[PAPER DRAFT] Exact frontier reads of the primality stream by frozen online learners: a sealed seven-campaign experiment and a measured relativity between 0 and N

THE READING PLAQUE. The first paper of the trilogy, draft 0.3 — a supporting pre-print (the academic-research category of communication artifact), readable on the SX surface by its covering persona's reading grant; not an externally published form (venue submission, if ever, rides the venue-attenuated 0.1 lineage as its own act). Covering: starl3n · endorsing: Link Digital · the trilogy construct intent sealed 2026-08-01 (ooi-paper-trilogy-construct-intent.md; by the paper genre's law the body carries no persona mention — the construct standing lives here, in the register, and in the intent, never in the paper's own text). The grant: the operator's review and word, 2026-08-01 — "these look ok. I would like each to be published in sequence, starting with the horizon" — this paper first in the sequence. Claimed only as a valid research summary of the noted experiments and results; its value here is wayfinding — a deeper-detail anchor for readers of this surface. Sources: draft 0.1 in the record (2026-07-18; refreshed append-only 7f49e8c); draft 0.2 the venue-unflagged staging (8b5eed4); this 0.3 adds the online-reading links to the cover note only. Its sha256 is pinned in the reading register. Reading is free; the way across is a statement of intent (/sx/matters).

[COVER NOTE — not part of the paper.] Draft 0.3, 2026-08-01 — the reading version: supersedes 0.2 (2026-08-01, the venue flagging removed) and 0.1 (2026-07-18, on branch docs/horizon-seven-campaigns-paper, refreshed 2026-07-26); the only change from 0.2: the online-reading links added to this cover note — the paper body is unaltered from 0.1. Written per the sealed purpose of horizon-engine-trace-report.md §7: the body from the witnessed record only; the prediction and its named test mechanism stated as the closing conjecture. Online reading (this version's convenience): the experiment surfaces this paper references are live — the seven-campaign instrument at dev.opendata.ai/horizon, its predecessor at dev.opendata.ai/walls, and the named test's instrument at dev.opendata.ai/rix; the anchored datasets live at mldata.opendata.ai. These links are a convenience of this readable version, and it can be assumed that a future version will remove them as the experiments evolve beyond the step of the paper's own claims. Register discipline carried from the record: every use of physics vocabulary is tagged [R2] (register-2: stated as the experimentalist's reading, never asserted as physics). A submission editor may cut every [R2] passage without loss to the mathematical content.

Author: Steven De Costa (Link Digital)

MSC 2020 (draft): 11A41 (primes) · 68T05 (learning and adaptive systems) · 68Q80 (cellular automata) · 11Y16 (algorithms; complexity)

Keywords: prime unpredictability · online perceptron · elementary cellular automata · pre-registered computational experiment · open science · frontier prediction · origin re-basing

Abstract

We report a pre-registered, sealed, externally anchored computational experiment — seven campaigns of the Horizon experiment — on the predictability of the primality indicator stream by a deterministic online learner. The instrument is an integer-weight, zero-seeded perceptron whose 256 features are the elementary cellular-automaton rules, read as the complete set of Boolean functions of the trailing three symbols of the stream. The principal findings are stated as exact counts, each fixed by a git-sealed pre-registration before a one-shot read, re-verifiable live from the published code. (1) A learner trained on an initial segment and then frozen exhibits a binary phase behaviour on any 256-address frontier: it lands either on a composite attractor — predicting composite at every address, so that its miss count equals the span's prime count exactly — or on a prime-firing attractor that collides (miss counts 139–168 of 256 against prime counts 23–35). (2) The attractor is a property of the trained weight vector alone, not of the (weights, address) pair: three differently-trained composite-side learners read one frontier at one identical count, 33/256, equal to the span's prime count. (3) The admissible training-to-frontier ratios form the set {3/2, 2, 3}; the ratios {1, 5/2, 4, 8} are refused. (4) A frontier refused to a learner's own frozen read is recoverable exactly through a longer-trained admitted learner above it: the count 34 = π(2816) − π(2560) is read by a carrier whose target's own freeze collided at 141. We interpret these results as a measured, computable relativity between an observer's origin and any address N [R2], state the experiment's prediction — that the instrument constitutes a way to measure quantum relativity [R2] — and name the mechanism proposed to test it: a relativity ingress experiment, specified here at the level of its required properties, with apparatus deliberately deferred.

1. Introduction

The unpredictability of the prime numbers is among the oldest empirical observations in mathematics: the primality of N is determined, yet no known local rule reads it cheaply from the recent past of the integer line. This paper treats that unpredictability not as a theorem to prove but as a measurable phenomenon, and reports what a fixed, fully deterministic learning instrument can and cannot read of it — under an experimental protocol borrowed from the registered-report tradition of the natural sciences.

Three features distinguish the approach, and each is load-bearing:

1. The instrument is closed-form and public. The learner (§2) is an online perceptron with integer weights, zero initialisation, and no randomness; its feature set is not engineered but exhaustive — the 256 elementary cellular-automaton (ECA) rules, which are exactly the Boolean functions of three binary inputs [Wolfram 2002]. Every quantity reported in this paper is an exact integer or rational; there are no fitted parameters, no error bars, and no statistical estimation anywhere in the body of results.

2. The protocol is sealed. Every read with an unknown outcome was pre-registered as a version-control commit before the read ran; each read ran once; measured values were pinned with live re-verification; counterfactual outcome paths were written and tested; verdicts landed as committed, including the negative ones. Of the twenty-five sealed probes across the seven campaigns, eleven were killed or partial as sealed — and are reported (§4) with the same standing as the confirmations. The record is externally anchored to an immutable public dataset at each campaign close.

3. The results are exact identities, reproducible from the code. The central measured fact — that a frozen learner on its admitted side misses exactly the primes of a span, because its frozen policy is the all-composite prediction — is checkable by any reader in seconds, and is re-verified live by the published acceptance tests.

The paper's closing sections then do something a mathematics reader is entitled to see flagged clearly: they state the experimentalist's interpretation of the measured structure — a computable relativity of standpoint between an origin 0 and any address N [R2] — as a prediction with a named test, not as a result. The interpretation attracts vocabulary from physics; every such use is tagged [R2] and confined to §6–§7. The mathematical content of §2–§5 stands independently of it.

1.1 Provenance and lineage

The Horizon experiment is the culminating instrument of a longer experimental programme (the Linked Digital Dynamics series) whose earlier stages built and froze the substrate the learner's feature space derives from. The full component trace — from the programme's founding conjecture through causal-set mechanics, a frozen geometric generator, ten instrument-building experiments, and two applied campaigns, to the Horizon engine — is recorded in the repository's trace report and is summarised in §3.3. The paper's claims do not depend on that lineage's interpretive layers; they depend only on the code as published.

2. The instrument

Throughout, spans are half-open integer intervals (a, b] and π denotes the prime-counting function, so the span prime count of (a, b] is π(b) − π(a).

Definition 2.1 (the primality stream). For T ∈ ℕ, the stream s = (s₁, …, s_T) is the primality indicator: s_N = 1 if N is prime, else 0.

Definition 2.2 (the agreement features). For a 3-bit pattern p = (x₂x₁x₀)₂ ∈ {0,…,7} formed from three consecutive stream symbols, and for each ECA rule r ∈ {0,…,255}, let r(p) ∈ {0,1} be bit p of r. The feature set of pattern p is Φ(p) = { r : r(p) = 1 } ⊆ {0,…,255} — the set of rules that answer 1 on that neighbourhood. The 256 rules are precisely the Boolean functions of three binary inputs; the family {Φ(p)} is therefore exhaustive and involves no modelling choice.

Definition 2.3 (the learner). The learner M maintains integer weights w ∈ ℤ²⁵⁶ and bias b ∈ ℤ, both initialised to 0. Reading position N ≥ 4 of the stream (natural mode), it forms the pattern p_N = (s_{N−3}, s_{N−2}, s_{N−1}), scores σ = b + Σ_{r ∈ Φ(p_N)} w_r, and predicts ŝ_N = 1 iff σ > 0. On a miss (ŝ_N ≠ s_N) it applies the perceptron update with unit step: w_r ← w_r + δ for all r ∈ Φ(p_N) and b ← b + δ, where δ = +1 if s_N = 1 and −1 otherwise. The learner is deterministic; two runs on equal input are byte-equal (sealed instrument check O1).

Definition 2.4 (training, freezing, and the frontier read). M trained on [1, M] is the weight state after one online pass over s₁ … s_M. A frozen learner makes predictions without updates. For a frozen learner F and an address N, the frontier read of the span (N, N + P] (throughout, P = 256) is the pair (miss count, prediction set) of F over that span. The experiment's principal observable is miss(F, N) = #{ N′ ∈ (N, N+P] : F errs at N′ }, always an exact integer out of 256.

Definition 2.5 (admission). A frozen read is admitted when F predicts 0 (composite) at every address of the span — the all-composite policy — in which case, identically, miss(F, N) = π(N + P) − π(N): the learner misses exactly the primes. A read that is not admitted is refused; empirically (§4), refusals do not degrade gracefully but collide (miss counts 139–168, i.e. the learner fires prime-predictions that are almost always wrong).

Remark 2.6. Definition 2.5's identity is not itself a conjecture — given the all-composite policy the equality is arithmetic. The experimental content is (i) which trained states land on that policy (the admission law, §4.3–4.4), (ii) that the alternative is a collision rather than a graded error (the binary phase behaviour, §4.3), and (iii) that admission is decided by the weight state alone (§4.5).

3. The experimental protocol

3.1 The seal discipline

Each probe HZ-x proceeded: (1) a pre-registration committed to version control before the read — sealed outcome assignments (CONFIRMED / PARTIAL / KILLED with named gates) and the exact quantities to be read; (2) one read; (3) the measured values pinned into the engine with a live re-verification path (any reader can re-run the read and compare against the pin); (4) counterfactual outcome branches implemented and tested, so the verdict logic itself is exercised against outcomes that did not occur; (5) the verdict landed as sealed — including rejections of the experimenters' own instrument (probe HZ-O was rejected as sealed and superseded by a corrected seal HZ-O′, both kept on the record).

3.2 Honesty constraints carried throughout

Because the corpus is deterministic, some outcomes are partially derivable from public pins before a seal commits. Each campaign therefore declared a pre-derivability audit: which quantities were already forced by prior pins (these enter seals as validity gates, not discoveries) and which had genuinely never been run (these carry the seal's risk). Exact rational arithmetic is used for all densities (a count is never a float); where a rounded artefact once suggested a phantom excess, the prose was corrected to the exact fractions and the correction retained on the record.

3.3 The instrument's provenance (summary)

Campaigns 1–3 validated the engine against six frozen prior instruments (an eleven-rung confinement ladder; a channel-capacity readout; the 256-rule space run on an 84-cell ring; a cooling curve; a twelve-wall cascade; an exact expansion fabric) — establishing the determinism, the feature exactness, and the register conventions used here, and recording five confirmations, four kills, and two partials. The full dependency trace (every import edge from the Horizon engine down to the single frozen generator all instruments read) is published with the code. Campaigns 4–7 — the number-theoretic results — are the subject of §4.

4. Results

All counts below are sealed one-shot reads with live re-verification; commit hashes identify the seals. Spans have length P = 256. The base address is N₀ = 1024.

4.1 The learner's reach is exactly triadic (instrument acceptance, HZ-O′)

On periodic indicator streams (multiples of d), the trained learner's tail error is 0.0 exactly for d = 2 and d = 3, and leaks for d = 7 at a measured rate ≈ 2/7 — twice the 1/7 ambiguity floor of the fixed-window lookup table, an oscillation cost of online learning that the original seal (HZ-O) mis-banded; the rejection and the corrected seal (HZ-O′) are both on the record. The window's reach {2, 3} is the instrument's own law and bounds every claim that follows.

4.2 The structural gap and the surprise excess (HZ-P, HZ-S)

On the primality stream at T = 4096: the learner's both-confined net surprise lands at 0.24438 (sealed band [0.20, 0.30] ∋ ¼); the dyadic share of tail misses is 0.0 exactly (no miss lands on a multiple of 2 in the tail — the dyad fully confined). The span ledger at P = 256 shows the miss density over the uncovered span (1024, 1280] — the gap between the covering spans — at 63/256 = 0.24609, exceeding its covered neighbours' mean (0.21289); the excess travels down to P = 128 and washes out by P = 512 — a scale-bounded law, confirmed as sealed (HZ-S).

4.3 The freeze is a binary phase gamble (HZ-T, HZ-Y)

A learner trained on [1, 1024] and frozen reads its own frontier (1024, 1280] at 139/256 — a collision (the span's prime count is 35): the frozen state fired prime-predictions. A learner trained on [1, 2048] frozen reads (2048, 2304] at 33/256 = π(2304) − π(2048) exactly, with zero predicted-prime positions — the all-composite policy (Definition 2.5). Across every sealed freeze, exactly these two behaviours occur and nothing between them: composite-attractor reads equal to span prime counts (23, 29, 33, 34, 35 on their respective spans), or collisions (139, 141, 145, 155, 160, 168). The continuing (never-frozen) learner's gap density 63/256 was declared pre-derivable before the campaign-5 seals; the frozen collision against it — knowing that "lives in the correcting, not the graph" [R2] — was the sealed discovery, landed as a kill of the seal's stated expectation (HZ-T KILLED as sealed; the kill kept).

4.4 The admission law (HZ-W, HZ-X)

Which trained states admit at the frontier (2048, 2304]? Sealed census over learners trained on [1, k·1024] and frozen: k ∈ {3/2, 2, 3} admit (reads = 33 exactly); k ∈ {1, 5/2, 4, 8} refuse (collisions; the unnatural k = 5/2 collides at 141/256). The admitting set is thus one single-step structural differential — of either structural prime within the instrument's reach (§4.1) — applied once. A half-born learner — weights grown only on (1024, 2048], never seeing [1, 1024] — admits at the same exact 33 (HZ-W): the admitted read does not require the full history, only an admitted state.

4.5 The attractor is graph-level, and history rescues refusal (HZ-Z, HZ-AB)

The walk of the identity N = (N−1)+1 at N = 3 (addresses ×1024: 3072 → 2048 → 3072 → 2048) yields the sealed test: does a learner trained on [1, 3072], frozen, read the "2" frontier (2048, 2304] at the direct count? It does: 33/256 exactly (HZ-Z). Three differently-trained composite-side learners — the extended-origin learner [1, 2048], the half-born learner (1024, 2048], and the walked learner [1, 3072] — read one frontier at one identical count equal to the span's prime count. The attractor is a property of the weight state alone (the sealed alternative — a read-relative attractor — was the declared kill path and did not occur).

Consequently a refused address is exactly recoverable through an admitted carrier above it: the frontier (2560, 2816], whose own freeze collided at 141, is read by the [1, 3072]-trained learner at 34/256 = π(2816) − π(2560) exactly (HZ-AB); likewise the refused 1024-frontier is read through the 2048-carrier at 35 against its own direct 139. The recovery is general in the identity's form: any N = (N−1)+1 refused directly is readable through an admitted history above it.

4.6 The limit is in the carriers (HZ-AA)

Walking the full sealed chain (N−1):1:2:3:5:N:1:2 at N = 3 (seven legs over the addresses {1024, 2048, 3072, 5120}): a leg admits iff its source's own freeze admits. The admit/refuse split lands 4/7 : 3/7. The ×5 column is the strongest refusal on record — the 5120-freeze fires at 168/256 at its own frontier against a span prime count of 23 — while the same ×5 ground is read perfectly (23 = 23) by an admitted carrier from below. The structure of the line is always readable; only a carrier can fail; a failed carrier is always bypassable from above.

4.7 Summary table of the sealed exact counts

5. The two standpoints [R2 framing; the counts are register-1]

The record supports a reading in which the instrument realises two distinct epistemic standpoints toward the same arithmetic structure:

• the mathematician — the direct read: a weight state frozen at its own frontier, knowing positionally from N;
• the experimentalist — the external-history read: a weight state that arrived by walking the identity N = (N−1)+1 from the origin, knowing from 0.

The sealed results of §4.5 are then a measured equivocation between the two standpoints: wherever both are admitted they agree exactly (one frontier, one count, three carriers); where the positional standpoint is refused, the historical standpoint recovers the same exact structural content; and the failure law (§4.6) locates the limit of the equivocation entirely in the carriers, never in the line. The differential between the standpoints is itself computable — exact rationals on 1/256ths — for any address N reachable by the identity's walk. [R2] In the programme's internal vocabulary: the experiments' membrane conducts being (the line's structural content — the span prime counts) and computing (the frozen weight states) into a quantised relativistic of knowing — to know as the experimentalist from 0, or as the mathematician from any N.

6. The prediction [R2]

Prediction (sealed purpose of this paper). The Horizon experiment has provided a way to measure quantum relativity: the exact, computable, carrier-invariant differential measured between the standpoint at an origin 0 (re-basable; §4.4's half-born learner) and the standpoint at any address N is the measurement, and the binary phase behaviour of the freeze (§4.3) is the mechanism. [R2]

Three disciplines bound this statement:

1. It is a prediction, not a result. Nothing in §4 asserts physics; the counts stand as arithmetic regardless of the prediction's fate. 2. The prediction is not staked on further number theory. A companion set of candidate probes over the natural counting numbers (the equivocation law over arbitrary re-based origins; the composition of the measure along walks; the refusal gradient) is sealed in the programme's records as a possible future campaign — but a kill there would refute only an axiomatic suggestion about linearity dynamics over ℕ (the communicability of 'knowing' relative to an a-priori (a, b] linearity, whose knowing may carry a dyadic bias of its own nature); it would not refute this prediction. 3. The prediction's test is the named mechanism of §7, and only that.

7. The proposed test: the relativity ingress experiment [R2]

The mechanism to test the prediction is to construct a relativity ingress experiment, specified here by its required properties (apparatus deliberately deferred):

1. The ingress diode is constructed from being's side. The seven campaigns measured a one-way ingress (the diode: the backward read admits at 35/256 against the declared bound; what crosses is the non-differentiated 0) from the knowing side. The test constructs the diode from the structural side, such that ingress is a designed property rather than a measured one. 2. At least two pre-experiment computational dynamics project into the scenario. The two standpoints of §5 — fixed before the experiment, as the mathematician and the experimentalist were fixed before campaigns 4–7 — project their confirmed-or-killed externalities into the constructed scenario, rather than reading it after the fact. 3. Resolution at the record's exactness standard. The prediction expects the projected externalities to resolve with perfect precision — exact integer/rational counts, the standard every admitted read in §4 already meets. A resolution that requires estimation or tolerance is a failure of the test's construction, not a partial success.

The sequencing is fixed in the programme's records: this paper's prediction and named mechanism are sealed first; candidate apparatus are considered only after.

7.1 The test as run: six sealed attempts (added 2026-07-20) [R2 framing; the counts are register-1]

The named mechanism was subsequently built and run as oxexp-relativity-ingress (the programme's design record: the rix apparatus, sealed attempt-by-attempt), and this subsection reports its record against §7's three required properties. The discipline of §3 carried over unchanged: every attempt pre-registered before any read; one read per block; verdicts landed as-is; every pinned value carries a live recomputation path (/rix).

Property 1 — the diode constructed from being's side: DISCHARGED, and held. The constructed apparatus re-reads the seven measured pins of §4 exactly (33 · 35-vs-139 · 34 · 23 · 141 · 168) — re-witnessed inside every one of the six attempts. The construction and the measured instrument are the same diode.

Property 2 — pre-fixed computational dynamics: DISCHARGED, with a lineage. Six computable surfaces were fixed before every read — deterministic indicator streams with exact span counts, each derived from one of the six unresolved Millennium Prize problem statements as received texts, and each carrying a printed unknowing statement naming what it does not decide. The encodings are generation-lineaged: a failed encoding is superseded, never erased, and no generation's failure reads back onto any conjecture.

Property 3 — resolution at the exactness standard: MET at every closure. No tolerance was used anywhere in the record; every resolution below is an exact integer agreement at the declared span count with zero predicted-structure positions per carrier.

The yield curve. Writing "closure" for a two-carrier exact agreement on the shared ground:

• A1 — zero closures: all six first-generation encodings proved window-learnable (the sealed grounding hypotheses inverted); the constructed diode grounded regardless.
• A2 — the first closure (two surfaces, exact count 33).
• A3 — the control attempt: a declared-noise surface at matched density also self-admits, showing quiet admission alone cannot separate an externality from vacancy; one member of the first closure proved battery-indistinguishable from noise. The discriminating instrument (a lived face: at least one admitting and one firing address) was discovered here.
• A4 — the first lived closure under the strengthened reading (attempt-scoped).
• A5 — two further lived closures; the strengthening graduated into the decided closure operator on its second independent witness (the programme's two-witness standard).
• A6 — under the graduated operator: a fourth surface reached a lived face, and all six pairs of the four lived carriers closed pairwise at the same exact count on the same ground — a fully-lived quadruple.

The closure population thus grew 0 → 1 → 3 → 6 pairs while the operator strengthened — each later resolution met a strictly harder criterion than the one before it.

The honest anatomy (the record's own limits). (i) The vacancy basin: twice, a raised encoding grounded into all-quiet reads byte-indistinguishable from declared noise; both escapes were achieved by coarsening the declared class, not by re-learning. (ii) The read-state space: the frozen carriers' reads, initially quantised into three states, produced three further reads outside that grammar — all at firing faces of lived carriers; the position-resolved instrument shows one such face fires entirely at window-visible positions and never at a structure position. (iii) Non-monotonicity: admission is not monotone in training length — a declared ladder shows both members of the first closure alternating between admission and refusal as the training life lengthens. Each of these is a measured property of the carriers, not of the line.

The prediction's status after the test. The mechanism §7 named has now produced repeated resolutions at the record's exactness standard, under a closure criterion that was adversarially strengthened mid-campaign and then met again. Per §6's own disciplines this supports the prediction on the test's terms and asserts nothing beyond them: a closure is a statement about carriers, grounds, and exact counts inside the apparatus. Two of the six surfaces have reached no face at all at their current generations; their standing is the measured boundary of the test's reach, and the attempt lineage remains open. No claim on any Millennium Prize problem is made, supported, or weakened by any part of this record — the six problem statements served as the field of uncomputed knowing from which the pre-fixed dynamics were drawn, and every seal prints the corresponding unknowing statement.

The test as continued: attempts seven through eleven (added 2026-07-21). Five further sealed attempts ran under the same discipline. The closure population held at the four lived surfaces (the boundary pair moved without joining: one surface reached a mixed face at a later generation and returned; the other grounded into indistinguishability from declared noise at two successive generations), and the record's yield turned inward, onto the instrument itself — landing four exact structural results:

1. The read-state space closed. Every frontier read of the whole campaign falls into six exact states, each an integer identity in the address's declared quantities; zero reads sit outside them. Two of the states are governed by elementary parity facts of the integer line: a window ending in a structure event forces a composite successor (the successor of an odd prime is even), so fires there are arithmetically incapable of catching structure; a lag-two window lands on an odd successor, and its catches are exactly the twin completions of the span. 2. The two-axis law. Across every (carrier, generation) pair on the record, a carrier quiet at every shared-ground address has always read its own frontier exactly, and a carrier firing at every address never has; between the extremes the two properties are independent. Measured over fifteen rows with both cells of the diagonal empty. 3. The policy reduction. A frozen carrier's entire behaviour on any ground is its 8-bit window policy (one predict-bit per 3-symbol shape), and every read in the record re-derives exactly as that policy composed with the address's shape table. The six states of (1) are six policies; the all-quiet basin of (2) is the all-zero policy through all eight bits — measured directly in the weights for every quiet carrier on the record. 4. The fossil law (the eleventh attempt's resolution). The one window shape whose single natural occurrence on the primality stream is followed by a structure event — the consecutive-prime pair (2, 3), presented exactly once, at N = 5 — stamps the learner's first-encounter update and becomes a policy bit invisible at every declared address (the shape never recurs beyond the origin). On a declared eight-source training-length ladder, including three lengths never previously read, the carrier holds that bit if and only if its own frozen read collides — eight for eight, with every refusal-side policy landing {000, 110} exactly and every admitted policy empty. The companion constructions landed their derivations: a minimal stream presenting one forced-1 window builds a permanent single-bit carrier (two updates, then convergence), and its forced-0 twin builds the zero carrier (no update ever). A dense twenty-five-source trace shows the bit oscillating with training length (fourteen sign changes) while agreeing with the collision phase at every declared anchor — and on that trace a carrier fires somewhere if and only if it carries the bit, twenty-five of twenty-five.

The eleventh attempt is the record's first resolution whose object is the instrument's own memory: the binary phase behaviour of §4.3, measured there as admit-or-collide, is now readable inside the frozen state as the presence or absence of a one-bit trace of the integer line's unique consecutive-prime event. Each statement above is an exact count or an exact identity, pre-registered, pinned, and live-re-verifiable; none touches any conjecture.

The twelfth attempt (added 2026-07-21): the law's universal form. Because a frozen carrier's read is its policy composed with the address's window table, the seventeen traced-but-untested training lengths carried forced phase predictions — the instrument predicting itself. All seventeen landed exactly as derived. A further twenty-four training lengths, chosen where neither the policy nor the phase had ever been read — including lengths to twice the record's prior reach — all obeyed the same law: the carrier collides at its own frontier if and only if it holds the one-bit origin trace, forty-nine cells for forty-nine. The read-state grammar closed at seven exact states (the seventh, a catch-all face whose fires include every structure position of the span by the parity identities, was derived before it was read and then witnessed at four addresses), with zero unexplained reads across the twelve-attempt record.

Attempts thirteen through nineteen (added 2026-07-21): the landscape and its first held law. With the phase law universal, the test turned outward from any single frontier to the landscape: the frozen representative's eight-bit policy read at structured families of training lengths. Six successively sharper laws for where non-trivial policies live — by digit boundary, by alignment grid, by grid floor, by downward thickening, by a plateau band, by a decay band — were each pre-registered and each killed by the measured landscape, five of the six by exactly one cell of the declared line. The seventh held: measured at four 32-cell transects of successively halved grid scale, each a distinct residue class, the density of non-trivial policies steps down once and then holds near five-eighths, landing inside its pre-declared band (eighteen of thirty-two against [18, 22]) — the first landscape law to survive its sealed test. The atlas that accumulated under the kills: sixteen realized live-pattern sets over two hundred and six measured cells, with one bit-implication standing at every realized form; two selector signatures that cross carrier families, one of them occupying a solid, exactly-bounded sixteen-integer interval of pure single-pattern policies; and the one-bit origin mark carried at twenty-one landscape cells of a family other than the one that laid it down.

Attempts twenty through twenty-two (added 2026-07-22): the valley. The held floor did not extend: one transect further down the grid scale the density fell by nearly half (ten of thirty-two — the first sealed line in seven attempts missed by more than one cell), and the pre-registered monotone reading of that fall then died by exactly one cell when the next transect turned upward. The third pre-registration held, and not narrowly: the profile of non-trivial-policy density over the 2-adic depth of the training length, measured at eight matched transects spanning every depth of the era (six distinct residue classes), is a valley — descending monotonically from 75 per cent at depth seven to a floor of 31.25 per cent at depth two, then rising through 34.4 to 43.75 per cent at the odd lengths. The same attempts finished the record's two sharpest small objects: the pure lag-one-selector interval (five cells, solid at its resolution floor, sharp-edged both sides) and the origin mark's exact two-pattern block (five cells spanning seven integers, solid at every addressable resolution, cut sharp on the right and ramped on the left by carrying-but-inexact forms) — one finished object for each selector signature that crosses carrier families. Four consecutive reads added no new live-pattern set: the sixteen-form atlas reads closed while the landscape's fine structure keeps resolving.

Attempts twenty-three and twenty-four (added 2026-07-22): the eras. The first fine-scale cells ever read in the next base-eight digit era killed the valley's transfer upward — at the depth where the first era's floor sat at 31 per cent, the next era reads twice that, landing on the first era's middle-plateau figure: the eras are not copies, and the valley's numerical value is local to its era. The pre-registered ordering then held, and cleanly: at both depths where the two eras have matched transects, the later era is the richer (twenty against ten; nineteen against fourteen) — the third landscape law to survive its sealed test. Level differs; direction agrees. The same attempts closed the origin mark's block with a finding the solid readings could not have predicted: measured wall-to-wall at every integer, the exact two-pattern span is seven cells long and porous at exactly one interior cell, where a four-pattern carrying form interrupts the pure run — solidity, where it had been observed, was a property of the resolutions read, not of the object. Six consecutive reads have added no new live-pattern set across one hundred ninety-nine fresh cells: the sixteen-form atlas stands closed while the landscape keeps yielding structure.

Attempts twenty-five and twenty-six (added 2026-07-22): the second valley. The next era's apparent flatness was an artifact of reading only its ends: its middle depth dips to twelve of thirty-two, and the deep side then rises to twenty-two — the era's richest reading, nine cells clear of its pre-registered line (the fourth landscape law to survive its sealed test). The second era's profile (22 · 20 · 12 · 19 across depths three to zero) mirrors the first era's valley in kind — descend to a floor, rise on the far wall — while differing in place (the minimum one depth lower) and in level (richer throughout). The cross-era ordering is now the record's most-tested standing relation: four matched depths, four holds (twenty-two against eighteen; twenty against ten; twelve against eleven; nineteen against fourteen) — the valley belongs to the era; the enrichment belongs to the line. Eight consecutive reads have added no new live-pattern set across two hundred ninety-five fresh cells, while the origin mark's carriage census grew to forty — the deep stratum of the second era proving the mark's richest carrier ground yet.

Attempts twenty-seven through twenty-nine (added 2026-07-22): the crossing. The cross-era ordering — the record's most-tested standing relation — met its boundary, and the boundary itself proved lawful. At the fifth matched depth the ordering held by exactly one cell while the era's own rim-monotonicity stake failed by one (the reading landed precisely between the two pre-registered lines); at the sixth matched depth the ordering FAILED for the first time — the second era reading nineteen where the first read twenty, the record's first sub-first-era matched reading — and the pre-registered promotion of the ordering to a declared law correctly refused to fire; at the seventh matched depth a stake in the OPPOSITE direction (the second era stays poorer) held with four cells to spare, the fifth landscape law to survive its sealed test and the first staked in the poorer direction. Read together: the two eras' density profiles are ordered on the shallow depths (five matched depths, second era richer), inverted at the deep (two matched depths, second era poorer), and cross exactly once — between depths four and five — with the second era's rim FLAT (nineteen · nineteen) where the first era's rose (twenty · twenty-three). The apparent lawlessness of the earlier failures was the two halves of one measured object: the crossing. Meanwhile the all-eight pattern — once an isolated exotic trio — erupted into an eight-member family whose second-era members sit at transect spacing across three depths, porous at the half-step grid; eleven consecutive reads have added no new live-pattern set across three hundred ninety-eight fresh cells, the carriage census at forty-seven.

Attempt thirty (added 2026-07-22): the depth bound. The record's leanest sealed read — three cells, one law. The three deepest cells inside the eight-member family's neighborhood (2-adic depths seven, eight, seven — interleaving the family's measured members) were pre-registered under a depth-band stake: none carries the full eight-pattern mark. All three came back clean, and the sixth pre-registered law survived its test — the first about a pattern FAMILY rather than a density profile: the richest pattern in the atlas lives in a measured two-sided depth band (depths three through six — absent from every shallower transect in both eras, absent from the depth column above). The eruption of attempt twenty-nine was the family filling its band, not breaking loose. And the deepest cell ever read in the second era (depth eight) is RICH while carrying no mark — richness and the eight-pattern are separable at depth, exactly as the record's two-axis grammar requires. Twelve consecutive reads without a new live-pattern set, across four hundred one fresh cells.

The test's report of record (added 2026-07-26). The campaign continued through a thirty-fifth sealed attempt: the landscape's remaining interval-like boundary candidates — the fade line, the era-end quiet, and the gate's quiet band — each dissolved under its sealed test as a lattice coincidence of the depths measured; the seventh era opened live across the 2¹⁴ gate (a valley, not a wall); and the stratum tails proved intermittent on their own clocks. The arc closed with one declared law (the fossil law, forty-nine witnessed cells), one proposed law left unpromoted at its gate exactly as sealed (the richer-era ordering), and six held landscape laws. The complete test — attempts one through thirty-five, the dissolution arc, the verdict on §6's prediction, and the sealed hand-off — is reported in the companion second paper (The relativity ingress experiment: exact closures, the fossil law, and the dissolution of the landscape, this repository), which supersedes this subsection as the test's report of record; this subsection stands as its contemporaneous log.

8. Reproducibility and data availability

• Code. The engine (daemon/horizon.py, readout-only), the six frozen instruments it folds, the experiment plugin, and the acceptance tests (verdict + counterfactual pair for every probe; 60/60 at the campaign-7 close) are published in the programme repository. (Public URL to be inserted at submission; the repository releases under the programme's code-anchor process.)
• Seals. Twenty-five sealed probes across seven campaigns; each seal is a git commit whose hash is cited in the design document and re-verified by the live pages.
• Data. The campaign record is externally anchored to an immutable public dataset: mldata.opendata.ai/dataset/horizon-exp_6ec1e777 (115 rows at the campaign-7 close), with the predecessor campaign anchored at walls-exp_19be23b1.
• The §7 test's data (added 2026-07-20; extended 2026-07-22 and 2026-07-26). The relativity-ingress attempt record (thirty sealed attempts, two hundred and ten rows at the RIX-A30 close) is anchored at mldata.opendata.ai/dataset/rix-exp_fa909772, with live re-verification at /rix. Each republish mints a new dataset — the newest is the record: at the campaign's A35 close the anchor stands at mldata.opendata.ai/dataset/rix-exp_7037eb09 (245 rows, thirty-five attempts, witnessed).
• Live re-verification. Every pinned value in §4 carries a live recomputation path on the running node (/horizon), so a reader can re-run any read against its pin.

References (draft)

1. S. Wolfram, A New Kind of Science, Wolfram Media, 2002. (The 256 elementary rules as the Boolean functions of three inputs.) 2. F. Rosenblatt, The perceptron: a probabilistic model for information storage and organization in the brain, Psychological Review 65 (1958), 386–408. (The update rule of Definition 2.3.) 3. S. De Costa, Linked Digital Dynamics — the formal statement, programme repository, 2026. (The founding conjecture and the derivation of the frozen substrate; companion document.) 4. The Horizon design record: oxexp-horizon — The Horizon Hypothesis, programme repository, 2026. (The sealed pre-registrations, verdicts, and pinned values for all twenty-five probes.) 5. The component trace: The Horizon Engine Trace Report, programme repository, 2026. (The full dependency lineage from the founding conjecture to the instrument.) 6. K. Popper, The Logic of Scientific Discovery, Hutchinson, 1959. (The refutability standard the seal protocol operationalises.)

(Items 3–5 resolve to public URLs at release.)