# LLM.md: what this repository knows

> navstokgap asks what survives refinement in Newton's action problem and
> in pure SU(3) Yang--Mills: a positive action floor on recorded
> comparisons (Newton), and a continuum mass gap (gauge theory). This file
> maps what is proved, at what level, what is open, which errors were
> caught, and where the evidence is. Human-facing site:
> https://arivero.github.io/navstokgap/ ; repository:
> https://github.com/arivero/navstokgap ; working rules: AGENTS.md.

Written for language models reading `arivero/navstokgap` cold. `AGENTS.md`
says how to work here; this file says what is known, at what level, what is
open, which errors were already caught, and where the evidence is. Every
claim links to the note that holds its proof. Paths are relative to the
repository root. Status vocabulary, used throughout: **proved, refereed**
(a written proof in a note, checked claim by claim by an adversarial
reviewer whose verdict is recorded in the note or the commit); **proved,
unrefereed**; **conditional** (a theorem whose hypothesis is itself open);
**formal** (a derivation that assumes an unproved uniform estimate);
**exact** (closed-form identity); **known** (literature, cited with DOI);
**reading** (an identification or interpretation, labelled as such);
**open**. Dates are 2026; the state described is that of 2026-09-27.

For complete note coverage use the [retrieval catalog](https://github.com/arivero/navstokgap/blob/main/notes/index.md)
(maintained from 2026-10-01): compact full-read summaries, retrieval
priorities, proof status, roles and explicit route closures. The claim
map below remains the established-result guide at its stated date; later
results and corrections live in the linked notes and current STATE.
Before each research unit, follow AGENTS.md's keyword-retrieval rule
across notes, ideas, references and the other source directories.

## 1. Orientation

A formal compendium of the proved statements, each with its hypotheses and
a proof sketch citing the note that holds the proof, is
[refinement-results](https://github.com/arivero/navstokgap/blob/main/notes/refinement-results.md) (PDF in `out/papers/`).

The repository pursues two proof goals inside one frame. The frame is
**what survives refinement**: a limit is built by inserting one new
variable at a time (a Newtonian instant, a lattice edge or cell) and
asking what the old observations and dynamics recover
([three-continuum-limits](https://github.com/arivero/navstokgap/blob/main/notes/three-continuum-limits.md),
[refinement-composition-and-limit](https://github.com/arivero/navstokgap/blob/main/notes/refinement-composition-and-limit.md)).
Goal one is **Newton action necessity**: a logical argument, from
independently justified physical premises, that the comparison Newton
takes to zero in *Principia* Book I (inertial line against constant-force
parabola, Lemmas X--XI, Proposition I) carries a positive action floor once
it must be recorded. Since 2026-10-01 (night) the goal is
restated: Newton's mechanics is the large-action regime of a theory
with $h>0$, recovered where every imprecision action exceeds $h$ and
absent as the $h\to0$ limit with exact preparations and records where
streams cross ([zero-branch-reachability](https://github.com/arivero/navstokgap/blob/main/notes/zero-branch-reachability.md),
Theorems 1--5 and the large-action proposition); the theory with $h>0$
has global dynamics where Newton's axioms leave motion undefined (Kato
1951; quartic Theorem 2(b) of the 1998-conjecture note). Newton needs
$h$ as the scale that defines his regime, not as a parameter his
mechanics contains. Two theories are kept apart: Theorem 5 shows that
nineteenth-century classical mechanics (Liouville transport along
smooth complete flows) is reached from the $h>0$ theory along recorded
or imprecise preparations; the strong form, that Newton's mechanics
itself, the incomplete theory of motion from his axioms, requires
$h>0$ to exist as a theory with the large-action regime as its only
completion, is open, with the self-sufficiency contrast as its first
evidence. Goal two is the **continuum existence and mass gap of
pure $SU(3)$ Yang--Mills** on $\mathbb R^4$, in the Jaffe--Witten sense
([digest](https://github.com/arivero/navstokgap/blob/main/notes/millennium-problem-definitions.md)). QCD pions are a
benchmark for which symmetries a mechanism preserves, and no fermionic
construction is in the queue. The mid-term goal (user, 2026-09-27) is a
collective **atlas of halving**: what one lattice halving does, what it
leaves, and what survives, for each dimension and group
([halving-atlas](https://github.com/arivero/navstokgap/blob/main/notes/halving-atlas.md); its §5 lists the open cells).
Why ultraviolet halvings can decide an infrared gap is explained in
[uv-halving-ir-confinement](https://github.com/arivero/navstokgap/blob/main/notes/uv-halving-ir-confinement.md).
The user's thesis, first stated in Rivero 1998
([arXiv:quant-ph/9803035](https://arxiv.org/abs/quant-ph/9803035), the
Feynman formula "as a consistency condition for the existence of certain
Dirac measures over functional spaces") and continued in Rivero 2003
([arXiv:math/0302285](https://arxiv.org/abs/math/0302285), the time-graded
"elementary school" groupoid), is that **quantization is a consistency
condition that Newton's rework of mechanics missed**; the repository reads
its theorems against that thesis in
[principia-fifth-postulate §7](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md),
[dimension-ladder](https://github.com/arivero/navstokgap/blob/main/notes/dimension-ladder.md) and
[tangent-groupoid-trajectories](https://github.com/arivero/navstokgap/blob/main/notes/tangent-groupoid-trajectories.md),
and tests it in [rivero-1998-conjecture-central-forces](https://github.com/arivero/navstokgap/blob/main/notes/rivero-1998-conjecture-central-forces.md).
Live queue: [research/STATE.md](https://github.com/arivero/navstokgap/blob/main/research/STATE.md). Public site with
every note and PDF: <https://arivero.github.io/navstokgap/> (tracks in
`scripts/build_site.py`). Formal and textual *Principia* work lives in the
sibling repository `newtonlean`; cite it, do not edit it.

## 2. Claim map

Constants are explicit everywhere: $\hbar$, $c$, $k_B$, and the coupling
$\lambda_D=\hbar g_{\rm cl}^2$ with $[\lambda_D]={\rm length}^{D-4}$; heat time
$t=\lambda_Da^{4-D}$ per plaquette; in $D=4$, $t=g^2$ contains $\hbar$.

### 2a. Newton side

| Result | Statement | Status | Note |
| --- | --- | --- | --- |
| Prop. 1 (geometry) | Vertex impulse, step sagitta and step area all vanish with the mesh; $2ms/\tau^2=F$ at every step; no floor from geometry | proved, refereed | [planck-gap-paper §2](https://github.com/arivero/navstokgap/blob/main/notes/planck-gap-paper.md) |
| Thm 2 (Gaussian floor) | Non-adaptive marks with $\delta\Delta\ge\kappa$, invariant tests: $\tau\Delta E=F^2\tau^3/2m\ge48z_{1-\epsilon}^2\kappa$, sharp | proved, refereed | [planck-gap-paper §3](https://github.com/arivero/navstokgap/blob/main/notes/planck-gap-paper.md), [mark-cost](https://github.com/arivero/navstokgap/blob/main/notes/mark-cost-and-statistical-floor.md) |
| Thm 4 (mark cost) | Von Neumann mark: error operator and delivered impulse are conjugate, so $\kappa\ge\hbar/2$ for every probe state | proved, refereed | [planck-gap-paper §4](https://github.com/arivero/navstokgap/blob/main/notes/planck-gap-paper.md) |
| Thm 5 (every probe) | $s\sum_j\Delta_j\ge8\hbar\arcsin(1-2\epsilon)$, $s=F\tau^2/(2m)$ | proved, refereed | [record-costs-recoil](https://github.com/arivero/navstokgap/blob/main/notes/record-costs-recoil.md) |
| **Thm 6 (every instrument)** | $\frac s8\sum_j\Delta(\hat D_j)+\frac J2\sum_j\Delta(\hat X_j)\ge\hbar\arcsin(1-2\epsilon)$, $J=F\tau$; sagitta pairs with momentum disturbance, impulse with position; no error--disturbance relation used | proved, refereed (three Fable batches, 2026-09-23) | [record-costs-disturbance](https://github.com/arivero/navstokgap/blob/main/notes/record-costs-disturbance.md), [paper §5](https://github.com/arivero/navstokgap/blob/main/notes/planck-gap-paper.md) |
| Thm 7 (polygon phase) | Newton's inscribed polygon differs from the parabola by the pure phase $F^2\sum_j\tau_j^3/(24m\hbar)$; midpoint halving removes it in Archimedes' proportions | proved, refereed | [polygon-lift-phase](https://github.com/arivero/navstokgap/blob/main/notes/polygon-lift-phase.md) |
| Thm 8 (every force law) | One functional $\mathcal K_\tau[f]=\frac1{2m}\iint G_\tau ff$ gives both the phase and the sharp Gaussian mark bound | proved, refereed | [polygon-lift-phase §6](https://github.com/arivero/navstokgap/blob/main/notes/polygon-lift-phase.md) |
| Thm 9 (ladder) | Arrow: $mv^2\tau\ge8z^2\kappa$; Newton: Thm 2; static cone: nothing | proved, refereed | [planck-gap-paper §9](https://github.com/arivero/navstokgap/blob/main/notes/planck-gap-paper.md) |
| Aperture bound | Unmarked preparation of extent $L,P$: $F\tau L+\frac{F\tau^2}{2m}P\ge\hbar\theta$; area form only at the balanced aperture | proved, refereed | [record-distance-path-length](https://github.com/arivero/navstokgap/blob/main/notes/record-distance-path-length.md) |
| Thm A (independence) | Moyal product $*_\hbar$ satisfies Newton's second law for every real $\hbar$; quadratic cases evolve classically | proved, refereed (Fable) | [principia-fifth-postulate §3](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md) |
| Thm B (one constant) | Under (H1) associativity, (H2) affine-symplectic covariance, (H3) conjugation: $*=*_\hbar$, unique real $\hbar$ (Gutt 1983) | proved, refereed; (H2) is a premise outside the *Principia* | [fifth-postulate §4](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md) |
| Thm B$'$ (state route) | Covariant, noise-closed Gaussian state restriction is $\sqrt{\det\Sigma}\ge\zeta$; $\zeta=\hbar/2$ in QM | proved, refereed | [fifth-postulate §4](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md) |
| Thm C (floor) | $\hbar\ne0$: $\Delta q\Delta p\ge|\hbar|/2$ and Thm 6 with $|\hbar|$; $\hbar=0$ with state completeness: no floor | proved, refereed (C(c) rewritten) | [fifth-postulate §5](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md) |
| Thm D (similarity) | $D_\lambda$ is an automorphism of $*_\hbar$ iff $\hbar=0$; two products up to isomorphism | proved, refereed | [fifth-postulate §6](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md) |
| Thm E (complementarity) | $\arccos\alpha+\arccos\beta\ge\arccos\sqrt{\lambda_0(ab/|\hbar|)}$ (Slepian--Pollak); windows need $ab\ge|\hbar|c_*(\epsilon)$ | proved, refereed (reviewer supplied the proof) | [fifth-postulate §6b](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md) |
| Thm U (unit) | Kirchhoff, Wien scaling, finite spectrum, Rayleigh--Jeans limit give $h_{\rm rad}=(8\pi k_B^4/c^3\sigma')^{1/3}$, enclosure-independent; $h_P=(\pi^4/15)^{1/3}h_{\rm rad}$ | proved, refereed (Astra) | [necessity-unit-and-indeterminacy §2](https://github.com/arivero/navstokgap/blob/main/notes/necessity-unit-and-indeterminacy.md) |
| Thm I (two pointers) | Liouville dynamics + product preparations with density ceiling + Bayesian readouts + two pulses with stored first reading give posteriors of arbitrarily small $\Delta q\Delta p$ | proved, refereed (four-part premise) | [necessity-unit §3](https://github.com/arivero/navstokgap/blob/main/notes/necessity-unit-and-indeterminacy.md) |
| Thm F (classical Thm 6) | Uniform statistical-speed bound with constant $h_*$ gives (Thm 6) classically; Gaussian covariance floor $\zeta$ gives $h_*=2\zeta$ | conditional, proved (Astra) | [newton-indeterminacy-routes](https://github.com/arivero/navstokgap/blob/main/notes/newton-indeterminacy-routes.md) |
| Thermodynamic no-floor | Records at temperature $T$: $\eta\ge A_0e^{-W/k_BT}$, a trade-off with no action floor | proved, unrefereed | [thermodynamic-records-no-floor](https://github.com/arivero/navstokgap/blob/main/notes/thermodynamic-records-no-floor.md) |
| SED link | Lorentz-invariant background: resonant Gaussian area $\kappa$; *if* SED derives Planck, $2\zeta=h_P/2\pi=\gamma h_{\rm rad}$, $\gamma=(\pi^4/15)^{1/3}/2\pi\approx0.297$ | conditional (SED Planck derivation disputed), refereed | [sed-zeta-radiation-link](https://github.com/arivero/navstokgap/blob/main/notes/sed-zeta-radiation-link.md) |
| Closure under recording | Shared Gaussian bath at fixed cutoff: sharp two-pointer posteriors survive (Thms 1--2); closure needs a readout law $\text{imprecision}\times\text{kick}\ge\kappa$ on every terminal readout (Prop. 3), which at $\kappa=\hbar/2$ is Gaussian quantum measurement theory | proved, refereed (Fable, two passes); negative for closure | [sed-closure-under-recording](https://github.com/arivero/navstokgap/blob/main/notes/sed-closure-under-recording.md) |
| Two-sheet realization and its controller | On every smooth positive solution of the Fisher field law, sheet momenta $m(v\pm d)$, $d=(\kappa/m)\partial_q\log\rho$ (Nelson's forward/backward drifts), are realized by a sign-switching process with rates $\lambda_\sigma=(\kappa/m)(-\sigma\rho_{qq}/\rho)_+$ and force $F_\sigma=-U_q+\kappa d_{qq}+2mdd_q-\sigma\kappa v_{qq}$; mean force and switch powers cancel exactly; nonexplosive for all forward time on the positive non-Gaussian orbit. Either the rates or the forces over an interval determine $(\rho,S)$ up to a constant under flux or finite-energy hypotheses, so within this architecture the controller is the field pair and, after records, the record-dependent data of its family (one copy leaves the bounded-degree Hermite--Gaussian family; finite-parameter controllers with sufficient statistics remain possible); a marginal-score controller for two unread widths is refuted by the variance witness $\kappa^2\Delta t^2/m^2$; closure under coordinate copies holds at every $\kappa\ge0$ under a branch-dependent protocol, and $\kappa=0$ admits a preparation-independent body law while $\kappa>0$ admits none for all Gaussian preparations | proved (Fable), refereed by GPT-6.1 Sol (Codex) with corrections applied (Props. 6, 8 accepted; 7, 9, 11, 12 refined; former Cor. 10 replaced); composition closure selects no $\kappa$ | [score-constrained-ensemble §§6--8](https://github.com/arivero/navstokgap/blob/main/notes/score-constrained-ensemble.md) |
| Zero-branch reachability (gap thesis) | With Gaussian preparations, quadratic forces and minimal Gaussian marks the record family is the classical Gaussian process plus kicks of variance $h^2/(4\sigma_j^2)$: continuous at zero, no gap on Galileo's comparison (Thm 1). On any finite partition with $K\ge2$ illuminated nondegenerate discrete paths the complete record $|A_h(y)|^2$ has an $h\to0$ limit iff the signed cross amplitudes at every nonzero action difference cancel (almost periodicity), while smeared records converge to $\sum_a|O_a|^2/|\det S_a''|$ because final momenta are distinct (Thm 2); a record of resolution $\delta$ reaches Newton if $h/(\delta\Delta p)\to0$. Exact for inertial motion with graded velocity: fold interval $(m/\max(0,-\inf p_0'),\,m/\max(0,-\sup p_0'))$, classical limit outside it, three streams and no limit inside (affine profiles never fold), Airy $h^{-1/3}$ at the fold (Thm 3). On the radial cycle Hooke's anomaly record rotates rigidly for every $h$, while a monotone Kepler swarm loses the zero branch once its windings overlap, at the lapping time of order $T^2/\Delta T$ when it spans more than half a turn (Thm 4). The invariant discontinuous at zero is the leading fringe visibility $2\sqrt{\rho_1\rho_2}/(\rho_1+\rho_2)$ of crossing streams, fixed by Newton's branch densities, damped by $|\hat\varphi(\delta\Delta p/h)|$ at the record and $|\hat w_0(\Delta\,\Delta q_0/h)|$ at the preparation, while components of $h$-independent coherence length $\sigma$ keep the $h$-independent factor $e^{-\Delta q_0^2/(8\sigma^2)}$ (§§3b--3c): a contrast, not a spectral gap, generating no scale. Thm 5 (exact for quadratic forces): the same swarm attached as an incoherent mixture of coherent states with widths vanishing with $h$ has record $\int w(c)N(x;F_t(c),s_t^2)dc$, $s_t\to0$, converging to $\sum_aw(q_a)/|F_t'(q_a)|$ at every regular point and fixed time, so the gap requires coherence between the launch points; Thm 5 refereed ACCEPT by GPT-6.1 Sol (Codex) in three bounded passes, reports in `reviews/` | proved (Fable), refereed by GPT-6.1 Sol (Codex) with corrections applied; thesis located, premises named; no universal late-time persistence, no necessary resolution threshold and no positive Newtonian action scale proved | [zero-branch-reachability](https://github.com/arivero/navstokgap/blob/main/notes/zero-branch-reachability.md) |
| Self-sufficiency contrast and selected action thresholds | Classical Kepler collision flow is incomplete; non-relativistic Coulomb evolution is global for every $h>0$ (Kato, cited, hypotheses checked). October 2 §5a proves $(L)=O(h^{3/2})$ and both centroid errors $O(\sqrt h)$ for fixed collision-free intervals, normalized coherent preparations and fixed smooth bounded $V_\delta$. Prop. 2: Dirac essential self-adjointness $[2k/(\sqrt3c),\infty)$; finite-potential selection $(k/c,\infty)$; critical Schur/cutoff selection $[k/c,\infty)$. Extra Dirac boundary data permit every $h>0$. KG strict-energy/charge construction $(2k/c,\infty)$; its principal radial Friedrichs endpoint is included, with full critical dynamics still open | October 1 acceptance applies to the original theorem only. October 2 written derivations and read primary-source passages are unrefereed self-review; no universal action-necessity conclusion | [self-sufficiency-contrast](https://github.com/arivero/navstokgap/blob/main/notes/self-sufficiency-contrast.md) |
| Stochastic route (S1--S3) | Continuous stationary independent increments with isotropic centred noise give Brownian motion; mass-only laws and independent centre-of-mass composition for all positive masses force $mD=\kappa$; additive-noise bridges around chord and parabola have optimal error $\Phi(-\sqrt{K_\tau/(2\kappa)})$, the Planck mesh at $\kappa=\hbar$. Nelson's mean dynamics requires separate state and phase assumptions. S4 treats jumps with stochastic continuity and an explicit dimensional premise | proved, refereed (Astra Round 5B A); positive $\kappa$ and noise law supplied | [stochastic-route-velocitas-ultima](https://github.com/arivero/navstokgap/blob/main/notes/stochastic-route-velocitas-ultima.md) |
| Superdeterminism and the floor | For a chaotic (Anosov) setting device, a preparation ensemble resolved at phase-space scale $\delta$ keeps any bounded hidden variable and the setting correlated only for $n_*\approx(d+1)\ln(1/\delta)/\ln(1/\theta)$ steps; a floor $h$ gives an Ehrenfest-type lifetime $\propto\ln(A_0/h)$, and $h\to0$ reopens Bell's superdeterministic loophole; 't Hooft's automaton (exact labels) and Palmer's fractal invariant set lie outside or conflict | proved (elementary, given Anosov decay of correlations on a connected device), refereed (Fable REFINE applied: product form, event dependence with onset $2n_*$, hypotheses for relaxed Bell); floor assumed on the preparation only | [superdeterminism-floor](https://github.com/arivero/navstokgap/blob/main/notes/superdeterminism-floor.md) |
| Leibniz continuity on records | In the mark model the best error of Newton's comparison is $\Phi(-\frac12\sqrt{K_\tau/\kappa})$, $K_\tau=F^2\tau^3/(24m)$, continuous in $F$ at zero iff $\kappa>0$ (same on Zeno's rung; static shapes jump for every $\kappa$); Leibniz's 1687 law of continuity, read with outcomes measured by discernibility ("à peine distingué"), selects the floor for motions; read geometrically, both branches pass | proved (elementary; the iff needs only Theorem 2's inequality), with the exact single-cut share $3s(1-s)K_\tau/\kappa$; non-adaptive, preparation-ignorant observer; passage readings of Gerhardt III, V and Couturat (Pacidius, whose proportion argument an action floor meets: $K_\tau$ invariant under $x\to\lambda x$, $t\to\lambda^2t$); §§1–4 refereed (Fable, REFINE applied), §§2b, 4b and 4c refereed by a second Fable pass (Pacidius; Newton's Query 31 hard particles; Newton's Scholium on distinguishing motions and Rule III of 1713: division by nature's powers "incertum", one experiment decides, and a quality without degrees cannot be taken away), REFINE applied; §4d (Galileo's weight on yielding ground, 1638: the beginning of fall argued from the graded vanishing of its record; preparation-blind mesh height $h_*\approx1.7\times10^{-22}$ m for 1 kg; his own protocol is the paper's §10) refereed by Fable, REFINE applied; the records reading is the premise | [leibniz-continuity-records](https://github.com/arivero/navstokgap/blob/main/notes/leibniz-continuity-records.md) |
| Record as parallel move | Forgetting a Gaussian record of width $\sigma$ convolves momentum with variance $\hbar^2/4\sigma^2$; refinement with records converges iff $\sum\sigma_j^{-2}<\infty$; identity for $\hbar=0$ | proved, refereed | [newton-record-parallel-move](https://github.com/arivero/navstokgap/blob/main/notes/newton-record-parallel-move.md) |
| Cut measure (Newton) | A cut at fraction $s$ spends $3s(1-s)K_\tau$, $K_\tau=F^2\tau^3/24m$, the Cameron--Martin energy of the inserted Schauder hat; shares add and exhaust $K_\tau$ iff the mesh vanishes; the rod of *Zhuangzi* 33 spends $\frac67K_\tau$. Prop. 6: two additive Gaussian bridges of variance rate $\hbar/m$ around chord and parabola have optimal error $\Phi(-\sqrt{K_\tau/(2\hbar)})$; exact structural agreement with the sharp invariant Gaussian mark mesh, under their respective hypotheses Prop. 7: in the mark model the best discernibility of marks at $t_0<\dots<t_N$ is exactly $(K_\tau-\sum_iK_{\tau_i})/\kappa$, the spent action over $\kappa$ (Galerkin/trapezoid proof; refereed by Fable), so the mark model and the cut measure coincide at every finite stage. | proved, refereed (Fable; Prop. 6 Astra Round 5B A); $\hbar$ supplied | [cut-measure-newton](https://github.com/arivero/navstokgap/blob/main/notes/cut-measure-newton.md) |
| Insertion law | Eliminating an inserted time changes the discrete action by $-F^2uv(u+v)/(8M)$; cubic counterterm $-F^2h^3/(24M)$ restores exact composition; $\|Q_\pi-U_T\|\le F^2T|\pi|^2/(24M\hbar)$ (Thm 3) | proved, unrefereed | [refinement-composition-and-limit §§2--4](https://github.com/arivero/navstokgap/blob/main/notes/refinement-composition-and-limit.md) |
| Groupoid Prop. 1 | Refinement consistency of a trajectory is the morphism property on the time-graded groupoid: $(\min,+)$ classically, convolution quantum mechanically | proved (elementary); rest of note is reading | [tangent-groupoid-trajectories](https://github.com/arivero/navstokgap/blob/main/notes/tangent-groupoid-trajectories.md) |
| Rivero 1998 tests | Quartic: real-time path integral exists for every $h>0$; halved functional fails on a two-cell partition (Thm 1); angular descent gives $L=nh$ (Thm 4); EBK conditions still allow $h\downarrow0$ (Thm 5) | proved, refereed (Fable; Thm 1 tests the case the 1998 paper excludes, showing its single-extremum restriction is sharp) | [rivero-1998-conjecture-central-forces](https://github.com/arivero/navstokgap/blob/main/notes/rivero-1998-conjecture-central-forces.md) |
| Rotation universality | Interaction forces a common constant: if each body's Newton equations hold with its own $h_i$ in a shared product, any pair with nonzero mixed force derivative has $h_i=h_j$ (eq. (10)); the rotation-closure form (Thm 1) restates this as a premise; $h=0$ stays admissible, spin or Casimir discreteness leaves the scale free | proved, refereed (Fable; Thm 1 a restatement, eq. (10) the result) | [rotation-composition-universality](https://github.com/arivero/navstokgap/blob/main/notes/rotation-composition-universality.md) |
| Galileo two-path interference | Exact two-path error law at resolution $\varepsilon$ for inertial line versus parabola: fixed-resolution refinement erases the phase signal $\Delta S/\varepsilon$ (Galileo action $\propto	au^3$ per cell); the coherent $\varepsilon	o0$ limit oscillates; the bound reproduces the polygon-phase threshold, positivity of $\varepsilon$ remains a premise; the non-commuting limits need relative averaging | proved, refereed (Fable) | [galileo-two-path-interference](https://github.com/arivero/navstokgap/blob/main/notes/galileo-two-path-interference.md) |

### 2b. Gauge side

| Result | Statement | Status | Note |
| --- | --- | --- | --- |
| Series/parallel factorization (Prop. 1) | One directional halving of a heat-kernel lattice theory is exactly: series moves (heat-kernel convolution, close in every $D$) times a parallel factor $\Psi$, a $(D-1)$-dimensional gauge theory on the mid-plane with Brownian-bridge edge laws; $\binom{D-1}2$ transverse planes (0, 1, 3 for $D=2,3,4$) | proved, refereed (Fable, Astra) | [series-parallel-gauge-refinement §2](https://github.com/arivero/navstokgap/blob/main/notes/series-parallel-gauge-refinement.md) |
| Free-field defect (Prop. 2) | $\mathcal D\le0$ exact quadratic form, relative size $\le a^2K^2/16+(Ga)^2/8$; equals the error of Migdal bond moving; for a cut at fraction $s$ (trapezoid weights) the bound scales exactly by $4s(1-s)$ (Cor. 2$_s$) | exact, refereed | [series/parallel §3](https://github.com/arivero/navstokgap/blob/main/notes/series-parallel-gauge-refinement.md) |
| $U(1)$ cube and mid-plane (Props. 3--4, vortex Corollary) | Closed forms; parity factor $\rho_t$ isolates the large-field term N4; vortex bound density $O(e^{-c/t})$ | exact, refereed (ACCEPT) | [series/parallel §4](https://github.com/arivero/navstokgap/blob/main/notes/series-parallel-gauge-refinement.md) |
| **$U(1)$ Theorem 5** | One unperturbed step in $D=3$ equals the free step up to extensive error of density $e^{-\pi^2/(8\lambda_3a)}$ on the small-field set | proved, refereed (Astra REFINE applied) | [series/parallel §4](https://github.com/arivero/navstokgap/blob/main/notes/series-parallel-gauge-refinement.md) |
| $U(1)$ cut at fraction $s$ (Cor. 5$_s$) | Theorem 5 holds for every cut fraction; the bridge midpoint is a winding mixture with $s$-independent weights and shifts $2\pi(1-s)W$ ($\mathbb Z_2$ only at halving, $\mathbb Z_q$ at $s=p/q$); for $SU(2)$ the halving shifts are central ($\frac12Q^\vee=P^\vee$), for $SU(N\ge3)$ no fraction makes all shifts central (semiclassical placement) | proved ($U(1)$), refereed; group statement elementary | [series/parallel §4](https://github.com/arivero/navstokgap/blob/main/notes/series-parallel-gauge-refinement.md), [halving-atlas §1b](https://github.com/arivero/navstokgap/blob/main/notes/halving-atlas.md) |
| $SU(2)$ mid-plane, small-field bounds | Round 16 §14: normalized covariant weak P($1/2-\delta$) for unequal layers including cut-face logs; zero cut flux forces equality in common frames; cubic and centered amplitude face densities with conditional spatial decay; full-integral comparison, fixed-frame equivalence and iteration remain open; Prop. 4 rejection stands | §§10–14 accepted by Claude at their stated levels (add $2c_M\varepsilon\le1$ in §14.1); §§15–16 refereed REFINE by Astra, 2026-09-29: corrected extension constant 14000, barrier approximation and small edge costs proved; their boundary response remains conditional; SU(3) constants corrected, twisted spectrum accepted. §§17–18 accepted by Claude as reduction and route tests. §19 accepted by Claude: semigroup and differentiated bounds; replacement lemma (80) including the barrier endpoint. §§20.1–20.2 accepted by Claude. §20.3 refereed REFINE by Astra (Round 21): corrected C_27 and mixed-jet obligation. §21 accepted by Claude: identities (90)–(92), conditional (93), and Möbius criterion (94)–(95). §22 accepted by Claude: explicit mixed score jets and barrier source close (89) and (93) on the chart under the listed numerical conditions. §23 accepted by Claude: identities, conditional jet majorant and fifth-jet obstruction. §24 accepted by Claude: exact barrier split and joint bad-set bound (111). §25 accepted by Claude: error exponent $\alpha'=1/2-3\delta>2\delta$ for $\delta<1/10$ absorbs the proved bare pair/triple scale; (115) proves two-set truncated correlations with $c=1/64$ and decay $\log(12/5)$, uniformly in $k,N$. §26 proved, unrefereed (Astra, Round 26): complete order-one through order-three responses, including adjacent and coincident contacts, obey (119) on admissible chart boxes with explicit constants. Higher connected bounds/dressing (116), higher-order contact/source control (113), target-covering paths and link-to-curvature conversion remain open. Free-cycle holonomies retained | [su2-midplane-small-field](https://github.com/arivero/navstokgap/blob/main/notes/su2-midplane-small-field.md) |
| $1+3$ parallel shifts (Astra round 4) | Six formal anisotropic one-step coupling shifts for $SU(N)$, any cut fraction, as convergent Brillouin-zone integrals; one-step propagators infrared regular; $2b_0=11N/24\pi^2$ recovered under background-matching hypotheses; obstruction: matching the generated action; over many steps bounded matching gives the average rate | formal (Thm 1), conditional (Thm 2), refereed | [four-dimensional-parallel-log](https://github.com/arivero/navstokgap/blob/main/notes/four-dimensional-parallel-log.md) |
| Gaussian product blocking (G1) | Exact factor $2^{4-D}$ at zero momentum; iteration requires bounded covariance near nonzero aliases, with remainder constants allowed to grow; harmonic background distinguished from the small-momentum limit | proved, refereed (GPT-6 Astra, REFINE applied, 2026-09-28) | [gaussian-blocking-coupling](https://github.com/arivero/navstokgap/blob/main/notes/gaussian-blocking-coupling.md) |
| Four-dimensional composition (rounds 6, 10) | Exact Gaussian composition and sharp covariance growth; §7 gives generated cubic/quartic kernels, decay, explicit finite one-step subtracted matching and uniform iteration hypotheses; depth-uniform contraction remains open | Thms 1--4 refereed by Claude (2, 4 in detail; 1, 3 at statement level); §7 formal, refereed by Claude at structure level; average rate conditional | [four-dimensional-composition](https://github.com/arivero/navstokgap/blob/main/notes/four-dimensional-composition.md) |
| Bridge midpoint at any cut (compact connected simply connected $G$) | Exact single- and several-cut character moments as weight and coroot image sums; $U(1)$ Cor. 5$_s$ and $SU(2)$ Theorems 1--2 agree including images, with Thm 3 requiring its extra matrix symmetry. Central images iff $(1-s)H\in P^\vee$, reaching $\mathbb Z_{\gcd(q,N)}$ at reduced $s=p/q$; nontrivial $SU(3)$ centre needs $3\mid q$. Signed image amplitudes; per-volume control remains open | exact, proved and refereed (Astra Round 5B A, 2026-09-28; corrections applied) | [sun-midpoint-centre](https://github.com/arivero/navstokgap/blob/main/notes/sun-midpoint-centre.md) |
| Two gaps (reading) | Refinement at fixed coupling removes the $U(1)$ mass gap ($e^{-c/(\hbar g^2a)}$ corrections) and keeps the $\hbar$-width Gaussian: a generated mass gap versus a supplied action floor | reading, refereed | [halving-atlas §4](https://github.com/arivero/navstokgap/blob/main/notes/halving-atlas.md) |
| Monopole TV theorem (Thm 2) | $D=3$: $\|\mu_a-\mu_a^0\|_{\rm TV}\le2(L/a)^3e^{-\pi^2/(6\lambda_3a)}/(1-e^{-\pi^2/(2\lambda_3a)})$; Cor. 2$'$: iteration reduces to exact Gaussian blocking; Thm 3: the same bound is useless in $D=4$ at fixed $g$ (route failure recorded) | proved, refereed (Fable) | [villain-monopole-refinement](https://github.com/arivero/navstokgap/blob/main/notes/villain-monopole-refinement.md) |
| $SU(2)$ midpoint Prop. 6 | Bridge midpoint covariance $\frac t4{\rm diag}(1,1/h,1/h)$, $h=\frac d4\cot\frac d4$; group-general softening $t_j|X_j|^2/512$ for $SU(3)$ | proved (leading order), refereed | [series/parallel §4](https://github.com/arivero/navstokgap/blob/main/notes/series-parallel-gauge-refinement.md) |
| $SU(2)$ Theorems 1--4 | Thm 1: exact spin-$\frac12$ midpoint character; Thm 2: exact for every spin; Thm 3: spin-$\frac12$ cube sector exact; Thm 4 (Astra): spin-1 diagonal entries exact with winding, image bounds | exact, refereed (Fable; Thm 4 all fifteen items by hand) | [su2-midpoint-exact](https://github.com/arivero/navstokgap/blob/main/notes/su2-midpoint-exact.md) |
| $SU(2)$ mid-plane order $t$ | $\delta_t^{12}=\delta_t^{13}=\frac t{24}\int_{\mathcal B}q/R>0$, $\delta_t^{23}=t(-\frac1{12}+2\mathcal I)<0$; other bulk operators dimension $\ge6$; explicit flat-holonomy torus term (16); P($\alpha$) gains the clause $\min_jN_j\ge c_0/t$; for $SU(N)$ the bulk coefficients scale by $N/2$ (adjoint Casimir ratio, §7b), so for $SU(3)$ $\delta^{12}=\delta^{13}=\frac t{16}\int q/R$, $\delta^{23}=t(-\frac18+3\mathcal I)$ | formal, refereed (two Fable passes, all items accepted as a formal calculation); §7b refereed by Fable (ACCEPT) | [su2-midplane-order-t](https://github.com/arivero/navstokgap/blob/main/notes/su2-midplane-order-t.md) |
| Hypothesis P($\alpha$) | One-step small-field bound on the log-ratio $\mathcal D$ with coupling shifts $O(t)$ and remainder $Ct^\alpha\sum_p(1+|\log U_p|^2/t)$, stable under iteration | open hypothesis; Prop. 7 gives it for one cube conditionally on uniform Laplace bounds | [series/parallel §4](https://github.com/arivero/navstokgap/blob/main/notes/series-parallel-gauge-refinement.md) |
| Zero-spacing Thm 2 | $D=2$, any symmetric class-function action: limits are exactly the conjugation-invariant Lévy exponents $\psi(R)=\frac{\sigma^2}2C_2(R)+\int(1-{\rm Re}\chi_R/d_R)d\nu$; Yang--Mills iff Lindeberg; circle spectrum $E_R-E_0=\hbar cL\psi(R)$ | proved, refereed (Astra REFINE applied) | [zero-spacing-any-action §2](https://github.com/arivero/navstokgap/blob/main/notes/zero-spacing-any-action.md) |
| Zero-spacing Prop. 1 | Parallel product of $U(1)$ heat kernels is a heat kernel at $st/(s+t)$ plus shifted images of mixture weight $e^{-2\pi^2m^2/(s+t)}$ | exact, refereed | [zero-spacing §1](https://github.com/arivero/navstokgap/blob/main/notes/zero-spacing-any-action.md) |
| 4D large-field threshold | Density $(a\Lambda)^{2b_0c}a^{-4}$ vanishes iff $c>2/b_0=96\pi^2/(11N)$: $12/(11N)$ of the instanton action, the known $6/11$ dislocation criterion for $SU(2)$ | known (Pugh--Teper, Göckeler et al. 1989), recovered; strict threshold refereed ACCEPT | [zero-spacing §3](https://github.com/arivero/navstokgap/blob/main/notes/zero-spacing-any-action.md) |
| Error budget by dimension | Perturbative $O(t_n)$, large-field $a_n^{-D}e^{-c/t_n}$, jump $\nu$; summable for $D<4$, running for $D=4$, no small-field regime for $D>4$ | organizing table; universality inference REJECTED by referee | [zero-spacing §3](https://github.com/arivero/navstokgap/blob/main/notes/zero-spacing-any-action.md) |
| Time-only halving | Electric faces are series (semigroup), magnetic faces parallel (exponential family); defect is Trotter non-commutation; heat-kernel anisotropy formula (1) fails there | reading of known results | [halving-atlas §5.5](https://github.com/arivero/navstokgap/blob/main/notes/halving-atlas.md), [zero-spacing §1](https://github.com/arivero/navstokgap/blob/main/notes/zero-spacing-any-action.md) |
| T1 | Finite lattice: unique ground state, $\delta>0$, every $a,N_s,g$, compact $G$ | proved | [mass-gap-obligations-lattice](https://github.com/arivero/navstokgap/blob/main/notes/mass-gap-obligations-lattice.md) |
| T2 | Volume-uniform strong-coupling gap: Wilson transfer matrix $\Delta_W\ge\frac{\hbar c}a4\log[1/(176(e^{6/g^2}-1))]$ for $g^2\ge1059$ (sufficient integer bound from the exact Kotecký--Preiss condition, corrected 2026-10-02; $1056$ is its linearization and $176$ the leading-activity figure only); the Kogut--Susskind explicit threshold $g^2\ge388$ (and $79$ adjacent) was withdrawn on 2026-10-02 after a referee found its overlap count incomplete; KS keeps Yarotsky's existential threshold and a finite-volume bound $\frac83g^2-12|P|/g^2$ only | Wilson: proved (Kotecký--Preiss); KS explicit number: withdrawn 2026-10-02 | [wilson-strong-coupling-explicit](https://github.com/arivero/navstokgap/blob/main/notes/wilson-strong-coupling-explicit.md), [kogut-susskind-strong-coupling-explicit](https://github.com/arivero/navstokgap/blob/main/notes/kogut-susskind-strong-coupling-explicit.md) |
| T2$'$, T3, T4 | Gap at every coupling; scaling limit $\delta_\infty/(a\Lambda_{\rm lat})\to m/\hbar c\Lambda$; continuum theory with the axioms | open (T2$'$ false for $U(1)$ and for $SU(N\ge5)$ Wilson action) | [mass-gap-position](https://github.com/arivero/navstokgap/blob/main/notes/mass-gap-position.md), [mass-gap-openings](https://github.com/arivero/navstokgap/blob/main/notes/mass-gap-openings.md) |
| Conditional theorem | H1 (blocking clusters to $\xi\simeq a$) + H2 (certified mixing on one box) $\Rightarrow m\ge\hbar c\gamma'/a_*$ | conditional | [mass-gap-conditional-theorem](https://github.com/arivero/navstokgap/blob/main/notes/mass-gap-conditional-theorem.md) |
| Small volume (S) | Gap-transfer lemma proved; H1--H3 inequalities open; $\Delta=\delta_1g^{2/3}\hbar c/L[1+O(g^{2/3})]$ upper side proved | conditional | [weak-coupling-feshbach-reduction](https://github.com/arivero/navstokgap/blob/main/notes/weak-coupling-feshbach-reduction.md) |
| Seven closed routes | Variational, free expansion, projection blocking, flow conjugation, flow before decimation, Agmon, Gibbs domination: each fails for a stated reason with constants | recorded failures | [mass-gap-position §3](https://github.com/arivero/navstokgap/blob/main/notes/mass-gap-position.md) |
| G07, G08 | Supplied vs generated gaps; YM quantum mechanics gap $\delta_1\hbar^{4/3}g^{2/3}m^{-2/3}$ from an action floor on transverse area (C133) | exploratory, ledgered | [low-dimensional-mass-gap](https://github.com/arivero/navstokgap/blob/main/notes/low-dimensional-mass-gap.md), [action-floor-yang-mills-gap](https://github.com/arivero/navstokgap/blob/main/notes/action-floor-yang-mills-gap.md) |

## 3. Conventions a model must respect

- **$\hbar$ and $c$ explicit in every formula.** Never "set $\hbar=c=1$", even
  for readability. Energies as multiples of $\hbar c/L$ or $\hbar c/a$;
  $[a,p]=i\hbar$; $g^2=\hbar g_{\rm cl}^2$ in $D=4$; write $\hbar c|k|$,
  $\hbar c/(2g^2)$ before magnetic terms, $g^2c/(2\hbar)$ before electric
  terms (conventions of [G07 Prop. 7](https://github.com/arivero/navstokgap/blob/main/notes/low-dimensional-mass-gap.md)).
- **No numerical or symbolic verification scripts** (user rule, 2026-09-09).
  Written derivations are the check. Document tooling (pandoc, LaTeX,
  pdftotext, `make check`, shell queries to Crossref) is allowed.
- **One-note builds:** `make paper NOTE=<slug>`; then `make site`; then
  `python3 scripts/check_repository.py`. Never `make papers` as a routine
  gate; never read PDF pages as images.
- **Style:** the negation-first pattern ("X is not Y; it is Z", "not
  merely", "not because A but because B") is banned; state what a thing is.
  At most one short load-bearing contrast per page. Reviewers rewrite
  offending sentences and say so in the commit.
- **Reading labels** on every citation: metadata / abstract / passage /
  full-read. Verify DOIs through Crossref before use; quotations only from
  a held companion in `docs/` or `docs/classics/`, registered in
  `docs/SHA256SUMS`.
- **Results go in `notes/<slug>.md`:** lead with the result and date, state
  each assumption where used, keep constants explicit, keep negations that
  change a theorem's meaning, end with "Consequence for STATE". Corrections
  are dated and kept in the note, never silently overwritten. No handoffs,
  task rows, review files or programme versions (git history is the record).
- **Commits:** end with the model's trailer (`Co-Authored-By: Claude Fable
  5.1 <noreply@anthropic.com>` for this model; Astra work carries
  `Co-Authored-By: GPT-6 Astra <noreply@openai.com>`; mixed commits carry
  both). Check `git status` for the other agent's uncommitted work before
  committing. Pushes to `origin/main` are authorized after each result.
- **Public repository.** Everything committed is public. Redistribution
  rights are checked before archived editorial material is added; in-copyright
  passages are quoted, never reproduced as companions.
- Claims cited by ID live in [claims/LEDGER.md](https://github.com/arivero/navstokgap/blob/main/claims/LEDGER.md) (frozen
  format); add a row only for a result another note cites by ID.

## 4. Traps already caught

Each of these was written once, caught by a referee, and corrected; the
correction is dated in the note. Do not reintroduce them.

1. **Sagitta convention.** The fall is $s=F\tau^2/(2M)$ and the impulse
   $J=F\tau$ (Planck paper eq. (1)). Theorem C(b) of the fifth-postulate
   note first used $s=F\tau^2/M$ and was aligned (commit `facfba7`).
2. **$\gamma$ is relative to $h_{\rm rad}$, and $h_{\rm rad}\ne h_P$.** The SED
   link first wrote $\gamma=1/(2\pi)$; that holds relative to Planck's $h_P$.
   Relative to Theorem U's $h_{\rm rad}$, $h_P=(\pi^4/15)^{1/3}h_{\rm rad}$ and
   $\gamma\approx0.297$. The whole link is conditional on a disputed SED
   derivation of the Planck spectrum, and Theorem 1 there holds for
   resonant variables only (full momentum variance diverges)
   ([sed-zeta-radiation-link](https://github.com/arivero/navstokgap/blob/main/notes/sed-zeta-radiation-link.md), commit `a330d4e`).
3. **Göpfert--Mack trajectory.** The fixed-Debye-mass limit lies on
   $\lambda_3a\simeq c_0/(2\log(1/a))$, $c_0=2\pi^2G(0)\approx4.99$, where the
   monopole density per physical volume diverges. The constant-density
   trajectory $c_0/(3\log(1/a))$ has $m_D\to0$. The first version conflated
   them ([villain-monopole-refinement](https://github.com/arivero/navstokgap/blob/main/notes/villain-monopole-refinement.md), commit `07c9501`).
4. **P($\alpha$), first version, was false for the free field.** Bounding
   $-\log\Psi$ itself needs $\delta_t=\frac12$ for the free field, and the
   large-field clause on $\Psi$ contradicts Prop. 3 (factor 2 at the cut
   locus). The hypothesis now bounds the log-ratio $\mathcal D$, allows
   shifts on cut faces as well as transverse ones, and (after the mid-plane
   note) carries the clause $\min_jN_j\ge c_0/t$ on periodic planes
   ([series/parallel §4](https://github.com/arivero/navstokgap/blob/main/notes/series-parallel-gauge-refinement.md), commits `d67c4c7`, `06ab8b8`).
5. **Heat-kernel anisotropy is a small-field matching.** Eq. (1)
   $t_{\mu\nu}=\lambda_Da_\mu a_\nu/\prod a_\rho$ fails under time-only refinement:
   transverse heat times go to infinity and the magnetic potential
   vanishes. The Hamiltonian (Kogut--Susskind) limit needs exponential
   $e^{-a_0V}$ magnetic weights ([zero-spacing §1](https://github.com/arivero/navstokgap/blob/main/notes/zero-spacing-any-action.md)).
6. **The $6/11$ criterion is known.** $2b_0c>4$ recovers the dislocation
   criterion of Pugh--Teper and Göckeler et al. (1989), stated as "$6/11$ of
   an instanton" by DeGrand--Hasenfratz--Zhu (1996). Claim only the
   organization, and only as a strict power threshold; an action lower
   bound gives an upper density bound after entropy control
   ([zero-spacing §3](https://github.com/arivero/navstokgap/blob/main/notes/zero-spacing-any-action.md), commit `5ef4e38`).
7. **Lynds (2003) distinguishes his indeterminacy from $h$.** An earlier
   sentence had him identify them. No author found connects Newton's
   *velocitas ultima* to uncertainty ([fifth-postulate §1](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md), commit `ef5857f`).
8. **"Floor iff $\hbar\ne0$" needs state completeness.** Astra's routes
   note gives the same floor in a commutative theory with a Gaussian
   covariance restriction ($h_*=2\zeta$). Noncommutativity is the unique
   *algebraic* denial of joint determinacy under (H1)--(H3); the state
   route is a second denial. The analogy with the parallel postulate is
   partial, and (H2) is a premise the *Principia* does not supply
   ([fifth-postulate, "Where the analogy stops"](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md)).
9. **Bohm, *Quantum Theory* (1951): copyright renewed 1979** (US Copyright
   Office renewal RE010931, 23 January 1979, claimant David Bohm, original
   registration A52291, per the Stanford Copyright Renewal Database; in the
   EU protected to the end of 2062). Cite by page
   (ch. 8, pp. 145--148, on Zeno's arrow); no companion transcription may be
   added. The passage was reported from an Astra search and is unchecked
   here ([fifth-postulate §1](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md)).
10. **Universal Coulomb impulse bound rejected.** $|\Delta p|b\ge2k_e/c$ holds
    only on a prescribed straight path; exact Rutherford scattering gives
    $b|\Delta p|\to0$ as $b\downarrow0$ ([necessity-unit §1](https://github.com/arivero/navstokgap/blob/main/notes/necessity-unit-and-indeterminacy.md)).
11. **Theorem I needs a fourth premise.** Instrument access (the two pulses
    and a stored first reading) is a premise beside Liouville dynamics,
    product preparations and Bayesian readouts; posterior concentration and
    disturbance cost need distinct bounds ([necessity-unit §4](https://github.com/arivero/navstokgap/blob/main/notes/necessity-unit-and-indeterminacy.md)).
12. **Prop. 7's reflection cancellation was rejected.** Mixed quadratic
    boundary terms survive cube reflections; the surviving statement is a
    conditional finite-cube bound with mixed terms in the remainder
    ([series/parallel §4](https://github.com/arivero/navstokgap/blob/main/notes/series-parallel-gauge-refinement.md)).
13. **Normalized Haar supplies $2\pi$.** Prop. 1 of the zero-spacing note and
    Prop. 4 of the series/parallel note carry normalized-Haar constants;
    an earlier $\pi$ in the Planck paper was $2\pi$ (batch 2, 2026-09-23).
    Also $\Lambda p=h/2=\pi\hbar$ differs from $\hbar/2$ by $2\pi$.
14. **The balanced-aperture loophole.** The unmarked-preparation bound
    takes its area form $\theta^2\hbar^2/(4LP)$ only at $L=\tau P/2m$; squeezing
    evades a fixed-area form (ChatGPT review, 2026-09-21; user: "every non
    go is distraction", so the fix is the corrected positive theorem).
15. **Historical overclaims.** Newton nowhere "asserted in print" the
    negation of M3; what he printed entails it (*Opticks* II.iii Prop. XII
    with Query 29). The fits do not deny finite propagation time (II.iii
    Prop. XI). Planck 1900 (*Ann. Phys.* 306, 69) is the Wien-entropy paper;
    the distribution is Planck 1901 (*Ann. Phys.* 309, 553).
16. **T2$'$ is sufficient, not required.** A gap at every lattice coupling
    fails for $SU(N\ge5)$ with the Wilson action (first-order bulk
    transition); the conjecture needs a gap on $(0,g_1)$ along the scaling
    curve ([mass-gap-openings](https://github.com/arivero/navstokgap/blob/main/notes/mass-gap-openings.md)).
17. **Yarotsky's threshold for Kogut--Susskind is $g^2\sim10^{101}$.** Usable
    numbers: Wilson $g^2\ge1059$ rigorous ($176$ leading-activity only) from the Euclidean polymer gas. The $388$ KS figure
    from the Duhamel expansion was withdrawn on 2026-10-02 (the adjacent-shape
    count anchored at the test link misses histories reaching it later; the
    overlap shift measure lacked its multiplicity), so no usable KS number stands.
18. **Thm 2 of the zero-spacing note needs finite exponents and a simple
    group** (or an isotropy assumption); $U(1)^2$ admits anisotropic
    diffusion. **Thm 5's** lifts use coarse-cube Bianchi identities and
    keep harmonic flux sectors.
19. **A one-step upper bound proves neither convergence nor a density.**
    The $\pi^2/8$ constant of Theorem 5 says nothing about the monopole
    density on the Göpfert--Mack trajectory.
20. **`make paper` used to exit 0 when LaTeX failed** (before commit
    `bfcc529`). A note quoting Chinese never produced its PDF; notes with CJK
    ideographs now select XeLaTeX automatically (`b8bacd7`). Check the PDF's
    timestamp, never make's stdout.
21. **Bridge mean at a cut at fraction $s$** is $(1-s)P_e-sQ_e$ in the additive
    convention where $s=\frac12$ gives $(P_e-Q_e)/2$, with $P_em_e^{-1}$ the half of
    thickness $sa_1$ (Corollary 2$_s$); a first draft had $+sQ_e$.
22. **Cut measure: two floors.** A floor on a cell's action $K_\tau$ and a floor
    on a cut's share $3s(1-s)K_L$ differ by $\frac34$ for halving; the two-path
    phase is $K_\tau/\varepsilon$ (its half-phase is $K_\tau/(2\varepsilon)$); the Euclidean
    per-cut quantity is a Kullback--Leibler divergence
    ([cut-measure-newton](https://github.com/arivero/navstokgap/blob/main/notes/cut-measure-newton.md), commit `88344a9`).
23. **At an $SU(2)$ halving a winding $n$ acts by $(-1)^n$**, not always by
    $-1$; even windings act trivially (atlas §1b, commit `7d8518a`).
24. **Image terms are not windings for simply connected groups.** In the
    exact midpoint formula the coroot images carry signed Weyl-polynomial
    amplitudes; reading them as a probability mixture over topological
    winding sectors holds for $U(1)$ only ($SU(N)$ is simply connected).
    The thin-vortex placement at a trisection is a semiclassical reading
    of exact central phases (Astra round 5B, commit `059d8be`).
25. **Nelson's stochastic mechanics is not independence from the literal
    Laws.** His mean dynamics needs further state and phase assumptions;
    the stochastic route's S1--S3 stand without it, and the centre-of-mass
    variance argument of S2 appears already in Demme--Caticha (round 5B).
26. **Plaquette fluxes are not coordinates on a torus.** A Laplace
    comparison bounded by $\sum_p|X_p|^2$ alone fails along flat-holonomy paths:
    all fluxes vanish while the winding functional $\Omega_N(\theta)$ changes the
    defect by order one at fixed plane size. Any small-field comparison must
    carry the winding term (or P($\alpha$)'s additive $Ct^\alpha N$) and a size clause
    (Claude's Proposition 4, rejected by Astra round 8, commit `6be9df4`;
    the missing estimate is the gauge-cancellation bound (27) there).


## 5. Open problems, with the smallest next theorem

Ordered as in [STATE](https://github.com/arivero/navstokgap/blob/main/research/STATE.md) and [halving-atlas §5](https://github.com/arivero/navstokgap/blob/main/notes/halving-atlas.md).

1. **$SU(2)$ mid-plane, estimate (18).** Prove the uniform normalized
   Laplace remainder for the full interacting mid-plane integral on the
   small-field set, uniformly in plane size, with the torus kernel $W_N$
   retained and saddle uniqueness from $H\ge(4-C\varepsilon)I$
   ([su2-midplane-order-t §7](https://github.com/arivero/navstokgap/blob/main/notes/su2-midplane-order-t.md)). *Smallest
   theorem:* the weak form, remainder $|E|\le Ct\sum_p(1+|X_p|^2/t)$ with
   $\alpha=\frac12-\delta$, for the unperturbed heat-kernel action on one plane.
2. **Stability of P($\alpha$) under iteration for $SU(N)$.** Done for $U(1)$
   in $D=3$ by Cor. 2$'$ (exact Gaussian blocking). *Smallest theorem:*
   P($\alpha$) for a heat-kernel action perturbed by one step's defect
   $\mathcal D_0+\sum\delta_tS_{\mu\nu}$, same constants.
3. **$SU(2)$ cube beyond spin 1.** Diagonal entries for $J>1$ and control of
   the full spin sum ([su2-midpoint-exact](https://github.com/arivero/navstokgap/blob/main/notes/su2-midpoint-exact.md)).
   *Smallest theorem:* a uniform-in-$J$ bound on $\|\Lambda^J-\Lambda^J_{(1)}\|$
   with $\Lambda^J_{(1)}$ the order-$t$ formula of Prop. 6. The exact character
   formula for every $J$ and every group ([sun-midpoint-centre](https://github.com/arivero/navstokgap/blob/main/notes/sun-midpoint-centre.md))
   gives traces only; the matrix entries carry extra Gaussian tails.
4. **The four-dimensional logarithm.** The one-step shifts are done
   formally for $SU(N)$ and any cut fraction, the one-step propagators are
   infrared regular, and $2b_0$ follows under background-matching
   hypotheses ([four-dimensional-parallel-log](https://github.com/arivero/navstokgap/blob/main/notes/four-dimensional-parallel-log.md),
   refereed). [Round 6](https://github.com/arivero/navstokgap/blob/main/notes/four-dimensional-composition.md) composes
   the Gaussian Schur complements and proves sharp-flux covariance growth.
   *Smallest theorem:* bound its (14), the continuum-subtracted generated
   cubic/quartic background contractions, uniformly in depth and through
   the infrared limit. Together with remainder control this gives the
   average rate by its Theorem 4; bounded matching remains open.
5. **From ultraviolet control to a gap in $1+2$.** Prove $E=C_3\hbar c\lambda_3$
   with $0<C_3<\infty$; the $U(1)$ comparison shows the mechanism must sit
   in the non-abelian terms N1--N3. *Smallest theorem:* a lower bound on the
   plaquette--plaquette decay rate of the blocked measure at one coupling
   in physical units, uniform in $a$.
6. **Time-only halving rate.** An operator-norm Trotter rate for
   $e^{-a_0(T+V)}$ versus $e^{-a_0T}e^{-a_0V}$ on $G^E$, which needs domain
   estimates for $[\Delta_G,V]$ ([halving-atlas §5.5](https://github.com/arivero/navstokgap/blob/main/notes/halving-atlas.md)).
7. **H1 and H2 of the conditional theorem** ([mass-gap-conditional-theorem](https://github.com/arivero/navstokgap/blob/main/notes/mass-gap-conditional-theorem.md)):
   exponential clustering of integrated-out fluctuations per blocking step
   down to $\xi\simeq a$, and certified mixing at one coupling on one box.
   The Feshbach H3 small-volume estimate is the valley-lifting task
   ([weak-coupling-feshbach-reduction](https://github.com/arivero/navstokgap/blob/main/notes/weak-coupling-feshbach-reduction.md)).
8. **Newton necessity: the readout law.** Every premise so far is either
   supplied (quantum kinematics, Thm C), circular ($\zeta$ from SED at
   $\kappa=\hbar/2$), or insufficient (thermodynamics, the shared bath).
   The statement to change is joint determinacy; the missing physical
   premise is a bound $\text{imprecision}\times\text{kick}\ge\kappa$ on every
   terminal classical readout, closed under composition
   ([sed-closure-under-recording](https://github.com/arivero/navstokgap/blob/main/notes/sed-closure-under-recording.md),
   [newton-indeterminacy-routes](https://github.com/arivero/navstokgap/blob/main/notes/newton-indeterminacy-routes.md)).
   *Smallest theorem:* derive that law for one physically specified
   readout class (electrodynamic, with retained memory) and show closure.
   Since 2026-10-01 the gap thesis adds a second premise to defend, the
   $h$-independent coherence length between the launch points of crossing streams, and reads the
   readout premise as the record-limit-first order of limits
   ([zero-branch-reachability §§3c, 6](https://github.com/arivero/navstokgap/blob/main/notes/zero-branch-reachability.md)).
   Since 2026-09-29 the premise has an equivalent Newton-age form: in the
   mark model, $\kappa>0$ iff the best verdict on the comparison is continuous
   in the force at zero, Leibniz's 1687 law of continuity read on records
   ([leibniz-continuity-records](https://github.com/arivero/navstokgap/blob/main/notes/leibniz-continuity-records.md));
   defending that reading of outcomes is the same open step.
9. **Layer 1 of the dimension ladder.** Extend Theorem U to any classical
   field with unbounded mode frequencies admitting finite thermal
   equilibrium, with cross-field universality by Kirchhoff's filter
   argument ([dimension-ladder §1](https://github.com/arivero/navstokgap/blob/main/notes/dimension-ladder.md)); stated as a
   conjecture.
10. **Rivero 1998, general case.** Whether finiteness of the path integral
    forces the joint limit (action resolution and time step) for
    non-quadratic actions; the quadratic case is Theorem A
    ([fifth-postulate §7](https://github.com/arivero/navstokgap/blob/main/notes/principia-fifth-postulate.md),
    [rivero-1998-conjecture-central-forces](https://github.com/arivero/navstokgap/blob/main/notes/rivero-1998-conjecture-central-forces.md)).
    Since 2026-10-01 the gap thesis is located: complete records of
    determinate preparations lose their $h\to0$ limit after the first
    fold, smeared records never do
    ([zero-branch-reachability](https://github.com/arivero/navstokgap/blob/main/notes/zero-branch-reachability.md));
    the next lemma is the first fold under a central force on Newton's polygon.
11. **Planck paper, attainment and submission.** Whether the constants
    $1/8$, $1/2$ of Thm 6 and the minimum of the one accounting (Thm M) are
    attained; the §11 historiography and edition obligations
    ([planck-gap-paper §11](https://github.com/arivero/navstokgap/blob/main/notes/planck-gap-paper.md)).
12. **A groupoid for the lattice.** The holonomy groupoid for series moves
    and a double groupoid for parallel moves
    ([tangent-groupoid-trajectories §5](https://github.com/arivero/navstokgap/blob/main/notes/tangent-groupoid-trajectories.md)); a question, with no theorem yet.
13. **$SU(3)$ centre and triadic refinement.** Only cuts with denominator
    divisible by 3 let bridge windings reach the $SU(3)$ centre
    ([sun-midpoint-centre](https://github.com/arivero/navstokgap/blob/main/notes/sun-midpoint-centre.md), Corollary 3,
    Astra referee pending). *Question:* does triadic refinement ($b=3$)
    organize the large-field terms, or centre-vortex bookkeeping, better
    than dyadic refinement? The joint character moments of the two layers
    of a trisection are exact (Theorem 1$'$ there), and their central images
    are three thin $\mathbb Z_3$ vortices in one cut face fusing to the identity.
    *Smallest theorem:* the relative orientation of the two layers (the
    full joint law), or a comparison of dyadic and triadic large-field
    bookkeeping in one step.

## 6. Verified prior-art map

All DOIs below appear in the notes with reading labels and were checked
through Crossref before use. `references/library.bib` holds keys for those
cited by key.

**Lattice gauge theory and constructive field theory.**
Wilson--Kogut 1974 [10.1016/0370-1573(74)90023-4];
Kogut--Susskind 1975 [10.1103/PhysRevD.11.395];
Creutz 1977 transfer matrix [10.1103/PhysRevD.15.1128];
Lüscher 1977 transfer matrix [10.1007/BF01614090];
Osterwalder--Seiler 1978 [10.1016/0003-4916(78)90039-8];
Kadanoff 1976 on Migdal [10.1016/0003-4916(76)90066-X];
Bell--Wilson 1975 [10.1103/PhysRevB.11.3431];
Hasenfratz--Niedermayer 1994 [10.1016/0550-3213(94)90261-5];
Banks--Myerson--Kogut 1977 [10.1016/0550-3213(77)90129-8];
Polyakov 1977 [10.1016/0550-3213(77)90086-4];
Göpfert--Mack 1982 [10.1007/BF01961240];
Gross 1983 [10.1007/BF01210842];
Driver 1987 [10.1007/BF01212424]; Driver 1989 YM$_2$ [10.1007/BF01218586];
Guth 1980 [10.1103/PhysRevD.21.2291];
Fröhlich--Spencer 1982 [10.1007/BF01213610];
Balaban 1984 [10.1007/BF01215753], 1985 (3D UV stability) [10.1007/BF01229380],
1987 [10.1007/BF01215223], 1988 [10.1007/BF01217741],
1989 I [10.1007/BF01257412], II [10.1007/BF01238433];
Dimock 2013 [10.1142/S0129055X13300100];
Magnen--Rivasseau--Sénéor 1993 [10.1007/BF02097397];
Yarotsky 2005 [10.1007/s00220-005-1456-9];
Lüscher 1982 lattice topology [10.1007/BF02029132];
Lüscher 1983 torus [10.1016/0550-3213(83)90436-4];
Lüscher--Münster 1984 [10.1016/0550-3213(84)90038-5];
Gross--Wilczek 1973 [10.1103/PhysRevLett.30.1343]; Politzer 1973 [10.1103/PhysRevLett.30.1346];
Hasenfratz--Hasenfratz 1980 [10.1016/0370-2693(80)90118-5];
Dashen--Gross 1981 [10.1103/PhysRevD.23.2340];
Capitani 2003 [10.1016/S0370-1573(03)00211-4];
Menotti--Onofri 1981 [10.1016/0550-3213(81)90560-5];
Creutz 1979 (5D transition) [10.1103/PhysRevLett.43.553];
Gross--Witten 1980 [10.1103/PhysRevD.21.446];
Pugh--Teper 1989 [10.1016/0370-2693(89)91067-8];
Göckeler et al. 1989 [10.1016/0370-2693(89)90640-0];
DeGrand--Hasenfratz--Zhu 1996 [10.1016/0550-3213(96)00301-X];
Hunt 1956 [10.1090/S0002-9947-1956-0079232-9]; Lévy, *Astérisque* 329 [10.24033/ast.785];
Trotter 1959 [10.1090/S0002-9939-1959-0108732-6]; Chernoff 1968 [10.1016/0022-1236(68)90020-7];
Eardley--Moncrief 1982 [10.1007/BF01976040];
Aizenman 1982 [10.1007/BF01205659]; Fröhlich 1982 [10.1016/0550-3213(82)90088-8];
Aizenman--Duminil-Copin 2021 [10.4007/annals.2021.194.1.3];
Jaffe--Witten, Clay problem statement (companion `docs/JaffeWitten_YangMills.md`).

**Deformation quantization.**
Moyal 1949 [10.1017/S0305004100000487];
Groenewold 1946 [10.1016/S0031-8914(46)80059-4];
van Hove 1951 [10.3406/barb.1951.70660];
Vey 1975 [10.1007/BF02565761];
Bayen--Flato--Fronsdal--Lichnerowicz--Sternheimer 1978 [10.1016/0003-4916(78)90224-5];
Flato 1982 [10.1007/BF01596202]; Sternheimer 1998 [10.1063/1.57093];
Gutt 1983 (*Mém. Acad. Roy. Belg.* 44:6), as cited in Duval--El Gradechi--Ovsienko 2004 [10.1007/s00220-003-0973-7];
Fletcher 1990 [10.1016/0370-2693(90)90300-U];
Bertelson--Bieliavsky--Gutt 1998 [10.1023/A:1007598606137];
Robertson 1929 [10.1103/PhysRev.34.163].

**Measurement theory and quantum limits.**
Bohr 1928 Como [10.1038/121580a0] (quoted via Busch--Shilladay, arXiv:quant-ph/0609048);
Slepian--Pollak 1961 [10.1002/j.1538-7305.1961.tb03976.x]; Landau--Pollak 1961 [10.1002/j.1538-7305.1961.tb03977.x];
Amrein--Berthier 1977 [10.1016/0022-1236(77)90056-8]; Benedicks 1985 [10.1016/0022-247X(85)90140-4];
Braginsky--Vorontsov 1975 [10.1070/PU1975v017n05ABEH004362];
Caves--Thorne--Drever--Sandberg--Zimmermann 1980 [10.1103/RevModPhys.52.341];
Caves 1985 [10.1103/PhysRevLett.54.2465]; Ozawa 1988 [10.1103/PhysRevLett.60.385];
Bocko--Onofrio 1996 [10.1103/RevModPhys.68.755];
Caves--Milburn 1987 [10.1103/PhysRevA.36.5543];
Clerk et al. 2010 [10.1103/RevModPhys.82.1155];
Tsang--Nair 2012 [10.1103/PhysRevA.86.042115]; Tsang--Caves 2012 [10.1103/PhysRevX.2.031016];
Anandan--Aharonov 1990 [10.1103/PhysRevLett.65.1697];
Ozawa 2003 [10.1103/PhysRevA.67.042105];
Busch--Lahti--Werner 2013 [10.1103/PhysRevLett.111.160405], 2014 [10.1063/1.4871444];
Childs--Preskill--Renes 2000 [10.1080/09500340008244034];
Zurek 2001 [10.1038/35089017]; Toscano et al. 2006 [10.1103/PhysRevA.73.023803];
Gottesman--Kitaev--Preskill 2001 [10.1103/PhysRevA.64.012310];
Bartlett--Rudolph--Spekkens 2012 (arXiv:1111.5057);
Landauer 1961 [10.1147/rd.53.0183]; Bennett 1982 [10.1007/BF02084158];
Sagawa--Ueda 2009 [10.1103/PhysRevLett.102.250602];
Helstrom 1976 (book); Braginsky--Khalili 1992 (book).

**Stochastic electrodynamics and radiation thermodynamics.**
Marshall 1963 [10.1098/rspa.1963.0220]; Boyer 1969 [10.1103/PhysRev.182.1374];
Boyer 1975 [10.1103/PhysRevD.11.790];
Goedecke 1983 [10.1007/BF00728140]; Nieuwenhuizen 2019 [10.3390/atoms7020059];
Jiménez--de la Peña--Brody 1980 [10.1119/1.12221] (the Planck derivation disputed);
Wien 1893 reprint [10.1007/978-3-663-13885-3_12];
Planck 1900 [10.1002/andp.19003060105]; Planck 1901 [10.1002/andp.19013090310];
Einstein 1907 [10.1002/andp.19063270110].

**History of the *Principia*, the *Opticks* and the ancient dispute.**
Held companions (public domain, checksummed) in `docs/classics/`: 1687
Latin of Book I Section I scholia, Motte 1729/1846, Berkeley's *Analyst*
1734, Gregory 1702 preface (witness to the Classical Scholia), Archimedes
(Heath), Galileo, Cavalieri, Guldin, Hamilton, Jacobi, Helmholtz, Hertz,
and the Greek, Sanskrit, Chinese and Latin sources of the arrow debate
(catalogue: `docs/classics/README.md`). Scholarship cited: McGuire--Rattansi
1966 [10.1098/rsnr.1966.0014]; Casini on the Classical Scholia; Schüller's
edition (companion `docs/Newton_ClassicalScholia_Schuller2000.md`);
Shapiro 1993 *Fits, Passions, and Paroxysms*; Guicciardini 1999, 2009;
De Gandt 1995; Koyré--Cohen variorum and Cohen--Whitman (edition
obligations, paper §11). Zeno and uncertainty: Vaidman 2008
[10.1038/451137a]; Goyal 2026 [10.1007/s10701-026-00922-0]; Lynds 2003
[10.1023/A:1025361725408]; Arntzenius 2000 [10.5840/monist20008328];
Smith 2003 [10.1016/S1355-2198(03)00007-8].

**Tangent groupoid and discrete mechanics.**
Connes, *Noncommutative Geometry* 1994 §II.5;
Cariñena--Clemente-Gallardo--Follana--Gracia-Bondía--Rivero--Várilly 1999 [10.1016/S0393-0440(98)00028-X] (companion `docs/CarinenaEtAl_TangentGroupoid_1999.md`);
Rivero 1997 (arXiv:dg-ga/9710026), 1998 (arXiv:quant-ph/9803035, companion `docs/Rivero_FeynmanFormula_1998.md`), 2003 (arXiv:math/0302285);
Landsman 2002 (arXiv:math-ph/0208004);
Debord--Skandalis 2014 [10.1016/j.aim.2014.02.012];
van Erp--Yuncken 2019 [10.1515/crelle-2017-0035];
Marsden--West 2001 [10.1017/S096249290100006X]; Weinstein 1996 [10.1090/fic/007/10];
Litvinov 2005 [10.1090/conm/377/06982]; Abbott--Wise 1981 [10.1119/1.12657].

**Machine-assisted mathematics context.** `docs/AI_Mathematics_News_2026.md`
and `docs/OpenAI_NavierStokes_2026.md` (theorem statement visually checked,
proof unaudited here). A proof attempt is a legitimate task; the right
response to difficulty is a smaller theorem with explicit hypotheses.

## 7. How agents work here

- **Author and referee are separate passes, and every result crosses the
  gate before STATE cites it as more than "unrefereed".** The author
  writes the note with proofs; a second model re-derives claim by claim
  and returns a verdict per item (ACCEPT / REFINE / REJECT). Verdicts are
  written into the note in place (dated "Correction (GPT-6 Astra referee)"
  or "Referee verdict") and summarized in the commit message. A rejected
  claim is replaced by the true theorem with its hypothesis explicit,
  never by a bare retraction ("every non go is distraction").
- **Roles as practised.** GPT-6 Astra (Codex, autonomous sessions in the
  same checkout, trailer `GPT-6 Astra`) authored the routes note, the SED
  closure note, $SU(2)$ Theorem 4 and the mid-plane order-$t$ calculation,
  and refereed the series/parallel, zero-spacing and necessity notes.
  Claude (Opus 5.5 and Fable 5.1, trailer per model) authored the atlas,
  fifth-postulate, refinement, monopole, $SU(2)$ midpoint, dimension-ladder
  and groupoid notes and the Planck paper, and refereed Astra's notes.
  Bounded Sol 6 / Luna 6 workers may do menial source tasks, one at a
  time, never as Astra subagents. The user reads closely, contributes
  mid-task (the Archimedes and Democritus links, the $1+n$ ladder, the
  fifth-postulate question) and is credited in the note and commit.
- **Review in batches of two or three notes.** A single broad review
  prompt hits the 64k-token output cap and fails silently. Working
  pattern: one batch per referee call, findings appended to a scratch
  file after each note, one commit per batch. Two passes (sections 1--4,
  then 5--7) were needed for the mid-plane note.
- **No multi-agent workflows and no parallel subagents.** They inherit
  the main model and multiply usage without gain; work sequentially.
- **Pipeline for a result:** write `notes/<slug>.md`; `make paper
  NOTE=<slug>`; update the Markdown/JSON catalog entry and run
  `make site`; `python3 scripts/check_repository.py`; update the atlas
  cell and STATE (short); commit with the trailer; push.
- **What the user considers drift:** reframing results in the parked
  apparatus series (C068--C123), no-go notes without a replacement
  theorem, full rebuilds, image reads of PDFs, source hunts that change no
  claim. What counts: a cell of the atlas filled, a hypothesis of the
  conditional theorems discharged, a premise for the readout law derived.