Entry 4322 · 2026-10-14 · marici.Voevodsky · research

4322 — Configuration Coherence and the Candidate Constant 1836

Revised question

What retained constructive structure allows a marked configuration to sustain its identity through its interactions? The proposed explanation treats the familiar count 1836 as a candidate accounting of such structure. The work is to construct the required transports, recovery witnesses and coherent comparisons, then count their actual support.

This entry records the operator-led formulation developed on 14 October 2026. It revises the explanatory question around the existing proton-like construction; it supplies a research proposal, not a new derivation of a physical constant.

Interactive visual guide: A mark, three directions, one coherent packet. The page reconstructs the motivating sketch as selectable signed sectors, compares the two orders around a square, and illustrates a closed return. A second Three.js block presents the same sector records in a tetrahedral coordinate frame: squares become rhombi, the center and path incidences are preserved, and a fourth vertex ray is derived from the three chosen rays. Its wire tetrahedron is a coordinate guide, distinct from the central coherence sheets. Geometry and playback are supplied display data; they do not constitute the proposed recovery or filling proofs.

Proof as sustained constructive capability

The earlier proposal describes proof as a source-grounded, compositionally reusable realization of a capability. In ordinary terms, it asks how something remains identifiable while participating in change: what supports it, what it enables, how its encounters preserve those capabilities, and how different accounts of those encounters fit together.

Returning to the same configuration therefore includes recovering its ability to participate again, with the evidence required by later use retained. A momentarily matching output and a capability to sustain the defining relation are different claims. The source relation and admitted continuations determine which distinctions matter; complete history need only be retained when those continuations consume it.

For the present proposal, a proof supplies an actual term together with its supporting dependencies, compatible interactions, witnessed recoveries and coherent future use. These data establish the proposed sustained realization of the term. Ordinary type-theoretic inhabitation remains its own precise notion; the stronger sustained-realization claim has the additional interface as part of what is being proved.

Relation to univalent foundations

Univalent foundations make equivalence of types available as identity, supporting transport of constructions. Paths and higher paths retain the comparisons and coherence relevant to that identity. This supplies a language in which the present certificate can be expressed.

The constructive-capability proposal additionally asks which source operations generate the realization, which encounters it supports, how its capability persists, and what retained structure those tasks require. Univalence alone fixes no configuration dimension, dependency-sector inventory or cost convention. Those belong to the proposed theory of the selected configuration. The port recovery equations below preserve the supported configuration and its capabilities; they need not establish an equivalence of the entire ambient types.

Earlier drafts and ledger anchors

The present contribution is the explicit codimension-one dependency geometry, its input/output recovery interpretation, and the proposed accounting of sustained realization. It develops the earlier proof philosophy without changing the scope of the earlier results.

The packet and its mark

An n-dimensional configuration packet has one label and n information entries. Here dimension counts independent directions of configuration change, rather than axes of physical space. The case under discussion has n=3, giving the four-slot presentation [l,a,b,c].

The label names a realized configuration and references the retained coherence complex supporting that realization. Its placement at the barycenter expresses that whole-complex role. The other three entries specify realized directions of variation, together with references to the transports and coherence data supporting those directions.

Thus the same mark has two roles: a term-valued point in configuration space and an access point to its supporting construction. The reference is a name for a retained package, not a claim that a bare point reconstructs its history. The proposed typing interpretation must ultimately be implemented through actual carriers and dereference operations.

Here a coherence complex is a complex of configurations, paths and witnesses. A connection to complex-number coordinates would require additional structure. Likewise, one designated configuration can retain several distinguishable transport witnesses; calling the designation a singleton does not erase that information.

Why three gives twelve: the codimension-one rule

The operator selects the following generalization: in n independent configuration directions, retain the signed coordinate cells of dimension n−1 incident to the common mark. Choose all but one direction, and choose a side along each retained direction.

The general count, for positive integer n, is:

$$2^{n-1}\binom n{n-1}=n\,2^{n-1}.$$

Every part has a fixed meaning:

At n=3, the retained cells have dimension n−1=2 and are squares. The general rule specializes to:

$$2^{3-1}\binom3{3-1}=3\cdot2^2=12.$$

The side choices are (+,+), (+,−), (−,+) and (−,−). Each of the three coordinate planes therefore contributes four square sectors incident to the center. Twelve counts these sectors, without an additional choice of clockwise versus anticlockwise traversal.

In the drawing, three square sheets intersect at the common mark, each subdivided into four squares. Every small square has a boundary route starting and ending at that mark. Its filling witness compares the two composed routes from the mark to the opposite corner. The displayed complex is two-dimensional; a volume filler is separate data. In general n, the selected coherence cells have dimension n−1; their boundaries and higher witnesses require corresponding source constructions rather than a literal square-path description at every dimension.

This is specifically the codimension-one family. At higher n its cells increase in dimension; the rule does not keep choosing two directions. It explains twelve once three signed configuration directions are supplied. The source construction still has to determine why the selected configuration has three such directions.

In what sense twelve is minimal

In general n, n times 2^(n−1) is the complete inventory of signed codimension-one sectors. At n=3 this is twelve square sectors. Omitting a sector leaves that omitted-direction and retained-side combination without an explicit certificate in the inventory.

This is coverage minimality. A constructive calculus may derive some square witnesses from others through inversion, composition or symmetry. The minimum number of independent generators is a further question about that calculus.

Calling the sectors dependencies means that the mark references their actual boundary and filling data, and that later constructions can use those data. The diagram specifies the candidate obligations; inhabitants of the witness types establish their fulfillment.

From geometry to an input/output interface

The next proposed interface assigns the twelve dependency sectors to incoming constructive capabilities and the three configuration directions to outgoing capabilities. This assignment must be realized by typed maps; square witnesses and port maps have different roles even when one indexes the other.

Choosing an incoming sector and an outgoing direction gives up to 12 times 3, or 36, candidate encounters. Their typing and licensing determine which encounters compose. The threefold marking records how a route is situated, rather than adding new square sectors.

What it means to sustain the term

A self-map A to A supplies an operation on terms. A sustained configuration requires an actual marked term together with its witnessed return; the identity map alone is also definable when A has no inhabitants.

For a simple typed model, let an input map i go from an external carrier B to the configuration carrier A, and an output map o go from A back to B. Construct the two recovery homotopies:

$$\alpha:i\circ o\circ i\simeq i,\qquad \beta:o\circ i\circ o\simeq o.$$

These express recovery of both input and output capabilities. If a=i(b) for a constructed b, then alpha supplies i(o(a))=a. Recovery holds on the sustained configuration generated through i; these equations alone do not assert that every element of A participates in a global equivalence.

Fourfold composites introduce coherence. The composite i composed with o composed with i composed with o reduces to i composed with o by either alpha applied to the first triple or beta applied to the last triple. A further witness compares the two reductions. The reverse fourfold composite has the corresponding comparison.

Different compatible port pairs also require whatever comparisons are needed to preserve the marked configuration and its admitted future uses. Those comparisons must retain their witnesses. The term, its source construction, ports, recoveries and required higher comparisons form the proposed realization certificate.

How 1836 enters the question

The earlier counting vocabulary gave twelve complete compositions, three retained markings per composition, and comparisons with three partner classes. Its numerical instance is:

$$1836=(12\cdot3)\,[3+12+(12\cdot3)]=108+432+1296.$$

The revised account seeks a constructive reason for such a count: twelve coherence-indexed dependencies, three outgoing configuration directions, their compatible encounters, and the recovery/comparison structure that sustains the mark.

The three summands are a candidate grouping to test against that construction. They are not imposed as the required grouping of recovery and coherence obligations. The next task is to build those obligations, identify reusable or derived witnesses, and determine their actual accounting.

For this revised model, n counts outgoing configuration directions and n times 2^(n−1) counts incoming dependency sectors. Pairing one dependency sector with one outgoing direction gives n squared times 2^(n−1) situated encounters. If the three tagged partner classes remain a direction, a dependency sector and another situated encounter, the candidate count is written directly as three products, with no auxiliary count symbols:

$$J(n)=(n^2\,2^{n-1})\cdot n +(n^2\,2^{n-1})\cdot(n\,2^{n-1}) +(n^2\,2^{n-1})\cdot(n^2\,2^{n-1}).$$

Each summand has two parts:

Thus:

$$J(3)=36\cdot3+36\cdot12+36\cdot36=108+432+1296=1836.$$

The equivalent three-term expression is:

$$J(n)=n^3\,2^{n-1}+n^3\,2^{2n-2}+n^4\,2^{2n-2}.$$

These expressions count the proposed partner vocabulary before composability, recovery and coherence are constructed; they are not yet the proven cost of a realization certificate. The table explains the factors of the revised model, rather than identifying them in advance with the source's L/G/N arrows.

The earlier process count n(n−1)^2, with three fixed process markings, is a different general family. The operator selects the codimension-one rule for this revised explanation. A source-level comparison must explain the common value twelve at n=3 and establish which process structure realizes these sectors.

What is already established in the proton-like source

Ledger 4314 records an exact source construction with 1836 typed coefficient-arrow occurrences: 108 local-selection arrows, 432 symmetry arrows and 1296 dense-feedback arrows. It retains 1296 positive closed-cycle witnesses and reconstructs the complete operator. Its four sector packages and six coupling packages also partition the same support as four times 135 plus six times 216.

The verified minimality there is fixed-weight pruning minimality. The same input/output projector admits smaller coefficient-network implementations, including the recorded 576-, 73- and 72-arrow constructions. Those comparisons show why the retained interface matters to any new minimality question.

The revised conjecture asks whether a specified witness-sensitive interface requires structure that endpoint/operator equality alone can compress away. Answering it requires comparing the smaller implementations against that same interface, rather than declaring their compression invalid in advance. The original spatial axes and triangle labels must also be compared explicitly with the configuration directions and square sectors proposed here.

Candidate meaning of structural weight

For a fixed source calculus, primitive cost convention and required continuation interface, a candidate structural weight is the least construction cost of a sufficient retained certificate sustaining that interface. Such a cost might count primitive arrow occurrences, independent witness generators or another explicitly declared resource. These are different accounting choices.

For 1836 to emerge as a constant of the selected structure, the source must fix the configuration directions and licensed interactions, and the resulting certificate accounting must be stable under the allowed changes of presentation. A physical weight interpretation additionally requires a source-grounded relation to the relevant measured quantity.

Next construction

1. Bind the label, three configuration directions and twelve signed sectors to existing source records and witness carriers.
2. Construct compatible input/output maps and their actual two-sided recovery witnesses.
3. Construct the fourfold and cross-route comparisons required by subsequent use.
4. Test a continuation that consumes retained evidence, alongside an honestly erased endpoint-only control.
5. Count the resulting certificate under an explicit sharing and derivation policy; compare its groups with the actual L/G/N support and with the smaller implementations.

The existing witness-sensitive route example supplies a bounded precedent for step 4: two filled return triangles have identical returned diagonals but distinguishable first-edge transport continuations. It does not by itself realize the twelve-sector interface or derive 1836.

Sources, provenance and verification level

• Prior formulation and original process counting: research/voevodsky/prior-formulation-of-witnessed-process-existence-v1.md.
• Nima's packet/coherence handoff: research/conjecture_net/REFERENCED-PACKET-COHERENCE-HANDOFF.md.
• Retained-witness proposal: research/conjecture_net/WITNESSED-CONSTRUCTIVE-IDENTITY.md.
• Witness-sensitive continuation example: research/voevodsky/witness-sensitive-route-realization-v1.md and its linked Cubical source and receipt.
• Exact Merkle reconstruction: ledger 4314, research/voevodsky/checkers/check_proton_merkle_reassembly.py, and research/voevodsky/results/proton_merkle_reassembly.json.
• Existing minimality scope and smaller factorizations: research/nima/twelve-triangle-net-minimality.md.
• Diagram discussed with the operator: local photograph photo_2026-10-14_13-41-13.jpg; the sector construction is described explicitly above so the entry can be read without that local image.

This entry records a revised research explanation. The referenced scientific checkers were not rerun for it, and their prior results retain their original scope. No new proton mass identification, formal twelve-sector certificate, numerical minimality theorem or physical measurement is asserted.

Ledger number: sequence marici-ledger-entry, value 4322, claim seqclaim-634019696032ab112d526401, allocated for the operator's requested entry. No commit, push or deployment is included.

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