OOI · SXThe dyadic bias measured: exact laws of composition on a frozen carrier over the integer line (the third paper)

[PAPER DRAFT] The dyadic bias measured: exact laws of composition on a frozen carrier over the integer line

THE READING PLAQUE. The third 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 third, granted after the first and second papers' readable surfaces stood, so that this version can reference them both. Claimed only as a valid research summary of the noted experiments and results. Sources: draft 0.1 in the record (sealed 4a30b40, 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 third paper of the programme, companion to the first (horizon-first-paper-draft.md) and second (rix-second-paper-draft.md). The first paper's §6 named, beside its prediction, a companion campaign over the natural counting numbers whose knowing "may carry a dyadic bias of its own nature"; this paper reports that campaign as actually run — three structurally external follow-on experiments (matters, notes, chords; twenty-four sealed attempts) consuming their predecessors only through published records and witnessed anchors. The body is drawn from the witnessed record only. Online reading (this version's convenience): the experiment surfaces this paper references are live — the three composition instruments at dev.opendata.ai/mtx, dev.opendata.ai/ntx, and dev.opendata.ai/chx, and the parent test's instrument at dev.opendata.ai/rix; the anchored datasets live at mldata.opendata.ai. The first and second papers are readable on this surface at the readable first paper and the readable second paper — the first names the test, the second reports it, and this paper reports the companion campaign the first pre-committed. 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: physics and cosmology vocabulary appears only in quotation from the sealed records and is tagged [R2] (stated as the experimentalist's reading, never asserted); an editor may cut every [R2] passage without loss to the mathematical content. The experiments' firewalls are carried verbatim in §2 and §9 and bind 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: frozen online learner · window policy · exact composition laws · 2-adic structure · pre-registered computational experiment · inter-experiment externality · open science

Abstract

The first paper of this programme reported a frozen online learner's exact frontier reads of the primality stream and closed with a prediction and a named test; the second paper reported that test run to its measured boundary. Beside the prediction, the first paper's §6 named a second direction — a companion campaign over the natural counting numbers, pre-committed so that a kill there "would refute only an axiomatic suggestion about linearity dynamics over ℕ," whose knowing "may carry a dyadic bias of its own nature." This paper reports that campaign: three follow-on experiments — matters (six sealed attempts), notes (ten), and chords (eight) — each structurally external to its predecessors, consuming them only through sealed published records and externally anchored datasets, never by code import. The principal findings, each an exact count or exact identity, pre-registered before its read: (1) every structural, form, and ordering regularity received from the predecessor record proved LOCAL on the new instruments — while one mechanism held without exception: rich cells coincide with the one-bit origin mark at every virgin cell ever read (208/208 on one instrument, 473/473 on a second, with the third keeping its own accord at 256/256); (2) THE DUO LAW — every measured (odd, even) adjacent cell pair carries one window policy — was declared after a 118-duo re-derivation and held at one hundred sixty-six own-pin duos with zero exceptions, surviving assertion at a never-read 2-adic depth; (3) THE ACCUMULATOR LAW — exact dyadic accumulator identities T/4 and T/4 + 1 at every composed length from 512 to 65536 — held at eight octaves; (4) THE TAIL LAW — the composed read of an address-order concatenation at whole-period length equals the final member's own composition — held fifteen-for-fifteen at five member values and four scales, with two further numeric identities (accumulators T/2 and T/4) landing exactly at an adversarially chosen fourth scale; and (5) every extrapolated pattern died at its first staked prediction — an alternation with five confirming terms bought no sixth, on two instruments independently. The dyadic bias the first paper hedged in one parenthesis is measured here into a family of exact laws; the generalisations that reached beyond their scopes dissolved, exactly as the second paper's landscape did one scale up. No claim on any Millennium Prize problem, and no physical claim of any kind, is made, supported, or weakened by any part of this record.

1. The inheritance

The first paper's §6, having stated its prediction, pre-committed a second direction with its own kill-scope (quoted):

A companion set of candidate probes over the natural counting numbers … 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.

This paper reports that campaign as it was actually run. Three experiments were opened in sequence, each at a sealed position of its predecessors' records, each inheriting the programme's protocol unchanged: pre-registration committed before every read; one read per block; measured values pinned with live re-verification; counterfactual verdict branches implemented and tested; verdicts landed as committed, kills kept with the same standing as confirmations; the external anchor republished and witnessed at every close. The kill-scope pre-commitment is honoured throughout: nothing below bears on the first paper's prediction, which was tested only by the second paper's mechanism; everything below is a measurement of what an a-priori linearity's knowing carries [R2] — and what it measurably carries is dyadic.

2. The method: externality as an instrument

The campaign's methodological contribution is the discipline by which the three experiments consume their predecessors. Each declared, at its outline seal, a structural externality clause of the same four-part form (quoted here in the form the first follow-on sealed it; the later experiments extend it to every predecessor):

1. Zero predecessor edits, ever — no follow-on act touches any predecessor file; the predecessor is consumed only through its published record: the sealed design text and the externally anchored dataset. Never by code import (enforced by structural no-import tests). 2. No back-citation — the predecessor never cites the follow-on; the causal note lives in the follow-on only, dated and hash-pinned. 3. No forward guidance — no follow-on content is an input to any predecessor attempt; overlaps are typed coincidence-of-position and adjudicated by commit-hash order. 4. Kill semantics — no follow-on outcome refutes, weakens, or re-opens anything in the predecessor record; a failed prediction is a fact about the follow-on's mapping, never about the law it consumed.

Two further instruments carry the honesty burden. The sealed hand-off: the predecessor record was digested once, as restatement by citation only — no promotion, no demotion, no fresh claim; the design text governs at any divergence; the digest guides nothing. The witnessed anchor: every attempt close republishes the record to an immutable public dataset, and every subsequent seal witnesses the republish before acting (each publish mints a new dataset; the newest is the record). The three experiments' anchors were re-witnessed live at this paper's drafting (§8).

The campaign also audited itself, and the audits are part of the record: the third experiment's honesty ledger runs four entries deep, evolving a standing rule — from a virgin audit (which cells has any predecessor already pinned?), through a complete inventory (full read-lists including elimination values; aggregate-only cells typed received, not citable), to a three-surface inventory (every predecessor's own far reads carried, including its census and flank reads; no truncated search output ever). Under the final audit, several of the third experiment's early "discoveries" were re-typed as received content and the corrections owned on the record with the same standing as results — the surviving discoveries are the ones the audits could not take away.

The firewalls. The union of the three experiments' firewall stacks binds this paper as it bound every seal: no claim about physical cosmology or astronomy, ever; no claim about physical quantum mechanics or any physical measurement problem, ever; no claim about consciousness or phenomenal experience, ever; no claim about physical complex systems, natural emergence, or collective phenomena, ever; no claim about artificial learning systems, ever; no claim about human knowledge, groups, or society, ever; no observational dataset as input, ever; no claim on any Millennium Prize problem, ever. Every read is a statement about carriers, grounds, and exact counts inside its apparatus, and about nothing else.

3. The matters experiment: everything local but the direct line

The first follow-on (six sealed attempts) tested the predictive value of regularities received from the predecessor record against fresh territory, under a declared reading of the record's eras and marks as address-structure events (its cosmological vocabulary register-2 throughout, quoted from its GIVEN and never asserted [R2]). Its corpus was five received exemplars; its instrument, the same frozen-carrier readout at declared families of cells.

The result is a clean split. Every structural, form, and ordering regularity received from the record proved LOCAL on the new instrument: the received atlas grain-local (first attempt); the received population depth-local (third); the received ordering k-local, through inversion to an exact tie (second and third); the era residue depth-local. In the record's own words at the third close: the one linear component with an unbroken direct line of effect is the origin-mark mechanism (the received fossil class); everything else is situational.

The direct line, by contrast, never broke. THE COINCIDENCE — a cell reads rich if and only if it carries the '110' origin mark — held at every cell ever read: 208 of 208 across six attempts, four 2-adic depths, two eras, and all form classes (the 208-cell census: 123 quiet · 74 rich · 11 seven-bit). Its promotion to a declared law stands recorded as ripe and deliberately unacted — the operator's call, reserved exactly as the programme's promotion grammar requires.

And the instrument's own extrapolation died the same death as the received ones. A five-term alternation pattern, measured on this instrument itself, was staked on its sixth term — and failed (era five persisted, 21 against 19). The close states the lesson: a pattern measured on this instrument is exactly as local as a pattern received from the record; five confirming terms bought no sixth; only the direct line predicts here. The sixth attempt also resolved a positive object: the seven-bit population is an eleven-member, four-depth, two-era family, contiguous in its band and era-asymmetric (seven members to four) — a measured object, not a law.

4. The notes experiment: the duo and the accumulator

The second follow-on (ten attempts) descended to the micro grain: the note — a single cell's resolved-only readout (its observable, communicable, and effectual components), with probability discarded for situational clarity and the carrier's mid-walk state never read (a structural rule of the apparatus, tested at import). Its campaign opened with two open-edge closes — each C3 kill naming a richer object than the stake — and then ran eight consecutive full conjunctions.

Its two laws:

THE DUO LAW (declared). Every measured (odd, even) adjacent duo carries ONE policy. One hundred sixty-six own-pin duos, zero exceptions ever; three staked windows, one pointwise sweep spanning five orders of magnitude, two asserted windows, and one won divergence. Every future read asserts it; a violation refutes the law, never the instrument.

The declaration followed the programme's promotion grammar exactly: proposed under the precedent of the second paper's fossil law, gated on a 118-duo re-derivation from the instrument's own pins plus a sixteen-duo fresh stake, both held — and then asserted against territory the law itself named, including a cell at a never-read ninth 2-adic depth (v₂ = 8), which landed as named.

THE ACCUMULATOR LAW. acc({000}, T) = T/4 and acc(ALL-BUT-111, T) = T/4 + 1, exact at every composed length from 512 to 65536 — eight octaves deep.

The campaign's standing objects at its pause: the family of twenty-four members in twelve oriented pairs, whose three-duo motif is confined to [4129..4340] as read, with the far territory carrying member-free pure-F duo runs; the schedule grammar — every transition event ever measured lands at a duo's opening odd cell, with the fine-grain surplus ten for ten; the power ladder intermittent through 2²²; and THE ACCORD — the origin-mark coincidence read on this instrument's own tally — at 473/473 across nine depths, every rung, every form class (681 cumulative with the sibling's 208; the promotion gate recorded, unacted). The honesty ledger records one seal-time derivation error, owned; and the alternation died on this instrument too — un-staked: the flag discipline held it as data outside every seal, it failed on its own terms, and a framing error about it at the tenth close was owned on the record. Across the two instruments the alternation is thus dead twice over — once at a staked term, once free.

5. The chords experiment: composition, and the first macro-object

The third follow-on (eight attempts) raised the grain: the chord — a declared span of notes read as ONE resolved object (the span, its member notes in address order, the census multiset, the schedule signature), existing only after every member resolves. Its corpus was three 'how'-shaped externalities under a new rule of the house grammar: a 'how' question names a hypothesis family, never a stake — only exact-identity cards land. Its campaign opened with two open-edge closes and then ran six consecutive full conjunctions. The composed grain produced the campaign's sharpest early results:

Its law:

THE TAIL LAW (declared). For every declared address-order concatenation chord-stream ground read at a whole-period length T, the composed note equals the k = 1 composition of the ground's final member. Standing fifteen-for-fifteen at five member values and four scales (512..4096); every read asserts it; a violation refutes the law, never the instrument.

At the eighth attempt the campaign staked its first numeric accumulator identities — deliberately at the fourth scale, the same scale-count at which the sibling instrument's alternation had died, so that the stake was adversarial by construction — and both landed exactly: compose(F, 4096) with accumulator 2048 = T/2, and the [∅, F] pair at 4096 with accumulator 1024 = T/4, each pattern four-for-four across four octaves. Their promotion stands gated and unacted — the operator's call.

The same campaign resolved the programme's first macro-object: a connected rich region at [~8263..8320], crossing three 32-cell window seams and ending at 8321 — resolved across four windows that each saw only a sample of it (the region, not the window, is the natural object at this grain). Two full six-cell motifs were completed beyond the received confinement — one staked cell-by-cell on three virgin cells and landing three-for-three; the clock grammar ran forty-four transition events for forty-four at odd cells; and the accord's own tally closed at 256/256.

6. The synthesis: what binds, and what refuses to

Read together, the twenty-four attempts sort every measured regularity into two kinds.

What binds — the laws, and they are dyadic. The duo (a two-cell pairing with one policy); the accumulators (T/4 and T/4 + 1 at eight octaves; T/2 and T/4 again at the composed grain); the tail (composition collapsing to the final member at whole periods); the origin-mark coincidence (208 + 473 + 256 on three independent tallies, zero exceptions anywhere). Each is exact, each was pre-registered against a named kill, and each carries an explicitly dyadic form — powers of two in the scales, halves and quarters in the accumulators, 2-adic depth in every profile. The first paper's parenthesis — that the a-priori linearity's knowing "may carry a dyadic bias of its own nature" — is no longer a hedge: the bias is the measured content [R2].

What refuses to bind — every extrapolated shape. The received atlas, population, ordering, and residue regularities: local. The alternation: dead at its first staked term on one instrument, dead free on the second. Solidity of measured blocks: a property of the resolutions read (the porous cell). The confinement of the motif: a property of where the record had looked (two completions beyond it). This is the second paper's terminal lesson recurring one grain down: there, nothing interval-like bound across strata while local laws held; here, nothing pattern-like binds across scopes while exact laws hold. The programme's grammar states both at once: laws are exact and local to their declared scopes; shapes extrapolated past their scopes die at their first stake.

7. The standing gates and the open edges

The campaign closes nothing permanently; it pauses with its gates recorded and its continuations sealed:

8. Reproducibility and data availability

9. Closing register

The three experiments asked what a frozen carrier's knowing carries with it across the integer line, and the answer landed in two registers at once: a small family of exact dyadic laws that have never missed, and a graveyard of extrapolations that never survived their first stake. The programme takes both as the result.

The firewalls stand over this paper as over every seal it reports: no claim about physical cosmology or astronomy, ever; no claim about physical quantum mechanics or any physical measurement problem, ever; no claim about consciousness or phenomenal experience, ever; no claim about physical complex systems, natural emergence, or collective phenomena, ever; no claim about artificial learning systems, ever; no claim about human knowledge, groups, or society, ever; no claim on any Millennium Prize problem, ever — every read is a statement about carriers, grounds, and exact counts inside its apparatus, and about nothing else.

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 prediction, and the §6 pre-commitment this paper reports against.) 2. S. De Costa, The relativity ingress experiment: exact closures, the fossil law, and the dissolution of the landscape, programme repository, 2026. (The second paper: the named test; the promotion grammar and the dissolution lesson cited here.) 3. The design records: oxexp-matters — the Matters Experiment; oxexp-notes — the Notes Experiment; oxexp-chords — the Chords Experiment, programme repository, 2026. (The sealed pre-registrations, verdicts, and pinned values for all twenty-four attempts; the sole sources of truth.) 4. The sealed digest: oxexp-relativity-ingress — THE HARVEST (A1..A35), programme repository, 2026. 5. S. Wolfram, A New Kind of Science, Wolfram Media, 2002. 6. F. Rosenblatt, The perceptron: a probabilistic model for information storage and organization in the brain, Psychological Review 65 (1958), 386–408. 7. K. Popper, The Logic of Scientific Discovery, Hutchinson, 1959.

(Items 1–4 resolve to public URLs at release.)