A classification of one dimensional almost periodic tilings arising from the projection method
Bibliographic record
Abstract
For each irrational number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with continued fraction expansion <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 semicolon a 1 comma a 2 comma a 3 comma ellipsis right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>;</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0; a_1, a_2,a_3, \dots ]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we classify, up to translation, the one dimensional almost periodic tilings which can be constructed by the projection method starting with a line of slope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The invariant is a sequence of integers in the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript alpha Baseline equals left-brace left-parenthesis x Subscript i Baseline right-parenthesis Subscript i equals 1 Superscript normal infinity Baseline bar x Subscript i Baseline element-of StartSet 0 comma 1 comma 2 comma ellipsis comma a Subscript i Baseline EndSet"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mi> α </mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msubsup> <mml:mo> ∣ </mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X_\alpha = \{(x_i)_{i=1}^\infty \mid x_i \in \{0,1,2, \dots ,a_i\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript i plus 1 Baseline equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">x_{i+1} = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whenever <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript i Baseline equals a Subscript i Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x_i = a_i\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> modulo the equivalence relation generated by tail equivalence and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis a 1 comma 0 comma a 3 comma 0 comma ellipsis right-parenthesis tilde left-parenthesis 0 comma a 2 comma 0 comma a 4 comma ellipsis right-parenthesis tilde left-parenthesis a 1 minus 1 comma a 2 minus 1 comma a 3 minus 1 comma ellipsis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∼
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".