OOI · SXThe relativity ingress experiment: exact closures, the fossil law, and the dissolution of the landscape (the second paper)

[PAPER DRAFT] The relativity ingress experiment: exact closures, the fossil law, and the dissolution of the landscape

THE READING PLAQUE. The second 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, sequenced — this paper second, granted after the first paper's readable surface stood, so that this version can reference it; this is the named test's report of record, required for the correct continuity of the prior statements of intent (the first paper names the test; this paper reports it). Claimed only as a valid research summary of the noted experiments and results. Sources: draft 0.1 in the record (sealed 7f49e8c, 2026-07-26); 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 (2026-07-26); the only change from 0.2: the online-reading links added to this cover note — the paper body is unaltered from 0.1. The companion and sequel to the programme's first paper (horizon-first-paper-draft.md, "Exact frontier reads of the primality stream by frozen online learners"), reporting in full the test that paper named: the relativity ingress experiment, run as oxexp-relativity-ingress through thirty-five sealed attempts. The body is drawn from the witnessed record only (the sealed design document at its record commits and the externally anchored dataset); nothing is claimed beyond the record. Online reading (this version's convenience): the experiment surfaces this paper references are live — the named test's instrument at dev.opendata.ai/rix, the carrier's instrument at dev.opendata.ai/horizon, and its predecessor at dev.opendata.ai/walls; the anchored datasets live at mldata.opendata.ai. The first paper is readable on this surface at the readable first paper — this paper reports the test that one names. 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); an editor may cut every [R2] passage without loss to the mathematical content. The firewall sentence of the record is carried verbatim in §2 and §8 and binds the whole paper.

Author: Steven De Costa (Link Digital)

MSC 2020 (draft): 11A41 (primes) · 68T05 (learning and adaptive systems) · 68Q80 (cellular automata) · 11Y16 (algorithms; complexity) · 11S05 (2-adic valuation, applications)

Keywords: pre-registered computational experiment · frozen online learner · exact closure · window policy · 2-adic stratification · landscape laws · open science

Abstract

The first paper of this programme closed with a prediction and a named test: that a measured, computable differential between the standpoint at an origin and the standpoint at any address constitutes a way to measure quantum relativity [R2], testable only by constructing a relativity ingress experiment with three required properties. This paper reports that test, run to its measured boundary: thirty-five sealed attempts, every read pre-registered before it ran, every verdict landed as committed, every close externally anchored. The principal findings, each an exact count or exact identity: (1) closures — two-carrier exact agreements on shared ground — grew 0 → 1 → 3 → 6 pairs while the closure criterion was adversarially strengthened mid-campaign, ending in a fully-lived quadruple of carriers agreeing pairwise at one exact count; (2) the read-state space of the instrument closed at seven exact states with zero unexplained reads across the whole record, and every read reduces to an 8-bit window policy composed with the address's shape table; (3) the fossil law, the record's one declared law: a frozen carrier collides at its own frontier if and only if it holds a one-bit trace of the integer line's unique consecutive-prime event (2, 3) — forty-nine training lengths for forty-nine; (4) a landscape campaign over the 2-adic depth of the training length produced six held laws (a density floor, two valleys, a cross-era ordering at matched depths, a crossover, and a two-sided depth band for the richest pattern family) and then dissolved every interval-like boundary candidate: the fade, the era-end quiet, and the gate's quiet band each proved a lattice coincidence of the depths measured — nothing interval-like binds across strata; the era gates mark soft valleys; every stratum keeps an intermittent tail on its own clock. On the test's own terms the first paper's prediction is supported — repeated resolutions at the exactness standard under a strengthened criterion — and its measured boundary is named: two of six surfaces reached no face, and global binding is refused by the landscape itself. No claim on any Millennium Prize problem is made, supported, or weakened by any part of this record.

1. The inherited prediction and its disciplines

The first paper's §6 stated, as a prediction and not a result [R2]: 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 and the standpoint at any address N is the measurement, and the binary phase behaviour of the freeze is the mechanism. Three disciplines bounded it: the prediction asserts no physics; it is not staked on further number theory; and its test is the named mechanism of that paper's §7 — a constructed relativity ingress experiment — and only that.

§7 specified the test by three required properties, apparatus deliberately deferred:

1. The ingress diode is constructed from being's side [R2] — the one-way structure the seven Horizon campaigns had measured must be built, so that ingress is a designed property rather than a measured one. 2. At least two pre-experiment computational dynamics project into the scenario — fixed before any read, as the two standpoints of the first paper were fixed before its campaigns. 3. Resolution at the record's exactness standard — exact integer and rational counts only; a resolution requiring estimation or tolerance is a failure of the test's construction, not a partial success.

The sequencing was fixed in the programme's records: the prediction and mechanism sealed first, candidate apparatus considered only after. This paper reports the apparatus as built and the test as run — thirty-five sealed attempts under the first paper's protocol, unchanged: pre-registration committed before every read; one read per block; measured values pinned with live re-verification paths; counterfactual verdict branches implemented and tested; verdicts landed as committed, kills kept with the same standing as confirmations; the external anchor republished at every close.

2. The apparatus

The constructed diode. The apparatus re-reads the first paper's seven measured pins exactly — 33 · 35-vs-139 · 34 · 23 · 141 · 168 — re-witnessed inside every one of the thirty-five attempts by a calibration harness (seven for seven, every attempt). The constructed instrument and the measured instrument are the same diode; property 1 was discharged at the first attempt and held at every subsequent one.

The six surfaces. Property 2 was 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. The six problem statements served as the field of uncomputed knowing [R2] from which the pre-fixed dynamics were drawn — and the record's firewall binds every section of this paper as it bound every seal: no claim on any Millennium Prize problem, ever; every read is a statement about carriers, grounds, and exact counts inside the apparatus, and about nothing else.

The verdict grammar. Each attempt sealed its outcome assignments before reading — RESOLVED against named gates, OPEN-EDGE otherwise — and closed witnessed either way. Of the thirty-five attempts, the record contains both resolutions and open edges in roughly equal measure; the kills below are reported with the confirmations, as sealed.

3. The closure yield (attempts one through six)

Writing "closure" for a two-carrier exact agreement on shared ground:

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, recorded alongside. (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. (iii) Non-monotonicity: admission is not monotone in training length — both members of the first closure alternate between admission and refusal as the training life lengthens. Each of these is a measured property of the carriers, not of the line.

4. The inward turn (attempts seven through eleven)

Five further attempts held the closure population 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 (provisionally at six). Every frontier read of the campaign to that point falls into six exact states, each an integer identity in the address's declared quantities, zero reads outside them. Two 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 — and re-witnessed at every subsequent attempt through A35. 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 discovery (the eleventh attempt). 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 was the record's first resolution whose object is the instrument's own memory: the binary phase behaviour the first paper measured as admit-or-collide is readable inside the frozen state as the presence or absence of a one-bit trace of the integer line's unique consecutive-prime event.

5. The universal form, and the declaration (attempts twelve and thirteen)

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 — 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 — a closure that has held through all thirty-five attempts.

At the thirteenth attempt's close the promotion fired under its sealed conditional (the forty-nine-cell re-derivation held), and the law entered the record DECLARED — the campaign's one declared law:

THE FOSSIL LAW. bit-'110' ⟺ ¬admit (the D family: the (2, 3) origin mark decides the phase). Declared at RIX-A13; forty-nine witnessed cells.

6. The landscape (attempts thirteen through thirty)

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, profiled over the 2-adic depth v₂ of the training length and over successive base-eight digit eras.

Six kills, then a floor. 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: 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 (eighteen of thirty-two against the pre-declared band [18, 22]) — the first held landscape law.

The first valley. The floor did not extend downward: one transect further the density fell to ten of thirty-two, and the pre-registered monotone reading of the fall died by one cell when the next transect turned upward. The third pre-registration held, and not narrowly: the density profile over v₂, at eight matched transects spanning every depth of the era, 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 second held law.

The eras, the ordering, the crossing. The next base-eight digit era is not a copy: at the depth where the first era's floor sat at 31 per cent, the second era reads twice that. The pre-registered cross-era ordering held cleanly at the matched depths (twenty against ten; nineteen against fourteen) — the third held law — and the second era's own profile (22 · 20 · 12 · 19 across its measured depths) mirrors the first era's valley in kind while differing in place and level: the fourth held law (the wall stake, rich 22/32, nine clear of its one-sided line). Then the ordering met its boundary, and the boundary proved lawful: at the fifth matched depth it held by exactly one cell while a rim stake failed by one; at the sixth matched depth the ordering failed for the first time (nineteen against 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 held with four cells to spare — the fifth held law, the first staked in the poorer direction. Read together: the two eras' density profiles are ordered on the shallow depths (five matched depths), inverted at the deep (two), and cross exactly once — between depths four and five. The proposed law remains on the record unpromoted:

THE ORDERING LAW (PROPOSED, undeclared). The richer-era ordering: at every matched depth the sixth era reads strictly richer than the fifth (the enrichment belongs to the line; the shape belongs to the era). Proposed at RIX-A28; the promotion gate refused at the sixth matched depth, exactly as sealed.

The depth band. The richest pattern in the atlas — the all-eight policy — erupted from an isolated trio into a nine-member family, and the record's leanest sealed read (three cells, one law) bounded it: the family lives in a measured two-sided depth band, v₂ ∈ [3, 6] — absent from every shallower transect in both eras, absent from the depth column above — the sixth held law, the first about a form family rather than a density profile. The eruption was the family filling its band, not breaking loose; and the deepest cell ever read (v₂ = 8) is rich while carrying no mark — richness and the all-eight are separable at depth, exactly as the two-axis grammar requires.

The atlas. Under the kills and holds an atlas accumulated: sixteen realized live-pattern sets over hundreds of measured cells, with one bit-implication standing at every realized form; two selector signatures crossing carrier families — one occupying a solid, exactly-bounded sixteen-integer interval of pure single-pattern policies, later finished as a five-cell object sharp-edged both sides; and the origin mark's exact two-pattern block, finished wall-to-wall at seven cells and porous at exactly one interior cell — solidity, where it had been observed, was a property of the resolutions read, not of the object. The atlas closed early and stayed closed: by the campaign's end, seventeen consecutive reads had added no new live-pattern set.

7. The dissolution (attempts thirty-one through thirty-five)

The landscape campaign left three interval-like boundary candidates standing: a fade line where richness appeared to stop, the apparent quiet approaching the era's end, and the era gate at 2¹⁴. Five sealed attempts then killed each of them — and the kills, read together, are the campaign's terminal result.

The record's closing grammar, quoted from the thirty-fifth attempt as landed: the campaign's boundary story is now fully consistent: NOTHING interval-like binds across strata — not the fade, not the end zone, not the gate's quiet band — while the era gates mark soft valleys and every stratum keeps an intermittent tail on its own clock. Seventeen closure-clean reads; the sixteen-form atlas held still across the dissolution's hundreds of fresh cells; the origin-mark carriage census closed at fifty-one; the all-eight family at nine members, depth-banded [3, 6].

8. The verdict on the prediction

Per the first paper's own disciplines, the verdict is stated on the test's terms and on no other.

Supported, on the test's terms. The named mechanism produced repeated resolutions at the record's exactness standard — exact integer agreements at declared span counts, zero tolerance anywhere in thirty-five attempts — under a closure criterion that was adversarially strengthened mid-campaign and then met again, ending in a fully-lived quadruple closing pairwise at one exact count. The three required properties of the test were discharged and held: the diode constructed and re-witnessed at every attempt; the pre-fixed dynamics projected with their unknowing statements printed at every seal; the exactness standard met at every closure.

The boundary, named. Two of the six surfaces reached no face at any generation read; their standing is the measured boundary of the test's reach, and the attempt lineage remains open. And the landscape campaign's terminal shape bounds the interpretation from the other side: the instrument's laws are local — a declared phase law inside the frozen state, held profile laws inside eras and strata — while every global, interval-like binding candidate dissolved under its sealed test. In the first paper's grammar, its §4.6 lesson — the limit is in the carriers, never in the line — recurs here one scale up: the limit is in the strata and the lattices measured, never in the line [R2]; what the relativity measures is exact wherever a carrier stands, and refuses to globalise [R2].

What is not claimed. A closure is a statement about carriers, grounds, and exact counts inside the apparatus. No claim on any Millennium Prize problem is made, supported, or weakened by any part of this record — the six problem statements served as received texts from which the pre-fixed dynamics were drawn, and every seal prints the corresponding unknowing statement.

9. The standing laws and the open edge

The record's law ledger at the thirty-fifth close:

The continuation is recorded, not acted: the thirty-sixth attempt's hypotheses stand sealed in the design record — the seventh era's own profile (its first transect, the thirty-two virgin v₂ = 5 cells in [16384, 20480), with the band sealed from the sixth era's matched reading), and the tail-shape question (is intermittence the general stratum-tail grammar?) typed, not staked. The lineage is open; nothing runs before its pre-registration seals.

10. Reproducibility and data availability

References (draft)

1. S. De Costa, Exact frontier reads of the primality stream by frozen online learners: a sealed seven-campaign experiment and a measured relativity between 0 and N, programme repository, 2026. (The first paper: the instrument, the seven campaigns, the prediction, and the named mechanism this paper reports.) 2. The design record: oxexp-relativity-ingress — the Relativity Ingress Experiment, programme repository, 2026. (The sealed pre-registrations, verdicts, and pinned values for all thirty-five attempts; the sole source of truth.) 3. The sealed digest: oxexp-relativity-ingress — THE HARVEST (A1..A35), programme repository, 2026. (The record's consumable hand-off; restatement by citation only.) 4. S. Wolfram, A New Kind of Science, Wolfram Media, 2002. (The 256 elementary rules as the Boolean functions of three inputs.) 5. F. Rosenblatt, The perceptron: a probabilistic model for information storage and organization in the brain, Psychological Review 65 (1958), 386–408. 6. K. Popper, The Logic of Scientific Discovery, Hutchinson, 1959. (The refutability standard the seal protocol operationalises.)

(Items 1–3 resolve to public URLs at release.)