# [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](https://dev.opendata.ai/horizon), its predecessor at
[dev.opendata.ai/walls](https://dev.opendata.ai/walls), and the named test's
instrument at [dev.opendata.ai/rix](https://dev.opendata.ai/rix); the anchored
datasets live at [mldata.opendata.ai](https://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

| Read | Span | Count | Span π-count | Status |
|---|---|---|---|---|
| direct freeze at 2048 | (2048, 2304] | 33 | 33 | admitted, exact |
| half-born (N observer) | (2048, 2304] | 33 | 33 | admitted, exact |
| walked [1, 3072] ("2-via-3") | (2048, 2304] | 33 | 33 | admitted, exact |
| carrier [1, 3072] at refused 2560 | (2560, 2816] | 34 | 34 | rescued, exact |
| carrier [1, 2048] at refused 1024 ("1-via-2") | (1024, 1280] | 35 | 35 | rescued, exact |
| admitted far-forward onto ×5 ground | (5120, 5376] | 23 | 23 | admitted, exact |
| direct freeze at 1024 | (1024, 1280] | 139 | 35 | refused (collision) |
| direct freeze at 2560 (its own frontier) | (2560, 2816] | 141 | 34 | refused (collision) |
| 5120's own freeze | (5120, 5376] | 168 | 23 | refused — strongest on record |

---

## 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](https://mldata.opendata.ai/dataset/horizon-exp_6ec1e777)
  (115 rows at the campaign-7 close), with the predecessor campaign anchored at
  [walls-exp_19be23b1](https://mldata.opendata.ai/dataset/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](https://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](https://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.)*