# [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](https://dev.opendata.ai/mtx),
[dev.opendata.ai/ntx](https://dev.opendata.ai/ntx), and
[dev.opendata.ai/chx](https://dev.opendata.ai/chx), and the parent 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). The first and second papers are
readable on this surface at [the readable first paper](/sx/reading/horizon-first-paper)
and [the readable second paper](/sx/reading/rix-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:

- **Interference is real at k = 2.** The composed chord over [F, ∅] collapses to ∅
  (accumulator 258): the empty member is not invisible in composition.
- **Order is real.** The reversed pair [∅, F] lands {000, 010} with accumulator
  256 = T/4 exactly — same members, same density, opposite outcomes from member order
  alone; the density reading died at its first stake.

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:

- **The promotion gates, recorded and unacted** — the matters coincidence (208/208,
  typed beyond ripe); the notes accord (473/473); the chords accumulator identities
  (T/2 and T/4, four-for-four each). Each promotion is reserved to the operator's
  explicit call at a seal of the operator's naming — the same grammar under which the
  duo and tail laws were declared and the second paper's ordering law was refused.
- **The recorded continuations** — each experiment's next attempt stands sealed as
  hypotheses and unacted: the matters member-era asymmetry census; the notes eleventh
  window (fully virgin, with the declared law naming its opening cell); the chords
  ninth attempt (the accumulator gate; the region as a declared object). The lineages
  are append-only; nothing runs before its pre-registration seals.

## 8. Reproducibility and data availability

- **Code.** The three engines (`daemon/mtx.py`, `daemon/ntx.py`, `daemon/chx.py`,
  each readout-only over the frozen house carrier), their experiment plugins, and
  their acceptance tests are published in the programme repository, with live
  recomputation paths on the running node (`/mtx`, `/ntx`, `/chx`). *(Public URL at
  submission; the repository releases under the programme's code-anchor process.)*
- **Seals.** Twenty-four sealed attempts across the three experiments; each seal is a
  git commit whose hash is cited in its design document (the sole source of truth for
  its record).
- **Data.** Each experiment's record is externally anchored to an immutable public
  dataset, republished at every close and witnessed before every subsequent seal (each
  publish mints a new dataset — the newest is the record). Re-witnessed live at this
  paper's drafting:
  [mldata.opendata.ai/dataset/mtx-exp_52b4bb62](https://mldata.opendata.ai/dataset/mtx-exp_52b4bb62)
  (42 rows, six attempts) ·
  [mldata.opendata.ai/dataset/ntx-exp_e4c9a847](https://mldata.opendata.ai/dataset/ntx-exp_e4c9a847)
  (70 rows, ten attempts) ·
  [mldata.opendata.ai/dataset/chx-exp_f16412dd](https://mldata.opendata.ai/dataset/chx-exp_f16412dd)
  (56 rows, eight attempts).
- **The predecessor's hand-off.** The predecessor record enters this campaign only
  through its sealed digest (*oxexp-relativity-ingress — THE HARVEST (A1..A35)*,
  commit `940891b` over the design record at `7fbbd6a`) and its anchored datasets, as
  reported in the second paper.

## 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.)*