**Traditional approach.** Notice that what is considered in [DMW00; DFM03] and elsewhere to quantize the C-field flux $G_4$ is not just ordinary cohomology, but ordinary cohomology with bells and whistles added as need be:

Foremost there is the half-integral quantization of the $G_4$-flux, mentioned as (3.2) in [DMW00]. Ordinary cohomology may be modified ("shifted") to accommodate this, as maybe first formalized in [HS02] and used in [DFM03].

Beyond this, [DMW00] argue that the "M-theory path integral" imposes a further condition on the shifted integral cohomology class of $G_4$, namely that it be in the kernel of the Steenrod operation $Sq^3$. Whether this "integral equation of motion" is enforced by flux quantization in some further modification of ordinary cohomology is not discussed there.

Instead, the observation made is that the constraint $Sq^3 [F_4] = 0$ is also the first condition that appears in the Atiyah-Hirzebruch spectral sequence for lifting an ordinary cohomology class $[F_4]$ to complex topological K-theory, as demanded by the widely accepted conjecture that RR-field fluxes $F_{n}$ in string theory are quantized in topological K-theory.

Since, moreover, $F_4$ is meant to come from $G_4$ as one lifts string theory to M-theory, the observation of [DMW00] is hence that the "integral equation of motion" in M-theory reproduces one of the constraints on one of the RR-field components given by flux quantization in K-theory.

Suggestive as this is, this consistency check is arguably not a "derivation of K-theory" in string theory. In fact, the conjecture that string theoretic RR-flux really is quantized in K-theory remains itself being debated (for instance, it's not clear how to make it compatible with S-duality). The problem here is that little is known with certainty about non-perturbative string theory beyond a web of interlocking conjectures.

**Hypothesis H.** In view of this situation it seems worthwhile to try a strict top-down approach where a unified generalized cohomology theory in M-theory is postulated and its consequences on flux quantization rigorously derived.

There are good hints what this M-theoretic cohomology theory ought to be: Its image in rational cohomology must see the trivialization of the cup square of the $G_4$-flux demanded by 11d supergravity -- this being the M-theoretic lift of the twisted Bianchi identities that motivate the twisted K-theory conjecture [MSa 03, Sec. 4.2; FSS 16, Sec. 3]. The condition happens to be exactly the relation that identifies the Sullivan model of the 4-sphere, thus suggesting that M-brane charge is quantized in 4-Cohomotopy theory; due to [Sa13, Sec. 2.5].

Indeed, cohomotopical charge quantization in M-theory, on the rational level, follows from a first-principles analysis of the super $p$-brane scan, and is as such the direct M-theoretic analogue of a computation that derives the twisted K-theory classification of D-brane charge ([FSS15], reviewed in [FSS19a, Sec. 7]).

This means that any cohomology theory which quantizes M-brane charge should rationally coincide with Cohomotopy theory. Accordingly, it is quite natural to consider the hypothesis (dubbed "Hypothesis H") that M-brane charge is in Cohomotopy theory itself [Sa13] (suitably twisted by the tangent bundle [FSS19b; FSS19c]).

**Implications.** Indeed, one finds that the assumption of Hypothesis H, that M-brane charge is in J-twisted Cohomotopy theory, readily implies both the half-integral shifted flux quantization on $G_4$ as well as the "integral equation of motion" -- together with a list of further expected constraints [FSS19b, Table 1].

This way Hypothesis H implies as much "derivation of K-theory"; though one should push further to an actual derivation of the implied flux quantization of RR-fields. Derivation from Hypothesis H of more of the fine-print in the K-theory conjecture is in [SS19a; BSS18; BMSS18].

Moreover, Hypothesis H sees the Hořava-Witten Green-Schwarz mechanism in the presence of M5-branes [FSS19d; SS20b], reveals fine-print in the M5-brane anomaly cancellation argument [FSS19c; SS20a], and seems to see a zoo of subtle brane charge effects expected in Hanany-Witten systems [SS19b].

Certainly none of these effects follows from flux quantization in just ordinary cohomology (nor in K-theory, for that matter).

By way of outlook, we think we see now that there is a natural chromatic character map on twisted Cohomotopy which exhibits the M5-brane partition function as charge-quantized in elliptic cohomology, matching traditional discussion of M5-brane ellitptic genera. This is work in progress.

**Conclusion.** In summary, rigorous derivation of the implications of Hypothesis H suggests that twisted Cohomotopy theory sees a fair number of subtle effects that have previously been argued informally to appear in the elusive non-perturbative completion of string theory. This may be indication that, going beyond the traditional approach of hard-coding M-theoretic folklore into a putative "C-field model", charge quantization in twisted Cohomotopy might get to the native heart of the elusive mathematical nature of "M-theory". But, of course, more analysis is necessary.

**Exposition.** More exposition of motivation and development of Hypothesis H may be found in talk notes of the recent meeting at NYU AD *M-Theory and Mathematics*: [Sa20; Sc20].

**References.**

**[DMW00]** E. Diaconescu, G. Moore, E. Witten, *$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory*, Adv. Theor. Math. Phys. 6:1031-1134, 2003 (arXiv:hep-th/0005090), summarised in: *A Derivation of K-Theory from M-Theory* (arXiv:hep-th/0005091)

**[HS02]** M Hopkins, I. Singer, *Quadratic Functions in Geometry, Topology,and M-Theory*, J. Differential Geom. Volume 70, Number 3 (2005), 329-452 (arXiv:math.AT/0211216, euclid:1143642908)

**[DFM03]** E. Diaconescu, D. Freed, G. Moore, *The $M$-theory 3-form and $E_8$-gauge theory*, chapter in: *Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues*, Cambridge University Press 2007 (arXiv:hep-th/0312069, doi:10.1017/CBO9780511721489)

**[MSa03]** V. Mathai, H. Sati, *Some Relations between Twisted K-theory and E8 Gauge Theory*, JHEP 0403:016, 2004 (arXiv:hep-th/0312033, doi:10.1088/1126-6708/2004/03/016)

**[Sa13]** H. Sati, *Framed M-branes, corners, and topological invariants*, J. Math. Phys. 59 (2018), 062304 (arXiv:1310.1060)

**[FSS15]** D. Fiorenza, H. Sati, U. Schreiber, *The WZW term of the M5-brane and differential cohomotopy*, J. Math. Phys. 56, 102301 (2015) (arXiv:1506.07557, doi:10.1063/1.4932618)

**[FSS16]** D. Fiorenza, H. Sati, U. Schreiber, *Rational sphere valued supercocycles in M-theory and type IIA string theory*, J. Geom. Phys., Vol 114 (2017) (arXiv:1606.03206, doi:10.1016/j.geomphys.2016.11.024)

**[BMSS18]** V. Braunack-Mayer, H. Sati, U. Schreiber, *Gauge enhancement of super M-branes via parametrized stable homotopy theory*, Comm. Math. Phys. 371: 197 (2019) (arXiv:1806.01115, doi:10.1007/s00220-019-03441-4)

**[BSS18]** S. Burton, H. Sati, U. Schreiber, *Lift of fractional D-brane charge to equivariant Cohomotopy theory*, J. Geom. Phys., 2020 (in print) (arXiv:1812.09679)

**[FSS19a]** D. Fiorenza, H. Sati, U. Schreiber, *The rational higher structure of M-theory*, in: *Proceedings of Higher Structures in M-Theory 2018*, Fortsch. Phys. 2019 (arXiv:1903.02834, doi:10.1002/prop.201910017)

**[FSS19b]** D. Fiorenza, H. Sati, U. Schreiber, *Twisted Cohomotopy implies M-Theory anomaly cancellation on 8-manifolds*, Comm. Math. Phys. 377(3), 1961-2025 (2020) (arXiv:1904.10207, doi:10.1007/s00220-020-03707-2)

**[FSS19c]** D. Fiorenza, H. Sati, U. Schreiber, *Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane*, Comm. Math. Phys. 2020 (in print) (arXiv:1906.07417)

**[FSS19d]** D. Fiorenza, H. Sati, U. Schreiber, *Twistorial Cohomotopy Implies Green-Schwarz anomaly cancellation* (arXiv:2008.08544)

**[SS19a]** H. Sati, U. Schreiber, *Equivariant Cohomotopy implies orientifold tadpole cancellation* J. Geom. Phys. Vol 156, 2020, 103775 (arXiv:1909.12277, doi:10.1016/j.geomphys.2020.103775)

**[SS19b]** H. Sati, U. Schreiber, *Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams* (arXiv:1912.10425)

**[SS20a]** H. Sati, U. Schreiber, *Twisted Cohomotopy implies M5-brane anomaly cancellation* (arXiv:2002.07737)

**[SS20b]** H. Sati, U. Schreiber, *The character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes* (arXiv:2011.06533)

**[Sa20]** H. Sati, *M-theory and cohomotopy*, talk at *M-Theory and Mathematics*, NYUAD 2020 (pdf)

**[Sc20]** U. Schreiber, *Microscopic brane physics from Cohomotopy theory*, talk at *M-Theory and Mathematics*, NYUAD 2020 (pdf)