# Sundog Capset Ledger
Working hook:
> Cap-set is the workbench. Unit-distance is the horizon. The apparatus
> on trial is not discrete geometry — it is AI mathematics under
> traceability.
Sundog Capset is the staging ledger for the program's coupling between
Sundog's existing alignment apparatus — Bayes comparator, three-gate
failure taxonomy, evidence tiers, falsifier-first discipline — and the
2026 wave of AI-produced mathematics, beginning with OpenAI's May 2026
disproof of the planar unit-distance conjecture.
This document is not a roadmap. It is a holding pattern for ambition.
None of the candidates below have run. Each, if executed, would either
ratchet the coupling claim into earned language or push it back to the
"out-of-scope analogy" pile in `presentation/claims-and-scope.md`.
The one existing anchor is the cap-set workbench at `/capset`
([`capset.html`](../capset.html), shipped 2026-05-21). It is a hands-on
primer for the 2016 Croot–Lev–Pach / Ellenberg–Gijswijt polynomial-method
breakthrough — the cleanest historical precedent for "an algebraic
substrate suddenly resolves a problem nobody knew was algebraic" — and
its narrative cards informally point at the 2026 unit-distance result as
the same shape of breakthrough. This ledger is where that informal
pointer is staged for later, defensible coupling work.
### Promotions
No candidates have been promoted out of this ledger yet. The block is
retained for symmetry with [`SUNDOG_V_GRAVITY.md`](SUNDOG_V_GRAVITY.md)
and will be populated when a candidate graduates into a committed
roadmap (see Promotion Criteria below).
## Claim Boundary
This document does **not** claim that Sundog has produced original
mathematics on either the cap-set problem or the unit-distance problem.
It claims that:
1. there is a coherent structural argument — that Sundog's apparatus,
built to evaluate indirect-signal claims under partial observability,
is well-shaped to evaluate AI-produced mathematics, which arrives with
the same epistemic structure (a constructed object proposed by a
system whose internal reasoning is partially opaque);
2. that argument is currently defended by analogy and by the cap-set
workbench as a hands-on primer, not by a Sundog-authored note,
reading, or evaluator output on either result;
3. the proof targets that would test the argument are non-trivial enough
that they need to live in a ledger before they live in a roadmap.
The cap-set workbench is a primer, not an evaluator. The narrative cards
on `/capset` say cap-set and unit-distance "rhyme" — that is a public
analogy, not a Sundog-original technical claim, and the ledger does not
upgrade it.
If a candidate below is promoted into a roadmap document, it leaves this
ledger.
## The Coupling Claim
The coupling between Sundog's apparatus and the 2026 wave of AI-produced
mathematics is staged on two fronts.
### Front A — Evaluator (defensible)
The OpenAI unit-distance disproof is interesting precisely because it
survived external mathematician verification: Gowers, Alon, Shankar,
Tsimerman wrote a companion paper; Will Sawin extracted an explicit
exponent (δ ≥ 0.014) the original proof did not state. The result was
not "trust the AI" — it was "AI proposes, formal mathematics disposes,
human refinement deepens." That is the same epistemic shape Sundog
already imposes on its own work: claim hygiene, evidence tiers, named
falsifiers, residual-coverage-detection failure attribution.
Front A stages the claim that Sundog's apparatus can be turned outward
— used to *read* AI-mathematics results, not just internal Sundog
experiments. Concrete candidates apply specific Sundog instruments
(three-gate failure taxonomy; Bayes-comparator framing; evidence-tier
overlay) to the OpenAI result and its surrounding discourse.
This front is defensible now in the sense that no Sundog-original
mathematics is claimed. The product is a reading, not a proof.
### Front B — Substrate analogue (horizon)
A more ambitious coupling: the parhelion case (Sundog's own
`SUNDOG_V_GEOMETRY.md` Phase 10 single-handle verdict for the
parhelion-offset inverse) and the unit-distance case (algebraic number
fields beating the rescaled square grid) share a structural shape —
"deep substrate reveals geometric content that the naïve view discards."
Cap-set 2016 is the third instance in the same shape.
Front B stages the claim that this structural rhyme is real, not
decorative — that the three cases can be named under a shared frame
without overreach. This is the gravity-level ambition for the ledger.
It is *not* defensible now; the candidates below describe what would
need to happen for it to become defensible.
## Falsification Surface
The coupling claim can fail in four named modes:
1. **Front-A vacuity** — Sundog's instruments, when applied to the
OpenAI result, produce readings that are either (a) what any careful
mathematician would say, or (b) so abstract that no specific Sundog
discipline is doing the work. The instrument has no edge over a good
editorial.
2. **Front-A miscalibration** — the three-gate taxonomy or Bayes
comparator, when applied outside Sundog's home domain, lands on
confidently wrong readings (e.g., classifies the 78-year stalemate as
a coverage failure when working mathematicians say it was a detection
failure, or vice versa). Cross-domain transfer was illegitimate.
3. **Front-B reach** — the three-substrate rhyme (parhelion / cap-set /
unit-distance) turns out to be a surface analogy: the cases share a
slogan but the substrates work by structurally different mechanisms,
and a careful mathematician reading the Sundog claim flags it as
pareidolia.
4. **Workbench misrepresents** — the `/capset` workbench, as a primer,
leaves the curious-audience reader with a wrong picture of either the
polynomial method or the unit-distance result. The primer is itself
the failure surface, not the ledger.
Each candidate below has to declare which mode it attacks.
## Evaluation Criteria
A candidate is worth pursuing if it:
- attacks at least one failure mode above with a falsifiable artifact;
- can be written without claiming Sundog-original mathematics, *or* has
a clear external-review path if it does;
- produces a deliverable (note, overlay, exhibit) that could plausibly
land on the eventual `/geometry.html` page alongside the cap-set
workbench rather than a standalone diversion;
- is honest about its dependence on the OpenAI proof, the companion
paper, and any chain-of-thought material released alongside.
## Shortlist Recommendation
As of 2026-05-21, the recommended sequencing is:
1. **Candidate 1** (three-gate reading of the 78-year stalemate) — lowest
cost, highest leverage on Front A. A short note, mathematician
sanity-check, then a `/geometry.html` rail position alongside the
cap-set workbench.
2. **Candidate 3** (evidence-tier overlay on the OpenAI announcement) —
demonstrates the apparatus on a fresh major result; pairs well with
the chat-widget's claim-hygiene posture in
[`SUNDOG_V_CHAT.md`](SUNDOG_V_CHAT.md).
3. **Candidate 5** (substrate-rhyme note) — Front B, but staged
explicitly as a reach claim with a falsifier built in. Only if
Candidates 1 and 3 land cleanly.
Candidates 2, 4, 6 are held until the first three resolve.
---
## Candidate 1 — Three-Gate Reading of the Unit-Distance Stalemate
Working hook:
> A 78-year stalemate is a failure attribution problem. Sundog already
> has a taxonomy for that.
### Why it is strong
The three-gate failure taxonomy (residual / coverage / detection) is one
of Sundog's most portable instruments — it was developed for the mesa
empirical front and has already been applied across mesa, geometry, and
perception. The OpenAI result invites exactly the question the taxonomy
answers: *which gate failed for 78 years, and why did 2026 unblock it?*
A clean reading would say something like:
- **Residual gate:** had any prior construction left value on the table?
Yes — Erdős's original lower bound already secretly used the Gaussian
integers; nobody pushed the construction to richer number fields. The
residual was non-zero and visible in hindsight.
- **Coverage gate:** did the field's search space exclude the
construction that eventually won? Partly — combinatorial intuition
kept attention on lattice-like point configurations; class field
towers and Golod–Shafarevich live in a different research neighborhood
that combinatorial geometers do not typically reach for.
- **Detection gate:** even if the construction lay within reach, was the
bridge invisible? Yes — *that* the algebraic-number-theory machinery
could be wielded for this specific bound was a non-obvious move that
the AI's chain of thought (abridged) appears to have surfaced.
The Sundog reading would then say: the stalemate was *primarily* a
detection-gate failure with a coverage-gate component. That is a
specific, falsifiable claim, distinct from "nobody had the right
construction" (residual) or "the field was looking in the wrong place"
(pure coverage).
### Why it is weaker
The three-gate taxonomy was forged in Sundog's home domain (mesa
controller envelopes and halo geometry). Transferring it to a different
field's intellectual history risks projecting Sundog's vocabulary onto
disciplinary judgments that are not Sundog's to make. The note must be
shown to working combinatorial geometers — not just framed by them —
before it claims edge over a normal historical reading.
### Sundog variant
Write a short note (`docs/notes/CAPSET_THREEGATE_READING.md` or similar)
that:
- restates the three-gate taxonomy in self-contained terms;
- applies it to the unit-distance stalemate with explicit citations to
the Spencer–Szemerédi–Trotter upper bound history, the
Pach–Raz–Solymosi line, and the Matoušek / Alon–Bucić–Sauermann
non-Euclidean evidence Erdős's conjecture cited;
- declares the reading (detection-primary, coverage-secondary,
residual-nontrivial-but-not-decisive);
- carries its own counterexamples — what evidence would force the
reading toward residual-primary, for example.
### Sundog expression
- **Hidden target:** the gate attribution of a multi-decade open problem.
- **Indirect signal:** the historical record (published constructions,
upper-bound improvements, conjectured forms).
- **Transformation:** three-gate taxonomy as the decoder.
- **Actionable output:** a falsifiable attribution claim, citable in the
workbench narrative cards and on `/geometry.html`.
- **Failure boundary:** mathematicians outside Sundog reject the gate
attribution as a category error.
### Falsification target
Mode (2) — Front-A miscalibration. A null result is "two of three
combinatorial-geometry working mathematicians flag the gate attribution
as misapplied." A positive result is sign-off (informal is enough at
this stage) from at least one practicing combinatorial geometer that the
reading is at minimum coherent.
### Current recommendation
First-priority. Cheapest path to Front-A defense. Note plus external
sanity-check, then a rail position on `/geometry.html` adjacent to the
cap-set workbench.
---
## Candidate 2 — Bayes-Comparator on AI Construction Search
Working hook:
> When an AI surfaces a number-theoretic construction, what does the
> Bayes comparator say versus what does Sundog say?
### Why it is strong
[`SUNDOG_V_BAYES.md`](SUNDOG_V_BAYES.md) frames the Bayes-vs-Sundog
split as "Bayes turns evidence into belief; Sundog turns response into
control." Applied to AI-produced mathematics, the comparator becomes
something different: Bayes asks *given the corpus prior, how likely was
this construction?*, Sundog asks *what signal in the search trace
actually moved the model toward number fields rather than lattice
constructions?*
The OpenAI announcement linked to an abridged chain of thought. That
makes the comparator non-vacuous — there is something to read.
### Why it is weaker
This candidate is contingent on the chain-of-thought material being
substantive enough to support either reading. Abridged transcripts
notoriously elide the actually-decisive moves. If the abridgment hides
the search trace, the comparator has nothing to compare.
It also drifts close to the kind of LLM-introspection claim Sundog has
been careful not to make. The note would have to be framed as "structural
comparator," not "we read the model's mind."
### Sundog variant
Write a comparator note that:
- summarizes the two priors (Bayes: prior over construction families;
Sundog: prior over which search-trace signals are control-relevant);
- reads the abridged chain of thought through both lenses;
- declares which lens fits the visible record better, with explicit
caveat that abridgment limits the reading;
- ships the note as a `/geometry.html` exhibit, not a load-bearing
Sundog result.
### Sundog expression
- **Hidden target:** the decisive search-trace moment that surfaced
number-fields over lattices.
- **Indirect signal:** the abridged chain of thought published by
OpenAI.
- **Transformation:** Bayes-vs-Sundog comparator from
`SUNDOG_V_BAYES.md`.
- **Actionable output:** a comparator note that is either a clean lens-fit
call or a clean abridgment-limited inconclusive.
- **Failure boundary:** chain of thought abridged past the decisive
moves; comparator has nothing to bite on.
### Falsification target
Mode (1) — Front-A vacuity. Null result: the comparator produces a
reading no different from any thoughtful mathematician's editorial.
Positive result: the comparator picks out a specific signal class (e.g.,
"the search ranked richness-of-symmetry as a target before generating
candidate constructions") that the standard reading misses.
### Current recommendation
Second-priority. Run after Candidate 1 has established that Sundog's
apparatus can be turned outward without immediate miscalibration.
Contingent on chain-of-thought material being substantive.
---
## Candidate 3 — Evidence-Tier Overlay on the OpenAI Announcement
Working hook:
> A major AI result is a claim with mixed evidence tiers. Sundog already
> has a tier overlay.
### Why it is strong
Sundog's evidence-tier discipline (in the chat widget, in the
applications gallery, in `presentation/claims-and-scope.md`) sorts
claims into tiers — formally proved, externally verified, observational,
editorial, aspirational. The OpenAI announcement is a mixed-tier object:
- *formally proved*: the construction exists; external mathematicians
verified the proof; Sawin extracted δ ≥ 0.014;
- *observational*: the proof came from a general-purpose model under a
specific scaffolding, on a specific evaluation slate;
- *editorial*: "first time AI has resolved a central conjecture
autonomously" is a framing claim, not a verified one;
- *aspirational*: extrapolations to biology, physics, materials science
in the "Why this matters" section.
A Sundog tier overlay would sort the announcement claim-by-claim, with
the same discipline the chat widget enforces on Sundog's own promotional
copy.
### Why it is weaker
This drifts close to media-criticism of OpenAI's framing, which is not
what Sundog is for. The overlay has to be substantive about evidence
tiers, not a contrarian re-write of the press release. The deliverable
must read as an apparatus demonstration on a public exhibit, not a
take.
### Sundog variant
Produce an overlay artifact:
- a side-by-side rendering of the announcement with tier badges per
claim;
- a short rubric explaining what would move each claim up or down a
tier;
- the overlay lives on `/geometry.html` as an exhibit, with the same
visual treatment as the structural-failure boundary-map page.
### Sundog expression
- **Hidden target:** the evidence-tier distribution of a fresh major
result.
- **Indirect signal:** the published announcement and companion paper.
- **Transformation:** Sundog tier rubric.
- **Actionable output:** a public, browsable tier overlay; rubric is
reusable for the next AI-mathematics result.
- **Failure boundary:** overlay reads as editorial axe-grinding rather
than apparatus demonstration.
### Falsification target
Mode (1) — Front-A vacuity. Null result: the overlay's tier assignments
match what a careful reader of the announcement would write
unprompted. Positive result: the overlay surfaces at least one
non-obvious tier disagreement that holds up under a second pass.
### Current recommendation
Second-priority alongside Candidate 2. The overlay is the artifact most
directly suited to a `/geometry.html` rail position next to the cap-set
workbench, because it is visual and self-contained.
---
## Candidate 4 — Workbench Extension to F_3^5 and F_3^6
Working hook:
> The Ellenberg–Gijswijt bound starts mattering at moderate n. The
> workbench should too.
### Why it is strong
The current `/capset` workbench supports n=2, 3, 4. The
Ellenberg–Gijswijt bound (3·2.7551^n) only becomes tighter than the
trivial 3^n around n ≈ 13. Between n=4 and n=13, the bound is the
right object but the workbench cannot show it — the readout currently
displays "— (asymptotic)" to avoid showing a number larger than 3^n.
Extending the workbench to n=5 (243 cells, max cap = 45) and n=6 (729
cells, max cap = 112) would let the bound chart's gap close visibly and
would let visitors *see* the polynomial-method curve descend below the
trivial.
### Why it is weaker
Visualization scales badly: 243 cells is borderline for a clean grid
layout; 729 is hostile. The greedy fill becomes useless (greedy in F_3^6
rarely finds anything near 112). The "Load a maximum cap" preset for
n=6 would need to be a known construction (BCH-derived or Calderbank–
Fishburn–Rains) rather than a search result.
### Sundog variant
- add n=5 (3×3 grid of 3×3 grids of 3×3 cells, scaled small);
- ship a known max cap for n=5 (Calderbank–Fishburn–Rains style, 45
points) baked into the workbench, verified violation-free;
- defer n=6 unless visualization design lands cleanly;
- update the bound chart to include up to n=10 even though the
workbench grid only goes to n=5, so the asymptotic gap is visible.
### Sundog expression
- **Hidden target:** the n where polynomial-method bound visibly bites.
- **Indirect signal:** size of known maxes plotted against 3^n and
3·2.7551^n.
- **Transformation:** log-scale bound chart with extended range.
- **Actionable output:** workbench extension; chart update; verified
presets.
- **Failure boundary:** visualization at n=5 is unreadable; the
extension is graphical clutter, not pedagogy.
### Falsification target
Mode (4) — Workbench misrepresents. Null result: visitors find the n=5
grid harder to parse than n=4 and the bound chart's extended range
confuses rather than illuminates. Positive result: the extended chart
makes the "gap that the polynomial method opened" visually obvious to a
curious-audience visitor.
### Current recommendation
Third-priority. Pure workbench-side improvement; no Sundog-original
claim attached. Only worth doing once the Front-A candidates have given
the workbench a defensible editorial frame on `/geometry.html`.
---
## Candidate 5 — Substrate-Rhyme Note (Parhelion / Cap-set / Unit-Distance)
Working hook:
> Three breakthroughs, one shape: a substrate reveals geometric content
> the naïve view discarded.
### Why it is strong
This is the Front-B claim made explicit. The Sundog
parhelion-offset Phase 10 single-handle verdict (see
`SUNDOG_V_GEOMETRY.md`) is the project's own clearest example of "the
substrate (geometric optics) carries content that pure-geometry
intuition misses." Cap-set 2016 and unit-distance 2026 are external
examples of the same shape, in finite-field combinatorics and
combinatorial geometry respectively.
A careful note tracing the structural analogy — with explicit
disanalogies — would give Sundog a coherent way to talk about the 2026
AI-mathematics wave without overclaiming.
### Why it is weaker
This is the candidate most exposed to the "decorative analogy"
failure. Three different fields, three different substrates, three
different proof styles. The note has to be careful enough that a
mathematician reading it does not flag it as pareidolia, and concrete
enough that it does work no looser editorial would do.
There is also reflexive risk: if the note is too good at the analogy,
it overclaims that Sundog has a unified theory of "substrate beats
geometry." That is exactly the kind of universal Sundog has been careful
to avoid.
### Sundog variant
Write a short note that:
- names the three cases (parhelion-offset, cap-set, unit-distance);
- for each, identifies the substrate (optics; finite-field polynomials;
algebraic number fields);
- for each, identifies what "naïve geometric intuition" looked like
before the breakthrough (no parhelion-offset model; rescaled square
grid; cap density near 3^n);
- explicitly identifies the disanalogies — what makes the three cases
*not* the same — and weights them against the analogy;
- carries a falsifier: "what fourth case in this shape would falsify
the rhyme by being unambiguously distinct from the other three?"
### Sundog expression
- **Hidden target:** the structural-rhyme claim's defensible scope.
- **Indirect signal:** the three case studies side-by-side.
- **Transformation:** structural-shape rubric with named substrates and
failure modes.
- **Actionable output:** a note that either passes mathematician review
as coherent or is downgraded to "informal Sundog editorial."
- **Failure boundary:** mathematician reviewer calls the analogy
pareidolia; note is downgraded or shelved.
### Falsification target
Mode (3) — Front-B reach. Null result: the rhyme is decorative, the
disanalogies dominate. Positive result: at least one practicing
mathematician (in either combinatorial geometry or algebraic number
theory) signs off informally that the structural-shape claim is at
minimum coherent — not endorsed, but not rejected as category error.
### Current recommendation
Reach. Run only after Candidates 1 and 3 land cleanly. This is the
gravity-level horizon claim for the ledger and should not be attempted
without Front A already defended.
---
## Candidate 6 — `/geometry.html` Rail Position for the Coupling
Working hook:
> The workbench is one exhibit. The coupling needs an editorial frame.
### Why it is strong
The forthcoming `/geometry.html` page (see
`SUNDOG_V_GEOMETRY.md`) is staged to present multiple
geometry-adjacent workbenches and exhibits. The cap-set workbench is
one of them. Without an editorial frame, it sits there as a clever
math toy; with a frame, it becomes the entry point for whichever Front-A
candidates have landed.
### Why it is weaker
This is downstream of the other candidates. Until at least one Front-A
candidate has produced an artifact, there is nothing for the rail to
point at beyond the workbench itself.
### Sundog variant
Once Candidates 1 / 3 have artifacts:
- add a `/geometry.html` rail card titled something like "AI mathematics
under traceability" or "When the algebra reveals the geometry";
- the card links to the cap-set workbench, the three-gate reading note
(Cand. 1), the evidence-tier overlay (Cand. 3), and the substrate-
rhyme note (Cand. 5) if it lands;
- the rail card carries the same claim-hygiene posture as the rest of
the page — what is Sundog-original, what is primer / reading, what is
reach.
### Sundog expression
- **Hidden target:** the editorial frame that makes the cap-set
workbench part of a coherent geometry-page narrative rather than a
standalone diversion.
- **Indirect signal:** the artifacts produced by Candidates 1, 3, and
optionally 5.
- **Transformation:** `/geometry.html` rail card and editorial copy.
- **Actionable output:** the rail card.
- **Failure boundary:** the rail reads as a press-release roundup, not a
Sundog-discipline frame.
### Falsification target
Mode (1) — Front-A vacuity, at the page level. Null result: visitors
land on the rail and read it as "Sundog has opinions about AI math."
Positive result: visitors land on the rail and use the artifacts as
worked examples of Sundog's apparatus.
### Current recommendation
Last. Depends on Candidates 1 and 3 having landed.
---
## Promotion Criteria
A candidate graduates from this ledger to a committed roadmap (analogue
of Mesa's promotion of Gravity Candidates 1/2/7) when:
1. there is a defensible Sundog-original contribution — not a primer, a
reading, or an overlay, but a claim that takes responsibility for a
technical position;
2. external review (informal is enough at first) has been taken on the
contribution and not rejected;
3. the `/geometry.html` page has a rail position for the contribution as
first-class content, not as a footnote to the cap-set workbench;
4. a concrete first phase is scoped with a deliverable and a gate.
Until then, this ledger holds candidates and the workbench. If none of
the candidates here ever clear those four criteria, the right disposition
is to retire the ledger and keep the workbench as a primer.
## Horizon
What would tip the ledger toward a committed roadmap:
- A second comparable AI-mathematics result lands (a different field,
different machinery) and the three-gate reading from Candidate 1
transfers cleanly. Two data points for a Sundog instrument are far
more defensible than one.
- The substrate-rhyme note from Candidate 5 survives mathematician
review as coherent. That would license Front B as something Sundog
can talk about publicly without overclaiming.
- The `/geometry.html` page reaches the point where exhibits demand
editorial unification, and the coupling claim is the most natural
frame for the cap-set workbench's adjacency.
## Cross-references
- [`SUNDOG_V_GEOMETRY.md`](SUNDOG_V_GEOMETRY.md) — the geometry roadmap
and the eventual host page for the cap-set workbench and any
candidates promoted from this ledger.
- [`SUNDOG_V_GRAVITY.md`](SUNDOG_V_GRAVITY.md) — the ledger pattern this
document mirrors. Gravity's Promotions section and candidate template
are the structural precedent.
- [`SUNDOG_V_MESA.md`](SUNDOG_V_MESA.md) — the three-gate failure
taxonomy used in Candidate 1 lives here; the Formal Separability
Theorem appendix is the precedent for "ledger candidate becomes
appendix in a roadmap."
- [`SUNDOG_V_BAYES.md`](SUNDOG_V_BAYES.md) — the Bayes-vs-Sundog
comparator used in Candidate 2.
- [`SUNDOG_V_CHAT.md`](SUNDOG_V_CHAT.md) — the claim-hygiene posture and
evidence-tier discipline applied in Candidate 3.
- [`presentation/claims-and-scope.md`](presentation/claims-and-scope.md)
— the claims-policy document any promotion has to be consistent with.
- [`../capset.html`](../capset.html) — the workbench artifact, the one
current anchor.
## Inspection Trail
- OpenAI · An OpenAI model has disproved a central conjecture in
discrete geometry · 2026-05-20 ·
- Croot, Lev, Pach · Progression-free sets in ℤ_4^n are exponentially
small · 2016 ·
- Ellenberg, Gijswijt · On large subsets of F_q^n with no three-term
arithmetic progression · 2016 ·
- Cap set (overview) ·