# Coarse-Graining Proof Roadmap > The staged proof path for **Postulate 1** — the Coarse-Graining Postulate, the > theorem the project has been circling for a year without naming. Source and > ranking: [`internal/theory/postulations.md`](../internal/theory/postulations.md) ▸ > Postulate 1. This roadmap turns that conjecture into a sequenced campaign of > analytical and empirical phases, each with entry/exit criteria and a > pre-registered negative, so the result can be re-audited cleanly and so the > grand-frame claim stops resting on a shared adjective. > > **Working hook.** Sundog does not need to invert the world. It needs the > signature to be a *sufficient statistic for the optimal action* even when the > signature cannot reconstruct the world. If that is a theorem, the > cross-substrate "convergence" is a shared **operator**, not a shared word — > and the pre-registered failure boundary is *predicted*, not patched. > > **Status as of 2026-05-31:** Phase 0 definitions lock landed at > [`proof/POSTULATE1_DEFINITIONS.md`](proof/POSTULATE1_DEFINITIONS.md); > Phase 1 LQG proof reviewed and closed **positive** at > [`proof/PHASE1_LQG.md`](proof/PHASE1_LQG.md); Phase 2 finite-MDP proof > reviewed and closed **positive** at > [`proof/PHASE2_MDP.md`](proof/PHASE2_MDP.md); Phase 3 boundary theorem > reviewed and closed **positive** at > [`proof/PHASE3_BOUNDARY.md`](proof/PHASE3_BOUNDARY.md); Phase 4 three-body > spec is drafted at > [`proof/PHASE4_THREEBODY.md`](proof/PHASE4_THREEBODY.md); the Bayesian-floor > controller buildout is staged and BF-4 smoke-passed at > [`proof/PHASE4_BAYESIAN_FLOOR_BUILDOUT.md`](proof/PHASE4_BAYESIAN_FLOOR_BUILDOUT.md); > BF-4b off-set calibration first receipt failed criterion (2) (cell `off`, > regret CI `[0, 0]`), but the satisfiability probe validated the pre-registered > cell (oracle-signature headroom CI `[0.064, 0.499]`). The compute-unconstrained > Information-Accessibility Diagnostic **closed 2026-05-31 with Privileged-only > verdict on three-body** (8 seeds × 512 particles × 800-step horizon × 800-step > hold; mean regret `0.0213`, 95% CI `[-0.00055, 0.052]` — includes 0; only ~7% > of oracle headroom recovered; 4/8 negative-regret rate): the off-set headroom > is NOT recoverable from `Φ` history on the three-body cell-`Φ` triple. > Per the Privileged-only branch, BF-5 is NOT the path forward; substrate > re-scope launched. Balance substrate-empirical leg **closed 2026-05-31 > Φ-ACCESSIBLE** at > [`proof/PHASE4_BALANCE.md`](proof/PHASE4_BALANCE.md): > 8 seeds × 512 particles × 0.5s horizon on `near_fall × light_elevation=8`, > mean regret `0.4089`, 95% CI `[0.1040, 0.7132]` — CI lower well above 0, > 0/8 negative-regret rate, 153% of Phase-15 reference recovered. **Phase 4 > closes positive overall on Balance with the three-body negative documented as > a substrate boundary;** Phase 5 unblocked (Balance + Mesa give the ≥2 > substrates needed); > Phase 6 > lambda-confound control **closed 2026-05-29 with Branch 3 verdict** at > [`proof/PHASE6_LAMBDA_CONTROL.md`](proof/PHASE6_LAMBDA_CONTROL.md): > Control A (no-op) HELD (`old_basin_pref` 0.003 / 5.367 at λ=0.95 / 0.97 delta — > cliff still localized to (0.95, 0.97]); Control B (rescale) did NOT follow the > predicted `lambda_eff` map (κ=2 cliff sits in (0.90, 0.91], not at predicted > 0.909, with the {0.90, 0.92} bracket inverted; κ=0.5 bracket {0.97, 0.98} all > protected after midpoint at 0.975 anchored it). Branch 1 ruled out → Mesa > survives as Phase-5 substrate, Postulate 4 stays alive, anniversary cliff > citable as operating-envelope reference only; Postulate-2 capacity-law > promotion blocked pending optimizer diagnosis. > This is a > theory-track roadmap, not an operating-envelope workbench — it is the *trunk* > of which the Formal Separability Theorem appendix > ([`SUNDOG_V_MESA.md`](SUNDOG_V_MESA.md), promoted from > [`SUNDOG_V_GRAVITY.md`](SUNDOG_V_GRAVITY.md) Candidate 1) is a corollary; it > does **not** replace or fork that front (see §2). No claim here is > public-facing until Phase 5 lands and the §6 gating control clears. Nothing in > this roadmap unblocks or re-strengthens any coupled surface on its own. ## 1. Research Object Stated to slot directly into [`SCIENTIFIC_CRITERIA.md`](SCIENTIFIC_CRITERIA.md) ▸ "Research Object." The object is **not** the broad theorem in full generality. It is the measurability predicate: > Let world microstate `x ∈ X`, signature map `Φ : X → Σ`, signature > `σ = Φ(x)`, true objective `J`, optimal policy `π*`. Let `𝓕_σ` be the > σ-algebra generated by `Φ`. A task is **Sundog-solvable** iff `π*` is > `𝓕_σ`-measurable — i.e. `π*(x) = g(Φ(x))` a.e. for some measurable `g` — > equivalently, `Φ` is **Blackwell-sufficient for the control objective** (not > for state reconstruction). That object is small enough to prove in the linear-Gaussian and finite-MDP cases (Phases 1–2), to falsify on a known substrate (Phase 4), and to test for cross-substrate sameness (Phase 5). The nonlinear/learned general case is **not** in scope and is named as future work, per house posture. ## 2. Relationship to the existing formal front (no fork) The Gravity Ledger already pairs an empirical mesa front with a **Formal Separability Theorem appendix** ([`SUNDOG_V_MESA.md`](SUNDOG_V_MESA.md)). That appendix names the structural condition under which a signature-trained agent does *not* grow an internal reward proxy. The Coarse-Graining Postulate is the **generalization**: separability is the special case of `𝓕_σ`-measurability where the agent class is bounded so it cannot invert `Φ` (this is Postulate 2's capacity relativity, [`internal/theory/postulations.md`](../internal/theory/postulations.md) ▸ Postulate 2). To avoid a parallel theory track: - The separability appendix remains the **mesa-specific** formal statement. - This roadmap proves the **substrate-independent** trunk and shows separability falls out as a corollary in Phase 2. - The mesa↔geometry crossover ([`MESA_CROSSOVER_NOTE.md`](MESA_CROSSOVER_NOTE.md)) is the in-vitro/in-the-wild pair this roadmap formalizes in Phase 5; its post-audit hedged strength is inherited unchanged, not re-opened. ## 3. Phases Each phase: **Entry** (what must be true to start) · **Work** · **Exit** (the artifact that closes it) · **Pre-registered negative** (the outcome that stops or redirects the campaign, fixed *before* the phase runs, per AGENTS.md ▸ "Pre-register the negative"). ### Phase 0 — Definitions lock - **Entry.** This roadmap filed. - **Work.** Pin `X, Φ, Σ, J, π*, 𝓕_σ` and the Sundog-solvable predicate as a single definitions section; reconcile notation with the founding `H(x) = ∂S/∂τ` (the Postulate 6 thread) and with [`SCIENTIFIC_CRITERIA.md`](SCIENTIFIC_CRITERIA.md) ▸ Research Object. No new numbers introduced; existing locked figures (λ≈0.952588 cliff; core U=526, p=0.26, n=30; 16× acquisition; ρ≈0.086 parhelic-belt) are *referenced*, never re-derived or restated at a different value (spec self-consistency rule). - **Exit.** `docs/proof/POSTULATE1_DEFINITIONS.md` reviewed for internal consistency (every symbol used once, defined once). **Landed 2026-05-16** at [`proof/POSTULATE1_DEFINITIONS.md`](proof/POSTULATE1_DEFINITIONS.md). - **Pre-registered negative.** If the predicate cannot be stated without reference to an unmeasurable quantity in *every* candidate substrate, the postulate is not operational — stop, and demote Postulate 1 to speculative in [`internal/theory/postulations.md`](../internal/theory/postulations.md). ### Phase 1 — Linear-Gaussian existence proof (the computable case) - **Entry.** Phase 0 exit. - **Work.** Prove the schema where everything is computable: linear dynamics, Gaussian noise, quadratic cost (LQG). The separation principle already gives a low-dimensional control-sufficient statistic (the Kalman state estimate); recast it as `𝓕_σ`-measurability of `π*` and show signature-only control achieves the Bayes-optimal cost. Run the **Postulate 6 toy check** here: compute `∂S/∂τ` at the optimum and the Fisher information of `Φ`; record whether they are proportional. - **Exit.** A written proof (LQG) + the recorded Postulate-6 proportionality result (pass/fail), both in `docs/proof/PHASE1_LQG.md`. **Reviewed and closed positive 2026-05-16** at [`proof/PHASE1_LQG.md`](proof/PHASE1_LQG.md). - **Pre-registered negative.** If signature-only LQG control does **not** reach Bayes-optimal cost on the `𝓕_σ`-measurable set, the schema is false in its easiest case — halt the whole roadmap and file the falsification against [`SCIENTIFIC_CRITERIA.md`](SCIENTIFIC_CRITERIA.md) ▸ Falsifiable Expectations. ### Phase 2 — Finite-MDP sufficiency + separability corollary - **Entry.** Phase 1 exit (positive). - **Work.** Extend to finite MDPs: prove `Φ` is control-sufficient iff `π*` is constant on `Φ`-fibers over the occupancy support, via policy equivalence on `Φ`-fibers (Markov quotient in the lumpable special case). Derive the **Formal Separability** statement (§2) as the bounded-agent-class corollary that *subsumes* the appendix, leaving it in place (no replace). - **Exit.** Proof + a *constructed* finite-MDP counterexample where sufficiency fails, characterizing the boundary, in `docs/proof/PHASE2_MDP.md`. **Reviewed and closed positive 2026-05-16** at [`proof/PHASE2_MDP.md`](proof/PHASE2_MDP.md). - **Pre-registered negative.** If the separability corollary does **not** drop out of the trunk, this roadmap has forked the theory rather than generalized it (§2 violated) — stop, reconcile with the mesa appendix before continuing. ### Phase 3 — The boundary theorem (predict the pushable-occluder) - **Entry.** Phase 2 exit. - **Work.** Prove the negative direction: if the control-relevant bit is not `𝓕_σ`-measurable (it exists only *after* a preparatory action), no `𝓕_σ`-policy is optimal. Map this onto the pre-registered pushable-occluder design ([`PHASE2_BLOCKS_DESIGN.md`](PHASE2_BLOCKS_DESIGN.md)). - **Exit.** Proof + an explicit statement that the pushable-occluder failure is now a *theorem-predicted* boundary, cross-filed in [`PHASE2_BLOCKS_DESIGN.md`](PHASE2_BLOCKS_DESIGN.md) and the "river and dam" entry of [`internal/theory/analogies.md`](../internal/theory/analogies.md). **Reviewed and closed positive 2026-05-16** at [`proof/PHASE3_BOUNDARY.md`](proof/PHASE3_BOUNDARY.md). - **Pre-registered negative.** If a flat `𝓕_σ`-controller *does* solve a task whose decisive bit is provably not `𝓕_σ`-measurable, the boundary is wrong and Postulate 1's failure prediction is falsified — record, do not rescue. ### Phase 4 — Measured substrate leg (three-body, against the Bayesian floor) - **Entry.** Phase 3 exit. Bayesian-floor baseline available in the chosen workbench (per the "Bayesian floor" standing commitment in [`internal/feedback/quarantine/attack_vectors.md`](../internal/feedback/quarantine/attack_vectors.md) — a baseline *inside* the workbench, **not** a `bayes_v_sundog` track). - **Work.** Use the planar restricted three-body near-escape pocket, where `Φ` (the guarded accelerometer-proxy signature) is available and the Bayesian-floor buildout pins the tractable `π*` proxy path. Measure signature-only regret vs the Bayes-optimal controller once that floor exists. - **Exit.** Regret curve + the pre-registered gate result, in `results/proof/phase4/` with a summary in `docs/proof/PHASE4_THREEBODY.md`. **Spec drafted 2026-05-16** at [`proof/PHASE4_THREEBODY.md`](proof/PHASE4_THREEBODY.md); empirical entry advanced through the BF-4 Bayesian-floor smoke: the current three-body workbench now has a smoke-passed separate `bayes_floor_particle_mpc` evaluator under the signature information regime, while the privileged oracle / `forward_oracle_strict` references remain yardsticks only. BF-4b off-set calibration first receipt failed criterion (2): the cell classified `off`, floor sanity passed, but the off-set regret CI was `[0, 0]`. Follow-up diagnostic work rejected multiplier-only tuning, retained the energy-trend terminal value, and validated the pre-registered cell by the satisfiability probe (oracle-signature headroom mean `0.310`, CI `[0.064, 0.499]`). **BF-4b Information-Accessibility Diagnostic closed 2026-05-31 with Privileged-only verdict** (`_bf4b-accessibility`, 8 seeds × 512 particles × 800-step horizon × 800-step hold, sharded 3-wide via [`scripts/threebody-phase4-iad-{shard,concurrent,merge}.mjs`](../scripts); mean off-class regret `0.0213`, 95% CI `[-0.00055, 0.052]` — includes 0 by the strict pre-registered rule; ~7% of oracle headroom recovered; 4/8 negative-regret rate; `floor_status = non_decisive_floor_repair_required`). **BF-5 is NOT the next step** per the pre-registered Privileged-only branch — more floor engineering will not make the off-set arm satisfiable on this cell. The Phase-4 substrate-empirical leg as currently scoped needs re-scope (different cell, different `Φ`, different substrate, or accept the Phases 1-3 analytical trunk without a positive Phase-4 substrate leg). See [`proof/PHASE4_BAYESIAN_FLOOR_BUILDOUT.md`](proof/PHASE4_BAYESIAN_FLOOR_BUILDOUT.md) ▸ §IAD Receipt for full per-seed table and run inventory. - **Pre-registered negative.** Gate, fixed now: signature-only regret vs Bayes must → 0 (within bootstrap CI) **on** the `𝓕_σ`-measurable cell set and stay bounded away from 0 (CI excludes 0) **off** it. If regret is bounded-away *on* the measurable set, sufficiency-for-control is empirically false on a real substrate → halt and falsify. If regret → 0 *off* the measurable set, the boundary (Phase 3) is wrong → reopen Phase 3. - **ON-arm populated (2026-06-06).** The committed three-body receipt had `on=0` (the gate's ON arm never populated; Balance measured only the OFF-direction "Φ-accessible" positive). The ON arm is now populated on the substrate where measurability is *provable* (Phase-2 Theorem 1): on a constructed finite POMDP the two-sided gate **fires** — `on` regret `0.0`, CI `[0,0]`; `off` regret `0.3525`, CI `[0.326, 0.380]` (excludes 0); verdict `on_arm_pass_two_sided_gate_fired`. FROZEN slate + receipt: [`proof/PHASE4_ON_ARM_FINITE_POMDP_SLATE.md`](proof/PHASE4_ON_ARM_FINITE_POMDP_SLATE.md), `results/proof/phase4-on-arm/`. **Scope:** this is the *sufficiency* ON-arm on a constructed substrate (provable measurability, not fit); it does NOT establish body-resistance/regime-2, a real/high-dim substrate, or a trained body. The real-substrate ON cell (the thing three-body could not produce) remains open. - **Compute discipline (AGENTS.md ▸ ~10-minute rule, load-bearing).** This leg exceeds the ~10-min inline budget. Do **not** run it inline. Stage the exact invocation in `docs/proof/PHASE4_THREEBODY.md` as runnable PowerShell anchored to the existing single-threaded threebody envelope harness (the `threebody:phaseNN` pattern; sequential `envelopeCases`, no workers — a many-core host gives *offload, not speedup*), with: a measured per-cell rate from a capped probe (probe itself ≤ ~10 min), the extrapolated full wall-clock, resume-safety notes, the `results/proof/phase4/manifest.json` `startedAt`/`completedAt` read-back path, and the branch each gate outcome selects. Per Runs 4–5 in AGENTS.md, a smoke that will not fit a ~1 h interactive session moves to a long-budget runner (`workflow_dispatch`), not an agent session. Exact module/flags are pinned in the Phase 4 spec at spec time; this roadmap does not invent them. ### Phase 5 — Cross-substrate sameness (the anti-equivocation leg) - **Entry.** Phase 4 exit (positive). - **Work.** Show the **same** coarse-graining operator structure — the *measured* sufficient statistic, not the eyeballed forward-rich/inverse-narrow shape — in ≥2 substrates: mesa in-vitro (the λ-cliff subspace at `net.7`) and geometry in-the-wild (the promoted parhelion-offset handle on the strict 3-photo subset, at its settled post-audit strength — inherited unchanged from [`MESA_CROSSOVER_NOTE.md`](MESA_CROSSOVER_NOTE.md), **not** re-opened). Optional third: the vortex/wishing-well toy ([`internal/theory/postulations.md`](../internal/theory/postulations.md) ▸ Postulate 5). - **Exit.** A cross-substrate operator-identity table (same `Φ` up to isomorphism of the coarse-graining) in `docs/atlas/ATLAS_PHASE5_CROSS_SUBSTRATE.md`, hedged at settled strength. - **Pre-registered negative.** If the substrates do **not** share the operator structure under measurement, the cross-substrate convergence *was* the equivocation of [`internal/feedback/quarantine/attack_vectors.md`](../internal/feedback/quarantine/attack_vectors.md) ▸ #1 — concede it in those words; the grand frame loses its empirical support and Postulate 1 stays a single-substrate result. ### Phase 6 — Gating control: the λ-confound (not optional, runs before any public cite) - **Entry.** May run in parallel with Phases 1–3; **must** clear before the λ-cliff is cited in any anniversary surface or before Postulates 2/4 are promoted out of speculative. - **Work.** Show the Mesa cliff is **invariant** under a transformation that should not move it and **moves predictably** under reward-gradient rescale (the [`internal/feedback/quarantine/attack_vectors.md`](../internal/feedback/quarantine/attack_vectors.md) ▸ #4 defense — an experiment, not an argument). - **Exit.** `docs/proof/PHASE6_LAMBDA_CONTROL.md` with the invariance/scaling result. **Spec staged 2026-05-16; empirical run 2026-05-29; closed 2026-05-29 with Branch 3 verdict** at [`proof/PHASE6_LAMBDA_CONTROL.md`](proof/PHASE6_LAMBDA_CONTROL.md): no-op invariance held (Control A: `old_basin_pref` 0.003 protected at λ=0.95 delta, 5.367 collapsed at λ=0.97 delta — cliff still localized to (0.95, 0.97]); reward-rescale did not follow the predicted `lambda_eff` map (Control B: κ=2 cliff in (0.90, 0.91] not at predicted 0.909, with {0.90, 0.92} bracket inverted; κ=0.5 bracket {0.97, 0.98} all protected after midpoint at 0.975 anchored it). Anniversary cliff language survives as operating-envelope reference; Postulate-2 capacity-law promotion blocked pending optimizer diagnosis (reward normalization, value-loss scale, gradient norms per Branch 3). **Substrate split close 2026-05-31**: three-body IAD closed Privileged-only at [`proof/PHASE4_BAYESIAN_FLOOR_BUILDOUT.md`](proof/PHASE4_BAYESIAN_FLOOR_BUILDOUT.md) ▸ §IAD Receipt (mean regret 0.0213, CI `[-0.00055, 0.052]` — includes 0; ~7% reference recovered; 4/8 negative-regret rate); Balance substrate-leg re-scope closed Φ-ACCESSIBLE at [`proof/PHASE4_BALANCE.md`](proof/PHASE4_BALANCE.md) (mean regret 0.4089, CI `[0.1040, 0.7132]` — lower well above 0; 0/8 negative-regret rate; 153% of Phase-15 reference recovered on the `near_fall × light_elevation=8` cell with `sundog_shadow`-signature vs `bayes_floor_shadow_particle`). **Phase 4 substrate-empirical leg closes positive on Balance with three-body documented as a substrate boundary case;** Phase 5 entry unblocked. - **Pre-registered negative.** If the cliff moves under the no-op transform, λ is an optimizer artifact: Mesa loses its standing as a Phase-5 substrate, Postulate 4 dies, Postulate 2's cliff-prediction is unsupported, and the anniversary cliff language is pulled. This is the cheapest result that can collapse the grand frame — run it early. ## 4. Falsifiable Expectations (mirrors SCIENTIFIC_CRITERIA posture) The postulate is **weakened / dead** if: Phase 1 fails (false in the computable case); Phase 2 fails to yield separability (theory forked); Phase 4 regret is bounded-away on the measurable set; Phase 6 cliff moves under the no-op. It is **strengthened** if: Phases 1–3 prove cleanly; Phase 4 gate passes on a real substrate; Phase 5 shows a *measured* shared operator across ≥2 substrates; the Postulate 6 **metric/Jacobian interpretation** holds in the LQG toy — i.e. Fisher information is the noise-weighted contraction of the signature Jacobian (a *qualified* pass, recorded 2026-05-16 in [`proof/PHASE1_LQG.md`](proof/PHASE1_LQG.md)). Literal signed-scalar `I_Φ ∝ ∂S/∂τ` is **not** the strengthening condition and is known to fail. This qualified pass holds under the torque-coordinate reading only; it does **not** by itself close the year-one proper-time `H(x)`-was-an-entropy arc, and that stronger line stays unshipped regardless of Phase 1. ## 5. Compute & spec discipline - AGENTS.md ▸ "Workflow — Running Experiments": analytical phases (0–3) are desk work; empirical phases (4, 6) obey the ~10-minute rule — capped probe, extrapolate, stage the long run for the operator/runner, record measured rates in the phase doc, pre-register the negative before running. - Spec self-consistency: derived numbers reconciled across phases; the gate in a phase's pre-registered negative is the *exact* unchanged condition the phase runs; ablation metrics are stated against the privileged truth (Bayes-optimal), not a tuned baseline. - Artifacts live under `docs/proof/` and `results/proof/`; only `dist/` is ever published (AGENTS.md ▸ Publishing Shape) — this roadmap is research-internal. ## 6. Cross-references - **Origin / ranking:** [`internal/theory/postulations.md`](../internal/theory/postulations.md) ▸ Postulate 1 (trunk), 2 (capacity relativity), 3 (conservation law), 4 (5D invariant), 5 (substrates), 6 (founding `H`). - **The analogy this formalizes:** [`internal/theory/analogies.md`](../internal/theory/analogies.md) ▸ "The thermometer / statistical mechanics." - **The attack this answers:** [`internal/feedback/quarantine/attack_vectors.md`](../internal/feedback/quarantine/attack_vectors.md) ▸ #1 (equivocation), with the Bayesian-floor standing defense feeding Phase 4. - **Existing formal front (not forked):** [`SUNDOG_V_MESA.md`](SUNDOG_V_MESA.md) (Separability appendix), [`SUNDOG_V_GRAVITY.md`](SUNDOG_V_GRAVITY.md) (ledger), [`MESA_CROSSOVER_NOTE.md`](MESA_CROSSOVER_NOTE.md) (in-vitro/in-the-wild pair). - **PDE sidecar:** [`proof/PDE_DETERMINING_MODES_POSTULATE1.md`](proof/PDE_DETERMINING_MODES_POSTULATE1.md) drafts the Navier-Stokes Candidate 1 reading: determining modes imply Postulate-1 sufficiency when they reconstruct state, while the stricter Sundog question is whether objective-specific control sufficiency can hold on coarser fibers. - **Boundary substrate:** [`PHASE2_BLOCKS_DESIGN.md`](PHASE2_BLOCKS_DESIGN.md) (pushable occluder = the Phase 3 predicted boundary). - **Scope guardrails:** [`SCIENTIFIC_CRITERIA.md`](SCIENTIFIC_CRITERIA.md), `presentation/claims-and-scope.md`. - **Deferred siblings:** the publication-gating wording fixes remain in [`internal/roadmaps/fix_roadmap.md`](../internal/roadmaps/fix_roadmap.md) (owner-deferred until the brainstorm docs are massaged and staged).