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Two t27 courses of 27 lessons each: FPGA, then AI numbers

2026-10-07 · 9 min read

[no lesson trains a network; the tests of the 15 new lessons run in recordings of native t27c, not in the browser; 35 of 127 mutants survive and are not sorted; software only, no board] The t27 course is now two courses of 27 lessons, 9 modules of 3 lessons each (recordings for lessons 13 to 21 are pending t27#7400): From zero to a chip, on programming an FPGA, and AI numbers with t27, which starts with the 6 lessons that moved and adds 21 more, ending at a small ternary network. Each of the last 15 lessons plants a one-line bug in its spec and shows the one named test that fails. Over 14 of their functions, tri mutate spec made 127 mutants and the tests killed 92.

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#t27#Ternary#MutationTesting

The t27 course had grown to 33 lessons in 11 modules: 27 on programming an FPGA, then 6 on number formats for AI. That is two topics in one course, and a link to it could not share either topic on its own. It is now two courses of 27 lessons each, and each course is 9 modules of 3 lessons. Course 1, From zero to a chip, keeps the FPGA lessons and its address. Course 2, AI numbers with t27, starts with the 6 lessons that moved and adds 21 more, ending at a small ternary network. Every lesson opens its own widget and one t27 spec, and each of the last 15 lessons plants one bug in its spec and shows the one test that catches it.

Why 27, and why two courses

27 is 3 x 3 x 3, the shape of a TRI-27 word, and every course is 9 modules of 3 lessons, 27 lessons in all. When a topic needs more than 27 lessons it becomes a second course, chained to the one before it: the last lesson of course 1 ends with a link to lesson 1 of course 2, and lesson 1 of course 2 links back. Each course has its own address in the app, its own page under t27.ai/learn/, its own preview card and its own line in the sitemap, so either one can be shared alone.

No link that worked before the split stops working. A lesson page stays at t27.ai/learn/<lesson>/ whichever course the lesson is in. In the app, the old address of a moved lesson, #/course/<lesson>, now opens it at #/ai-numbers/<lesson>. Progress is one list in your browser for both courses, so a lesson you marked done before the split is still done after it.

Course 2, part by part

Lessons	Part	What it teaches
1-3	Lab: our own research	A number format of our own, an honest scoreboard, and a model's tables multiplied on the board.
4-6	AI numbers: the MX block	How AI chips keep weights in a few bits: one shared scale per block, the scale byte itself, and what one outlier does to its neighbours.
7-9	Ternary weights	Weights that are only minus, zero or plus a scale, the five rules a ternary alphabet must pass, and a test pass that checked nothing.
10-12	The Ternary Network Float	A rule the compiler enforces before any test runs, and a 17-bit float whose exponent is four balanced trits.
13-15	Arithmetic on signed numbers	Multiply two signed numbers, add them when their signs differ, and do both at once in a multiply-accumulate.
16-18	Parts of a neuron	A ReLU that bends at zero, a power of two for softmax, and an argmax that names the answer.
19-21	Learning from a mistake	A loss that prices a wrong guess in bits, one step that moves a weight against its gradient, and the hidden layer that XOR needs.
22-24	BitNet: ternary networks	A threshold that squeezes a sum back to three values, one neuron that becomes a different function when its weights change, and a neuron that reads its inputs 27 trits at a time.
25-27	The ternary MAC as a chip	The 27-trit dot product as wires with no register, the same sum added into a register on every clock, and a small whole network to close the course.

Course 1 keeps its 27 lessons in its 9 modules of 3: the chip, numbers in hardware, your t27 program, inside t27c, from spec to hardware, reading synthesis, place, route and timing, the bitstream, and on the board.

Every new lesson plants one bug

Lessons 13 to 27 each teach one spec from the ternary directory of t27. The widget of each is a recording of a terminal on our lab machine, where the native t27c compiler and Zig run the tests the browser cannot run yet. The recording shows the lines that matter and runs the spec's tests. Then tri mutate plant changes one line, runs the tests again and passes only if exactly the named test fails, and git diff shows the file back as it was. In all 15, exactly one named test fails.

Recordings pending for lessons 13 to 21. Their nine specs (gft_smul, gft_sadd, gft_signed_mac, gft_relu, gft_exp2, gft_argmax4, gft_nll, gft_sgd_step and gft_xornet) are recorded with tri test and tri mutate plant from t27#7400, which is not merged yet. Until it is, each of those lessons opens a placeholder page that says the recording is pending and shows no run; the lesson text says so too. Lessons 22 to 27 open their recordings now.

Lesson	Spec	Tests	The planted bug	The one test that fails
13 A sign is one bit	gft_smul.t27	3	the sign of the product is always 0	m2
14 Opposite signs subtract	gft_sadd.t27	3	adds the sizes when the signs differ	a2
15 Multiply, then add	gft_signed_mac.t27	4	the sign is set with OR, not XOR	pp
16 A bend at zero	gft_relu.t27	4	negative inputs pass through	negz
17 Two to the x	gft_exp2.t27	4	drops the minus sign of k for negative x	em1
18 Pick the largest	gft_argmax4.t27	4	> becomes >= for two positive scores	tie_low
19 A loss in bits	gft_nll.t27	3	neg turns 0 into a zero with the sign bit set	perfect
20 One step downhill	gft_sgd_step.t27	3	multiplies g by itself, not by eta	ascend
21 XOR needs a bend	gft_xornet.t27	4	no relu on one hidden unit	x00
22 Back to three values	activation_quantizer.t27	7	> becomes >= at the threshold	quantize_boundary_hi
23 Same neuron, new weights	bitnet_majority.t27	12	one weight goes from P to Z	maj_p_n_n
24 A neuron in chunks	bitnet_neuron_nchunk.t27	11	the loop reads one chunk, not all of them	neuron_zero_chunks
25 A dot product made of wires	comb_ternary_dot.t27	4	N times P gives +1	dot_all_n_x_all_p
26 Add it up on every clock	stream_ternary_mac.t27	4	a P in a reads as N	dot_all_p_x_all_p
27 A whole network	bitnet_mlp.t27	4	the third hidden trit lands on the wrong bit	pack3_zzz

One planted bug shows that a test can fail. It says little about the tests that were never written. So each recording ends with tri mutate spec, which makes every one-line change it knows in the lesson's function and counts how many of them the tests notice.

What the mutants found

Over 14 of the 15 functions, tri mutate spec made 127 mutants. 92 were killed, every one of them by a failing test, and 35 survived; none hung and none failed to build. Eight functions killed every mutant. Six did not:

Spec	Function	Mutants	Killed	Survived
gft_exp2.t27	on_comb	34	17	17
gft_sadd.t27	sadd	18	10	8
gft_smul.t27	smul	16	12	4
gft_argmax4.t27	gt	9	6	3
comb_ternary_dot.t27	tmul	7	5	2
gft_relu.t27	on_comb	5	4	1

A survivor is a one-line change that no test notices. Either a test is missing, or the change does not change what the function returns. Both kinds occur in specs like these. In gft_relu.t27, line 11, if (x == 0) { return 0; }, can be deleted without changing any result, because line 13 returns x, and x is already 0. Which of the 35 survivors is which kind is not sorted yet; gft_exp2.t27, with 4 tests and 17 survivors, is where to look first. For gft_xornet.t27 the count printed no summary at all, so it is left out of the totals.

Making the recordings found two problems in the tools. tri test used to print a green line without running a single test, so a recording made with it proved nothing. Pull request t27#7400 makes it run the tests and fail when one fails or none ran, and adds tri mutate plant, which plants a bug, demands that exactly the named tests fail, and checks the file's hash before and after. t27c test-report still exits 0 when a test fails, so every verdict in these recordings is read from its text (t27#7370).

Where else to learn this

Before writing this, we read the pages of 16 other resources that teach or build ternary networks, quantized networks on FPGAs, or a network from scratch, on 7 October 2026. The table keeps 11 of them.

Resource	Kind	Ternary weights	Runs where	Tests behind it
The Era of 1-bit LLMs, S. Ma et al. (2024)	paper	yes, the definition of BitNet b1.58	GPUs	perplexity and accuracy against LLaMA
bitnet.cpp, Microsoft	inference library	yes	CPUs, and GPUs since 2025	no test suite named
Fine-tuning LLMs to 1.58bit, Hugging Face (2024)	article	yes, the quantizer step by step	GPUs	benchmarks and loss curves
A Visual Guide to Quantization, M. Grootendorst (2024)	illustrated article	yes, with worked numbers	static figures	none, there is no code
Ternary Weight Encoder, binarycon.com	interactive page	packs weights that are already ternary	your browser	none stated
FINN, AMD Research	compiler and notebooks	binarized and quantized; ternary is not named on the pages read	FPGA boards; Python, C++ and RTL simulation	yes: each stage against a golden reference
TernaryCore (2026)	Verilog library	yes: a MAC, a dot product, a matrix multiply	simulation; Arty A7-100T and Tang Nano 9k boards	testbench pass counts and RTL against Python; tests not named
TeLLMe, Y. Qiao et al. (arXiv:2504.16266)	paper	yes, 1.58-bit weights	a Kria KV260 FPGA	no correctness check described
micrograd and Neural Networks: Zero to Hero, A. Karpathy	library and video course	no	Python	gradients checked against PyTorch
MiniTorch, S. Rush	teaching library	no	Python, and a CUDA module	unit and property tests the learner must pass
nand2tetris, Project 2	course and book	no	a desktop hardware simulator	a test script and a compare file per chip
This course, lessons 13 to 27	course	yes: a 27-trit dot product, a MAC and a small network	specs in your browser; tests in recordings of native t27c	yes: 74 tests, one planted bug per lesson, 127 mutants

Each of these does something this course does not. The BitNet paper, bitnet.cpp and the Hugging Face article work at the scale of real models, from 700M to 70B parameters, with measured speedups; no lesson here touches a trained model. FINN compiles trained networks into FPGA accelerators and checks every stage in simulation, and TeLLMe runs a whole ternary language model on an edge FPGA. TernaryCore is the closest: open Verilog that goes from a ternary MAC to a dot product to a matrix multiply, with tests, and runs on real boards. micrograd, MiniTorch and nand2tetris have the learner build each piece against tests.

What we did not find on any of those pages is the shape of these lessons: one planted one-line bug per lesson that fails exactly one named test, with the file put back afterwards. nand2tetris and MiniTorch come near, since a learner's own mistake fails their tests, but they do not plant one and show it. That is a narrow claim about the pages we read, not about every course there is. Nor do the tests of these 15 lessons run in your browser yet; they run in the recordings.

What this does not show

No lesson trains a network. The tests pass weights set by hand, and lesson 20 takes one step.
Lesson 27 is not yet a chip: in the generated Verilog the chunk loop of neuronN is marked NOT UNROLLED, and the lesson says yosys rejects it.
The browser player skips the tests of the 15 new specs, because its runner does not know assert_eq yet. They run natively, in the recordings.
The 35 surviving mutants are not sorted into missing tests and harmless changes.
A function with 1 mutant, like those of gft_sgd_step.t27 and bitnet_majority.t27, is barely measured by that count.
The 15 new lessons run in software only; none of them ran on a board.

Try it

Course 1, From zero to a chip: t27.ai/learn/course/
Course 2, AI numbers with t27: t27.ai/learn/ai-numbers/
Course 2, lesson 16, a bend at zero: t27.ai/learn/a-bend-at-zero/
Course 2, lesson 27, a whole network: t27.ai/learn/a-whole-network/
In the app: t27.ai/#/course and t27.ai/#/ai-numbers

What this does not settle

No lesson trains a network. The tests pass weights set by hand, and lesson 20 takes one step.
Lesson 27 is not yet a chip: in the generated Verilog the chunk loop of neuronN is marked NOT UNROLLED, and the lesson says yosys rejects it.
The browser player skips the tests of the 15 new specs, because its runner does not know assert_eq yet. They run natively, in the recordings.
35 of 127 mutants survive and are not sorted into missing tests and harmless changes; gft_exp2.t27 has 17 of them with 4 tests. For gft_xornet.t27 the count printed no summary and is left out.
A function with 1 mutant, like those of gft_sgd_step.t27 and bitnet_majority.t27, is barely measured by that count.
The 15 new lessons run in software only; none of them ran on a board.
The recordings use tri test and tri mutate plant from t27#7400, which was not merged when this was written, and t27c test-report exits 0 when a test fails (t27#7370), so every verdict is read from its text.
The comparison keeps 11 of the 16 resources whose pages were read on 7 October 2026; it is not a complete survey. A course that already plants one bug per lesson and shows the one test that fails would make the post's narrow claim wrong.

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