From cone sections to time refinement
The bibliography supplies a concrete experiment: insert an intermediate point, eliminate it by composition, and identify which data survive. For the free particle, this operation closes exactly on a family of Gaussian kernels with an arbitrary positive action scale. The next research question is the physical selection of a member of that family.
The maintained calculation is Time Refinement and an Action Scale. Proof and literature status are recorded separately in the claim ledger.
What each source puts into the model
| Source | Idea to retain | Calculation or research obligation |
|---|---|---|
| Plutarch in Xylander, 1570 | Compare the faces produced by a cut | Specify separated faces and their limiting object |
| Rivero, cone, 1999 | Compare refinements at a common scale | Retain a rescaled difference when raw differences vanish |
| Rivero, path integral, 1998 | Eliminate inserted positions and track regulators | Write and test a normalized finite blocking map |
| Brouder, 1999 | Rooted-tree algebra for composed numerical flows and renormalisation | Organise nonlinear corrections after the Gaussian calculation |
| Cariñena et al., 1999 | Quantum and classical structures in a single geometric construction | Audit the precise quantum-completion and classical-limit theorem |
The Wilson–Kogut connection now has a direct §12.2 source reading: compare theories at a common physical correlation length. M05 implements that idea by fixing a reference bridge variance and running the bare kinetic coefficient. Brouder supplies the specific Butcher/Connes–Kreimer algebraic link. These are separate source links, each with a mathematical operation to investigate.
Spatial and temporal cuts
For a smooth cone profile with area \(A(z)\), a modern difference quotient retains \([A(z+\epsilon)-A(z)]/\epsilon\) as the two faces approach. Its limit is \(A'(z)\). The area’s units are length squared; this is ordinary spatial geometry.
Time slicing introduces a different operation. At fixed endpoints \(x,z\), an inserted position \(y\) is integrated over:
\[K_{s+t}(z,x)=\int_{\mathbb R}K_t(z,y)K_s(y,x)\,dy.\]
Thus temporal refinement involves both subdivision and a rule for eliminating the new degree of freedom. For real positive Gaussian kernels the integral is ordinary; for oscillatory free kernels it is defined through Fourier operators or a stated regularisation. Calling the positive-kernel model “Euclidean” refers to its quantum imaginary-time counterpart, a different use of the word from Euclidean cone geometry.
The first completed test and the next selection question
For \(m>0\), time \(t>0\) and action parameter \(\kappa>0\), the free positive kernel is
\[G_t^\kappa(x)=\sqrt{\frac{m}{2\pi\kappa t}} \exp\!\left(-\frac{mx^2}{2\kappa t}\right).\]
Its variance is \(\kappa t/m\). Convolution adds variances, so integrating out any number of inserted points leaves the same \(\kappa\). A Gaussian bridge between fixed endpoints concentrates on the straight Newtonian path as \(\kappa\to0\). These standard facts give an exact setting in which mesh refinement and the action-scale limit can be examined independently.
The general time-homogeneous Gaussian family has variance \(a t\), with \(a\ge0\). The action scale is \(\kappa=ma\). Time composition and continuity leave \(a\) free; \(a=0\) is the deterministic member. Demanding a positive-width density selects \(a>0\) by assumption. The strong target requires an independently motivated physical principle that produces that selection and explains universality across interacting systems. A lower bound on \(\kappa\) over an entire model class would be a stronger statement, with its own quantifiers and units.
The corresponding oscillatory family supplies unitary free evolution for each \(\kappa>0\). Converting the positive kernel to it is an explicitly chosen analytic-continuation step. A physical justification for that step belongs alongside, rather than downstream of, scale selection.
Reception history: a testable question
Xylander’s Latin Moralia contains the cone dilemma on printed pp. 823–824 in 1570. The H04 batch establishes availability and records a bounded reception search. Modern historical discussion is present: Auffret examines the cone in relation to limits in a chapter on Leibniz and Chrysippus. His direct Leibniz example concerns a rolling cylinder/cone and free will, which is a distinct passage.
The next historical test is author-specific: did a mathematician invoke the section dilemma in work on indivisibles or the continuum? Begin with Newton’s Plutarch references in the existing dossier and an independently indexed Leibniz corpus. Search title variants and Latin phrases as well as names; record ownership, quotation and mathematical use separately. The hypothesis of disciplinary separation between philosophical reception and mathematical practice can then be tested against those records.
Refinement milestone and continuation
M05/B07a/B08 completed the quadratic normalization, exact scaling map and source audit. The new two-regulator note records the finite action-defect theorem and routes it into M06’s dynamical test. B07b retains the tangent-groupoid theorem’s hypotheses and limit topology; match these to the strong target’s quantum-completion branch.
Each source task must end with a premise to test, a construction to implement, or a proof obligation to resolve. Use one small worker at a time and review its handoff before resuming coordinator work. The spectral laboratory M03 and Classical Scholia H02/H03 remain separate unfinished tasks.