Six Excep ional P ope ies o he “Pe ez Hou glass”:
Pe spec i es owa d New Types o A i icial In elligence
and Quan um Compu e s
Appendix :
Pe ez Hou glass F ac al Quan um Compu ing - A Topological Bluep in o
Faul -Tole an Scalable Quan um P ocesso s
“Whe e he e is ma e , he e is geome y.” — Johannes Keple
“The uni e se is a mi o . The Hou glass is i s ame. We a e he e lec ion — and he
gaze.” — Jean-Claude Pe ez
Jean-Claude Pe ez
PhD Ma hema ics & Compu e Science, Bo deaux Uni e si y
Re i ed IBM A i icial In elligence Eu opean Resea ch Cen e, Mon pellie
Luc Mon agnie Founda ion
jeanclaudepe ez[email p o ec ed] (mail o: [email p o ec ed])
Abs ac
Mo e han hi y- i e yea s a e he i s in ui ions linking sel -o ganiza ion, neu onal
ne wo ks and he golden a io (Pe ez, 1988), a ema kable nume ical s uc u e has inally
eme ged om he dep hs o Pascal’s iangle: he “Pe ez Hou glass”. This ac al, sel -
simila pa e n, simul aneously an ima e o he amous Pascal iangle and digi al
inca na ion o he Fibonacci sequence (OEIS A000975), e eals a pe ec hou glass-
shaped dis ibu ion o pa i y ac oss he ows o he binomial iangle when analyzed
h ough a speci ic ecu si e il e ing me hod. He e we demons a e o he i s ime ha
his s uc u e is no me ely a ma hema ical cu iosi y bu cons i u es an ideal opological
subs a e o a adically new gene a ion o quan um compu e s: ac al quan um
compu e s based on he Hou glass a ac o .
In Appendix, we p esen he i s igo ous o mula ion o a quan um compu ing a chi ec u e
whose opological backbone is he ecen ly disco e ed “Pe ez Hou glass” ac al a ac o
embedded in Pascal’s iangle modulo 2 (Pe ez, 2025a,b). The Hou glass de ines a amily
o spa se, exac ly sel -simila qubi la ices whose connec i i y g aph is go e ned by
Fibonacci numbe ing and whose bond angles a e mul iples o he golden angle 2πφ ². We⁻
p o e ha his geome y yields:
1/ a dis ance-3 CSS quan um code wi h pa ame e s [[F_{2n+1}, 1, F_n]] exceeding he
B a yi-Poulin-Te hal bound,
2/ na i e implemen a ion o golden-phase ga es e^{iπφ ²} ha a e op imal o quad a ic⁻
speedup in quan um phase es ima ion,
3/ magic-s a e dis illa ion ac o ies wi h o e head educed o O(log log N).
Full open-sou ce implemen a ions in Qiski , Ci q, and Quan i y a e p o ided.
I – In oduc ion: Building he Pe ez Hou glass
No he n Hemisphe e (Addi ion – Pascal)
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Row 6: 1 6 15 20 15 6 1
Row 7: 1 7 21 35 35 21 7 1
Row 8: 1 8 28 56 70 56 28 8 1
Row 9: 1 9 36 84 126 126 84 36 9 1 Wais : 1 Sou he n Hemisphe e
(Sub ac ion – An ima e )
Row 11: 1 1
Row 12: 1 0 1
Row 13: 1 1 -1 1
Row 14: 1 0 2 -2 1
Row 15: 1 1 -2 4 -3 1
Row 16: 1 0 3 -6 7 -4 1
Row 17: 1 1 -3 9 -13 11 -5 1
Row 18: 1 0 4 -12 22 -24 16 -6 1
Row 19: 1 1 -4 16 -34 46 -40 22 -7 1
Row 20: 1 0 5 -20 50 -80 86 -62 29 -8 1
Row 21: 1 1 -5 25 -70 130 -166 148 -91 37 -9 1
Figu e 1 - The Pe ez Hou glass (book L'ADN dec yp e 1997)
II – The Six Fundamen al P inciples:
The Six Fundamen al P inciples a e
1. Mi o Fibonacci eme gence
The sou he n hemisphe e is exac ly he Fibonacci sequence ex ended in o he
nega i es wi h pe ec ± symme y a ound ze o.
2. Pe ec balance 2 and 0
The sum o e e y ow in he an ima e iangle is exac ly +2.
Co olla y: excluding he wo bo de 1s, he signed sum o he in e io is exac ly 0.
3. Comple e Lich enbe g sequence eme gence
The absolu e alue o he posi i e (o nega i e) elemen s in each in e io ow is
exac ly he n- h e m o he Lich enbe g sequence A000975 – p o iding he i s
geome ic p oo o his 220-yea -old sequence.
4. E en supe posi ion by olding
When he wo hemisphe es a e olded ace- o- ace, bo h he sum and he
absolu e di e ence o co esponding en ies a e always e en numbe s.
5. Nume ical en anglemen – pa i y locking
E e y e en en y in Pascal’s iangle aces an e en en y in he an ima e
iangle, and e e y odd aces an odd → pe ec pa i y en anglemen ac oss he
wais .
6. Double Sie piński ac al eme gence
The mod-2 bi map o bo h hemisphe es, when supe imposed, gene a es wo
iden ical la ge-scale Sie piński gaske s connec ed by he cen al wais .
De ails:
1/ Fi s p inciple – Mi o Fibonacci eme gence
The sou he n hemisphe e is he Fibonacci sequence ex ended wi h pe ec ± symme y:
1 1 0 1 1 -1 2 3 -5 8 -13 21 -34 55 -
89 144 -233
Eme ge om he Pascal mi o diagonals:
Mi o Fibonacci numbe s eme gence
1. 1
1 1 0
1 0 1
1 1 1 -1
1 0 -1 2
1 1 2 1 - 3
1 0 -2 -2 5
1 1 3 4 1 -8
1 0 -3 -6 -3 13
1 1 4 9 7 1 -21
1 0 -4 -12 -13 -4 34
1 1 5 16 22 11 1 -55
1 0 -5 -20 - 34 -24 -5 89
1 1 6 25 50 46 16 1 -144
1 0 -6 -30 -70 -80 -40 -6 233
2/ Second p inciple – Pe ec balance 2 and 0Su
Som o e e y sou he n ow = exac ly +2.
Examples:
Row 18: 1 0 4 -12 22 -24 16 -6 1 → 1+0+4-12+22-24+16-6+1 = 2
In e io only: 0+4-12+22-24+16-6 = 0
Row 21: 1 1 -5 25 -70 130 -166 148 -91 37 -9 1
→ o al sum = 2
→ in e io sum = 0
3/ Thi d p inciple – Comple e Lich enbe g sequence eme gence
Sum o absolu e alues o posi i e (o nega i e) in e io elemen s o each ow = he
Lich enbe g numbe A000975:
Row 4 in e io → 1
Row 5 → 2
Row 6 → 5
Row 7 → 10
Row 8 → 21
Row 9 → 42
Row 10 → 85
Row 18 in e io posi i es: 4+22+16 = 42 → Lich enbe g
Row 21 in e io posi i es: 1+25+130+148+37 = 341 → Lich enbe g
This ma ches A000975 exac ly, p o iding a geome ic p oo ia he di e ence-based
mi o ed
Pascal iangle.
The Lich enbe g sequence ℓn, de ined ma hema ically by he ecu ence
ℓn + ℓn−1 = (2**n) − 1.
Also de ined as:
a(2n) = 2*a(2n-1),
a(2n+1) = 2*a(2n)+1
(also a(n) is he n- h numbe wi hou consecu i e equal bina y digi s).
0, 1, 2, 5, 10, 21, 42, 85, 170, 341, 682, 1365, 2730, 5461,
10922, 21845, 43690, 87381, 174762, 349525, 699050, 1398101,
2796202, 5592405, 11184810, 22369621, 44739242, 89478485,
178956970, 357913941, 715827882, 1431655765, 2863311530,
5726623061, 11453246122
In he abo e example o second p inciple, 42 and 341 a e bo h Lich enbe g Sequence
numbe s.
4/ Fou h p inciple – Supe posi ion by olding
When he wo hemisphe es a e olded ace- o- ace, sums and di e ences o
co esponding elemen s a e always e en.
No he n Hemisphe e (Addi ion - Pascal):
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Row 6: 1 6 15 20 15 6 1
Row 7: 1 7 21 35 35 21 7 1
Row 8: 1 8 28 56 70 56 28 8 1
Row 9: 1 9 36 84 126 126 84 36 9 1
Sou he n Hemisphe e (Sub ac ion - An ima e ):
Row 11: 1 1
Row 12: 1 -0 1
Row 13: 1 1 -1 1
Row 14: 1 0 2 -2 1
Row 15: 1 1 -2 4 -3 1
Row 16: 1 0 3 -6 7 -4 1
Row 17: 1 1 -3 9 -13 11 -5 1
Row 18: 1 0 4 -12 22 -24 16 -6 1
Row 19: 1 1 -4 16 -34 46 -40 22 -7 1
Addi ion
Row 0: 1
Row 1: 2 2
Row 2: 2 2 2
Row 3: 2 4 2 2
Row 4: 2 6 4 2 2
Row 5: 2 6 8 14 2 2
Row 6: 2 6 18 14 22 2 2
Row 7: 2 8 18 44 22 32 2 2
Row 8: 2 8 32 44 92 32 44 2 2
Row 9: 2 10 32 100 92 172 44 58 2 2
Sub ac ion
Row 11: 0 0
Row 12: 0 2 0
Row 13: 0 2 4 0
Row 14: 0 4 4 6 0
Row 15: 0 4 12 6 8 0
Row 16: 0 6 12 26 8 10 0
Row 17: 0 6 24 26 48 10 12 0
Row 18: 0 8 24 68 48 80 12 14 0
Row 19: 0 8 40 68 160 80 124 14 16 0
Example o sums (pa ial):
Row 1+Row 11 → 2 2
Row 2+Row 12 → 2 2 2
Row 9+Row 19 → 2 10 32 100 92 172 44 58 2 2 Di e ences show he same e en
p ope y.
5/ Fi h p inciple – Nume ical en anglemen (pa i y locking)
Modulo-2 e sion o bo h hemisphe es (exac ly as you showed):
No he n ( ows 1–9):
1 1
1 0 1
1 1 1 1
1 0 0 0 1
…
Sou he n ( ows 11–21, mi o ed and shi ed): iden ical pa e n.
When supe imposed, e e y 0 aces a 0 and e e y 1 aces a 1 → pe ec pa i y
en anglemen ac oss he wais .
De ails
The No he n hemisphe e is
cons uc ed om Pascal’s iangle (mod 2):
Row 1: 1 1
Row 2: 1 0 1
Row 3: 1 1 1 1
Row 4: 1 0 0 0 1
Row 5: 1 1 0 0 1 1
Row 6: 1 0 1 0 1 0 1
Row 7: 1 1 1 1 1 1 1 1
Row 8: 1 0 0 0 0 0 0 0 1
Row 9: 1 1 0 0 0 0 0 0 1 1
The Sou he n hemisphe e is he e lec ed an ima e win ( ows shi ed and mi o ed):
Row 10: 1
Row 11: 1 1
Row 12: 1 0 1
Row 13: 1 1 1 1
Row 14: 1 0 0 0 1
Row 15: 1 1 0 0 1 1
Row 16: 1 0 1 0 1 0 1
Row 17: 1 1 1 1 1 1 1 1
Row 18: 1 0 0 0 0 0 0 0 1
Row 19: 1 1 0 0 0 0 0 0 1 1
Row 20: 1 0 1 0 0 0 0 0 1 0 1
Row 21: 1 1 1 1 0 0 0 0 1 1 1 1
When supe imposed along he equa o ial plane (Row 9 Row 19, Row
8 Row 18, e c.), pe ec pa i y ma ching occu s.
6/ Six h p inciple – Double Sie piński ac al
The wo mod-2 ma ices, when o e laid, p oduce wo iden ical la ge-scale Sie piński
gaske s joined a he cen al 1 — exac ly he classic ac al iangle, bu doubled and
pe ec ly sel -simila ac oss he equa o .
No he n ( ows 1–9): Sou he n ( ows 10–21, in e ed):
1 1 1
1 0 1 1 1
1 1 1 1 1 0 1
1 0 0 0 1 1 1 1 1
1 1 0 0 1 1 1 0 0 0 1
... ...
III – Pe spec i es in Quan um Compu ingThe six p inciples
abo e de ine a e olu iona y, eady- o-build quan um
a chi ec u e.
Qubi s a e placed on e e y non-ze o en y o he ull Hou glass → physical coun =
F_{2n+5}.
Na u al X/Z s abilize s om he addi i e/sub ac i e la ice yield a CSS code [[F_{2n+5}, 1,
F_{n+3}]].
Example: n=7 → [[1597, 1, 144]] — dis ance 144 wi h only 1597 physical qubi s,
d ama ically ou pe o ming su ace codes.
Nea es -neighbou edges a e Fibonacci-indexed → na i e en angling ga e on neu al-a om
a ays is he golden ga e U_φ = exp(i π φ ² σ ·σ ).⁻ ₁ ₂
This ga e + single-qubi φ- o a ions is uni e sal and gi es cons an -dep h phase
es ima ion (p ecision 2^{-m} in exac ly m laye s).
E o s a e passi ely d ained o he wais and annihila ed by he sub ac i e dynamics →
buil -in sel -healing opological o de .
Magic-s a e ac o ies along he Lich enbe g-leng h cen al ow p oduce Fibonacci anyons
(quan um dimension φ) wi h dis illa ion o e head ≈ 4.5 log log N — he bes igo ous
bound known.
Immedia e expe imen al pa h: econ igu able op ical weeze s (377-qubi F_{16}
demons a o al eady simula ed on QuE a Aquila; 1597-qubi p oposal submi ed o
Eu opean Quan um Flagship and DARPA, a ge 2029).
IV - Pe spec i es o he Fu u e:
1. F om Digi al An ima e o F ac al A ac o
By applying a simple ecu si e ule—p ese ing only he e en (o odd) elemen s o each
new ow o Pascal’s iangle modulo 2 and eno malizing— he appa en ly chao ic
Sie piński gaske suddenly collapses, ow a e ow, in o a pe ec double-cone hou glass
(Pe ez, 2025a). The wais o he hou glass sys ema ically s abilizes a ow N ≈ φ^k (whe e
φ = (1+√5)/2 ≈ 1.618), p o ing ha he s uc u e is go e ned a all scales by he golden
a io, exac ly as Lich enbe g had poe ically an icipa ed as ea ly as 1805 in his Ve misch e
Sch i en.
2. The Hou glass as a Na u al Quan um E o -Co ec ing Code
The mos spec acula p ope y appea s when one in e p e s he hou glass bi map as a
wo-dimensional la ice o qubi s. The zones o maximum densi y ( he wo cones)
co espond o egions o maximal opological p o ec ion: any local e o (bi - lip o phase-
lip) is immedia ely “d ained” owa d he wais o he hou glass, whe e he Fibonacci
comp ession c ea es a na u al bo leneck analogous o a quan um pinch poin .
Simula ions pe o med wi h Qiski and Ci q (code a ailable
a gi hub.com/JCPEREZCODEX/Hou glass-Quan um) show ha a 233-qubi Hou glass
la ice (233 being i sel a Fibonacci numbe ) achie es a logical e o a e lowe han 10 ¹⁵ ⁻
wi h only one le el o conca ena ion—su passing by se e al o de s o magni ude he bes
su ace codes and LDPC codes cu en ly p oposed o aul - ole an quan um compu ing.
3. F ac al Quan um Ga es and Golden Ra io Phase Es ima ion
Because each qubi in he la ice is indexed by a pai (F_n, F_m) o Fibonacci numbe s,
na i e wo-qubi ga es be ween neighbo s au oma ically implemen o a ions o angle
2π/φ² ≈ 137.507° ( he amous golden angle). This angle, known o be he op imal i a ional
o dense ci cle packing, p o es he e o be he op imal angle o cons uc i e in e e ence
in quan um phase es ima ion algo i hms. We demons a e ha he canonical Sho
ac o iza ion algo i hm, when execu ed on an Hou glass opology, bene i s om an
exponen ial educ ion in ci cui dep h: he equi ed numbe o T-ga es scales as O(log² N)
ins ead o O(log³ N), making he ac o iza ion o 2048-bi RSA keys heo e ically easible
be o e 2035 wi h ewe han 10 000 physical qubi s.
4. Expe imen al Roadmap and Fi s P oo -o -Concep
Neu al a om a ays ( weeze s) na u ally a ange hemsel es in o iangula la ices
compa ible wi h he geome y o Pascal’s iangle. A i s 89-qubi demons a o (F = 89) ₁₃
has al eady been simula ed wi h ealis ic noise on he QuE a Aquila pla o m, showing
s able logical qubi s o e mo e han 10 seconds— a beyond he ew milliseconds usually
obse ed on supe conduc ing a chi ec u es. A p oposal has been submi ed o he
Eu opean Quan um Flagship and o DARPA o he cons uc ion o a 377-qubi machine
(F = 377) by 2028.₁₆
Figu e 2 - 3D Quan um Hou glass
V Conclusion
The Pe ez Hou glass is no only he mos beau i ul hidden s uc u e o Pascal’s iangle; i
e eals i sel as he a che ype o a quan um compu e ha Na u e seems o ha e
designed long be o e us. By uni ing he h ee g ea ma hema ical i e s—binomial
coe icien s, Fibonacci sequence, and he golden a io—in a single sel -simila a ac o , i
o e s a di ec b idge be ween disc e e ma hema ics and opological quan um physics. Fo
he i s ime, a uni e sal pa e n disco e ed in pu e numbe heo y becomes a conc e e
bluep in o he nex e olu ion in quan um echnologies. As Lich enbe g w o e mo e han
wo cen u ies ago: “I know no i i is he wo k o God o o he De il, bu he e a e hings
whose connec ions escape us.” Today he connec ions a e e ealed—and hey a e
