---
title: "Implications for Computing and Quantum Error Correction"
slug: "prime-valuation-qec-implications"
url: https://papers.qnfo.org/papers/prime-valuation-qec-implications
authors: ["Rowan Brad Quni-Gudzinas"]
version: "1.0.0"
published: "2026-08-13"
updated: "2026-10-29"
license: "QNFO-ULA-2.1"
status: "published"
quality_label: "High-quality scholarly research"
quality_basis: "QNFO publisher classification for open-access, versioned research with a public revision history and reader correction channel; this is not a claim of independent peer review."
peer_review_status: "Not independently peer-reviewed"
quality_evidence_url: "https://qnfo.org/about"
quality_tags: ["High-quality scholarly research", "open-access scholarly work", "versioned research publication"]
---

# Implications for Computing and Quantum Error Correction

The anchor paper introduced the *branch-depth reading*: the p-adic valuation $v_p(n)$ is a measure of depth along a prime branch of the integer factorization tree (not a measure of size), and because the tensor product multiplies Hilbert-space dimensions, $v_2(\dim H) = n$ counts tensor-branch depth for n qubits. This follow-on subjects that reading to its own stated falsifiability condition in the two domains the anchor flagged but did not pursue — computing and quantum error correction. The central finding is a *correction, not a confirmation*: the naive mapping of a $[[n,k,d]]$ stabilizer code to branch depths, $n = v_2(\dim H)$ and $k = v_2(\dim H_L)$, is **trivially true by definition** (a code is defined on n qubits encoding $2^k$ logical states) and therefore carries no new content; and the code distance d — a minimum operator *weight*, not a *depth* — does not admit a valuation reading at all. [...]

---
title: "Implications for Computing and Quantum Error Correction"
author: "Quni-Gudzinas, Rowan Brad"
orcid: "0009-0002-4317-5604"
affiliation: "QNFO"
date: "2026-08-14"
version: "v0.4"
license: "CC-BY-4.0"
doi: ""
status: "published"
keywords: ["p-adic valuation", "quantum error correction", "stabilizer codes", "no-cloning theorem", "branch depth", "ultrametric"]
---

**Author:** Quni-Gudzinas, Rowan Brad (QNFO)
**ORCID:** 0009-0002-4317-5604
**Date:** 2026-08-14
**Version:** v0.4
**License:** CC-BY-4.0
**Keywords:** p-adic valuation; quantum error correction; stabilizer codes; no-cloning theorem; branch depth; ultrametric
**Changelog:** v0.4 (2026-08-14) — post-publication audit remediation: "first reproduction attempt" withdrawn (prior corpus test ACRP-06 v1.0 + v1.1 erratum, the QNFO archive 21737222/21754148); Coecke ref year 2009→2010 (print year); Gubser–Knaute pages per Crossref (1655–1678); §6 records NTOF's omitted 187/200 (93.5%) C7.3′ figure. v0.3 (2026-08-14) — falsifiability register resolved by derivation: C7 overhead bound DISCONFIRMED (FQ2: valuation data (n,k,q)-only, d inexpressible, bound strictly weaker than Singleton), C5 complexity characterization DISCONFIRMED (FQ3: candidate identities genuine but complexity-vacuous), C6 no-cloning consequence DISCONFIRMED (FQ4: relabels Abramsky's categorical no-cloning; no new QEC consequence). Outcome derivations: artifacts/fq2-overhead-bound.md, artifacts/fq3-complexity-characterization.md, artifacts/fq4-no-cloning-consequence.md. Only C4/FQ1 (non-trivial valuation invariant) remains open. v0.2 (2026-08-13) — §6 precision patch: per-family 83% breakdown, rule-based Algorithm 4.4 reframing, first reproduction attempt documented (Mahler leg not reproduced at n ≤ 18; K–N leg blocked by under-specification), reference corrections (Gubser–Knaute; Bhattacharyya).

**Anchor:** *Prime Valuation Depth: Multiplication as Branching, the Calculus of Indications, and the Structural No-Cloning Reading* (https://papers.qnfo.org/papers/prime-valuation-depth/).

---

## Abstract

The anchor paper introduced the *branch-depth reading*: the p-adic valuation $v_p(n)$ is a measure of depth along a prime branch of the integer factorization tree (not a measure of size), and because the tensor product multiplies Hilbert-space dimensions, $v_2(\dim H) = n$ counts tensor-branch depth for n qubits. This follow-on subjects that reading to its own stated falsifiability condition in the two domains the anchor flagged but did not pursue — computing and quantum error correction. The central finding is a *correction, not a confirmation*: the naive mapping of a $[[n,k,d]]$ stabilizer code to branch depths, $n = v_2(\dim H)$ and $k = v_2(\dim H_L)$, is **trivially true by definition** (a code is defined on n qubits encoding $2^k$ logical states) and therefore carries no new content; and the code distance d — a minimum operator *weight*, not a *depth* — does not admit a valuation reading at all. The genuinely open question is whether there exists a **non-trivial** valuation invariant of stabilizer codes beyond the definitional n and k, and it is here — not in the naive mapping — that the existing "83% classification accuracy" claim (Kodaira–Néron classifier, https://papers.qnfo.org/papers/number-theoretic-ultrametric-foundations/) lives and must be independently reproduced; two attempts now exist (this pipeline's §6 run and the sibling ACRP-06 extension with its v1.1 erratum), both failing to support the claim at the reported sizes. The structural no-cloning reading is retained but repositioned honestly: it is a known categorical result (Abramsky 2009; Coecke 2009), which the branch-depth vocabulary re-expresses rather than discovers. Every non-empirical claim below carries an explicit falsifiability condition.

---

## 1. Introduction and Positioning

### 1.1 What the anchor paper established

*Prime Valuation Depth* developed three claims, each with a stated falsifiability condition:

1. **[TERRITORY — established, Ostrowski 1916]** Every positive integer is a finite product of prime powers; $v_p(n)$ is depth along the prime-p branch, not size.
2. **[MAP — interpretive]** The valuation-as-depth reading is a bridge between the calculus of indications (branching distinctions) and number theory (the prime divisor tree).
3. **[MAP — interpretive]** The tensor product multiplies dimensions, so prime factorization of dim H labels branch types and $v_p(\dim H)$ counts branch depth; the no-cloning theorem is then the impossibility of a linear diagonal map — cloning is nonlinear in the amplitudes, and quantum evolution preserves only linear structure.

### 1.2 The honest starting point of this paper

The anchor paper labels its own quantum reading as "interpretive rather than new physics" and attaches the falsifiability condition: *disconfirmed if the reading yields no explanatory or predictive content beyond the standard formalism — i.e., if it is pure relabeling.* This follow-on takes that condition seriously and applies it to the two domains the anchor flagged for later work. The result, reported here without flinching, is that **the most obvious extension — reading $[[n,k,d]]$ parameters as valuations — is largely pure relabeling.** This is not a failure of the program; it is the falsifiability machinery doing its job, and it sharpens the question to the one place where the branch-depth vocabulary could carry real weight.

### 1.3 External precedent (confirmation-bias correction)

The anchor's quantum reading does not occur in a vacuum. The categorical content — that quantum processes form a compact-closed (monoidal but non-Cartesian) category, and that no-cloning follows from this — is established in the categorical quantum mechanics literature [Abramsky & Coecke 2004; Abramsky 2009; Coecke 2009; Coecke & Duncan 2011]. Independently, the connection between p-adic geometry and quantum error-correcting codes is established in the holographic tensor-network literature [Heydeman, Marcolli, Saberi & Stoica 2018; Bhattacharyya, Hung, Lei & Li 2018; Gubser & Knaute 2017]. The present paper does not claim priority on either bridge; its narrow contribution is the specific *branch-depth vocabulary* (valuation as depth, in the calculus-of-indications sense) applied to code parameters, and its consequence for how QEC limits are framed.

---

## 2. The Branch-Depth Reading, Restated

Let $n \in \mathbb{Z}^+$ factor as $n = \prod p_i^{a_i}$. The anchor's reading: each prime $p_i$ is a *branch type*, and the exponent $a_i = v_{p_i}(n)$ is the *depth* of nesting along that branch. A tensor product of Hilbert spaces multiplies dimensions: $\dim(H_1 \otimes H_2) = \dim(H_1)\cdot\dim(H_2)$. Hence for n qubits, $\dim H = 2^n$ and

$$v_2(\dim H) = n.$$

This is the single quantitative bridge the anchor supplies. The question this paper asks is: **what does it buy us?**

---

## 3. The [[n,k,d]] Mapping — and Why It Is Mostly Relabeling

### 3.1 The definitional facts

A $[[n,k,d]]$ stabilizer code is a $2^k$-dimensional subspace (the codespace) of an n-qubit Hilbert space $H \cong (\mathbb{C}^2)^{\otimes n}$, with $\dim H = 2^n$. Two identities follow immediately:

$$n = v_2(\dim H) = v_2(2^n) = n,$$
$$k = v_2(\dim H_L) = v_2(2^k) = k.$$

**Claim [MAP — to be tested]:** these identities constitute a "branch-depth reading" of the code parameters.

**Finding [SELF-CORRECTION]:** the identities are *definitional*. n is the number of qubits *by construction*, and k is the number of logical qubits *by construction*. Restating them as $v_2$ of a dimension adds nothing: the valuation is doing no work that the exponent of 2 in "$2^n$" and "$2^k$" was not already doing. This is the anchor's own falsifiability condition, triggered: **the n and k mappings are pure relabeling.**

### 3.2 The distance d does not have a valuation reading

The code distance d is the minimum weight (number of non-identity Pauli factors) of a non-trivial logical operator. It is a **Hamming weight** — a count of tensor factors an error touches — not a p-adic depth. There is no integer whose valuation yields d in the way $2^n$ yields n. The tempting phrase "d = branch-crossing error weight" is a metaphor, not a valuation identity, and must not be presented as mathematics. **[Falsifiability condition: disconfirmed if any specific valuation identity for d is claimed; d is a weight, and the burden of proof for any valuation-based bound on d is on the claimer.]**

### 3.3 What is left after the relabeling is stripped away

Stripping the definitional n and k mappings and the unavailable d mapping leaves a single, precise, and genuinely open question:

> **Does there exist a non-trivial valuation invariant of a stabilizer code — one not equal to $v_2(\dim H)$ or $v_2(\dim H_L)$ — that carries classification or predictive power across code families?**

This is where the existing internal result attaches.

---

## 4. Computing as Path-Tracing: A Scoped, Falsifiable Claim

The anchor's computing leg was left as a flagged direction. The honest scope is narrow:

**Claim [MAP — interpretive, scoped]:** reversible classical computation and Clifford quantum computation are *path-tracing through a branching state space*, and the p-adic valuation is the natural depth coordinate on that tree, in the specific sense that the Hensel-code arithmetic of exact rational computation [cf. the p-adic arithmetic literature] is computation *on* the branch tree.

**Falsifiability condition:** disconfirmed if no computational task can be shown to have a *valuation-based* complexity characterization that differs from (or tightens) its standard characterization. The claim is advanced only as a research program, not as an established result. **[CONTESTED] — no such characterization has yet been produced; this section is explicitly promissory.]**

---

## 5. Structural No-Cloning and the Necessity of QEC

### 5.1 The honest attribution

The claim "cloning would require a nonlinear diagonal map, and quantum evolution preserves only linear structure" is a known categorical result. In the categorical quantum mechanics framework, a compact-closed category is **non-Cartesian**: the monoidal product $\otimes$ is not a categorical product, so there is no natural diagonal $\Delta: A \to A \otimes A$ that is linear (a morphism in the category). No-cloning is the physical statement of the absence of this linear diagonal [Abramsky 2009; Coecke 2009]. The anchor paper's contribution is the *vocabulary* — expressing the same fact as "multiplicative branching cannot be linearly duplicated" — not the *result*.

**Claim [MAP — re-expression]:** the branch-depth vocabulary re-expresses no-cloning as: *a branch (tensor factor) cannot be duplicated by a linear process, so redundancy — the resource QEC consumes — must be carried in non-orthogonal (entangled) configurations, never in clones.*

**Falsifiability condition:** disconfirmed if this re-expression cannot be shown to yield at least one new, checkable consequence for QEC limits that the standard no-cloning statement does not already imply. **[Risk: HIGH — this is the relabeling risk again, and it is flagged as such.]**

### 5.2 The one place the framing could bite

A candidate consequence, stated as a falsifiable hypothesis: **if redundancy is non-cloneable, then the overhead (physical-to-logical qubit ratio) of a QEC code is bounded below by a function of the code's own valuation structure.** Whether this bound is tighter than, equivalent to, or weaker than the quantum Singleton bound $n - k \ge 2(d-1)$ is the test. **[RESOLVED — DISCONFIRMED (FQ2, 2026-08-14): the bound has been derived and is strictly weaker than Singleton. The valuation data of a $[[n,k,d]]$ code — $v_2(\dim H)$, $v_2(\dim H_L)$, $v_2(\|S\|)$ — reduce to functions of (n, k, q) only; the Singleton input d is a Hamming weight with no valuation reading (C3, rejected); the sharpest valuation-only bound is $n/k \ge 1$, strictly weaker than Singleton's $n/k \ge n/(n-2(d-1))$ for every code with $d \ge 2$ (378/378 codes in scan). Full derivation: artifacts/fq2-overhead-bound.md.]**

---

## 6. The 83% Classification Claim: Status and Reproduction Requirement

A prior internal report (NTOF, https://papers.qnfo.org/papers/number-theoretic-ultrametric-foundations/) claims that a **rule-based** Kodaira–Néron-fiber
classifier (Algorithm 4.4: binary symplectic form H → Cox ring R_C → Weierstrass coefficients →
degenerate loci → fiber type) assigns code families with **166/200 (83%)** correct on 50 test codes
per family: Surface 46/50 (92%), CSS 39/50 (78%), Optimal 45/50 (90%), Random 36/50 (72%), with
Mahler $v_p^{max} = 28$ for optimal versus $v_p^{max} = 4$ for random ensembles. The source itself records
a documented partial failure — "FAIL for surface codes (systematic mismatch in the I_n*
classification boundaries)" — so the aggregate 83% conceals a known per-family defect. The NTOF
record ships no dataset, no implementation, and no baseline; the classifier is deterministic
rule-based, not a learned ML classifier. A program-registry summary's phrase "p-adic valuations
classify QEC codes at 83% accuracy" is therefore a compressed (and partially misleading)
rendering of this state of affairs.

**Status [UNVERIFIED-INTERNAL]:** this is an internal report. It has not, to this author's knowledge, been independently reproduced on a fresh, held-out test set with a stated leakage-control protocol. Until it is, the 83% figure is a claim to be tested, not a result to be built upon.

**This pipeline's first reproduction attempt (2026-08-13, P4.2) — not the corpus's first:** the Mahler
spectral leg ($C7.3'$) was independently implemented and run on 55 verified codes (CSS
$[[7,1,3]]$, $[[15,7,3]]$; toric surface L=2,3; optimal $[[5,1,3]]$; 50 random $[[10,4]]$). The
claimed separation ($v_p^{max}$ optimal ~ 28 vs random ~ 4, gap >= 10) did **not** reproduce
at $n \le 18$ under the weight-enumerator Mahler normalization: observed optimal 4, random
median 3 (max 6, which exceeds optimal). Two source under-specifications block a decisive
test: the Mahler target function is undefined in NTOF, and Algorithm 4.4's Cox-ring ideal
$I_C$ is unspecified. See the reproduction report in the companion artifacts. C8 remains
[UNVERIFIED-INTERNAL] pending source clarification.

**Prior-art correction (2026-08-14):** the v0.2 text called this the "first reproduction
attempt"; it was not. Sibling project ACRP-06 (the QNFO archive 21737222 v1.0, 2026-08-01) tested
$C7.3'$ on 8 additional families, initially confirmed $v_p^{max}=28$ on Golay-type self-dual
codes, then issued an erratum (v1.1, 21754148, 2026-08-02) correcting Golay $v_p^{max}$ from
28 to 2/4. The 28 has now failed two independent reproductions plus an erratum chain, and the
"first attempt" wording is withdrawn. NTOF's own text ("up to n=64 qubit codes") also removes
the n≤18 size premise that anchored an earlier impossibility argument in the companion
reproduction report. The source also reports a C7.3' spectral-gap verification figure of
187/200 (93.5%) that v0.2 §6 did not mention; it is recorded here for completeness.

**Reproduction requirement (this paper's P4 critical path):** because the classifier is
rule-based and the source ships no implementation, the reproduction is a **re-implementation from
specification** (Algorithm 4.4) plus **fresh code-family generation** (50 per family; surface, CSS,
optimal, random) with (i) a stated generation protocol and seeds, (ii) an independent computation
of the $v_p$-spectral invariant, (iii) baselines (majority-class and random-assignment) reported
alongside, and (iv) the source's documented surface-code boundary defect tested explicitly.
Full protocol: companion artifacts. Success is reproduction *within stated
confidence* (pre-registered acceptance criteria); failure is reported honestly either way.

**Falsifiability condition:** the "83% accuracy" claim is disconfirmed if independent reproduction fails to exceed a stated baseline by a pre-registered margin. **[Status: first reproduction attempt executed (P4.2, 2026-08-13) did not reproduce the claimed spectral separation at $n \le 18$; the decisive test remains blocked by source under-specification. This is disclosed, not hidden.]**

---

## 7. Complete Falsifiability Register

| # | Claim | Type | Falsifiability condition | Current status |
|:--|:------|:-----|:--------------------------|:---------------|
| C1 | $v_p(n)$ is depth along a prime branch | established (Ostrowski) | — | established |
| C2 | $n = v_2(\dim H)$, $k = v_2(\dim H_L)$ is a "branch-depth reading" of $[[n,k,d]]$ | MAP | disconfirmed (pure relabeling) | **SELF-CORRECTED** |
| C3 | d admits a valuation reading | MAP | no valuation identity exists | **REJECTED** |
| C4 | a non-trivial valuation invariant of codes exists | open question | no such invariant found ⇒ program fails | OPEN |
| C5 | computing = path-tracing, with valuation-based complexity content | MAP | no valuation-based complexity characterization produced | **DISCONFIRMED** (FQ3, 2026-08-14 — candidate identities genuine but complexity-vacuous; artifacts/fq3-complexity-characterization.md) |
| C6 | no-cloning re-expressed as non-cloneable redundancy | re-expression | no new checkable consequence ⇒ relabeling | **DISCONFIRMED** (FQ4, 2026-08-14 — relabels Abramsky's categorical no-cloning; no new QEC consequence; artifacts/fq4-no-cloning-consequence.md) |
| C7 | overhead bounded by valuation structure | hypothesis | compare to Singleton bound | **DISCONFIRMED** (FQ2, 2026-08-14 — valuation data (n,k,q)-only, d inexpressible, bound strictly weaker than Singleton; artifacts/fq2-overhead-bound.md) |
| C8 | Kodaira–Néron classifier is 83% accurate | empirical (internal) | independent reproduction vs baseline; first attempt (P4.2) not reproduced at $n \le 18$ | UNVERIFIED-INTERNAL |

---

## 8. Relation to Prior Work (External)

- **Categorical QM / no-cloning:** Abramsky & Coecke (2004) *A categorical semantics of quantum protocols*; Abramsky (2009) *No-Cloning in Categorical Quantum Mechanics*; Coecke (2009) *Quantum Pictorialism*; Coecke & Duncan (2011) *Interacting Quantum Observables*. The no-cloning content of §5 is theirs; this paper adds only vocabulary.
- **p-adic holographic QEC:** Heydeman, Marcolli, Saberi & Stoica (2018) *Tensor networks, p-adic fields, and algebraic curves* (ATMP 22:93); Bhattacharyya, Hung, Lei & Li (2018) *Tensor network and (p-adic) AdS/CFT* (JHEP 2018(1):139, DOI 10.1007/jhep01(2018)139); Gubser & Knaute (2017) *A p-adic version of AdS/CFT* (ATMP 21(7):1655–1678, DOI 10.4310/atmp.2017.v21.n7.a3). The p-adic/QEC connection is theirs; this paper's branch-depth framing is a distinct, narrower lens.
- **Stabilizer methods:** Bravyi, Browne, Calpin, Campbell, Gosset & Howard (2019) *Simulation of quantum circuits by low-rank stabilizer decompositions* — the working background for §3–§4.

**Relation to the internal corpus:** the p-adic QEC space is densely worked internally (see the due-diligence appendix for the DOI list). This paper's narrow differentiation is the branch-depth vocabulary and the self-correction of §3, which the existing corpus (which treats valuation as a *classifier weight*, not a *depth*) does not perform.

---

## 9. Conclusion

The branch-depth reading, applied honestly to computing and quantum error correction, yields a **negative result and a sharpened open question**. The negative result: the obvious extension — reading $[[n,k,d]]$ parameters as valuations — is definitional for n and k and unavailable for d; it is, in the anchor's own terms, pure relabeling, and this paper says so rather than papering over it. The three falsifiable consequences of the reading have now been derived and each is **disconfirmed**: the valuation-based overhead bound is strictly weaker than the quantum Singleton bound (FQ2), the valuation-based complexity characterization is complexity-vacuous (FQ3), and the no-cloning re-expression yields no new checkable consequence beyond the standard categorical result (FQ4). What remains is the single precise open question — *does a non-trivial valuation invariant of stabilizer codes exist?* — where the 83% classification claim (Kodaira–Néron classifier, https://papers.qnfo.org/papers/number-theoretic-ultrametric-foundations/) is the testable hypothesis at the center. Two independent reproduction attempts now exist — this pipeline's §6 run (failed at $n \le 18$) and the sibling ACRP-06 extension with its v1.1 erratum (Golay 28 corrected to 2/4) — and the decisive test remains blocked by source under-specification, so the claim must still be independently reproduced before it is treated as established. This is the honest state of the program: a family of falsifiable claims tested, a sharp question remaining, and a prior claim awaiting verification.

---

## Declarations

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

**Data Availability:** Reproduction scripts, raw results, and the reproduction report are included as companion artifacts of this record (rq3-mahler-reproduction.py, rq3-results.json, rq3-reproduction-report.md, rq3-reproduction-protocol.md).

**Code Availability:** The Mahler spectral reproduction implementation is available in the project repository under notebooks/rq3-mahler-reproduction.py.

**Author Contributions:** Sole author.

**Ethics Approval:** Not applicable (no human or animal subjects; mathematical and philosophical research).

**Consent to Participate:** Not applicable.

**Consent for Publication:** Not applicable.

**Acknowledgements:** The author thanks the reviewers of the companion paper whose falsifiability conditions this work takes up directly.

---

## References

1. Abramsky, S., & Coecke, B. (2004). A categorical semantics of quantum protocols. *Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS)*.
2. Abramsky, S. (2009). No-Cloning in Categorical Quantum Mechanics. In *Semantic Techniques in Quantum Computation*, Cambridge University Press. DOI 10.1017/cbo9781139193313.002.
3. Coecke, B. (2010). Quantum Pictorialism. *Contemporary Physics* 51(1):59–83. DOI 10.1080/00107510903257624 (first published online 2009).
4. Coecke, B., & Duncan, R. (2011). Interacting Quantum Observables: Categorical Algebra and Diagrammatics. *New Journal of Physics* 13:043016. DOI 10.1088/1367-2630/13/4/043016.
5. Heydeman, M., Marcolli, M., Saberi, I., & Stoica, B. (2018). Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS$_3$/CFT$_2$ correspondence. *Adv. Theor. Math. Phys.* 22(1):93–176. DOI 10.4310/atmp.2018.v22.n1.a4. arXiv:1605.07639.
6. Bhattacharyya, A., Hung, L.-Y., Lei, Y., & Li, W. (2018). Tensor network and (p-adic) AdS/CFT. *JHEP* 2018(1):139. DOI 10.1007/jhep01(2018)139. arXiv:1703.05445.
7. Gubser, S. S., & Knaute, J. (2017). A p-adic version of AdS/CFT. *Adv. Theor. Math. Phys.* 21(7):1655–1678. DOI 10.4310/atmp.2017.v21.n7.a3.
8. Bravyi, S., Browne, D., Calpin, P., Campbell, E., Gosset, D., & Howard, M. (2019). Simulation of quantum circuits by low-rank stabilizer decompositions. *Quantum* 3:181. DOI 10.22331/q-2019-09-02-181.
9. Ostrowski, A. (1916). Über einige Lösungen der Funktionalgleichung φ(x)φ(y)=φ(xy). *Acta Mathematica* 41:271–284.
10. Quni-Gudzinas, R. B. (2026). Prime Valuation Depth. the QNFO archive. https://papers.qnfo.org/papers/prime-valuation-depth/.
11. QNFO (2026). Number-Theoretic Ultrametric Foundations. the QNFO archive. https://papers.qnfo.org/papers/number-theoretic-ultrametric-foundations/.