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RSS Feed for Wolfram Community showing any discussions tagged with Discrete Mathematics sorted by active.A short arithmetic program producing four CODATA-level constants
https://community.wolfram.com/groups/-/m/t/3716598
I would like to share a compact deterministic arithmetic program that produces four numerical values lying within the CODATA 2022 one-standard-uncertainty windows for:
- the inverse fine-structure constant
- the proton/electron mass ratio
- the muon/electron mass ratio
- the tau/electron mass ratio
The code was motivated by a broader physical framework, but the point of this post is not to argue for that framework.
The narrower question is this:
Can this compact computational object be explained as ordinary numerical fitting, and if so, where is the fitted information stored?
The program contains:
- no empirical target constants as inputs
- no decimal fit coefficients
- no continuous optimization loop
- no output-specific adjustable parameters
In addition, the three mass-ratio outputs share the same denominator ladder.
Wolfram Language code:
```wolfram
u = 1;
r = 3;
A = r*(r - u);
B = 2;
n = A + B - u;
rel = A + B + r;
ph = n + B;
h = (rel + u) + n;
nVal = r + rel + ph + h;
cVal = 2*nVal;
atten = 1 - A/n^2;
Phi = nVal*r + rel + r/cVal + (rel - ph)/cVal^2 + (18/(n*Pi))*atten/cVal^3 + (4*Pi)*atten/cVal^4;
d1 = r + ph + u;
d2 = h*(nVal - (r + u));
d3 = d1*d2;
d4 = d2*(nVal - (r + u))*A*(rel - r);
Psi1 = cVal*(h + r) - (ph + r) + (rel - ph)/d1 - u/d2 + (rel - ph)/d3 - u/d4;
Psi2 = h*rel - r + (rel - u)/d1 - u/d2 + (r + u)/d3 + (h - r - u)/d4;
Psi3 = nVal*(cVal - u) - ph + (rel - ph)/d1 - (rel - ph)/d2 + (r + u)/d3 - (r*h)/d4;
NumberForm[N[{Phi, Psi1, Psi2, Psi3}, 15], 15]
Output:
{137.035999165800, 1836.15267343627, 206.768283284455, 3477.15145895483}
For comparison, the CODATA 2022 reference values are approximately:
alpha^-1 = 137.035999177(21)
proton/electron = 1836.152673426(32)
muon/electron = 206.7682827(46)
tau/electron = 3477.23(23)
All four outputs lie within the corresponding one-standard-uncertainty windows.
Under a conservative unit-range normalization, the four simultaneous window hits roughly correspond to ~66 bits of target-window information.
So my question is simple:
**If this is ordinary numerical fitting, what is the information channel?**
For completeness, I include a short note containing the description-length analysis:
[https://doi.org/10.5281/zenodo.20158423][1]
[1]: https://doi.org/10.5281/zenodo.20158423Tetsuya Momose2026-05-15T11:09:21ZHat tile dynamics from dual-grid projection onto a fractal-subdivided torus
https://community.wolfram.com/groups/-/m/t/3715016
![Hat tile dynamics from dual-grid projection onto a fractal-subdivided torus][1]
&[Wolfram Notebook][2]
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=titleImage.png&userId=2932486
[2]: https://www.wolframcloud.com/obj/644687a3-8e85-4505-b3bc-cc74fbc04ee9Johannes Martin2026-05-12T09:53:01ZProgress in the no-three-in-line-problem
https://community.wolfram.com/groups/-/m/t/3714366
![Progress in the no-three-in-line-problem][1]
&[Wolfram Notebook][2]
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=693022037_35804801055800585_4027629096720536130_n.jpg&userId=20103
[2]: https://www.wolframcloud.com/obj/7a439e86-9be9-457a-94b2-d58c415fb284Ed Pegg2026-05-11T15:13:27ZTwo open problems about Leibnizian strings
https://community.wolfram.com/groups/-/m/t/3712556
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/f5ffb538-d69e-402b-8c12-c0def144513bFurkan Semih Dündar2026-05-06T07:48:16ZThe abc eraser: algebraic filtration and the emergence of universality
https://community.wolfram.com/groups/-/m/t/3695987
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/12016646-05c7-4e0f-aafc-a180d14e3a8eRamón Eduardo Chan López2026-04-19T00:41:46ZInformation symmetry in complementary lattice dynamics
https://community.wolfram.com/groups/-/m/t/3696660
A chromatic cellular automaton on {0,1}³ generates a cruciform pattern from a single seed (established in Post 3). At every timestep, the null-space complement of the pattern — obtained by mapping each state s → 7−s — is computed and compared to the positive space using five independent complexity measures: Shannon entropy, joint entropy of adjacent pairs, spatial mutual information, 2×2 block entropy, and Kolmogorov complexity approximation via compression.
Result: the first four measures show exact equality C(P,t) = C(N,t) at every step. The fifth (Kolmogorov complexity approximated via compression) shows near-equality with fractional difference under 3%, consistent with encoder-dependent variation in a computable upper bound rather than genuine structural asymmetry. The structural complexity of the complement matches the complexity of the pattern — not approximately, but identically — across all information-theoretic measures tested.
The complement operator also reveals a vertex absent from the automaton's lifecycle. The lifecycle visits Red, Green, Blue, Yellow, and Magenta. Cyan (0,1,1) never appears. In the complement, Red maps to Cyan. Since Red is the most frequently renewed state (Dead → Idea), Cyan becomes the dominant active state in the null space. The vertex absent from the dynamics is the most prominent vertex of the shadow.
Four questions for the community:
1. Continuous extension. The information symmetry C(P,t) = C(N,t) holds by construction for any bijective complement on a finite vertex set. Does it extend to continuous state spaces on [0,1]³?
2. Cyan dominance. The complement reveals Cyan (0,1,1) as the dominant active state of the null space. If Cyan were introduced into the lifecycle, would the complement become impoverished at a corresponding vertex?
3. Simultaneous threshold crossing. The null-space complement achieves identical structural complexity without its own dynamics. In any system where the positive space crosses a self-referential complexity threshold, does the complement cross simultaneously?
4. Universality challenge. Five complexity measures all show C(P,t) = C(N,t). Is there ANY computable complexity measure for which the equality fails?
Notebook Attached.
https://www.wolframcloud.com/obj/f17551bc-aec2-415b-9323-ad8865b5d46dDustin Sprenger2026-04-20T02:45:27ZRule 110: conditional dynamics, the OR-XOR switch, and the structure of irreducibility
https://community.wolfram.com/groups/-/m/t/3673620
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/ad350d33-f532-4a74-ba44-050ece07c9fbRamón Eduardo Chan López2026-04-03T07:56:17ZRule 30 exact binomial-Lucas lifting: boolean logic to integer coefficients, Stirling & support sets
https://community.wolfram.com/groups/-/m/t/3647733
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/b04b6551-fecf-465d-b02d-63d95abd751cTigran Nersissian2026-03-02T11:53:13ZHidden symmetry in permutation space
https://community.wolfram.com/groups/-/m/t/3640755
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/62b1d34a-2b56-4242-9809-6049ce46064fDavid Vasholz2026-02-16T20:10:48ZFrom integers to art: fractal geometry of digit-sum parities
https://community.wolfram.com/groups/-/m/t/3639196
![From integers to art: fractal geometry of digit-sum parities][1]
&[Wolfram Notebook][2]
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=FromIntegerstoArt-fractalGeometryofDigit-SumParities.png&userId=20103
[2]: https://www.wolframcloud.com/obj/618232ce-99e9-4c82-8bb7-a7824760b505Sergio Quiroga2026-02-12T20:31:54ZNumerically 2026 is unremarkable yet happy: semiprime with primitive roots
https://community.wolfram.com/groups/-/m/t/3594686
[![Numerically 2026 is unremarkable yet happy: semiprime with primitive roots][1]][2]
&[Wolfram Notebook][3]
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Numerically2026isunremarkableyethappysemiprimewithprimitiveroots.png&userId=20103
[2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Numerically2026isunremarkableyethappysemiprimewithprimitiveroots.png&userId=20103
[3]: https://www.wolframcloud.com/obj/69781695-a7d5-4734-bf59-80ba72537964Anton Antonov2025-12-21T20:34:09ZExtended Rule-110 family via ANF over GF(2): prior art and validation of a factor onto Rule 110
https://community.wolfram.com/groups/-/m/t/3634252
Hi everyone,
My name is Tigran. I am working with the Wolfram Language on cellular automata and would like to ask whether a particular larger-neighborhood extension of Rule 110, defined in algebraic normal form (ANF) over GF(2), is already known, and how best to validate it rigorously.
⸻
Rule family (ANF over GF(2))
For each neighborhood size m >= 3, I define a Boolean local rule
f_m(x1, …, xm) =
x_{m-1} + x_m + x_{m-1}x_m + (x1x2*…*xm) (mod 2)
where:
• addition is XOR (mod 2),
• multiplication is AND.
For m = 3, this becomes
f_3(x1, x2, x3) =
x2 + x3 + x2x3 + x1x2*x3 (mod 2)
which corresponds exactly to Wolfram Rule 110.
Converting the ANF to a lookup table and then to a Wolfram rule index gives, for example:
m = 3 -> 110
m = 4 -> 28398
m = 6 -> 7993589098607472366
⸻
Evolution setting
I evolve a one-sided (causal) wedge with boundary zeros and a single-seed initial condition:
a(0,0) = 1
a(0,t > 0) = 0
Out-of-range indices are treated as zero.
The update rule is:
a(n+1, t) =
f_m( a(n, t-m+1), …, a(n, t) )
This is a triangular / wedge evolution rather than a standard bi-infinite CA.
⸻
Observation
Empirically, for certain neighborhood sizes — most notably
m = 2^j + 2
the resulting spacetime diagrams show persistent activity and localized structures strongly reminiscent of Rule 110, including glider-like behavior. This does not appear to be a trivial left-right symmetry or a simple shift.
⸻
Questions
1. Prior art
Is this specific ANF extension
x_{m-1} + x_m + x_{m-1}x_m + (x1x2*…*xm)
already known or studied (in Wolfram’s work, the CA literature, or related discussions of Rule-110 generalizations)?
2. Validation / factorization
What are the standard techniques to show that a larger-neighborhood Boolean CA factors onto Rule 110, for example via:
• a sliding block code,
• coarse-graining or renormalization,
• substitution or rescaling,
• or a proof of a semiconjugacy to Rule 110?
Does the use of a wedge evolution change what would count as a valid factor or simulation?
3. Practical workflow
If the right goal is to exhibit an explicit map pi such that
pi( evolution under f_m )
= evolution under Rule 110 applied to pi(state),
what would be a reasonable computational strategy (for example, searching small block maps) to test this in Wolfram Language?
If useful, I can share a minimal notebook or GitHub repository
Thank you very much for your time and for any pointers or references.
Tigran
ClearAll[NumberCellularAutomatonFastC];
NumberCellularAutomatonFastC[nMin_Integer, nMax_Integer, kMin_Integer,
kMax_Integer, ruleIndex_Integer, M_Integer /; M > 0] :=
Module[{bits, tMin = nMin, tMax = nMax, nMaxIter = kMax, outAll,
slice, cf},
bits = Reverse@IntegerDigits[ruleIndex, 2, 2^M];
cf = Compile[{{b, _Integer, 1}, {tmax, _Integer}, {nmax, _Integer}, {m, _Integer}},
Module[{prev, curr, out, n, t, j, idx, pattern, val},
prev = ConstantArray[0, tmax + 1];
prev[[1]] = 1;
out = ConstantArray[0, {nmax + 1, tmax + 1}];
out[[1]] = prev;
For[n = 1, n <= nmax, n++,
curr = ConstantArray[0, tmax + 1];
For[t = 0, t <= Min[tmax, (m - 1) n], t++,
pattern = 0;
(* pattern built from prev[t-(m-1)]..prev[t] *)
For[j = 1, j <= m, j++,
idx = t - m + j;
val = If[idx >= 0, prev[[idx + 1]], 0];
pattern = 2 pattern + val;
];
curr[[t + 1]] = b[[pattern + 1]];
];
prev = curr;
out[[n + 1]] = prev;
];
out
],
CompilationTarget -> "C", RuntimeOptions -> "Speed"
];
outAll = cf[bits, tMax, nMaxIter, M];
slice = outAll[[kMin + 1 ;; kMax + 1, tMin + 1 ;; tMax + 1]];
Transpose[slice]
]
XX[n_] :=
Symbol["x" <> ToString[n]] + Symbol["x" <> ToString[n - 1]] +
Symbol["x" <> ToString[n - 1]]*Symbol["x" <> ToString[n]] +
Product[Symbol["x" <> ToString[j]], {j, 1, n}];
Table[
{anfToRule[XX[(2^j + 2)], (2^j + 2)],
XX[(2^j + 2)],
ArrayPlot[
NumberCellularAutomatonFastC[
0, 10*(2^j + 2)^2 + 100,
0, 10*(2^j + 2)^2 + 100,
anfToRule[XX[(2^j + 2)], (2^j + 2)], (2^j + 2)],
ColorRules -> {0 -> Black, 1 -> LightBlue}]
},
{j, 0, 3}
]Tigran Nersissian2026-02-04T13:27:33ZMGroups package: teaching finite group theory
https://community.wolfram.com/groups/-/m/t/3623800
&[Wolfram Notebook][1]
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![enter image description here][1]
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https://community.wolfram.com/groups/-/m/t/3608683
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https://community.wolfram.com/groups/-/m/t/3604657
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https://community.wolfram.com/groups/-/m/t/3598035
![Visual explanation of the problem Putnam 2025 A3][1]
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https://community.wolfram.com/groups/-/m/t/3597439
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