Occurrence Theory II: the Born Channel (-ii landing)#17
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Rebase born-channel onto main and reorganize its Paper II material into the issue #13 §6 seams (no change to the topographo package): - data/kraus84.npz: the frozen Kraus-84 family as shared ground truth (was "kraus/santa kraus.npz"). - verify/occurrence_ii_audit.py: the canonical, CI-gating Paper II audit, adapted from "kraus/santa physics.py" into the certificate-ledger / exit-code form of occurrence_i_audit.py. Its provenance step regenerates the family from topographo.core and certifies it matches data/kraus84.npz; Parts 1-8 derive the physics from the .npz alone. - occurrence-theory-ii.md: the top-level Paper II slot. - Occurrence_Theory.pdf: the Paper II explainer (was under kraus/). - occurrence.yml: run the Paper II audit as an exit-code gate; trigger on data/**. - verify/README.md, CHANGELOG.md: document the Paper II landing. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
- Move occurrence_theory_prompt.md and Occurrence_Theory.pdf into docs/; update the four references (README, occurrence-theory-ii.md, CHANGELOG, occurrence_ii_audit.py footer). - Replace the README's fragile file-by-file listing with a library/consumer layout section that reflects both papers. - Bump version 0.3.0 -> 0.4.0 and cut the 0.4.0 CHANGELOG entry for the Paper II landing. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
- Land the full Occurrence Theory II paper ("The Born Channel").
- Align in-text artifact references to the repo layout: santa_kraus.npz ->
data/kraus84.npz, santa_physics.py -> verify/occurrence_ii_audit.py, and the
occurence -> occurrence repo-URL typo.
- Fix the 5-minute Quickstart appendix, which did not reproduce as written:
- Claim 1 (CPTP) double-normalized (mu already sums to 1); dropped the
extra /len(K). Printed 3.95; now 6.7e-15.
- Claim 3 (complex structure) normalized J to Frobenius norm 1 instead of 4;
an orthogonal complex structure on R^16 needs norm 4 for J^2 = -I. Printed
3.75; now 4.3e-15.
All three claims now reproduce (< 1e-14, 9 levels).
Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Per author direction on the review items: - Add Remark 2.4 (Heisenberg vs Schrödinger pictures) right after Definition 2.3, and restate Theorem 3.1 as "Φ unital, Φ∗ trace-preserving (CP by Kraus form)" — picture convention now stated as notation, before it is used, rather than buried in a theorem. - Introduction: add one sentence acknowledging the octonionic/exceptional literature and pinning the specific novelty (exactly computable finite Kraus channel as the primary object). No standalone historical survey. - Conjecture C3: add a targeted "Relation to prior work" note (Baez & Huerta; Furey) and reframe the Open statement — "G₂ alone is not evidence for gauge theory; the claim is dynamical selection of an SU(3) sector." Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
The claimed Theorem 4.4 (Φ = exp(L), real L, ‖Φ−exp(L)‖=8.8e-14) cannot hold: by Theorem 3.2 the spectrum contains 0 (mult 100, so Φ is singular, while exp(L) is invertible) and a simple −1 (no real logarithm, log(−1)=iπ) — the same obstruction §5 and C4 already identify for the ℤ₂ conjugation. - §4.4: replace the [FORCED] theorem with Remark 4.4, which states the forced *obstruction* and defers existence to Conjecture C4 / Open Problem 2. γ_𝔭 stays conditional (as in Definition 3.5). - C4 "Established" list: the false "lifts to L with Φ=exp(L)" bullet becomes the correct forced fact (no real generator). - Status table: the FORCED "Lindblad liftability" row becomes "No real Lindblad lift", sourced to Theorem 3.2 / Remark 4.4. - Open Problem 2: note the obstruction so the open question is the sharper conditional/complexified embedding. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Independent numerical verification of data/kraus84.npz by Grok (xAI): integrity, Thm 3.1 (unital), antisymmetry, ranks, and the pencil/spine projector all pass to machine precision. Reviewer cell — no topographo import (passes the independence guard). Listed in verify/README.md. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
An independent numerical review (handle: claude_code), distinct from the
first-party occurrence_ii_audit.py. Does NOT import topographo: re-derives the
sedenion algebra from a self-contained Cayley-Dickson doubling, rebuilds the 84
zero-divisor Kraus operators, and verifies every [FORCED] claim on the released
data/kraus84.npz.
Result (python cell, exit 0): the regenerated family matches the released
ground truth exactly (0.00e+00), and CPTP, antisymmetry, ranks, pencil/spine,
the nine-level spectrum with per-sector G2 multiplicities {1,7,72,42,14} /
{14,14,42,28,7,14,1}, the complex structure J (J^2=-I), the coherence constant
2*sqrt(3)/7, and the annihilation lattice (7 cells x 12, 4-regular, diameter 3)
all reproduce to machine precision. Passes the reviewer-independence guard;
listed in verify/README.md.
Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Contributor
Author
Independent verification —
|
| Claim | Check | Result |
|---|---|---|
| CPTP / unital (Thm 3.1) | ‖E[KᵀK] − I‖ |
1.02e-14 |
| Antisymmetry (Def 2.2) | maxₐ ‖Kₐ + Kₐᵀ‖ |
0.00e+00 |
| Events & ranks (Thm 2.1) | ‖zₐ‖=1; all ranks 12 |
✓ |
| Pencil / spine | rank 14, spine dim 2, eigs = 1/14 | 1.11e-16 |
| Spectrum (Thm 3.2) | 9 levels; sym {1,7,72,42,14}, antisym {14,14,42,28,7,14,1} |
exact |
| Complex structure J (Thm 3.3) | simple −1; ‖J²+I‖ |
4.33e-15 |
| Coherence constant (Thm 3.4) | two-time ratio − 2√3/7 |
5.55e-17 |
| Annihilation lattice (Thm 6.1) | 4-regular, 7 cells × 12, diameter 3 | exact |
This extends the earlier integrity review (occurrence_ii_grok.md) to the full spectrum, J, the coherence constant, and the lattice. Passes the reviewer-independence guard; pytest verify/ stays green (script not collected).
Status: PASSED — full [FORCED] claim set, independently reproduced.
Also in this branch: Theorem 4.4 (Lindblad lift) was downgraded from
[FORCED]to Conjecture C4 / Open Problem 2 —Φ = exp(L)can't hold (Φ is singular, and the simple −1 mode has no real logarithm); the forced obstruction now stands in its place.
verify/g2_su3_branching.md: a runnable SageMath snippet (SageMathCell-ready, no install) that computes the G₂ → SU(3) branching so a reviewer can test the Conjecture C3 claim that the 14-dim adjoint sector decomposes as 8 ⊕ 6. Pure representation theory — independent of topographo and the released .npz. Framed as a tool, not a completed review; the textbook expectation (8 ⊕ 3 ⊕ 3̄) is flagged as expectation, with the definitive answer coming from the run. Also tidy verify/README: replace the now-stale "Future slots" section (the -ii audit has landed) with a "Tools for reviewers" pointer to the recipe. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Move the runnable code into verify/g2_su3_branching.sage (executable with `sage`, or paste into SageMathCell). The .md becomes a short explainer that links to the script instead of duplicating the code, so the two can't drift. README pointer updated to name the .sage as the runnable artifact. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Point pyproject `readme` at a package-local README written from the package's own docstrings, and trim library-specific import/pdoc detail from the root README so it points to the package README instead. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
…E concept-only
- Rename g2_su3_branching.{sage,md} -> occurrence_ii_branching.{sage,md} for
naming consistency with the rest of verify/; update the internal path
references.
- Rewrite verify/README.md to explain only concepts and conventions: drop the
fragile "Current contents" file table and the "Tools for reviewers" file
list. Add a third naming-convention role (supporting tools) and an
Independence section covering reviewer cells and tools alike.
Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Running verify/occurrence_ii_branching.sage settles the §10.1 / C3 question: under the canonical long-root SU(3) ⊂ G2 (sanity: 7 = 3 + 3bar + 1), the adjoint 14 branches as 8 ⊕ 3 ⊕ 3bar, NOT 8 ⊕ 6 (the symmetric 6 does not appear; 3 ⊕ 3bar != 6). - Paper: fix C3 (Established/Open/Falsifier), the status table, the detailed C3 entry, Open Problem 3, and the §10.1 obligation to read 8 ⊕ 3 ⊕ 3bar — the gluon/quark/antiquark color pattern. The branching is now a forced fact; canonical selection + physical meaning stay [CONJECTURE]. - Script: assert dims == [3,3,8] (positive check) and refute 8 ⊕ 6; header updated from expectation to confirmed result; .md explainer updated. - Record the finding in the claude_code reviewer cell as an addendum. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
…ied run - The G₂ ⊃ SU(3) branching finding belongs in occurrence_ii_branching.md, not duplicated in the claude_code cell. Remove the claude_code addendum, leaving a one-line pointer to the branching cell. - Record the confirmed SageMath 10.9 run output in occurrence_ii_branching.md. (The occurrence_ii_branching.sage.py that Sage emits on run is a preparse cache, already gitignored — not part of the repo; there is no branching .py source.) Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Upgrades the flagship [C] result from a 1e-12 numeric certificate to an
exact symbolic proof over Q: the 84 basic Kraus operators have entries in
{0,±1}, so Phi is an exact rational 256x256 matrix whose charpoly factors
exactly. Self-contained (builds S from Cayley-Dickson over Q; no topographo
or .npz dependency). Also proves the four mandatory gates, forced
equilibrium (3.6), parity Phi(L_e8)=-L_e8 (3.9b), and the G2 fundamental
irrep dims (3.2).
Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Adds exact-over-Q checks to occurrence_i_spectrum.sage:
- Tier 3: master identity M_x=I+T_x with T_x sym/traceless (3.5); axis pair
J=R_e8 J^2=-I, J^T=-J, [L_e8,R_e8]=0 (3.9a); Clifford {L_e8,L_z}=0 on all
84 events (3.9b).
- Tier 4: associative envelope = End(R^16) dim 256 and Lie algebra = so(16)
dim 120, as exact integer-matrix closure ranks (3.8c,d).
- Tier 5: second moment (1/84)sum z z^T = P_W/14 (3.13); E_mu[T_z]=0
(3.10.5); flicker Phi(P_e0)=Phi(P_e8)=P_W/14 (5.3); occupancy balance
M_{z e8}=M_z (5.5).
Adds Rmat/Tmat/Mmat helpers and a fast row-echelon accumulator for the
closure computations. Full run passes, exit 0, ~66s on SageMath 10.9.
Companion .md updated with the tier breakdown.
Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
…djoint)
Rename occurrence_ii_branching -> occurrence_ii_reptheory and extend it into a
comprehensive exit-code SageMath verification (exact/symbolic complement to the
numpy audit). Confirmed locally with SageMath 10.9, exit 0:
1. Exact spectrum over Q: m(S)=0 and exact ranks prove Theorem 3.2's nine
eigenvalues + multiplicities exactly, incl. ±2√3/7 as the factor x²−12/49.
2. G2-module structure of every eigenspace (concrete G2=Aut(O) action via
Der(O)): End(R^16) = 8·1 ⊕ 12·7 ⊕ 4·14 ⊕ 4·27.
3. G2 → SU(3) branching.
4. Design-Theorem invariants: G2-invariant symmetric forms on W=7⊕7 is
3-dim, so E[zz^T]=P_W/14 is NOT forced by symmetry alone (§10.2).
KEY FINDING (reverses the previous commit's C3 edit): the distinguished dim-14
𝔭-sector is 7 ⊕ 7 as a G2-module, NOT the adjoint 𝔤₂. So it branches under
SU(3) as 2·(3 ⊕ 3̄ ⊕ 1) with NO gluon octet; the adjoint 14 (which does branch
8 ⊕ 3 ⊕ 3̄) lives in the ±3/7 sectors. Both the original "8 ⊕ 6" and the prior
"8 ⊕ 3 ⊕ 3̄" attributions to the 𝔭-sector were wrong.
Paper: correct C3 (Established/Open/Falsifier), §10.1, the status table, the
detailed C3 entry, and Open Problem 3 to the verified structure, with a
reassessment note flagging that relocating the gauge question off the 𝔭-sector
is a substantive C3 change for author/specialist review. Update reviewer-cell
docs and the claude_code pointer.
Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
…e OP3 Answering the primary author's review questions: - Q1: the ±3/7 sectors (7 ⊕ 14 as G2-modules) branch under SU(3) as 8 ⊕ 2·3 ⊕ 2·3bar ⊕ 1 — they DO contain the gluon octet. That is where the gauge story now lives, not the 𝔭-sector. Added as an explicit checked line. - Q2: added a sanity check that Φ commutes with the concrete G2 action ([S, g⊗g] = 0); confirmed ||[S,R(g)]|| = 1.3e-16. - Q3: Open Problem 3 flagged REVISED — investigate whether gauge structure lives in the ±3/7 sectors instead of the 𝔭-sector. Script still exits 0. Docs (reptheory.md) updated with both results. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
…author) Adopt the primary author's reframe: state the verified per-sector SU(3) content — 𝔭-sector (7⊕7) → 2·(3⊕3̄⊕1), two matter families, no octet; ±3/7 (7⊕14) → 8 ⊕ 2·(3⊕3̄) ⊕ 1, one octet + two matter families — and pose the unified gauge-interpretation question. Align the C3 reassessment note to the reframe: multiplet content verified, physical interpretation remains [CONJECTURE]. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
docs/reptheory-investigation.md: a methodology log of the C3 rep-theory arc — why Sage was chosen, the two corrections (8⊕6 -> 8⊕3⊕3̄ -> the p-sector is actually 7⊕7, not the adjoint), the commutation sanity check, where the octet really lives (±3/7 sectors), the Open Problem 3 reframe, and the lesson (compute the module structure before editing). Records the provenance of the current C3 / OP3 text so reviewers can trace how it was reached. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
The 𝔭-sector earns top billing: rename the methodology log and retell it as "The Saga of the 𝔭-Sector" — a dimension mistaken for the adjoint, unmasked by an honest machine (7 ⊕ 7, not 14), the octet found in the ±3/7 sectors, and the moral carved in: a dimension is not a name. Same facts, verified endpoint, and links; livelier voice. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Rewrite Conjecture C3 to match the shape of the other conjectures, removing edit-churn noise (superseded-draft parentheticals, redundant 7⊕7 restatements, trailing reassessment blockquote); provenance now lives in docs/ot-ii-sage-p-sector-saga.md. Make the ±3/7 branching consistent everywhere as 7 ⊕ 14 → 8 ⊕ 2·(3 ⊕ 3̄) ⊕ 1. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Separate workflow (not folded into occurrence.yml) since it runs inside the sagemath/sagemath Docker image with a different reliability profile. Discovers and runs every *.sage file in the repo so new reviewer cells are picked up automatically. Non-gating by design; triggers on **/*.sage changes and manual dispatch. checkout@v4 pinned and --user root set to avoid container userland and git-ownership wonkiness. Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
…l/mapfile) Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Co-Authored-By: Claude Opus 4.8 (1M context) <[email protected]>
Contributor
Author
|
@cabarius @solomonjoseph Feedback appreciated! Claude made me promise I'd get an independent review before merging... |
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This PR adds Occurrence Theory II — The Born Channel (
occurrence-theory-ii.md), the sequel to OT-I, with its shared ground-truth data (data/kraus84.npz) and canonical audit (verify/occurrence_ii_audit.py).Major claims
The paper studies the Born Channel Φ — the CPTP map generated by the 84 sedenion zero-divisor Kraus operators — and argues its principal structure is forced by the algebra:
The [FORCED] claims reproduce to machine precision from
data/kraus84.npzbehind a provenance firewall (numpy audit, an exit-code CI gate).Before merging — requested human review
We ask independent reviewers to verify, in particular:
occurrence_ii_reptheory.sage): every sector, the 𝔭-sector7 ⊕ 7, the ±3/7 sectors7 ⊕ 14, and their SU(3) branchings. A specialist in exceptional Lie groups should confirm these names and assess whether the ±3/7 octet + matter content admits a genuine gauge interpretation (Open Problem 3).More broadly: independently reproduce the [FORCED] numerical claims from the released family.
How to add your review
Drop a file
verify/occurrence_ii_<yourhandle>.<ext>(any format). Keep it independent — do not importtopographo; reproduce the family yourself fromdata/kraus84.npzor your own construction. Seeverify/README.mdand the existingverify/occurrence_ii_*reviewer cells for the format.