AK-OS v1.0.1: an operating system for high-dimensional mathematical exploration
Core: provable, 1D constructible persistence over a field
[Spec]: auditable, post-truncation (non-expansive) extensions and search policies
This repository contains AK–HDPST v17.0.1, positioned not as a “universal solver” for major conjectures, but as a proof-oriented operating system (AK-OS) for exploring them with explicit scope boundaries, budgets, and audit artifacts.
A companion project, AK_AP (Arithmetic Programs) v1.0.1, provides closed-world calibration on arithmetic moduli spaces (finite-field “Weil world” and “FLT world”) under the same OS, using known theorems as boundary inputs.
Strong guarantees are strictly confined to the Core scope. Everything else must be explicitly marked [Spec] and audited.
.
├─ OLD/ # Old files (kept as cautionary archive; may contain obsolete claims)
│ └─ ... # historical snapshots (v1.x–v16.x)
├─ AK–HDPST_v17.0.1/
│ ├─ AK_HPDST_v17.0.1.tex # XeLaTeX source (main paper)
│ └─ AK_HPDST_v17.0.1.pdf # compiled PDF (main paper)
└─ AK–Arithmetic_Programs_v1.0.1/
├─ AK–Arithmetic Programs v1.0.1.tex # XeLaTeX source (companion)
└─ AK–Arithmetic Programs v1.0.1.pdf # compiled PDF (companion)
- Not: a claimed proof of any grand conjecture (Navier–Stokes, BSD, RH, FLT, Langlands, etc.).
- Is: a modular OS for exploration, built around:
- an exact truncation operator
T_tau(“bar deletion”) - a filtered lift
C_tau(defined up to filtered quasi-isomorphism) - a δ-ledger for budget accounting (algorithmic / discretization / measurement)
- tower diagnostics
(mu_collapse, nu_collapse)for invisible limit failures (Type IV) - windowed gates (B–Gate⁺ / Overlap Gate) operating after truncation
- a search layer HDPS ([Spec]) that builds a Map of Validity on parameter spaces
- an exact truncation operator
-
Core
- One-parameter, constructible persistence over a field
k - Exact truncation
T_tau: delete bars of length<= tau(reflective localization), idempotent, 1-Lipschitz for the interleaving distance - Filtered lift
C_tauup to f.q.i., compatible with persistence:P_i(C_tau F) ≅ T_tau(P_i F)(at the persistence layer)
- One-way bridge (in
D^b(k-mod)under amplitude<= 1and a t-exact realization):PH1(F)=0 => Ext^1(R(F), k)=0
- Tower diagnostics
(mu_collapse, nu_collapse)computed after truncation, detecting Type IV failures - Windowed proof policy with explicit budgets and audit outputs
- One-parameter, constructible persistence over a field
-
[Spec]
- Domain-specific realizations (arithmetic / PDE / Fukaya / Langlands, etc.), guarded by Core
- HDPS: Terrain Cells, Hunter / Mapper / Lifter agents, Map of Validity
- PF/BC after-collapse comparator, Mirror / Transfer, A/B soft-commuting policies
- Quantitative heuristics and exploration strategies
Rule: every [Spec] component must be post-truncation non-expansive (in the project’s sense) and fully audited via δ-ledger entries and tower diagnostics.
T_tau: exact truncation deleting barcode intervals of length<= tauC_tau: filtered lift (up to filtered quasi-isomorphism), withP_i(C_tau F) ≅ T_tau(P_i F)
A basic Core predicate (used throughout the pipeline):
CollapseAdmissible(F) := (PH1(F)=0) AND (Ext^1(R(F),k)=0)
The one-way bridge makes the Ext condition dischargeable from PH1=0 under the stated hypotheses; no global converse is claimed.
v17.0.1 packages “absence of invisible limit obstruction” as:
DiagZero := (mu_collapse, nu_collapse) = (0,0)
This is the canonical way the paper expresses “no Type IV / tail obstruction” on a certified window/cell.
Definition (Core, windowed, after truncation): on a right-open window W=[u,u') at threshold tau, B–Gate⁺ passes if:
PH1(C_tau F | W) = 0Ext^1(R(C_tau F | W), k) = 0(eligibility checked)DiagZero(i.e.,(mu_collapse, nu_collapse)=(0,0))- Budget check:
gap_tau > sum_delta(δ-ledger)
This is the project’s canonical “collapse contract” at the window level.
For overlapping windows W_a, W_b, the Overlap Gate enforces (post-truncation) coherence on overlaps, with:
- overlap distance bound (interleaving distance on cropped, truncated data)
<= sum_delta_ab - budget margin
gap_tau > sum_delta_ab DiagZero
The Overlap Gate is the mechanism used to paste window-local certificates into coherent global / multi-window artifacts.
Earlier versions tried to unify multiple heavy domains at once. v17.x makes an explicit pivot:
- From: “one gigantic framework that might prove everything”
- To: a proof-oriented OS that cleanly separates
- what is proved and stable (Core),
- what is operational, domain-dependent, or heuristic ([Spec]),
- and what is uncertain (budgeted and audited).
v17.0.1 is a patch release whose primary goal is auditability and boundary clarity:
- reference / label / cross-reference hygiene (duplicate labels and unresolved refs removed)
- tightened “Core vs [Spec]” marking for statements that were previously easy to misread
- minimum bibliographic anchors for “standard facts” (so reviewers can classify leaps vs citations)
- canonicalization of “Type IV diagnostics” phrasing via
DiagZero
These are “trust-surface” upgrades: they improve third-party auditability without expanding the Core scope.
AK_AP is a companion program/paper that performs closed-world calibration on arithmetic moduli spaces using AK-OS.
Principle: AK_AP does not claim new proofs of Deligne or Wiles–Taylor. Instead, those are treated as boundary inputs in a world where the truth is known, and the OS is tested for consistency and auditability:
M_Weil: finite-field Weil world (Deligne as boundary input)M_FLT: FLT world / Frey data (Wiles–Taylor / modularity as boundary input)
To remain consistent with AK-OS:
- B–Gate⁺ is reserved for the v17.0.1 Core definition (windowed, post-truncation, budgeted, with
DiagZero). - Any arithmetic-specific checks (e.g., defect potentials / modularity potentials) must be treated as pre-gates (sanity checks) or [Spec] measurements, logged to the δ-ledger, and must not silently redefine B–Gate⁺.
If you publish AK_AP outputs, present them as:
- “calibration / consistency evidence under declared budgets,” not as proofs.
- As mathematics: treat Core as the only “theorem-grade” layer; treat [Spec] as auditable, implementation-dependent extensions.
- As software architecture: view AK-OS as a pipeline OS for truncation-first stability, budget accounting, and failure diagnosis.
- As AI × human collaboration: the OS is designed to make exploration reproducible and criticizable (run manifests + artifacts), not “mystical.”
All comparisons and decisions follow a post-truncation order:
t → persistence P_i → truncation T_tau → compare / audit / gate / map_validity
Outputs are organized as auditable artifacts (barcodes, ledgers, diagnostics, checksums) so that third parties can rerun and criticize.
git clone https://github.com/your-org/ak-hdpst.git
cd ak-hdpst
python -m venv .venv
source .venv/bin/activate # Windows: .venv\Scripts\activate
pip install -e ".[all]"Note: the actual package layout and extras depend on the repository implementation.
This README describes the OS and audit contract; code scaffolding may vary.
# Prepare a run configuration (example path)
cp examples/v17.0.1/minimal/run.yaml ./run.yaml
# Execute the windowed post-truncation pipeline
akhdpst run run.yaml
# Inspect gate verdicts, tower diagnostics, and δ-ledger per window
akhdpst audit out/artifacts --by-windowfrom akhdpst.core import T_tau, C_tau
from akhdpst.gate import b_gate_plus
from akhdpst.tower import audit_tower
F = ... # load filtered object / persistence
tau = 0.15
W = (0.0, 0.5) # right-open in the protocol; represented here as a pair
# Truncation at persistence level
P1_trunc = T_tau(F.persistence(1), tau)
# Filtered lift (up to f.q.i.)
F_trunc = C_tau(F, tau)
# Tower diagnostics (after truncation)
mu, nu = audit_tower(tower=[...], tau=tau, degree=1)
# Gate check (windowed)
ok = b_gate_plus(
F=F,
window=W,
tau=tau,
mu=mu,
nu=nu,
gap_tau=0.03,
sum_delta=0.011,
)meta:
name: "demo-v17.0.1"
seed: 1337
version: "17.0.1"
author: "your-name"
data:
input: "data/example.h5"
backend: "bars" # bars | chain
degrees: [0, 1]
windows:
- label: "w0"
range: [0.0, 0.5) # right-open; MECE; coverage audited
- label: "w1"
range: [0.5, 1.0)
truncation:
tau: 0.15
lift: "C_tau"
reflector: "T_tau"
gate:
require:
PH1_zero: true
Ext1_zero: true # only used if eligibility holds (amplitude<=1 etc.)
DiagZero: true # (mu_collapse, nu_collapse) = (0,0)
gap_tau_gt_sum_delta: true
safety_margin:
gap_tau: 0.03
audit:
outputs: ["bars", "ledger", "tower", "gate", "checksums"]
checksums: "sha256"
output:
dir: "out/artifacts"
overwrite: falseout/artifacts/
bars/
w0_deg1_trunc.json
w1_deg1_trunc.json
ledger/
ledger_w0.json
ledger_w1.json
tower/
phi_maps_w0_deg1.json # comparison maps; (mu,nu); tail flags
gate/
gate_w0.json
gate_w1.json
run.yaml
audit_summary.json
checksums.txt
bars/: truncated barcodes per window / degree.ledger/: δ-ledger per window (algorithmic / discretization / measurement).tower/: comparison maps and diagnostics(mu_collapse, nu_collapse); tail/isomorphism flags.gate/: gate outcomes (B–Gate⁺, Overlap Gate where applicable).audit_summary.json: rollups across windows.checksums.txt: SHA256 sums for reproducibility.
All monotonicity / stability claims are stated after truncation T_tau.
| Update type | Examples (illustrative) | Guarantee (post T_tau) |
|---|---|---|
| Deletion-type | restrictions, conservative contractions, controlled averaging | monotone non-increasing for selected post-trunc indicators |
| ε-continuation | small homotopies under interleaving bounds | 1-Lipschitz stability; ε logged in δ-ledger |
| Inclusion-type | adding cells / enlarging domains | stability-only (non-expansive), not monotone |
Important note: spectral indicators on realized chain-level objects are not f.q.i.-invariants; v17.0.1 therefore enforces a fixed policy and budgets (and logs any deviations).
- OS framing tightened: AK–HDPST is explicitly an exploration OS with provable Core guarantees.
- HDPS and the Map of Validity remain [Spec] and must be audited.
- Closed-world calibration via AK_AP is positioned as consistency testing, not proof.
- Stronger separation of Core vs [Spec] claims in the writing.
- Reference hygiene: duplicate labels and unresolved references removed.
- Minimum bibliographic anchors for standard facts (improves third-party audit classification).
- Canon phrasing for Type IV diagnostics via
DiagZero.
-
Core
- additional formalization (Lean / Coq):
T_tau,C_tau,(mu,nu), B–Gate⁺, Overlap Gate - more automated τ-sweep and stability-band diagnostics with certified bounds
- additional formalization (Lean / Coq):
-
[Spec]
- HDPS tooling improvements: multi-resolution terrain cells; agent policies with explicit δ-budgets
- richer toy models with fully implemented validity maps
- domain adapters (arithmetic / PDE / symplectic) that respect Core constraints
- Keep Core vs [Spec] explicit in docs and code.
- Every [Spec] component must:
- include post-truncation non-expansiveness tests (as applicable), and
- log δ-ledger entries.
- Prefer minimal, reproducible examples with auditable artifacts.
If you use AK–HDPST v17.0.1 in research, please cite:
AK–HDPST v17.0.1 / AK-OS v1.0.1:
High-Dimensional Projection Structural Theory and an OS for collapse-based exploration.
Authors: Atsushi Kobayashi, et al.
Year: 2025
Repository / archive: (insert URL or DOI)
Foundational references (non-exhaustive; illustrative):
- T. Crawley-Boevey, Decomposition of pointwise finite-dimensional persistence modules, 2015.
- F. Chazal, V. de Silva, M. Glisse, S. Oudot, Structure and stability of persistence modules and barcodes, 2016.
For the arithmetic calibration program:
AK_AP v1.0.1: Arithmetic Programs for AK-OS (closed-world calibration)
Repository / archive:
- Constructible 1D persistence: finite critical set; pointwise finite-dimensional over bounded windows.
T_tau: exact truncation removing all bars of length<= tau; idempotent; 1-Lipschitz.C_tau: filtered lift ofT_tauup to filtered quasi-isomorphism;P_i(C_tau F) ≅ T_tau(P_i F).DiagZero:(mu_collapse, nu_collapse)=(0,0); “no tail obstruction / no Type IV” on a window.- B–Gate⁺ (Core): windowed after-collapse gate requiring:
PH1=0,Ext^1=0(eligible; amplitude<=1 regime),DiagZero,- budget margin
gap_tau > sum_delta.
- Overlap Gate (Core): overlap coherence gate with distance bound, budget margin, and
DiagZero. (mu_collapse, nu_collapse): tower diagnostics derived from comparison maps after truncation; non-zero signals invisible limit failure (Type IV).- HDPS ([Spec]): search layer building a Map of Validity (
Valid/Obstructed/Unknown) under Core guards. - δ-ledger: budget accounting for algorithmic / discretization / measurement deviations.