# Sundog vs. ARC-AGI Abstraction > **Cross-substrate failure-map entry:** CONVERGENCE-TO-NULL (Phase 3 converged > deterministically; reopening needs a reframing, not another sweep) — see > [`CROSS_SUBSTRATE_NOTES.md`](CROSS_SUBSTRATE_NOTES.md) > "Cross-Substrate Generality Failure Map". Working hook: > ARC-AGI tests whether a system can abstract and reason on novel tasks > without reward hacking or memorization. Sundog's shadow-coupling > machinery — gauge-invariant local projections, Blackwell-sufficient > signatures, 5D mesa subspace — is a geometric alternative to > high-dimensional pattern matching. This roadmap finds out whether that > machinery earns its keep on a public benchmark. This roadmap is filed as Gravity Ledger Candidate 14 and promoted directly into a committed execution plan targeting the ARC Prize 2026 Paper Track (Theory category). It couples the existing shadow-projection, signature-sufficiency, and mesa-subspace machinery to the ARC-AGI task family — the first Sundog roadmap with an external submission deadline and a public benchmark deliverable alongside the internal receipt chain. The pairing of theory paper and Kaggle entry is deliberate. A paper without an entry can be dismissed as arm-waving. An entry without a paper can be dismissed as architecture search. Run together, the geometric argument (why shadow coupling should help with abstraction) and the benchmark artifact (does it actually help) each discipline the other. The theory-side differentiator is the "why" — Blackwell sufficiency of the signature for core-knowledge abstraction tasks, with the 5D mesa subspace as the dimensionality-reduction mechanism. This is not a new architecture paper; it is a structural argument about what makes abstraction work, tested against the benchmark that was designed to measure it. **Official ARC Prize 2026 anchors (checked 2026-05-28):** competition submissions are due November 2, 2026; papers are due November 8, 2026; Paper Prize submissions must link to a working Kaggle code submission for ARC-AGI-2 or ARC-AGI-3; the static ARC-AGI-2 track scores exact task matches with two predictions per test input and no internet during Kaggle evaluation. The first Sundog lane targets ARC-AGI-2 because its static grid format is the cleanest bridge to shadow-projected signatures. ARC-AGI-3 is out of scope unless a later amendment opens an interactive-agent branch. > **Gravity Ledger dependency.** This roadmap inherits directly from > the mesa lane's operating-envelope result (Phases 0–7, especially > v3.1–v3.8 5D subspace finding) and the geometry lane's forward-rich / > inverse-narrow field-shape pattern. The shadow-projection operator > (Phase 2 below) ports the same gauge-cocycle / Procrustes machinery > used in the three-body workbench and photometric alignment. No new > physics or simulator infrastructure is required. ## Why ARC Is the Right Next Coupling The gravity claim names three falsification modes: 1. field manipulation is cheap; 2. the signature is a reward in costume; 3. mesa-optimization re-emerges. Mesa (`SUNDOG_V_MESA.md`) attacks modes (2) and (3) in a synthetic environment. The geometry lane attacks the forward/inverse field-shape pattern in the wild. ARC-AGI attacks a fourth surface the other roadmaps do not reach: 4. **abstraction failure** — shadow coupling produces competent navigation in continuous-field environments but does not generalize to the discrete, novel-task abstraction regime that core knowledge priors are supposed to enable. ARC-AGI is purpose-built to test exactly this: can a system solve tasks it has never seen, using core knowledge (objectness, counting, symmetry, geometry) rather than pattern-matching over a training distribution? If the shadow-signature machinery genuinely captures something about abstraction — if the Blackwell-sufficient statistic carries the right information — it should show up here. If it does not, the gravity claim is bounded to continuous-field navigation and the "gravity for agents" hook requires a named retreat. ARC is also the most externally legible venue for the Sundog program. The Paper Track Theory category rewards exactly the kind of structural argument Sundog already makes — "why does abstraction work?" — and the receipt-heavy, pre-registered, falsification-first methodology maps directly onto what the prize committee is looking for. ## What's Honest vs. What's Reach **Honest:** - A clean algebraic demonstration that the Sundog signature is Blackwell sufficient for a registered, falsifiable subset of ARC-AGI tasks emphasizing core knowledge and spatial abstraction. - Empirical confirmation that the effective representation collapses toward a low-dimensional invariant structure (analogous to the 5D mesa result) under shadow coupling on those tasks. - A measured boundary where performance collapses — tasks requiring information outside the signature's capacity — with named failure modes. - A working Kaggle entry (even modest-score) demonstrating the operator, not as a competitive leaderboard bid, but as an existence proof. **Reach (do not claim):** - "Sundog solves ARC-AGI." - "Shadow coupling achieves human-level abstraction." - "The signature replaces learned representations for novel tasks." - "This approach scales to the 85% regime without additional machinery." - "Low-dimensional structured fields are sufficient for all core knowledge priors." The danger of this work is the same as the mesa roadmap's: a weak benchmark result could be misread as a falsification of the entire gravity program. It is not. A structural argument about *why* shadow coupling helps on a characterized task subset — with an explicit boundary where it doesn't — is itself a novel, publishable result in the Theory category. The roadmap is structured so that either outcome ratchets the program forward. ## Ratified Hook Language Safe hook: > Sundog vs. ARC-AGI asks whether shadow-coupled agents, using local > gauge-invariant projections and Blackwell-sufficient signatures, can > solve novel abstraction tasks without full state reconstruction — and > whether the effective representation collapses toward the known 5D mesa > subspace structure under this coupling. Short version: > The field knows enough. The agent doesn't need to reconstruct > everything — just the signature. Avoid: - "Sundog solves ARC." - "Shadow coupling is all you need for abstraction." - "Our approach beats transformers / foundation models on ARC." - "The 5D subspace is universal for core knowledge." - "Geometric methods are superior to learned representations." ## Claim Boundary This document does **not** claim that Sundog has demonstrated novel-task abstraction, competitive ARC-AGI performance, or that shadow coupling replaces learned representations. It claims that: 1. there is a coherent structural argument — signature as Blackwell-sufficient statistic under gauge-invariant shadow projection — for why shadow-coupled agents should generalize on core-knowledge abstraction tasks without reward hacking or full state reconstruction; 2. that argument is currently defended by the mesa lane's 5D subspace finding and the geometry lane's forward-rich / inverse-narrow field-shape pattern, not by any abstraction benchmark result; 3. the proof targets that would test the argument (ARC-AGI task subset, Blackwell sufficiency audit, dimensionality collapse check) are concrete enough to live in a roadmap rather than a ledger. If the Phase 3 Blackwell sufficiency audit fails — if the signature is not sufficient for the registered task class — the honest response is a named quarantine on "signature sufficiency for discrete abstraction," not a program retreat. The mesa and geometry receipts stand independently. ## Pre-Registered Hypothesis (v0.1) In the registered class of ARC-AGI tasks (core knowledge / abstraction without heavy reward signal), a shadow-coupled agent using: - Sundog-style local shadow projections (gauge-invariant, Procrustes-aligned), - signature as Blackwell sufficient statistic, and - the known 5D mesa subspace structure achieves non-trivial abstraction performance on held-out tasks with the signature alone being sufficient (no need for full state reconstruction). Performance degrades predictably outside the registered domain (e.g., when signature capacity is exceeded). ## Falsification Surface The ARC coupling claim can fail in five named modes: 1. **Representation vacuity** — the shadow projection, applied to ARC grid representations, produces features that are either (a) what any convolutional encoder would produce, or (b) so abstract that no Sundog-specific discipline is doing the work. The gauge invariance has no edge over a good embedding layer. 2. **Sufficiency failure** — the signature is not Blackwell sufficient for the registered task class. Core-knowledge tasks require mutual information that the signature discards. The dimensionality reduction is lossy in a load-bearing way. 3. **Subspace non-collapse** — the effective representation under shadow coupling does not collapse toward a low-dimensional invariant structure. The 5D mesa pattern is environment-specific (continuous fields) and does not transfer to discrete grids. 4. **Boundary absence** — the agent's performance does not degrade predictably at the registered boundary. Either it fails everywhere (no signal) or it succeeds everywhere (no falsifiable boundary), and neither outcome sharpens the gravity claim. 5. **Benchmark disconnect** — the Kaggle entry score is so low that the paper's structural argument cannot be grounded in any empirical artifact, and the "even modest" framing reads as excuse-making. Modes (1) and (2) are the most informative failures — they would say something specific about the limits of the shadow-coupling approach. Mode (5) is the most embarrassing but least informative. Mode (3) would be the most consequential for the broader Sundog program because it would bound the mesa subspace finding to continuous domains. ## Starting Assets These exist and are portable: - **Shadow projection + Faraday clean zero receipt** — the gauge- invariant local-projection operator and its zero-residual baseline from the three-body workbench. - **Gauge cocycle / Procrustes machinery** — the alignment operator that makes shadow projections comparable across reference frames. Requires adaptation from continuous 2D fields to discrete grids. - **5D mesa localization proof + signature sufficiency postulates** — the v3.1–v3.8 finding that the basin-attractor mechanism lives in an entangled 5D subspace at `net.7`. The sufficiency postulates are stated but untested outside the mesa environment family. - **Three-body invariants and isotrophy operators** — the existing invariant-subspace machinery ports cleanly as a toy-model comparison or warmup. - **Three-gate failure taxonomy** — residual / coverage / detection gates for classifying route failures. Directly applicable to the falsification battery (Phase 5). - **Pre-registration infrastructure** — the `docs/prereg/` ladder discipline and append-only methodology. ## Roadmap ### Phase 0 — Registered Task Subset Definition Goal: curate a falsifiable, pre-registered subset of ARC-AGI tasks that emphasizes core knowledge and spatial abstraction, and pin the exclusion criteria before any operator design. Deliverables: - A registered task subset drawn from the public ARC-AGI training and evaluation splits, with explicit inclusion criteria (core knowledge priors exercised: objectness, counting, symmetry, spatial reasoning, geometric transformation) and exclusion criteria (tasks requiring extensive sequential reasoning, language-like pattern matching, or information-theoretic capacity exceeding a stated bound). - Pre-registration document in `docs/prereg/arc/` following the existing append-only ladder discipline. - Baseline performance numbers: random, brute-force enumeration, and a simple CNN/transformer reference on the registered subset. - Task taxonomy mapping each included task to the core-knowledge prior(s) it exercises, so Phase 3 can audit sufficiency per prior. Execution artifact: [`docs/prereg/arc/PHASE0_TASK_SUBSET_SPEC.md`](prereg/arc/PHASE0_TASK_SUBSET_SPEC.md). Phase 0 receipt: [`docs/prereg/arc/P0_BASELINES.md`](prereg/arc/P0_BASELINES.md) -- **ADMIT** as of 2026-05-28 after baseline expansion: 36-task public-training subset registered, leak-control passed, and all preregistered cheap baselines scored 0/36 exact. Phase 1 is admitted with `0/36` exact as the floor any Sundog-specific result must clear. Exit criterion: the task subset, exclusion criteria, baseline numbers, and public-evaluation leak-control policy are filed and pre-registered. No operator design has begun. ### Phase 1 — ARC Grid Representation as Shadow Domain Goal: define the formal correspondence between ARC grids and the shadow-field domain, establishing the gauge-invariant projection operator for discrete grid representations. Deliverables: - Formal definition of the ARC grid as a structured field: cell values as field intensities, spatial layout as the manifold, color channels as fiber degrees of freedom. - The local shadow-projection operator adapted from continuous 2D fields to discrete grids: probe geometry, neighborhood structure, gauge-covariance under grid symmetries (rotation, reflection, translation, color permutation). - Procrustes alignment operator for comparing shadow projections across input-output pairs within a single ARC task. - Validation that the projection operator recovers known grid invariants (e.g., translation invariance, rotation equivariance) on synthetic test grids before touching ARC tasks. Exit criterion: the projection operator is defined, implemented, and validated on synthetic grids. The gauge-covariance properties are algebraically verified. Execution artifact: [`docs/prereg/arc/PHASE1_SHADOW_DOMAIN_SPEC.md`](prereg/arc/PHASE1_SHADOW_DOMAIN_SPEC.md). Status (2026-05-28): synthetic validation passed on translation, rotation, reflection, color-role permutation, and a shape-mismatch negative; Phase 2 projection scaffold admitted, with no sufficiency or Kaggle claim. ### Phase 2 — Shadow Projection Operator on ARC Tasks Goal: apply the shadow-projection operator to the registered ARC task subset and produce the raw projected representations. Deliverables: - Shadow projections computed for every task in the registered subset (both training demonstrations and test inputs). - Per-task closure residuals: how much information the projection retains vs. discards, measured against the full grid representation. - Initial signal-to-noise characterization: on which tasks does the shadow projection concentrate signal (low residual, structured projection), and on which does it disperse (high residual, noisy projection)? - Comparison against naive baselines (raw pixel features, simple convolution features) on the same information-retention metric. Exit criterion: projections computed, residuals measured, initial signal characterization filed. No sufficiency claim yet. Execution artifact: [`docs/prereg/arc/PHASE2_PROJECTION_SPEC.md`](prereg/arc/PHASE2_PROJECTION_SPEC.md). Status (2026-05-28): projection measurement passed on the registered public-training subset (`36` tasks, `266` grids). Mean signature-collision residual `0.028571`; mean train-pair residual `0.594295`; signal labels `20` dispersed / `9` mixed / `7` compact. Phase 3 sufficiency-spec writing is admitted; no decoder, sufficiency, public-evaluation, or Kaggle claim is admitted. ### Phase 3 — Signature Sufficiency Audit Goal: formalize and test Blackwell sufficiency of the signature for the registered task class. Deliverables: - Algebraic formulation: under what conditions is the shadow-projected signature a Blackwell-sufficient statistic for the input→output mapping in the registered task class? State the theorem or conjecture with explicit assumptions. - Empirical test: train a simple decoder (linear or shallow MLP) from the signature alone to predict the correct output grid. Compare against a decoder from the full grid representation. If the signature decoder matches or exceeds the full-grid decoder on the registered subset, the sufficiency claim is supported. - Per-task and per-core-knowledge-prior breakdown: which priors does the signature capture (objectness? symmetry? counting?) and which does it miss? - Named quarantines: tasks or priors where sufficiency fails, with explicit failure-mode classification (residual / coverage / detection gate, per the three-gate taxonomy). Exit criterion: the sufficiency claim is either (a) supported on the registered subset with measured residuals and named quarantines outside, or (b) cleanly falsified with specific failure modes named. Either outcome is publishable. Execution artifact: [`docs/prereg/arc/PHASE3_SUFFICIENCY_SPEC.md`](prereg/arc/PHASE3_SUFFICIENCY_SPEC.md). Status (2026-05-28): spec filed; decoder-admission roadmap filed; Pass A representation/decoder contract filed; Pass B split/floor/discrimination contract filed; Pass C learner/metric contract filed; Pass D receipt/command contract filed; freeze-marker runner wiring implemented. Three deterministic low-capacity receipts are filed and all are verdict task-hardness / decoder-failure, not support or failure of signature sufficiency. Phase 2 baseline comparison sharpened the problem: `P_shadow_grid_v0` preserves distinctness but makes input-output pairs farther apart than raw-pixel and coarse-feature baselines. Phase 3 therefore tested learnability of the `input_rep -> output_rep` mapping from demonstrations, with `signature_palette` as the primary representation and `signature_only` as a stricter ablation. A Blackwell sufficiency lane is now filed: algebraic conditions, held-out-task splits, Branch A/B/C criteria, a frozen Python/PyTorch decoder, timing probe, and full clean receipt. The `blackwell_task_decoder_v1` receipt is Branch C bounded failure: `signature_palette` scored zero exact matches on both held-out lanes. The matched full-grid control also scored zero exact matches, so the receipt does not support sufficiency and should not be described as a full-grid superiority result. Subsequent full-grid controls did not open the comparison arena: V2 (`blackwell_publictrain_rawgrid_gate_v2`, freeze marker `79C5B060`) used the public-training auxiliary pool and filed `full_grid_control_floor`, with zero exact matches on both held-out lanes; the compact-7 diagnostic (`50EAEBBF`) narrowed to the Phase 2 compact-signal tasks and filed `compact_full_grid_control_floor`, also with zero exact matches. Compact-7 is qualitatively distinct because shape exact reached `1.000` while palette exact remained `0.000`; per-instance residuals show dominant-background predictions with at most two slot-1 colors even when targets use 3--9 colors. That failure is named **dominant-color mode collapse**. Branch A then filed `per_task_coord_mlp_v1`, a stochastic per-instance coordinate MLP trained from scratch on each held-out instance's conditioning demonstrations. Its 20-shard binding receipt returned **`branch_a_full_grid_floor`**: `raw_grid_per_task` scored zero exact tasks on both held-out lanes, so no `signature_palette_per_task` vs. `raw_grid_per_task` sufficiency comparison is licensed. The failure character is **conditioning starvation + shape-generalisation failure**, distinct from V1/V2's noise-dominated failure and compact-7's dominant-color mode collapse. Four full-grid-control receipts (V1, V2, compact-7, Phase 3A) now agree on the held-out exact-grid floor across two task distributions and two learner families. Branches A, B, and C are closed in their filed learner families; Branch D, a different framing such as structured edit or residual prediction, is the only remaining admissible Phase 3 reopen path. That path is now started as `structured_edit_residual_v1`: model the output as an input-derived baseline canvas plus a residual edit mask and edit colors. Its 20-shard binding receipt returned **`branch_d_full_grid_edit_floor`**: `raw_grid_edit` scored zero non-baseline exact tasks on both held-out lanes, so no `signature_palette_edit` vs. `raw_grid_edit` sufficiency comparison is licensed. The named failure character is **edit-color-rule failure**: the structured-edit framing found nontrivial shape/canvas and edit-mask signal, but the per-task edit-color learner did not recover exact output colors. Five Phase 3 full-grid controls now agree on the held-out exact-grid floor across two task distributions, two learner families, and two output framings. The first bottleneck-targeted Branch D variant, `structured_edit_color_rule_v2`, kept the baseline and edit-mask components fixed and replaced only the edit-color MLP with a deterministic conditioning-derived color-rule bank. Its binding receipt returned **`branch_d_color_rule_full_grid_floor`**: the raw-grid color-rule arena did not open, so no signature-vs-full-grid sufficiency comparison is licensed. Six Phase 3 full-grid controls now agree on the held-out exact-grid floor. The variant did, however, shift the bottleneck into measured slices: 41% edit-mask failure, 16% color-rule-selection failure, and 9% rule-bank-coverage failure. Future Phase 3 reopens now require a new pre-registered mask-targeted Branch D variant, selection-refinement Branch D variant, rule-bank extension Branch D variant, or Branch E spec. The mask-targeted variant is now filed as `structured_edit_mask_target_v3`: keep the baseline picker and deterministic edit-color rule bank fixed, and replace only the edit-mask predictor with a conditioning-derived mask-candidate bank. Its binding receipt returned **`branch_d_mask_target_full_grid_floor`**: the mask repair did not lift the floor, mask-stage labels still dominated, and the legacy learned mask candidate still won a non-trivial share of selections. Seven Phase 3 full-grid controls now agree on the floor. With both named structured-edit bottlenecks probed by deterministic banks and both floored, Branch E is the live frontier. The first Phase 3E spec is now filed as `PHASE3E_SIGNATURE_FIBER_CERTIFICATE_SPEC.md`: before training another solver, test whether registered ARC contexts contain exact or near `signature_palette` context-fiber collisions with incompatible required behavior. Its binding receipt returned **`phase3e_deferred_label_vacuity`** with a clean dual finding: no registered exact or near signature-fiber collision was found, and `signature_palette` strongly separates distinct registered tasks (minimum cross-task context distance `0.207`, above every frozen radius), but the Branch D-derived program-sketch oracle is prior-blind on 68% of primary contexts. The seven floors are therefore not explained by a registered signature collision, but Phase 3E also does not prove sufficiency or license a locality-positive Branch E selector. The v2 oracle repair then filed a binding receipt: **`phase3e_v2_deferred_sparse_fibers`**. It repairs and certifies the oracle defect (anti-vacuity 0.00, anti-prior-laundering 0% violations, anti-solver-leakage `unique_core=0.36` / `exact_lookup=0.04`; the four v1-blind priors are now non-vacuous with 6-7 of 9 facets populated) while leaving the frozen geometry byte-identical to v1 (0 collisions, 0 near pairs, minimum cross-task distance `0.207`, 0% fidelity-passing). The remaining Phase 3E deferral cause is therefore registered-context geometric sparsity, not label vacuity. The Phase 0 context expansion path then ran exactly the registered balanced expansion: **108 tasks, 18 per prior**, with verdict **`phase0_fiber_expansion_admit`**. The expanded Phase 3E v2 certificate returned **`phase3e_v2_expanded_oracle_regression`**: the anti-solver-leakage gate that was clean at 36 tasks failed on the larger register (`core_sketch_exact_lookup_fraction 0.04 -> 0.351`, threshold 0.20), while the fibers remained sparse despite `U_primary` growing 25 -> 336 (min cross-task distance `0.207 -> 0.196`, still 0 near pairs and 0% fidelity-passing). The expansion therefore neither promotes nor blocks signature sufficiency or Branch E; it says the absolute-epsilon fiber-collision approach is not populated by a 3x balanced register expansion, and the v2 oracle leakage calibration is not register-size robust. The relative-locality certificate then returned **`phase3e_relative_locality_negative`** on the same 108-task register: palette rank neighbors are statistically above prior-stratified random (`p=0.0002`), but the signal is metadata-tied (`delta=+0.006`), raw-grid-dominated (`delta=-0.036`), sign-unstable across k, below the random-margin bar, and 28.5% hard-incompatible. Across absolute and relative geometry, the Phase 3E certificate program finds no certifiable usable fiber or rank-local locality for `signature_palette_context`; Branch E now rests on capability grounds, not certified fiber geometry. Branch E then ran as a deterministic typed program-search solver over the registered primitive library, selecting programs only by train-pair consistency and never by signature geometry. Its binding verdict is **`branch_e_capability_demonstrated`**: the first non-zero exact-match result in the ARC lane, clearing the non-trivial floor with 2 distinct held-out tasks solved on both `test_lodo` and `pttest` (`be94b721` via `extract_largest_component`, `f25fbde4` via `crop_bbox >> scale`) plus two validation solves. The result is modest and capability-grounded; it does not prove Blackwell sufficiency, change the certificate verdicts, or make any public-evaluation/Kaggle claim. Branch E v2 then admitted the deferred intricate mask families, morphology cross-product, and depth-3 composition. Its binding verdict is **`branch_e_v2_capability_replicated`**: it retained the same two gated solves (`be94b721`, `f25fbde4`) on both `test_lodo` and `pttest`, solved zero new gated tasks, and dropped validation from 2 to 1 solved task per lane because the larger train-consistent program set can crowd the correct answer out of the top two. The deterministic program-search capability is therefore bounded near the v1 baseline; lifting it likely requires a different solver class rather than more deterministic primitives. ### Phase 4 — 5D / Low-Dimensional Collapse Check Goal: measure whether shadow coupling induces collapse toward a low-dimensional invariant structure analogous to the mesa 5D result. Deliverables: - Dimensionality analysis of the shadow-projected representation space: PCA / SVD spectrum, effective dimensionality measures, comparison against the full grid representation's dimensionality. - If collapse occurs: characterize the invariant subspace. Is it 5D or nearby? Does it share structural properties with the mesa `net.7` subspace (entangled, resists factorization)? - If collapse does not occur: characterize why. Is the ARC grid domain too heterogeneous? Is the registered subset too broad? Is the projection operator under-constrained? - Cross-reference with Phase 3: do the tasks where sufficiency holds coincide with the tasks where dimensionality collapse occurs? Exit criterion: the dimensionality question is answered with measured spectra and explicit comparison to the mesa result. The answer feeds the paper's central structural claim. ### Phase 5 — Falsification Battery Goal: run explicit falsifiers to characterize the boundary where shadow coupling fails. Deliverables: - **Non-local probes**: tasks requiring global information that local shadow projections cannot capture. Measure performance degradation and classify the failure mode. - **Capacity-exceeding tasks**: tasks where the information-theoretic content of the correct output exceeds the signature's capacity (as measured in Phase 3). Confirm that performance degrades predictably. - **Gauge-breaking perturbations**: tasks where the grid symmetries the projection operator exploits are violated or misleading. Measure whether the operator fails gracefully or catastrophically. - **Comparison with full-state baselines**: on the falsification tasks, does a full-state decoder succeed where the signature decoder fails? This confirms the failure is signature-specific, not task-impossibility. Exit criterion: a characterized boundary with named failure modes, publishable as the falsification receipt in the paper. ### Phase 6 — Kaggle Submission + Paper Goal: produce the ARC Prize 2026 Paper Track submission and a linked Kaggle entry. Deliverables: - **Paper** (Theory category): structured around the "why" — Blackwell sufficiency of the signature under gauge-invariant shadow coupling, with the 5D mesa subspace as the dimensionality-reduction mechanism. Sections: motivation (Goodhart sidestep applied to abstraction), formal setup (shadow projection, signature, gauge covariance), sufficiency result (Phase 3 receipt), dimensionality result (Phase 4 receipt), falsification boundary (Phase 5 receipt), connection to the broader Sundog program (mesa + geometry two-substrate convergence). Pre-registered hypothesis stated and adjudicated. - **Kaggle entry**: a working submission demonstrating the shadow-projection operator on the ARC-AGI evaluation set. Score is secondary to existence — the entry grounds the paper's structural argument in a runnable artifact. - **Pre-registration adjudication**: the Phase 0 pre-registered hypothesis is explicitly adjudicated — confirmed, partially confirmed with named quarantines, or falsified with named failure modes. Exit criterion: paper submitted, Kaggle entry linked, pre-registration adjudicated. ## Promotion Criteria This roadmap is self-contained. If Phase 3 produces a clean sufficiency result and Phase 4 confirms dimensionality collapse, the ARC coupling earns its own receipt in the Gravity Ledger's two-substrate convergence narrative (third substrate: discrete abstraction). If Phase 3 fails, the named quarantine is filed in the Gravity Ledger and the Capset Ledger's Front-A evaluator machinery is the natural fallback coupling surface. ## Timeline and External Constraints The ARC Prize 2026 Paper Track has a submission deadline. All phases are sequenced to deliver a submittable paper and linked Kaggle entry by that deadline. Phase 0 should be fast (task curation, not operator design). Phases 1–2 are the implementation core. Phases 3–4 are the measurement core. Phase 5 is the falsification discipline. Phase 6 is assembly. If the timeline compresses, Phase 5 can be scoped down to the two most informative falsifier classes (non-local probes and capacity-exceeding tasks) without compromising the paper's structural argument. Phase 4 can report preliminary dimensionality results if the full spectrum analysis is not complete. Phase 3 is non-negotiable — the sufficiency claim is the paper's differentiator.