-9, 1e-8, 3e-8, 1e-7; there is no measurement point between 1e-9 and 3e-9.
• B2 filled that gap with seven levels. The wrong answer appeared at sigma=1.6e-9 — a level B1 never measured.
• At the overlapping endpoints (1e-9, 3e-9), the 40 shared runs were verified to match B1 at object level (endpoint_matches: 41, including the zero-noise control).

The correct statement is therefore: "B1's ABSTAIN-only loss is an observation within B1's coarse grid and cannot be extended to the whole refined grid." B1's result is not wrong; its resolution covers a different range.

07The single false acceptance

Sources: 05_PHASE_B2/results/score_records.jsonl (run_index: 74), observer_records.jsonl

In the same run, delta_2 abstained with BIFURCATION_BRACKET_UNRESOLVED. According to the Observer's observable_audit, this run's b_hat_v was 1: null, 2: 0.5025, 3: 0.9582258064516129, 4: 0.9909677419354839 (against zero-noise A3's 1: 0.0799…, 2: 0.8054…, 3: 0.9581…, 4: 0.9910…).

For the limited replay / Decimal diagnostic on the stored input (output record result_41014.txt), see Part II.

7.1Fixed-runtime reproduction of the local acceptance path

On 2026-08-12 the local stored-data path

observed value pair -> period residual -> threshold decision -> period label -> bracket candidate -> delta_3

was re-executed once for this same stored input in the original fixed runtime (CPython 3.13.15, NumPy 2.3.2, frozen observer.py SHA-256 A72C880213D617C9479D8F4C9DFD3E7F277169CBB236FB9C4EDFEE0B800D7EB4). The incident input was reconstructed from the existing A3 payload and the existing seed-41014 noise slice; the truth and the Scorer were neither read nor used; no new seed and no new simulation were introduced. Source: SEED_41014_FIXED_RUNTIME_LOCAL_PATH_REPRODUCTION.txt.

This is not a new Phase, not a new simulation, not a formal third-party forensic examination, and not the discovery of a general failure mechanism.

All checks in that execution reported True, and the final status was PASS: the frozen Observer output matched the official raw Observer record; the frozen incident labels matched the stored B2 trace; the frozen zero-noise labels matched the stored zero reference; a small independent residual/label calculation matched the frozen Observer; independently enumerated bracket candidates matched the stored and frozen path; and the independent residual arrays matched the stored trace arrays.

Exactly four of the 6201 grid points changed period label relative to the zero-noise reference:

The threshold was tau = 1.00000000000000002e-08 for both the zero-noise and the incident array.

At the two points that govern the b_2 candidate set:

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At j=3116 the zero-noise contribution to the maximum pair is exactly 0, so the period-2 threshold crossing for that pair is essentially the noise difference alone. At j=4994 the observed signed difference is +1.01635164173607961e-8, of which the noise difference accounts for roughly 84.1%.

The b_2 candidate set moved as follows (a bookkeeping reconstruction from the two stored label arrays, not a separately simulated sequence):

For this one fixed stored input, the false b_2 candidate was created at j=3116, the zero-noise b_2 candidate was removed at j=4994, and the false candidate thereby became unique and was accepted. The same local path was reproduced in the original fixed runtime and matched the stored trace and the official raw Observer record. The resulting brackets b_2=0.5025, b_3=0.9582258064516129, b_4=0.9909677419354839 give delta_3=13.918719211822673, and an independent calculation from those reproduced bracket values returned the same number.

Two ancillary observations, recorded without causal claims: j=126 added a false 1 → 2 candidate, leaving two b_1 candidates, which is consistent with delta_2 abstaining; and j=5941 turned 8 → 0, changing the b_3 bracket from gap 0 to gap 1. j=5941 is not described here as a cause of the 13.9187 estimate.

7.2Read-only neighbourhood and fixed-20-seed candidate-set comparison

After the incident path had been reproduced, a separate lightweight comparison was made from the already sealed zero-noise and B2 trace arrays. It did not generate observations, rerun the Observer, change any stored artifact, add seeds, or add sigma values. The comparison used margin = residual - tau, so margin <= 0 is PASS, and retained the Observer's search order p=1,2,4,8,16. The term critical-margin predicate below means the predicate with minimum |margin| among those examined through the first PASS (or among all five periods if there is no PASS); it is not necessarily the first PASS itself.

For seed 41014, the 6,201 grid points were partitioned mechanically as follows:

B=0 is a consequence of this first-PASS label rule, not a newly observed robustness property: changing a sign at or before the first PASS necessarily changes where the first PASS occurs. Within C, 29 points had |critical margin| <= 1e-9 in the incident trace. These are near-threshold non-crossings, not 29 observed label-flip events. The six D points were j=419, 498, 501, 1242, 2329, 3480; they account for the period-predicate changes hidden behind an earlier PASS.

The same calculation was then applied at the same stored level, sigma=1.6e-9, to all 20 fixed B2 noise fields:

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Across these fixed 20 fields, A ranged from 2 to 8 (median 6), the near-threshold C subset from 14 to 29 (median 23), D from 6 to 18 (median 11.5), and all predicate crossings from 11 to 26 (median 18.5). Seed 41014 had the largest near-threshold C subset, but it had only four label changes, the fewest D points, and the fewest predicate crossings; its maximum critical-margin movement ranked seventh. Thus the stored comparison does not support describing 41014 as simply the most broadly disrupted field.

The four incident positions also separated across seeds. j=5941 changed 8 → 0 in all 20 fields at this sigma and was therefore not incident-specific. The label changes at j=126, j=3116, and j=4994 occurred only in seed 41014. At j=4994, 14 other seeds changed only a later predicate (class D), while five retained all predicate signs.

The bracket-candidate comparison used the frozen rule of locating a left label 2^(ell-1), skipping at most four zero labels, and accepting a following label 2^ell. The zero-noise candidate sets were unique:

Because the common j=5941 change extends the ell=3 bracket to (5940, 5942, 1), exact tuple identity alone would misleadingly call the same local boundary a removed and newly false candidate. For this comparison, a current candidate was therefore called zero-reference-associated when its inclusive [left,right] interval overlapped the zero-noise candidate interval; candidates without such overlap were called non-reference candidates. This is an operational association rule, not access to physical truth.

Across the 80 seed-by-ell cells, there were 104 exact candidate-tuple additions and 22 exact removals. Of the removals, 21 were shifts or extensions of the same zero-reference-associated boundary. There were 83 non-reference candidate additions, 35 direct unique → nonunique changes, no nonunique → unique changes, 79/80 cells retaining a zero-reference-associated candidate, and one cell containing only a non-reference candidate. No cell had an empty candidate set. The candidate uniqueness and midpoint reconstructed from the stored label arrays matched the stored Observer audit in 80/80 cells.

The unique exceptional cell was seed 41014 at ell=2:

zero noise: {(4993, 4994, gap=0)}
seed 41014: {(3115, 3116, gap=0)}

At ell=2, 18 other seeds retained the zero-reference-associated candidate while adding one or more non-reference candidates and therefore became nonunique; seed 41017 retained the reference candidate alone; only seed 41014 lost the reference-associated candidate while retaining a different candidate uniquely. For seed 41014, ell=1 was nonunique because a new candidate was added while the reference-associated candidate remained, ell=3 remained uniquely associated with the same boundary after its one-point gap extension, and ell=4 was unchanged. The resulting delta_3 input topology was therefore a unique non-reference ell=2 candidate plus unique reference-associated ell=3 and ell=4 candidates.

The directly observed endpoint change at ell=2 is unique reference-only → unique non-reference-only. The intermediate sequence {reference} → {reference, non-reference} → {non-reference} in §7.1 is explicitly a bookkeeping decomposition obtained by applying the two stored label changes in a chosen order. It is not an observed nonunique → unique transition in the zero and final traces, and it does not establish temporal or causal order.

These comparisons are descriptive results for one stored sigma level, one fixed payload, one Observer, and 20 fixed noise fields selected before B2. They do not turn the fields into a population sample, estimate an error probability, show that near-threshold counts caused the wrong estimate, establish a general failure mechanism, or validate a guard.

08Exploratory follow-up (Phase C; limited evidential weight)

8.1Question and fixed design

Phase C moved away from the logistic-map finite-level quantities. It asked a narrower exploratory comparative question: when two separately developed and sealed Observers receive the same anonymised noisy curve, do their final decisions support a pre-specified Observer-specific pattern, a cross-Observer pattern, or neither?

This is motivated by the observation-space/target-structure distinction in the discussion note, but it addresses that distinction only indirectly. It changes both the target system and the task; it does not test the B2 seed-41014 input with a second detector.

The private Generator uses

f_mu(x) = mu*x*(1-x^2)
signal(mu) = 1 - |3 - 2*mu|
u_i = i/400,  mu_i = 1.63 + 0.71*u_i,  i=0,...,400

The signal is the local-stability margin of the registered fixed-point branch. On the registered coordinate interval it has one positive-to-nonpositive transition, at the private truth u = 37/71 = 0.5211267605633803…. The Observers were not given the map, formula, original parameter, truth, noise construction, sigma, scoring tolerance, or relationships among cases.

• 20 fixed deterministic Rademacher (+1/-1) noise fields generated by SplitMix64.
• Sigma grid [0, 0.01, 0.025, 0.05, 0.10]; the same field is reused across sigma for a paired design.
• 100 targets (20 fields × 5 sigma levels), each containing 401 ordered observation points. At sigma=0, the 20 targets are repeated copies of the same clean curve under different opaque IDs; they are not 20 independent clean realisations.
• Exactly two sealed Observers, O1 and O2-independent-multiscale-v1, each run in a fresh process per target. They received byte-identical public specifications and payload bytes.
• An ESTIMATE is RECOVERED when |estimate - truth| <= 0.015; another valid estimate is WRONG; a valid abstention is ABSTAIN; invalid or failed execution is TECHNICAL_FAILURE.
• Two valid estimates have LOCATION_AGREEMENT when |e_1-e_2| <= 0.02. Two abstentions are status-concordant but their locations are not comparable.

The fixed noise fields are descriptive repetitions, not a probability sample. The tolerance and comparison cutoffs are operational choices, not confidence bounds.

8.2Pre-registered experiment-level rule

Before the Phase C payload was generated or either Observer was run, Stage 1 v4 fixed three possible conclusions: OBSERVER_SPECIFIC_PATTERN, CROSS_OBSERVER_PATTERN, or INCONCLUSIVE. In abbreviated form:

• OBSERVER_SPECIFIC_PATTERN required an absolute O1/O2 estimate-rate or recovery-rate difference of at least 0.25 after the technical-validity gate.
• CROSS_OBSERVER_PATTERN required both recovery rates to be at least 0.60, both rate differences to be at most 0.15, status concordance to be at least 0.70, and location agreement among joint estimates to be at least 0.70.
• Otherwise the result was INCONCLUSIVE (with additional technical gates for excessive failures, too few comparable targets, or too few joint estimates).

These thresholds are fixed decision rules, not statistically calibrated hypothesis tests. Passing or failing them has no p-value interpretation.

8.3Reported Run 4 results (descriptive only)

Source: Phase C Run 4 STAGE6/scoring_output.json (SHA-256 bd0c275caa84c544eaa9f8b293a6bd614ba9d2fc20b25185938fdd2c90b6e297; schema_version: phase-c-score-v4, 200 scored rows, experiment conclusion INCONCLUSIVE). The Stage 6 completion marker SHA-256 is 61bf5dc0db3f8ae7a57c4dc3fd6a4d1bcf21f69e4b99f5bf4c50dd45bbc87c03.

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Across all 100 targets, there were 47 joint estimates, all 47 satisfying the fixed location-agreement rule. Status concordance was 0.83; O2's estimate and recovery rates exceeded O1's by 0.07 each. The 0.07 difference did not meet the 0.25 Observer-specific threshold, while the two recovery rates (0.50, 0.57) did not meet the 0.60 cross-Observer requirement. Even within Phase C's own rules, the result is INCONCLUSIVE. It is not a win for O2, not a confirmation of an Observer-independent pattern, and not evidence used by the main record.

8.4Conditional comparison with Phases A/B1/B2

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The absolute sigma values, recovery rates, and reason codes must not be compared numerically across the two studies. The generated signal, observation scale, noise distribution, target quantity, tolerance, and decision rules all differ. Phase C is therefore an exploratory related exercise, not a replication of the B1/B2 threshold. The table records a possible qualitative analogy, not additional evidence for the primary result.

Conditionally, the tendency is compatible with the discussion note's decomposition

noise realisation + finite observation + decision rules -> operational structure

The suggestive feature is that clear cases were reportedly located similarly, whereas all four Phase C false acceptances were paired with abstention by the other rule. If reliable, that would fit the hypothesis that an ambiguous pattern becomes certified structure through its interaction with a decision rule. Phase C does not demonstrate this proposition: it did not test the seed-41014 input with another Observer, and its own evidential limitations prevent promotion of the tendency beyond hypothesis formation.

8.5Post-hoc description of the four Phase C wrong estimates

A read-only analysis of the sealed Stage 6 output found four WRONG rows, all at sigma=0.05: two from O1 and two from O2. In every one of those four targets, the paired Observer abstained. The analysis artifact has SHA-256 4f954dd8944d750031bfca2edfa0ff64a1840295e091a0049d52ce91afe39d2a. The 100-target joint classification was:

Thus the four false acceptances did not recur as false acceptances under the other Phase C decision rule on the same targets. This is a post-hoc descriptive result, not part of the pre-registered experiment conclusion. It does not show that one Observer can generally guard the other, nor does it establish that the B2 incident and the Phase C errors share a mechanism.

8.6Execution history and strictness qualifications

Phase C contains useful mechanical controls: the Stage 1 v4 generator, truth, fixed cases, scoring rule, comparison rule, and experiment-level conclusion were sealed before payload generation and before Observer outputs; the Observer inference implementations were sealed before payload release; and Run 4 raw outputs were sealed before truth was opened. Run 4 completed 100/100 targets for each Observer with no retry, no technical failure, no raw modification, and no post-result threshold change. These controls support the integrity of the reported Run 4 files; they do not make the exercise confirmatory or cure the broader reliability limitations below.

It is nevertheless not a pristine single-shot independent replication. Three earlier run attempts were preserved and failed for operational reasons:

1. Run 1: O1's seal metadata duplicated entry-point arguments, so its estimator never started; O2 completed.
2. Run 2: both batches completed, but Stage 5 rejected all O2 rows because the seal identity O2 did not match the raw identity O2-independent-multiscale-v1. The public output schema allowed either string; the exact internal identity-binding requirement had not been caught in preflight.
3. Run 3: O1 stopped before its first process because an attempt parent directory was absent; O2 completed 37 recorded targets before an external 120-second caller limit terminated the batch.

Before Run 4, the execution infrastructure was revised and tested on dummy inputs: command construction and identity binding were checked, target and batch timeouts were separated, the caller timeout requirement was raised, and a versioned infrastructure manifest was sealed. The scientific design, payload, truth, Scorer, scoring thresholds, and Observer inference code were reported unchanged; only seal metadata and execution infrastructure were corrected. The same fixed payload bytes were then used in Run 4. No truth was opened and no scoring was performed in Runs 1–3.

This history does not by itself overturn the Run 4 counts, but it weakens any claim that Phase C was an untouched one-opportunity trial. The infrastructure manifest was recorded in Run 4 batch evidence but was not itself bound inside STAGE3_COMPLETE. In addition, the two Observers were separate implementations produced in related AI-assisted workflows, not independent laboratories; the degree of algorithmic independence was not quantified. All seals and most cross-checks remain within the same implementation and operator lineage.

The appropriate reading is therefore modest: Run 4 preserves a fixed-design descriptive output after infrastructure repair, while its formal conclusion remains INCONCLUSIVE and its evidential weight is insufficient for the primary claims. Hash and completion checks should not be confused with scientific validation.

09Exploratory Phase D0: fresh-seed recurrence probe at the 41014 condition

9.1Fixed question and design

Phase D0 asked one deliberately narrow question: does the unchanged Observer produce another WRONG classification for delta_3 on previously unused noise fields at the exact B2 incident condition? It was not designed to map a sigma-dependent failure region, identify a cause, evaluate a guard, compare Observers, or estimate a general error probability.

Before execution, the following were fixed and hash-sealed:

• the original A3 payload, frozen Phase A/B Observer, frozen Scorer and private truth;
• sigma=1.6e-9 only;
• fresh seeds 41021–41060, with the B2 seeds 41001–41020 excluded;
• 40 Observer runs and 160 decisions, with N_WRONG_DELTA3 as the sole primary outcome;
• no retry, replacement seed, adaptive sigma, threshold change, Observer change, Scorer change, or automatic extension;
• raw Observer records must be sealed before truth access and scoring; and
• CPython 3.13.15, NumPy 2.3.2, executable SHA-256 72B29481593C5DA37C99248C82777FBFB56217EA7809B771BC760D0A9ECB179B.

The probe's standing was fixed as EXPLORATORY_PRE_SPECIFIED_RECURRENCE_PROBE. The pre-run plan, design and runner hashes were subsequently reproduced exactly by the raw seal.

9.2Result

The recorded execution completed all 40 fixed seeds and all 160 decisions with retry_count=0 and replacement_seed_count=0. The delta_3 primary rows were:

The one recovered row was seed 41042: estimate 4.667487684729068, truth 4.656251017651357, relative error 0.0024132434 (about 0.241%). All 39 abstentions used BIFURCATION_BRACKET_UNRESOLVED. Because there were no WRONG rows, there was no fresh wrong case on which to observe the pre-specified 41014-shaped downstream fingerprint; path_resembles_41014_count=0 should not be read as a separate negative mechanism test.

9.3Seal and audit checks

The 40 raw records contain exactly seeds 41021–41060, 40 distinct noise-field hashes, one common sigma, and four Observer decisions per seed. Their SHA-256 is 148A738D984CAF5592B80E7C1845A43518AA06E30A92BC1CE959BEDA0E1F22B4. RAW_SEAL.json records truth_opened:false and binds that raw file to the sealed plan, design, runner, A3 payload, Observer source and runtime executable; its own SHA-256 is D9CFA8E46F4F9F92F992F9F83AE24418D9567BF6AFEE17F4F5D024FFE01DF89B. The truth-access record references this raw-seal hash before scoring. The final score-record and summary hashes are 413A0F66E5BE613E15EF052CA0B678A66FEA1AF46F8B9C480F23867038912866 and 7084B7240A191042E469FBDC558FD511C5A9C43FFF06583DA1354E947AD14730, respectively; all current files reproduce the hashes recorded in COMPLETE.json.

These checks support identity and execution-order consistency within the same local operator and implementation lineage. They are not an independent replication, a formal tamper-proof audit, or scientific validation of the Observer.

9.4Post-hoc derived candidate-set reconstruction (not part of the original D0 seal)

These traces were reconstructed post hoc from the fixed Phase D0 inputs and frozen generation/Observer rules. They were not part of the original sealed Phase D0 output artifacts.

The reconstruction regenerated seeds 41021–41060 at sigma=1.6e-9 from the sealed A3 identity and fixed SeedSequence([20260811, seed]) / PCG64DXSM rule. Before candidate analysis, all 40 regenerated noise-field SHA-256 values matched the hashes in the original D0 raw records. The frozen Observer outputs, reconstructed b_hat_v, and labels then matched the stored records or frozen label function 40/40. The derived arrays contain one (6201,) label vector and one (6201,5) residual array for periods 1,2,4,8,16 per seed.

The candidate rule and reference-association rule were unchanged from the fixed-20-seed comparison in §7.2: a candidate is a left label 2^(ell-1), at most four intervening zero labels, and a right label 2^ell; a current candidate is reference-associated when its inclusive index interval overlaps the zero-reference candidate interval. Reference here denotes the zero-noise computational reference, not physical truth.

For ell=2, the zero-reference candidate was (4993,4994,0). The 40 reconstructed cells were:

All 39 delta_3 ABSTAIN rows had the same sufficient ell=2 condition: the reference-associated candidate remained, but one or more alternatives were also present, so the candidate set was nonunique. Their ell=2 candidate counts ranged from 2 to 8. The sole RECOVERED row, seed 41042, retained exactly (4993,4994,0) with midpoint 0.8054032258064516 and no alternative candidate.

Across all 40 seeds × four levels = 160 cells:

→ table scrolls sideways

The 160 cells comprised 67 exact reference-only unique states, 21 reference-associated unique states with a changed tuple, and 72 reference-plus-alternative nonunique states. There were 203 strict tuple additions and 40 strict tuple removals. All 40 removals occurred at ell=3; the replacement interval still overlapped the zero-reference interval, so reference association was retained in every case.

Unlike seed 41014's stored ell=2 endpoint — where (4993,4994,0) was absent and the alternative (3115,3116,0) remained uniquely — the fresh 40 contained no reference-lost or unique false-only ell=2 cell. The bounded conclusion is: no unique false-only ell=2 candidate state was observed among the 40 post-hoc reconstructed fresh-seed traces. This does not establish non-recurrence, a probability, safety, or a general causal mechanism.

The derived package is stored separately. Its MANIFEST_SHA256.txt SHA-256 is FC435F96D0D961BFE399F54ABD66A08AD6BD4757D50BD2BC4BE9EA9919FBBFD4; derived_trace_arrays.npz is 979047706699A1A33B4ACD2B7BD934A2C313B420CEEC4F9CA312DD558287F7F5; and summary.json is 8E0770646FFA82104D3E3C16CC0D916EC0751CE88C772ADC9A04382A3F55D499. The original D0 source tree remained byte-identical before and after reconstruction.

10Reproducibility, sealing, and audit boundaries

10.1Execution order

1. Phase A: 4 deterministic runs → 16 decisions → truth opened → Scorer scoring.
2. Phase B1: all 16 Phase A files were sealed once as an immutable manifest and re-checked before and after benchmarking and after the full run, each time into a separate audit file (expected=16 / observed=16 / missing=0 / extra=0 / mismatch=0; the post-full-run audit reports pass: true and 27/27 agreement on the Phase A implementation manifest). The truth was opened only after every Observer output was final.
3. Phase B2:
   • The sigma=0 zero-noise complete trace reference was created and sealed first.
   • For each of the 180 positive runs, the frozen Observer output was committed write-once first, and only then was the diagnostic tracer run.
   • For all 181 items (180 traces + 1 reference), the tracer's reconstructed decisions were compared with the frozen Observer at object level (reconstruction_matches: 181).
   • The raw Observer outputs and all traces were hash-sealed in observer_trace_seal_manifest.json (truth_opened: false) before the truth was opened and the Scorer run (truth_access_record.json: purpose: SCORER_ONLY, referencing raw_seal_sha256).
   • Full-run verification: PHASE_B2_FULL_RUN_VERIFICATION_V1.json reports status: PASS, implementation manifest 67/67, run artifacts 16/16, baseline checks 17/17, endpoint matches 41, and no error ledger.
4. Exploratory Phase C Run 4 (not part of the primary evidential chain):
   • Stage 1 v4 sealed the private design, Generator, Truth, Scorer, comparison rules, experiment-conclusion rules, and public interface before payload generation.
   • O1 and O2 were separately developed and sealed before the 100-target release. The release was byte-identical for both.
   • After the failed Runs 1–3 (§8.6), a sealed Execution Infrastructure v2 was regression-tested on dummy targets. Run 4 then started 200 fresh Observer processes, producing 100/100 valid raw outputs for each Observer with no retry.
   • Stage 5 verified and sealed all 200 raw outputs before truth access. Stage 6 materialised truth and invoked the frozen Scorer once. The Stage 6 marker records no raw-hash mismatch, modification, regeneration, Observer rerun, technical failure, or post-result threshold addition.
5. Exploratory Phase D0 (not part of the primary evidential chain): the plan, fixed design and runner were sealed before the run; exactly 40 fresh seeds at one fixed sigma produced 160 decisions; the raw records were sealed with truth_opened:false; the truth-access record bound that raw seal; and the frozen Scorer then produced 160 classifications. COMPLETE.json records status:COMPLETE, zero retries and zero replacement seeds.
6. Post-hoc D0 candidate reconstruction (not part of the original D0 seal): after D0 was complete, the same 40 deterministic inputs were regenerated in a separate output tree. Noise hashes and Observer outputs matched 40/40 before the reconstructed labels, residuals and candidate sets were used descriptively. No original D0 artifact was modified.

Each Phase's PLAN is a pre-registration document fixing the permitted scope and stop state at the time of writing; it was not rewritten to COMPLETE afterwards. Benchmarking and full runs were each authorised by a separate, subsequent approval artifact (in the original environment: Phase A PHASE_A_FULL_RUN_APPROVAL_V1.json; B1 PHASE_B1_FULL_RUN_APPROVAL_V1.json = 7CCBD452…; B2 FULL_RUN_APPROVAL_V1.json = C93AD681…). The B2 approval authorises fixed_design_sha256=5B564B6E…, 181 runs / 724 decisions / 180 positive traces / 1 zero reference / 724 classifications, with no retries and no substitute seeds. The approval documents themselves are not included in the distribution package.

10.2Runtime

Phases A/B1/B2 and the original D0 run used CPython 3.13.15 / NumPy 2.3.2 (Windows, AMD64). Both B1 and B2 require exact version agreement as a precondition for execution (runtime_fingerprint in run_provenance.json); D0 additionally bound the executable SHA-256 in its pre-run and raw seals. The later post-hoc D0 candidate reconstruction used CPython 3.12.13 / NumPy 2.3.5 with the same hash-bound A3 payload and Observer source; its regenerated noise hashes and complete stored Observer outputs nevertheless matched 40/40. Phase C O1/O2 were sealed to CPython 3.14.5 on the same local Windows host; the executable path, binary hash, and version-output hash were recorded in each Observer seal.

10.3Audit boundaries (what is not guaranteed)

• All sealing and cross-checking was generated and verified within the same implementation lineage. There has been no re-verification by an independent third-party implementation and no adversarial tamper-resistance testing.
• A narrow independent calculation reproduced the local residual/label/bracket path for the stored 41014 input (§7.1), but the Observer as a whole has not been independently reimplemented or third-party verified. That checker recomputes one narrow path in separate code; it is not a complete independent reimplementation of the Observer and not a forensic examination carrying formal guarantees.
• The diagnostic tracer is a separate module reading the same public inputs as the Observer; it is not an independent implementation.
• The forensic framework carrying formal guarantees (a design including Stage 1–4 approvals, a manifest chain, and a self-verifier) repeatedly ended in FAIL / HOLD and was abandoned before the incident run was ever executed. It was then replaced by a single-shot script with a deliberately narrowed claim (§7, Part II). The number of tests that passed inside the abandoned framework is not added to the scientific credibility of the incident.
• The roughly 2 GB of per-run diagnostic trace (diagnostic_trace_index.jsonl) is not included in the distribution package. Its hash is recorded in observer_trace_seal_manifest.json.
• Phase C's audit trail is internally detailed but remains same-lineage. The same operator environment built the design, coordinated the separate Observer work, repaired the execution infrastructure, and ran the final scoring. Hash consistency demonstrates artifact identity, not institutional independence or freedom from coordinated design bias.
• Phase D0 reused the same Observer, Scorer, payload family, local host and operator lineage as A/B. Fresh seeds add new noise realisations, not an independent implementation or laboratory.
• The post-hoc D0 reconstruction used the same Observer implementation and deterministic input rules. Its exact matches demonstrate reproducibility of these 40 computational paths across the recorded environments, not independent algorithmic or institutional replication.

11Limitations and explicit non-claims

11.1Design limitations

• A single map family (the logistic map's principal cascade), a single Observer implementation, a single payload (A3), fixed 20 paired noise fields, and fixed sigma grids.
• The 20 seeds are a descriptive fixed repetition, not a population sample. No p-values, confidence intervals, FPR guarantees, multiple-comparison corrections, or curve fitting of a continuous boundary were performed.
• The 5% relative error is a pre-registered operational threshold, not a statistical guarantee. No sensitivity analysis was performed.
• Observation noise only. Process noise, parameter noise, coloured noise, quantisation, missing data, and outliers are out of scope.
• Sigma-boundary resolution extends only to the grid spacing. No extrapolation to a true threshold in continuous sigma.
• Phase C adds one second map-derived curve, not a broad family of systems. All 100 targets share one underlying noiseless curve and one truth location; only the fixed field and sigma vary.
• Phase C's 20 deterministic fields are not a population sample, its 20 sigma=0 cases are duplicate clean curves, and its operational cutoffs (0.015, 0.02, and the experiment-level rate thresholds) were not statistically calibrated.
• Phase C used two distinct sealed implementations, but did not quantify their algorithmic dependence or establish independence in the stronger sense of separate laboratories, teams, or implementation lineages.
• Phase D0 used 40 pre-fixed fresh seeds at one incident-matched sigma. They are a descriptive fixed set, not a calibrated population sample; the probe has no resolution across sigma and no independent Observer. The later candidate-set reconstruction is post hoc and adds mechanism-level description of the same 40 deterministic inputs, not 40 additional observations.

11.2Explicit non-claims

This study claims none of the following.

• "An AI discovered Feigenbaum universality" / "discovered an unknown law". The subject is measurement of the recoverability of already-known finite-level quantities.
• "The Observer's safety was proved." If anything, WRONG=1 is an observation in the opposite direction.
• "A universal limit of noise tolerance was measured." What was measured is the breakdown curve of one fixed procedure.
• "Noise induced a genuinely new bifurcation." What was observed is a failure of the detector's operational labelling.
• "B2's diagnostic quantities would enable a better ABSTAIN rule." Rule design was outside B2's scope and remains untested.
• "Phase C replicated the logistic-map threshold." Its observation and sigma scales are different and not numerically comparable.
• "Phase C proved a cross-Observer pattern" or "O2 won". The pre-registered conclusion is INCONCLUSIVE.
• "The four Phase C false acceptances share the B2 incident's mechanism." No common mechanism was tested.
• "Phase D0 proved that 41014 was unique or non-recurrent." It observed no additional delta_3 wrong answer in one fixed set of 40 seeds; the post-hoc reconstruction additionally observed no unique false-only ell=2 state in those same 40 inputs. Neither zero count establishes zero risk.

11.3Not executed

Phase D0 supplies one narrow fresh-seed evaluation at sigma=1.6e-9; a broader Phase B3 across pre-fixed sigma levels remains unexecuted. A third Observer, an independent-laboratory replication, and the design, implementation, or evaluation of plausibility guards or cross-series consistency guards are also unexecuted. Phase C itself is complete only as an exploratory execution record in the qualified Run 4 sense described in §8.6; it is not a completed confirmatory study.

12File guide

(Relative paths are from the root of the distribution package.)

→ table scrolls sideways

Phase C artifacts (fixed design, public contract, Run 4 scoring and completion, post-hoc wrong-case analysis, execution history) and Phase D0 artifacts (pre-run specification, runner, raw output and seal, truth-access record, scoring and summary, completion) currently reside outside the original distribution package in the local working tree, together with the separately stored post-hoc D0 candidate reconstruction (derived arrays, candidate records, outcome join, verification, summary and integrity manifest).

13Conclusion

Under ideal conditions the fixed blind Observer recovered all four finite-level quantities within the 5% threshold from anonymised finite observations alone (Phase A, 16/16). Under observation Gaussian noise, on a coarse grid all four targets lost recovery inside 1e-9 < sigma <= 3e-9, and within that grid the loss carried no wrong answers (Phase B1, WRONG=0). Refining the same interval showed that the direct cause of the breakdown was not a shared period-recognition layer but split into delta-side bracket uniqueness and alpha-side superstable geometry, with 33 of 80 rows failing simultaneously at the same depth (Phase B2, PERIOD_COMMON=0). And one false acceptance appeared that the coarse grid had not revealed.

The read-only comparison at sigma=1.6e-9 further located that B2 case within the fixed 20 fields. Seed 41014 was not the field with the most label or predicate changes. Its distinguishing stored bracket topology was narrower: among 80 seed-by-level cells, its ell=2 cell alone replaced the zero-reference-associated unique candidate with a unique non-reference candidate, while the comparison supplied no evidence that this topology is a population-level risk marker or general causal mechanism.

The pre-specified exploratory D0 probe then applied the unchanged Observer to 40 unused seeds at that same A3/sigma condition. It observed delta_3: RECOVERED=1 / WRONG=0 / ABSTAIN=39; no second 41014-type false acceptance was found. A separate post-hoc reconstruction showed that all 39 abstentions retained the reference-associated ell=2 candidate but also contained alternatives and were therefore nonunique; seed 41042 alone retained the exact reference-only unique tuple and was recovered. All 40 retained a reference-associated ell=2 candidate, and none had the unique false-only endpoint observed for 41014. This makes the correct record "no recurrence observed in these fixed 40 seeds, and no unique false-only ell=2 state observed in their post-hoc reconstructed traces," not "the incident cannot recur." The result supplies no population rate and does not alter the primary B2 fact that a false acceptance occurred.

The primary judgment is therefore twofold: operational recoverability was demonstrated under the finite clean conditions tested, and added observation noise was shown to move the fixed procedure from recovery toward abstention while also permitting a false acceptance. In particular, an ESTIMATE is an algorithmic certification, not proof that target-system structure was recovered.

These primary findings remain tied to one logistic-map payload, one Observer, fixed noise fields, operational thresholds, and fixed sigma grids. Nothing in the primary study demonstrates an information-theoretic limit, a population error rate, universal safety, or a new general theory.

14Reader notes (outside the main record): remaining limitations

The following are not claims of the research record; they are notes to prevent over-reading the results.

1. The precursor onset is left-censored. B2's period-layer precursor was already detected at 1e-9, the smallest positive sigma measured, so the true onset is <= 1e-9. The same traces were not recorded at B1's smaller sigma values.
2. The 41014 replay / Decimal diagnostic and the 41014 fixed-runtime local-path reproduction are two different things, and neither is a formal forensic examination. The earlier one (result_41014.txt, Part II) is a single-shot, unaudited reference diagnostic that replayed the stored input and re-evaluated the downstream arithmetic at Decimal 50/100/200 digits; it concludes nothing about cause or safety, and it is not promoted to a formal forensic result by anything in §7.1. The later one (§7.1) re-executed the local path in the original fixed runtime and cross-checked it against a narrow independent calculation, the stored B2 trace, and the official raw record. Both are confined to the same one fixed stored input.
3. §7.1 records a local reproducible path, not a cause. It does not establish a root cause, a general failure mechanism, a frequency, or an Observer-wide safety conclusion, and no guard or repair was validated.
4. Same-Observer recurrence now has one narrow fresh-seed probe plus a post-hoc decomposition of the same inputs, not a frequency study. The earlier read-only comparison found the unique non-reference-only ell=2 topology only for seed 41014 among the original 20 fields. Phase D0 then tested 40 unused seeds at the same sigma and observed WRONG=0. Their later reconstructed traces retained the reference-associated ell=2 candidate 40/40; 39 were nonunique because alternatives were also present, while seed 41042 remained exact reference-only unique. This did not add seeds, sigma values, detectors, or a probability model, and it does not establish non-recurrence or safety.
5. Phase C's clean count is repeated-case bookkeeping. The 20 targets at sigma=0 contain identical observation values because the noise field is multiplied by zero. 20/20 confirms stable handling across opaque target IDs and isolated processes; it is not evidence from 20 distinct clean signals.
6. Phase C's INCONCLUSIVE result has priority over attractive descriptive patterns. The 47/47 joint-estimate location agreement and the qualitative clean-to-abstain curve are reported fixed-sample summaries, but neither overrides the fixed experiment-level rule or the decision to treat Phase C as non-confirmatory.
7. Phase C's wrong-case join is post-hoc. The fact that all four wrong estimates were paired with abstention, with both WRONG=0, was extracted after Stage 6 for comparison with B2. It may motivate a later registered hypothesis, but it is not a Phase C primary conclusion.
8. Run 4 followed infrastructure repair. Runs 1–3 remain part of the record. The repairs were confined to seal identity/arguments and execution infrastructure, with the scientific design and Observer inference code reported unchanged, but this history still prevents presentation as a pristine single-attempt replication.

i.1What happened (plain summary)

In one Phase B2 run, the Observer did not abstain on delta_3 and instead emitted 13.918719211822673. The truth is 4.656251017651357, so the relative error is about 1.99 (roughly 199%). In the same run, delta_2 had abstained for want of a bracket.

Afterwards, the stored input file and the frozen Observer were used to re-execute the case exactly once, and the same input was then promoted to high precision (Decimal) to redo only the downstream computation. The results were:

• The same 13.918719211822673 was reproduced, matching the official raw record at object level.
• At Decimal 50, 100, and 200 digits alike, not a single period label changed, the candidate bracket endpoints were identical, and the decision status was unchanged. The value moved only by about -2.66e-14 in its trailing digits.

The explanation "binary64 rounding was the main cause" is therefore weak within the scope of this diagnostic. Numerically, the observations are consistent with the first bracket b_2 used by delta_3 being accepted at a position far to the left of where it should be, with that displacement then amplified by the ratio computation.

i.2The single case as a formal result

Sources: 05_PHASE_B2/results/score_records.jsonl (run_index: 74), observer_records.jsonl

The run's observable_audit.b_hat_v (the bifurcation-point candidates the Observer accepted on the anonymised coordinate v):

The zero-noise reference is 03_PHASE_A/results/A3.json.

i.3The diagnostic performed

Script used: 06_INCIDENT_41014/diagnose_41014.py (SHA-256 E54AD11346099676F60E3F57A14BF149EA820C90D52E1E5FF98EF0079CB42ADC, confirmed to match the file in this distribution package).

The script references absolute paths in the original environment and is a single-shot script; it cannot be re-executed from this distribution package alone, which is a README-writing excerpt. The A3 payload, the noise fields, the roughly 2 GB of B2 traces, and the pinned runtime are deliberately not bundled.

The script does only two things.

1. Frozen replay: reconstruct the binary64 y64 from the stored A3 payload and B1's noise field, run the frozen Observer once, and compare against the official B2 raw output at object level.
2. Decimal re-evaluation: promote the same stored binary64 input to Decimal 50 / 100 / 200 digits and recompute only the period-label decision, the bracket candidates, and the downstream delta_3 arithmetic.

The script reads neither the truth nor the Scorer. It states its own scope explicitly: REFERENCE DIAGNOSTIC ONLY; NO FORENSIC RELIABILITY CLAIM at the top, and one fixed stored input; no cause or safety conclusion at the end.

Provenance of the output

The numbers come from the text file result_41014.txt (3,753 bytes, SHA-256 F2EDE3AF717889347A5DA224CC20AF9DCE90C5B6AE3200CB1F65A9EAD4FDA7CA).

Environment and input hashes at diagnostic time (as recorded in the same file):

These agree with the values sealed in B1 and B2.

i.4Diagnostic results (observed facts)

binary64 frozen replay

Decimal 50 / 100 / 200 digits

All three precisions gave:

Bracket values recorded at Decimal 100 / 200 digits:

b_2 = 0.502500000000000002220446049250313080847263336181640625
b_3 = 0.958225806451612871494916134906816296279430389404296875
b_4 = 0.99096774193548387010821443254826590418815612792968750

delta_3 = (b_3 - b_2) / (b_4 - b_3) = 13.91871921182264593805302584132463227397693395010716...

Final label

Only b_2 has moved far to the left of its proper position (about 0.8054 in the zero-noise case). The combination of a large numerator and a small denominator amplifies the ratio to roughly 13.9.

Incidental observation: the only unresolved point inside the ell=3 candidate bracket is grid index j=5941, the same index that appears in first_precursor_nodes in all 80 rows of Phase B2. This is a juxtaposition of observed facts, not a claim of causal connection.

i.5Limited interpretation

Proximate mechanism at the implementation level (descriptive)

delta_3 could reach ESTIMATE in the same run in which delta_2 abstained because the Observer decides each k independently, and only the local conditions required for delta_3 were satisfied (one bracket candidate each for ell=2,3,4, positive denominator). There was no guard requiring cross-series consistency — no coupling to whether delta_2 succeeded, no check that the bracket spacings form a plausible series, no re-check when the ratio falls far outside an expected range.

This is a description of the implementation, not a claim that adding such guards would improve anything. The effectiveness of any new guard is entirely untested.

On b_1=null

In run 74, b_hat_v[1] was unresolved, which directly explains delta_2's ABSTAIN. delta_3 does not use b_1, so it reached ESTIMATE on its own local conditions.

Whether the unresolved b_1 and the false b_2 acceptance arose from the same noise-induced label breakdown cannot be determined from the present materials. No common cause is asserted. This is recorded only as one instance in which cross-series consistency was not checked.

i.6What this note does not show

• The dynamical root cause of the noise: no analysis was performed tracing back to the individual observations to determine which values created the apparent 2→4 transition at v≈0.5025.
• The existence of a genuinely new bifurcation: it is not shown that noise induced a real period doubling in the underlying map. What was observed is a failure of the Observer's operational labelling of noisy finite observations.
• General causation: it is not shown that false acceptance necessarily occurs at sigma=1.6e-9, nor with what probability. All that follows is that it occurred for this one noise realization (seed 41014).
• Whether this is specific to a single realization: whether the same kind of false acceptance recurs with other seeds or other detectors is unverified.
• A defect of the Observer as a whole: no investigation was made of whether comparable failures occur at other seeds, other sigma values, or for other target quantities.
• Third-party auditability: the forensic framework carrying formal guarantees ended in FAIL / HOLD and was abandoned before the production run. The tests that passed inside it are not counted toward the credibility of this diagnostic. This diagnostic is a single script, a single execution, unaudited.

i.7Limitations of the diagnostic itself

• It starts from the stored binary64 input; it does not redo the generation of the observations at high precision. "High precision" here means downstream arithmetic on promoted stored binary64 input.
• The precision-stability decision (precision_paths_stable) consists of matching labels, matching candidates, matching status, and agreement of delta_3 to roughly the first 30 digits. It is not a check of exact agreement across 50/100/200 digits.
• The correspondence between noise fields and seeds is determined by array index (fields[seed - 41001]); it was not cross-checked against the manifest's seed ordering. That said, a wrong correspondence would immediately break the object-level match between the frozen replay and the official B2 output, so this functions as a practical consistency check.
• Executed once only. No repeated execution, no reproduction in a different environment, no verification by a different implementation.
• The output is standard output saved after the fact, not an artifact sealed at execution time.

i.8If a third party wishes to reproduce this

Passing the seed and sigma and regenerating the noise will not produce the same noise sequence under a different RNG lineage, so 13.9187 is not guaranteed. To attempt reproduction, one must be given not the seed but the following:

• the stored A3 payload (SHA-256 14D404ACC81690174682CA5B31C758A40B465DCD81104307E7A5652C6BD04D18)
• B1's noise fields (SHA-256 0EAB212E1EC8CC001DFBEF3CF8E5FAD4A15540754109B737FC0404E1861E2B3E) and the correspondence for the relevant seed
• the Observer source (SHA-256 A72C880213D617C9479D8F4C9DFD3E7F277169CBB236FB9C4EDFEE0B800D7EB4)
• runtime: CPython 3.13.15 / NumPy 2.3.2
• expected value: 13.918719211822673

d.1What happened

The subject is the following single case from Phase B2.

• condition: A3
• seed: 41014
• observation-noise scale: sigma = 1.6e-9
• estimand: delta_3
• Observer estimate: 13.918719211822673
• truth: 4.656251017651357
• Scorer classification: WRONG

What the noise was added to in this experiment was not the internal state of the logistic map. Writing y_0 for the noise-free generated observation series, the values handed to the Observer were

The map's parameter, its initial state, its internal states, and the truth were all left unchanged.

No new 2→4 bifurcation therefore occurred in the map itself. What did occur is that the noisy observation series actually handed to the Observer contained a local arrangement that the frozen Observer certified as a 2→4 boundary.

d.2Within the observation series, it was structure

The brackets the Observer adopted here were approximately

which yielded

The anomalous estimate did not erupt out of unrelated arithmetic runaway. There is a computational path: the Observer accepted a candidate well to the left as b_2, and the subsequent ratio computation greatly amplified that displacement in position.

In a replay using the stored binary64 input, the official raw result was reproduced. Furthermore, within the scope of promoting that input to Decimal 50, 100, and 200 digits and re-evaluating the downstream arithmetic, the period labels, candidate endpoints, and status did not change, and the numerical difference in delta_3 stayed at roughly 2.66e-14 in the trailing digits.

In this sense, for seed 41014's stored observation series, the local pattern the Observer certified was stable. The same input combined with the same decision rules reproduced the same structural certification and the same erroneous estimate.

d.3"False" and "real" are not in contradiction

This case shows the need to separate three layers.

The structure is therefore not a hallucination the machine conjured out of nothing. The Observer was reading an actual arrangement of input values. But the meaning it assigned to that arrangement did not correspond to the structure of the noise-free map that was supposed to be recovered.

One can put it this way: as a feature of the observation space it was real; as a correspondence to the estimand it was false.

d.4Stable reproduction does not guarantee truth

Ordinarily, if a result reproduces and the decision does not change as computational precision is raised, confidence in that result increases. Here, however, what was stable was not "recovery of the correct structure" but "the same structural certification for the same observation series".

That is, the following two are distinct.

1. That the decision path is computationally stable
2. That the certified structure corresponds correctly to what was to be recovered

For seed 41014, (1) was confirmed while (2) was denied by the Scorer.

This does not mean reproducibility should be discounted. It means the object that reproducibility guarantees has to be delimited precisely. Read the same input under the same rules, and it is not only correct certifications that reproduce — stable misidentifications reproduce too.

d.5What did the Observer "create"?

The Observer did not generate the observation values themselves. The noisy values were produced by combining the noise-free observation series with a noise field.

On the other hand, which local arrangement gets carved out as period-2, as period-4, or as a transition bracket is determined by the Observer's rules. It is therefore more accurate to regard the structure in this case as arising within the relation

rather than as belonging to either the observation series alone or the Observer alone.

The Observer did not fabricate values that were nowhere in the input. But it selected a pattern in the input that fit its local conditions, and certified that pattern as structure of the same kind as the recovery target. This "selection and assignment of meaning" is what led to the false acceptance.

d.6Local consistency and global consistency

The Observer here did check individual local conditions: that the required period labels exist, that bracket candidates are obtained, that the denominator of the ratio is positive, and so on.

At least along the path taken here, however, cross-series questions like the following were not among its decision conditions.

• Are b_2, b_3, b_4 consistent as a single bifurcation series?
• Is it acceptable to pass delta_3 alone as ESTIMATE when delta_2 is ABSTAIN?
• Is the resulting ratio extreme compared with the estimate at the adjacent level?

As a result, it is possible that no distinction was drawn between each part satisfying its local conditions and their combination constituting a meaningful finite-level structure.

This is not a result demonstrating the effectiveness of additional guards. Which consistency conditions ought to be introduced, and whether they would send correct estimates too readily to ABSTAIN, remain untested.

d.7What is currently unresolved

The following cannot be settled from this single incident.

• Whether the candidate substitution observed for seed 41014 is specific to that stored realization or recurs beyond the fixed fields examined so far
• How often comparable candidate substitutions occur at the same noise scale or near the same v; the fixed-seed records do not define a population probability
• Whether a different period detector, or an independent feature, would also detect structure at the same location
• Which local condition contributed most to the acceptance of the false b_2
• Whether a cross-series guard could safely turn this one WRONG into an ABSTAIN

In particular, repeatedly computing on the same stored input is a different thing from the phenomenon recurring under a different noise realization. The former was confirmed. A later pre-specified Phase D0 probe observed no additional delta_3 WRONG among 40 fixed fresh seeds at the same A3/sigma condition. A separate post-hoc reconstruction of those same inputs found no unique false-only ell=2 candidate state. Those are bounded fixed-set observations, not a recurrence probability or a safety result.

d.8Research memo: propagation and preservation under local perturbations

The motivating surprise is two-sided. It is not only that a very small perturbation could propagate so far. It is also that, across many other perturbed inputs, so much of the reference-associated structure was preserved well enough for ambiguity to remain visible.

This shifts the emphasis away from treating seed 41014 as the research object in itself. Seed 41014 is useful as a contrast case because it exposes two questions at once.

1. Propagation: how can a local threshold crossing alter a period label, change candidate identity, move a bracket far along the observation axis, and then be amplified by a ratio into a large final error?
2. Preservation: when local labels and candidate sets are being perturbed elsewhere, what preserves a reference-associated candidate or otherwise keeps the disturbance visible as nonuniqueness, so that the Observer can ABSTAIN rather than certify a unique alternative?

The fixed candidate-set comparisons make the second question concrete. In the original 20 fields at sigma=1.6e-9, 18 ell=2 cells retained a reference-associated candidate while adding one or more alternatives and becoming nonunique. Seed 41017 remained exact reference-only unique. Seed 41014 alone lost the reference-associated candidate and retained a unique non-reference alternative. In the later post-hoc reconstruction of the 40 D0 inputs, the reference-associated ell=2 candidate was retained in 40/40: 39 also contained alternatives and were nonunique, corresponding to all 39 delta_3 ABSTAIN outcomes, while seed 41042 remained exact reference-only unique and was RECOVERED.

These observations suggest a useful descriptive distinction:

This is not yet a general mechanism. The first line is a compact description of the observed ell=2 candidate states in the fixed comparisons, and the second line describes the stored endpoint for seed 41014. It does not show that every reference-retaining perturbation is safe, that every unique alternative is wrong, or that candidate identity alone determines the final decision at all levels.

The open question is therefore not simply "why did 41014 fail?" It is:

Equivalently, why did the disturbance propagate as far as it did in the incident path, and why was comparable disturbance absorbed or exposed as ambiguity in so many other paths?

The existing records narrow the question but do not answer it. The number of local label or predicate changes alone is insufficient: seed 41014 was not the most broadly perturbed field. Maximum movement or near-threshold counts alone are also not established separators. What may matter is the configuration — where a transition occurs, which period predicate changes, in which direction, and how those changes jointly map onto reference identity, alternative creation, and candidate-set cardinality.

For that reason, ABSTAIN is not merely a failed performance outcome in this memo. It is evidence that ambiguity remained observable to the decision rule. The object of interest becomes the transformation

This is a question prompted by the completed fixed studies, not an additional result or a proposal that has already been tested. It does not authorise added seeds, new sigma levels, a guard, or a causal claim.

d.9The central question this case raises

What matters in this case is not "did noise create a real bifurcation". The more direct question is:

In Phase A, under ideal conditions, the Observer's operational structure agreed with the truth. In B1, as the noise grew stronger, many decisions moved to ABSTAIN. In B2's seed 41014, the structural certification on the observation series remained stable while its correspondence with the truth came apart.

Taken together, the results give a concrete instance of the boundary lying between "structure being visible" and "having correctly recovered the structure of the target".

d.10Provisional summary

Seed 41014's noisy finite observation series contained a local transition-like pattern that the frozen Observer consistently certified as a 2→4 transition. That certification was stable under replay of the stored input and under downstream re-evaluation at Decimal 50, 100, and 200 digits.

That stability, however, did not mean the finite-level structure of the noise-free map had been correctly recovered. A pattern genuinely present in the observation series was certified by the Observer's local rules as a meaningful bifurcation series, and became a reproducible erroneous estimate by way of the ratio computation.

What this case suggests is not simply that the procedure "was weak against noise". It is that when structure is read out of finite observations, three things have to be handled separately: regular features present in the input, the operational structure the detector constructs, and the structure one wishes to claim about the target system. The later fixed-seed comparisons add a complementary question: not only how local perturbation can propagate into a stable misidentification, but how candidate identity and visible ambiguity can remain preserved despite widespread local disturbance.

This is, however, a discussion at the present stage, not a general conclusion that has passed through repetition at other seeds, confirmation by an independent detector, or evaluation of additional guards.