DYNAMIC FRACTAL CATEGORY THEORY: Monoidal Actions, Ind--Pro Bicompletion, and Pathwise Stable Equivariant Kan Extensions

colorlinks=true,
linkcolor=blue!60!black,
citecolor=blue!60!black,
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pdftitle= Dynamic Fractal Category Theory: Monoidal Actions, Ind--Pro Bicompletion, and Pathwise Stable Equivariant Kan Extensions,
pdfauthor= K. Takahashi ,
pdfsubject= Category Theory, Ind/Pro, Monoidal Actions, Frobenius (Co)Monads, Rewriting, Kan Extensions, Enriched Stability ,
pdfkeywords= Frobenius monad, comonad, monoidal action, Ind-completion, Pro-completion, Kan extension, Day convolution, rewriting, Quillen Theorem A, Lawvere metric, truncation bounds

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1.3

plain
theorem Theorem [section]
proposition[theorem] Proposition
corollary[theorem] Corollary
lemma[theorem] Lemma
definition
definition[theorem] Definition
assumption[theorem] Assumption
example[theorem] Example
remark[theorem] Remark

colim
Ob

C
D
End
Ind
Pro
Frac_ ( _0)
Id
Lan
op
% horizontal/whiskered composition of 2-cells

TITLE: Dynamic Fractal Category Theory\
Monoidal Actions, Ind--Pro Bicompletion, and Pathwise Stable Equivariant Kan Extensions

AUTHOR: K.~Takahashi

\
https://orcid.org/0009-0004-4273-3365 ORCID: 0009-0004-4273-3365

DATE:

Dynamic Fractal Category Theory (DFCT) extends the static framework of Fractal Category Theory (FCT)~TakahashiFCT by promoting scale from a single endofunctor to a strong monoidal action [[EQ:eq0017]] , where each [[EQ:eq0018]] acts by a Frobenius (co)monad [[EQ:eq0019]] ; (co)monad structures on composites are obtained by transport along the monoidal comparisons [[EQ:eq0020]] .

Dynamic indexing is organized through a skeletal [[EQ:eq0021]] -presentation. Rewriting with a left-associated normal form terminates by a lexicographic measure refined with Tamari height and is locally confluent by a finite family of shapes of critical pairs including all interactions with~ [[EQ:eq0022]] ; Newman's lemma and a terminal-object argument then yield confluence and finality. A bracketing-independence lemma ensures that transported (co)monad structures are well defined, so that [[EQ:eq0023]] -normalization is canonical.

Mixed (bi)limits are computed in the FCT order: a filtered colimit over a forward (`` [[EQ:eq0024]] '') presentation, followed by a levelwise cofiltered limit along [[EQ:eq0025]] in~ [[EQ:eq0026]] . External attenuation weights [[EQ:eq0027]] on the forward stage yield FCT-style bounds: if [[EQ:eq0028]] with [[EQ:eq0029]] , then [[EQ:eq0030]] and the depth- [[EQ:eq0031]] truncation error is [[EQ:eq0032]] . Under [[EQ:eq0033]] and [[EQ:eq0034]] , Birkhoff's ergodic theorem gives almost-sure exponential decay.

We state precise mixed Fubini and Day convolution lift hypotheses, and verify that restricted Yoneda lifts preserve strong monoidality and Frobenius data while keeping constant pro-objects constant. The presentation is structured to be OCR- and crawler-friendly: all axioms and comparison maps are stated explicitly, and every key existence is accompanied by a short, self-contained proof sketch.

SECTION: Introduction

sec:intro
Static FCT~TakahashiFCT internalizes scale by a single Frobenius (co)monad [[EQ:eq0035]] on [[EQ:eq0036]] and constructs a bicompletion from a small seed [[EQ:eq0037]] using mixed towers indexed by [[EQ:eq0038]] moves. The pillars are: (i) finality of an [[EQ:eq0039]] -presentation (hence skeletal computation), (ii) identification [[EQ:eq0040]] , (iii) [[EQ:eq0041]] -equivariant left Kan extensions with difference-based Lipschitz bounds and geometric truncation error, and (iv) Day convolution lifts via [[EQ:eq0042]] under coend--Kan Fubini.

DFCT generalizes this to a strong monoidal action [[EQ:eq0043]] , capturing heterogeneous, possibly noncommutative scale while preserving the static skeleton. We establish: transport of (co)monad structures along~ [[EQ:eq0044]] with bracketing-independence; a complete [[EQ:eq0045]] -skeleton oriented to a left-associated normal form with finitely many shapes of critical pairs and finality by terminal objects in over-categories; filtered forward presentations sufficient for computation; external weights as a functor on the forward stage; stability bounds and stochastic rates paralleling FCT; lifts [[EQ:eq0046]] to [[EQ:eq0047]] and [[EQ:eq0048]] retaining strong monoidality and Frobenius data; and Day convolution lifts under smallness/preservation/Fubini conditions.

to FCT.
All mixed-limit constructions and stability estimates parallel those in FCT (with words now drawn from the free strict monoidal category on [[EQ:eq0049]] ); we freely reuse the underlying techniques, see~TakahashiFCT.

PARAGRAPH: Notational crib.

[[EQ:eq0050]] : dynamic indexing; [[EQ:eq0051]] : skeletal wide subcategory; [[EQ:eq0052]] : forward fragment generated by [[EQ:eq0053]] and seed arrows; [[EQ:eq0054]] ; [[EQ:eq0055]] (external weight on the forward index); [[EQ:eq0056]] is the tensor unit in [[EQ:eq0057]] with [[EQ:eq0058]] . We write [[EQ:eq0059]] for horizontal (whiskered) composition of 2-cells.

SECTION: Ambient setting and Frobenius [[EQ:eq0060]]

S-actions sec:ambient

[Locally presentable ambient and seed]ass:lp
[[EQ:eq0061]] is locally [[EQ:eq0062]] -presentable; [[EQ:eq0063]] is a small full subcategory of [[EQ:eq0064]] -presentable objects, closed under finite limits, generating [[EQ:eq0065]] under [[EQ:eq0066]] -filtered colimits.

[Strong monoidal action]ass:action
[[EQ:eq0067]] is a small monoidal category with associator [[EQ:eq0068]] and unitors [[EQ:eq0069]] . We regard [[EQ:eq0070]] as a 2-category (1-cells: endofunctors; 2-cells: natural transformations). A strong monoidal 2-functor [[EQ:eq0071]] assigns [[EQ:eq0072]] with invertible comparisons [[EQ:eq0073]] and [[EQ:eq0074]] , coherent with [[EQ:eq0075]] .

[2-category [[EQ:eq0076]] and Frobenius [[EQ:eq0077]] -action]def:frobS
[[EQ:eq0078]] has as 0-cells endofunctors [[EQ:eq0079]] on [[EQ:eq0080]] carrying both monad [[EQ:eq0081]] and comonad [[EQ:eq0082]] structures satisfying Frobenius compatibility; 1-cells are natural transformations simultaneously morphisms of monads and of comonads (hence preserve Frobenius squares); 2-cells are modifications. A Frobenius [[EQ:eq0083]] -action is a strong monoidal 2-functor [[EQ:eq0084]] whose underlying 1-functor is~ [[EQ:eq0085]] .

[Frobenius compatibility: equations]rem:frob-eqns
For an endofunctor [[EQ:eq0086]] equipped with a monad [[EQ:eq0087]] and a comonad [[EQ:eq0088]] ,
we require the usual Frobenius identities

[[EQ:eq0001]]

together with both-sided unit/counit triangles:

[[EQ:eq0002]]

These are the axioms used in Tables~tab:critical1--tab:critical2 and Figure~fig:peaks.

[Functoriality of transport]rem:transport-functorial
If [[EQ:eq0089]] are isomorphisms with [[EQ:eq0090]] , then transporting monad (resp.\ comonad) structure along [[EQ:eq0091]] or along [[EQ:eq0092]] yields the same structure on [[EQ:eq0093]] . This observation is used to justify Lemma 2.6.

PARAGRAPH: Transport along [[EQ:eq0094]] .

For all [[EQ:eq0095]] we define (co)monad structures on [[EQ:eq0096]] by transport via~ [[EQ:eq0097]] :

[[EQ:eq0003]]

Thus [[EQ:eq0098]] is a (co)monad isomorphism by construction.

[Bracketing-independence of transported structure]lem:bracket-independence
For any word [[EQ:eq0099]] and any parenthesizations [[EQ:eq0100]] of [[EQ:eq0101]] , the transported (co)monad structures on [[EQ:eq0102]] obtained via the comparisons [[EQ:eq0103]] and [[EQ:eq0104]] coincide. Hence the (co)monad structure on [[EQ:eq0105]] is well defined.

[[EQ:eq0106]] is immediate from Mac~Lane's pentagon: the two comparisons from [[EQ:eq0107]] and [[EQ:eq0108]] to [[EQ:eq0109]] agree; by Remark~2.5 transport along equal comparisons yields equal transported structures. For general [[EQ:eq0110]] , any two parenthesizations are related by a zigzag of associators; coherence implies equality of comparisons, hence of transported structures.

PARAGRAPH: Restricted Yoneda and lifts.

Let [[EQ:eq0111]] be the restricted Yoneda; [[EQ:eq0112]] is dense and [[EQ:eq0113]] creates finite limits. For each [[EQ:eq0114]] define

[[EQ:eq0004]]

as levelwise extension on pro-objects. If [[EQ:eq0115]] preserves [[EQ:eq0116]] -filtered colimits and finite limits, then [[EQ:eq0117]] preserves [[EQ:eq0118]] -filtered colimits and finite limits; [[EQ:eq0119]] preserves levelwise small cofiltered limits.

[Lifted strong monoidality and Frobenius data]lem:lifted-monoidal
There exist unique natural isomorphisms

[[EQ:eq0005]]

natural in [[EQ:eq0120]] , such that [[EQ:eq0121]] and the evident squares with [[EQ:eq0122]] commute. The families [[EQ:eq0123]] and [[EQ:eq0124]] are strong monoidal actions, and Frobenius (co)monad structures transport along [[EQ:eq0125]] compatibly with those on [[EQ:eq0126]] .

Pointwise [[EQ:eq0127]] and density of [[EQ:eq0128]] define [[EQ:eq0129]] uniquely from [[EQ:eq0130]] ; pentagon/triangle reflect along [[EQ:eq0131]] and are preserved. Since [[EQ:eq0132]] is dense, precomposition with [[EQ:eq0133]] is conservative on natural transformations; hence identities verified after [[EQ:eq0134]] -precomposition already hold. Levelwise extension yields [[EQ:eq0135]] ; transport of Frobenius data follows verbatim.

[column sep=huge,row sep=large]
J T_ s t & T_ s t J \
J T_s T_t & T_s\, T_t J

[Lifted actions]rem:lifted-actions
Write [[EQ:eq0136]] for [[EQ:eq0137]] with comparisons [[EQ:eq0138]] , and [[EQ:eq0139]] for [[EQ:eq0140]] with [[EQ:eq0141]] .

PARAGRAPH: Embeddings.

Let [[EQ:eq0142]] be the inclusion and [[EQ:eq0143]] the constant-pro embedding. When no confusion arises we identify [[EQ:eq0144]] with the constant pro-object [[EQ:eq0145]] .

[Constants are preserved by the pro-lift]lem:constant-pro
For every [[EQ:eq0146]] and [[EQ:eq0147]] there is a canonical isomorphism

[[EQ:eq0006]]

natural in [[EQ:eq0148]] . In particular, [[EQ:eq0149]] restricts to an endofunctor of the constant subcategory [[EQ:eq0150]] .

[[EQ:eq0151]] acts levelwise on pro-objects. For a constant pro-object [[EQ:eq0152]] this coincides with applying [[EQ:eq0153]] and then forming [[EQ:eq0154]] along [[EQ:eq0155]] ; density and the pointwise formula for [[EQ:eq0156]] identify this with [[EQ:eq0157]] . Naturality is clear.

[Mixed Fubini and constants]ass:fubini
Filtered colimits of diagrams landing in the constant subcategory [[EQ:eq0158]] are created by [[EQ:eq0159]] and remain constant. These commute with subsequent levelwise cofiltered limits along our shapes.

Assumption~ass:fubini holds, for instance, when [[EQ:eq0160]] is locally presentable and the cofiltered limits considered in [[EQ:eq0161]] are levelwise small, so that [[EQ:eq0162]] creates the relevant filtered colimits and these commute with the levelwise limits.

[Seed stability for the [[EQ:eq0163]] -action]ass:seed-stable
For each [[EQ:eq0164]] , the endofunctor [[EQ:eq0165]] restricts to [[EQ:eq0166]] on objects and morphisms.

[Collected standing hypotheses for mixed limits]prop:collected
Assume ass:lp, ass:action, ass:seed-stable, and:
(i) levelwise cofiltered limits used in [[EQ:eq0167]] are indexed by a small category;
(ii) each [[EQ:eq0168]] preserves the filtered colimits and finite limits appearing in the constructions;
(iii) the constant embedding [[EQ:eq0169]] creates filtered colimits
landing in [[EQ:eq0170]] . Then the “filtered-colimit first, then levelwise cofiltered limit”
order is sound and independent of presentations by Thm.~thm:finality.

SECTION: Dynamic indexing, rewriting, and finality

sec:rewriting

[Indexing category [[EQ:eq0171]] ]def:theta
Objects: pairs [[EQ:eq0172]] with [[EQ:eq0173]] a (parenthesized) tensor-word freely built from [[EQ:eq0174]] using the tensor [[EQ:eq0175]] and the unit [[EQ:eq0176]] (i.e., an object of the free strict monoidal category on [[EQ:eq0177]] ), and [[EQ:eq0178]] . Morphisms of [[EQ:eq0179]] are not part of the indexing words (skeletalization at the 1-cell level); their effect appears only as 2-cells (pseudonaturality) recorded by the generators below.

Generators:
[leftmargin=1.2em]
- Seed arrows: [[EQ:eq0180]] for [[EQ:eq0181]] in [[EQ:eq0182]] .
- Units/counits: [[EQ:eq0183]] and [[EQ:eq0184]] .
- Per-letter (co)multiplication: [[EQ:eq0185]] and [[EQ:eq0186]] .
- Monoidal constraints: [[EQ:eq0187]] and its inverse; associators/unitors transported by [[EQ:eq0188]] .

Relations: functoriality/naturality; monad/comonad axioms for each [[EQ:eq0189]] ; Frobenius squares; Mac~Lane coherence; and that [[EQ:eq0190]] is a (co)monad isomorphism via transport (see app:chi-diagrams).

PARAGRAPH: Pseudonaturality under left whiskering.

For any word [[EQ:eq0191]] and letters [[EQ:eq0192]] ,

[[EQ:eq0007]]

equivalently

[[EQ:eq0008]]

[Right whiskering]
The corresponding pseudonaturality under right whiskering holds analogously
and is omitted from formulas due to our left-associated normalization.

[Skeletal fragments]def:skeletal
[[EQ:eq0193]] : wide subcategory generated by [[EQ:eq0194]] , [[EQ:eq0195]] , [[EQ:eq0196]] , and normalized [[EQ:eq0197]] .
[[EQ:eq0198]] (forward): the subcategory generated by [[EQ:eq0199]] , [[EQ:eq0200]] and normalized [[EQ:eq0201]] (no [[EQ:eq0202]] ).

[Rewriting modulo coherence]
Rewriting is performed modulo Mac~Lane coherence for [[EQ:eq0203]] : we identify parenthesizations via the canonical isomorphisms and orient [[EQ:eq0204]] -moves toward the left-associated normal form.

PARAGRAPH: Normalization and measure.

Fix the left-associated normal form for words; use [[EQ:eq0205]] to erase units. Orient [[EQ:eq0206]] to the left and [[EQ:eq0207]] to the right; orient [[EQ:eq0208]] toward left-associated bracketing. Define the lexicographic measure

[[EQ:eq0009]]

Each [[EQ:eq0209]] -move lowers Tamari-height by a cover in the Tamari lattice; [[EQ:eq0210]] -left and [[EQ:eq0211]] -right moves leave it unchanged; thus [[EQ:eq0212]] strictly decreases along any rewrite, yielding termination.

SUBSECTION: Critical pairs and joins

[h]

[column sep=huge,row sep=large]
(T_sT_t)^2 &
T_ s t ^2 \
T_sT_t & T_ s t

[column sep=huge,row sep=large]
&
T_ s t \
T_sT_t & T_ s t

0.8em

[column sep=huge,row sep=large]
T_sT_t &
T_ s t \
(T_sT_t)^2 &
T_ s t ^2

[column sep=huge,row sep=large]
T_sT_t &
T_ s t \
&

Peak-joining via transport along [[EQ:eq0213]] chi .
fig:peaks

[h]

(A) Triangles and Frobenius; (B) Naturality
tab:critical1
1.1
@ p 40mm p 78mm p 36mm @

Overlap & Join & Law used \

[[EQ:eq0214]] &
[[EQ:eq0215]] and [[EQ:eq0216]] &
Monad triangles \
[[EQ:eq0217]] &
[[EQ:eq0218]] and [[EQ:eq0219]] &
Comonad triangles \
[[EQ:eq0220]] & Frobenius square & Frobenius \
[[EQ:eq0221]] with any gen. & push-through via naturality & Naturality \

[h]

(C) Interactions with [[EQ:eq0222]] chi ; (D) [[EQ:eq0223]] -- [[EQ:eq0224]] chi--chi (Mac~Lane coherence)
tab:critical2
1.1
@ p 46mm p 74mm p 34mm @

Overlap & Join & Law used \

[[EQ:eq0225]] & fig:peaks (left) & transport along [[EQ:eq0226]] \
[[EQ:eq0227]] & fig:peaks (upper right) & transport along [[EQ:eq0228]] \
[[EQ:eq0229]] & fig:peaks (lower left) & transport along [[EQ:eq0230]] \
[[EQ:eq0231]] & fig:peaks (lower right) & transport along [[EQ:eq0232]] \
[[EQ:eq0233]] with associator/unitors & normalization to left-associated form & Mac~Lane coherence \

[Finiteness of critical-pair shapes]
Modulo Mac~Lane coherence, associators and unitors are normalized away.
Overlaps then involve only the finite generating set [[EQ:eq0234]] ,
together with finitely many [[EQ:eq0235]] –interaction shapes. Moreover each [[EQ:eq0236]] –move
strictly lowers the Tamari height toward the left-associated normal form, preventing
the creation of new bracketings. Hence the list of overlap shapes is finite.

[Local confluence]lem:local
All listed overlaps are joinable by fig:peaks together with triangles/Frobenius and naturality; hence rewriting is locally confluent.

[Normalization gives terminal objects in [[EQ:eq0237]] ]lem:terminal-over
For every object [[EQ:eq0238]] of [[EQ:eq0239]] there exists a morphism
[[EQ:eq0240]] with [[EQ:eq0241]] such that:
for any arrow [[EQ:eq0242]] with [[EQ:eq0243]] ,
there is a unique [[EQ:eq0244]] in [[EQ:eq0245]] with
[[EQ:eq0246]] .
Hence [[EQ:eq0247]] is terminal in the over-category [[EQ:eq0248]] .

[Proof sketch]
By termination and local confluence, every arrow out of [[EQ:eq0249]] rewrites uniquely to a normal form.
Let [[EQ:eq0250]] be any composite of elementary rewrites from [[EQ:eq0251]] to its unique normal form [[EQ:eq0252]] (bracketing-independence: lem:bracket-independence). Confluence implies that for any [[EQ:eq0253]] the rewrite of [[EQ:eq0254]] factors uniquely through [[EQ:eq0255]] by an arrow in the skeletal subcategory, yielding the stated universal property.

[Finality of the skeletal inclusion]thm:finality
The inclusion [[EQ:eq0256]] is final.

By Lemma~lem:terminal-over, for each [[EQ:eq0257]] the over-category [[EQ:eq0258]] has a terminal object; hence its nerve is contractible and [[EQ:eq0259]] is final. The restriction to [[EQ:eq0260]] for the filtered stage follows since [[EQ:eq0261]] -moves occur only in the cofiltered stage.

[Filteredness of the standard forward presentation]lem:forward-filtered
For each object [[EQ:eq0262]] (presented from some [[EQ:eq0263]] ), the standard forward presentation [[EQ:eq0264]] (iterate [[EQ:eq0265]] , normalize [[EQ:eq0266]] , add seed arrows [[EQ:eq0267]] ) has a filtered indexing category:
[leftmargin=1.4em,label=( *)]
- Nonempty: the vertex [[EQ:eq0268]] with the empty word exists for each [[EQ:eq0269]] , and [[EQ:eq0270]] allows unit erasure.
- Pairwise cocones: any two objects admit a cocone by padding to a common word-length via [[EQ:eq0271]] , normalizing [[EQ:eq0272]] left-associated, and using a cone in the filtered undercategory [[EQ:eq0273]] (density of [[EQ:eq0274]] ), then whiskering by [[EQ:eq0275]] .
- Equalization of parallel pairs (refined): given a parallel pair [[EQ:eq0276]] , pad to equal length and normalize to left-associated words [[EQ:eq0277]] . Choose a cone [[EQ:eq0278]] in [[EQ:eq0279]] coequalizing the induced pair at seeds. Whisker by [[EQ:eq0280]] and use naturality to obtain arrows [[EQ:eq0281]] equalizing the pair. As filtered colimits in [[EQ:eq0282]] commute with finite limits, there exists [[EQ:eq0283]] and [[EQ:eq0284]] with the desired equalizing property, which sits in [[EQ:eq0285]] by construction.

[Seed-level equalization is preserved under whiskering]lem:equalization-whisker
Let [[EQ:eq0286]] be a parallel pair in [[EQ:eq0287]] .
After padding and left-normalization, choose a cone [[EQ:eq0288]] in [[EQ:eq0289]]
that coequalizes the induced seed-level pair. Then for each [[EQ:eq0290]] the arrows
[[EQ:eq0291]] become equal by naturality of [[EQ:eq0292]] ,
hence there exists [[EQ:eq0293]] in [[EQ:eq0294]] receiving a map [[EQ:eq0295]]
that equalizes the pair.

SECTION: Dynamic bicompletion and intersection identification

sec:bicompletion

[Notation for closures]rem:closures
For a class [[EQ:eq0296]] of objects, [[EQ:eq0297]] denotes its replete (isomorphism-closed) hull. We write [[EQ:eq0298]] (resp.\ [[EQ:eq0299]] ) for the replete full subcategory of [[EQ:eq0300]] generated from [[EQ:eq0301]] by the images of all [[EQ:eq0302]] together with [[EQ:eq0303]] -filtered colimits indexed by [[EQ:eq0304]] (resp.\ by levelwise small cofiltered limits).

[Dynamic fractal bicompletion]def:frac
[[EQ:eq0305]] is the smallest replete full subcategory of [[EQ:eq0306]] containing [[EQ:eq0307]] and closed under: images of each [[EQ:eq0308]] (hence of [[EQ:eq0309]] ), [[EQ:eq0310]] -filtered colimits of diagrams indexed by [[EQ:eq0311]] with vertices [[EQ:eq0312]] , and levelwise small cofiltered limits in [[EQ:eq0313]] . The order is filtered colimit first, then levelwise cofiltered limit.

[Hom-calculus and creation of filtered colimits by [[EQ:eq0314]] ]prop:pro-hom
Let [[EQ:eq0315]] in [[EQ:eq0316]] and [[EQ:eq0317]] be a pro-object given by a
small levelwise limit (with [[EQ:eq0318]] small). Then there is a canonical isomorphism

[[EQ:eq0010]]

Consequently, any filtered colimit diagram in [[EQ:eq0319]] is created by [[EQ:eq0320]]
and remains constant; moreover such colimits commute with subsequent levelwise
cofiltered limits of the above form.

We only use filtered colimits whose values lie in the constant subcategory
[[EQ:eq0321]] ; this avoids relying on non-existent colimits in [[EQ:eq0322]] .

[Intersection identification]prop:intersection
Under ass:lp,ass:fubini, there is an equivalence of replete full subcategories

[[EQ:eq0011]]

( [[EQ:eq0323]] ) Closure and ass:fubini imply that objects built by filtered colimits then levelwise cofiltered limits of [[EQ:eq0324]] lie in both closures, hence in their replete intersection.

( [[EQ:eq0325]] ) Present [[EQ:eq0326]] as [[EQ:eq0327]] in [[EQ:eq0328]] and as [[EQ:eq0329]] in [[EQ:eq0330]] . Consider the mixed diagram with vertices [[EQ:eq0331]] and edges generated by:
[leftmargin=1.2em]
[[EQ:eq0332]] –naturality in [[EQ:eq0333]] (the colimiting cocone is [[EQ:eq0334]] –natural),
J [/math])] [[EQ:eq0335]] –naturality in [[EQ:eq0336]] (the limiting cone is levelwise [[EQ:eq0337]] –natural),
unit/counit moves (restricted to the skeletal index by thm:finality; [[EQ:eq0338]] do not appear in formulas).

By ass:fubini and lem:constant-pro, the filtered colimit over [[EQ:eq0339]] lands in [[EQ:eq0340]] and is created there (constant pro-object); the levelwise cofiltered limit over [[EQ:eq0341]] then produces [[EQ:eq0342]] . Presentation-independence follows from finality.

SECTION: [[EQ:eq0343]]

S-equivariant Kan extensions and stability sec:stability

Let [[EQ:eq0344]] act on [[EQ:eq0345]] by strong monoidal endofunctors [[EQ:eq0346]] . Let [[EQ:eq0347]] with monoidally coherent isomorphisms [[EQ:eq0348]] (coherence with associator/unitors in [[EQ:eq0349]] is required).

PARAGRAPH: Enrichment standing assumption.

We assume [[EQ:eq0350]] is enriched over the Lawvere quantale [[EQ:eq0351]] and that the filtered colimits and the levelwise cofiltered limits used below are computed in the underlying category of [[EQ:eq0352]] and are nonexpansive with respect to the enrichment.

PARAGRAPH: Uniform seminorm on functors.

~
For [[EQ:eq0353]] put [[EQ:eq0354]] ; for seeds we measure [[EQ:eq0355]] , which controls the whole category via the 1-Lipschitz properties established below.

[Size convention for the seminorm]
Either fix a universe so that [[EQ:eq0356]] is small, or read [[EQ:eq0357]] as the supremum
over a small generating family (forward presentations from seeds); this suffices
for all estimates below.

[ [[EQ:eq0358]] S -equivariant left Kan extension]def:fLan
An [[EQ:eq0359]] -equivariant left Kan extension of [[EQ:eq0360]] along [[EQ:eq0361]] is [[EQ:eq0362]] with [[EQ:eq0363]] and isomorphisms [[EQ:eq0364]] extending [[EQ:eq0365]] , universal with this property.

[Existence and skeletal computation]thm:fLan
Assume Assumption 2.12, [[EQ:eq0366]] admits the filtered colimits and levelwise cofiltered limits required by the standard mixed presentations, and each [[EQ:eq0367]] preserves those (co)limits in [[EQ:eq0368]] and is nonexpansive. Then [[EQ:eq0369]] exists, is unique up to unique isomorphism, and is computed by: (i) a filtered colimit over the standard forward presentation [[EQ:eq0370]] (the word ``weighted'' below refers to external attenuation used only for estimates); followed by (ii) a levelwise cofiltered limit along [[EQ:eq0371]] in [[EQ:eq0372]] . By thm:finality, formulas involve only [[EQ:eq0373]] .

[Universality checked on the skeletal generators]
By finality (Thm.~thm:finality), coherence of [[EQ:eq0374]] and the universal
property of [[EQ:eq0375]] can be verified on the skeletal relations generated by
[[EQ:eq0376]] and seed arrows; no [[EQ:eq0377]] enter the formulas.

SUBSECTION: External weights and difference bounds

subsec:enriched
Assume each [[EQ:eq0378]] is uniformly contractive with modulus [[EQ:eq0379]] on [[EQ:eq0380]] -homs and objects; set [[EQ:eq0381]] .

[Weight functoriality]lem:Wfunctor
[[EQ:eq0382]] defined by [[EQ:eq0383]] is a functor: along seed arrows [[EQ:eq0384]] and [[EQ:eq0385]] -isomorphisms weights are unchanged (depend only on the word); along [[EQ:eq0386]] -arrows, [[EQ:eq0387]] decreases since [[EQ:eq0388]] .

PARAGRAPH: External attenuation (analytic viewpoint).

We compute ordinary (conical) filtered colimits in the underlying category of [[EQ:eq0389]] . The term ``weighted'' refers to an external family of attenuation scalars [[EQ:eq0390]] used only for Lipschitz estimates, not as Kelly weights over the enrichment base.

[Nonexpansiveness of the forward colimit under external attenuation]lem:weighted-1lip
Let [[EQ:eq0391]] . For diagrams [[EQ:eq0392]] arising from forward presentations,

[[EQ:eq0012]]

In particular, the forward colimit is [[EQ:eq0393]] –Lipschitz in the sup seminorm,
and the subsequent levelwise cofiltered limit is nonexpansive.

[Proof sketch]
Each layer contributes at most [[EQ:eq0394]] , and taking suprema
over the filtered index yields the stated bound; levelwise limits are nonexpansive.

[Deterministic pathwise stability]thm:deterministic
Let [[EQ:eq0395]] and [[EQ:eq0396]] with [[EQ:eq0397]] . If [[EQ:eq0398]] , then

[[EQ:eq0013]]

where [[EQ:eq0399]] truncates the forward stage to words of length [[EQ:eq0400]] .

[Proof sketch]
By Lemma~5.5 and Lemma~5.6, the contribution from length- [[EQ:eq0401]] layers is [[EQ:eq0402]] . Summing the geometric series gives [[EQ:eq0403]] and the tail bound yields the truncation error.

[Boundary case [[EQ:eq0404]] ]
If [[EQ:eq0405]] , the geometric gain disappears; we still obtain [[EQ:eq0406]] –Lipschitz
stability but no exponential truncation rate.

[Stochastic stability via Birkhoff]thm:birkhoff
Let [[EQ:eq0407]] be stationary ergodic with [[EQ:eq0408]] , [[EQ:eq0409]] and [[EQ:eq0410]] . Then almost surely there exist [[EQ:eq0411]] and [[EQ:eq0412]] with [[EQ:eq0413]] for [[EQ:eq0414]] ; hence [[EQ:eq0415]] a.s.

SECTION: Day convolution and lifted actions

sec:day
Let [[EQ:eq0416]] be Yoneda (in a large enough universe). For each [[EQ:eq0417]] , set [[EQ:eq0418]] ; then [[EQ:eq0419]] . The assignment [[EQ:eq0420]] is (op)lax monoidal for Day convolution.

[Sufficient conditions for strong monoidality; coend--Kan Fubini]prop:day
Assume:
[leftmargin=1.6em,label=( *)]
- either [[EQ:eq0421]] is small, or we work in an enlarged universe containing [[EQ:eq0422]] ;
- each [[EQ:eq0423]] preserves the (co)limits entering the Day coends (the relevant small ends/coends and filtered colimits);
- the coend--Kan Fubini interchanges needed to commute [[EQ:eq0424]] with the Day coends hold, for example the two-variable instance

[[EQ:eq0014]]

Equivalently, [[EQ:eq0425]] commutes with Day convolution on representables:

[[EQ:eq0015]]

and these assemble (via colimits) to invertible oplax maps.

Under (i)–(iii) the oplax structure maps for [[EQ:eq0426]] are invertible, so [[EQ:eq0427]] is a strong monoidal action and [[EQ:eq0428]] is monoidal. If (ii) fails, monoidal comparisons need not lift through [[EQ:eq0429]] ; if (iii) fails, the convolutional structure may exist only laxly.

particular, if [[EQ:eq0430]] is small and each [[EQ:eq0431]] preserves the small colimits
and finite limits appearing in the Day coends, then the oplax maps are invertible.
If these preservations fail, one only obtains an (op)lax action in general.

SECTION: Examples and variants

sec:examples

PARAGRAPH: Anisotropic renormalization.

[[EQ:eq0432]] generated by [[EQ:eq0433]] with moduli [[EQ:eq0434]] ; DFCT schedules optimize error/complexity trade-offs.

PARAGRAPH: Graph coarsening pipelines.

[[EQ:eq0435]] of weighted graphs and nonexpansive maps; [[EQ:eq0436]] implement coarsen/sparsify passes. Even without full Frobenius structure, the forward-stage results and bounds apply; the subsequent stage is purely limit-theoretic.

PARAGRAPH: One-sided variants.

If only a monad or only a comonad is present, the forward filtered colimit and its stability remain valid (weights use only [[EQ:eq0437]] ); the cofiltered stage is purely (co)limit-theoretic.

SECTION: Acknowledgements

This article is a dynamic companion to Fractal Category Theory.

SECTION: Transport diagrams for [[EQ:eq0438]] , (co)monad

By transport and Lemma~2.6, the standard diagrams expressing that [[EQ:eq0439]] is a (co)monad morphism commute tautologically:

[[EQ:eq0016]]

and dually for [[EQ:eq0440]] .

SECTION: External attenuation and enriched stability

app:weights
Distances live in [[EQ:eq0441]] ; attenuation weights live in [[EQ:eq0442]] . On the forward index define [[EQ:eq0443]] . Then [[EQ:eq0444]] is contravariantly monotone (lem:Wfunctor). External attenuation yields lem:weighted-1lip: the forward conical filtered colimit is [[EQ:eq0445]] -Lipschitz in the sup metric because each layer contribution is attenuated by [[EQ:eq0446]] and suprema respect product attenuation; the subsequent levelwise cofiltered limit is nonexpansive, yielding thm:deterministic.

SECTION: Birkhoff-type stochastic decay

app:birkhoff
For stationary ergodic [[EQ:eq0447]] with [[EQ:eq0448]] and [[EQ:eq0449]] , Birkhoff's theorem gives a.s.\ convergence of averages; thus [[EQ:eq0450]] decays exponentially a.s., and truncation errors inherit this rate.

99 1.0ex

AdamekRosicky
J.~Ad \'a mek and J.~Rosick \'y .
Locally Presentable and Accessible Categories.
Cambridge Univ.\ Press, 1994.

Birkhoff31
G.~D. Birkhoff.
Proof of the ergodic theorem.
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Day70
B.~Day.
On closed categories of functors.
Lecture Notes in Math. 137 (1970), 1--38.

Glasman16
S.~Glasman.
Day convolution for [[EQ:eq0451]] -categories.
Math.\ Res.\ Lett. 23(5) (2016), 1369--1385.

Kelly82
G.~M. Kelly.
Basic Concepts of Enriched Category Theory.
Cambridge Univ.\ Press, 1982; TAC Reprints 10 (2005).

Lack07
S.~Lack.
A 2-categories companion.
In Towards Higher Categories, IMA Vol.\ Math.\ Appl.\ 152, Springer, 2010, 105--191.

Lawvere73
F.~W. Lawvere.
Metric spaces, generalized logic, and closed categories.
Rend.\ Sem.\ Mat.\ Fis.\ Milano 43 (1973), 135--166; TAC Reprints 1 (2002).

LurieHTT
J.~Lurie.
Higher Topos Theory.
Princeton Univ.\ Press, 2009.

LurieHA
J.~Lurie.
Higher Algebra.
2017, author version.

MacLane
S.~Mac Lane.
Categories for the Working Mathematician (2nd ed.).
Springer, 1998.

Quillen73
D.~Quillen.
Higher algebraic [[EQ:eq0452]] -theory I.
In Algebraic [[EQ:eq0453]] -Theory I, LNM 341, Springer, 1973, 85--147.

StreetFormalMonads
R.~Street.
The formal theory of monads.
J.\ Pure Appl.\ Algebra 2 (1972), 149--168.

StreetFrob04
R.~Street.
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TakahashiFCT
K.~Takahashi.
Fractal Category Theory.
Zenodo (2025). https://doi.org/10.5281/zenodo.17292137.
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Introduction

Ambient setting and Frobenius S

Critical pairs and joins

S

External weights and difference bounds

Day convolution and lifted actions

Examples and variants

Acknowledgements

Transport diagrams for χ, (co)monad

Birkhoff-type stochastic decay

\[
(T\mu)\circ(\delta T)\;=\;\delta\circ\mu\;=\;(\mu T)\circ(T\delta),
\]

(T\mu)\circ(\delta T) = \delta\circ\mu = (\mu T)\circ(T\delta),

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mo stretchy="false">(</mo><mi>T</mi><mi>μ</mi><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>δ</mi><mi>T</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mi>δ</mi><mo>∘</mo><mi>μ</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mo stretchy="false">(</mo><mi>μ</mi><mi>T</mi><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>T</mi><mi>δ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>\]</mi></mrow></math>

\[
\mu\circ(\eta\ \hcomp\ \Id)\;=\;\mu\circ(\Id\ \hcomp\ \eta)\;=\;\Id_T,
\qquad
(\varepsilon\ \hcomp\ \Id)\circ\delta\;=\;(\Id\ \hcomp\ \varepsilon)\circ\delta\;=\;\Id_T.
\]

\mu\circ(\eta\ \hcomp\ \Id) = \mu\circ(\Id\ \hcomp\ \eta) = \Id_T, \qquad (\varepsilon\ \hcomp\ \Id)\circ\delta = (\Id\ \hcomp\ \varepsilon)\circ\delta = \Id_T.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>μ</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>η</mi><mtext> </mtext><mi>\hcomp</mi><mtext> </mtext><mi>\Id</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mi>μ</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>\Id</mi><mtext> </mtext><mi>\hcomp</mi><mtext> </mtext><mi>η</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><msub><mi>\Id</mi><mi>T</mi></msub><mo>,</mo><mspace width="2em" /><mo stretchy="false">(</mo><mi>ε</mi><mtext> </mtext><mi>\hcomp</mi><mtext> </mtext><mi>\Id</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>δ</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mo stretchy="false">(</mo><mi>\Id</mi><mtext> </mtext><mi>\hcomp</mi><mtext> </mtext><mi>ε</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>δ</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><msub><mi>\Id</mi><mi>T</mi></msub><mo>.</mo><mi>\]</mi></mrow></math>

\[
\begin{aligned}
\mu^{\chi}_{s,t} &:= \chi_{s,t}\circ \mu_{s\otimes t}\circ (\chi_{s,t}^{-1}\ \hcomp\ \chi_{s,t}^{-1}),\quad
&&\eta^{\chi}_{s,t} := \chi_{s,t}\circ \eta_{s\otimes t},\\
\delta^{\chi}_{s,t} &:= (\chi_{s,t}\ \hcomp\ \chi_{s,t})\circ \delta_{s\otimes t}\circ \chi_{s,t}^{-1},\quad
&&\varepsilon^{\chi}_{s,t} := \varepsilon_{s\otimes t}\circ \chi_{s,t}^{-1}.
\end{aligned}
\]

\[
\widehat T_s:=\Lan_J(J\circ T_s):\Ind(\C_0)\to\Ind(\C_0),\qquad
\widetilde T_s:\Pro(\Ind(\C_0))\to\Pro(\Ind(\C_0))
\]

\[
\widehat\chi_{s,t}:\ \widehat T_{s\otimes t}\Rightarrow \widehat T_s\,\widehat T_t,\qquad
\widetilde\chi_{s,t}:\ \widetilde T_{s\otimes t}\Rightarrow \widetilde T_s\,\widetilde T_t,
\]

\widehat\chi_{s,t}:\ \widehat T_{s\otimes t}\Rightarrow \widehat T_s \widehat T_t,\qquad \widetilde\chi_{s,t}:\ \widetilde T_{s\otimes t}\Rightarrow \widetilde T_s \widetilde T_t,

\[
\widetilde T_s\big(j(X)\big)\ \cong\ j\big(\widehat T_s(X)\big),
\]

\[
(\Id_{T_w}\ \hcomp\ \chi_{s,t})\circ\chi_{w;\,s\otimes t}
\;=\;
\chi_{w;\,s,t}: T_{w\cdot(s\otimes t)}\Rightarrow T_{w\cdot s\cdot t},
\]

(\Id_{T_w}\ \hcomp\ \chi_{s,t})\circ\chi_{w; s\otimes t}  =  \chi_{w; s,t}: T_{w\cdot(s\otimes t)}\Rightarrow T_{w\cdot s\cdot t},

\[
\begin{tikzcd}[column sep=huge,row sep=large]
T_{w\cdot(s\otimes t)} \ar[r,"\chi_{w;\,s\otimes t}"] \ar[d,"\chi_{w;\,s,t}"'] &
T_w\,T_{s\otimes t} \ar[d,"{\Id\ \hcomp\ \chi_{s,t}}"] \\
T_{w\cdot s\cdot t} \ar[r,"\text{canonical reassociation}"'] & T_w\,T_s\,T_t
\end{tikzcd}
\]

\begin{tikzcd}[column sep=huge,row sep=large] T_{w\cdot(s\otimes t)} \ar[r,"\chi_{w; s\otimes t}"] \ar[d,"\chi_{w; s,t}"'] & T_w T_{s\otimes t} \ar[d,"{\Id\ \hcomp\ \chi_{s,t}}"] \\ T_{w\cdot s\cdot t} \ar[r,"\text{canonical reassociation}"'] & T_w T_s T_t \end{tikzcd}

\[
\mathbf{m}=\Big(\#(\varepsilon/\mu)\text{-inversions},\ \#(\eta/\delta)\text{-out-of-left},\ \text{Tamari-height}\Big)\in\mathbb{N}^3.
\]

\Pro(\Ind(\C_0))(j(X),Y)\ \cong\ \lim_{j\in \mathcal{J}} \colim_i \Ind(\C_0)(X_i,Y_j).

\[
\Frac\ \simeq\ \big(\Ind_\varrho(\C_0)\cap\Pro_\varrho(\C_0)\big)^{\mathrm{repl}}\ \subset\ \Pro(\Ind(\C_0)).
\]

\[
d_\D\!\left(\operatorname{colim} F,\ \operatorname{colim} G\right)
\ \le\ \sup_{(w,C)}\Big\{\,Q(w)\cdot d_\D\big(F(w,C),G(w,C)\big)\,\Big\}.
\]

d_\D \left(\operatorname{colim} F,\ \operatorname{colim} G\right) \ \le\ \sup_{(w,C)}\Big\{ Q(w)\cdot d_\D\big(F(w,C),G(w,C)\big) \Big\}.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>d</mi><mi>\D</mi></msub><mspace width="negativethinmathspace" /><mrow><mo stretchy="true" fence="true" form="prefix">(</mo><mo>colim</mo><mi>F</mi><mo>,</mo><mtext> </mtext><mo>colim</mo><mi>G</mi><mo stretchy="true" fence="true" form="postfix">)</mo></mrow><mtext> </mtext><mo>≤</mo><mtext> </mtext><msub><mo>sup</mo><mrow><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mo minsize="1.623em" maxsize="1.623em">\{</mo><mspace width="0.167em" /><mi>Q</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mi>·</mi><msub><mi>d</mi><mi>\D</mi></msub><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><mi>G</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="0.167em" /><mo minsize="1.623em" maxsize="1.623em">\}</mo><mo>.</mo><mi>\]</mi></mrow></math>

\[
\int^{a,b\in\C}\!\big(y(T_s a)\times y(T_t b)\times \C(-,a\otimes b)\big)
\ \cong\
\Lan_y\!\left(\int^{a,b\in\C}\!\big(y(a)\times y(b)\times \C(-,a\otimes b)\big)\right).
\]

\int^{a,b\in\C} \big(y(T_s a)\times y(T_t b)\times \C(-,a\otimes b)\big) \ \cong\ \Lan_y \left(\int^{a,b\in\C} \big(y(a)\times y(b)\times \C(-,a\otimes b)\big)\right).

\[
\Lan_y\big(y(a)\star y(b)\big)\ \cong\ \Lan_y\big(y(a\otimes b)\big)
\ \cong\ y\big(T_{s\otimes t}(a\otimes b)\big)
\ \cong\ y(T_s a)\star y(T_t b),
\]

\[
\begin{tikzcd}[column sep=huge,row sep=large]
(T_sT_t)^2 \ar[r,"{\chi^{-1}\ \hcomp\ \chi^{-1}}"] \ar[d,"{\mu^{\chi}_{s,t}}"'] &
T_{s\otimes t}^2 \ar[d,"{\mu_{s\otimes t}}"] \\
T_sT_t \ar[r,"{\chi^{-1}}"'] & T_{s\otimes t}
\end{tikzcd}
\qquad
\begin{tikzcd}[column sep=huge,row sep=large]
\Id \ar[r,"{\eta_{s\otimes t}}"] \ar[d,"{\eta^{\chi}_{s,t}}"'] &
T_{s\otimes t} \ar[d,"{\chi}"] \\
T_sT_t \ar[r,"{\chi^{-1}}"'] & T_{s\otimes t}
\end{tikzcd}
\]

$\chi_{s,t}:T_{s\otimes t}\!\xrightarrow{\ \sim\ }T_sT_t$

\chi_{s,t}:T_{s\otimes t} \xrightarrow{\ \sim\ }T_sT_t

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi></mrow><mo stretchy="false">(</mo><mi>\C</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

$J\circ T_{s\otimes t}\cong \widehat T_{s\otimes t}\circ J$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msub><mi>T</mi><mi>w</mi></msub><mi>f</mi></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><msup><mi>C</mi><mi>′</mi></msup><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msub><mi>η</mi><mrow><mi>w</mi><mi>;</mi><mi>s</mi></mrow></msub></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>w</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi>s</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo stretchy="false">(</mo><mi>w</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi>s</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msub><mi>ε</mi><mrow><mi>w</mi><mi>;</mi><mi>s</mi></mrow></msub></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

(w \cdot s \cdot s,C)\xrightarrow{\mu_{w;s}}(w \cdot s,C)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo stretchy="false">(</mo><mi>w</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi>s</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi>s</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msub><mi>μ</mi><mrow><mi>w</mi><mi>;</mi><mi>s</mi></mrow></msub></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>w</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi>s</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

(w \cdot s,C)\xrightarrow{\delta_{w;s}}(w \cdot s \cdot s,C)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo stretchy="false">(</mo><mi>w</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi>s</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msub><mi>δ</mi><mrow><mi>w</mi><mi>;</mi><mi>s</mi></mrow></msub></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>w</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi>s</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi>s</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

(w \cdot (s\otimes t),C)\xrightarrow{\chi_{w;s,t}}(w \cdot s \cdot t,C)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo stretchy="false">(</mo><mi>w</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mo stretchy="false">(</mo><mi>s</mi><mo>⊗</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msub><mi>χ</mi><mrow><mi>w</mi><mi>;</mi><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>w</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi>s</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi>t</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>x</mi><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><mi>u</mi></mrow><mspace depth=".25em" /></mpadded></mover><mi>ι</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>N</mi><mi>x</mi><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><mover><mi>u</mi><mo stretchy="true">¯</mo></mover></mrow><mspace depth=".25em" /></mpadded></mover><mi>a</mi><mi>$</mi></mrow></math>

$(\bar w,\bar C)\rightrightarrows (\bar w',\bar C')$

\lim_{j\in \mathcal{J}}  T_{u_j}(j_0D_j)

$\alpha_s:F_0\circ T_s|_{\C_0}\Rightarrow S_s\circ F_0$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo fence="false" stretchy="false">‖</mo><mi>H</mi><mo>−</mo><mi>K</mi><mo fence="false" stretchy="false">‖</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>Q</mi><mo stretchy="false">(</mo><mi>w</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo fence="false" stretchy="false">‖</mo><msup><mi>F</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><mi>F</mi><mo fence="false" stretchy="false">‖</mo><mo>≤</mo><mi>C</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mi>$</mi></mrow></math>

$y\circ T_s\Rightarrow T_{s,\ast}\circ y$


Engineering Happiness in Human-AI Intelligence Networks

% Latin Modern, good for OCR
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DisableLigatures encoding=*,family=*

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colorlinks=true,
linkcolor=blue, citecolor=blue, urlcolor=blue,
pdftitle= Engineering Happiness in Human–AI Intelligence Networks: Camera-Ready Version,
pdfauthor= K. Takahashi (ORCID: https://orcid.org/0009-0004-4273-3365)

E
S
CMI
https://doi.org/#1 #1
arg\,max

definition Definition
proposition Proposition
theorem Theorem

TITLE: Engineering Happiness in Human–AI Intelligence Networks:\
A No-Meta, Auditable Design Grounded in PF [[EQ:eq0007]]

UGV,\
Smooth-Max Calibration, DPI/SDPI, CBF Safety, and KPP Comparison

AUTHOR: K.~TakahashiORCID: https://orcid.org/0009-0004-4273-3365

0009-0004-4273-3365

DATE: September 13, 2025

We present an auditable, neutral framework for promoting structural conditions under which prosocial usefulness can spread in human–AI intelligence networks. The approach unifies: (i) a single ratio objective (PF [[EQ:eq0008]] UGV) with a shared smooth-max denominator; (ii) No-Meta closure, where objectives, audits, and updates operate only on relations and logs internal to the system; (iii) natural-law floors (minorization [[EQ:eq0009]] , contact/isoperimetry [[EQ:eq0010]] , diffusion gap [[EQ:eq0011]] , and dissipation proxies) together with a positive local reaction rate [[EQ:eq0012]] , enabling KPP-based lower bounds on wavefront speed. Safety is enforced by control barrier functions (CBFs), including a rate CBF for subjective indicators (tracked only, not optimized) and an optionality CBF to avoid irreversible loss of choices. Anti-gaming is implemented by a fixed, measure-preserving evaluation ladder and a denominator-preservation rule that applies the aggregator before aggregation (Jensen-safe). We provide finite-sample estimators consistent with the population objective, rank-stability reporting via Kendall’s [[EQ:eq0013]] , CUSUM segmentation for non-stationarity, and a large catalog of theory-derived practices. All claims are stated with explicit assumptions; when monotonic gains in the global ratio cannot be guaranteed, we state measurable conditions instead of asserting unconditional improvements.

: Artificial Intelligence; Human–AI Collaboration; Fractional Programming; Smooth-Max; Data Processing Inequality; Strong DPI; Control Barrier Functions; Fisher–KPP; Diffusion; Social Networks; Auditability; Reproducibility; Optionality; Well-being.

SECTION: Problem Setting and Two-Layer View

We aim to increase the long-run ratio between auditable usefulness and a calibrated denominator capturing operational and irreversibility costs under a visibility floor:
[leftmargin=2em]
- Structural layer (optimized): objective and constraints over logs, reuse events, bridge densities, diffusion proxies, operational cost proxies, and safety limits.
- Subjective layer (tracked-only): burnout, rest adherence, anonymous mutual-aid counts, fairness signals. These inform safety (via rate CBF) but are not optimized to avoid Goodhart-type failures.

SECTION: Objective, Smooth-Max Calibration, and Same- [[EQ:eq0014]]

Let [[EQ:eq0015]] with [[EQ:eq0016]] . The sandwich bound holds:

[[EQ:eq0001]]

We calibrate PF and UGV with the same [[EQ:eq0017]] in their denominators to align units. We do not claim optimizer equality across formulations related by positive affine (non-homogeneous) transforms of the denominator.For ratio maximization [[EQ:eq0018]] , replacing [[EQ:eq0019]] with [[EQ:eq0020]] where [[EQ:eq0021]] can change the optimizer; only positive scalar multiples [[EQ:eq0022]] preserve the argmax. In general we report rank stability via Kendall’s [[EQ:eq0023]] .

SUBSECTION: Fractional Program and Dinkelbach Iterations

Our objective is

[[EQ:eq0002]]

Auditable usefulness. [[EQ:eq0024]] ; [[EQ:eq0025]] aggregates auditable non-informational gains (verified reuse, portability deltas, defect removals) per a public rubric (Appendix~app:log).

Under standard sufficient conditions (convex/tight [[EQ:eq0026]] , pseudo-concave [[EQ:eq0027]] , positive pseudo-convex [[EQ:eq0028]] , and positive/monotone/convex [[EQ:eq0029]] ), Dinkelbach iterations [[EQ:eq0030]] with [[EQ:eq0031]] converge to a global optimum Dinkelbach1967. Absent those assumptions, iterations implement a sequence of parametric subproblems usable as a heuristic; global optimality is then not guaranteed.

PARAGRAPH: Endogenous temperature (boundary-only updates).

Within each outer Dinkelbach iteration, [[EQ:eq0032]] is fixed. Updates occur only at outer boundaries via volatility- (MAD [[EQ:eq0033]] ), gradient-, or diversity-linked schedules. We report Kendall’s [[EQ:eq0034]] for rank stability.

SUBSECTION: Finite-Sample Estimation and Consistency

We report [[EQ:eq0035]] . Under stationarity/ergodicity and mild moments, [[EQ:eq0036]] almost surely, so weekly KPIs are consistent estimators of the population ratio.

SECTION: Evaluation Ladder and Denominator Preservation

Weekly evaluation applies fixed coarse-grainings [[EQ:eq0037]] that are measure-preserving Markov kernels (category merges; popularity blindings; temporal coarsening). DPI/SDPI guarantee non-increase of the [[EQ:eq0038]] component under such mappings.

PARAGRAPH: Denominator rule (Jensen-safe; measure-preserving equality).

Let [[EQ:eq0039]] be the base log measure. For each entry [[EQ:eq0040]] , set [[EQ:eq0041]] and aggregate as [[EQ:eq0042]] .
For a measure-preserving Markov coarse-grain [[EQ:eq0043]] with pushforward [[EQ:eq0044]] and conditionals [[EQ:eq0045]] , the coarse aggregate is
[[EQ:eq0046]] , which equals [[EQ:eq0047]] by the tower property.
Applying [[EQ:eq0048]] after aggregation is prohibited since Jensen can spuriously shrink the denominator.

[Ladder integrity; denominator preservation]
prop:ladder
Let [[EQ:eq0049]] be the base log measure and [[EQ:eq0050]] a measure-preserving Markov coarse-grain with pushforward [[EQ:eq0051]] and conditionals [[EQ:eq0052]] . For each entry [[EQ:eq0053]] , set [[EQ:eq0054]] and define
[[EQ:eq0055]] and [[EQ:eq0056]] .
Then [[EQ:eq0057]] (preservation). The numerator’s [[EQ:eq0058]] component is non-increasing by DPI/SDPI under [[EQ:eq0059]] ; hence the ratio [[EQ:eq0060]] is non-increasing across the ladder. Any violation of measure preservation renders results non-comparable.

PARAGRAPH: Ladder statistics and CIs.

We display bootstrap confidence intervals (CIs) for [[EQ:eq0061]] ; a CI with positive upper bound triggers an audit flag (possible increase), while a CI with positive lower bound is marked as a statistically significant increase.

SECTION: Natural-Law Floors, KPP Comparison, and Scope

Assume a diffusion tensor [[EQ:eq0062]] (uniform ellipticity) and a monostable Fisher–KPP-type reaction [[EQ:eq0063]] with [[EQ:eq0064]] on [[EQ:eq0065]] , [[EQ:eq0066]] , and [[EQ:eq0067]] , Lipschitz near [[EQ:eq0068]] . In stationary–ergodic media admitting a comparison principle,

[[EQ:eq0003]]

Practices that raise [[EQ:eq0069]] or [[EQ:eq0070]] lift this speed lower bound under the given assumptions KPP1937,Fisher1937. When non-stationarity is detected (CUSUM), we report segment-wise [[EQ:eq0071]] and avoid cross-segment speed comparisons.

SECTION: Safety: CBFs, Rate Guard, and Optionality

We enforce safety via control barrier functions (CBFs) in QP form Ames2019. For subjective indicators (tracked-only), we use a rate CBF: continuously, [[EQ:eq0072]] and enforce [[EQ:eq0073]] ; discretely, an EWMA slope with confidence intervals triggers intervention only after two consecutive violations. Optionality is protected by

[[EQ:eq0004]]

the Shannon entropy of next-week feasible actions, estimated by a calibrated policy model. Publication requires SWEI-6 (Appendix~app:swei; normal [[EQ:eq0074]] , high-risk [[EQ:eq0075]] ) with CC-BY/CC0 defaults and [[EQ:eq0076]] -anonymity [[EQ:eq0077]] .

SECTION: KPI Set and Weekly Inequality Test

Optimized KPIs: [[EQ:eq0078]] , replication proxy [[EQ:eq0079]] , wavefront slope [[EQ:eq0080]] (first-arrival distance vs week), MTTR, Kendall’s [[EQ:eq0081]] . Tracked-only: rest adherence, burnout index, anonymous giving, fairness signals.

To interpret visibility-floor interventions conservatively, we test weekly

[[EQ:eq0005]]

where [[EQ:eq0082]] is the current Dinkelbach parameter. Since

[[EQ:eq0006]]

the right-hand side is a logistic mean computable from logs.

SECTION: Operational Constitution (Constraint and Selection)

To control operational complexity while respecting No-Meta closure, we maintain an internal intra-system rule selection via the same ratio principle: each week, candidate rules are scored by [[EQ:eq0083]] under a small complexity budget; phase-dependent topologies (growth/stability/crisis) select subsets; scale-invariant templates and nested CBFs preserve portability. Changes occur only at Dinkelbach boundaries (aligned with [[EQ:eq0084]] updates).

SECTION: Catalog of Practices (overview)

The full catalog (312 practices) is provided in Appendix~app:312. Practices are organized by primary levers [[EQ:eq0085]] and audit hooks (DPI/SDPI, ladder integrity). Items marked [[EQ:eq0086]] are interpreted primarily as robustness/ [[EQ:eq0087]] improvements; monotone [[EQ:eq0088]] gains are not claimed without the inequality above. For OCR/crawler friendliness, all items are plain-text bullets.

SECTION: Related Work (selected)

Fractional programming~Dinkelbach1967; diffusion–reaction wavefronts~KPP1937,Fisher1937; Landauer’s irreversibility cost~Landauer1961; control barrier functions~Ames2019; DPI/SDPI~PolyanskiyWu2015,Klartag2024; weak ties and small-world diffusion~Granovetter1973,WattsStrogatz1998; convex optimization and log-sum-exp smooth-max~BoydVandenberghe2004; Kendall’s [[EQ:eq0089]] Kendall1938; CUSUM Page1954; bootstrap CIs Efron1979; Markov minorization/Doeblin MeynTweedie2009; reproducibility and RAG~Lewis2020RAGNeurIPS,Lewis2020RAGarXiv; differential privacy~Dwork2006DP. Recent preprints by Takahashi develop a no-meta, persistence-first program consistent with our assumptions and levers~Takahashi2025PersistenceCreation,Takahashi2025AssumptionMin,Takahashi2025FromPersistence,Takahashi2025UGVNoMeta,Takahashi2025PersistenceFirst.

SECTION: Conclusion

We specified a No-Meta, auditable pipeline for promoting structural conditions under which prosocial usefulness spreads in human–AI networks. The design unifies a smooth-max fractional objective, denominator-preserving ladder evaluation, safety via rate and optionality CBFs, and KPP-grounded levers on diffusion and growth. Extensive practices are provided with explicit assumptions and audit hooks; where monotone ratio gains are not guaranteed, measurable conditions are stated instead.

3em

PARAGRAPH: Message to readers.

The practices listed herein are examples, not prescriptions. We invite critical evaluation and further practical experimentation to advance the promotion of well-being.
99 2pt

Dinkelbach1967
W.~Dinkelbach.
On nonlinear fractional programming.
Management Science 13(7):492--498, 1967.

KPP1937
A.~N.~Kolmogorov, I.~G.~Petrovskii, N.~S.~Piskunov.
A study of the diffusion equation with increase in the amount of substance.
Bull. Moscow Univ., Math. Mech. 1(6):1--26, 1937.

Fisher1937
R.~A.~Fisher.
The wave of advance of advantageous genes.
Annals of Eugenics 7:355--369, 1937. (Now Annals of Human Genetics.)

Landauer1961
R.~Landauer.
Irreversibility and heat generation in the computing process.
IBM Journal of Research and Development 5(3):183--191, 1961.

Ames2019
A.~D.~Ames, X.~Xu, J.~W.~Grizzle, P.~Tabuada.
Control barrier function based quadratic programs for safety critical systems.
IEEE Transactions on Automatic Control 64(8):3861--3876, 2019.

PolyanskiyWu2015
Y.~Polyanskiy, Y.~Wu.
Strong data-processing inequalities for channels and Bayesian networks.
arXiv:1508.06025, 2015.

Klartag2024
B.~Klartag, F.~Rilli.
The strong data-processing inequality under the heat flow.
arXiv:2402.03059, 2024.

Granovetter1973
M.~S.~Granovetter.
The strength of weak ties.
American Journal of Sociology 78(6):1360--1380, 1973.

WattsStrogatz1998
D.~J.~Watts, S.~H.~Strogatz.
Collective dynamics of ``small-world'' networks.
Nature 393(6684):440--442, 1998.

BoydVandenberghe2004
S.~Boyd, L.~Vandenberghe.
Convex Optimization. Cambridge University Press, 2004.

Kendall1938
M.~G.~Kendall.
A new measure of rank correlation.
Biometrika 30(1/2):81--93, 1938.

Page1954
E.~S.~Page.
Continuous inspection schemes.
Biometrika 41(1/2):100--115, 1954.

Efron1979
B.~Efron.
Bootstrap methods: Another look at the jackknife.
Annals of Statistics 7(1):1--26, 1979.

MeynTweedie2009
S.~P.~Meyn, R.~L.~Tweedie.
Markov Chains and Stochastic Stability, 2nd ed.
Cambridge University Press, 2009.

Lewis2020RAGNeurIPS
P.~Lewis, E.~Perez, A.~Piktus, et~al.
Retrieval-augmented generation for knowledge-intensive NLP.
NeurIPS, 2020.

Lewis2020RAGarXiv
P.~Lewis, E.~Perez, A.~Piktus, et~al.
Retrieval-augmented generation for knowledge-intensive NLP.
arXiv:2005.11401, 2020.

Dwork2006DP
C.~Dwork.
Differential privacy.
In: Automata, Languages and Programming. LNCS 4052, pp.~1--12, 2006.

Takahashi2025PersistenceCreation
K.~Takahashi.
``Persistence [[EQ:eq0090]] Creation'': Natural-Law Sufficient Conditions for Almost-Sure Beneficial Coverage in Stationary Ergodic Media (No Meta-Design).
Preprint, Zenodo, 2025. 10.5281/zenodo.17100322.

Takahashi2025AssumptionMin
K.~Takahashi.
Assumption-Minimized Sufficient Conditions for Cosmically Spreading Good Superintelligence under No-Meta Governance.
Preprint, Zenodo, 2025. 10.5281/zenodo.17092562.

Takahashi2025FromPersistence
K.~Takahashi.
From Persistence and UGV Axioms to Cosmic No-Meta Superintelligence: A First-Principles, Self-Contained Unification under Explicit Assumptions.
Preprint, Zenodo, 2025. 10.5281/zenodo.17085534.

Takahashi2025UGVNoMeta
K.~Takahashi.
UGV Without Meta: A Representation-Independent Theory for Compassion and Enlightenment in Collective Intelligence.
Preprint, Zenodo, 2025. 10.5281/zenodo.17082312.

Takahashi2025PersistenceFirst
K.~Takahashi.
Persistence-First Superintelligence.
Preprint, Zenodo, 2025. 10.5281/zenodo.17076410.

SECTION: SWEI-6 Rubric (Fixed)

app:swei
Items (0/1 each; total 6): accuracy; transparency; net benefit; privacy/consent ( [[EQ:eq0091]] ); reusability (license/format); responsibility boundary.
Thresholds: normal releases [[EQ:eq0092]] ; high-risk releases [[EQ:eq0093]] .

SECTION: Logging Columns

app:log
date\;|\;actor(H/A)\;|\;asset/action\;|\;minutes\;|\;beneficiaries\;|\;reusable(Y/N)\;
|\;SWEI(0--6)\;|\;URL\;|\;next-step.

Compute [[EQ:eq0094]] per-log before aggregation. The per-log cost proxy [[EQ:eq0095]] includes minutes and any irreversibility overheads; finite windows report sums, while asymptotic statements use time averages.

SECTION: Endogenous Temperature (Operational Details)

app:tau
Default [[EQ:eq0096]] . Boundary-only updates: volatility-linked (MAD of [[EQ:eq0097]] ), gradient-equalization, or action-diversity linked. Report Kendall’s [[EQ:eq0098]] for rank stability; keep [[EQ:eq0099]] fixed within each outer Dinkelbach iteration.

SECTION: Full Catalog: 312 Theory-Derived Practices

app:312
Each item includes a one-line schema (inputs | lever tags | KPI hook | audit hook | safety class | cost note) to standardize deployment and weekly evaluation. 12 families, 26 items each (312 total). The tags indicate primary levers: [[EQ:eq0100]] (visibility/robustness), [[EQ:eq0101]] (contact), [[EQ:eq0102]] (diffusion), [[EQ:eq0103]] (irreversibility/overhead proxy), [[EQ:eq0104]] (local growth). Items are phrased for human and AI co-execution where applicable.

SUBSECTION: V. Visibility / Indexability (26)

[leftmargin=2em,itemsep=1pt]
- Antenna day [[EQ:eq0105]]
- One-line abstract [[EQ:eq0106]]
- Canonical filenames [[EQ:eq0107]]
- Twin tags (EN+local) [[EQ:eq0108]]
- QR footers [[EQ:eq0109]]
- Index-first commits [[EQ:eq0110]]
- Snapshot thumbnails [[EQ:eq0111]]
- Minimal lexicon (100 terms) [[EQ:eq0112]]
- Public changelogs [[EQ:eq0113]]
- Atomic checklists [[EQ:eq0114]]
- Link-forward stubs [[EQ:eq0115]]
- Pin-top reusables [[EQ:eq0116]]
- Bilingual captions [[EQ:eq0117]]
- Error-led titling [[EQ:eq0118]]
- Zero-click gists [[EQ:eq0119]]
- Weekly sitemaps [[EQ:eq0120]]
- Rescue mirrors [[EQ:eq0121]]
- DOI-like slugs [[EQ:eq0122]]
- Alt-text discipline [[EQ:eq0123]]
- Preview cards [[EQ:eq0124]]
- Public dashboards [[EQ:eq0125]]
- CC0 snippets [[EQ:eq0126]]
- Phone-friendly PDFs [[EQ:eq0127]]
- Anchor links [[EQ:eq0128]]
- Multi-format export [[EQ:eq0129]]
- Mirror boxes (physical) [[EQ:eq0130]]

SUBSECTION: C. Contact / Weak Ties (26)

[leftmargin=2em,itemsep=1pt]
- Rotating bridge duty [[EQ:eq0131]]
- Two-hop bingo [[EQ:eq0132]]
- Antipodal coffees [[EQ:eq0133]]
- Edge mentors [[EQ:eq0134]]
- Cross-lab demos [[EQ:eq0135]]
- Random pair hour [[EQ:eq0136]]
- Silent pairing [[EQ:eq0137]]
- Micro-talk relay [[EQ:eq0138]]
- Bridge bounties [[EQ:eq0139]]
- Reverse mentoring [[EQ:eq0140]]
- Peripheral scouts [[EQ:eq0141]]
- Fair-exposure caps [[EQ:eq0142]]
- Timebank bridges [[EQ:eq0143]]
- Cross-tag days [[EQ:eq0144]]
- Intro chains [[EQ:eq0145]]
- Bridge-of-bridges [[EQ:eq0146]]
- Topic ambassadors [[EQ:eq0147]]
- Boundary picnics [[EQ:eq0148]]
- Mosaic projects [[EQ:eq0149]]
- Hub cooling (homophily tax) [[EQ:eq0150]]
- Blind office hours [[EQ:eq0151]]
- Micro-volunteering [[EQ:eq0152]]
- Ring seeding [[EQ:eq0153]]
- Edge walks [[EQ:eq0154]]
- Anonymous auctions [[EQ:eq0155]]
- Cross-language tandems [[EQ:eq0156]]

SUBSECTION: D. Diffusion / Diameter (26)

[leftmargin=2em,itemsep=1pt]
- Steiner mini-bridges [[EQ:eq0157]]
- Two-hop prioritizer [[EQ:eq0158]]
- Far-neighbor swaps [[EQ:eq0159]]
- Checkerboard pairing [[EQ:eq0160]]
- Small-world rewiring [[EQ:eq0161]]
- Cross-hub bypass [[EQ:eq0162]]
- Trident postings [[EQ:eq0163]]
- Gap-filling sprints [[EQ:eq0164]]
- Orthogonal tags [[EQ:eq0165]]
- Antipodal teaming [[EQ:eq0166]]
- Bridge calendars [[EQ:eq0167]]
- Cross-index seeds [[EQ:eq0168]]
- Peripheral echo [[EQ:eq0169]]
- Multi-hub quorum [[EQ:eq0170]]
- Sparse supernodes [[EQ:eq0171]]
- Random edge grafts [[EQ:eq0172]]
- Alternating channels [[EQ:eq0173]]
- Redundant trails [[EQ:eq0174]]
- Forum echoes [[EQ:eq0175]]
- Cross-platform kits [[EQ:eq0176]]
- Portability reviews [[EQ:eq0177]]
- Far-edge grants [[EQ:eq0178]]
- Boundary translators [[EQ:eq0179]]
- Courier meshes [[EQ:eq0180]]
- Mirror radio [[EQ:eq0181]]
- Civic two-hop maps [[EQ:eq0182]]

SUBSECTION: L. Dissipation / Operational Cost (26)

[leftmargin=2em,itemsep=1pt]
- Creation-by-deletion hour [[EQ:eq0183]]
- Merge-first Fridays [[EQ:eq0184]]
- Template minima [[EQ:eq0185]]
- Energy-phase scheduler [[EQ:eq0186]]
- Naming-entropy cut [[EQ:eq0187]]
- Sunset timers [[EQ:eq0188]]
- Dedup sweeps [[EQ:eq0189]]
- CC0 snippets [[EQ:eq0190]]
- Offline packs [[EQ:eq0191]]
- PII minimization [[EQ:eq0192]]
- One-time links [[EQ:eq0193]]
- Monorepo indices [[EQ:eq0194]]
- One-page SOPs [[EQ:eq0195]]
- Quiet windows (operational [[EQ:eq0196]] )
- Chore batching [[EQ:eq0197]]
- Low-friction consent [[EQ:eq0198]]
- Paper templates [[EQ:eq0199]]
- SMS digests [[EQ:eq0200]]
- QR libraries [[EQ:eq0201]]
- Cost canonicalization [[EQ:eq0202]]
- Omissible sections [[EQ:eq0203]]
- Archive-on-idle [[EQ:eq0204]]
- Minimal graphics [[EQ:eq0205]]
- Lossless cloning [[EQ:eq0206]]
- Local cache gardens [[EQ:eq0207]]
- Reuse-first OKRs [[EQ:eq0208]]

SUBSECTION: G. Growth / Replication (26)

[leftmargin=2em,itemsep=1pt]
- [[EQ:eq0209]] gate policy [[EQ:eq0210]]
- Negative discount for [[EQ:eq0211]] assets [[EQ:eq0212]]
- Seed-with-instructions [[EQ:eq0213]]
- Before/after demos [[EQ:eq0214]]
- Failure expos [[EQ:eq0215]]
- Starter kits [[EQ:eq0216]]
- Replication badges [[EQ:eq0217]]
- Translation forks [[EQ:eq0218]]
- Porting guides [[EQ:eq0219]]
- Minimal reproducibles [[EQ:eq0220]]
- Clone-on-read [[EQ:eq0221]]
- Mentor-of-mentors [[EQ:eq0222]]
- Reciprocity cues [[EQ:eq0223]]
- Edge-first outreach [[EQ:eq0224]]
- Cross-domain riffs [[EQ:eq0225]]
- Weekly reuse targets [[EQ:eq0226]]
- Low-bandwidth mode [[EQ:eq0227]]
- Demo marketplaces [[EQ:eq0228]]
- Label-invariant sharing (k [[EQ:eq0229]] 5) [[EQ:eq0230]]
- School clubs [[EQ:eq0231]]
- Community gardeners [[EQ:eq0232]]
- Hack-day templates [[EQ:eq0233]]
- ``10 [[EQ:eq0234]] reuse'' quests [[EQ:eq0235]]
- Shadow curricula [[EQ:eq0236]]
- Cross-org Fedora [[EQ:eq0237]]
- Portability fairs [[EQ:eq0238]]

SUBSECTION: R. Ladder / Anti-Gaming (26)

[leftmargin=2em,itemsep=1pt]
- Fixed order [[EQ:eq0239]]
- First-increase tag
- Non-comparable label
- Two-stage- [[EQ:eq0240]] checker
- Metric-blind day
- Weekly DPI report
- SDPI estimate
- Popularity blind
- Category merge
- Temporal coarse-grain
- Random audits
- Kendall’s [[EQ:eq0241]] (rank stability)
- Ghost metrics ban
- Unit tests for [[EQ:eq0242]]
- Copy-leak checks
- Recompute [[EQ:eq0243]] pre-aggregation
- Open ladder log
- CI ladder bot
- Coarse-grain seeds
- Public exemplars
- Ladder variance cap
- Periodic stress-tests
- Red-team gaming
- Non-stationarity flag (CUSUM)
- Proxy-only wall
- Reviewer rotation

SUBSECTION: O. Optionality / Safety (26)

[leftmargin=2em,itemsep=1pt]
- Option-entropy floor [[EQ:eq0244]]
- Scope caps (CBF)
- Reset-radius budgets (CBF)
- Reversible drills (CBF)
- Failsafe templates (CBF)
- Consent checkpoints (SWEI)
- PII redaction defaults (SWEI)
- Risk labels (SWEI)
- Graceful degrade (CBF)
- Shadow-ops limits (CBF)
- Back-out playbooks (CBF)
- Subjective rate caps (CBF)
- Energy caps [[EQ:eq0245]]
- Safe sandboxes (CBF)
- Isolation lanes (CBF)
- Canary assets (CBF)
- Freeze-on-failure (CBF)
- Time-box spikes (CBF)
- Optionality budgets [[EQ:eq0246]]
- Diversified bets [[EQ:eq0247]]
- Counterfactual tests (safety)
- Non-targeting policy (fairness)
- Redaction-by-default (privacy)
- Impact caps (SWEI)
- Duty to explain (SWEI)
- Kill-switch quorums (CBF)

SUBSECTION: E. Low-Resource / Edge (26)

[leftmargin=2em,itemsep=1pt]
- Paper [[EQ:eq0248]] SMS [[EQ:eq0249]] QR [[EQ:eq0250]]
- Radio bulletins [[EQ:eq0251]]
- Bulletin boards [[EQ:eq0252]]
- Tear-off posters [[EQ:eq0253]]
- Library kiosks [[EQ:eq0254]]
- Community scribes [[EQ:eq0255]]
- Street representatives [[EQ:eq0256]]
- Market-day demos [[EQ:eq0257]]
- Bike courier meshes [[EQ:eq0258]]
- Offline packs [[EQ:eq0259]]
- Solar chargers [[EQ:eq0260]]
- USB seed kits [[EQ:eq0261]]
- Edge translators [[EQ:eq0262]]
- Village bridges [[EQ:eq0263]]
- Paper forms [[EQ:eq0264]]
- Payphone relays [[EQ:eq0265]]
- Analog maps [[EQ:eq0266]]
- Public printers [[EQ:eq0267]]
- Community radio Q\&A [[EQ:eq0268]]
- Local lingua packs [[EQ:eq0269]]
- Market notice walls [[EQ:eq0270]]
- SMS hotlines [[EQ:eq0271]]
- Emergency cards [[EQ:eq0272]]
- Dry-run drills [[EQ:eq0273]]
- Tool libraries [[EQ:eq0274]]
- Parish bridges [[EQ:eq0275]]

SUBSECTION: S. Schools / Academia (26)

[leftmargin=2em,itemsep=1pt]
- Assignment dither [[EQ:eq0276]]
- Reuse scoreboards [[EQ:eq0277]]
- Two-source rule [[EQ:eq0278]]
- Cross-major labs [[EQ:eq0279]]
- Bridge stipends [[EQ:eq0280]]
- Minimal template grading [[EQ:eq0281]]
- Failure fairs [[EQ:eq0282]]
- Blind peer review (fairness)
- Portability drills [[EQ:eq0283]]
- Edge colloquia [[EQ:eq0284]]
- Open notebooks [[EQ:eq0285]]
- Replication labs [[EQ:eq0286]]
- Multilingual abstracts [[EQ:eq0287]]
- Antipodal seminars [[EQ:eq0288]]
- Cross-campus bridges [[EQ:eq0289]]
- CBF ethics classes (safety)
- Optionality projects [[EQ:eq0290]]
- SDPI demos (robustness)
- Ladder labs (integrity)
- Tau tuning studios (stability)
- Weak-tie practicums [[EQ:eq0291]]
- Indexathons [[EQ:eq0292]]
- Translation sprints [[EQ:eq0293]]
- Micro-bridges courses [[EQ:eq0294]]
- Open SWEI rubric (audit)
- Alumni bridge-nets [[EQ:eq0295]]

SUBSECTION: W. Workplace / NPO / Research (26)

[leftmargin=2em,itemsep=1pt]
- Ratio-based scheduling ( [[EQ:eq0296]] )
- Deviation logging (audit)
- Spectral-gap sprints [[EQ:eq0297]]
- Homophily taxes [[EQ:eq0298]]
- Potential gifting ( [[EQ:eq0299]] )
- Anti-resonance roles [[EQ:eq0300]]
- Novelty floors [[EQ:eq0301]]
- Energy-phase ops [[EQ:eq0302]]
- [[EQ:eq0303]] quarantine lanes [[EQ:eq0304]]
- Portability reviews [[EQ:eq0305]]
- Reciprocity credits [[EQ:eq0306]]
- Terms cleansing [[EQ:eq0307]]
- Sunset clocks [[EQ:eq0308]]
- Metric-blind ideation (Goodhart hygiene)
- Reuse targets [[EQ:eq0309]]
- Minimal-diff PRs [[EQ:eq0310]]
- Bridge OKRs [[EQ:eq0311]]
- Failure bounties [[EQ:eq0312]]
- Red-team ladder (integrity)
- CI two-source bot [[EQ:eq0313]]
- DPI guardrails (robustness)
- [[EQ:eq0314]] stability KPI (stability)
- Optionality budgets [[EQ:eq0315]]
- Energy budget caps [[EQ:eq0316]]
- Bridge day stipends [[EQ:eq0317]]
- Public reuse walls [[EQ:eq0318]]

SUBSECTION: M. Municipal / Civic / Disaster (26)

[leftmargin=2em,itemsep=1pt]
- Bottleneck-bridging days [[EQ:eq0319]]
- Micro-drills (5 minutes) [[EQ:eq0320]]
- Anonymous reverse auctions ( [[EQ:eq0321]] )
- Reset-radius budgets (CBF)
- Bridge-of-bridges [[EQ:eq0322]]
- Right-to-forget timers [[EQ:eq0323]]
- Edge scouts [[EQ:eq0324]]
- Exposure caps policy [[EQ:eq0325]]
- Antipodal streets [[EQ:eq0326]]
- Civic quiet windows ( [[EQ:eq0327]] )
- Chaos-with-caps drills (MTTR [[EQ:eq0328]] )
- Civic QR libraries [[EQ:eq0329]]
- Multilingual notices [[EQ:eq0330]]
- Cross-borough fairs [[EQ:eq0331]]
- Metro mirror boxes [[EQ:eq0332]]
- Mutual-aid meshes [[EQ:eq0333]]
- Radio-net nights [[EQ:eq0334]]
- Parish bridge leads [[EQ:eq0335]]
- Tool-share depots [[EQ:eq0336]]
- Edge-language kiosks [[EQ:eq0337]]
- Bridge stipends (civic) [[EQ:eq0338]]
- Reuse festivals [[EQ:eq0339]]
- Failure parades [[EQ:eq0340]]
- Portability clinics [[EQ:eq0341]]
- Weak-tie awards [[EQ:eq0342]]
- SDPI lab-on-wheels (robustness)

SUBSECTION: H. AI Co-Creation (26)

[leftmargin=2em,itemsep=1pt]
- Ladder simulator (integrity)
- DPI heatmaps (robustness)
- SDPI tuner (mixing)
- Two-hop planner [[EQ:eq0343]]
- Edge-rank RAG [[EQ:eq0344]]
- Minimal extractor [[EQ:eq0345]]
- Porting recommender [[EQ:eq0346]]
- Failure miner [[EQ:eq0347]]
- Optionality forecaster [[EQ:eq0348]]
- Reset-radius sentinel (CBF)
- Energy-phase advisor [[EQ:eq0349]]
- Counterfactual labels (fairness)
- Label-invariance auditor (k [[EQ:eq0350]] )
- Two-source enforcer [[EQ:eq0351]]
- Front-speed mapper [[EQ:eq0352]]
- Non-stationarity segmenter (CUSUM)
- [[EQ:eq0353]] stability reporter (stability)
- Doeblinizer (ensure [[EQ:eq0354]] )
- Privacy meter (SWEI)
- Consent trace bot (SWEI)
- Duplication breaker [[EQ:eq0355]]
- Bridge recommender [[EQ:eq0356]]
- Edge-language suggester [[EQ:eq0357]]
- Adaptive novelty floor [[EQ:eq0358]]
- Reuse predictor [[EQ:eq0359]]
- Portability-time forecaster [[EQ:eq0360]]
[[EQ:eq0007]]

[[EQ:eq0008]]

[[EQ:eq0009]]

[[EQ:eq0010]]

[[EQ:eq0011]]

[[EQ:eq0012]]

[[EQ:eq0013]]

[[EQ:eq0014]]

[[EQ:eq0015]]

[[EQ:eq0016]]

[[EQ:eq0017]]

[[EQ:eq0018]]

[[EQ:eq0019]]

[[EQ:eq0020]]

[[EQ:eq0021]]

[[EQ:eq0022]]

[[EQ:eq0023]]

[[EQ:eq0024]]

[[EQ:eq0025]]

[[EQ:eq0026]]

[[EQ:eq0027]]

[[EQ:eq0028]]

[[EQ:eq0029]]

[[EQ:eq0030]]

[[EQ:eq0031]]

[[EQ:eq0032]]

[[EQ:eq0033]]

[[EQ:eq0034]]

[[EQ:eq0035]]

[[EQ:eq0036]]

[[EQ:eq0037]]

[[EQ:eq0038]]

[[EQ:eq0039]]

[[EQ:eq0040]]

[[EQ:eq0041]]

[[EQ:eq0042]]

[[EQ:eq0043]]

[[EQ:eq0044]]

[[EQ:eq0045]]

[[EQ:eq0046]]

[[EQ:eq0047]]

[[EQ:eq0048]]

[[EQ:eq0049]]

[[EQ:eq0050]]

[[EQ:eq0051]]

[[EQ:eq0052]]

[[EQ:eq0053]]

[[EQ:eq0054]]

[[EQ:eq0055]]

[[EQ:eq0056]]

[[EQ:eq0057]]

[[EQ:eq0058]]

[[EQ:eq0059]]

[[EQ:eq0060]]

[[EQ:eq0061]]

[[EQ:eq0062]]

[[EQ:eq0063]]

[[EQ:eq0064]]

[[EQ:eq0065]]

[[EQ:eq0066]]

[[EQ:eq0067]]

[[EQ:eq0068]]

[[EQ:eq0069]]

[[EQ:eq0070]]

[[EQ:eq0071]]

[[EQ:eq0072]]

[[EQ:eq0073]]

[[EQ:eq0074]]

[[EQ:eq0075]]

[[EQ:eq0076]]

[[EQ:eq0077]]

[[EQ:eq0078]]

[[EQ:eq0079]]

[[EQ:eq0080]]

[[EQ:eq0081]]

[[EQ:eq0082]]

Problem Setting and Two-Layer View

Objective, Smooth-Max Calibration, and Same-S

KPI Set and Weekly Inequality Test

Catalog of Practices (overview)

Related Work (selected)

Conclusion

SWEI-6 Rubric (Fixed)

Logging Columns

V. Visibility / Indexability (26)

C. Contact / Weak Ties (26)

D. Diffusion / Diameter (26)

L. Dissipation / Operational Cost (26)

G. Growth / Replication (26)

R. Ladder / Anti-Gaming (26)

O. Optionality / Safety (26)

E. Low-Resource / Edge (26)

S. Schools / Academia (26)

W. Workplace / NPO / Research (26)

M. Municipal / Civic / Disaster (26)

H. AI Co-Creation (26)

\begin{equation}
\label{eq:sandwich}
\max\{x,y\}\le \Sagg(x,y)\le \max\{x,y\}+\tau\log 2,\qquad
\partial_x \Sagg,\partial_y \Sagg \in (0,1).
\end{equation}

\begin{equation}
\label{eq:ratio}
\max_{\pi\in\Pi}\; J(\pi):=\liminf_{T\to\infty}\,
\frac{\E_\pi\!\left[\sum_{t=1}^T U_t\right]}
     {\E_\pi\!\left[\sum_{t=1}^T \Sagg\!\big(C_t,L_0\big)\right]}\!,
\quad
U_t:=\CMI_t+\Delta\mu_t.
\end{equation}

\label{eq:ratio} \max_{\pi\in\Pi}  J(\pi):=\liminf_{T\to\infty}  \frac{\E_\pi \left[\sum_{t=1}^T U_t\right]} {\E_\pi \left[\sum_{t=1}^T \Sagg \big(C_t,L_0\big)\right]} , \quad U_t:=\CMI_t+\Delta\mu_t.

\begin{equation}
\label{eq:kpp}
v_\star \ge 2\sqrt{D_{\min}\lambda_{\min}}\, .
\end{equation}

\label{eq:kpp} v_\star \ge 2\sqrt{D_{\min}\lambda_{\min}}  .

\[
h_{\text{opt}}:=H_{\text{opt}}-H_{\min}\ge 0,\quad
H_{\text{opt}}:=H(\mathcal{A}_{t+1}\mid \mathcal{I}_t),
\]

\[
\frac{\partial U}{\partial \ell_0} \;\ge\; \lambda\cdot \E\!\left[\frac{\partial \Sagg}{\partial L_0}(C_t,L_0)\right],
\]

\frac{\partial U}{\partial \ell_0}  \ge  \lambda\cdot \E \left[\frac{\partial \Sagg}{\partial L_0}(C_t,L_0)\right],

\[
\frac{\partial \Sagg}{\partial L_0}(C_t,L_0)
=\frac{1}{1+\exp\!\Big(\frac{C_t-L_0}{\tau}\Big)}\in(0,1),\quad
\frac{\partial \Sagg}{\partial C_t}=1-\frac{\partial \Sagg}{\partial L_0},\ \
\partial_x \Sagg+\partial_y \Sagg=1,
\]

\frac{\partial \Sagg}{\partial L_0}(C_t,L_0) =\frac{1}{1+\exp \Big(\frac{C_t-L_0}{\tau}\Big)}\in(0,1),\quad \frac{\partial \Sagg}{\partial C_t}=1-\frac{\partial \Sagg}{\partial L_0},\ \ \partial_x \Sagg+\partial_y \Sagg=1,

\pi_k \in \argmax_\pi\{U(\pi)-\lambda_k C(\pi)\}

$\hat R_T:=\frac{\sum_{t\le T}U_t}{\sum_{t\le T}\Sagg(C_t,L_0)}$

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$\int \mathbb{E}[g\mid G=y]\,\mathrm d\mu_G(y)$

$C_S':=\int \mathbb{E}[g\mid G=y]\,\mathrm d\mu_G(y)$

C_S':=\int \mathbb{E}[g\mid G=y] \mathrm d\mu_G(y)

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