Google Search AI — 26 October 2026
Results of the Convergence Hypothesis from the Planck Scale
| Field | What to record |
|---|---|
| Record ID | 81018-PR-001 |
| Date | 26 October 2026 |
| System | Google Search AI |
| Purpose | Identify reproducible octahedral and section-based dimensionless quantities |
| Status | Exploratory; prompted and iteratively corrected |
| Materials | Exact prompts and complete responses, in chronological order |
| Corrections | Editorial annotations (kept separate from the quoted answers) |
| Outcome | Corrected geometric inventory; no derivation of supplied |
26 October 2026 — Geometric inquiry recorded. A separate response log documents an exploratory exchange with Google Search AI concerning the regular octahedron and its four central face-parallel hexagonal sections. The exchange produced a corrected inventory of several geometric quantities, with remaining qualifications noted editorially. It did not supply a derivation of . H1 is therefore not established by this response; the response is not a confirmatory outcome for H1a. The original pre-registration is preserved unchanged.
This is an exploratory AI-generated response, not a verified geometric account. Subsequent checking identified incorrect radius ratios and unsupported interpretations concerning fundamental constants. Corrections were requested, and separate follow-up responses are documented separately.
Correction and Reconciliation of Radius Blunder
In the initial structural analysis, an algebraic error was made in computing the inradius (\(R_{i}\)), leading to an inconsistency between the radius subsection and the later volume calculation.
For a regular octahedron defined by the boundary \(\vert{}x\vert{} + \vert{}y\vert{} + \vert{}z\vert{} \le \sqrt{2}\), the 6 vertices sit exactly at a distance of \(\sqrt{2}\) from the origin. The distance between any two adjacent vertices—such as \((\sqrt{2}, 0, 0)\) and \((0, \sqrt{2}, 0)\)—is \(\sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = 2\). Therefore, the baseline edge length is exactly \(a = 2\).
The exact geometric derivation of the three concentric radii from the origin \((0,0,0)\) proceeds as follows:
- Circumradius (\(R_{c}\)): Evaluated from the origin to any vertex, such as \((\sqrt{2}, 0, 0)\).
\(R_{c}=\sqrt{(\sqrt{2})^{2}+0^{2}+0^{2}}=\sqrt{2}\) - Midsphere Radius (\(R_{m}\)): Evaluated from the origin to any edge midpoint, such as the midpoint between \((\sqrt{2}, 0, 0)\) and \((0, \sqrt{2}, 0)\), which is \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 0)\).
\(R_{m}=\sqrt{\left(\frac{\sqrt{2}}{2}\right)^{2}+\left(\frac{\sqrt{2}}{2}\right)^{2}+0^{2}}=\sqrt{\frac{1}{2}+\frac{1}{2}}=1\) - Inradius (\(R_{i}\)): Evaluated from the origin to any face centroid, such as the face bounded by \((\sqrt{2}, 0, 0)\), \((0, \sqrt{2}, 0)\), and \((0, 0, \sqrt{2})\), which yields a centroid located at \((\frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{3})\).
\(R_{i}=\sqrt{3\times \left(\frac{\sqrt{2}}{3}\right)^{2}}=\sqrt{3\times \frac{2}{9}}=\frac{\sqrt{6}}{3}\)
The corrected independent radius ratios are:
\(\frac{R_{c}}{R_{m}}=\sqrt{2},\quad \frac{R_{m}}{R_{i}}=\sqrt{\frac{3}{2}},\quad \frac{R_{c}}{R_{i}}=\sqrt{3}\)
The value of \(R_i = \sqrt{3}/3\) used in the earlier response was incorrect because it assumed an octahedron with a circumradius of 1 rather than \(\sqrt{2}\). Using \(R_i = \sqrt{6}/3\) matches the face-pyramid volume calculation perfectly.
Dimensionless Geometric Quantities of the Hexarchon System
Canonical Criterion: An entry is defined as canonical if it represents an exact, scale-invariant ratio or angular boundary derived exclusively from the intrinsic coordinates and symmetries of the specified figure, without requiring external parameters or arbitrary scaling choices.
| Named Geometric Objects | Exact Dimensionless Value | Derivation & Reproducible Construction | Required Assumptions / Added Objects |
|---|---|---|---|
| A. Octahedron Itself | |||
| Radius Boundary Ratio (\(R_c / R_m\)) | \(\sqrt{2}\) | Ratio of circumradius to midsphere radius: \(\frac{\sqrt{2}}{1}\) | None (Intrinsic metric) |
| Radius Boundary Ratio (\(R_m / R_i\)) | \(\sqrt{\frac{3}{2}}\) | Ratio of midsphere radius to inradius: \(1 / \frac{\sqrt{6}}{3}\) | None (Intrinsic metric) |
| Adjacent Face Dihedral Angle | \(\arccos\left(-\frac{1}{3}\right)\) | Angle between normal vectors of adjacent triangular faces | None (Intrinsic metric) |
| B. Specified Sections | |||
| Plate Edge to Face Edge | \(\frac{1}{2}\) | Side length of regular hexagon (\(1\)) divided by face side length (\(2\)) | Slicing planes: \(\pm x \pm y \pm z = 0\) |
| Individual Area Footprint | \(\frac{3}{2}\) | Area of one plate (\(\frac{3\sqrt{3}}{2}\)) divided by area of one outer face (\(\sqrt{3}\)) | Slicing planes: \(\pm x \pm y \pm z = 0\) |
| Inter-Plate Dihedral Angle | \(\arccos\left(\frac{1}{3}\right)\) | Angle between normal axes of the 4 defined section planes | Slicing planes: \(\pm x \pm y \pm z = 0\) |
| C. Additional Constructions | |||
| Inscribed/Circumscribed Volumes | \(\frac{\pi }{6}\) | Volume of the inscribed sphere (\(V=\frac{4}{3}\pi R_i^3\)) divided by bounding cube volume (\(8R_{m}^{3}\)) | External concentric sphere and bounding cube |
Status of Transcendentals and Irrationals
- \(\sqrt{2}\) and \(\sqrt{3}\): Derived intrinsically from the coordinate axes, radii, and triangular face heights.
- \(\pi \): Absent from the raw polyhedral constraints; it can only be introduced if one adds non-polyhedral objects, such as calculating the surface areas or volumes of the circumscribed, midsscribed, or inscribed concentric spheres.
- \(\phi \)
PreRegistration
1. Background
An exploratory numerical survey (81018, October 2026; “Base-2 Planck Notation and the Four Selected Irrationals”) identified a near-coincidence between the muon reduced Compton wavelength and the fractional target (2φ/5) mod 1, at Δ ≈ 1.26 × 10⁻⁴. The survey correctly reported this as a numerical near-coincidence and did not establish statistical significance, noting that the illustrative chance probability under an independence assumption is approximately 42%.
This document pre-registers a confirmatory test of that near-coincidence, structured so that the test can fail. The test has three components: a geometric derivation of the rational factor 2/5 from the Hexarchon, performed without reference to the muon; an inventory of physical scales assembled by a rule that does not permit knowledge of the muon result to influence inclusion; and a null model and decision criterion fixed in advance.
2. Hypothesis
H1 (primary). The rational factor 2/5 that appears in the muon near-coincidence is derivable from the cross-sectional geometry of the Hexarchon, independently of any particle-physics data. The derivation must be completed and dated before inventory construction (Section 3) and before the null model is applied (Section 4).
H1a (secondary). If H1 holds, then the inventory of independently established physical scales (Section 3) will contain a number of matches to fractional targets of the form (p/q × c) mod 1, with p/q = 2/5 and c ∈ {π, φ, e, √2}, that exceeds the expectation under the null model (Section 4) at the decision threshold specified in Section 5.
H1b (exploratory, not confirmatory). The inventory may contain matches at other rational factors. These will be reported but will not count toward the confirmatory test. These are exploratory observations.
What would falsify H1. If the factor 2/5 cannot be derived from the Hexarchon geometry without using the muon mass or the muon Compton wavelength, H1 fails and the near-coincidence remains a coincidence candidate with no geometric support.
What would falsify H1a. If the inventory contains no excess of matches at (2/5 × c) mod 1 above the null expectation at the decision threshold, H1a fails.
3. Inventory construction rule
The inventory for this test will be constructed under the following rule, fixed in advance:
- Source restriction. Entries must be drawn from NIST CODATA 2022 and the Particle Data Group (PDG) review, or from standard textbook conversions of quantities listed in those sources. No entry may be drawn from a source that discusses the muon/φ near-coincidence or the 81018 framework.
- Blind construction. The inventory will be assembled by a procedure that does not permit the constructor to know the muon result. Acceptable procedures: (a) construction by a person who has not been shown the muon result and does not know the target (2/5 × φ) mod 1; (b) construction by an automated script that draws a pre-specified list of quantities from CODATA/PDG tables in a fixed order, without reference to the target. The procedure used must be documented.
- Pre-specified classes. Each entry must be classified in advance as one of: (a) measured characteristic scale; (b) defined quantum-mechanical length; (c) derived electromagnetic length; (d) model-dependent scale; (e) macro/astrophysical control. The classification must be assigned before any match is computed.
- Minimum size. The inventory must contain at least [N — recommend 100] entries, of which at least [N_meas — recommend 20] are class (i) measured characteristic scales.
- No post-hoc exclusion. No entry may be removed after matches are computed, except on grounds of a documented error in the source value. Removals must be logged with the reason.
- Dating. The inventory file will be dated and hash-stamped before the match computation is run.
4. Null model
The null model must account for the heterogeneity of the inventory and the dependence among comparisons. It will be specified as follows:
- Type B comparisons only. The confirmatory test uses Type B (fractional) comparisons only, because the exploratory result was a Type B match and because Type A comparisons introduce a different dependence structure. Type A results will be reported as exploratory.
- Dependence-aware null. The null distribution will be estimated by Monte Carlo simulation under a model in which the inventory lengths are drawn from a distribution that reproduces the observed clustering of particle-physics scales in n. The specific model must be specified before the simulation is run. Acceptable options include: (a) a kernel density estimate of the observed n-distribution, resampled with replacement; (b) a log-uniform distribution over the span of the inventory, with the particle-physics cluster modeled as a separate component; (c) a permutation null in which the labels of inventory entries are permuted while their n-values are held fixed. The choice must be justified in writing before the simulation.
- Number of simulations. At least 10,000 Monte Carlo realizations. The null distribution of the count of matches at each Δ threshold (0.0001, 0.0005, 0.001, 0.002, 0.005) will be reported.
- Target set. The confirmatory target set is {(2/5 × c) mod 1 : c ∈ {π, φ, e, √2}}. That is four targets. The exploratory target set of 60 fractional targets is not used for the confirmatory test; it is reported as exploratory.
5. Decision criterion
The confirmatory test will be evaluated at a pre-specified significance level of α = 0.01 (one-sided). The decision rule is:
- Confirm H1a if the observed count of matches to the confirmatory target set at Δ < 0.001 exceeds the 99th percentile of the null distribution.
- Fail to confirm H1a otherwise.
- Report as inconclusive if the observed count falls between the 95th and 99th percentiles; in that case the test is repeated on an independent inventory constructed under the same rule.
The threshold Δ < 0.001 is chosen because it is the smallest threshold at which the exploratory survey found more than one match (6 matches at Δ < 0.001 across the 60-target exploratory set). The confirmatory target set is four times smaller, so the expected count under the null is correspondingly smaller; the decision criterion is calibrated to the confirmatory set, not the exploratory one.
6. What counts as a positive result
A positive result is: (i) a completed geometric derivation of 2/5 from the Hexarchon, dated before inventory construction; (ii) an inventory constructed under the rule in Section 3; (iii) an observed count of matches to the confirmatory target set that exceeds the null expectation at the decision threshold in Section 5.
A positive result would not, by itself, establish that the convergence hypothesis is correct. It would establish that the specific prediction made by H1 and H1a is confirmed at the pre-registered threshold. Further confirmation would require independent replication on a second inventory and, ideally, a prediction of a new scale not in the original inventory.
7. What counts as a negative result
A negative result is: (i) a failure to derive 2/5 from the Hexarchon without using the muon, or (ii) an observed count of matches that does not exceed the null expectation at the decision threshold.
A negative result would not, by itself, refute the broader three-system convergence hypothesis. It would refute the specific claim that the muon near-coincidence is geometric rather than coincidental. That is the claim this pre-registration is designed to test.
8. Reporting
Results will be reported in full, including: the inventory as constructed (with classifications and sources), the geometric derivation of 2/5 (with the date it was completed), the null model specification, the Monte Carlo output, the observed counts, and the decision reached. Negative and inconclusive results will be reported with the same prominence as positive results.
9. Registration
This document is registered on [23 October 2026] at 81018.com/pre-registration/. The version hash is v.1. Any subsequent revision will be logged with date and reason; the original version will remain accessible.